# Matlab Control System Toolbox User S Guide

### Matlab-Control-System-Toolbox-User-s-Guide

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Control System Toolbox™ User's Guide R2018b How to Contact MathWorks Latest news: www.mathworks.com Sales and services: www.mathworks.com/sales_and_services User community: www.mathworks.com/matlabcentral Technical support: www.mathworks.com/support/contact_us Phone: 508-647-7000 The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 Control System Toolbox™ User's Guide © COPYRIGHT 2001–2018 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from The MathWorks, Inc. FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by, for, or through the federal government of the United States. By accepting delivery of the Program or Documentation, the government hereby agrees that this software or documentation qualifies as commercial computer software or commercial computer software documentation as such terms are used or defined in FAR 12.212, DFARS Part 227.72, and DFARS 252.227-7014. Accordingly, the terms and conditions of this Agreement and only those rights specified in this Agreement, shall pertain to and govern the use, modification, reproduction, release, performance, display, and disclosure of the Program and Documentation by the federal government (or other entity acquiring for or through the federal government) and shall supersede any conflicting contractual terms or conditions. If this License fails to meet the government's needs or is inconsistent in any respect with federal procurement law, the government agrees to return the Program and Documentation, unused, to The MathWorks, Inc. Trademarks MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See www.mathworks.com/trademarks for a list of additional trademarks. Other product or brand names may be trademarks or registered trademarks of their respective holders. Patents MathWorks products are protected by one or more U.S. patents. Please see www.mathworks.com/patents for more information. Revision History June 2001 July 2002 June 2004 March 2005 September 2005 March 2006 September 2006 March 2007 September 2007 March 2008 October 2008 March 2009 September 2009 March 2010 September 2010 April 2011 September 2011 March 2012 September 2012 March 2013 September 2013 March 2014 October 2014 March 2015 September 2015 March 2016 September 2016 March 2017 September 2017 March 2018 September 2018 Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only Online only New for Version 5.1 (Release 12.1) Revised for Version 5.2 (Release 13) Revised for Version 6.0 (Release 14) Revised for Version 6.2 (Release 14SP2) Revised for Version 6.2.1 (Release 14SP3) Revised for Version 7.0 (Release 2006a) Revised for Version 7.1 (Release 2006b) Revised for Version 8.0 (Release 2007a) Revised for Version 8.0.1 (Release 2007b) Revised for Version 8.1 (Release 2008a) Revised for Version 8.2 (Release 2008b) Revised for Version 8.3 (Release 2009a) Revised for Version 8.4 (Release 2009b) Revised for Version 8.5 (Release 2010a) Revised for Version 9.0 (Release 2010b) Revised for Version 9.1 (Release 2011a) Revised for Version 9.2 (Release 2011b) Revised for Version 9.3 (Release 2012a) Revised for Version 9.4 (Release 2012b) Revised for Version 9.5 (Release 2013a) Revised for Version 9.6 (Release 2013b) Revised for Version 9.7 (Release 2014a) Revised for Version 9.8 (Release 2014b) Revised for Version 9.9 (Release 2015a) Revised for Version 9.10 (Release 2015b) Revised for Version 10.0 (Release 2016a) Revised for Version 10.1 (Release 2016b) Revised for Version 10.2 (Release 2017a) Revised for Version 10.3 (Release 2017b) Revised for Version 10.4 (Release 2018a) Revised for Version 10.5 (Release 2018b) Contents Linear System Modeling 1 Linear System Model Objects What Are Model Objects? . . . . . . . . . . . . . . . . . . . . . . . . . . Model Objects Represent Linear Systems . . . . . . . . . . . . About Model Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2 1-2 1-2 Control System Modeling with Model Objects . . . . . . . . . 1-4 Types of Model Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-7 Dynamic System Models . . . . . . . . . . . . . . . . . . . . . . . . . . 1-10 Static Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-12 Numeric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numeric Linear Time Invariant (LTI) Models . . . . . . . . . Identified LTI Models . . . . . . . . . . . . . . . . . . . . . . . . . . Identified Nonlinear Models . . . . . . . . . . . . . . . . . . . . . 1-13 1-13 1-14 1-14 Generalized Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized and Uncertain LTI Models . . . . . . . . . . . . . Control Design Blocks . . . . . . . . . . . . . . . . . . . . . . . . . Generalized Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 1-16 1-16 1-16 1-18 Models with Tunable Coefficients . . . . . . . . . . . . . . . . . . Tunable Generalized LTI Models . . . . . . . . . . . . . . . . . . Modeling Tunable Components . . . . . . . . . . . . . . . . . . . Modeling Control Systems with Tunable Components . . Internal Structure of Generalized Models . . . . . . . . . . . 1-19 1-19 1-19 1-20 1-20 v 2 vi Contents Using Model Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-24 Model Creation Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transfer Function Representations . . . . . . . . . . . . . . . . . Commands for Creating Transfer Functions . . . . . . . . . . Create Transfer Function Using Numerator and Denominator Coefficients . . . . . . . . . . . . . . . . . . . . . . Create Transfer Function Model Using Zeros, Poles, and Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-3 2-3 2-4 State-Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . State-Space Model Representations . . . . . . . . . . . . . . . . Explicit State-Space Models . . . . . . . . . . . . . . . . . . . . . . Descriptor (Implicit) State-Space Models . . . . . . . . . . . . Commands for Creating State-Space Models . . . . . . . . . . Create State-Space Model From Matrices . . . . . . . . . . . . 2-6 2-6 2-6 2-7 2-7 2-7 2-4 2-5 Frequency Response Data (FRD) Models . . . . . . . . . . . . . Frequency Response Data . . . . . . . . . . . . . . . . . . . . . . Commands for Creating FRD Models . . . . . . . . . . . . . . Create Frequency-Response Model from Data . . . . . . . . 2-10 2-10 2-10 2-11 Proportional-Integral-Derivative (PID) Controllers . . . . Continuous-Time PID Controller Representations . . . . . Create Continuous-Time Parallel-Form PID Controller . . Create Continuous-Time Standard-Form PID Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-13 2-13 2-14 Two-Degree-of-Freedom PID Controllers . . . . . . . . . . . . . Continuous-Time 2-DOF PID Controller Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-DOF Control Architectures . . . . . . . . . . . . . . . . . . . . 2-16 Discrete-Time Numeric Models . . . . . . . . . . . . . . . . . . . . Create Discrete-Time Transfer Function Model . . . . . . . 2-23 2-23 2-14 2-16 2-18 Other Model Types in Discrete Time Representations . . 2-23 Discrete-Time Proportional-Integral-Derivative (PID) Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discrete-Time PID Controller Representations . . . . . . . Create Discrete-Time Standard-Form PID Controller . . . Discrete-Time 2-DOF PI Controller in Standard Form . . 2-24 2-24 2-26 2-26 MIMO Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . Concatenation of SISO Models . . . . . . . . . . . . . . . . . . . Using the tf Function with Cell Arrays . . . . . . . . . . . . . 2-28 2-28 2-29 MIMO State-Space Models . . . . . . . . . . . . . . . . . . . . . . . . MIMO Explicit State-Space Models . . . . . . . . . . . . . . . . MIMO Descriptor State-Space Models . . . . . . . . . . . . . State-Space Model of Jet Transport Aircraft . . . . . . . . . 2-31 2-31 2-32 2-33 MIMO Frequency Response Data Models . . . . . . . . . . . . 2-37 Select Input/Output Pairs in MIMO Models . . . . . . . . . . 2-39 Time Delays in Linear Systems . . . . . . . . . . . . . . . . . . . . . First Order Plus Dead Time Model . . . . . . . . . . . . . . . . Input and Output Delay in State-Space Model . . . . . . . . Transport Delay in MIMO Transfer Function . . . . . . . . . Discrete-Time Transfer Function with Time Delay . . . . . 2-40 2-40 2-41 2-43 2-43 Closing Feedback Loops with Time Delays . . . . . . . . . . . 2-45 Time-Delay Approximation . . . . . . . . . . . . . . . . . . . . . . . . Time-Delay Approximation in Discrete-Time Models . . . 2-48 2-48 Time-Delay Approximation in Continuous-Time Open-Loop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-50 Time-Delay Approximation in Continuous-Time ClosedLoop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-55 Approximate Different Delays with Different Approximation Orders . . . . . . . . . . . . . . . . . . . . . . . . . . 2-60 Convert Time Delay in Discrete-Time Model to Factors of 1/z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-64 vii viii Contents Frequency Response Data (FRD) Model with Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-69 Internal Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Internal Delays Are Necessary . . . . . . . . . . . . . . . Behavior of Models With Internal Delays . . . . . . . . . . . Inside Time Delay Models . . . . . . . . . . . . . . . . . . . . . . . Functions That Support Internal Time Delays . . . . . . . . Functions That Do Not Support Internal Time Delays . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-73 2-73 2-74 2-75 2-76 2-76 2-77 Create Tunable Low-Pass Filter . . . . . . . . . . . . . . . . . . . . 2-78 Create Tunable Second-Order Filter . . . . . . . . . . . . . . . . 2-79 Create State-Space Model with Both Fixed and Tunable Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-81 Control System with Tunable Components . . . . . . . . . . . 2-83 Control System with Multichannel Analysis Points . . . . 2-85 Mark Signals of Interest for Control System Analysis and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analysis Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specify Analysis Points for MATLAB Models . . . . . . . . . Specify Analysis Points for Simulink Models . . . . . . . . . Refer to Analysis Points for Analysis and Tuning . . . . . . 2-89 2-89 2-90 2-91 2-95 Model Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Are Model Arrays? . . . . . . . . . . . . . . . . . . . . . . . Uses of Model Arrays . . . . . . . . . . . . . . . . . . . . . . . . . Visualizing Model Arrays . . . . . . . . . . . . . . . . . . . . . . Visualizing Selection of Models From Model Arrays . . 2-100 2-100 2-100 2-101 2-101 Select Models from Array . . . . . . . . . . . . . . . . . . . . . . . . 2-103 Query Array Size and Characteristics . . . . . . . . . . . . . . 2-106 Linear Parameter-Varying Models . . . . . . . . . . . . . . . . . What are Linear Parameter-Varying Models? . . . . . . . . Regular vs. Irregular Grids . . . . . . . . . . . . . . . . . . . . . 2-109 2-109 2-112 Use Model Arrays to Create Linear Parameter-Varying Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximate Nonlinear Systems using LPV Models . . . Applications of Linear Parameter-Varying Models . . . . Using LTI Arrays for Simulating Multi-Mode Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-114 2-115 2-116 2-118 Working with Linear Models 3 Data Manipulation Store and Retrieve Model Data . . . . . . . . . . . . . . . . . . . . . . Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specify Model Properties at Model Creation . . . . . . . . . . Examine and Change Properties of an Existing Model . . . 3-2 3-2 3-2 3-3 Extract Model Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . Functions for Extracting Model Coefficients . . . . . . . . . . Extracting Coefficients of Different Model Type . . . . . . . Extract Numeric Model Data and Time Delay . . . . . . . . . Extract PID Gains from Transfer Function . . . . . . . . . . . 3-6 3-6 3-6 3-6 3-8 Attach Metadata to Models . . . . . . . . . . . . . . . . . . . . . . . . . 3-9 Specify Model Time Units . . . . . . . . . . . . . . . . . . . . . . . . 3-9 Interconnect Models with Different Time Units . . . . . . . . 3-9 Specify Frequency Units of Frequency-Response Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10 Extract Subsystems of Multi-Input, Multi-Output (MIMO) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10 Specify and Select Input and Output Groups . . . . . . . . . 3-11 Query Model Characteristics . . . . . . . . . . . . . . . . . . . . . . 3-14 Customize Model Display . . . . . . . . . . . . . . . . . . . . . . . . . Configure Transfer Function Display Variable . . . . . . . . 3-17 3-17 ix Configure Display Format of Transfer Function in Factorized Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 x Contents 3-18 Model Interconnections Why Interconnect Models? . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 Catalog of Model Interconnections . . . . . . . . . . . . . . . . . . Model Interconnection Commands . . . . . . . . . . . . . . . . . Arithmetic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3 4-3 4-4 Numeric Model of SISO Feedback Loop . . . . . . . . . . . . . . 4-6 Control System Model With Both Numeric and Tunable Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8 Multi-Loop Control System . . . . . . . . . . . . . . . . . . . . . . . . 4-10 Mark Analysis Points in Closed-Loop Models . . . . . . . . . 4-13 MIMO Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19 MIMO Feedback Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-22 How the Software Determines Properties of Connected Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-27 Rules That Determine Model Type . . . . . . . . . . . . . . . . . . 4-29 Recommended Model Type for Building Block Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-31 5 Model Transformation Conversion Between Model Types . . . . . . . . . . . . . . . . . . . Explicit Conversion Between Model Types . . . . . . . . . . . Automatic Conversion Between Model Types . . . . . . . . . Recommended Working Representation . . . . . . . . . . . . . 5-2 5-2 5-2 5-3 Convert From One Model Type to Another . . . . . . . . . . . . 5-4 Get Current Value of Generalized Model by Model Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6 Decompose a 2-DOF PID Controller into SISO Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8 Discretize a Compensator . . . . . . . . . . . . . . . . . . . . . . . . . 5-13 Improve Accuracy of Discretized System with Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-19 Convert Discrete-Time System to Continuous Time . . . . 5-23 Continuous-Discrete Conversion Methods . . . . . . . . . . . Choosing a Conversion Method . . . . . . . . . . . . . . . . . . . Zero-Order Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First-Order Hold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Impulse-Invariant Mapping . . . . . . . . . . . . . . . . . . . . . . Tustin Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . Zero-Pole Matching Equivalents . . . . . . . . . . . . . . . . . . Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-26 5-26 5-27 5-28 5-29 5-30 5-34 5-34 Upsample Discrete-Time System . . . . . . . . . . . . . . . . . . . 5-36 Choosing a Resampling Command . . . . . . . . . . . . . . . . . . 5-40 xi 6 xii Contents Model Simplification Model Reduction Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . When to Reduce Model Order . . . . . . . . . . . . . . . . . . . . . Choosing a Model Reduction Method . . . . . . . . . . . . . . . 6-2 6-2 6-4 Reduce Model Order Using the Model Reducer App . . . . 6-6 Balanced Truncation Model Reduction . . . . . . . . . . . . . . Balanced Truncation in the Model Reducer App . . . . . . Approximate Model by Balanced Truncation at the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compare Truncated and DC Matched Low-Order Model Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximate Model with Unstable or Near-Unstable Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency-Limited Balanced Truncation . . . . . . . . . . . . 6-17 6-17 6-26 6-30 6-35 6-40 Pole-Zero Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . Pole-Zero Simplification in the Model Reducer App . . . . Pole-Zero Cancellation at the Command Line . . . . . . . . 6-47 6-47 6-53 Mode-Selection Model Reduction . . . . . . . . . . . . . . . . . . Mode Selection in the Model Reducer App . . . . . . . . . . Mode Selection at the Command Line . . . . . . . . . . . . . . 6-57 6-57 6-63 Visualize Reduced-Order Models in the Model Reducer App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6-67 6-67 6-68 6-70 6-72 Linear Analysis 7 Time Domain Analysis Plotting System Responses . . . . . . . . . . . . . . . . . . . . . . . . . 7-2 Time-Domain Responses . . . . . . . . . . . . . . . . . . . . . . . . . . 7-20 Time-Domain Response Data and Plots . . . . . . . . . . . . . . 7-21 Time-Domain Characteristics on Response Plots . . . . . . 7-24 Numeric Values of Time-Domain System Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7-29 Time-Domain Responses of Discrete-Time Model . . . . . . 7-31 Time-Domain Responses of MIMO Model . . . . . . . . . . . . 7-34 Time-Domain Responses of Multiple Models . . . . . . . . . 7-36 Joint Time-Domain and Frequency-Domain Analysis . . . 7-40 Response from Initial Conditions . . . . . . . . . . . . . . . . . . . 7-45 Import LTI Model Objects into Simulink . . . . . . . . . . . . . Simulate LTI Model in Simulink . . . . . . . . . . . . . . . . . . Import MIMO LTI Model into Simulink . . . . . . . . . . . . . 7-48 7-48 7-50 Analysis of Systems with Time Delays . . . . . . . . . . . . . . . Considerations to Keep in Mind when Analyzing Systems with Internal Time Delays . . . . . . . . . . . . . . . . . . . . . 7-53 7-56 xiii 8 9 Frequency Domain Analysis Frequency-Domain Responses . . . . . . . . . . . . . . . . . . . . . . 8-2 Frequency Response of a SISO System . . . . . . . . . . . . . . . 8-4 Frequency Response of a MIMO System . . . . . . . . . . . . . . 8-6 Frequency-Domain Characteristics on Response Plots . 8-10 Numeric Values of Frequency-Domain Characteristics of SISO Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8-13 Pole and Zero Locations . . . . . . . . . . . . . . . . . . . . . . . . . . 8-16 Assessing Gain and Phase Margins . . . . . . . . . . . . . . . . . 8-19 Analyzing Control Systems with Delays . . . . . . . . . . . . . . 8-32 Analyzing the Response of an RLC Circuit . . . . . . . . . . . 8-50 Sensitivity Analysis Model Array with Single Parameter Variation . . . . . . . . . . 9-2 Model Array with Variations in Two Parameters . . . . . . . . 9-6 Study Parameter Variation by Sampling Tunable Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9 Sensitivity of Control System to Time Delays . . . . . . . . . xiv Contents 9-12 Passivity and Conic Sectors 10 About Passivity and Passivity Indices . . . . . . . . . . . . . . . 10-2 About Sector Bounds and Sector Indices . . . . . . . . . . . . 10-9 Passivity Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10-19 Parallel Interconnection of Passive Systems . . . . . . . . . 10-24 Series Interconnection of Passive Systems . . . . . . . . . . 10-27 Feedback Interconnection of Passive Systems . . . . . . . 10-31 Control Design 11 PID Controller Design PID Controller Design at the Command Line . . . . . . . . . 11-2 Designing Cascade Control System with PI Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-9 Tune 2-DOF PID Controller (Command Line) . . . . . . . . 11-15 Tune 2-DOF PID Controller (PID Tuner) . . . . . . . . . . . . 11-21 PID Controller Types for Tuning . . . . . . . . . . . . . . . . . . . Specifying PID Controller Type . . . . . . . . . . . . . . . . . . 1-DOF Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-DOF Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-DOF Controllers with Fixed Setpoint Weights . . . . . 11-31 11-31 11-33 11-33 11-34 PID Controller Tuning in Simulink . . . . . . . . . . . . . . . . 11-37 xv Design PID Controller Using Estimated Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-46 Design Family of PID Controllers for Multiple Operating Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-56 Design PID Controller Using Simulated I/O Data . . . . . 12 xvi Contents 11-65 Classical Control Design Choosing a Control Design Approach . . . . . . . . . . . . . . . 12-2 Control System Designer Tuning Methods . . . . . . . . . . . Graphical Tuning Methods . . . . . . . . . . . . . . . . . . . . . . Automated Tuning Methods . . . . . . . . . . . . . . . . . . . . . Effective Plant for Tuning . . . . . . . . . . . . . . . . . . . . . . . Select a Tuning Method . . . . . . . . . . . . . . . . . . . . . . . . 12-4 12-4 12-5 12-6 12-7 Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . Add Design Requirements . . . . . . . . . . . . . . . . . . . . . Edit Design Requirements . . . . . . . . . . . . . . . . . . . . . Root Locus and Pole-Zero Plot Requirements . . . . . . . Open-Loop and Closed-Loop Bode Diagram Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open-Loop Nichols Plot Requirements . . . . . . . . . . . . Step and Impulse Response Requirements . . . . . . . . . 12-9 12-10 12-13 12-14 Feedback Control Architectures . . . . . . . . . . . . . . . . . . . 12-21 Design Multiloop Control System . . . . . . . . . . . . . . . . . 12-24 Multimodel Control Design . . . . . . . . . . . . . . . . . . . . . . Control Design Overview . . . . . . . . . . . . . . . . . . . . . . Model Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nominal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Controller for Multiple Plant Models . . . . . . . . 12-35 12-35 12-35 12-37 12-39 12-40 12-16 12-17 12-18 Bode Diagram Design . . . . . . . . . . . . . . . . . . . . . . . . . . . Tune Compensator For DC Motor Using Bode Diagram Graphical Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . 12-49 12-49 Root Locus Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-64 Tune Electrohydraulic Servomechanism Using Root Locus Graphical Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . 12-64 Nichols Plot Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tune Compensator For DC Motor Using Nichols Plot Graphical Design . . . . . . . . . . . . . . . . . . . . . . . . . . 12-80 Edit Compensator Dynamics . . . . . . . . . . . . . . . . . . . . . . Compensator Editor . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical Compensator Editing . . . . . . . . . . . . . . . . . Poles and Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lead and Lag Networks . . . . . . . . . . . . . . . . . . . . . . . Notch Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-93 12-93 12-96 12-97 12-97 12-98 12-80 Design Compensator Using Automated Tuning Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Select Tuning Method . . . . . . . . . . . . . . . . . . . . . . . Select Compensator and Loop to Tune . . . . . . . . . . . PID Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimization-Based Tuning . . . . . . . . . . . . . . . . . . . . LQG Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loop Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal Model Control Tuning . . . . . . . . . . . . . . . . . 12-100 12-100 12-101 12-102 12-108 12-110 12-112 12-113 Analyze Designs Using Response Plots . . . . . . . . . . . . Analysis Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Editor Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . Plot Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Requirements . . . . . . . . . . . . . . . . . . . . . . . . 12-117 12-117 12-120 12-121 12-122 12-124 Compare Performance of Multiple Designs . . . . . . . . 12-127 Design Hard-Disk Read/Write Head Controller . . . . . 12-133 Design Compensator for Plant Model with Time Delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-147 xvii Design Compensator for Systems Represented by Frequency Response Data . . . . . . . . . . . . . . . . . . . . . 12-155 Design Internal Model Controller for Chemical Reactor Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-161 13 Design LQG Tracker Using Control System Designer 12-176 Export Design to MATLAB Workspace . . . . . . . . . . . . . 12-188 Generate Simulink Model for Control Architecture . . 12-191 Tune Simulink Blocks Using Compensator Editor . . . 12-193 Single Loop Feedback/Prefilter Compensator Design 12-201 Cascaded Multiloop Feedback Design . . . . . . . . . . . . . 12-211 Reference Tracking of DC Motor with Parameter Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12-222 State-Space Control Design Extended and Unscented Kalman Filter Algorithms for Online State Estimation . . . . . . . . . . . . . . . . . . . . . . . . Extended Kalman Filter Algorithm . . . . . . . . . . . . . . . . Unscented Kalman Filter Algorithm . . . . . . . . . . . . . . . xviii Contents 13-2 13-2 13-5 Generate Code for Online State Estimation in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tunable and Nontunable Object Properties . . . . . . . . . 13-11 13-13 Validate Online State Estimation at the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examine Output Estimation Error . . . . . . . . . . . . . . . . Examine State Estimation Error for Simulated Data . . 13-15 13-15 13-16 Validate Online State Estimation in Simulink . . . . . . . . Examine Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-18 13-18 Examine State Estimation Error for Simulated Data . . Compute Residuals and State Estimation Errors . . . . . 13-19 13-19 Troubleshoot Online State Estimation . . . . . . . . . . . . . . 13-22 Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter . . . . . . . . . . . . . . . . . . . . . . . 13-24 Estimate States of Nonlinear System with Multiple, Multirate Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13-45 Regulate Pressure in Drum Boiler . . . . . . . . . . . . . . . . . 13-59 Control System Tuning 14 Control System Tuning Automated Tuning Overview . . . . . . . . . . . . . . . . . . . . . . . 14-3 Choosing an Automated Tuning Approach . . . . . . . . . . . 14-5 Automated Tuning Workflow . . . . . . . . . . . . . . . . . . . . . . . 14-7 Specify Control Architecture in Control System Tuner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-9 About Control Architecture . . . . . . . . . . . . . . . . . . . . . . 14-9 Predefined Feedback Architecture . . . . . . . . . . . . . . . . 14-9 Arbitrary Feedback Control Architecture . . . . . . . . . . 14-11 Control System Architecture in Simulink . . . . . . . . . . 14-12 Open Control System Tuner for Tuning Simulink Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Command-Line Equivalents . . . . . . . . . . . . . . . . . . . . 14-13 14-14 xix Specify Operating Points for Tuning in Control System Tuner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . About Operating Points in Control System Tuner . . . . Linearize at Simulation Snapshot Times . . . . . . . . . . . Compute Operating Points at Simulation Snapshot Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compute Steady-State Operating Points . . . . . . . . . . . 14-17 14-21 Specify Blocks to Tune in Control System Tuner . . . . . 14-24 View and Change Block Parameterization in Control System Tuner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . View Block Parameterization . . . . . . . . . . . . . . . . . . . Fix Parameter Values or Limit Tuning Range . . . . . . . Custom Parameterization . . . . . . . . . . . . . . . . . . . . . . Block Rate Conversion . . . . . . . . . . . . . . . . . . . . . . . . 14-26 14-26 14-28 14-30 14-31 Setup for Tuning Control System Modeled in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-35 How Tuned Simulink Blocks Are Parameterized . . . . . . Blocks With Predefined Parameterization . . . . . . . . . . Blocks Without Predefined Parameterization . . . . . . . . View and Change Block Parameterization . . . . . . . . . . 14-36 14-36 14-37 14-38 Specify Goals for Interactive Tuning . . . . . . . . . . . . . . . 14-39 14-15 14-15 14-15 Quick Loop Tuning of Feedback Loops in Control System Tuner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-48 xx Contents Quick Loop Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedback Loop Selection . . . . . . . . . . . . . . . . . . . . . . Desired Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-58 14-58 14-58 14-58 14-59 14-60 14-61 Step Tracking Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step Response Selection . . . . . . . . . . . . . . . . . . . . . . . Desired Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-62 14-62 14-62 14-63 14-64 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-65 14-66 Step Rejection Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Step Disturbance Response Selection . . . . . . . . . . . . . Desired Response to Step Disturbance . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-68 14-68 14-68 14-69 14-70 14-71 14-72 Transient Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . Initial Signal Selection . . . . . . . . . . . . . . . . . . . . . . . . Desired Transient Response . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-74 14-74 14-74 14-75 14-76 14-76 14-77 14-78 14-79 LQR/LQG Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Signal Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LQG Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-81 14-81 14-81 14-82 14-83 14-84 14-84 14-85 Gain Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I/O Transfer Selection . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-86 14-86 14-86 14-87 14-88 14-90 Variance Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I/O Transfer Selection . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-92 14-92 14-92 14-92 14-93 14-94 xxi Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-95 Reference Tracking Goal . . . . . . . . . . . . . . . . . . . . . . . . . 14-97 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-97 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-97 Response Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-98 Tracking Performance . . . . . . . . . . . . . . . . . . . . . . . . 14-99 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-100 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-102 xxii Contents Overshoot Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response Selection . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-104 14-104 14-104 14-105 14-106 14-107 Disturbance Rejection Goal . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbance Scenario . . . . . . . . . . . . . . . . . . . . . . . . Rejection Performance . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-109 14-109 14-109 14-110 14-111 14-112 14-112 Sensitivity Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Evaluation . . . . . . . . . . . . . . . . . . . . . . . . Sensitivity Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-114 14-114 14-114 14-115 14-116 14-116 14-117 Weighted Gain Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I/O Transfer Selection . . . . . . . . . . . . . . . . . . . . . . . Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-119 14-119 14-119 14-119 14-120 14-121 14-122 Weighted Variance Goal . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-124 14-124 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I/O Transfer Selection . . . . . . . . . . . . . . . . . . . . . . . Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-124 14-124 14-125 14-126 14-127 14-127 Minimum Loop Gain Goal . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open-Loop Response Selection . . . . . . . . . . . . . . . . . Desired Loop Gain . . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-129 14-129 14-129 14-131 14-132 14-132 14-133 Maximum Loop Gain Goal . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open-Loop Response Selection . . . . . . . . . . . . . . . . . Desired Loop Gain . . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-135 14-135 14-135 14-137 14-138 14-138 14-139 Loop Shape Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open-Loop Response Selection . . . . . . . . . . . . . . . . . Desired Loop Shape . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-141 14-141 14-141 14-143 14-144 14-144 14-146 Margins Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedback Loop Selection . . . . . . . . . . . . . . . . . . . . . Desired Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-148 14-148 14-148 14-149 14-150 14-150 14-151 Passivity Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I/O Transfer Selection . . . . . . . . . . . . . . . . . . . . . . . 14-153 14-153 14-153 14-154 xxiii Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-155 14-156 Conic Sector Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I/O Transfer Selection . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-158 14-158 14-158 14-159 14-160 14-161 14-162 Weighted Passivity Goal . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I/O Transfer Selection . . . . . . . . . . . . . . . . . . . . . . . Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-165 14-165 14-165 14-166 14-167 14-168 14-169 Poles Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Feedback Configuration . . . . . . . . . . . . . . . . . . . . . . Pole Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-171 14-171 14-171 14-172 14-173 14-174 14-174 Controller Poles Goal . . . . . . . . . . . . . . . . . . . . . . . . . . Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constrain Dynamics of Tuned Block . . . . . . . . . . . . . Keep Poles Inside the Following Region . . . . . . . . . . Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-176 14-176 14-176 14-177 14-177 14-178 Manage Tuning Goals . . . . . . . . . . . . . . . . . . . . . . . . . . 14-179 Generate MATLAB Code from Control System Tuner for Command-Line Tuning . . . . . . . . . . . . . . . . . . . . . . . 14-181 Interpret Numeric Tuning Results . . . . . . . . . . . . . . . . Tuning-Goal Scalar Values . . . . . . . . . . . . . . . . . . . . Tuning Results at the Command Line . . . . . . . . . . . . Tuning Results in Control System Tuner . . . . . . . . . . xxiv Contents 14-184 14-184 14-185 14-185 Improve Tuning Results . . . . . . . . . . . . . . . . . . . . . . 14-187 Visualize Tuning Goals . . . . . . . . . . . . . . . . . . . . . . . . . Tuning-Goal Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference Between Dashed Line and Shaded Region Improve Tuning Results . . . . . . . . . . . . . . . . . . . . . . 14-189 14-189 14-191 14-197 Create Response Plots in Control System Tuner . . . . 14-198 Examine Tuned Controller Parameters in Control System Tuner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-205 Compare Performance of Multiple Tuned Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-207 Create and Configure slTuner Interface to Simulink Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-212 Stability Margins in Control System Tuning . . . . . . . . Gain and Phase Margins . . . . . . . . . . . . . . . . . . . . . . Combined Gain and Phase Variations . . . . . . . . . . . . Interpreting the Gain and Phase Margin Plot . . . . . . Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-218 14-219 14-219 14-220 14-222 Tune Control System at the Command Line . . . . . . . . 14-223 Speed Up Tuning with Parallel Computing Toolbox Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-225 Validate Tuned Control System . . . . . . . . . . . . . . . . . . Extract and Plot System Responses . . . . . . . . . . . . . Validate Design in Simulink Model . . . . . . . . . . . . . . 14-227 14-227 14-230 Extract Responses from Tuned MATLAB Model at the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14-232 xxv 15 16 Loop-Shaping Design Structure of Control System for Tuning With looptune . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15-2 Set Up Your Control System for Tuning with looptune . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Set Up Your Control System for looptunein MATLAB . . . Set Up Your Control System for looptune in Simulink . . 15-4 15-4 15-4 Tune MIMO Control System for Specified Bandwidth . . 15-6 Tuning Feedback Loops with LOOPTUNE . . . . . . . . . . . 15-13 Decoupling Controller for a Distillation Column . . . . . 15-20 Tuning of a Digital Motion Control System . . . . . . . . . . 15-32 Gain-Scheduled Controllers Gain Scheduling Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . Gain Scheduling in Simulink . . . . . . . . . . . . . . . . . . . . . Tune Gain Schedules . . . . . . . . . . . . . . . . . . . . . . . . . . 16-2 16-2 16-3 Model Gain-Scheduled Control Systems in Simulink . . . 16-4 Model Scheduled Gains . . . . . . . . . . . . . . . . . . . . . . . . 16-4 Gain-Scheduled Equivalents for Commonly Used Control Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-7 Custom Gain-Scheduled Control Structures . . . . . . . . 16-12 Tunability of Gain Schedules . . . . . . . . . . . . . . . . . . . 16-13 xxvi Contents Tune Gain Schedules in Simulink . . . . . . . . . . . . . . . . . Workflow for Tuning Gain Schedules . . . . . . . . . . . . . 16-15 16-15 Plant Models for Gain-Scheduled Controller Tuning . . Obtaining the Family of Linear Models . . . . . . . . . . . . 16-18 16-19 Set Up for Gain Scheduling by Linearizing at Design Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample System at Simulation Snapshots . . . . . . . . . . . Sample System at Varying Parameter Values . . . . . . . . Eliminate Samples at Unneeded Design Points . . . . . . LPV Plants in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . 16-20 16-23 16-23 16-24 16-25 Multiple Design Points in slTuner Interface . . . . . . . . . Block Substitution for Plant . . . . . . . . . . . . . . . . . . . . Multiple Block Substitutions . . . . . . . . . . . . . . . . . . . . Substituting Blocks that Depend on the Scheduling Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resolving Mismatches Between a Block and its Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Block Substitution for LPV Blocks . . . . . . . . . . . . . . . . 16-26 16-26 16-26 Parameterize Gain Schedules . . . . . . . . . . . . . . . . . . . . . Basis Function Parameterization . . . . . . . . . . . . . . . . . Tunable Gain Surfaces . . . . . . . . . . . . . . . . . . . . . . . . Tunable Gain With Two Independent Scheduling Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tunable Surfaces in Simulink . . . . . . . . . . . . . . . . . . . Tunable Surfaces in MATLAB . . . . . . . . . . . . . . . . . . . 16-32 16-32 16-35 16-36 16-38 16-40 Change Requirements with Operating Condition . . . . . Define Variable Tuning Goal . . . . . . . . . . . . . . . . . . . . Enforce Tuning Goal at Subset of Design Points . . . . . Exclude Design Points from systune Run . . . . . . . . . . 16-42 16-42 16-44 16-45 Validate Gain-Scheduled Control Systems . . . . . . . . . . Examine Tuned Gain Surfaces . . . . . . . . . . . . . . . . . . Visualize Tuning Goals . . . . . . . . . . . . . . . . . . . . . . . . Check Linear Performance . . . . . . . . . . . . . . . . . . . . . Validate Gain Schedules in Nonlinear System . . . . . . . 16-46 16-46 16-46 16-49 16-50 Gain-Scheduled Control of a Chemical Reactor . . . . . . 16-52 Tuning of Gain-Scheduled Three-Loop Autopilot . . . . . 16-70 Trimming and Linearization of the HL-20 Airframe . . . 16-86 Angular Rate Control in the HL-20 Autopilot . . . . . . . . 16-95 16-28 16-29 16-30 xxvii 17 Attitude Control in the HL-20 Autopilot - SISO Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-104 Attitude Control in the HL-20 Autopilot - MIMO Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16-116 MATLAB Workflow for Tuning the HL-20 Autopilot . . 16-127 Control System Tuning Examples - Generalized LTI Models Tuning Control Systems with SYSTUNE . . . . . . . . . . . . . 18 Building Tunable Models . . . . . . . . . . . . . . . . . . . . . . . . 17-10 Active Vibration Control in Three-Story Building . . . . 17-18 Vibration Control in Flexible Beam . . . . . . . . . . . . . . . . 17-31 Passive Control with Communication Delays . . . . . . . . 17-44 Control System Tuning Examples Tuning Multiloop Control Systems . . . . . . . . . . . . . . . . . . xxviii Contents 17-2 18-2 PID Tuning for Setpoint Tracking vs. Disturbance Rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-13 Time-Domain Specifications . . . . . . . . . . . . . . . . . . . . . . 18-25 Frequency-Domain Specifications . . . . . . . . . . . . . . . . . 18-31 Loop Shape and Stability Margin Specifications . . . . . 18-42 System Dynamics Specifications . . . . . . . . . . . . . . . . . . 18-49 Configuring Design Requirements . . . . . . . . . . . . . . . . . 18-52 Validating Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-54 Tune Control Systems in Simulink . . . . . . . . . . . . . . . . . 18-63 Tune a Control System Using Control System Tuner . . 18-73 Using Parallel Computing to Accelerate Tuning . . . . . . 18-94 Control of a Linear Electric Actuator . . . . . . . . . . . . . . 18-99 Control of a Linear Electric Actuator Using Control System Tuner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18-109 Multi-Loop PI Control of a Robotic Arm . . . . . . . . . . . 18-141 Control of an Inverted Pendulum on a Cart . . . . . . . . 18-163 Digital Control of Power Stage Voltage . . . . . . . . . . . . 18-173 MIMO Control of Diesel Engine . . . . . . . . . . . . . . . . . . 18-184 Tuning of a Two-Loop Autopilot . . . . . . . . . . . . . . . . . . 18-199 Multiloop Control of a Helicopter . . . . . . . . . . . . . . . . 18-217 Fixed-Structure Autopilot for a Passenger Jet . . . . . . 18-226 Fault-Tolerant Control of a Passenger Jet . . . . . . . . . . 18-240 Passive Control of Water Tank Level . . . . . . . . . . . . . . 18-251 Tuning for Multiple Values of Plant Parameters . . . . . 18-270 xxix Customization 19 20 Preliminaries Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-2 Property and Preferences Hierarchy . . . . . . . . . . . . . . . . 19-3 Ways to Customize Plots . . . . . . . . . . . . . . . . . . . . . . . . . . 19-5 Setting Toolbox Preferences Toolbox Preferences Editor . . . . . . . . . . . . . . . . . . . . . . . . Overview of the Toolbox Preferences Editor . . . . . . . . . Opening the Toolbox Preferences Editor . . . . . . . . . . . . Units Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Style Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Options Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control System Designer Pane . . . . . . . . . . . . . . . . . . . 21 Setting Tool Preferences Linear System Analyzer Preferences Editor . . . . . . . . . . Opening the Linear System Analyzer Preference Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Units Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Style Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Options Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameters Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx Contents 20-2 20-2 20-2 20-3 20-5 20-6 20-7 21-2 21-2 21-3 21-5 21-6 21-7 22 Customizing Response Plot Properties Customize Response Plots Using the Response Plots Property Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-2 Opening the Property Editor . . . . . . . . . . . . . . . . . . . . . 22-2 Overview of Response Plots Property Editor . . . . . . . . . 22-3 Labels Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-5 Limits Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-6 Units Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-6 Style Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-16 Options Pane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22-18 Editing Subplots Using the Property Editor . . . . . . . . 22-22 Customizing Response Plots Using Plot Tools . . . . . . . Properties You Can Customize Using Plot Tools . . . . . . Opening and Working with Plot Tools . . . . . . . . . . . . . Example of Changing Line Color Using Plot Tools . . . . 23 22-24 22-24 22-24 22-25 Customizing Response Plots from the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of Customizing Plots from the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Obtaining Plot Handles . . . . . . . . . . . . . . . . . . . . . . . . Obtaining Plot Options Handles . . . . . . . . . . . . . . . . . Examples of Customizing Plots from the Command Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties and Values Reference . . . . . . . . . . . . . . . . 22-28 22-31 22-32 Build GUI With Interactive Response-Plot Updates . . . 22-53 22-28 22-34 22-38 Design Case Studies Design Yaw Damper for Jet Transport . . . . . . . . . . . . . . . . . . . Overview of this Case Study . . . . . . . . . . . . . . . . . . . . . . . . . Creating the Jet Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computing Open-Loop Poles . . . . . . . . . . . . . . . . . . . . . . . . . Open-Loop Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Root Locus Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-2 23-2 23-2 23-3 23-4 23-8 xxxi Washout Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 LQG Regulation: Rolling Mill Case Study . . . . . . . . . . . . . . . Overview of this Case Study . . . . . . . . . . . . . . . . . . . . . . . . Process and Disturbance Models . . . . . . . . . . . . . . . . . . . . LQG Design for the x-Axis . . . . . . . . . . . . . . . . . . . . . . . . . . LQG Design for the y-Axis . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-Coupling Between Axes . . . . . . . . . . . . . . . . . . . . . . . MIMO LQG Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-19 23-19 23-19 23-22 23-28 23-30 23-33 Kalman Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23-37 Reliable Computations Scaling State-Space Models . . . . . . . . . . . . . . . . . . . . . . . . . . . Why Scaling Is Important . . . . . . . . . . . . . . . . . . . . . . . . . . . When to Scale Your Model . . . . . . . . . . . . . . . . . . . . . . . . . . Manually Scale Your Model . . . . . . . . . . . . . . . . . . . . . . . . . . 25 23-13 24-2 24-2 24-2 24-3 Linear System Analyzer Linear System Analyzer Overview . . . . . . . . . . . . . . . . . . . . . . 25-2 Using the Right-Click Menu in the Linear System Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of the Right-Click Menu . . . . . . . . . . . . . . . . . . . . . Setting Characteristics of Response Plots . . . . . . . . . . . . . . . 25-4 25-4 25-4 Importing, Exporting, and Deleting Models in the Linear System Analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-9 Importing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-9 Exporting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-10 Deleting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-10 xxxii Contents Selecting Response Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods for Selecting Response Types . . . . . . . . . . . . . . . . Right Click Menu: Plot Type . . . . . . . . . . . . . . . . . . . . . . . . Plot Configurations Window . . . . . . . . . . . . . . . . . . . . . . . . Line Styles Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-12 25-12 25-12 25-12 25-14 Analyzing MIMO Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of Analyzing MIMO Models . . . . . . . . . . . . . . . . . Array Selector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I/O Grouping for MIMO Models . . . . . . . . . . . . . . . . . . . . . Selecting I/O Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25-17 25-17 25-18 25-19 25-20 Customizing the Linear System Analyzer . . . . . . . . . . . . . . . Overview of Customizing the Linear System Analyzer . . . . . Linear System Analyzer Preferences Editor . . . . . . . . . . . . 25-22 25-22 25-22 xxxiii Linear System Modeling 35 1 Linear System Model Objects • “What Are Model Objects?” on page 1-2 • “Control System Modeling with Model Objects” on page 1-4 • “Types of Model Objects” on page 1-7 • “Dynamic System Models” on page 1-10 • “Static Models” on page 1-12 • “Numeric Models” on page 1-13 • “Generalized Models” on page 1-16 • “Models with Tunable Coefficients” on page 1-19 • “Using Model Objects” on page 1-23 • “References” on page 1-24 1 Linear System Model Objects What Are Model Objects? Model Objects Represent Linear Systems In Control System Toolbox, System Identification Toolbox™, and Robust Control Toolbox™ software, you represent linear systems as model objects. In System Identification Toolbox, you also represent nonlinear models as model objects. Model objects are specialized data containers that encapsulate model data and other attributes in a structured way. Model objects allow you to manipulate linear systems as single entities rather than keeping track of multiple data vectors, matrices, or cell arrays. Model objects can represent single-input, single-output (SISO) systems or multiple-input, multiple-output (MIMO) systems. You can represent both continuous- and discrete-time linear systems. The main families of model objects are: • Numeric Models — Basic representation of linear systems with fixed numerical coefficients. This family also includes identified models that have coefficients estimated with System Identification Toolbox software. • Generalized Models — Representations that combine numeric coefficients with tunable or uncertain coefficients. Generalized models support tasks such as parameter studies or compensator tuning. About Model Data The data encapsulated in your model object depends on the model type you use. For example: • Transfer functions store the numerator and denominator coefficients • State-space models store the A, B, C, and D matrices that describe the dynamics of the system • PID controller models store the proportional, integral, and derivative gains Other model attributes stored as model data include time units, names for the model inputs or outputs, and time delays. For more information about setting and retrieving model attributes, see “Model Attributes”. 1-2 See Also Note All model objects are MATLAB® objects, but working with them does not require a background in object-oriented programming. To learn more about objects and object syntax, see “Role of Classes in MATLAB” (MATLAB). See Also More About • “Control System Modeling with Model Objects” on page 1-4 • “Types of Model Objects” on page 1-7 1-3 1 Linear System Model Objects Control System Modeling with Model Objects Model objects can represent individual components of a control architecture, such as the plant, actuators, sensors, or controllers. You can connect model objects to build aggregate models of block diagrams that represent the combined response of multiple elements. For example, the following control system contains a prefilter F, a plant G, and a controller C, arranged in a single-loop configuration. The model also includes a representation of sensor dynamics, S. You can represent each of the components as a model object. You do not need to use the same type of model object for each component. For example, represent the plant G as a zero-pole-gain (zpk) model with a double pole at s = -1; C as a PID controller, and F and S as transfer functions: G C S F = = = = zpk([],[-1,-1],1); pid(2,1.3,0.3,0.5); tf(5,[1 4]); tf(1,[1 1]); You can then combine these elements build models that represent your control system or the control system as a whole. For example, create the open-loop response SGC: open_loop = S*G*C; To build a model of the unfiltered closed-loop response, use the feedback command: T = feedback(G*C,S); To model the entire closed-loop system response from r to y, combine T with the filter transfer function: Try = T*F; 1-4 Control System Modeling with Model Objects The results open_loop, T, and Try are also linear model objects. You can operate on them with Control System Toolbox™ control design and analysis commands. For example, plot the step response of the entire system: stepplot(Try) When you combine Numeric LTI models, the resulting Numeric LTI model represents the aggregate system. The resulting model does not retain the original data from the combined components. For example, T does not separately keep track of the dynamics of the components G, C, and S that are combined to create T. 1-5 1 Linear System Model Objects See Also feedback Related Examples • “Numeric Model of SISO Feedback Loop” on page 4-6 • “Multi-Loop Control System” on page 4-10 • “MIMO Control System” on page 4-19 More About • 1-6 “Types of Model Objects” on page 1-7 Types of Model Objects Types of Model Objects The following diagram illustrates the relationships between the types of model objects in Control System Toolbox, Robust Control Toolbox, and System Identification Toolbox software. Model types that begin with id require System Identification Toolbox software. Model types that begin with u require Robust Control Toolbox software. All other model types are available with Control System Toolbox software. 1-7 1 Linear System Model Objects The diagram illustrates the following two overlapping broad classifications of model object types: 1-8 See Also • Dynamic System Models vs. Static Models — In general, Dynamic System Models represent systems that have internal dynamics, while Static Models represent static input/output relationships. • Numeric Models vs. Generalized Models — Numeric Models are the basic numeric representation of linear systems with fixed coefficients. Generalized Models represent systems with tunable or uncertain components. See Also More About • “What Are Model Objects?” on page 1-2 • “Dynamic System Models” on page 1-10 • “Static Models” on page 1-12 • “Numeric Models” on page 1-13 • “Generalized Models” on page 1-16 1-9 1 Linear System Model Objects Dynamic System Models Dynamic System Models generally represent systems that have internal dynamics or memory of past states such as integrators, delays, transfer functions, and state-space models. Most commands for analyzing linear systems, such as bode, margin, and linearSystemAnalyzer, work on most Dynamic System Model objects. For Generalized Models, analysis commands use the current value of tunable parameters and the nominal value of uncertain parameters. Commands that generate response plots display random samples of uncertain models. The following table lists the Dynamic System Models. Model Family Model Types Numeric LTI models — Basic numeric representation of linear systems (requires Control System Toolbox) tf zpk ss frd pid pidstd pid2 pidstd2 Identified LTI models — Representations of linear systems with tunable coefficients, whose values can be identified using measured input/output data. (requires System Identification Toolbox) idtf idss idfrd idgrey idpoly idproc Identified nonlinear models — Representations of nonlinear systems with tunable coefficients, whose values can be identified using input/output data. Limited 1-10 idnlarx idnlhw See Also Model Family support for commands that analyze linear systems. (requires System Identification Toolbox) Model Types idnlgrey Generalized LTI models — Representations of systems that include tunable or uncertain coefficients (tunable models require Control System Toolbox; uncertain models require Robust Control Toolbox) genss Dynamic Control Design Blocks — Tunable, uncertain, or switch analysis points for constructing models of control systems (tunable Control Design Blocks and analysis points require Control System Toolbox; uncertain Control Design Blocks require Robust Control Toolbox) tunableGain genfrd uss ufrd tunableTF tunableSS tunablePID tunablePID2 ultidyn udyn AnalysisPoint See Also More About • “Numeric Linear Time Invariant (LTI) Models” on page 1-13 • “Identified LTI Models” on page 1-14 • “Identified Nonlinear Models” on page 1-14 • “Generalized and Uncertain LTI Models” on page 1-16 • “Control Design Blocks” on page 1-16 1-11 1 Linear System Model Objects Static Models Static Models represent static input/output relationships and generalize the notions of matrix and numeric array to parametric or uncertain arrays. You can use static models to create parametric or uncertain expressions, and to construct Generalized LTI models whose coefficients are parametric or uncertain expressions. The Static Models family includes: • Tunable parameters (realp objects) • Generalized matrices (genmat objects) • Uncertain parameters and matrices (ureal, ucomplex, ucomplexm) (requires Robust Control Toolbox software) • Uncertain matrices (umat) objects (requires Robust Control Toolbox software) For more information about using these objects to create parametric models, see “Models with Tunable Coefficients” on page 1-19. For information about creating uncertain static models, see “Uncertain Real Parameters” (Robust Control Toolbox) and “Uncertain Matrices” (Robust Control Toolbox). 1-12 Numeric Models Numeric Models Numeric Linear Time Invariant (LTI) Models Numeric LTI models are the basic numeric representation of linear systems or components of linear systems. Use numeric LTI models for modeling dynamic components, such as transfer functions or state-space models, whose coefficients are fixed, numeric values. You can use numeric LTI models for linear analysis or control design tasks. The following table summarizes the available types of numeric LTI models. Model Type Description tf Transfer function model in polynomial form zpk Transfer function model in zero-pole-gain (factorized) form ss State-space model frd Frequency response data model pid Parallel-form PID controller pidstd Standard-form PID controller pid2 Parallel-form two-degree-of-freedom (2-DOF) PID controller pidstd2 Standard-form 2-DOF PID controller Creating Numeric LTI Models For information about creating numeric LTI models, see: • “Transfer Functions” on page 2-3 • “State-Space Models” on page 2-6 • “Frequency Response Data (FRD) Models” on page 2-10 • “Proportional-Integral-Derivative (PID) Controllers” on page 2-13 Applications of Numeric LTI Models You can use Numeric LTI models to represent block diagram components such as plant or sensor dynamics. By connecting Numeric LTI models together, you can derive Numeric LTI models of block diagrams. Use Numeric LTI models for most modeling, analysis, and control design tasks, including: 1-13 1 Linear System Model Objects • Analyzing linear system dynamics using analysis commands such as bode, step, or impulse. • Designing controllers for linear systems using the Control System Designer app or the PID Tuner GUI. • Designing controllers using control design commands such as pidtune, rlocus, or lqr/lqg. Identified LTI Models Identified LTI Models represent linear systems with coefficients that are identified using measured input/output data (requires System Identification Toolbox software). You can specify initial values and constraints for the estimation of the coefficients. The following table summarizes the available types of identified LTI models. Model Type Description idtf Transfer function model in polynomial form, with identifiable parameters idss State-space model, with identifiable parameters idpoly Polynomial input-output model, with identifiable parameters idproc Continuous-time process model, with identifiable parameters idfrd Frequency-response model, with identifiable parameters idgrey Linear ODE (grey-box) model, with identifiable parameters Identified Nonlinear Models Identified Nonlinear Models represent nonlinear systems with coefficients that are identified using measured input/output data (requires System Identification Toolbox software). You can specify initial values and constraints for the estimation of the coefficients. The following table summarizes the available types of identified nonlinear models. 1-14 Numeric Models Model Type Description idnlarx Nonlinear ARX model, with identifiable parameters idnlgrey Nonlinear ODE (grey-box) model, with identifiable parameters idnlhw Hammerstein-Wiener model, with identifiable parameters 1-15 1 Linear System Model Objects Generalized Models Generalized and Uncertain LTI Models Generalized LTI Models represent systems having a mixture of fixed coefficients and tunable or uncertain coefficients. Generalized LTI models arise from combining numeric LTI models with Control Design Blocks. For more information about tunable Generalized LTI models and their applications, see “Models with Tunable Coefficients” on page 1-19. Uncertain LTI Models are a special type of Generalized LTI model that include uncertain coefficients but not tunable coefficients. For more information about using uncertain models, see “Uncertain State-Space Models” (Robust Control Toolbox) and “Create Uncertain Frequency Response Data Models” (Robust Control Toolbox). Family Model Type Description Generalized LTI Models genss Generalized LTI model arising from combination of Numeric LTI models (except frd models) with Control Design Blocks genfrd Generalized LTI model arising from combination frd models with Control Design Blocks uss Generalized LTI model arising from combination of Numeric LTI models (except frd models) with uncertain Control Design Blocks ufrd Generalized LTI model arising from combination frd models with uncertain Control Design Blocks Uncertain LTI Models (requires Robust Control Toolbox software) Control Design Blocks Control Design Blocks are building blocks for constructing tunable or uncertain models of control systems. Combine tunable Control Design Blocks with numeric arrays or Numeric LTI models to create Generalized Matrices or Generalized LTI models that include both fixed and tunable components. 1-16 Generalized Models Tunable Control Design Blocks include tunable parameter objects as well as tunable linear models with predefined structure. For more information about using tunable Control Design Blocks, see “Models with Tunable Coefficients” on page 1-19. If you have Robust Control Toolbox software, you can use uncertain Control Design Blocks to model uncertain parameters or uncertain system dynamics. For more information about using uncertain blocks, see “Uncertain LTI Dynamics Elements” (Robust Control Toolbox), “Uncertain Real Parameters” (Robust Control Toolbox), and “Uncertain Complex Parameters and Matrices” (Robust Control Toolbox). The following tables summarize the available types of Control Design Blocks. Dynamic System Model Control Design Blocks Family Model Type Description Tunable Linear Components tunableGain Tunable gain block tunableTF SISO fixed-order transfer function with tunable coefficients tunableSS Fixed-order state-space model with tunable coefficients tunablePID One-degree-of-freedom PID controller with tunable coefficients tunablePID2 Two-degree-of-freedom PID controller with tunable coefficients Uncertain Dynamics (requires Robust Control Toolbox software) ultidyn Uncertain linear timeinvariant dynamics udyn Unstructured uncertain dynamics Analysis Point Block AnalysisPoint Points of interest for linear analysis or control system tuning 1-17 1 Linear System Model Objects Static Model Control Design Blocks Family Model Type Description Tunable Parameter realp Tunable scalar parameter or matrix Uncertain Parameters (requires Robust Control Toolbox software) ureal Uncertain real scalar ucomplex Uncertain complex scalar ucomplexm Uncertain complex matrix Generalized Matrices Generalized Matrices extend the notion of numeric matrices to matrices that include tunable or uncertain values. Create tunable generalized matrices by building rational expressions involving realp parameters. You can use generalized matrices as inputs to tf or ss to create tunable linear models with structures other than the predefined structures of the Control Design Blocks. Use such models for parameter studies or some compensator tuning tasks. If you have Robust Control Toolbox software, you can create uncertain matrices by building rational expressions involving uncertain parameters such as ureal or ucomplex. Model Type Description genmat Generalized matrix that includes parametric or tunable entries umat (requires Robust Control Toolbox software) Generalized matrix that includes uncertain entries For more information about generalized matrices and their applications, see “Models with Tunable Coefficients” on page 1-19. 1-18 Models with Tunable Coefficients Models with Tunable Coefficients Tunable Generalized LTI Models Tunable Generalized LTI models represent systems having both fixed and tunable (or parametric) coefficients. You can use tunable Generalized LTI models to: • Model a tunable (or parametric) component of a control system, such as a tunable lowpass filter. • Model a control system that contains both: • Fixed components, such as plant dynamics and sensor dynamics • Tunable components, such as filters and compensators You can use tunable Generalized LTI models for parameter studies. For an example, see “Study Parameter Variation by Sampling Tunable Model” on page 9-9. You can also use tunable Generalized LTI models for tuning fixed control structures using tuning commands such as systune or the Control System Tuner app. See “Multiloop, Multiobjective Tuning”. Modeling Tunable Components Control System Toolbox includes tunable components with predefined structure called “Control Design Blocks” on page 1-16. You can use tunable Control Design Blocks to model any tunable component that fits one of the predefined structures. To create tunable components with a specific custom structure that is not covered by the Control Design Blocks: 1 Use the tunable real parameter realp or the generalized matrix genmat to represent the tunable coefficients of your component. 2 Use the resulting realp or genmat objects as inputs to tf or ss to model the component. The result is a generalized state-space (genss) model of the component. For examples of creating such custom tunable components, see: • “Create Tunable Low-Pass Filter” on page 2-78 1-19 1 Linear System Model Objects • “Create Tunable Second-Order Filter” on page 2-79 • “Create State-Space Model with Both Fixed and Tunable Parameters” on page 2-81 Modeling Control Systems with Tunable Components To construct a tunable Generalized LTI model representing a control system with both fixed and tunable components: 1 Model the nontunable components of your system using numeric LTI models on page 1-13. 2 Model each tunable component using Control Design Blocks or expressions involving such blocks. See “Modeling Tunable Components” on page 1-19. 3 Use model interconnection commands such as series, parallel or connect, or the arithmetic operators +, -, *, /, \, and ^, to combine all the components of your system. The resulting model is: • A genss model, if none of the nontunable components is a frequency response data model (for example, frd) • A genfrd model, if the nontunable component is a frd model For an example of constructing a genss model of a control system with both fixed and tunable components, see “Control System with Tunable Components” on page 2-83. Internal Structure of Generalized Models A Generalized model separately stores the numeric and parametric portions of the model by structuring the model in Standard Form, as shown in the following illustration. 1-20 Models with Tunable Coefficients z w H y u B1 0 ... 0 . .. 0 B2 .. . 0 .. ... . 0 0 BN B w and z represent the inputs and outputs of the Generalized model. H represents all portions of the Generalized model that have fixed (non-parametric) coefficients. H is: • A state-space (ss) model, for genss models • A frequency response data (frd) model, for genfrd models • A matrix, for genmat models B represents the parametric components of the Generalized model, which are the Control Design Blocks B1, . . . , BN. The Blocks property of the Generalized model stores a list of the names of these blocks. If the Generalized model has blocks that occur multiple times in B1, . . . , BN, these are only listed once in the Blocks property. To access the internal representation of a Generalized model, including H and B, use the getLFTModel command. 1-21 1 Linear System Model Objects This Standard Form can represent any control structure. To understand why, consider the control structure as an aggregation of fixed-coefficient elements interacting with the parametric elements: external inputs w H Fixed system components (actuators, sensors, etc.) y1 uN B1 yN u1 y2 B2 external outputs z BN ... u2 To rewrite this in Standard Form, define u := [u1 ,…, uN ] y := [ y1 ,… , yN ] , and group the tunable control elements B1, . . . , BN into the block-diagonal configuration C. P includes all the fixed components of the control architecture—actuators, sensors, and other nontunable elements—and their interconnections. 1-22 Using Model Objects Using Model Objects After you represent your dynamic system as a model object, you can: • Attach additional information to the model using model attributes (properties). See “Model Attributes”. • Manipulate the model using arithmetic and model interconnection operations. See “Model Interconnection”. • Analyze the model response using commands such as bode and step. See “Linear Analysis”. • Perform parameter studies using model arrays. See “Model Arrays”. • Design compensators. You can: • Design compensators for systems specified as numeric LTI models. Available compensator design techniques include PID tuning, root locus analysis, pole placement, LQG optimal control, and frequency domain loop-shaping. See “PID Controller Tuning”, “Classical Control Design”, or “State-Space Control Design”. • Manually tune many control architectures using Control System Designer. See “Classical Control Design”. • Use tuning commands such as systune or Control System Tuner to automatically tune a control system that you represent as a genss model with tunable blocks. See “Multiloop, Multiobjective Tuning”. 1-23 1 Linear System Model Objects References [1] Dorf, R.C. and R.H. Bishop, Modern Control Systems, Addison-Wesley, Menlo Park, CA, 1998. 1-24 2 Model Creation • “Transfer Functions” on page 2-3 • “State-Space Models” on page 2-6 • “Frequency Response Data (FRD) Models” on page 2-10 • “Proportional-Integral-Derivative (PID) Controllers” on page 2-13 • “Two-Degree-of-Freedom PID Controllers” on page 2-16 • “Discrete-Time Numeric Models” on page 2-23 • “Discrete-Time Proportional-Integral-Derivative (PID) Controllers” on page 2-24 • “MIMO Transfer Functions” on page 2-28 • “MIMO State-Space Models” on page 2-31 • “MIMO Frequency Response Data Models” on page 2-37 • “Select Input/Output Pairs in MIMO Models” on page 2-39 • “Time Delays in Linear Systems” on page 2-40 • “Closing Feedback Loops with Time Delays” on page 2-45 • “Time-Delay Approximation” on page 2-48 • “Time-Delay Approximation in Continuous-Time Open-Loop Model” on page 2-50 • “Time-Delay Approximation in Continuous-Time Closed-Loop Model” on page 2-55 • “Approximate Different Delays with Different Approximation Orders” on page 2-60 • “Convert Time Delay in Discrete-Time Model to Factors of 1/z” on page 2-64 • “Frequency Response Data (FRD) Model with Time Delay” on page 2-69 • “Internal Delays” on page 2-73 • “Create Tunable Low-Pass Filter” on page 2-78 • “Create Tunable Second-Order Filter” on page 2-79 • “Create State-Space Model with Both Fixed and Tunable Parameters” on page 2-81 • “Control System with Tunable Components” on page 2-83 • “Control System with Multichannel Analysis Points” on page 2-85 • “Mark Signals of Interest for Control System Analysis and Design” on page 2-89 2 Model Creation • “Model Arrays” on page 2-100 • “Select Models from Array” on page 2-103 • “Query Array Size and Characteristics” on page 2-106 • “Linear Parameter-Varying Models” on page 2-109 • “Using LTI Arrays for Simulating Multi-Mode Dynamics” on page 2-118 2-2 Transfer Functions Transfer Functions Transfer Function Representations Control System Toolbox software supports transfer functions that are continuous-time or discrete-time, and SISO or MIMO. You can also have time delays in your transfer function representation. A SISO continuous-time transfer function is expressed as the ratio: G ( s) = N ( s) D( s ) , of polynomials N(s) and D(s), called the numerator and denominator polynomials, respectively. You can represent linear systems as transfer functions in polynomial or factorized (zeropole-gain) form. For example, the polynomial-form transfer function: G ( s) = s2 - 3 s - 4 s2 + 5 s + 6 can be rewritten in factorized form as: G ( s) = ( s + 1) (s - 4) . ( s + 2)( s + 3) The tf model object represents transfer functions in polynomial form. The zpk model object represents transfer functions in factorized form. MIMO transfer functions are arrays of SISO transfer functions. For example: Ès - 3˘ Ís + 4˙ G ( s) = Í ˙ Í s + 1˙ ÍÎ s + 2 ˙˚ is a one-input, two output transfer function. 2-3 2 Model Creation Commands for Creating Transfer Functions Use the commands described in the following table to create transfer functions. Command Description tf Create tf objects representing continuous-time or discretetime transfer functions in polynomial form. zpk Create zpk objects representing continuous-time or discretetime transfer functions in zero-pole-gain (factorized) form. filt Create tf objects representing discrete-time transfer functions using digital signal processing (DSP) convention. Create Transfer Function Using Numerator and Denominator Coefficients This example shows how to create continuous-time single-input, single-output (SISO) transfer functions from their numerator and denominator coefficients using tf. Create the transfer function G ( s ) = s 2 : s + 3s + 2 num = [1 0]; den = [1 3 2]; G = tf(num,den); num and den are the numerator and denominator polynomial coefficients in descending powers of s. For example, den = [1 3 2] represents the denominator polynomial s2 + 3s + 2. G is a tf model object, which is a data container for representing transfer functions in polynomial form. Tip Alternatively, you can specify the transfer function G(s) as an expression in s: 1 Create a transfer function model for the variable s. s = tf('s'); 2 2-4 Specify G(s) as a ratio of polynomials in s. See Also G = s/(s^2 + 3*s + 2); Create Transfer Function Model Using Zeros, Poles, and Gain This example shows how to create single-input, single-output (SISO) transfer functions in factored form using zpk. Create the factored transfer function G ( s ) = 5 Z P K G = = = = s ( s + 1 + i) ( s + 1 - i) ( s + 2) : [0]; [-1-1i -1+1i -2]; 5; zpk(Z,P,K); Z and P are the zeros and poles (the roots of the numerator and denominator, respectively). K is the gain of the factored form. For example, G(s) has a real pole at s = – 2 and a pair of complex poles at s = –1 ± i. The vector P = [-1-1i -1+1i -2] specifies these pole locations. G is a zpk model object, which is a data container for representing transfer functions in zero-pole-gain (factorized) form. See Also filt | tf | zpk Related Examples • “MIMO Transfer Functions” on page 2-28 • “State-Space Models” on page 2-6 • “Discrete-Time Numeric Models” on page 2-23 More About • “What Are Model Objects?” on page 1-2 • “Store and Retrieve Model Data” on page 3-2 2-5 2 Model Creation State-Space Models State-Space Model Representations State-space models rely on linear differential equations or difference equations to describe system dynamics. Control System Toolbox software supports SISO or MIMO state-space models in continuous or discrete time. State-space models can include time delays. You can represent state-space models in either explicit or descriptor (implicit) form. State-space models can result from: • Linearizing a set of ordinary differential equations that represent a physical model of the system. • State-space model identification using System Identification Toolbox software. • State-space realization of transfer functions. (See “Conversion Between Model Types” on page 5-2 for more information.) Use ss model objects to represent state-space models. Explicit State-Space Models Explicit continuous-time state-space models have the following form: dx = Ax + Bu dt y = Cx + Du where x is the state vector. u is the input vector, and y is the output vector. A, B, C, and D are the state-space matrices that express the system dynamics. A discrete-time explicit state-space model takes the following form: x [ n + 1] = Ax[ n] + Bu [ n] y [ n] = Cx [ n ] + Du [ n] where the vectors x[n], u[n], and y[n] are the state, input, and output vectors for the nth sample. 2-6 State-Space Models Descriptor (Implicit) State-Space Models A descriptor state-space model is a generalized form of state-space model. In continuous time, a descriptor state-space model takes the following form: E dx = Ax + Bu dt y = Cx + Du where x is the state vector. u is the input vector, and y is the output vector. A, B, C, D, and E are the state-space matrices. Commands for Creating State-Space Models Use the commands described in the following table to create state-space models. Command Description ss Create explicit state-space model. dss Create descriptor (implicit) state-space model. delayss Create state-space models with specified time delays. Create State-Space Model From Matrices This example shows how to create a continuous-time single-input, single-output (SISO) state-space model from state-space matrices using ss. Create a model of an electric motor where the state-space equations are: dx = Ax + Bu dt y = Cx + Du where the state variables are the angular position θ and angular velocity dθ/dt: Èq ˘ x = Í dq ˙ , Í ˙ ÎÍ dt ˙˚ 2-7 2 Model Creation u is the electric current, the output y is the angular velocity, and the state-space matrices are: È0 1˘ A=Í ˙, Î -5 -2 ˚ È0 ˘ B = Í ˙ , C = [ 0 1] , Î3 ˚ D = [ 0 ]. To create this model, enter: A = B = C = D = sys [0 1;-5 -2]; [0;3]; [0 1]; 0; = ss(A,B,C,D); sys is an ss model object, which is a data container for representing state-space models. Tip To represent a system of the form: E dx = Ax + Bu dt y = Cx + Du use dss. This command creates a ss model with a nonempty E matrix, also called a descriptor state-space model. See “MIMO Descriptor State-Space Models” on page 2-32 for an example. See Also delayss | dss | ss Related Examples 2-8 • “MIMO State-Space Models” on page 2-31 • “Transfer Functions” on page 2-3 • “Discrete-Time Numeric Models” on page 2-23 See Also More About • “What Are Model Objects?” on page 1-2 • “Store and Retrieve Model Data” on page 3-2 2-9 2 Model Creation Frequency Response Data (FRD) Models Frequency Response Data In the Control System Toolbox software, you can use frd models to store, manipulate, and analyze frequency response data. An frd model stores a vector of frequency points with the corresponding complex frequency response data you obtain either through simulations or experimentally. For example, suppose you measure frequency response data for the SISO system you want to model. You can measure such data by driving the system with a sine wave at a set of frequencies ω1, ω2, ,...,ωn, as shown: sini (wit) Gi (w) yi (t) At steady state, the measured response yi(t) to the driving signal at each frequency ωi takes the following form: yi ( t ) = a sin (wi t + b ) , i = 1,…, n. The measurement yields the complex frequency response G at each input frequency: G ( jwi ) = ae jb, i = 1,…, n. You can do most frequency-domain analysis tasks on frd models, but you cannot perform time-domain simulations with them. For information on frequency response analysis of linear systems, see Chapter 8 of [1]. Commands for Creating FRD Models Use the following commands to create FRD models. 2-10 Command Description frd Create frd objects from frequency response data. Frequency Response Data (FRD) Models Command Description frestimate Create frd objects by estimating the frequency response of a Simulink® model. This approach requires Simulink Control Design™ software. See “Frequency Response Estimation” (Simulink Control Design) for more information. Create Frequency-Response Model from Data This example shows how to create a single-input, single-output (SISO) frequency-response model using frd. A frequency-response model stores a vector of frequency points with corresponding complex frequency response data you obtain either through simulations or experimentally. Thus, if you measure the frequency response of your system at a set of test frequencies, you can use the data to create a frequency response model: 1 Load the frequency response data in AnalyzerData.mat. load AnalyzerData This command loads the data into the MATLAB workspace as the column vectors freq and resp. The variables freq and resp contain 256 test frequencies and the corresponding complex-valued frequency response points, respectively. Tip To inspect these variables, enter: whos freq resp 2 Create a frequency response model. sys = frd(resp,freq); sys is an frd model object, which is a data container for representing frequency response data. You can use frd models with many frequency-domain analysis commands. For example, visualize the frequency response data using bode. 2-11 2 Model Creation Tip By default, the frd command assumes that the frequencies are in radians/second. To specify different frequency units, use the TimeUnit and FrequencyUnit properties of the frd model object. For example: sys = frd(resp,freq,'TimeUnit','min','FrequencyUnit','rad/TimeUnit') sets the frequency units to radians/minute. See Also frd | frestimate Related Examples • “MIMO Frequency Response Data Models” on page 2-37 • “Discrete-Time Numeric Models” on page 2-23 More About 2-12 • “What Are Model Objects?” on page 1-2 • “Store and Retrieve Model Data” on page 3-2 Proportional-Integral-Derivative (PID) Controllers Proportional-Integral-Derivative (PID) Controllers You can represent PID controllers using the specialized model objects pid and pidstd. This topic describes the representation of PID controllers in MATLAB. For information about automatic PID controller tuning, see “PID Controller Tuning”. Continuous-Time PID Controller Representations You can represent continuous-time Proportional-Integral-Derivative (PID) controllers in either parallel or standard form. The two forms differ in the parameters used to express the proportional, integral, and derivative actions and the filter on the derivative term, as shown in the following table. Form Parallel (pid object) Formula C = Kp + Ki Kd s + , s Tf s + 1 where: • Kp = proportional gain • Ki = integrator gain • Kd = derivative gain • Tf = derivative filter time Standard (pidstd object) Ê ˆ Á Td s ˜ 1 C = K p Á1 + + ˜, Ti s Td Á s + 1 ˜˜ Á Ë N ¯ where: • Kp = proportional gain • Ti = integrator time • Td = derivative time • N = derivative filter divisor 2-13 2 Model Creation Use a controller form that is convenient for your application. For example, if you want to express the integrator and derivative actions in terms of time constants, use standard form. For information on representing PID Controllers in discrete time, see “Discrete-Time Proportional-Integral-Derivative (PID) Controllers” on page 2-24 Create Continuous-Time Parallel-Form PID Controller This example shows how to create a continuous-time Proportional-Integral-Derivative (PID) controller in parallel form using pid. Create the following parallel-form PID controller: C = 29 .5 + 26 .2 4.3 s . s 0 .06 s + 1 Kp = 29.5; Ki = 26.2; Kd = 4.3; Tf = 0.06; C = pid(Kp,Ki,Kd,Tf) C is a pid model object, which is a data container for representing parallel-form PID controllers. For more examples of how to create PID controllers, see the pid reference page. Create Continuous-Time Standard-Form PID Controller This example shows how to create a continuous-time Proportional-Integral-Derivative (PID) controller in standard form using pidstd. Ê ˆ Á 1 0 .15 s ˜ Create the following standard-form PID controller: C = 29 .5 Á 1 + + ˜. 1.13s 0 .15 s + 1 ˜ ÁÁ ˜ Kp = 29.5; Ë 2.3 ¯ Ti = 1.13; Td = 0.15; N = 2.3; C = pidstd(Kp,Ti,Td,N) 2-14 See Also C is a pidstd model object, which is a data container for representing standard-form PID controllers. For more examples of how to create standard-form PID controllers, see the pidstd reference page. See Also pid | pidTuner | pidstd | pidtune Related Examples • “Transfer Functions” on page 2-3 • “Discrete-Time Proportional-Integral-Derivative (PID) Controllers” on page 2-24 • “Two-Degree-of-Freedom PID Controllers” on page 2-16 More About • “What Are Model Objects?” on page 1-2 • “Store and Retrieve Model Data” on page 3-2 2-15 2 Model Creation Two-Degree-of-Freedom PID Controllers Two-degree-of-freedom (2-DOF) PID controllers include setpoint weighting on the proportional and derivative terms. A 2-DOF PID controller is capable of fast disturbance rejection without significant increase of overshoot in setpoint tracking. 2-DOF PID controllers are also useful to mitigate the influence of changes in the reference signal on the control signal. You can represent PID controllers using the specialized model objects pid2 and pidstd2. This topic describes the representation of 2-DOF PID controllers in MATLAB. For information about automatic PID controller tuning, see “PID Controller Tuning”. Continuous-Time 2-DOF PID Controller Representations This illustration shows a typical control architecture using a 2-DOF PID controller. The relationship between the 2-DOF controller’s output (u) and its two inputs (r and y) can be represented in either parallel or standard form. The two forms differ in the parameters used to express the proportional, integral, and derivative actions of the controller, as expressed in the following table. 2-16 Two-Degree-of-Freedom PID Controllers Form Parallel (pid2 object) Formula u = K p ( br - y ) + Ki K s ( r - y) + d ( cr - y) . s Tf s + 1 In this representation: • Kp = proportional gain • Ki = integrator gain • Kd = derivative gain • Tf = derivative filter time • b = setpoint weight on proportional term • c = setpoint weight on derivative term Standard (pidstd2 object) È ˘ Í ˙ Td s 1 u = K p Í( br - y ) + (r - y) + T ( cr - y ) ˙ . Ti s d Í ˙ s +1 ÍÎ ˙˚ N In this representation: • Kp = proportional gain • Ti = integrator time • Td = derivative time • N = derivative filter divisor • b = setpoint weight on proportional term • c = setpoint weight on derivative term Use a controller form that is convenient for your application. For instance, if you want to express the integrator and derivative actions in terms of time constants, use standard form. For examples showing how to create parallel-form and standard-form controllers, see the pid2 and pidstd2 reference pages, respectively. For information on representing PID Controllers in discrete time, see “Discrete-Time Proportional-Integral-Derivative (PID) Controllers” on page 2-24. 2-17 2 Model Creation 2-DOF Control Architectures The 2-DOF PID controller is a two-input, one output controller of the form C2(s), as shown in the following figure. The transfer function from each input to the output is itself a PID controller. Each of the components Cr(s) and Cy(s) is a PID controller, with different weights on the proportional and derivative terms. For example, in continuous time, these components are given by: Cr ( s ) = bK p + Ki s + cK d s Tf s + 1 , È K Kd s ˘ Cy ( s ) = - Í K p + i + ˙. s Tf s + 1 ˙˚ ÍÎ You can access these components by converting the PID controller into a two-input, oneoutput transfer function. For example, suppose that C2 is a 2-DOF PID controller, stored as a pid2 object. C2tf = tf(C2); Cr = C2tf(1); Cy = C2tf(2); Cr(s) is the transfer function from the first input of C2 to the output. Similarly, Cy(s) is the transfer function from the second input of C2 to the output. 2-18 Two-Degree-of-Freedom PID Controllers Suppose that G is a dynamic system model, such as a zpk model, representing the plant. Build the closed-loop transfer function from r to y. Note that the Cy(s) loop has positive feedback, by the definition of Cy(s). T = Cr*feedback(G,Cy,+1) Alternatively, use the connect command to build an equivalent closed-loop system directly with the 2-DOF controller C2. To do so, set the InputName and OutputName properties of G and C2. G.InputName = 'u'; G.OutputName = 'y'; C2.Inputname = {'r','y'}; C2.OutputName = 'u'; T = connect(G,C2,'r','y'); There are other configurations in which you can decompose a 2-DOF PID controller into SISO components. For particular choices of C(s) and X(s), each of the following configurations is equivalent to the 2-DOF architecture with C2(s). You can obtain C(s) and X(s) for each of these configurations using the getComponents command. Feedforward In the feedforward configuration, the 2-DOF PID controller is decomposed into a conventional SISO PID controller that takes the error signal as its input, and a feedforward controller. For a continuous-time, parallel-form 2-DOF PID controller, the components are given by: 2-19 2 Model Creation C ( s) = K p + Ki Kd s + , s Tf s + 1 X ( s) = ( b - 1 ) K p + ( c - 1 ) Kd s Tf s + 1 . Access these components using getComponents. [C,X] = getComponents(C2,'feedforward'); The following command constructs the closed-loop system from r to y for the feedforward configuration. T = G*(C+X)*feedback(1,G*C); Feedback In the feedback configuration, the 2-DOF PID controller is decomposed into a conventional SISO PID controller and a feedback controller. For a continuous-time, parallel-form 2-DOF PID controller, the components are given by: C ( s) = bK p + Ki cK d s + , s Tf s + 1 X ( s) = (1 - b ) K p + (1 - c ) Kd s Tf s + 1 . Access these components using getComponents. 2-20 Two-Degree-of-Freedom PID Controllers [C,X] = getComponents(C2,'feedback'); The following command constructs the closed-loop system from r to y for the feedback configuration. T = G*C*feedback(1,G*(C+X)); Filter In the filter configuration, the 2-DOF PID controller is decomposed into a conventional SISO PID controller and a prefilter on the reference signal. For a continuous-time, parallel-form 2-DOF PID controller, the components are given by: C ( s) = K p + X ( s) = Ki Kd s + , s Tf s + 1 (bK pTf + cK d ) s2 + ( bK p + K iTf ) s + K i . ( K pTf + Kd ) s2 + ( K p + KiTf ) s + Ki The filter X(s) can also be expressed as the ratio: –[Cr(s)/Cy(s)]. The following command constructs the closed-loop system from r to y for the filter configuration. T = X*feedback(G*C,1); For an example illustrating the decomposition of a 2-DOF PID controller into these configurations, see “Decompose a 2-DOF PID Controller into SISO Components” on page 5-8. The formulas shown above pertain to continuous-time, parallel-form controllers. Standard-form controllers and controllers in discrete time can be decomposed into 2-21 2 Model Creation analogous configurations. The getComponents command works on all 2-DOF PID controller objects. See Also getComponents | pid2 | pidTuner | pidstd2 | pidtune Related Examples • “Discrete-Time Proportional-Integral-Derivative (PID) Controllers” on page 2-24 • “Proportional-Integral-Derivative (PID) Controllers” on page 2-13 More About 2-22 • “What Are Model Objects?” on page 1-2 • “Store and Retrieve Model Data” on page 3-2 Discrete-Time Numeric Models Discrete-Time Numeric Models Create Discrete-Time Transfer Function Model This example shows how to create a discrete-time transfer function model using tf. Create the transfer function G ( z ) = z 2 with a sample time of 0.1 s. z - 2z - 6 num = [1 0]; den = [1 -2 -6]; Ts = 0.1; G = tf(num,den,Ts) num and den are the numerator and denominator polynomial coefficients in descending powers of z. G is a tf model object. The sample time is stored in the Ts property of G. Access the sample time Ts, using dot notation: G.Ts Other Model Types in Discrete Time Representations Create discrete-time zpk, ss, and frd models in a similar way to discrete-time transfer functions, by appending a sample time to the input arguments. For examples, see the reference pages for those commands. See Also frd | ss | tf | zpk More About • “What Are Model Objects?” on page 1-2 2-23 2 Model Creation Discrete-Time Proportional-Integral-Derivative (PID) Controllers All the PID controller object types, pid, pidstd, pid2, and pidstd2, can represent PID controllers in discrete time. Discrete-Time PID Controller Representations Discrete-time PID controllers are expressed by the following formulas. Form Parallel (pid) Formula C = K p + Ki IF ( z) + Kd , Tf + DF ( z ) where: • Kp = proportional gain • Ki = integrator gain • Kd = derivative gain • Tf = derivative filter time Standard (pidstd) Ê ˆ Á ˜ Td 1 C = K p Á 1 + IF ( z) + ˜, Td Ti Á + DF z ( ) ˜˜ Á Ë N ¯ where: • Kp = proportional gain • Ti = integrator time • Td = derivative time • N = derivative filter divisor 2-24 Discrete-Time Proportional-Integral-Derivative (PID) Controllers Form Formula 2-DOF Parallel (pid2) The relationship between the 2-DOF controller’s output (u) and its two inputs (r and y) is: u = K p ( br - y ) + Ki IF ( z ) ( r - y ) + Kd ( cr - y) . Tf + DF ( z ) In this representation: • Kp = proportional gain • Ki = integrator gain • Kd = derivative gain • Tf = derivative filter time • b = setpoint weight on proportional term • c = setpoint weight on derivative term 2-DOF Standard (pidstd2 object) È ˘ Í ˙ Td 1 u = K p Í( br - y ) + IF ( z) ( r - y) + cr - y) ˙ . ( Td Ti Í ˙ + DF ( z ) ÍÎ ˙˚ N In this representation: • Kp = proportional gain • Ti = integrator time • Td = derivative time • N = derivative filter divisor • b = setpoint weight on proportional term • c = setpoint weight on derivative term In all of these expressions, IF(z) and DF(z) are the discrete integrator formulas for the integrator and derivative filter, respectively. Use the IFormula and DFormula properties of the controller objects to set the IF(z) and DF(z) formulas. The next table shows available formulas for IF(z) and DF(z). Ts is the sample time. 2-25 2 Model Creation IFormula or DFormula IF(z) or DF(z) ForwardEuler (default) Ts z -1 BackwardEuler Ts z z -1 Trapezoidal Ts z + 1 2 z -1 If you do not specify a value for IFormula, DFormula, or both when you create the controller object, ForwardEuler is used by default. For more information about setting and changing the discrete integrator formulas, see the reference pages for the controller objects, pid, pidstd, pid2, and pidstd2. Create Discrete-Time Standard-Form PID Controller This example shows how to create a standard-form discrete-time Proportional-IntegralDerivative (PID) controller that has Kp = 29.5, Ti = 1.13, Td = 0.15 N = 2.3, and sample time Ts 0.1 : C = pidstd(29.5,1.13,0.15,2.3,0.1,... 'IFormula','Trapezoidal','DFormula','BackwardEuler') Ts z + 1 Tz and DF ( z ) = s . 2 z -1 z -1 You can set the discrete integrator formulas for a parallel-form controller in the same way, using pid. This command creates a pidstd model with IF ( z) = Discrete-Time 2-DOF PI Controller in Standard Form Create a discrete-time 2-DOF PI controller in standard form, using the trapezoidal discretization formula. Specify the formula using Name,Value syntax. Kp = 1; Ti = 2.4; Td = 0; N = Inf; 2-26 See Also b = 0.5; c = 0; Ts = 0.1; C2 = pidstd2(Kp,Ti,Td,N,b,c,Ts,'IFormula','Trapezoidal') C2 = 1 Ts*(z+1) u = Kp * [(b*r-y) + ---- * -------- * (r-y)] Ti 2*(z-1) with Kp = 1, Ti = 2.4, b = 0.5, Ts = 0.1 Sample time: 0.1 seconds Discrete-time 2-DOF PI controller in standard form Setting Td = 0 specifies a PI controller with no derivative term. As the display shows, the values of N and c are not used in this controller. The display also shows that the trapezoidal formula is used for the integrator. See Also pid | pid2 | pidstd | pidstd2 Related Examples • “Proportional-Integral-Derivative (PID) Controllers” on page 2-13 • “Two-Degree-of-Freedom PID Controllers” on page 2-16 More About • “What Are Model Objects?” on page 1-2 • “Store and Retrieve Model Data” on page 3-2 2-27 2 Model Creation MIMO Transfer Functions MIMO transfer functions are two-dimensional arrays of elementary SISO transfer functions. There are two ways to specify MIMO transfer function models: • Concatenation of SISO transfer function models • Using tf with cell array arguments Concatenation of SISO Models Consider the following single-input, two-output transfer function. È Í H ( s) = Í Í ÍÎ s2 s-1 s+1 s+ 2 ˘ ˙ ˙. ˙ + 4 s + 5 ˙˚ You can specify H(s) by concatenation of its SISO entries. For instance, h11 = tf([1 -1],[1 1]); h21 = tf([1 2],[1 4 5]); or, equivalently, s = tf('s') h11 = (s-1)/(s+1); h21 = (s+2)/(s^2+4*s+5); can be concatenated to form H(s). H = [h11; h21] This syntax mimics standard matrix concatenation and tends to be easier and more readable for MIMO systems with many inputs and/or outputs. Tip Use zpk instead of tf to create MIMO transfer functions in factorized form on page 2-5. 2-28 See Also Using the tf Function with Cell Arrays Alternatively, to define MIMO transfer functions using tf, you need two cell arrays (say, N and D) to represent the sets of numerator and denominator polynomials, respectively. See “What Is a Cell Array?” (MATLAB) for more details on cell arrays. For example, for the rational transfer matrix H(s), the two cell arrays N and D should contain the row-vector representations of the polynomial entries of È s-1 ˘ N ( s) = Í ˙, Îs + 2˚ È s +1 ˘ D( s) = Í ˙. Î s2 + 4 s + 5 ˚ You can specify this MIMO transfer matrix H(s) by typing N = {[1 -1];[1 2]}; % Cell array for N(s) D = {[1 1];[1 4 5]}; % Cell array for D(s) H = tf(N,D) Transfer function from input to output... s - 1 #1: ----s + 1 #2: s + 2 ------------s^2 + 4 s + 5 Notice that both N and D have the same dimensions as H. For a general MIMO transfer matrix H(s), the cell array entries N{i,j} and D{i,j} should be row-vector representations of the numerator and denominator of Hij(s), the ijth entry of the transfer matrix H(s). See Also tf | zpk Related Examples • “Transfer Functions” on page 2-3 2-29 2 Model Creation More About 2-30 • “What Are Model Objects?” on page 1-2 • “Store and Retrieve Model Data” on page 3-2 MIMO State-Space Models MIMO State-Space Models MIMO Explicit State-Space Models You create a MIMO state-space model in the same way as you create a SISO state-space model on page 2-7. The only difference between the SISO and MIMO cases is the dimensions of the state-space matrices. The dimensions of the B, C, and D matrices increase with the numbers of inputs and outputs as shown in the following illustration. # of columns = # of inputs # of rows = # of outputs A B C D In this example, you create a state-space model for a rotating body with inertia tensor J, damping force F, and three axes of rotation, related as: J dw + Fw = T dt y = w. The system input T is the driving torque. The output y is the vector of angular velocities of the rotating body. To express this system in state-space form: 2-31 2 Model Creation dx = Ax + Bu dt y = Cx + Du rewrite it as: dw = - J -1 Fw + J -1T dt y = w. Then the state-space matrices are: A = - J -1 F, B = J -1 , C = I , D = 0. To create this model, enter the following commands: J = [8 -3 -3; -3 8 -3; -3 -3 8]; F = 0.2*eye(3); A = -J\F; B = inv(J); C = eye(3); D = 0; sys_mimo = ss(A,B,C,D); These commands assume that J is the inertia tensor of a cube rotating about its corner, and the damping force has magnitude 0.2. sys_mimo is an ss model. MIMO Descriptor State-Space Models This example shows how to create a continuous-time descriptor (implicit) state-space model using dss. This example uses the same rotating-body system shown in “MIMO Explicit State-Space Models” on page 2-31, where you inverted the inertia matrix J to obtain the value of the B matrix. If J is poorly-conditioned for inversion, you can instead use a descriptor (implicit) state-space model. A descriptor (implicit) state-space model is of the form: 2-32 MIMO State-Space Models E dx = Ax + Bu dt y = Cx + Du Create a state-space model for a rotating body with inertia tensor J, damping force F, and three axes of rotation, related as: J dw + Fw = T dt y = w. The system input T is the driving torque. The output y is the vector of angular velocities of the rotating body. You can write this system as a descriptor state-space model having the following state-space matrices: A = - F, B = I, C = I, D = 0, E = J. To create this system, enter: J = [8 -3 -3; -3 8 -3; -3 -3 8]; F = 0.2*eye(3); A = -F; B = eye(3); C = eye(3); D = 0; E = J; sys_mimo = dss(A,B,C,D,E) These commands assume that J is the inertia tensor of a cube rotating about its corner, and the damping force has magnitude 0.2. sys is an ss model with a nonempty E matrix. State-Space Model of Jet Transport Aircraft This example shows how to build a MIMO model of a jet transport. Because the development of a physical model for a jet aircraft is lengthy, only the state-space equations are presented here. See any standard text in aviation for a more complete discussion of the physics behind aircraft flight. The jet model during cruise flight at MACH = 0.8 and H = 40,000 ft. is 2-33 2 Model Creation A = [-0.0558 0.5980 -3.0500 0 B = [ 0.0073 -0.4750 0.1530 0 C = [0 0 1 0 D = [0 0 0 0]; -0.9968 -0.1150 0.3880 0.0805 0.0802 -0.0318 -0.4650 1.0000 0.0415 0 0 0]; 0 0.0077 0.1430 0]; 0 0 0 1]; Use the following commands to specify this state-space model as an LTI object and attach names to the states, inputs, and outputs. states = {'beta' 'yaw' 'roll' 'phi'}; inputs = {'rudder' 'aileron'}; outputs = {'yaw rate' 'bank angle'}; sys_mimo = ss(A,B,C,D,'statename',states,... 'inputname',inputs,... 'outputname',outputs); You can display the LTI model by typing sys_mimo. sys_mimo a = beta yaw roll phi beta -0.0558 0.598 -3.05 0 yaw -0.9968 -0.115 0.388 0.0805 beta yaw roll rudder 0.0073 -0.475 0.153 aileron 0 0.0077 0.143 b = 2-34 roll 0.0802 -0.0318 -0.465 1 phi 0.0415 0 0 0 MIMO State-Space Models phi 0 0 yaw rate bank angle beta 0 0 yaw 1 0 rudder 0 0 aileron 0 0 c = roll 0 0 phi 0 1 d = yaw rate bank angle Continuous-time model. The model has two inputs and two outputs. The units are radians for beta (sideslip angle) and phi (bank angle) and radians/sec for yaw (yaw rate) and roll (roll rate). The rudder and aileron deflections are in degrees. As in the SISO case, use tf to derive the transfer function representation. tf(sys_mimo) Transfer function from input "rudder" to output... -0.475 s^3 - 0.2479 s^2 - 0.1187 s - 0.05633 yaw rate: --------------------------------------------------s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674 bank angle: 0.1148 s^2 - 0.2004 s - 1.373 --------------------------------------------------s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674 Transfer function from input "aileron" to output... 0.0077 s^3 - 0.0005372 s^2 + 0.008688 s + 0.004523 yaw rate: --------------------------------------------------s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674 bank angle: 0.1436 s^2 + 0.02737 s + 0.1104 --------------------------------------------------s^4 + 0.6358 s^3 + 0.9389 s^2 + 0.5116 s + 0.003674 2-35 2 Model Creation See Also ss Related Examples • “State-Space Models” on page 2-6 More About 2-36 • “What Are Model Objects?” on page 1-2 • “Store and Retrieve Model Data” on page 3-2 MIMO Frequency Response Data Models MIMO Frequency Response Data Models This example shows how to create a MIMO frequency-response model using frd. Frequency response data for a MIMO system includes a vector of complex response data for each of the input/output (I/O) pair of the system. Thus, if you measure the frequency response of each I/O pair of your system at a set of test frequencies, you can use the data to create a frequency response model: 1 Load frequency response data in AnalyzerDataMIMO.mat. load AnalyzerDataMIMO H11 H12 H21 H22 freq This command loads the data into the MATLAB workspace as five column vectors H11, H12, H21, H22, and freq. The vector freq contains 100 test frequencies. The other four vectors contain the corresponding complex-valued frequency response of each I/O pair of a two-input, two-output system. Tip To inspect these variables, enter: whos H11 H12 H21 H22 freq 2 Organize the data into a three-dimensional array. Hresp = zeros(2,2,length(freq)); Hresp(1,1,:) = H11; Hresp(1,2,:) = H12; Hresp(2,1,:) = H21; Hresp(2,2,:) = H22; The dimensions of Hresp are the number of outputs, number of inputs, and the number of frequencies for which there is response data. Hresp(i,j,:) contains the frequency response from input j to output i. 3 Create a frequency-response model. H = frd(Hresp,freq); H is an frd model object, which is a data container for representing frequency response data. You can use frd models with many frequency-domain analysis commands. For example, visualize the response of this two-input, two-output system using bode. 2-37 2 Model Creation Tip By default, the frd command assumes that the frequencies are in radians/second. To specify different frequency units, use the TimeUnit and FrequencyUnit properties of the frd model object. For example: H = frd(Hresp,freq,'TimeUnit','min','FrequencyUnit','rad/TimeUnit') sets the frequency units to in radians/minute. See Also frd Related Examples • “Frequency Response Data (FRD) Models” on page 2-10 More About 2-38 • “What Are Model Objects?” on page 1-2 • “Store and Retrieve Model Data” on page 3-2 Select Input/Output Pairs in MIMO Models Select Input/Output Pairs in MIMO Models This example shows how to select the response from the first input to the second output of a MIMO model. 1 Create a two-input, one-output transfer function. N = {[1 -1],[1];[1 2],[3 1 4]}; D = [1 1 10]; H = tf(N,D) Note For more information about using cell arrays to create MIMO transfer functions, see the tf reference page. 2 Select the response from the second input to the output of H. To do this, use MATLAB array indexing. H12 = H(1,2) For any MIMO system H, the index notation H(i,j) selects the response from the jth input to the ith output. See Also Related Examples • “MIMO Transfer Functions” on page 2-28 • “MIMO State-Space Models” on page 2-31 More About • “Store and Retrieve Model Data” on page 3-2 2-39 2 Model Creation Time Delays in Linear Systems Use the following model properties to represent time delays in linear systems. • InputDelay, OutputDelay — Time delays at system inputs or outputs • ioDelay, InternalDelay — Time delays that are internal to the system In discrete-time models, these properties are constrained to integer values that represent delays expressed as integer multiples of the sample time. To approximate discrete-time models with delays that are a fractional multiple of the sample time, use thiran. First Order Plus Dead Time Model This example shows how to create a first order plus dead time model using the InputDelay or OutputDelay properties of tf. To create the following first-order transfer function with a 2.1 s time delay: G ( s ) = e-2.1s 1 , s + 10 enter: G = tf(1,[1 10],'InputDelay',2.1) where InputDelay specifies the delay at the input of the transfer function. Tip You can use InputDelay with zpk the same way as with tf: G = zpk([],-10,1,'InputDelay',2.1) For SISO transfer functions, a delay at the input is equivalent to a delay at the output. Therefore, the following command creates the same transfer function: G = tf(1,[1 10],'OutputDelay',2.1) Use dot notation to examine or change the value of a time delay. For example, change the time delay to 3.2 as follows: 2-40 Time Delays in Linear Systems G.OutputDelay = 3.2; To see the current value, enter: G.OutputDelay ans = 3.2000 Tip An alternative way to create a model with a time delay is to specify the transfer function with the delay as an expression in s: 1 Create a transfer function model for the variable s. s = tf('s'); 2 Specify G(s) as an expression in s. G = exp(-2.1*s)/(s+10); Input and Output Delay in State-Space Model This example shows how to create state-space models with delays at the inputs and outputs, using the InputDelay or OutputDelay properties of ss. Create a state-space model describing the following one-input, two-output system: dx ( t ) dt = -2 x ( t ) + 3u ( t - 1 .5 ) È x ( t - 0 .7 ) ˘ y (t ) = Í ˙. Î - x (t ) ˚ This system has an input delay of 1.5. The first output has an output delay of 0.7, and the second output is not delayed. Note In contrast to SISO transfer functions, input delays are not equivalent to output delays for state-space models. Shifting a delay from input to output in a state-space model requires introducing a time shift in the model states. For example, in the model of this 2-41 2 Model Creation example, defining T = t – 1.5 and X(T) = x(T + 1.5) results in the following equivalent system: dX ( T ) dT = -2 X ( T ) + 3u ( T ) È X ( T - 2 .2 ) ˘ y (T ) = Í ˙. Î - X ( T - 1 .5 ) ˚ All of the time delays are on the outputs, but the new state variable X is time-shifted relative to the original state variable x. Therefore, if your states have physical meaning, or if you have known state initial conditions, consider carefully before shifting time delays between inputs and outputs. To create this system: 1 Define the state-space matrices. A B C D 2 = = = = -2; 3; [1;-1]; 0; Create the model. G = ss(A,B,C,D,'InputDelay',1.5,'OutputDelay',[0.7;0]) G is a ss model. Tip Use delayss to create state-space models with more general combinations of input, output, and state delays, of the form: N dx = Ax (t ) + Bu( t ) + Â ( Ajx (t - tj ) + Bju (t - t j)) dt j =1 N y (t ) = Cx (t ) + Du (t ) + Â ( Cjx( t - tj) + Dju (t - tj )) j =1 2-42 Time Delays in Linear Systems Transport Delay in MIMO Transfer Function This example shows how to create a MIMO transfer function with different transport delays for each input-output (I/O) pair. Create the MIMO transfer function: È - 0 .1 2 Íe s H (s ) = Í Í 10 ÍÎ s +1 ˘ s + 10 ˙ ˙. -0.2 s - 1 ˙ e s + 5 ˙˚ e - 0 .3 Time delays in MIMO systems can be specific to each I/O pair, as in this example. You cannot use InputDelay and OutputDelay to model I/O-specific transport delays. Instead, use ioDelay to specify the transport delay across each I/O pair. To create this MIMO transfer function: 1 Create a transfer function model for the variable s. s = tf('s'); 2 Use the variable s to specify the transfer functions of H without the time delays. H = [2/s (s+1)/(s+10); 10 (s-1)/(s+5)]; 3 Specify the ioDelay property of H as an array of values corresponding to the transport delay for each I/O pair. H.IODelay = [0.1 0.3; 0 0.2]; H is a two-input, two-output tf model. Each I/O pair in H has the time delay specified by the corresponding entry in tau. Discrete-Time Transfer Function with Time Delay This example shows how to create a discrete-time transfer function with a time delay. in In discrete-time models, a delay of one sampling period corresponds to a factor of the transfer function. For example, the following transfer function represents a discretetime SISO system with a delay of 25 sampling periods. 2-43 2 Model Creation To represent integer delays in discrete-time systems in MATLAB, set the 'InputDelay' property of the model object to an integer value. For example, the following command creates a tf model representing with a sampling time of 0.1 s. H = tf(2,[1 -0.95],0.1,'InputDelay',25) H = 2 z^(-25) * -------z - 0.95 Sample time: 0.1 seconds Discrete-time transfer function. If system has a time delay that is not an integer multiple of the sampling time, you can use the thiran command to approximate the fractional portion of the time delay with an all-pass filter. See “Time-Delay Approximation” on page 2-48. See Also Related Examples • “Closing Feedback Loops with Time Delays” on page 2-45 • “Convert Time Delay in Discrete-Time Model to Factors of 1/z” on page 2-64 More About • 2-44 “Time-Delay Approximation” on page 2-48 Closing Feedback Loops with Time Delays Closing Feedback Loops with Time Delays This example shows how internal delays arise when you interconnect models that have input, output, or transport time delays. Create a model of the following control architecture: r 0.5 + - 2.3 s C e-2.1s s +10 y G G is the plant model, which has an input delay. C is a proportional-integral (PI) controller. To create a model representing the closed-loop response of this system: 1 Create the plant G and the controller C. G = tf(1,[1 10],'InputDelay',2.1); C = pid(0.5,2.3); C has a proportional gain of 0.5 and an integral gain of 2.3. 2 Use feedback to compute the closed-loop response from r to y. T = feedback(C*G,1); The time delay in T is not an input delay as it is in G. Because the time delay is internal to the closed-loop system, the software returns T as an ss model with an internal time delay of 2.1 seconds. Note In addition to feedback, any system interconnection function (including parallel and series) can give rise to internal delays. T is an exact representation of the closed-loop response, not an approximation. To access the internal delay value, enter: T.InternalDelay A step plot of T confirms the presence of the time delay: 2-45 2 Model Creation step(T) Note Most analysis commands, such as step, bode and margin, support models with internal delays. The internal time delay is stored in the InternalDelay property of T. Use dot notation to access InternalDelay. For example, to change the internal delay to 3.5 seconds, enter: T.InternalDelay = 3.5 2-46 See Also You cannot modify the number of internal delays because they are structural properties of the model. See Also Related Examples • “Convert Time Delay in Discrete-Time Model to Factors of 1/z” on page 2-64 More About • “Internal Delays” on page 2-73 2-47 2 Model Creation Time-Delay Approximation Many control design algorithms cannot handle time delays directly. For example, techniques such as root locus, LQG, and pole placement do not work properly if time delays are present. A common technique is to replace delays with all-pass filters that approximate the delays. To approximate time delays in continuous-time LTI models, use the pade command to compute a Padé approximation. The Padé approximation is valid only at low frequencies, and provides better frequency-domain approximation than time-domain approximation. It is therefore important to compare the true and approximate responses to choose the right approximation order and check the approximation validity. Time-Delay Approximation in Discrete-Time Models For discrete-time models, use absorbDelay to convert a time delay to factors of 1/z where the time delay is an integer multiple of the sample time. Use the thiran command to approximate a time delay that is a fractional multiple of the sample time as a Thiran all-pass filter. For a time delay of tau and a sample time of Ts, the syntax thiran(tau,Ts) creates a discrete-time transfer function that is the product of two terms: • A term representing the integer portion of the time delay as a pure line delay, (1/z)N, where N = ceil(tau/Ts). • A term approximating the fractional portion of the time delay (tau - NTs) as a Thiran all-pass filter. Discretizing a Padé approximation does not guarantee good phase matching between the continuous-time delay and its discrete approximation. Using thiran to generate a discrete-time approximation of a continuous-time delay can yield much better phase matching. For example, the following figure shows the phase delay of a 10.2-second time delay discretized with a sample time of 1 s, approximated in three ways: • a first-order Padé approximation, discretized using the tustin method of c2d • an 11th-order Padé approximation, discretized using the tustin method of c2d • an 11th-order Thiran filter 2-48 See Also The Thiran filter yields the closest approximation of the 10.2-second delay. See the thiran reference page for more information about Thiran filters. See Also absorbDelay | pade | thiran Related Examples • “Time-Delay Approximation in Continuous-Time Open-Loop Model” on page 2-50 • “Convert Time Delay in Discrete-Time Model to Factors of 1/z” on page 2-64 • “Approximate Different Delays with Different Approximation Orders” on page 2-60 2-49 2 Model Creation Time-Delay Approximation in Continuous-Time OpenLoop Model This example shows how to approximate delays in a continuous-time open-loop system using pade. Padé approximation is helpful when using analysis or design tools that do not support time delays. 1 Create sample open-loop system with an output delay. 1 s 2 + 0.9s + 1 u e-2.6s y P s = tf('s'); P = exp(-2.6*s)/(s^2+0.9*s+1); P is a second-order transfer function (tf) object with a time delay. 2 Compute the first-order Padé approximation of P. Pnd1 = pade(P,1) Pnd1 = -s + 0.7692 ---------------------------------s^3 + 1.669 s^2 + 1.692 s + 0.7692 Continuous-time transfer function. This command replaces all time delays in P with a first-order approximation. Therefore, Pnd1 is a third-order transfer function with no delays. 3 Compare the frequency response of the original and approximate models using bodeplot. h = bodeoptions; h.PhaseMatching = 'on'; bodeplot(P,'-b',Pnd1,'-.r',{0.1,10},h) legend('Exact delay','First-Order Pade','Location','SouthWest') 2-50 Time-Delay Approximation in Continuous-Time Open-Loop Model The magnitude of P and Pnd1 match exactly. However, the phase of Pnd1 deviates from the phase of P beyond approximately 1 rad/s. 4 Increase the Padé approximation order to extend the frequency band in which the phase approximation is good. Pnd3 = pade(P,3); 5 Compare the frequency response of P, Pnd1 and Pnd3. bodeplot(P,'-b',Pnd3,'-.r',Pnd1,':k',{0.1 10},h) legend('Exact delay','Third-Order Pade','First-Order Pade',... 'Location','SouthWest') 2-51 2 Model Creation The phase approximation error is reduced by using a third-order Padé approximation. 6 Compare the time domain responses of the original and approximated systems using stepplot. stepplot(P,'-b',Pnd3,'-.r',Pnd1,':k') legend('Exact delay','Third-Order Pade','First-Order Pade',... 'Location','Southeast') 2-52 Time-Delay Approximation in Continuous-Time Open-Loop Model Using the Padé approximation introduces a nonminimum phase artifact (“wrong way” effect) in the initial transient response. The effect is quite pronounced in the firstorder approximation, which dips significantly below zero before changing direction. The effect is reduced in the higher-order approximation, which far more closely matches the exact system’s response. Note Using too high an approximation order may result in numerical issues and possibly unstable poles. Therefore, avoid Padé approximations with order N>10. 2-53 2 Model Creation See Also pade Related Examples • “Time-Delay Approximation in Continuous-Time Closed-Loop Model” on page 2-55 More About 2-54 • “Time-Delay Approximation” on page 2-48 • “Internal Delays” on page 2-73 Time-Delay Approximation in Continuous-Time Closed-Loop Model Time-Delay Approximation in Continuous-Time ClosedLoop Model This example shows how to approximate delays in a continuous-time closed-loop system with internal delays, using pade. Padé approximation is helpful when using analysis or design tools that do not support time delays. 1 Create sample continuous-time closed-loop system with an internal delay. r 0.06 + - 0.15 + 0.006s s e-4.2s C s2 s +1 + 0.68s + 1 y G Construct a model Tcl of the closed-loop transfer function from r to y. s = G = C = Tcl tf('s'); (s+1)/(s^2+.68*s+1)*exp(-4.2*s); pid(0.06,0.15,0.006); = feedback(G*C,1); Examine the internal delay of Tcl. Tcl.InternalDelay ans = 4.2000 2 Compute the first-order Padé approximation of Tcl. Tnd1 = pade(Tcl,1); Tnd1 is a state-space (ss) model with no delays. 3 Compare the frequency response of the original and approximate models. h = bodeoptions; h.PhaseMatching = 'on'; bodeplot(Tcl,'-b',Tnd1,'-.r',{.1,10},h); legend('Exact delay','First-Order Pade','Location','SouthWest'); 2-55 2 Model Creation The magnitude and phase approximation errors are significant beyond 1 rad/s. 4 Compare the time domain response of Tcl and Tnd1 using stepplot. stepplot(Tcl,'-b',Tnd1,'-.r'); legend('Exact delay','First-Order Pade','Location','SouthEast'); 2-56 Time-Delay Approximation in Continuous-Time Closed-Loop Model Using the Padé approximation introduces a nonminimum phase artifact (“wrong way” effect) in the initial transient response. 5 Increase the Padé approximation order to see if this will extend the frequency with good phase and magnitude approximation. Tnd3 = pade(Tcl,3); 6 Observe the behavior of the third-order Padé approximation of Tcl. Compare the frequency response of Tcl and Tnd3. bodeplot(Tcl,'-b',Tnd3,'-.r',Tnd1,'--k',{.1,10},h); legend('Exact delay','Third-Order Pade','First-Order Pade',... 'Location','SouthWest'); 2-57 2 Model Creation The magnitude and phase approximation errors are reduced when a third-order Padé approximation is used. Increasing the Padé approximation order extends the frequency band where the approximation is good. However, too high an approximation order may result in numerical issues and possibly unstable poles. Therefore, avoid Padé approximations with order N>10. See Also pade 2-58 See Also Related Examples • “Approximate Different Delays with Different Approximation Orders” on page 2-60 More About • “Time-Delay Approximation” on page 2-48 • “Internal Delays” on page 2-73 2-59 2 Model Creation Approximate Different Delays with Different Approximation Orders This example shows how to specify different Padé approximation orders to approximate internal and output delays in a continuous-time open-loop system. Load a sample continuous-time open-loop system that contains internal and output time delays. load(fullfile(matlabroot,'examples','control','PadeApproximation1.mat'),'sys') sys sys = A = x1 x2 x1 -1.5 1 x2 -0.1 0 B = x1 x2 u1 1 0 C = y1 x1 0.5 x2 0.1 D = y1 u1 0 (values computed with all internal delays set to zero) Output delays (seconds): 1.5 Internal delays (seconds): 3.4 Continuous-time state-space model. sys is a second-order continuous-time ss model with internal delay 3.4 s and output delay 1.5 s. 2-60 Approximate Different Delays with Different Approximation Orders Use the pade function to compute a third-order approximation of the internal delay and a first-order approximation of the output delay. P13 = pade(sys,inf,1,3); size(P13) State-space model with 1 outputs, 1 inputs, and 6 states. The three input arguments following sys specify the approximation orders of any input, output, and internal delays of sys, respectively. inf specifies that a delay is not to be approximated. The approximation orders for the output and internal delays are one and three respectively. Approximating the time delays with pade absorbs delays into the dynamics, adding as many states to the model as orders in the approximation. Thus, P13 is a sixth-order model with no delays. For comparison, approximate only the internal delay of sys, leaving the output delay intact. P3 = pade(sys,inf,inf,3); size(P3) State-space model with 1 outputs, 1 inputs, and 5 states. P3.OutputDelay ans = 1.5000 P3.InternalDelay ans = 0x1 empty double column vector P3 retains the output delay, but the internal delay is approximated and absorbed into the state-space matrices, resulting in a fifth-order model without internal delays. Compare the frequency response of the exact and approximated systems sys, P13, P3. h = bodeoptions; h.PhaseMatching = 'on'; bode(sys,'b-',P13,'r-.',P3,'k--',h,{.01,10}); legend('sys','approximated output and internal delays','approximated internal delay onl 'location','SouthWest') 2-61 2 Model Creation Notice that approximating the internal delay loses the gain ripple displayed in the exact system. See Also pade Related Examples • 2-62 “Time-Delay Approximation in Continuous-Time Open-Loop Model” on page 2-50 See Also More About • “Time-Delay Approximation” on page 2-48 • “Internal Delays” on page 2-73 2-63 2 Model Creation Convert Time Delay in Discrete-Time Model to Factors of 1/z This example shows how to convert a time delay in a discrete-time model to factors of 1/ _z_. In a discrete-time model, a time delay of one sampling interval is equivalent to a factor of 1/_z_ (a pole at z = 0) in the model. Therefore, time delays stored in the InputDelay, OutputDelay, or IODelay properties of a discrete-time model can be rewritten in the model dynamics by rewriting them as poles at z = 0. However, the additional poles increase the order of the system. Particularly for large time delays, this can yield systems of very high order, leading to long computation times or numerical inaccuracies. To illustrate how to eliminate time delays in a discrete-time closed-loop model, and to observe the effects of doing so, create the following closed-loop system: G is a first-order discrete-time system with an input delay, and C is a PI controller. G = ss(0.9,0.125,0.08,0,'Ts',0.01,'InputDelay',7); C = pid(6,90,0,0,'Ts',0.01); T = feedback(C*G,1); Closing the feedback loop on a plant with input delays gives rise to internal delays in the closed-loop system. Examine the order and internal delay of T. order(T) ans = 2 T.InternalDelay ans = 2-64 Convert Time Delay in Discrete-Time Model to Factors of 1/z 7 T is a second-order state-space model. One state is contributed by the first-order plant, and the other by the one pole of the PI controller. The delays do not increase the order of T. Instead, they are represented as an internal delay of seven time steps. Replace the internal delay by . Tnd = absorbDelay(T); This command converts the internal delay to seven poles at z = 0. To confirm this, examine the order and internal delay of Tnd. order(Tnd) ans = 9 Tnd.InternalDelay ans = 0x1 empty double column vector Tnd has no internal delay, but it is a ninth-order model, due to the seven extra poles introduced by absorbing the seven-unit delay into the model dynamics. Despite this difference in representation, the responses of Tnd exactly match those of T. stepplot(T,Tnd,'r--') legend('T','Tnd') 2-65 2 Model Creation bodeplot(T,Tnd,'r--') legend('T','Tnd') 2-66 See Also See Also pade Related Examples • “Time-Delay Approximation in Continuous-Time Open-Loop Model” on page 2-50 More About • “Time-Delay Approximation” on page 2-48 2-67 2 Model Creation • 2-68 “Internal Delays” on page 2-73 Frequency Response Data (FRD) Model with Time Delay Frequency Response Data (FRD) Model with Time Delay This example shows that absorbing time delays into frequency response data can cause undesirable phase wrapping at high frequencies. When you collect frequency response data for a system that includes time delays, you can absorb the time delay into the frequency response as a phase shift. Alternatively, if you are able to separate time delays from your measured frequency response, you can represent the delays using the InputDelay, OutputDelay, or ioDelay properties of the frd model object. The latter approach can give better numerical results, as this example illustrates. The frd model fsys includes a transport delay of 2 s. Load the model into the MATLAB® workspace and inspect the time delay. load(fullfile(matlabroot,'examples','control','frddelayexample.mat'),'fsys') fsys.IODelay ans = 2 A Bode plot of fsys shows the effect of the transport delay, causing the accumulation of phase as frequency increases. bodeplot(fsys) 2-69 2 Model Creation The absorbDelay command absorbs all time delays directly into the frequency response, resulting in an frd model with IODelay = 0. fsys2 = absorbDelay(fsys); fsys2.IODelay ans = 0 Comparing the two ways of representing the delay shows that absorbing the delay into the frequency response causes phase-wrapping. 2-70 See Also bode(fsys,fsys2) Phase wrapping can introduce numerical inaccuracy at high frequencies or where the , you frequency grid is sparse. For that reason, if your system takes the form might get better results by measuring frequency response data for G(s) and using InputDelay, OutputDelay, or ioDelay to model the time delay . See Also absorbDelay 2-71 2 Model Creation More About • 2-72 “Time-Delay Approximation” on page 2-48 Internal Delays Internal Delays Using the InputDelay, OutputDelay, and ioDelay properties, you can model simple processes with transport delays. However, these properties cannot model more complex situations, such as feedback loops with delays. In addition to the InputDelay and OutputDelay properties, state-space (ss) models have an InternalDelay property. This property lets you model the interconnection of systems with input, output, or transport delays, including feedback loops with delays. You can use InternalDelay property to accurately model and analyze arbitrary linear systems with delays. Internal delays can arise from the following: • Concatenating state-space models with input and output delays. • Feeding back a delayed signal. • Converting MIMO tf or zpk models with transport delays to state-space form. Using internal time delays, you can do the following: • In continuous time, generate approximate-free time and frequency simulations, because delays do not have to be replaced by a Padé approximation. In continuous time, this allows for more accurate analysis of systems with long delays. • In discrete time, keep delays separate from other system dynamics, because delays are not replaced with poles at z = 0, which boosts efficiency of time and frequency simulations for discrete-time systems with long delays. • Use most Control System Toolbox functions. • Test advanced control strategies for delayed systems. For example, you can implement and test an accurate model of a Smith predictor. See the example “Control of Processes with Long Dead Time: The Smith Predictor”. Why Internal Delays Are Necessary This example illustrates why input, output, and transport delays not enough to model all types of delays that can arise in dynamic systems. Consider the simple feedback loop with a 2 s. delay: 2-73 2 Model Creation e-2s s+2 The closed-loop transfer function is e- 2 s s + 2 + e- 2s The delay term in the numerator can be represented as an output delay. However, the delay term in the denominator cannot. In order to model the effect of the delay on the feedback loop, the InternalDelay property is needed to keep track of internal coupling between delays and ordinary dynamics. Typically, you do not create state-space models with internal delays directly, by specifying the A, B, C, and D matrices together with a set of internal delays. Rather, such models arise when you interconnect models having delays. There is no limitation on how many delays are involved and how the models are connected. For an example of creating an internal delay by closing a feedback loop, see “Closing Feedback Loops with Time Delays” on page 2-45. Behavior of Models With Internal Delays When you work with models having internal delays, be aware of the following behavior: • When a model interconnection gives rise to internal delays, the software returns an ss model regardless of the interconnected model types. This occurs because only ss supports internal delays. • The software fully supports feedback loops. You can wrap a feedback loop around any system with delays. • When displaying the A, B, C, and D matrices, the software sets all delays to zero (creating a zero-order Padé approximation). This approximation occurs for the display only, and not for calculations using the model. For some systems, setting delays to zero creates singular algebraic loops, which result in either improper or ill-defined, zero-delay approximations. For these systems: 2-74 Internal Delays • Entering sys returns only sizes for the matrices of a system named sys. • Entering sys.A produces an error. The limited display and the error do not imply a problem with the model sys itself. Inside Time Delay Models State-space objects use generalized state-space equations to keep track of internal delays. Conceptually, such models consist of two interconnected parts: • An ordinary state-space model H(s) with an augmented I/O set • A bank of internal delays. The corresponding state-space equations are: x& = Ax(t) + B1u(t) + B2w(t) y( t) = C1 x( t) + D11u( t) + D12 w(t) z(t) = C2 x(t) + D21u( t) + D22 w(t) w j (t) = z(t - t j ), j = 1,..., N 2-75 2 Model Creation You need not bother with this internal representation to use the tools. If, however, you want to extract H or the matrices A, B1, B2, ... , you can use getDelayModel, For the example: P = 5*exp(-3.4*s)/(s+1); C = 0.1 * (1 + 1/(5*s)); T = feedback(ss(P*C),1); [H,tau] = getDelayModel(T,'lft'); size(H) Note that H is a two-input, two-output model whereas T is SISO. The inverse operation (combining H and tau to construct T) is performed by setDelayModel. See [1], [2] for details. Functions That Support Internal Time Delays The following commands support internal delays for both continuous- and discrete-time systems: • All interconnection functions • Time domain response functions—except for impulse and initial • Frequency domain functions—except for norm Limitations on Functions that Support Internal Time Delays The following commands support internal delays for both continuous- and discrete-time systems and have certain limitations: • allmargin, margin—Uses interpolation, therefore these commands are only as precise as the fineness of the specified grid. • pole, zero—Returns poles and zeros of the system with all delays set to zero. • ssdata, get—If an SS model has internal delays, these commands return the A, B, C, and D matrices of the system with all internal delays set to zero. Use getDelayModel to access the internal state-space representation of models with internal delays. Functions That Do Not Support Internal Time Delays The following commands do not support internal time delays: 2-76 See Also • System dynamics—norm and isstable • Time-domain analysis—initial and initialplot • Model simplification—balreal, balred, and modred • Compensator design—rlocus, lqg, lqry, lqrd, kalman, kalmd, lqgreg, lqgtrack, lqi, and augstate. To use these functions on a system with internal delays, use pade to approximate the internal delays. See “Time-Delay Approximation” on page 2-48. References [1] P. Gahinet and L.F. Shampine, "Software for Modeling and Analysis of Linear Systems with Delays," Proc. American Control Conf., Boston, 2004, pp. 5600-5605 [2] L.F. Shampine and P. Gahinet, Delay-differential-algebraic Equations in Control Theory, Applied Numerical Mathematics, 56 (2006), pp. 574-588 See Also Related Examples • “Closing Feedback Loops with Time Delays” on page 2-45 2-77 2 Model Creation Create Tunable Low-Pass Filter This example shows how to create a low-pass filter with one tunable parameter a: You cannot use tunableTF to represent F, because the numerator and denominator coefficients of a tunableTF block are independent. Instead, construct F using the tunable real parameter object realp. Create a tunable real parameter with an initial value of 10. a = realp('a',10); Use tf to create the tunable filter F. F = tf(a,[1 a]); F is a genss object which has the tunable parameter a in its Blocks property. You can connect F with other tunable or numeric models to create more complex control system models. For example, see “Control System with Tunable Components” on page 2-83. See Also genss | realp | tunableTF More About 2-78 • “Models with Tunable Coefficients” on page 1-19 • “Create Tunable Second-Order Filter” on page 2-79 • “Create State-Space Model with Both Fixed and Tunable Parameters” on page 2-81 • “Control System with Tunable Components” on page 2-83 Create Tunable Second-Order Filter Create Tunable Second-Order Filter This example shows how to create a parametric model of the second-order filter: where the damping and the natural frequency are tunable parameters. Define the tunable parameters using realp. wn = realp('wn',3); zeta = realp('zeta',0.8); wn and zeta are realp parameter objects, with initial values 3 and 0.8, respectively. Create a model of the filter using the tunable parameters. F = tf(wn^2,[1 2*zeta*wn wn^2]); The inputs to tf are the vectors of numerator and denominator coefficients expressed in terms of wn and zeta. F is a genss model. The property F.Blocks lists the two tunable parameters wn and zeta. F.Blocks ans = struct with fields: wn: [1x1 realp] zeta: [1x1 realp] You can examine the number of tunable blocks in a generalized model using nblocks. nblocks(F) ans = 6 F has two tunable parameters, but the parameter wn appears five times - Twice in the numerator and three times in the denominator. To reduce the number of tunable blocks, you can rewrite F as: 2-79 2 Model Creation Create the alternative filter. F = tf(1,[(1/wn)^2 2*zeta*(1/wn) 1]); Examine the number of tunable blocks in the new model. nblocks(F) ans = 4 In the new formulation, there are only three occurrences of the tunable parameter wn. Reducing the number of occurrences of a block in a model can improve the performance of calculations involving the model. However, the number of occurrences does not affect the results of tuning the model or sampling it for parameter studies. See Also genss | nblocks | realp More About 2-80 • “Models with Tunable Coefficients” on page 1-19 • “Create Tunable Low-Pass Filter” on page 2-78 • “Create State-Space Model with Both Fixed and Tunable Parameters” on page 2-81 • “Control System with Tunable Components” on page 2-83 Create State-Space Model with Both Fixed and Tunable Parameters Create State-Space Model with Both Fixed and Tunable Parameters This example shows how to create a state-space genss model having both fixed and tunable parameters. where a and b are tunable parameters, whose initial values are -1 and 3, respectively. Create the tunable parameters using realp. a = realp('a',-1); b = realp('b',3); Define a generalized matrix using algebraic expressions of a and b. A = [1 a+b;0 a*b]; A is a generalized matrix whose Blocks property contains a and b. The initial value of A is [1 2;0 -3], from the initial values of a and b. Create the fixed-value state-space matrices. B = [-3.0;1.5]; C = [0.3 0]; D = 0; Use ss to create the state-space model. sys = ss(A,B,C,D) sys = Generalized continuous-time state-space model with 1 outputs, 1 inputs, 2 states, and a: Scalar parameter, 2 occurrences. b: Scalar parameter, 2 occurrences. Type "ss(sys)" to see the current value, "get(sys)" to see all properties, and "sys.Blo sys is a generalized LTI model (genss) with tunable parameters a and b. Confirm that the A property of sys is stored as a generalized matrix. 2-81 2 Model Creation sys.A ans = Generalized matrix with 2 rows, 2 columns, and the following blocks: a: Scalar parameter, 2 occurrences. b: Scalar parameter, 2 occurrences. Type "double(ans)" to see the current value, "get(ans)" to see all properties, and "ans See Also More About 2-82 • “Models with Tunable Coefficients” on page 1-19 • “Create Tunable Low-Pass Filter” on page 2-78 • “Create Tunable Second-Order Filter” on page 2-79 • “Control System with Tunable Components” on page 2-83 Control System with Tunable Components Control System with Tunable Components This example shows how to create a tunable model of the control system in the following illustration. The plant response is . The controller . The model of sensor dynamics is is a tunable PID controller, and the prefilter is a low-pass filter with one tunable parameter, . Create models representing the plant and sensor dynamics. Since the plant and sensor dynamics are fixed, represent them using numeric LTI models zpk and tf. G = zpk([],[-1,-1],1); S = tf(5,[1 4]); Create a tunable representation of the controller . C = tunablePID('C','PID'); C is a tunablePID object, which is a Control Design Block with a predefined proportional-integral-derivative (PID) structure. Create a model of the filter with one tunable parameter. a = realp('a',10); F = tf(a,[1 a]); a is a realp (real tunable parameter) object with initial value 10. Using a as a coefficient in tf creates the tunable genss model object F. Connect the models together to construct a model of the closed-loop response from to . 2-83 2 Model Creation T = feedback(G*C,S)*F T = Generalized continuous-time state-space model with 1 outputs, 1 inputs, 5 states, and C: Parametric PID controller, 1 occurrences. a: Scalar parameter, 2 occurrences. Type "ss(T)" to see the current value, "get(T)" to see all properties, and "T.Blocks" t T is a genss model object. In contrast to an aggregate model formed by connecting only Numeric LTI models, T keeps track of the tunable elements of the control system. The tunable elements are stored in the Blocks property of the genss model object. Display the tunable elements of T. T.Blocks ans = struct with fields: C: [1x1 tunablePID] a: [1x1 realp] You can use tuning commands such as systune to tune the free parameters of T to meet design requirements you specify. See Also Related Examples • “Create Tunable Low-Pass Filter” on page 2-78 • “Create Tunable Second-Order Filter” on page 2-79 • “Create State-Space Model with Both Fixed and Tunable Parameters” on page 2-81 More About • 2-84 “Models with Tunable Coefficients” on page 1-19 Control System with Multichannel Analysis Points Control System with Multichannel Analysis Points This example shows how to insert multichannel analysis points in a generalized statespace model of a MIMO control system. Consider the following two-input, two-output control system. The plant G has two inputs and two outputs. Therefore, the line marked y in the block diagram represents two signals, y(1) and y(2). Similarly, r and e each represent two signals. Suppose you want to create tuning requirements or extract responses that require injecting or measuring signals at the locations L and V. To do so, create an AnalysisPoint block and include it in the closed-loop model of the control system as shown in the following illustration. To create a model of this system, first create the numeric LTI models and control design blocks that represent the plant and controller elements. D is a tunable gain block, and C_L and C_V are tunable PI controllers. Suppose the plant model is the following: s = tf('s'); G = [87.8 -86.4 ; 108.2 -109.6]/(75*s+1); 2-85 2 Model Creation D = tunableGain('Decoupler',eye(2)); C_L = tunablePID('C_L','pi'); C_V = tunablePID('C_V','pi'); Create an AnalysisPoint block that bundles together the L and V channels. AP_1 = AnalysisPoint('AP_1',2) AP_1 = Multi-channel analysis point at locations: AP_1(1) AP_1(2) Type "ss(AP_1)" to see the current value and "get(AP_1)" to see all properties. For convenience, rename the channels to match the corresponding signals. AP_1.Location = {'L';'V'} AP_1 = Multi-channel analysis point at locations: L V Type "ss(AP_1)" to see the current value and "get(AP_1)" to see all properties. The following diagram illustrates the input names, output names, and channel names (locations) in the block AP_1. The input and output names of the AnalysisPoint block are distinct from the channel names. Use the channel names to refer to the analysis-point locations when extracting responses or defining design goals for tuning. You can use the input and output names AP_1.u and AP_1.y, for example, when interconnecting blocks using the connect command. 2-86 Control System with Multichannel Analysis Points You can now build the closed-loop model of the control system. First, join all the plant and controller blocks along with the first AnalysisPoint block. GC = G*AP_1*append(C_L,C_V)*D; Then, close the feedback loop. Recall that GC has two inputs and outputs. CL = feedback(GC,eye(2)); You can now use the analysis points for analysis or tuning. For example, extract the SISO closed-loop transfer function from 'L' to the first output. Assign a name to the output so you can reference it in analysis functions. The software automatically expands the assigned name 'y' to the vector-valued output signals {y(1),y(2)}. CL.OutputName = 'y'; TLy1 = getIOTransfer(CL,'L','y(1)'); bodeplot(TLy1); 2-87 2 Model Creation See Also AnalysisPoint More About • 2-88 “Mark Signals of Interest for Control System Analysis and Design” on page 2-89 Mark Signals of Interest for Control System Analysis and Design Mark Signals of Interest for Control System Analysis and Design Analysis Points Whether you model your control system in MATLAB or Simulink, use analysis points to mark points of interest in the model. Analysis points allow you to access internal signals, perform open-loop analysis, or specify requirements for controller tuning. In the block diagram representation, an analysis point can be thought of as an access port to a signal flowing from one block to another. In Simulink, analysis points are attached to the outports of Simulink blocks. For example, in the following model, the reference signal, r, and the control signal, u, are analysis points that originate from the outputs of the setpoint and C blocks respectively. Each analysis point can serve one or more of the following purposes: • Input — The software injects an additive input signal at an analysis point, for example, to model a disturbance at the plant input. • Output — The software measures the signal value at a point, for example, to study the impact of a disturbance on the plant output. • Loop Opening — The software inserts a break in the signal flow at a point, for example, to study the open-loop response at the plant input. You can apply these purposes concurrently. For example, to compute the open-loop response from u to y, you can treat u as both a loop opening and an input. When you use an analysis point for more than one purpose, the software applies the purposes in this sequence: output measurement, then loop opening, then input. 2-89 2 Model Creation Using analysis points, you can extract open-loop and closed-loop responses from a control system model. For example, suppose T represents the closed-loop system in the model above, and u and y are marked as analysis points. T can be either a generalized statespace model or an slLinearizer or slTuner interface to a Simulink model. You can plot the closed-loop response to a step disturbance at the plant input with the following commands: Tuy = getIOTransfer(T,'u','y'); stepplot(Tuy) Analysis points are also useful to specify design requirements when tuning control systems with the systune command. For example, you can create a requirement that attenuates disturbances at the plant input by a factor of 10 (20 dB) or more. Req = TuningGoal.Rejection('u',10); Specify Analysis Points for MATLAB Models Consider an LTI model of the following block diagram. 2-90 Mark Signals of Interest for Control System Analysis and Design G = tf(10,[1 3 10]); C = pid(0.2,1.5); T = feedback(G*C,1); With this model, you can obtain the closed-loop response from r to y. However, you cannot analyze the open-loop response at the plant input or simulate the rejection of a step disturbance at the plant input. To enable such analysis, mark the signal u as an analysis point by inserting an AnalysisPoint block between the plant and controller. AP = AnalysisPoint('u'); T = feedback(G*AP*C,1); T.OutputName = 'y'; The plant input, u, is now available for analysis. In creating the model T, you manually created the analysis point block AP and explicitly included it in the feedback loop. When you combine models using the connect command, you can instruct the software to insert analysis points automatically at the locations you specify. For more information, see connect. Specify Analysis Points for Simulink Models In Simulink, you can mark analysis points either explicitly in the block diagram, or programmatically using the addPoint command for slLinearizer or slTuner interfaces. To mark an analysis point explicitly in the model, right-click a signal and, under Linear Analysis Points, select an analysis point type. 2-91 2 Model Creation You can select any of the following closed-loop analysis point types, which are equivalent within an slLinearizer or slTuner interface; that is, they are treated the same way by analysis functions, such as getIOTransfer, and tuning goals, such as TuningGoal.StepTracking. 2-92 Mark Signals of Interest for Control System Analysis and Design • Input Perturbation • Output Measurement • Sensitivity • Complementary Sensitivity If you want to introduce a permanent loop opening at a signal as well, select one of the following open-loop analysis point types: • Open-Loop Input • Open-Loop Output • Loop Transfer • Loop Break When you define a signal as an open-loop point, analysis functions such as getIOTransfer always enforce a loop break at that signal during linearization. All openloop analysis point types are equivalent within an slLinearizer or slTuner interface. For more information on how the software treats loop openings during linearization, see “How the Software Treats Loop Openings” (Simulink Control Design). When you create an slLinearizer or slTuner interface for a model, any analysis points defined in the model are automatically added to the interface. If you defined an analysis point using: • A closed-loop type, the signal is added as an analysis point only. • An open-loop type, the signal is added as both an analysis point and a permanent opening. To mark analysis points programmatically, use the addPoint command. For example, consider the scdcascade model. open_system('scdcascade') 2-93 2 Model Creation To mark analysis points, first create an slTuner interface. ST = slTuner('scdcascade'); To add a signal as an analysis point, use the addPoint command, specifying the source block and port number for the signal. addPoint(ST,'scdcascade/C1',1); If the source block has a single output port, you can omit the port number. addPoint(ST,'scdcascade/G2'); For convenience, you can also mark analysis points using the: • Name of the signal. addPoint(ST,'y2'); • Combined source block path and port number. addPoint(ST,'scdcascade/C1/1') • End of the full source block path when unambiguous. addPoint(ST,'G1/1') You can also add permanent openings to an slLinearizer or slTuner interface using the addOpening command, and specifying signals in the same way as for addPoint. For 2-94 Mark Signals of Interest for Control System Analysis and Design more information on how the software treats loop openings during linearization, see “How the Software Treats Loop Openings” (Simulink Control Design). addOpening(ST,'y1m'); You can also define analysis points by creating linearization I/O objects using the linio command. io(1) = linio('scdcascade/C1',1,'input'); io(2) = linio('scdcascade/G1',1,'output'); addPoint(ST,io); As when you define analysis points directly in your model, if you specify a linearization I/O object with: • A closed-loop type, the signal is added as an analysis point only. • An open-loop type, the signal is added as both an analysis point and a permanent opening. When you specify response I/Os in a tool such as Linear Analysis Tool or Control System Tuner, the software creates analysis points as needed. Refer to Analysis Points for Analysis and Tuning Once you have marked analysis points, you can analyze the response at any of these points using the following analysis functions: • getIOTransfer — Transfer function for specified inputs and outputs • getLoopTransfer — Open-loop transfer function from an additive input at a specified point to a measurement at the same point • getSensitivity — Sensitivity function at a specified point • getCompSensitivity — Complementary sensitivity function at a specified point You can also create tuning goals that constrain the system response at these points. The tools to perform these operations operate in a similar manner for models created at the command line and models created in Simulink. To view the available analysis points, use the getPoints function. You can view the analysis for models created: 2-95 2 Model Creation • At the command line: • In Simulink: For closed-loop models created at the command line, you can also use the model input and output names when: • Computing a closed-loop response. ioSys = getIOTransfer(T,'u','y'); stepplot(ioSys) • Computing an open-loop response. loopSys = getLoopTransfer(T,'u',-1); bodeplot(loopSys) 2-96 Mark Signals of Interest for Control System Analysis and Design • Creating tuning goals for systune. R = TuningGoal.Margins('u',10,60); Use the same method to refer to analysis points for models created in Simulink. In Simulink models, for convenience, you can use any unambiguous abbreviation of the analysis point names returned by getPoints. ioSys = getIOTransfer(ST,'u1','y1'); sensG2 = getSensitivity(ST,'G2'); R = TuningGoal.Margins('u1',10,60); Finally, if some analysis points are vector-valued signals or multichannel locations, you can use indices to select particular entries or channels. For example, suppose u is a twoentry vector in a closed-loop MIMO model. 2-97 2 Model Creation G = ss([-1 0.2;0 -2],[1 0;0.3 1],eye(2),0); C = pid(0.2,0.5); AP = AnalysisPoint('u',2); T = feedback(G*AP*C,eye(2)); T.OutputName = 'y'; You can compute the open-loop response of the second channel and measure the impact of a disturbance on the first channel. L = getLoopTransfer(T,'u(2)',-1); stepplot(getIOTransfer(T,'u(1)','y')) When you create tuning goals in Control System Tuner, the software creates analysis points as needed. 2-98 See Also See Also AnalysisPoint | getIOTransfer | getPoints More About • “Control System with Multichannel Analysis Points” on page 2-85 • “Mark Analysis Points in Closed-Loop Models” on page 4-13 2-99 2 Model Creation Model Arrays What Are Model Arrays? In many applications, it is useful to consider collections of multiple model objects. For example, you may want to consider a model with a parameter that varies across a range of values, such as sys1 = tf(1, [1 1 1]); sys2 = tf(1, [1 1 2]); sys3 = tf(1, [1 1 3]); and so on. Model arrays are a convenient way to store and analyze such a collection. Model arrays are collections of multiple linear models, stored as elements in a single MATLAB array. For all models collected in a single model array, the following attributes must be the same: • The number of inputs and outputs • The sample time Ts • The time unit TimeUnit Uses of Model Arrays Uses of model arrays include: • Representing and analyzing sensitivity to parameter variations • Validating a controller design against several plant models • Representing linear models arising from the linearization of a nonlinear system at several operating points • Storing models obtained from several system identification experiments applied to one plant Using model arrays, you can apply almost all of the basic model operations that work on single model objects to entire sets of models at once. Functions operate on arrays model by model, allowing you to manipulate an entire collection of models in a vectorized fashion. You can also use analysis functions such as bode, nyquist, and step to model 2-100 Model Arrays arrays to analyze multiple models simultaneously. You can access the individual models in the collection through MATLAB array indexing. Visualizing Model Arrays To visualize the concept of a model array, consider the set of five transfer function models shown below. In this example, each model has two inputs and two outputs. They differ by parameter variations in the individual model components. Just as you might collect a set of two-by-two matrices in a multidimensional array, you can collect this set of five transfer function models as a list in a model array under one variable name, say, sys. Each element of the model array is a single model object. Visualizing Selection of Models From Model Arrays The following illustration shows how indexing selects models from a one-dimensional model array. The illustration shows a 1-by-5 array sysa of 2-input, 2-output transfer functions. 2-101 2 Model Creation The following illustration shows selection of models from the two-dimensional model array m2d. See Also Related Examples 2-102 • “Query Array Size and Characteristics” on page 2-106 • “Select Models from Array” on page 2-103 • “Model Array with Variations in Two Parameters” on page 9-6 Select Models from Array Select Models from Array This example shows how to select individual models or sets of models from a model array using array indexing. 1 Load the transfer function array m2d into the MATLAB workspace. load LTIexamples m2d 2 (Optional) Plot the step response of m2d. step(m2d) The step response shows that m2d contains six one-input, two-output models. The step command plots all of the models in an array on a single plot. 3 (Optional) Examine the dimensions of m2d. arraydim = size(m2d) This command produces the result: arraydim = 2 1 2 3 2-103 2 Model Creation • The first entries of arraydim, 2 and 1, show that m2d is an array of two-output, one-input transfer functions. • The remaining entries in arraydim give the array dimensions of m2d, 2-by-3. In general, the dimensions of a model array are [Ny,Nu,S1,...,Sk]. Ny and Nu are the numbers of outputs and inputs of each model in the array. S1,...,Sk are the array dimensions. Thus, Si is the number of models along the ith array dimension. 4 Select the transfer function in the second row, first column of m2d. To do so, use MATLAB array indexing. sys = m2d(:,:,2,1) Tip You can also access models using single index referencing of the array dimensions. For example, sys = m2d(:,:,4) selects the same model as m2d(:,:,2,1). 5 Select the array of subsystems from the first input to the first output of each model in m2d. m11 = m2d(1,1,:,:) 6 (Optional) Plot the step response of m11. step(m11) 2-104 See Also The step response shows that m11 is an array of six single-input, single-output (SISO) models. Note For frequency response data (FRD) models, the array indices can be followed by the keyword 'frequency' and some expression selecting a subset of the frequency points, as in: sys(outputs,inputs,n1,...,nk,'frequency',SelectedFreqs) See Also More About • “Model Arrays” on page 2-100 • “Query Array Size and Characteristics” on page 2-106 2-105 2 Model Creation Query Array Size and Characteristics This example shows how to query the size of model arrays, including the number of inputs and outputs of the models in the array, and the array dimensions. It also shows how to query characteristics of the models in the array, such as stability. Array Size Model arrays have two different sets of dimensions, the I/O dimensions and the array dimensions. The I/O dimensions are the numbers of inputs and outputs of the models in the array. (Each model in an array must have the same I/O dimensions.) The array dimensions are the dimensions of the array itself. Load a saved model array and query its dimensions. load(fullfile(matlabroot,'examples','control','queryexample.mat'),'sysarr') size(sysarr) 2x4 array of state-space models. Each model has 3 outputs, 1 inputs, and 3 states. When you use the size command on a model array with no output argument, the display shows the two sets of dimensions. To obtain the array dimensions as a numeric array, use size with an output argument. dims = size(sysarr) dims = 1×4 3 1 2 4 The first two entries in dims are the I/O dimensions of the models in sysarr, which each have three outputs and one input. The remaining entries in dims are the dimensions of the array itself. Thus, sysarr is a 2-by-4 array of models. To query the number of dimensions in the array, rather than the values of those dimensions, use ndims. dims = ndims(sysarr) dims = 4 2-106 Query Array Size and Characteristics In this case, sysarr has 4 = 2 + 2 dimensions: The I/O dimensions (outputs and inputs), and the array dimensions. Query the I/O dimensions alone using the iosize command. ios = iosize(sysarr) ios = 1×2 3 1 Query the total number of models in the array. N = nmodels(sysarr) N = 8 Because sysarr is a 2-by-4 array of models, this command returns a value of 2 × 4 = 8. Characteristics of Models in the Array Query commands such as isproper and isstable work on model arrays. For example, query whether the models in sysarr are stable. Bsiso = isstable(sysarr) Bsiso = logical 1 By default, isstable returns 1 (true) if all of the models in the array are stable. The commands returns 0 (false) if one or more of the models is not stable. To perform an element-by-element query of a model array, use the 'elem' option. Bsiso = isstable(sysarr,'elem') Bsiso = 2x4 logical array 1 1 1 1 1 1 1 1 Now isstable returns an array of Boolean values. The dimensions of this array match the array dimensions of sysarr. Each entry in the array Bsiso indicates whether the 2-107 2 Model Creation corresponding model of sysarr is stable. The 'elem' option works similarly for many query commands. See Also More About 2-108 • “Model Arrays” on page 2-100 • “Select Models from Array” on page 2-103 Linear Parameter-Varying Models Linear Parameter-Varying Models What are Linear Parameter-Varying Models? A linear parameter-varying (LPV) system is a linear state-space model whose dynamics vary as a function of certain time-varying parameters called scheduling parameters. In MATLAB, an LPV model is represented in a state-space form using coefficients that are parameter dependent. Mathematically, an LPV system is represented as: dx ( t ) = A ( p) x ( t ) + B ( p) u ( t ) y ( t ) = C ( p) x ( t ) + D ( p) u ( t) x ( 0 ) = x0 (2-1) where • u(t) are the inputs • y(t) the outputs • x(t) are the model states with initial value x0 • dx ( t ) is the state derivative vector x& for continuous-time systems and the state update vector x ( t + DT ) for discrete-time systems. ΔT is the sample time. • A(p), B(p), C(p) and D(p) are the state-space matrices parameterized by the scheduling parameter vector p. • The parameters p = p(t) are measurable functions of the inputs and the states of the model. They can be a scalar quantity or a vector of several parameters. The set of scheduling parameters define the scheduling space over which the LPV model is defined. Grid-Based LPV Model A common way of representing LPV models is as an interpolated array of linear statespace models. A certain number of points in the scheduling space are selected, usually forming a regular grid on page 2-112. An LTI system is assigned to each point, representing the dynamics in the local vicinity of that point. The dynamics at scheduling 2-109 2 Model Creation locations in between the grid points is obtained by interpolation of LTI systems at neighboring points. For example, the aerodynamic behavior of an aircraft is often scheduled over a grid of incidence angle (α) and wind speed (V) values. For each scheduling parameter, a range of values is chosen, such as α = 0:5:20 degrees, V = 700:100:1400 m/s. For each combination of (α,V) values, a linear approximation of the aircraft behavior is obtained. The local models are connected as shown in the following figure: Each donut represents a local LTI model, and the connecting curves represent the interpolation rules. The abscissa and ordinate of the surface are the scheduling parameters (α, V). This form is sometimes called the grid-based LPV representation. This is the form used by the LPV System block. For meaningful interpolations of system matrices, all the local models must use the same state basis. 2-110 Linear Parameter-Varying Models Affine Form of LPV Model The LPV system representation can be extended to allow offsets in dx, x, u and y variables. This form is known as affine form of the LPV model. Mathematically, the following represents an LPV system: ( dx ( t ) = A ( p) x ( t ) + B ( p) u ( t ) + dx ( p) - A ( p) x ( p) - B( p)u( p) ) y ( t ) = C ( p) x ( t ) + D ( p) u ( t) + ( y ( p) - C ( p) x ( p) - D( p)u ( p) ) x ( 0 ) = x0 (2-2) dx ( p) , x ( p) , u ( p) , y ( p) are the offsets in the values of dx(t), x(t), u(t) and y(t) at a given parameter value p = p(t). To obtain such representations of the linear system array, linearize a Simulink model over a batch of operating points (see “Batch Linearization” (Simulink Control Design).) The offsets correspond to the operating points at which you linearized the model. You can obtain the offsets by returning additional linearization information when calling functions such as linearize or getIOTransfer. You can then extract the offsets using getOffsetsForLPV. For an example, see “LPV Approximation of a Boost Converter Model” (Simulink Control Design). In the affine representation, the linear model at a given point p = p* in the scheduling space is: ( ) ( ) ( ) ( ) D y( t, p ) = C ( p* ) Dx ( t, p* ) + D ( p* ) D u ( t, p* ) dDx(t, p* ) = A p* Dx t, p* + B p* Du t, p* * The states of this linear model are related to the states of the overall LPV model ( ) Du ( t, p* ) = u ( t ) - u ( p* ) . (“Equation 2-2”) by ( ) . Similarly, Dy( t, p* ) = y ( t) - y ( p* ) and Dx t, p* = x ( t ) - x p* 2-111 2 Model Creation Regular vs. Irregular Grids Consider a system that uses two scheduling parameters, α and β. When α and β vary monotonically, a regular grid is formed, as shown in the next figure. The state space array contains a value at every combination of α and β values. Regular grid does not imply uniform spacing between values. When parameters co-vary, that is, α and β increase together, an irregular grid is formed. The system array parameters are available only along the diagonal in the parameter plane. 2-112 Linear Parameter-Varying Models If certain samples are missing from an otherwise regular grid, the grid is considered to be irregular. 2-113 2 Model Creation Use Model Arrays to Create Linear Parameter-Varying Models The array of state-consistent linear models that define an LPV model are represented by an array of state-space model objects. For more information on model arrays, see “Model Arrays”. The system array size is equal to the grid size in scheduling space. In the aircraft example, α takes 5 values in the 0–20 degrees range and V takes 8 values in the 700–1400 m/s range. If you define a linear model at every combination of (α,V) values (i.e., the grid is regular), the grid size is 5-by-8. Therefore, the model array size must be 5-by-8. The information about scheduling parameters is attached to the linear model array using its SamplingGrid property. The value of the SamplingGrid property must be a structure with as many fields as there are scheduling parameters. For each field, the value must be set to all the values assumed by the corresponding variable in the scheduling space. For the aircraft example, you can define the SamplingGrid property as: Alpha = 0:5:20; V = 700:100:1400; 2-114 Linear Parameter-Varying Models [Alpha_Grid,V_Grid] = ndgrid(Alpha, V); linsysArray.SamplingGrid = struct('Alpha',Alpha_Grid,'V',V_Grid); Approximate Nonlinear Systems using LPV Models In the same way as a linear model provides the approximation of system behavior at a given operating condition, an LPV model provides the approximation of the behavior over a span on operating conditions. A common approach for constructing the LPV model is by batch trimming and linearization, followed by stacking the local models in a state-space model array. Note When obtaining linear models by linearization, do not reduce or alter the state variables used by the models. The operating region is usually of a high dimension because it consists of all the input and state variables. Generating or interpolating local models in such high-dimensional spaces is usually infeasible. A simpler approach is to use a small set of scheduling parameters as a proxy for the operating space variables. The scheduling parameters are derived from the inputs and state variables of the original system. You must choose the values carefully so that for a fixed value of the scheduling parameters, the system behavior is approximately linear. This approach is not always possible. Consider a nonlinear system described by the following equations: x& 1 = x12 + x22 x& 2 = -2 x1 - 3 x2 + 2u y = x1 + 2 Suppose you use p( t) = x& 1 as a scheduling variable. At a given time instant t = t0, you have: x& 1 ª 2 x1 ( t0 ) x1 + 2 x2 ( t0 ) x2 - x&1 ( t0 ) x& 2 = -2 x1 - 3 x2 + 2u y = x1 + 2 2-115 2 Model Creation Thus, the dynamics are linear (affine) in the neighborhood of a given value of x& . The approximation holds for all time spans and values of input u as long as of x& does not deviate much from its nominal value at sampling point t0. Note that scheduling on input u or states x1 or x2 does not help locally linearize the system. Therefore, they are not good candidates for scheduling parameters. For an example of this approach, see “Approximating Nonlinear Behavior Using an Array of LTI Systems” (Simulink Control Design). Applications of Linear Parameter-Varying Models Modeling Multimode Dynamics You can use LPV models to represent systems that exhibit multiple modes (regimes) of operation. Examples of such systems include colliding bodies, systems controlled by operator switches, and approximations of systems affected by dry friction and hysteresis effects. For an example, see “Using LTI Arrays for Simulating Multi-Mode Dynamics” on page 2-118. Proxy Modeling for Faster Simulations This approach is useful for generating surrogate models that you can use in place of the original system for enabling faster simulations, reducing memory footprint of target hardware code, and hardware-in-loop (HIL) simulations. You can also use surrogate models of this type for designing gain-scheduled controllers and for initializing the parameter estimation tasks in Simulink. For an example of approximating a general nonlinear system behavior by an LPV model, see “Approximating Nonlinear Behavior Using an Array of LTI Systems” (Simulink Control Design). LPV models can help speed up the simulation of physical component based systems, such as those built using Simscape™ Multibody™ and Simscape Electrical™ Power Systems software. For an example of this approach, see “LPV Approximation of a Boost Converter Model” (Simulink Control Design). See Also LPV System | getOffsetsForLPV 2-116 See Also More About • “Using LTI Arrays for Simulating Multi-Mode Dynamics” on page 2-118 • “Approximating Nonlinear Behavior Using an Array of LTI Systems” (Simulink Control Design) • “LPV Approximation of a Boost Converter Model” (Simulink Control Design) 2-117 2 Model Creation Using LTI Arrays for Simulating Multi-Mode Dynamics This example shows how to construct a Linear Parameter Varying (LPV) representation of a system that exhibits multi-mode dynamics. Introduction We often encounter situations where an elastic body collides with, or presses against, a possibly elastic surface. Examples of such situations are: • An elastic ball bouncing on a hard surface. • An engine throttle valve that is constrained to close to no more than spring. using a hard • A passenger sitting on a car seat made of polyurethane foam, a viscoelastic material. In these situations, the motion of the moving body exhibits different dynamics when it is moving freely than when it is in contact with a surface. In the case of a bouncing ball, the motion of the mass can be described by rigid body dynamics when it is falling freely. When the ball collides and deforms while in contact with the surface, the dynamics have to take into account the elastic properties of the ball and of the surface. A simple way of modeling the impact dynamics is to use lumped mass spring-damper descriptions of the colliding bodies. By adjusting the relative stiffness and damping coefficients of the two bodies, we can model the various situations described above. Modeling Bounce Dynamics Figure 1 shows a mass-spring-damper model of the system. Mass 1 is falling freely under the influence of gravity. Its elastic properties are described by stiffness constant and damping coefficient . When this mass hits the fixed surface, the impact causes Mass 1 and Mass 2 to move downwards together. After a certain "residence time" during which the Mass 1 deforms and recovers, it loses contact with Mass 2 completely to follow a projectile motion. The overall dynamics are thus broken into two distinct modes - when the masses are not in contact and when they are moving jointly. Figure 1: Elastic body bouncing on a fixed elastic surface. 2-118 Using LTI Arrays for Simulating Multi-Mode Dynamics The unstretched (load-free) length of spring attached to Mass 1 is , while that of Mass 2 is . The variables and denote the positions of the two masses. When the masses are not in contact ("Mode 1"), their motions are governed by the following equations: with initial conditions , , , . is the height from which Mass 1 is originally dropped. is the initial location of Mass 2 which corresponds to an unstretched state of its spring. When Mass 1 touches Mass 2 ("Mode 2"), their displacements and velocities get interlinked. The governing equations in this mode are: with , where is the time at which Mass 1 first touches Mass 2. LPV Representation The governing equations are linear and time invariant. However, there are two distinct behavioral modes corresponding to different equations of motion. Both modes are governed by sets of second order equations. If we pick the positions and velocities of the masses as state variables, we can represent each mode by a 4th order state-space equation. In the state-space view, it becomes possible to treat the two modes as a single system whose coefficients change as a function of a certain condition which determines which mode is active. The condition is, of course, whether the two masses are moving freely or jointly. Such a representation, where the coefficients of a linear system are parameterized by an external but measurable parameter is called a Linear Parameter Varying (LPV) model. A common representation of an LPV model is by means of an array of linear statespace models and a set of scheduling parameters that dictate the rules for choosing the correct model under a given condition. The array of linear models must all be defined using the same state variables. 2-119 2 Model Creation For our example, we need two state-space models, one for each mode of operation. We also need to define a scheduling variable to switch between them. We begin by writing the above equations of motion in state-space form. Define values of masses and their spring constants. m1 = 7; k1 = 100; c1 = 2; % first mass (g) % spring constant for first mass (g/s^2) % damping coefficient associated with first mass (g/s) m2 = 20; k2 = 300; c2 = 5; % second mass (g) % spring constant for second mass (g/s^2) % damping coefficient associated with second mass (g/s) g = 9.81; % gravitational acceleration (m/s^2) a1 = 12; a2 = 20; % uncompressed lengths of spring 1 (mm) % uncompressed lengths of spring 2 (mm) h1 = 100; h2 = a2; % initial height of mass m1 (mm) % initial height of mass m2 (mm) First mode: state-space representation of dynamics when the masses are not in contact. A11 B11 C11 D11 = = = = [0 1; 0 0]; [0; -g]; [1 0]; 0; A12 B12 C12 D12 = = = = [0 1; -k2/m2, -c2/m2]; [0; -g+(k2*a2/m2)]; [1 0]; 0; A1 B1 C1 D1 = = = = blkdiag(A11, A12); [B11; B12]; blkdiag(C11, C12); [D11; D12]; sys1 = ss(A1,B1,C1,D1); Second mode: state-space representation of dynamics when the masses are in contact. A2 = [ 0 -k1/m1, 2-120 1, -c1/m1, 0, k1/m1, 0; ... c1/m1;... Using LTI Arrays for Simulating Multi-Mode Dynamics 0, k1/m2, 0, c1/m2, 0, -(k1+k2)/m2, 1; ... -(c1+c2)/m2]; B2 = [0; -g+k1*a1/m1; 0; -g+(k2/m2*a2)-(k1/m2*a1)]; C2 = [1 0 0 0; 0 0 1 0]; D2 = [0;0]; sys2 = ss(A2,B2,C2,D2); Now we stack the two models sys1 and sys2 together to create a state-space array. sys = stack(1,sys1,sys2); Use the information on whether the masses are moving freely or jointly for scheduling. Let us call this parameter "FreeMove" which takes the value of 1 when masses are moving freely and 0 when they are in contact and moving jointly. The scheduling parameter information is incorporated into the state-space array object (sys) by using its "SamplingGrid" property: sys.SamplingGrid = struct('FreeMove',[1; 0]); Whether the masses are in contact or not is decided by the relative positions of the two masses; when , the masses are not in contact. Simulation of LPV Model in Simulink The state-space array sys has the necessary information to represent an LPV model. We can simulate this model in Simulink using the "LPV System" block from the Control System Toolbox™'s block library. Open the preconfigured Simulink model LPVBouncingMass.slx open_system('LPVBouncingMass') open_system('LPVBouncingMass/Bouncing Mass Model','mask') 2-121 2 Model Creation The block called "Bouncing Mass Model" is an LPV System block. Its parameters are specified as follows: • For "State-space array" field, specify the state-space model array sys that was created above. • For "Initial state" field, specify the initial positions and velocities of the two masses. Note that the state vector is: . Specify its value as [h1 0 h2 0]'. • Under the "Scheduling" tab, set the "Interpolation method" to "Nearest". This choice causes only one of the two models in the array to be active at any time. In our example, the behavior modes are mutually exclusive. • Under the "Outputs" tab, uncheck all the checkboxes for optional output ports. We will be observing only the positions of the two masses. The constant block outputs a unit value. This serves as the input to the model and is supplied from the first input port of the LPV block. The block has only one output port which outputs the positions of the two masses as a 2-by-1 vector. The second input port of the LPV block is for specifying the scheduling signal. As discussed before, this signal represents the scheduling parameter "FreeMove" and takes discrete values 0 (masses in contact) or 1 (masses not in contact). The value of this parameter is computed as a function of the block's output signal. This computation is performed by the blocks with cyan background color. We take the difference between the two outputs (after demuxing) and compare the result to the unstretched length of spring attached to Mass 1. The resulting Boolean result is converted into a double signal which serves as the scheduling parameter value. We are now ready to perform the simulation. 2-122 Using LTI Arrays for Simulating Multi-Mode Dynamics open_system('LPVBouncingMass/Scope') sim('LPVBouncingMass') The yellow curve shows the position of Mass 1 while the magenta curve shows the position of Mass 2. At the start of simulation, Mass 1 undergoes free fall until it hits Mass 2. The collision causes the Mass 2 to be displaced but it recoils quickly and bounces Mass 1 back. The two masses are in contact for the time duration where . When the masses settle down, their equilibrium values are determined by the static settling due to gravity. For example, the absolute location of Mass 1 is 2-123 2 Model Creation Conclusions This example shows how a Linear Parameter Varying model can be constructed by using an array of state-space models and suitable scheduling variables. The example describes the case of mutually exclusive modes, although a similar approach can be used in cases where the dynamics behavior at a given value of scheduling parameters is influenced by several linear models. The LPV System block facilitates the simulation of parameter varying systems. The block also supports code generation for various hardware targets. 2-124 Working with Linear Models 125 3 Data Manipulation • “Store and Retrieve Model Data” on page 3-2 • “Extract Model Coefficients” on page 3-6 • “Attach Metadata to Models” on page 3-9 • “Query Model Characteristics” on page 3-14 • “Customize Model Display” on page 3-17 3 Data Manipulation Store and Retrieve Model Data Model Properties Model properties are the data fields that store all data about a dynamic system model. Data stored in model properties includes model dynamics, such as transfer-function coefficients, state-space matrices, and time delays. Model properties also let you specify other model attributes such as sample time, channel names, and state names. For information about the properties associated with each model type, see the corresponding reference page, such as tf, pid, or ss. Specify Model Properties at Model Creation When you create a dynamic system model, the software sets all property values. Properties that contain model dynamics are automatically set with the appropriate values. Other properties are set to default values. (See model reference pages for information about default property values.) You can specify other values for model properties at model creation using the Name,Value pair syntax of the model-creation command. In this syntax, you specify the name of the property you want to set, followed by the value. You can set multiple property values in one command. For example, assign a transport delay and input and output names to a new transfer function model. H = tf(1,[1 10],'IODelay',6.5,'InputName','torque','OutputName','velocity') H = From input "torque" to output "velocity": 1 exp(-6.5*s) * -----s + 10 Continuous-time transfer function. Some property values are reflected in the model display, such as the input and output names. You can use Name,Value pair syntax when creating any type of model. 3-2 Store and Retrieve Model Data Examine and Change Properties of an Existing Model Load an existing state-space (ss) model. load(fullfile(matlabroot,'examples','control','PadeApproximation1.mat'),'sys') sys sys = A = x1 x2 x1 -1.5 1 x2 -0.1 0 B = x1 x2 u1 1 0 C = y1 x1 0.5 x2 0.1 D = y1 u1 0 (values computed with all internal delays set to zero) Output delays (seconds): 1.5 Internal delays (seconds): 3.4 Continuous-time state-space model. The display shows that sys is a state-space model, and includes some of the property values of sys. To see all properties of sys, use the get command. get(sys) A: B: C: D: E: Scaled: [2x2 double] [2x1 double] [0.5000 0.1000] 0 [] 0 3-3 3 Data Manipulation StateName: StateUnit: InternalDelay: InputDelay: OutputDelay: Ts: TimeUnit: InputName: InputUnit: InputGroup: OutputName: OutputUnit: OutputGroup: Notes: UserData: Name: SamplingGrid: {2x1 cell} {2x1 cell} 3.4000 0 1.5000 0 'seconds' {''} {''} [1x1 struct] {''} {''} [1x1 struct] [0x1 string] [] '' [1x1 struct] Use dot notation to access the values of particular properties. For example, display the A matrix of sys. Amat = sys.A Amat = 2×2 -1.5000 1.0000 -0.1000 0 Dot notation also lets you change the value of individual model properties. sys.InputDelay = 4.2; sys.InputName = 'thrust'; sys.OutputName = 'velocity'; When you must change multiple property values at the same time to preserve the validity of the model, such as changing the dimensions of the state-space matrices, you can use the set command. For example, create a 1-state state-space model, and then replace the matrices with new values representing a 2-state model. sys2 Anew Bnew Cnew 3-4 = = = = rss(1); [-2, 1; 0.5 0]; [1; -1]; [0, -0.4]; See Also set(sys2,'A',Anew,'B',Bnew,'C',Cnew) sys2 sys2 = A = x1 x2 x1 -2 0.5 x2 1 0 B = x1 x2 u1 1 -1 C = y1 x1 0 x2 -0.4 D = y1 u1 0.3426 Continuous-time state-space model. Changing certain properties, such as Ts or TimeUnit, can cause undesirable changes in system behavior. See the property descriptions in the model reference pages for more information. See Also Related Examples • “Attach Metadata to Models” on page 3-9 • “Extract Model Coefficients” on page 3-6 3-5 3 Data Manipulation Extract Model Coefficients Functions for Extracting Model Coefficients Control System Toolbox software includes several commands for extracting model coefficients such as transfer function numerator and denominator coefficients, state-space matrices, and proportional-integral-derivative (PID) gains. The following commands are available for data extraction. Command Result tfdata Extract transfer function coefficients zpkdata Extract zero and pole locations and system gain ssdata Extract state-space matrices dssdata Extract descriptor state-space matrices frdata Extract frequency response data from frd model piddata Extract parallel-form PID data pidstddata Extract standard-form PID data get Access all model property values Extracting Coefficients of Different Model Type When you use a data extraction command on a model of a different type, the software computes the coefficients of the target model type. For example, if you use zpkdata on a ss model, the software converts the model to zpk form and returns the zero and pole locations and system gain. Extract Numeric Model Data and Time Delay This example shows how to extract transfer function numerator and denominator coefficients using tfdata. 1 Create a first-order plus dead time transfer function model. s = tf('s'); H = exp(-2.5*s)/(s+12); 3-6 Extract Model Coefficients 2 Extract the numerator and denominator coefficients. [num,den] = tfdata(H,'v') The variables num and den are numerical arrays. Without the 'v' flag, tfdata returns cell arrays. Note For SISO transfer function models, you can also extract coefficients using: num = H.Numerator{1}; den = H.Denominator{1}; 3 Extract the time delay. a Determine which property of H contains the time delay. In a SISO tf model, you can express a time delay as an input delay, an output delay, or a transport delay (I/O delay). get(H) Numerator: Denominator: Variable: IODelay: InputDelay: OutputDelay: Ts: TimeUnit: InputName: InputUnit: InputGroup: OutputName: OutputUnit: OutputGroup: Notes: UserData: Name: SamplingGrid: {[0 1]} {[1 12]} 's' 0 0 2.5000 0 'seconds' {''} {''} [1×1 struct] {''} {''} [1×1 struct] [0×1 string] [] '' [1×1 struct] The time delay is stored in the OutputDelay property. b Extract the output delay. delay = H.OutputDelay; 3-7 3 Data Manipulation Extract PID Gains from Transfer Function This example shows how to extract PID (proportional-integral-derivative) gains from a transfer function using piddata. You can use the same steps to extract PID gains from a model of any type that represents a PID controller, using piddata or pidstddata. 1 Create a transfer function that represents a PID controller with a first-order filter on the derivative term. Czpk = zpk([-6.6,-0.7],[0,-2],0.2) 2 Obtain the PID gains and filter constant. [Kp,Ki,Kd,Tf] = piddata(Czpk) This command returns the proportional gain Kp, integral gain Ki, derivative gain Kd, and derivative filter time constant Tf. Because piddata automatically computes the PID controller parameters, you can extract the PID coefficients without creating a pid model. See Also Related Examples • “Attach Metadata to Models” on page 3-9 More About • 3-8 “Store and Retrieve Model Data” on page 3-2 Attach Metadata to Models Attach Metadata to Models Specify Model Time Units This example shows how to specify time units of a transfer function model. The TimeUnit property of the tf model object specifies units of the time variable, time delays (for continuous-time models), and the sample time Ts (for discrete-time models). The default time units is seconds. Create a SISO transfer function model sys = 4s + 2 2 with time units in milliseconds: s + 3 s + 10 num = [4 2]; den = [1 3 10]; sys = tf(num,den,'TimeUnit','milliseconds'); You can specify the time units of any dynamic system on page 1-10 in a similar way. The system time units appear on the time- and frequency-domain plots. For multiple systems with different time units, the units of the first system are used if the time and frequency units in the “Toolbox Preferences Editor” on page 20-2 are auto. Note Changing the TimeUnit property changes the system behavior. If you want to use different time units without modifying system behavior, use chgTimeUnit. Interconnect Models with Different Time Units This example shows how to interconnect transfer function models with different time units. To interconnect models using arithmetic operations or interconnection commands, the time units of all models must match. 1 Create two transfer function models with time units of milliseconds and seconds, respectively. sys1 = tf([1 2],[1 2 3],'TimeUnit','milliseconds'); sys2 = tf([4 2],[1 3 10]); 3-9 3 Data Manipulation 2 Change the time units of sys2 to milliseconds. sys2 = chgTimeUnit(sys2,'milliseconds'); 3 Connect the systems in parallel. sys = sys1+sys2; Specify Frequency Units of Frequency-Response Data Model This example shows how to specify units of the frequency points of a frequency-response data model. The FrequencyUnit property specifies units of the frequency vector in the Frequency property of the frd model object. The default frequency units are rad/TimeUnit, where TimeUnit is the time unit specified in the TimeUnit property. Create a SISO frequency-response data model with frequency data in GHz. load AnalyzerData; sys = frd(resp,freq,'FrequencyUnit','GHz'); You can independently specify the units in which you measure the frequency points and sample time in the FrequencyUnit and TimeUnit properties, respectively. You can also specify the frequency units of a genfrd in a similar way. The frequency units appear on the frequency-domain plots. For multiple systems with different frequency units, the units of the first system are used if the frequency units in the “Toolbox Preferences Editor” on page 20-2 is auto. Note Changing the FrequencyUnit property changes the system behavior. If you want to use different frequency units without modifying system behavior, use chgFreqUnit. Extract Subsystems of Multi-Input, Multi-Output (MIMO) Models This example shows how to extract subsystems of a MIMO model using MATLAB indexing and using channel names. Extracting subsystems is useful when, for example, you want to analyze a portion of a complex system. 3-10 Attach Metadata to Models Create a MIMO transfer function. G1 = tf(3,[1 10]); G2 = tf([1 2],[1 0]); G = [G1,G2]; Extract the subsystem of G from the first input to all outputs. Gsub = G(:,1); This command uses MATLAB indexing to specify a subsystem as G(out,in), where out specifies the output indices and in specifies the input indices. Using channel names, you can use MATLAB indexing to extract all the dynamics relating to a particular channel. By using this approach, you can avoid having to keep track of channel order in a complex MIMO model. Assign names to the model inputs. G.InputName = {'temperature';'pressure'}; Because G has two inputs, use a cell array to specify the two channel names. Extract the subsystem of G that contains all dynamics from the 'temperature' input to all outputs. Gt = G(:,'temperature'); Gt is the same subsystem as Gsub. Note When you extract a subsystem from a state-space (ss) model, the resulting statespace model may not be minimal. Use sminreal to eliminate unnecessary states in the subsystem. Specify and Select Input and Output Groups This example shows how to specify groups of input and output channels in a model object and extract subsystems using the groups. Input and output groups are useful for keeping track of inputs and outputs in complex MIMO models. 3-11 3 Data Manipulation 1 Create a state-space model with three inputs and four outputs. H = rss(3,4,3); 2 Group the inputs as follows: • Inputs 1 and 2 in a group named controls • Outputs 1 and 3 to a group named temperature • Outputs 1, 3, and 4 to a group named measurements H.InputGroup.controls = [1 2]; H.OutputGroup.temperature = [1 3]; H.OutputGroup.measurements = [1 3 4]; InputGroup and OutputGroup are structures. The name of each field in the structure is the name of the input or output group. The value of each field is a vector that identifies the channels in that group. 3 Extract the subsystem corresponding to the controls inputs and the temperature outputs. You can use group names to index into subsystems. Hc = H('temperature','controls') Hc is a two-input, two-output ss model containing the I/O channels from the 'controls' input to the 'temperature' outputs. You can see the relationship between H and the subsystem Hc in this illustration. Hc controls 1 2 3 3-12 H 1 2 3 4 temperature See Also See Also Related Examples • “Store and Retrieve Model Data” on page 3-2 • “Extract Model Coefficients” on page 3-6 • “Query Model Characteristics” on page 3-14 3-13 3 Data Manipulation Query Model Characteristics This example shows how to query model characteristics such as stability, time domain, and number of inputs and outputs. You can use the techniques of this example on any type of dynamic system model. Load a saved state-space (ss) model. load(fullfile(matlabroot,'examples','control','queryexample.mat'),'T') Query whether T has stable dynamics. Bstab = isstable(T) Bstab = logical 1 The isstable command returns 1 (true) if all system poles are in the open left-half plane (for continuous-time models) or inside the open unit disk (for discrete-time models). Otherwise, isstable command returns 0 (false). Here, the result shows that the model is stable. Query whether T has time delays. Bdel = hasdelay(T) Bdel = logical 1 The returned value, 1, indicates that T has a time delay. For a state-space model, time delay can be stored as input delay, output delay, internal delay, or a combination. Use get(T) to determine which properties of T hold the time delay, and use dot notation to access the delay values. The hasInternalDelay command tells you whether there is any internal delay. Query whether T is proper. Bprop = isproper(T) Bprop = logical 1 3-14 See Also The returned value indicates that the system has relative degree less than or equal to 0. This is true of a SISO system when it can be represented as a transfer function in which the degree of the numerator does not exceed the degree of the denominator. Query the order of T. N = order(T) N = 5 For a state-space model, order returns the number of states, which is 5 in this case. For a tf or zpk model, the order is the number of states required for a state-space realization of the system. Query whether T is a discrete-time system. Bdisc = isdt(T) Bdisc = logical 1 The returned value indicates that T is a discrete-time model. Similarly, use isct to query whether T is a continuous-time model. Load a MIMO model and query the input/output dimensions. load(fullfile(matlabroot,'examples','control','queryexample.mat'),'Tmimo') ios = iosize(Tmimo) ios = 1×2 7 4 In the resulting array, the number of outputs is first. Therefore, Tmimo has 4 inputs and 7 outputs. See Also isproper | isstable | size 3-15 3 Data Manipulation Related Examples • “Select Models from Array” on page 2-103 More About • 3-16 “Store and Retrieve Model Data” on page 3-2 Customize Model Display Customize Model Display Configure Transfer Function Display Variable This example shows how to configure the MATLAB command-window display of transfer function (tf) models. You can use the same steps to configure the display variable of transfer function models in factorized form (zpk models). By default, tf and zpk models are displayed in terms of s in continuous time and z in discrete time. Use the Variable property change the display variable to 'p' (equivalent to 's'), 'q' (equivalent to 'z'), 'z^-1', or 'q^-1'. 1 Create the discrete-time transfer function H ( z) = z -1 2 z - 3z + 2 with a sample time of 1 s. H = tf([1 -1],[1 -3 2],0.1) H = z - 1 ------------z^2 - 3 z + 2 Sample time: 0.1 seconds Discrete-time transfer function. The default display variable is z. 2 Change the display variable to q^-1. H.Variable = 'q^-1' H = q^-1 - q^-2 ------------------1 - 3 q^-1 + 2 q^-2 3-17 3 Data Manipulation Sample time: 0.1 seconds Discrete-time transfer function. When you change the Variable property, the software computes new coefficients and displays the transfer function in terms of the new variable. The num and den properties are automatically updated with the new coefficients. Tip Alternatively, you can directly create the same transfer function expressed in terms of 'q^-1'. H = tf([0 1 -1],[1 -3 2],0.1,'Variable','q^-1'); For the inverse variables 'z^-1' and 'q^-1', tf interprets the numerator and denominator arrays as coefficients of ascending powers of 'z^-1' or 'q^-1'. Configure Display Format of Transfer Function in Factorized Form This example shows how to configure the display of transfer function models in factorized form (zpk models). You can configure the display of the factorized numerator and denominator polynomials to highlight: • The numerator and denominator roots • The natural frequencies and damping ratios of each root • The time constants and damping ratios of each root See the DisplayFormat property on the zpk reference page for more information about these quantities. 1 Create a zpk model having a zero at s = 5, a pole at s = –10, and a pair of complex poles at s = –3 ± 5i. H = zpk(5,[-10,-3-5*i,-3+5*i],10) H = 10 (s-5) ---------------------- 3-18 Customize Model Display (s+10) (s^2 + 6s + 34) Continuous-time zero/pole/gain model. The default display format, 'roots', displays the standard factorization of the numerator and denominator polynomials. 2 Configure the display format to display the natural frequency of each polynomial root. H.DisplayFormat = 'frequency' H = -0.14706 (1-s/5) ------------------------------------------(1+s/10) (1 + 1.029(s/5.831) + (s/5.831)^2) Continuous-time zero/pole/gain model. You can read the natural frequencies and damping ratios for each pole and zero from the display as follows: • Factors corresponding to real roots are displayed as (1 – s/ω0). The variable ω0 is the natural frequency of the root. For example, the natural frequency of the zero of H is 5. • Factors corresponding to complex pairs of roots are displayed as 1 – 2ζ(s/ω0) + (s/ω0)2. The variable ω0 is the natural frequency, and ζ is the damping ratio of the root. For example, the natural frequency of the complex pole pair is 5.831, and the damping ratio is 1.029/2. 3 Configure the display format to display the time constant of each pole and zero. H.DisplayFormat = 'time constant' H = -0.14706 (1-0.2s) ------------------------------------------(1+0.1s) (1 + 1.029(0.1715s) + (0.1715s)^2) Continuous-time zero/pole/gain model. You can read the time constants and damping ratios from the display as follows: • Factors corresponding to real roots are displayed as (τs). The variable τ is the time constant of the root. For example, the time constant of the zero of H is 0.2. 3-19 3 Data Manipulation • Factors corresponding to complex pairs of roots are displayed as 1 – 2ζ(τs) + (τs)2. The variable τ is the time constant, and ζ is the damping ratio of the root. For example, the time constant of the complex pole pair is 0.1715, and the damping ratio is 1.029/2. See Also tf | zpk Related Examples • 3-20 “Transfer Functions” on page 2-3 4 Model Interconnections • “Why Interconnect Models?” on page 4-2 • “Catalog of Model Interconnections” on page 4-3 • “Numeric Model of SISO Feedback Loop” on page 4-6 • “Control System Model With Both Numeric and Tunable Components” on page 4-8 • “Multi-Loop Control System” on page 4-10 • “Mark Analysis Points in Closed-Loop Models” on page 4-13 • “MIMO Control System” on page 4-19 • “MIMO Feedback Loop” on page 4-22 • “How the Software Determines Properties of Connected Models” on page 4-27 • “Rules That Determine Model Type” on page 4-29 • “Recommended Model Type for Building Block Diagrams” on page 4-31 4 Model Interconnections Why Interconnect Models? Interconnecting models of components allows you to construct models of control systems. You can conceptualize your control system as a block diagram containing multiple interconnected components, such as a plant or a controller. Using model arithmetic or interconnection commands, you combine models of each of these components into a single model representing the entire block diagram. For example, you can interconnect dynamic system models of a plant G(s), a controller C(s), sensor dynamics S(s), and a filter F(s) to construct a single model that represents the entire closed-loop control system in the following illustration: r F(s) + C(s) G(s) - S(s) See Also More About • 4-2 “Catalog of Model Interconnections” on page 4-3 y Catalog of Model Interconnections Catalog of Model Interconnections Each type of block diagram connection corresponds to a model interconnection command or arithmetic expression. The following tables summarize the block diagram connections with the corresponding interconnection command and arithmetic expression. Model Interconnection Commands Block Diagram Connection H1 u H2 H1 Arithmetic Expression series(H1,H2) H2*H1 parallel(H1,H2) H1+H2 parallel(H1,-H2) H1-H2 feedback(H1,H2) H1/(1+H2*H1) (not recommended) N/A H1/H2 (division) y + y u + H2 H1 + y u - H2 u Command + y H1 - H2 u H2-1 H1 y 4-3 4 Model Interconnections Block Diagram Connection u H2 H1-1 u H1-1 y w1 Arithmetic Expression N/A H1\H2 (left division) inv(H1) N/A lft(H1,H2,nu,ny) N/A y z1 H1 Command y u H2 z2 w2 Arithmetic Operations You can apply almost all arithmetic operations to dynamic system models, including those shown below. Operation Description + Addition - Subtraction * Multiplication .* Element-by-element multiplication / Right matrix divide \ Left matrix divide inv Matrix inversion ' Conjugate system. For a system G, the transfer function of G' is: • G(–s)T in continuous time. • G(1/z)T in discrete time. .' 4-4 Transposition See Also Operation Description ^ Powers of a dynamic system model, as in the following syntax for creating transfer functions: s = tf('s'); G = 25/(s^2 + 10*s + 25); In some cases, you might obtain better results using model interconnection commands, such as feedback or connect, instead of model arithmetic. For example, the command T = feedback(H1,H2) returns better results than the algebraic expression T = H1/ (1+H2*H1). The latter expression duplicates the poles of H1, which inflates the model order and might lead to computational inaccuracy. See Also connect | feedback | parallel | series Related Examples • “Numeric Model of SISO Feedback Loop” on page 4-6 • “Multi-Loop Control System” on page 4-10 • “MIMO Control System” on page 4-19 More About • “How the Software Determines Properties of Connected Models” on page 4-27 • “Recommended Model Type for Building Block Diagrams” on page 4-31 4-5 4 Model Interconnections Numeric Model of SISO Feedback Loop This example shows how to interconnect numeric LTI models on page 1-13 representing multiple system components to build a single numeric model of a closed-loop system, using model arithmetic and interconnection commands. Construct a model of the following single-loop control system. F(s) r + C(s) G(s) y - S(s) The feedback loop includes a plant G(s), a controller C(s), and a representation of sensor dynamics, S(s). The system also includes a prefilter F(s). 1 Create model objects representing each of the components. G C S F = = = = zpk([],[-1,-1],1); pid(2,1.3,0.3,0.5); tf(5,[1 4]); tf(1,[1 1]); The plant G is a zero-pole-gain (zpk) model with a double pole at s = –1. Model object C is a PID controller. The models F and S are transfer functions. 2 Connect the controller and plant models. H = G*C; To combine models using the multiplication operator *, enter the models in reverse order compared to the block diagram. Tip Alternatively, construct H(s) using the series command: H = series(C,G); 3 Construct the unfiltered closed-loop response T ( s ) = 4-6 H . 1 + HS See Also T = feedback(H,S); Caution Do not use model arithmetic to construct T algebraically: T = H/(1+H*S) This computation duplicates the poles of H, which inflates the model order and might lead to computational inaccuracy. 4 Construct the entire closed-loop system response from r to y. T_ry = T*F; T_ry is a Numeric LTI Model representing the aggregate closed-loop system. T_ry does not keep track of the coefficients of the components G, C, F, and S. You can operate on T_ry with any Control System Toolbox control design or analysis commands. See Also connect | feedback | parallel | series Related Examples • “Control System Model With Both Numeric and Tunable Components” on page 4-8 • “Multi-Loop Control System” on page 4-10 • “MIMO Control System” on page 4-19 More About • “Catalog of Model Interconnections” on page 4-3 4-7 4 Model Interconnections Control System Model With Both Numeric and Tunable Components This example shows how to create a tunable model of a control system that has both fixed plant and sensor dynamics and tunable control components. Consider the control system of the following illustration. Suppose that the plant response is dynamics is prefilter , and that the model of the sensor . The controller is a tunable PID controller, and the is a low-pass filter with one tunable parameter, a. Create models representing the plant and sensor dynamics. Because the plant and sensor dynamics are fixed, represent them using numeric LTI models. G = zpk([],[-1,-1],1); S = tf(5,[1 4]); To model the tunable components, use Control Design Blocks. Create a tunable representation of the controller C. C = tunablePID('C','PID'); C is a tunablePID object, which is a Control Design Block with a predefined proportional-integral-derivative (PID) structure. Create a model of the filter with one tunable parameter. a = realp('a',10); F = tf(a,[1 a]); a is a realp (real tunable parameter) object with initial value 10. Using a as a coefficient in tf creates the tunable genss model object F. 4-8 See Also Interconnect the models to construct a model of the complete closed-loop response from r to y. T = feedback(G*C,S)*F T = Generalized continuous-time state-space model with 1 outputs, 1 inputs, 5 states, and C: Parametric PID controller, 1 occurrences. a: Scalar parameter, 2 occurrences. Type "ss(T)" to see the current value, "get(T)" to see all properties, and "T.Blocks" t T is a genss model object. In contrast to an aggregate model formed by connecting only numeric LTI models, T keeps track of the tunable elements of the control system. The tunable elements are stored in the Blocks property of the genss model object. Examine the tunable elements of T. T.Blocks ans = struct with fields: C: [1x1 tunablePID] a: [1x1 realp] When you create a genss model of a control system that has tunable components, you can use tuning commands such as systune to tune the free parameters to meet design requirements you specify. See Also feedback | tunablePID More About • “Control Design Blocks” on page 1-16 • “Dynamic System Models” on page 1-10 4-9 4 Model Interconnections Multi-Loop Control System This example shows how to build an arbitrary block diagram by connecting models using connect. The system is a Smith Predictor, the single-input, single-output (SISO) multiloop control system shown in the following block diagram. u sum1 + e ysp ym C Gp y P yp Dp - y1 - + sum3 + sum2 + dp F dy For more information about the Smith Predictor, see “Control of Processes with Long Dead Time: The Smith Predictor”. The connect command lets you construct the overall transfer function from ysp to y. To use connect, specify the input and output channel names of the components of the block diagram. connect automatically joins ports that have the same name, as shown in the following figure. B1 error B1 connect(B1,B2) error B2 error B2 To build the closed loop model of the Smith Predictor system from ysp to y: 1 4-10 Create the components of the block diagram: the process model P, the predictor model Gp, the delay model Dp, the filter F, and the PI controller C. Specify names for the input and output channels of each model so that connect can automatically join them to build the block diagram. See Also s = tf('s'); P = exp(-93.9*s) * 5.6/(40.2*s+1); P.InputName = 'u'; P.OutputName = 'y'; Gp = 5.6/(40.2*s+1); Gp.InputName = 'u'; Gp.OutputName = 'yp'; Dp = exp(-93.9*s); Dp.InputName = 'yp'; Dp.OutputName = 'y1'; F = 1/(20*s+1); F.InputName = 'dy'; F.OutputName = 'dp'; C = pidstd(0.574,40.1); C.Inputname = 'e'; C.OutputName = 'u'; 2 Create the summing junctions needed to complete the block diagram. sum1 = sumblk('e = ysp - ym'); sum2 = sumblk('ym = yp + dp'); sum3 = sumblk('dy = y - y1'); The argument to sumblk is a formula that relates the input and output signals of the summing junction. sumblk creates a summing junction with the input and output signal names specified in the formula. For example, in sum1, the formula 'e = ysp - ym' specifies an output signal named e, which is the difference between input signals named ysp and ym. 3 Assemble the complete model from ysp to y. T = connect(P,Gp,Dp,C,F,sum1,sum2,sum3,'ysp','y'); You can list the models and summing junctions in any order because connect automatically interconnects them using their input and output channel names. The last two arguments specify the input and output signals of the multi-loop control structure. Thus, T is a ss model with input ysp and output y. See Also connect | sumblk 4-11 4 Model Interconnections Related Examples • “Control System Model With Both Numeric and Tunable Components” on page 4-8 • “MIMO Control System” on page 4-19 • “Mark Analysis Points in Closed-Loop Models” on page 4-13 More About • 4-12 “How the Software Determines Properties of Connected Models” on page 4-27 Mark Analysis Points in Closed-Loop Models Mark Analysis Points in Closed-Loop Models This example shows how to build a block diagram and insert analysis points at locations of interest using the connect command. You can then use the analysis points to extract various system responses from the model. For this example, create a model of a Smith predictor, the SISO multiloop control system shown in the following block diagram. Points marked by x are analysis points to mark for this example. For instance, if you want to calculate the step response of the closed-loop system to a disturbance at the plant input, you can use an analysis point at u. If you want to calculate the response of the system with one or both of the control loops open, you can use the analysis points at yp or dp. To build this system, first create the dynamic components of the block diagram. Specify names for the input and output channels of each model so that connect can automatically join them to build the block diagram. s = tf('s'); % Process model P = exp(-93.9*s) * 5.6/(40.2*s+1); P.InputName = 'u'; P.OutputName = 'y'; % Predictor model Gp = 5.6/(40.2*s+1); 4-13 4 Model Interconnections Gp.InputName = 'u'; Gp.OutputName = 'yp'; % Delay model Dp = exp(-93.9*s); Dp.InputName = 'yp'; Dp.OutputName = 'y1'; % Filter F = 1/(20*s+1); F.InputName = 'dy'; F.OutputName = 'dp'; % PI controller C = pidstd(0.574,40.1); C.Inputname = 'e'; C.OutputName = 'u'; Create the summing junctions needed to complete the block diagram. (For more information about creating summing junctions, see sumblk). sum1 = sumblk('e = ysp - ym'); sum2 = sumblk('ym = yp + dp'); sum3 = sumblk('dy = y - y1'); Assemble the complete model. input = 'ysp'; output = 'y'; APs = {'u','dp','yp'}; T = connect(P,Gp,Dp,C,F,sum1,sum2,sum3,input,output,APs) T = Generalized continuous-time state-space model with 1 outputs, 1 inputs, 4 states, and AnalysisPoints_: Analysis point, 3 channels, 1 occurrences. Type "ss(T)" to see the current value, "get(T)" to see all properties, and "T.Blocks" t When you use the APs input argument, the connect command automatically inserts an AnalysisPoint block into the generalized state-space (genss) model, T. The automatically generated block is named AnalysisPoints_. The three channels of AnalysisPoints_ correspond to the three locations specified in the APs argument to 4-14 Mark Analysis Points in Closed-Loop Models the connect command. Use getPoints to see a list of the available analysis points in the model. getPoints(T) ans = 3x1 cell array {'dp'} {'u' } {'yp'} Use these locations as inputs, outputs, or loop openings when you extract responses from the model. For example, extract and plot the response at the system output to a step disturbance at the plant input, u. Tp = getIOTransfer(T,'u','y'); stepplot(Tp) 4-15 4 Model Interconnections Similarly, calculate the open-loop response of the plant and controller by opening both feedback loops. openings = {'dp','yp'}; L = getIOTransfer(T,'ysp','y',openings); bodeplot(L) 4-16 See Also When you create a control system model, you can create an AnalysisPoint block explicitly and assign input and output names to it. You can then include it in the input arguments to connect as one of the blocks to combine. However, using the APs argument to connect as illustrated in this example is a simpler way to mark points of interest when building control system models. See Also AnalysisPoint | connect | sumblk 4-17 4 Model Interconnections Related Examples • “Control System with Multichannel Analysis Points” on page 2-85 More About • 4-18 “Mark Signals of Interest for Control System Analysis and Design” on page 2-89 MIMO Control System MIMO Control System This example shows how to build a MIMO control system using connect to interconnect Numeric LTI models on page 1-13 and tunable Control Design Blocks on page 1-16. Consider the following two-input, two-output control system. r e + - pL D CL pV CV L G y V The plant G is a distillation column with two inputs and two outputs. The two inputs are the reflux L and boilup V. The two outputs are the concentrations of two chemicals, represented by the vector signal y = [y1,y2]. You can represent this plant model as: G ( s) = 1 È 87 .8 -86 .4 ˘ Í ˙. 75s + 1 Î108.2 -109 .6 ˚ The vector setpoint signal r = [r1,r2] specifies the desired concentrations of the two chemicals. The vector error signal e represents the input to D, a static 2-by-2 decoupling matrix. CL and CV represent independent PI controllers that control the two inputs of G. To create a two-input, two-output model representing this closed-loop control system: 1 Create a Numeric LTI model representing the 2-by-2 plant G. s = tf('s','TimeUnit','minutes'); G = [87.8 -86.4 ; 108.2 -109.6]/(75*s+1); G.InputName = {'L','V'}; G.OutputName = 'y'; When you construct the closed-loop model, connect uses the input and output names to form connections between the block diagram components. Therefore, you must assign names to the inputs and outputs of the transfer function G in either of the following ways: . 4-19 4 Model Interconnections • You can assign input and output names to individual signals by specifying signal names in a cell array, as in G.InputName = {'L','V'} • Alternatively, you can use vector signal naming, which the software automatically expands. For example, the command G.OutputName = 'y' assigns the names 'y(1)' and 'y(2)' to the outputs of G. 2 Create tunable Control Design Blocks representing the decoupling matrix D and the PI controllers CL and CV. D = tunableGain('Decoupler',eye(2)); D.u = 'e'; D.y = {'pL','pV'}; C_L = tunablePID('C_L','pi'); C_L.u = 'pL'; C_L.y = 'L'; C_L.TimeUnit = 'minutes'; C_V = tunablePID('C_V','pi'); C_V.u = 'pV'; C_V.y = 'V'; C_V.TimeUnit = 'minutes'; Note u and y are shorthand notations for the InputName and OutputName properties, respectively. Thus, for example, entering: D.u = 'e'; D.y = {'pL','pV'}; is equivalent to entering: D.InputName = 'e'; D.OutputName = {'pL','pV'}; 3 Create the summing junction. The summing junction produces the error signals e by taking the difference between r and y. Sum = sumblk('e = r - y',2); Sum represents the transfer function for the summing junction described by the formula 'e = r - y'. The second argument to sumblk specifies that the inputs and outputs of Sum are each vector signals of length 2. The software therefore automatically assigns the signal names {'r(1)','r(2)','y(1)','y(2)'} to Sum.InputName and {'e(1)','e(2)'} to Sum.OutputName. 4-20 See Also 4 Join all components to build the closed-loop system from r to y. CLry = connect(G,D,C_L,C_V,Sum,'r','y'); The arguments to the connect function include all the components of the closed-loop system, in any order. connect automatically combines the components using the input and output names to join signals. The last two arguments to connect specify the output and input signals of the closed-loop model, respectively. The resulting genss model CLry has two-inputs and two outputs. See Also connect | sumblk Related Examples • “Control System Model With Both Numeric and Tunable Components” on page 4-8 • “Multi-Loop Control System” on page 4-10 • “MIMO Control System” on page 4-19 More About • “Catalog of Model Interconnections” on page 4-3 4-21 4 Model Interconnections MIMO Feedback Loop This example shows how to obtain the closed-loop response of a MIMO feedback loop in three different ways. In this example, you obtain the response from Azref to Az of the MIMO feedback loop of the following block diagram. You can compute the closed-loop response using one of the following three approaches: • Name-based interconnection with connect • Name-based interconnection with feedback • Index-based interconnection with feedback You can use whichever of these approaches is most convenient for your application. Load the plant Aerodyn and the controller Autopilot into the MATLAB® workspace. These models are stored in the datafile MIMOfeedback.mat. load(fullfile(matlabroot,'examples','control','MIMOfeedback.mat')) Aerodyn is a 4-input, 7-output state-space (ss) model. Autopilot is a 5-input, 1-output ss model. The inputs and outputs of both models names appear as shown in the block diagram. Compute the closed-loop response from Azref to Az using connect. T1 = connect(Autopilot,Aerodyn,'Azref','Az'); Warning: The following block inputs are not used: Rho,a,Thrust. 4-22 MIMO Feedback Loop Warning: The following block outputs are not used: Xe,Ze,Altitude. The connect function combines the models by joining the inputs and outputs that have matching names. The last two arguments to connect specify the input and output signals of the resulting model. Therefore, T1 is a state-space model with input Azref and output Az. The connect function ignores the other inputs and outputs in Autopilot and Aerodyn. Compute the closed-loop response from Azref to Az using name-based interconnection with the feedback command. Use the model input and output names to specify the interconnections between Aerodyn and Autopilot. When you use the feedback function, think of the closed-loop system as a feedback interconnection between an open-loop plant-controller combination L and a diagonal unity-gain feedback element K. The following block diagram shows this interconnection. L = series(Autopilot,Aerodyn,'Fin'); FeedbackChannels = {'Alpha','Mach','Az','q'}; K = ss(eye(4),'InputName',FeedbackChannels,... 'OutputName',FeedbackChannels); T2 = feedback(L,K,'name',+1); The closed-loop model T2 represents the positive feedback interconnection of L and K. The 'name' option causes feedback to connect L and K by matching their input and output names. 4-23 4 Model Interconnections T2 is a 5-input, 7-output state-space model. The closed-loop response from Azref to Az is T2('Az','Azref'). Compute the closed-loop response from Azref to Az using feedback, using indices to specify the interconnections between Aerodyn and Autopilot. L = series(Autopilot,Aerodyn,1,4); K = ss(eye(4)); T3 = feedback(L,K,[1 2 3 4],[4 3 6 5],+1); The vectors [1 2 3 4] and [4 3 6 5] specify which inputs and outputs, respectively, complete the feedback interconnection. For example, feedback uses output 4 and input 1 of L to create the first feedback interconnection. The function uses output 3 and input 2 to create the second interconnection, and so on. T3 is a 5-input, 7-output state-space model. The closed-loop response from Azref to Az is T3(6,5). Compare the step response from Azref to Az to confirm that the three approaches yield the same results. step(T1,T2('Az','Azref'),T3(6,5),2) 4-24 See Also See Also connect | feedback Related Examples • “Multi-Loop Control System” on page 4-10 • “MIMO Control System” on page 4-19 4-25 4 Model Interconnections More About • 4-26 “How the Software Determines Properties of Connected Models” on page 4-27 How the Software Determines Properties of Connected Models How the Software Determines Properties of Connected Models When you interconnect models, the operation and the properties of the models you are connecting determine the resulting model's properties. The following table summarizes some general rules governing how resulting model property values are determined. Property Expected Behavior Ts When connecting discrete-time models, all models must have identical or unspecified (sys.Ts = -1) sample time. The resulting model inherits the sample time from the connected models. InputName OutputName InputGroup InputGroup In general, the resulting model inherits I/O names and I/O groups from connected models. However, conflicting I/O names or I/O groups are not inherited. For example, the InputName property for sys1 + sys2 is left unspecified if sys1 and sys2 have different InputName property values. TimeUnit All connected models must have identical TimeUnit properties. The resulting model inherits its TimeUnit from the connected models. Variable A model resulting from operations on tf or zpk models inherits its Variable property value from the operands. Conflicts are resolved according the following rules: • For continuous-time models, 'p' has precedence over 's'. • For discrete-time models, 'q^-1' and 'z^-1' have precedence over 'q' and 'z', while 'q' has precedence over 'z'. Notes UserData Most operations ignore the Notes and UserData properties. These properties of the resulting model are empty. 4-27 4 Model Interconnections See Also More About • 4-28 “Rules That Determine Model Type” on page 4-29 Rules That Determine Model Type Rules That Determine Model Type When you combine numeric LTI models on page 1-13 other than frd models using connect, the resulting model is a state-space (ss) model. For other interconnection commands, the resulting model is determined by the following order of precedence: ss > zpk > tf > pid > pidstd For example, connect an ss model with a pid model. P = ss([-0.8,0.4;0.4,-1.0],[-3.0;1.4],[0.3,0],0); C = pid(-0.13,-0.61); CL = feedback(P*C,1) The ss model has the highest precedence among Numeric LTI models. Therefore, combining P and C with any model interconnection command returns an ss model. Combining Numeric LTI models with Generalized LTI models on page 1-16 or with Control Design Blocks on page 1-16 results in Generalized LTI models. For example, connect the ss model CL with a Control Design Block. F = tunableTF('F',0,1); CLF = F*CL CLF is a genss model. Any connection that includes a frequency-response model (frd or genfrd) results in a frequency-response model. Note The software automatically converts all models to the resulting model type before performing the connection operation. See Also connect | feedback | parallel | series Related Examples • “Numeric Model of SISO Feedback Loop” on page 4-6 4-29 4 Model Interconnections • “Multi-Loop Control System” on page 4-10 More About 4-30 • “How the Software Determines Properties of Connected Models” on page 4-27 • “Recommended Model Type for Building Block Diagrams” on page 4-31 Recommended Model Type for Building Block Diagrams Recommended Model Type for Building Block Diagrams This example shows how choice of model type can affect numerical accuracy when interconnecting models. You can represent block diagram components with any model type. However, certain connection operations yield better numerical accuracy for models in ss form. For example, interconnect two models in series using different model types to see how different representations introduce numerical inaccuracies. Load the models Pd and Cd. These models are ninth-order and second-order discrete-time transfer functions, respectively. load numdemo Pd Cd Compute the open-loop transfer function L = Pd*Cd using the tf, zpk, ss, and frd representations. Ltf = Pd*Cd; Lzp = zpk(Pd)*Cd; Lss = ss(Pd)*Cd; w = logspace(-1,3,100); Lfrd = frd(Pd,w)*Cd; Plot the magnitude of the frequency response to compare the four representations. bodemag(Ltf,Lzp,Lss,Lfrd) legend('tf','zpk','ss','frd') 4-31 4 Model Interconnections The tf representation has lost low-frequency dynamics that other representations preserve. See Also More About • 4-32 “Rules That Determine Model Type” on page 4-29 5 Model Transformation • “Conversion Between Model Types” on page 5-2 • “Convert From One Model Type to Another” on page 5-4 • “Get Current Value of Generalized Model by Model Conversion” on page 5-6 • “Decompose a 2-DOF PID Controller into SISO Components” on page 5-8 • “Discretize a Compensator” on page 5-13 • “Improve Accuracy of Discretized System with Time Delay” on page 5-19 • “Convert Discrete-Time System to Continuous Time” on page 5-23 • “Continuous-Discrete Conversion Methods” on page 5-26 • “Upsample Discrete-Time System” on page 5-36 • “Choosing a Resampling Command” on page 5-40 5 Model Transformation Conversion Between Model Types Explicit Conversion Between Model Types You can explicitly convert a model from one representation to another using the modelcreation command for the target model type. For example, convert to state-space representation using ss, and convert to parallel-form PID using pid. For information about converting to a particular model type, see the reference page for that model type. In general, you can convert from any model type to any other. However, there are a few limitations. For example, you cannot convert: • frd models to analytic model types such as ss, tf, or zpk (unless you perform system identification with System Identification Toolbox software). • ss models with internal delays to tf or zpk. You can convert between Numeric LTI models and Generalized LTI models. • Converting a Generalized LTI model to a Numeric LTI model evaluates any Control Design Blocks at their current (nominal) value. • Converting a Numeric LTI model to a Generalized LTI model creates a Generalized LTI model with an empty Blocks property. Automatic Conversion Between Model Types Some algorithms operate only on one type of model object. For example, the algorithm for zero-order-hold discretization with c2d can only be performed on state-space models. Similarly, commands such as tfdata or piddata expect a particular type of model (tf or pid, respectively). For convenience, such commands automatically convert input models to the appropriate or required model type. For example: sys = ss(0,1,1,0) [num,den] = tfdata(sys) tfdata automatically converts the state-space model sys to transfer function form to return numerator and denominator data. Conversions to state-space form are not uniquely defined. For this reason, automatic conversions to state space do not occur when the result depends on the choice of state 5-2 See Also coordinates. For example, the initial and kalman commands require state-space models. Recommended Working Representation You can represent numeric system components using any model type. However, Numeric LTI model types are not equally well-suited for numerical computations. In general, it is recommended that you work with state-space (ss) or frequency response data (frd) models, for the following reasons: • The accuracy of computations using high-order transfer functions (tf or zpk models) is sometimes poor, particularly for MIMO or high-order systems. Conversions to a transfer function representation can incur a loss of accuracy. • When you convert tf or zpk models to state space using ss, the software automatically performs balancing and scaling operations. Balancing and scaling improves the numeric accuracy of computations involving the model. For more information about balancing and scaling state-space models, see “Scaling State-Space Models” on page 24-2. In addition, converting back and forth between model types can introduce additional states or orders, or introduce numeric inaccuracies. For example, conversions to state space are not uniquely defined, and are not guaranteed to produce a minimal realization for MIMO models. For a given state-space model sys, ss(tf(sys)) can return a model with different state-space matrices, or even a different number of states in the MIMO case. See Also frd | pid | ss | tf | zpk Related Examples • “Convert From One Model Type to Another” on page 5-4 5-3 5 Model Transformation Convert From One Model Type to Another This example shows how to convert a numeric LTI model from one type (pid) to another type (tf). In general, you can convert a model from one type to another type using the modelcreation command for the target type. For example, you can use the tf command to convert an ss model to transfer function form, or use the ss command to convert a zpk model to state-space form. Create a PID controller. pid_sys = pid(1,1.5,3) pid_sys = 1 Kp + Ki * --- + Kd * s s with Kp = 1, Ki = 1.5, Kd = 3 Continuous-time PID controller in parallel form. Convert pid_sys to a transfer function model. C = tf(pid_sys) C = 3 s^2 + s + 1.5 --------------s Continuous-time transfer function. C is a tf representation of pid_sys. C has the same dynamics as pid_sys, but stores the dynamic parameters as transfer-function numerator and denominator coefficients instead of proportional, integral, and derivative gains. You can similarly convert transfer function models to pid models, provided the tf model object represents a parallel-form PID controller with 5-4 . See Also In general, you can use the technique of this example to convert any type of model to another type of model. For more specific information about converting to a particular model type, see the reference page for that model type. See Also frd | pid | ss | tf | zpk More About • “Conversion Between Model Types” on page 5-2 5-5 5 Model Transformation Get Current Value of Generalized Model by Model Conversion This example shows how to get the current value of a generalized model by converting it to a numeric model. This conversion is useful, for example, when you have tuned the parameters of the generalized model using a tuning command such as systune. Create a Generalized Model Represent the transfer function containing a real, tunable parameter, a, which is initialized to 10. a = realp('a',10); F = tf(a,[1 a]); F is a genss model parameterized by a. Tune the Model Typically, once of you have a generalized model, you tune the parameters of the model using a tuning command such as systune. For this example, instead of tuning the model, manually change the value of the tunable component of F. F.Blocks.a.Value = 5; Get the current value of the generalized model. Get the current value of the generalized model by converting it to a numeric model. F_cur_val = tf(F) F_cur_val = 5 ----s + 5 Continuous-time transfer function. 5-6 See Also tf(F) converts the generalized model, F, to a numeric transfer function, F_cur_val. To view the state-space representation of the current value of F, type ss(F). To examine the current values of the individual tunable components in a generalized model, use showBlockValue. See Also realp | showBlockValue | tf More About • “Models with Tunable Coefficients” on page 1-19 • “Conversion Between Model Types” on page 5-2 • “Convert From One Model Type to Another” on page 5-4 5-7 5 Model Transformation Decompose a 2-DOF PID Controller into SISO Components This example shows how to extract SISO control components from a 2-DOF PID controller in each of the feedforward, feedback, and filter configurations. The example compares the closed-loop systems in all configurations to confirm that they are all equivalent. Obtain a 2-DOF PID controller. For this example, create a plant model, and tune a 2-DOF PID controller for it. G = tf(1,[1 0.5 0.1]); C2 = pidtune(G,'pidf2',1.5); C2 is a pid2 controller object. The control architecture for C2 is as shown in the following illustration. This control system can be equivalently represented in several other architectures that use only SISO components. In the feedforward configuration, the 2-DOF controller is represented as a SISO PID controller and a feedforward compensator. Decompose C2 into SISO control components using the feedforward configuration. [Cff,Xff] = getComponents(C2,'feedforward') 5-8 Decompose a 2-DOF PID Controller into SISO Components Cff = 1 s Kp + Ki * --- + Kd * -------s Tf*s+1 with Kp = 1.12, Ki = 0.23, Kd = 1.3, Tf = 0.122 Continuous-time PIDF controller in parallel form. Xff = -10.898 (s+0.2838) -----------------(s+8.181) Continuous-time zero/pole/gain model. This command returns the SISO PID controller Cff as a pid object. The feedforward compensator X is returned as a zpk object. Construct the closed-loop system for the feedforward configuration. Tff = G*(Cff+Xff)*feedback(1,G*Cff); In the feedback configuration, the 2-DOF controller is represented as a SISO PID controller and an additional feedback compensator. Decompose C2 using the feedback configuration and construct that closed-loop system. [Cfb, Xfb] = getComponents(C2,'feedback'); Tfb = G*Cfb*feedback(1,G*(Cfb+Xfb)); 5-9 5 Model Transformation In the filter configuration, the 2-DOF controller is represented as a SISO PID controller and prefilter on the reference signal. Decompose C2 using the filter configuration. Construct that closed-loop system as well. [Cfr, Xfr] = getComponents(C2,'filter'); Tfr = Xfr*feedback(G*Cfr,1); Construct the closed-loop system for the original 2-DOF controller, C2. To do so, convert C2 to a two-input, one-output transfer function, and use array indexing to access the channels. Ctf = tf(C2); Cr = Ctf(1); Cy = Ctf(2); T = Cr*feedback(G,Cy,+1); Compare the step responses of all the closed-loop systems. stepplot(T,Tff,Tfb,Tfr) legend('2-DOF','feedforward','feedback','filter','Location','Southeast') 5-10 See Also The plots coincide, demonstrating that all the systems are equivalent. Using a 2-DOF PID controller can yield improved performance compared to a 1-DOF controller. For more information, see “Tune 2-DOF PID Controller (Command Line)” on page 11-15. See Also getComponents | pid2 | pidstd2 5-11 5 Model Transformation Related Examples • 5-12 “Two-Degree-of-Freedom PID Controllers” on page 2-16 Discretize a Compensator Discretize a Compensator This example shows how to convert a compensator from continuous to discrete time using several discretization methods, to identify a method that yields a good match in the frequency domain. You might design a compensator in continuous time, and then need to convert it to discrete time for a digital implementation. When you do so, you want the discretization to preserve frequency-domain characteristics that are essential to your performance and stability requirements. In the following control system, G is a continuous-time second-order system with a sharp resonance around 3 rad/s. One valid controller for this system includes a notch filter in series with an integrator. Create a model of this controller. notch = tf([1,0.5,9],[1,5,9]); integ = pid(0,0.34); C = integ*notch; bodeplot(C) 5-13 5 Model Transformation The notch filter centered at 3 rad/s counteracts the effect of the resonance in G. This configuration allows higher loop gain for a faster overall response. Discretize the compensator. Cdz = c2d(C,0.5); The c2d command supports several different discretization methods. Since this command does not specify a method, c2d uses the default method, Zero-Order Hold (ZOH). In the ZOH method, the time-domain response of the discretized compensator matches the continuous-time response at each time step. 5-14 Discretize a Compensator The discretized controller Cdz has a sample time of 0.5 s. In practice, the sample time you choose might be constrained by the system in which you implement your controller, or by the bandwidth of your control system. Compare the frequency-domain response of C and Cdz. bodeplot(C,Cdz) legend('C','Cdz'); , where is the sample time. Near The vertical line marks the Nyquist frequency, the Nyquist frequency, the response of the discretized compensator is distorted relative to the continuous-time response. As a result, the discretized notched filter may not properly counteract the plant resonance. 5-15 5 Model Transformation To fix this, try discretizing the compensator using the Tustin method and compare to the ZOH result. The Tustin discretization method often yields a better match in the frequency domain than the ZOH method. Cdt = c2d(C,0.5,'tustin'); plotopts = bodeoptions; plotopts.Ylim = {[-60,40],[-225,0]}; bodeplot(C,Cdz,Cdt,plotopts) legend('C','Cdz','Cdt') The Tustin method preserves the depth of the notch. However, the method introduces a frequency shift that is unacceptable for many applications. You can remedy the frequency shift by specifying the notch frequency as the prewarping frequency in the Tustin transform. 5-16 Discretize a Compensator Discretize the compensator using the Tustin method with frequency prewarping, and compare the results. discopts = c2dOptions('Method','tustin','PrewarpFrequency',3.0); Cdtp = c2d(C,0.5,discopts); bodeplot(C,Cdt,Cdtp,plotopts) legend('C','Cdt','Cdtp') To specify additional discretization options beyond the discretization method, use c2dOptions. Here, the discretization options set discopts specifies both the Tustin method and the prewarp frequency. The prewarp frequency is 3.0 rad/s, the frequency of the notch in the compensator response. 5-17 5 Model Transformation Using the Tustin method with frequency prewarping yields a better-matching frequency response than Tustin without prewarping. See Also c2d | c2dOptions More About 5-18 • “Continuous-Discrete Conversion Methods” on page 5-26 • “Improve Accuracy of Discretized System with Time Delay” on page 5-19 Improve Accuracy of Discretized System with Time Delay Improve Accuracy of Discretized System with Time Delay This example shows how to improve the frequency-domain accuracy of a system with a time delay that is a fractional multiple of the sample time. For systems with time delays that are not integer multiples of the sample time, the Tustin and Matched methods by default round the time delays to the nearest multiple of the sample time. To improve the accuracy of these methods for such systems, c2d can optionally approximate the fractional portion of the time delay by a discrete-time all-pass filter (a Thiran filter). In this example, discretize the system both without and with an approximation of the fractional portion of the delay and compare the results. Create a continuous-time transfer function with a transport delay of 2.5 s. G = tf(1,[1,0.2,4],'ioDelay',2.5); Discretize G using a sample time of 1 s. G has a sharp resonance at 2 rad/s. At a sample time of 1 s, that peak is close to the Nyquist frequency. For a frequency-domain match that preserves dynamics near the peak, use the Tustin method with prewarp frequency 2 rad/s. discopts = c2dOptions('Method','tustin','PrewarpFrequency',2); Gt = c2d(G,1,discopts) Warning: Rounding delays to the nearest multiple of the sampling period. For more accur Gt = 0.1693 z^2 + 0.3386 z + 0.1693 z^(-3) * -----------------------------z^2 + 0.7961 z + 0.913 Sample time: 1 seconds Discrete-time transfer function. The software warns you that it rounds the fractional time delay to the nearest multiple of the sample time. In this example, the time delay of 2.5 times the sample time (2.5 s) converts to an additional factor of z^(-3) in Gt. Compare Gt to the continuous-time system G. 5-19 5 Model Transformation plotopts = bodeoptions; plotopts.Ylim = {[-100,20],[-1080,0]}; bodeplot(G,Gt,plotopts); legend('G','Gt') There is a phase lag between the discretized system Gt and the continuous-time system G, which grows as the frequency approaches the Nyquist frequency. This phase lag is largely due to the rounding of the fractional time delay. In this example, the fractional time delay is half the sample time. Discretize G again using a third-order discrete-time all-pass filter (Thiran filter) to approximate the half-period portion of the delay. 5-20 Improve Accuracy of Discretized System with Time Delay discopts.FractDelayApproxOrder = 3; Gtf = c2d(G,1,discopts); The FractDelayApproxOrder option specifies the order of the Thiran filter that approximates the fractional portion of the delay. The other options in discopts are unchanged. Thus Gtf is a Tustin discretization of G with prewarp at 2 rad/s. Compare Gtf to G and Gt. plotopts.PhaseMatching = 'on'; bodeplot(G,Gt,Gtf,plotopts); legend('G','Gt','Gtf','Location','SouthWest') 5-21 5 Model Transformation The magnitudes of Gt and Gtf are identical. However, the phase of Gtf provides a better match to the phase of the continuous-time system through the resonance. As the frequency approaches the Nyquist frequency, this phase match deteriorates. A higherorder approximation of the fractional delay would improve the phase matching closer to the Nyquist frequencies. However, each additional order of approximation adds an additional order (or state) to the discretized system. If your application requires accurate frequency-matching near the Nyquist frequency, use c2dOptions to make c2d approximate the fractional portion of the time delay as a Thiran filter. See Also c2d | c2dOptions | thiran More About 5-22 • “Continuous-Discrete Conversion Methods” on page 5-26 • “Discretize a Compensator” on page 5-13 Convert Discrete-Time System to Continuous Time Convert Discrete-Time System to Continuous Time This example shows how to convert a discrete-time system to continuous time using d2c, and compares the results using two different interpolation methods. Convert the following second-order discrete-time system to continuous time using the zero-order hold (ZOH) method: G = zpk(-0.5,[-2,5],1,0.1); Gcz = d2c(G) Warning: The model order was increased to handle real negative poles. Gcz = 2.6663 (s^2 + 14.28s + 780.9) ------------------------------(s-16.09) (s^2 - 13.86s + 1035) Continuous-time zero/pole/gain model. When you call d2c without specifying a method, the function uses ZOH by default. The ZOH interpolation method increases the model order for systems that have real negative poles. This order increase occurs because the interpolation algorithm maps real negative poles in the domain to pairs of complex conjugate poles in the domain. Convert G to continuous time using the Tustin method. Gct = d2c(G,'tustin') Gct = 0.083333 (s+60) (s-20) ---------------------(s-60) (s-13.33) Continuous-time zero/pole/gain model. In this case, there is no order increase. 5-23 5 Model Transformation Compare frequency responses of the interpolated systems with that of G. bode(G,Gcz,Gct) legend('G','Gcz','Gct') In this case, the Tustin method provides a better frequency-domain match between the discrete system and the interpolation. However, the Tustin interpolation method is undefined for systems with poles at z = -1 (integrators), and is ill-conditioned for systems with poles near z = 1. 5-24 See Also See Also d2c | d2cOptions More About • “Continuous-Discrete Conversion Methods” on page 5-26 • “Discretize a Compensator” on page 5-13 5-25 5 Model Transformation Continuous-Discrete Conversion Methods Choosing a Conversion Method The c2d command discretizes continuous-time models. Conversely, d2c converts discretetime models to continuous time. Both commands support several discretization and interpolation methods, as shown in the following table. Discretization Method Use When “Zero-Order Hold” on page 5-27 You want an exact discretization in the time domain for staircase inputs. “First-Order Hold” on page 5-28 You want an exact discretization in the time domain for piecewise linear inputs. “Impulse-Invariant Mapping” on page 5-29 (c2d only) You want an exact discretization in the time domain for impulse train inputs. “Tustin Approximation” on page 5-30 • You want good matching in the frequency domain between the continuous- and discrete-time models. • Your model has important dynamics at some particular frequency. “Zero-Pole Matching Equivalents” on page 5-34 • You have a SISO model. “Least Squares” on page 5-34 (c2d only) • You have a SISO model. • You want good matching in the frequency domain between the continuous- and discrete-time models. • You want good matching in the frequency domain between the continuous- and discrete-time models. • You want to capture fast system dynamics but must use a larger sample time. 5-26 Continuous-Discrete Conversion Methods Zero-Order Hold The Zero-Order Hold (ZOH) method provides an exact match between the continuousand discrete-time systems in the time domain for staircase inputs. The following block diagram illustrates the zero-order-hold discretization Hd(z) of a continuous-time linear model H(s). The ZOH block generates the continuous-time input signal u(t) by holding each sample value u(k) constant over one sample period: u ( t ) = u [ k] , kTs £ t £ ( k + 1 ) Ts The signal u(t) is the input to the continuous system H(s). The output y[k] results from sampling y(t) every Ts seconds. Conversely, given a discrete system Hd(z), d2c produces a continuous system H(s). The ZOH discretization of H(s) coincides with Hd(z). The ZOH discrete-to-continuous conversion has the following limitations: • d2c cannot convert LTI models with poles at z = 0. • For discrete-time LTI models having negative real poles, ZOH d2c conversion produces a continuous system with higher order. The model order increases because a negative real pole in the z domain maps to a pure imaginary value in the s domain. Such mapping results in a continuous-time model with complex data. To avoid this issue, the software instead introduces a conjugate pair of complex poles in the s domain. See “Convert Discrete-Time System to Continuous Time” on page 5-23 for an example. 5-27 5 Model Transformation ZOH Method for Systems with Time Delays You can use the ZOH method to discretize SISO or MIMO continuous-time models with time delays. The ZOH method yields an exact discretization for systems with input delays, output delays, or transport delays. For systems with internal delays (delays in feedback loops), the ZOH method results in approximate discretizations. The following figure illustrates a system with an internal delay. H(s) e-ts For such systems, c2d performs the following actions to compute an approximate ZOH discretization: 1 Decomposes the delay τ as t = kTs + r with 0 £ r < Ts . 2 Absorbs the fractional delay r into H(s). 3 Discretizes H(s) to H(z). 4 Represents the integer portion of the delay kTs as an internal discrete-time delay z–k. The final discretized model appears in the following figure: H(z) e-sr H(s) z-k First-Order Hold The First-Order Hold (FOH) method provides an exact match between the continuousand discrete-time systems in the time domain for piecewise linear inputs. 5-28 Continuous-Discrete Conversion Methods FOH differs from ZOH by the underlying hold mechanism. To turn the input samples u[k] into a continuous input u(t), FOH uses linear interpolation between samples: u ( t ) = u [ k] + t - kTs (u [k + 1] - u [k]) , kTs £ t £ (k + 1) Ts Ts In general, this method is more accurate than ZOH for systems driven by smooth inputs. This FOH method differs from standard causal FOH and is more appropriately called triangle approximation (see [2], p. 228). The method is also known as ramp-invariant approximation. FOH Method for Systems with Time Delays You can use the FOH method to discretize SISO or MIMO continuous-time models with time delays. The FOH method handles time delays in the same way as the ZOH method. See “ZOH Method for Systems with Time Delays” on page 5-28. Impulse-Invariant Mapping The impulse-invariant mapping produces a discrete-time model with the same impulse response as the continuous time system. For example, compare the impulse response of a first-order continuous system with the impulse-invariant discretization: G = tf(1,[1,1]); Gd1 = c2d(G,0.01,'impulse'); impulse(G,Gd1) 5-29 5 Model Transformation The impulse response plot shows that the impulse responses of the continuous and discretized systems match. Impulse-Invariant Mapping for Systems with Time Delays You can use impulse-invariant mapping to discretize SISO or MIMO continuous-time models with time delays, except that the method does not support ss models with internal delays. For supported models, impulse-invariant mapping yields an exact discretization of the time delay. Tustin Approximation The Tustin or bilinear approximation yields the best frequency-domain match between the continuous-time and discretized systems. This method relates the s-domain and z-domain transfer functions using the approximation: 5-30 Continuous-Discrete Conversion Methods z=e sTs ª 1 + sTs / 2 . 1 - sTs / 2 In c2d conversions, the discretization Hd(z) of a continuous transfer function H(s) is: H d ( z ) = H ( s¢ ) , s¢ = 2 z-1 Ts z + 1 Similarly, the d2c conversion relies on the inverse correspondence H ( s ) = Hd ( z¢ ) , z¢ = 1 + sTs / 2 1 - sTs / 2 When you convert a state-space model using the Tustin method, the states are not preserved. The state transformation depends upon the state-space matrices and whether the system has time delays. For example, for an explicit (E = I) continuous-time model with no time delays, the state vector w[k] of the discretized model is related to the continuous-time state vector x(t) by: Ê T ˆ T T w [ kTs ] = Á I - A s ˜ x ( kTs ) - s Bu ( kTs ) = x ( kTs ) - s ( Ax ( kTs ) + Bu ( kTs ) ) . 2 ¯ 2 2 Ë Ts is the sample time of the discrete-time model. A and B are state-space matrices of the continuous-time model. Tustin Approximation with Frequency Prewarping If your system has important dynamics at a particular frequency that you want the transformation to preserve, you can use the Tustin method with frequency prewarping. This method ensures a match between the continuous- and discrete-time responses at the prewarp frequency. The Tustin approximation with frequency prewarping uses the following transformation of variables: H d ( z ) = H ( s¢ ) , s¢ = w z-1 tan (w Ts / 2 ) z + 1 5-31 5 Model Transformation This change of variable ensures the matching of the continuous- and discrete-time frequency responses at the prewarp frequency ω, because of the following correspondence: ( H ( jw ) = H d e jw Ts ) Tustin Approximation for Systems with Time Delays You can use the Tustin approximation to discretize SISO or MIMO continuous-time models with time delays. By default, the Tustin method rounds any time delay to the nearest multiple of the sample time. Therefore, for any time delay tau, the integer portion of the delay, k*Ts, maps to a delay of k sampling periods in the discretized model. This approach ignores the residual fractional delay, tau - k*Ts. You can to approximate the fractional portion of the delay by a discrete all-pass filter (Thiran filter) of specified order. To do so, use the FractDelayApproxOrder option of c2dOptions. See “Improve Accuracy of Discretized System with Time Delay” on page 519 for an example. To understand how the Tustin method handles systems with time delays, consider the following SISO state-space model G(s). The model has input delay τi, output delay τo, and internal delay τ. G(s) e-tis H(s) e-tos e-ts The following figure shows the general result of discretizing G(s) using the Tustin method. 5-32 Continuous-Discrete Conversion Methods Gd(z) z-mi Fi(z) Hd(z) z-m Fo(z) z-mo F(z) By default, c2d converts the time delays to pure integer time delays. The c2d command computes the integer delays by rounding each time delay to the nearest multiple of the sample time Ts. Thus, in the default case, mi = round(τi/Ts), mo = round(τo/Ts), and m = round(τ/Ts).. Also in this case, Fi(z) = Fo(z) = F(z) = 1. If you set FractDelayApproxOrder to a non-zero value, c2d approximates the fractional portion of the time delays by Thiran filters Fi(z), Fo(z), and F(z). The Thiran filters add additional states to the model. The maximum number of additional states for each delay is FractDelayApproxOrder. For example, for the input delay τi, the order of the Thiran filter Fi(z) is: order(Fi(z)) = max(ceil(τi/Ts), FractDelayApproxOrder). If ceil(τi/Ts) < FractDelayApproxOrder, the Thiran filter Fi(z) approximates the entire input delay τi. If ceil(τi/Ts) > FractDelayApproxOrder, the Thiran filter only approximates a portion of the input delay. In that case, c2d represents the remainder of the input delay as a chain of unit delays z–mi, where mi = ceil(τi/Ts) – FractDelayApproxOrder c2d uses Thiran filters and FractDelayApproxOrder in a similar way to approximate the output delay τo and the internal delay τ. When you discretizetf and zpk models using the Tustin method, c2d first aggregates all input, output, and transport delays into a single transport delay τTOT for each channel. c2d then approximates τTOT as a Thiran filter and a chain of unit delays in the same way as described for each of the time delays in ss models. 5-33 5 Model Transformation For more information about Thiran filters, see the thiran reference page and [4]. Zero-Pole Matching Equivalents This method of conversion, which computes zero-pole matching equivalents, applies only to SISO systems. The continuous and discretized systems have matching DC gains. Their poles and zeros are related by the transformation: zi = esi Ts where: • zi is the ith pole or zero of the discrete-time system. • si is the ith pole or zero of the continuous-time system. • Ts is the sample time. See [2] for more information. Zero-Pole Matching for Systems with Time Delays You can use zero-pole matching to discretize SISO continuous-time models with time delay, except that the method does not support ss models with internal delays. The zeropole matching method handles time delays in the same way as the Tustin approximation. See “Tustin Approximation for Systems with Time Delays” on page 5-32. Least Squares The least squares method minimizes the error between the frequency responses of the continuous-time and discrete-time systems up to the Nyquist frequency using a vectorfitting optimization approach. This method is useful when you want to capture fast system dynamics but must use a larger sample time, for example, when computational resources are limited. This method is supported only by the c2d function and only for SISO systems. As with Tustin approximation and zero-pole matching, the least squares method provides a good match between the frequency responses of the original continuous-time system and the converted discrete-time system. However, when using the least squares method with: 5-34 See Also • The same sample time as Tustin approximation or zero-pole matching, you get a smaller difference between the continuous-time and discrete-time frequency responses. • A lower sample time than what you would use with Tustin approximation or zero-pole matching, you can still get a result that meets your requirements. Doing so is useful if computational resources are limited, since the slower sample time means that the processor must do less work. References [1] Åström, K.J. and B. Wittenmark, Computer-Controlled Systems: Theory and Design, Prentice-Hall, 1990, pp. 48-52. [2] Franklin, G.F., Powell, D.J., and Workman, M.L., Digital Control of Dynamic Systems (3rd Edition), Prentice Hall, 1997. [3] Smith, J.O. III, "Impulse Invariant Method", Physical Audio Signal Processing, August 2007. https://www.dsprelated.com/dspbooks/pasp/ Impulse_Invariant_Method.html. [4] T. Laakso, V. Valimaki, "Splitting the Unit Delay", IEEE Signal Processing Magazine, Vol. 13, No. 1, p.30-60, 1996. See Also c2d | c2dOptions | d2c | d2cOptions | d2d | d2dOptions | thiran Related Examples • “Discretize a Compensator” on page 5-13 • “Improve Accuracy of Discretized System with Time Delay” on page 5-19 • “Convert Discrete-Time System to Continuous Time” on page 5-23 5-35 5 Model Transformation Upsample Discrete-Time System This example shows how to upsample a system using both the d2d and upsample commands and compares the results of both to the original system. Upsampling a system can be useful, for example, when you need to implement a digital controller at a faster rate than you originally designed it for. Create the discrete-time system with a sample time of 0.3 s. G = tf([1,0.4],[1,-0.7],0.3); Resample the system at 0.1 s using d2d. G_d2d = d2d(G,0.1) G_d2d = z - 0.4769 ---------z - 0.8879 Sample time: 0.1 seconds Discrete-time transfer function. By default, d2d uses the zero-order-hold (ZOH) method to resample the system. The resampled system has the same order as G. Resample the system again at 0.1 s, using upsample. G_up = upsample(G,3) G_up = z^3 + 0.4 --------z^3 - 0.7 5-36 Upsample Discrete-Time System Sample time: 0.1 seconds Discrete-time transfer function. The second input, 3, tells upsample to resample G at a sample time three times faster than the sample time of G. This input to upsample must be an integer. G_up has three times as many poles and zeroes as G. Compare the step responses of the original model G with the resampled models G_d2d and G_up. step(G,'-r',G_d2d,':g',G_up,'--b') legend('G','d2d','upsample','Location','SouthEast') 5-37 5 Model Transformation The step response of the upsampled model G_up matches exactly the step response of the original model G. The response of the resampled model G_d2d matches only at every third sample. Compare the frequency response of the original model with the resampled models. bode(G,'-r',G_d2d,':g',G_up,'--b') legend('G','d2d','upsample','Location','SouthWest') In the frequency domain as well, the model G_up created with the upsample command matches the original model exactly up to the Nyquist frequency of the original model. Using upsample provides a better match than d2d in both the time and frequency domains. However, upsample increases the model order, which can be undesirable. 5-38 See Also Additionally, upsample is only available where the original sample time is an integer multiple of the new sample time. See Also d2d | d2dOptions | upsample More About • “Choosing a Resampling Command” on page 5-40 5-39 5 Model Transformation Choosing a Resampling Command You can resample a discrete-time model using the commands described in the following table. To... Use the command... • Downsample a system. d2d • Upsample a system without any restriction on the new sample time. Upsample a system with the highest accuracy when: upsample • The new sample time is integer-valuetimes faster than the sample time of the original model. • Your new model can have more states than the original model. See Also d2d | d2dOptions | upsample Related Examples • 5-40 “Upsample Discrete-Time System” on page 5-36 6 Model Simplification • “Model Reduction Basics” on page 6-2 • “Reduce Model Order Using the Model Reducer App” on page 6-6 • “Balanced Truncation Model Reduction” on page 6-17 • “Pole-Zero Simplification” on page 6-47 • “Mode-Selection Model Reduction” on page 6-57 • “Visualize Reduced-Order Models in the Model Reducer App” on page 6-67 6 Model Simplification Model Reduction Basics Working with lower-order models can simplify analysis and control design, relative to higher-order models. Simpler models are also easier to understand and manipulate. Highorder models obtained by linearizing complex Simulink models or from other sources can contain states that do not contribute much to the dynamics of particular interest to your application. Therefore, it can be useful to reduce model order while preserving model characteristics that are important for your application. When to Reduce Model Order Cases where you might want to reduce model order include these situations: • You are working with a relatively high-order model obtained from linearizing a Simulink model, performing a finite-element calculation, interconnecting model elements, or other source. • You want to improve the simulation speed of a Simulink model at a certain operating point. In that case, you can linearize a portion of the model at that operating point and compute a reduced-order simplification or approximation of the linearized model. You can then replace the portion of the model with an LTI Block containing the reducedorder model. • You design a high-order controller that you want to implement as a lower-order controller, such as a PID controller. For example, controller design using LinearQuadratic-Gaussian methods or H∞ synthesis techniques can yield a high-order result. In this case, you can try reducing the plant order before synthesis, reducing the controller order after synthesis, or both. • You want to simplify a model obtained by identification with System Identification Toolbox software. The following diagram illustrates the relationship between model reduction and control design. 6-2 Model Reduction Basics G Plant reduction Controller design Controller design C GR Controller reduction Higher Order CR Lower Order In general, when designing a controller for a system represented by a high-order model, G, it is useful to start by simplifying the plant model. Then, design a relatively low-order controller, CR, for the lower-order plant model GR. After you design a controller for either the original or the reduced plant model, you can try to reduce the controller further. Reducing the plant or controller can include: • Discarding states that do not contribute to the system dynamics, such as structurally disconnected states or canceling pole-zero pairs. • Discarding low-energy states that contribute relatively little to system dynamics. • Focusing on a particular frequency region and discarding dynamics outside that region. For example, if your control bandwidth is limited by actuator dynamics, discard higher-frequency dynamics. In any case, when you reduce model order, you want to preserve model characteristics that are important for your application. Whenever you compute a reduced-order model, verify that the reduced model preserves time-domain or frequency-domain behavior that you care about. For example, for control design, it is useful to verify that the reduced closed-loop system is stable. It is also useful to check that the reduced open-loop transfer function CRGR adequately matches the original models where the open-loop gain GC is close to 1 (in the gain crossover region). 6-3 6 Model Simplification Choosing a Model Reduction Method To reduce the order of a model, you can either simplify your model, or compute a lowerorder approximation. The following table summarizes the differences among several model-reduction approaches. Approach Command Line Model Reducer App Simplification — Reduce • sminreal — Eliminate model order exactly by states that are canceling pole-zero pairs or structurally disconnected eliminating states that have from the inputs or no effect on the overall outputs. model response • minreal — Eliminate canceling or nearcanceling pole-zero pairs from transfer functions. Eliminate unobservable or uncontrollable states from state-space models. “Pole-Zero Simplification” on page 6-47 method — Eliminate: Approximation — Compute balred — Discard states a lower-order approximation that have relatively low of your model. effect on the overall model response. Balanced Truncation on page 6-17 method — Discard states that have relatively low effect on the overall model response. Mode Selection — Eliminate poles and zeros that fall outside a specific frequency range of interest. Mode Selection on page 657 method — Select frequency range of interest and discard dynamics outside that range. freqsep — Separate model into slow and fast dynamics around a specified cutoff frequency. • Structurally disconnected states • Unobservable or uncontrollable states from state-space models • Canceling or nearcanceling pole-zero pairs from transfer functions Sometimes, approximation can yield better results, even if the model looks like a good candidate for simplification. For example, models with near pole-zero cancellations are sometimes better reduced by approximation than simplification. Similarly, using balred to reduce state-space models can yield more accurate results than minreal. When you use a reduced-order model, always verify that the simplification or approximation preserves model characteristics that are important for your application. For example, compare the frequency responses of the original and reduced models using 6-4 See Also bodeplot or sigmaplot. Or, compare the open-loop responses for the original and reduced plant and controller models. See Also Apps Model Reducer Functions balred | freqsep | minreal | sminreal Related Examples • “Balanced Truncation Model Reduction” on page 6-17 • “Mode-Selection Model Reduction” on page 6-57 • “Pole-Zero Simplification” on page 6-47 6-5 6 Model Simplification Reduce Model Order Using the Model Reducer App This example shows how to reduce model order while preserving important dynamics using the Model Reducer app. This example illustrates the Balanced Truncation method, which eliminates states based on their energy contributions to the system response. Open Model Reducer With a Building Model This example uses a model of the Los Angeles University Hospital building. The building has eight floors, each with three degrees of freedom: two displacements and one rotation. The input-output relationship for any one of these displacements is represented as a 48state model, where each state represents a displacement or its rate of change (velocity). Load the building model and open Model Reducer with that model. load build.mat modelReducer(G) Select the model in the Data Browser to display some information about the model in the Preview section. Double-click the model to see more detailed information. 6-6 Reduce Model Order Using the Model Reducer App Open the Balanced Truncation Tab Model Reducer has three model reduction methods: Balanced Truncation, Mode Selection, and Pole/Zero Simplification. For this example, click Balanced Truncation. 6-7 6 Model Simplification Model Reducer opens the Balanced Truncation tab and automatically generates a reduced-order model. The top plot compares the original and reduced model in the frequency domain. The bottom plot shows the energy contribution of each state, where the states are sorted from high energy to low energy. The order of the reduced model, 14, is highlighted in the bar chart. In the reduced model, all states with lower energy contribution than this one are discarded. Compute Multiple Approximations Suppose that you want to preserve the first, second, and third peaks of the model response, around 5.2 rad/s, 13 rad/s, and 25 rad/s. Try other model orders to see whether 6-8 Reduce Model Order Using the Model Reducer App you can achieve this goal with a lower model order. Compute a 5th-order and a 10th-order approximation in one of the following ways: • In the Reduced model orders text box, enter [5 10]. • In the state-contribution plot, ctrl-click the bars for state 5 and state 10. Model Reducer computes two new reduced-order models and displays them on the response plot with the original model G. To examine the three peaks more closely, Zoom in on the relevant frequency range. The 10th-order model captures the three peaks successfully, while the 5th-order model only approximates the first two peaks. (For information about zooming and other interactions with the analysis plots, see “Visualize Reduced-Order Models in the Model Reducer App” on page 6-67.) 6-9 6 Model Simplification Compare Reduced Models With Different Visualizations In addition to the frequency response plot of all three models, Model Reducer lets you examine the absolute and relative error between the original and reduced models. Select Absolute error plot to see the difference between the building and reduced models. 6-10 Reduce Model Order Using the Model Reducer App The 5th-order reduced model has at most -60dB error in the frequency region of the first two peaks, below about 30 rad/s. The error increases at higher frequencies. The 10thorder reduced model has smaller error over all frequencies. Create Reduced Models in Data Browser Store the reduced models in the Data Browser by clicking Create Reduced Model. The 5th-order and 10th-order reduced models appear in the Data Browser with names GReduced5 and Greduced10. You can continue to change the model-reduction parameters and generate additional reduced models. As you do so, GReduced5 and Greduced10 remain unchanged in the Data Browser. 6-11 6 Model Simplification Focus on Dynamics at Particular Frequencies By default, balanced truncation in Model Reducer preserves DC gain, matching the steady-state response of the original and reduced models. Clear the Preserve DC Gain checkbox to better approximate high-frequency dynamics. Model Reducer computes new reduced models. The error in the high-frequency region is decreased at the cost of a slight increase in error at low frequencies. You can also focus the balanced truncation on the model dynamics in a particular frequency interval. For example, approximate only the second peak of the building model around 13 rad/s. First, select the Model response plot to see the Bode plots of models. Then check Select frequency range checkbox. Model Reducer analyzes state contributions in the highlighted frequency interval only. You can drag the boundaries to change the frequency range interactively. As you change the frequency interval, the Hankel Singular Value plot reflects the changes in the energy contributions of the states. Enter the frequency limits [10 22] into the text box next to Select frequency range. The 5th-order reduced model captures the essential dynamics. The 10th-order model has almost the same dynamics as the original building model within this frequency range. 6-12 Reduce Model Order Using the Model Reducer App Optionally, store these additional models in the Data Browser by clicking Create Reduced Model. Compare Models In Time Domain You can compare time-domain responses of the stored reduced models and the original in the Plots tab. In the Data Browser, control-click to select the models you want to compare, G, GReduced5, and GReduced10. Then, click Step. Model Reducer creates a step plot with all three models. 6-13 6 Model Simplification Zooming on the transient behavior of this plot shows that GReduced10 captures the time domain behavior of the original model well. However, the response of GReduced5 deviates from the original model after about 3 seconds. Export Model for Further Analysis Comparison of the reduced and original models in the time and frequency domains shows that GReduced10 adequately captures the dynamics of interest. Export that model to the MATLAB® workspace for further analysis and design. In the Model Reducer tab, click Export Model. Clear the check boxes for G and Greduced5, and click Export to export Greduced10. 6-14 See Also Greduced10 appears in the MATLAB workspace as a state-space (ss) model. See Also Model Reducer Related Examples • “Model Reduction Basics” on page 6-2 • “Balanced Truncation Model Reduction” on page 6-17 • “Pole-Zero Simplification” on page 6-47 6-15 6 Model Simplification • 6-16 “Mode-Selection Model Reduction” on page 6-57 Balanced Truncation Model Reduction Balanced Truncation Model Reduction Balanced truncation computes a lower-order approximation of your model by neglecting states that have relatively low effect on the overall model response. Using a lower-order approximation that preserves the dynamics of interest can simplify analysis and control design. In the balanced truncation method of model reduction, the software measures state contributions by Hankel singular values (see hsvd) and discards states with smaller values. There are two ways to compute a reduced-order model by balanced truncation: • At the command line, using the balred command. • In the Model Reducer, using the Balanced Truncation method. For more general information about model reduction, see “Model Reduction Basics” on page 6-2. Balanced Truncation in the Model Reducer App Model Reducer provides an interactive tool for performing model reduction and examining and comparing the responses of the original and reduced-order models. To approximate a model by balanced truncation in Model Reducer: 1 Open the app, and import an LTI model to reduce. For instance, suppose that there is a model named build in the MATLAB workspace. The following command opens Model Reducer and imports the model. modelReducer(build) 2 In the Data Browser, select the model to reduce. Click Balanced Truncation. 6-17 6 Model Simplification In the Balanced Truncation tab, Model Reducer displays a plot of the frequency response of the original model and a reduced version of the model. The frequency response is a Bode plot for SISO models, and a singular-value plot for MIMO models. The app also displays a Hankel singular-value plot of the original model. 6-18 Balanced Truncation Model Reduction The Hankel singular-value plot shows the relative energy contributions of each state in the system. Model Reducer computes an initial reduced-order model based on these values. The highlighted bar is the lowest-energy state in the initial reducedorder model. Model Reducer discards states that have lower Hankel singular values than the highlighted bar. 6-19 6 Model Simplification 3 Try different reduced-model orders to find the lowest-order model that preserves the dynamics that are important for your application. To specify different orders, either: • Enter model orders in the Reduced model orders field. You can enter a single integer or an array of integers, such as 10:14 or [8,11,12]. • Click a bar on the Hankel singular-value plot to specify the lowest-energy state of the reduced-order model. Ctrl-click to specify multiple values. When you change the specified reduced model order, Model Reducer automatically computes a new reduced-order model. If you specify multiple model orders, Model Reducer computes multiple reduced-order models and displays their responses on the plot. 6-20 Balanced Truncation Model Reduction 4 Optionally, examine the absolute or relative error between the original and reducedorder model, in addition to the frequency response. Select the error-plot type using the buttons on the Balanced Truncation tab. 6-21 6 Model Simplification For more information about using the analysis plots, see “Visualize Reduced-Order Models in the Model Reducer App” on page 6-67. 5 6-22 If low-frequency dynamics are not important to your application, you can clear the Preserve DC Gain checkbox. Doing so sometimes yields a better match at higher frequencies between the original and reduced-order models. Balanced Truncation Model Reduction When you check or clear the Preserve DC Gain checkbox, Model Reducer automatically computes new reduced-order models. For more information about this option, see “Compare Truncated and DC Matched Low-Order Model Approximations” on page 6-30. 6 Optionally, limit the Hankel singular-value computation to a specific frequency range. Such a limit is useful when the model has modes outside the region of interest to your particular application. When you apply a frequency limit, Model Reducer determines which states to truncate based on their energy contribution within the specified frequency range only. Neglecting energy contributions outside that range can yield an even lower-order approximation that is still adequate for your application. To limit the singular-value computation, check Select frequency range. Then, specify the frequency range by: 6-23 6 Model Simplification • In the text box, entering a vector of the form [fmin,fmax]. Units are rad/ TimeUnit, where TimeUnit is the TimeUnit property of the model you are reducing. • On the response plot or error plot, dragging the boundaries of the shaded region or the shaded region itself. Model Reducer analyzes the state contributions within the shaded region only. When you check or clear the Select frequency range checkbox or change the selected range, Model Reducer automatically computes new reduced-order models. Note Checking Select frequency range automatically clears Preserve DC Gain. To enforce a DC match even when using frequency limits, recheck Preserve DC Gain. 7 When you have one or more reduced models that you want to store and analyze further, click . The new models appear in the Data Browser. If you have specified multiple orders, each reduced model appears separately. Model names reflect the reduced model order. 6-24 Balanced Truncation Model Reduction After creating reduced models in the Data Browser, you can continue changing the reduction parameters and create reduced models with different orders for analysis and comparison. You can now perform further analysis with the reduced model. For example: • Examine other responses of the reduced system, such as the step response or Nichols plot. To do so, use the tools on the Plots tab. See “Visualize Reduced-Order Models in the Model Reducer App” on page 6-67 for more information. • Export reduced models to the MATLAB workspace for further analysis or control design. On the Model Reducer tab, click Export. Generate MATLAB Code for Balanced Truncation To create a MATLAB script you can use for further model-reduction tasks at the command line, click Create Reduced Model, and select Generate MATLAB Script. 6-25 6 Model Simplification Model Reducer creates a script that uses the balred command to perform model reduction with the parameters and options you have set on the Balanced Truncation tab. The script opens in the MATLAB editor. Approximate Model by Balanced Truncation at the Command Line At the command-line, use balred to compute a reduced-order approximation of a model. To do so, first examine the contribution of the various states to the overall model behavior. Choose the approximation order based on the number of states that make a significant contribution to the overall model behavior. For this example, load a high-order model. hplant is a 23rd-order SISO model. 6-26 Balanced Truncation Model Reduction load ltiexamples hplant order(hplant) ans = 23 Examine the relative amount of energy per state in hplant using a Hankel singular-value (HSV) plot. hsvplot(hplant) Small Hankel singular values indicate that the associated states contribute little to the behavior of the system. The plot shows that two states account for most of the energy in the system. Therefore, try simplifying the model to just first or second order. 6-27 6 Model Simplification opts = balredOptions('StateElimMethod','Truncate'); hplant1 = balred(hplant,1,opts); hplant2 = balred(hplant,2,opts); The second argument to balred specifies the target approximation order, so that hplant1 is a first-order approximation and hplant2 is a second-order approximation of hplant. By default, balred discards the states with the smallest Hankel singular values, and alters the remaining states to preserve the DC gain of the system. Setting the StateElimMethod option to Truncate causes balred to discard low-energy states without altering the remaining states. When working with reduced-order models, it is important to verify that the approximation does not introduce inaccuracies at frequencies that are important for your application. Therefore, compare the frequency responses of the original and approximated systems. For MIMO systems, use the sigmaplot command. For this SISO system, examine a Bode plot. bodeplot(hplant,hplant2,hplant1) legend('Original','2nd order','1st order') 6-28 Balanced Truncation Model Reduction The second-order approximation hplant2 matches the original 23rd-order system very well, especially at lower frequencies. The first-order system does not match as well. In general, as you decrease the order of the approximated model, the frequency response of the approximated model begins to differ from the original model. Choose an approximation that is sufficiently accurate in the bands that are important to you. For example, in a control system you might want good accuracy inside the control bandwidth. Accuracy at frequencies far above the control bandwidth, where the gain rapidly rolls off, might be less important. You can also validate the approximation in the time domain. For instance, examine the step responses of the original and reduced-order systems. 6-29 6 Model Simplification stepplot(hplant,hplant2,'r--',hplant1,'g--') legend('Original','2nd order','1st order','Location','SouthEast') This result confirms that the second-order approximation is a good match to the original 23rd-order system. Compare Truncated and DC Matched Low-Order Model Approximations This example shows how to compute a low-order approximation in two ways and compares the results. When you compute a low-order approximation by the balanced truncation method, you can either: 6-30 Balanced Truncation Model Reduction • Discard the states that make the smallest contribution to system behavior, altering the remaining states to preserve the DC gain of the system. • Discard the low-energy states without altering the remaining states. Which method you choose depends on what dynamics are most important to your application. In general, preserving DC gain comes at the expense of accuracy in higherfrequency dynamics. Conversely, state truncation can yield more accuracy in fast transients, at the expense of low-frequency accuracy. This example compares the state-elimination methods of the balred command, Truncate and MatchDC. You can similarly control the state-elimination method in the Model Reducer app, on the Balanced Truncation tab, using the Preserve DC Gain check box, as shown. Consider the following system. 6-31 6 Model Simplification Create a closed-loop model of this system from r to y. G = zpk([0 -2],[-1 -3],1); C = tf(2,[1 1e-2]); T = feedback(G*C,1) T = 2 s (s+2) -------------------------------(s+0.004277) (s+1.588) (s+4.418) Continuous-time zero/pole/gain model. T is a third-order system that has a pole-zero near-cancellation close to s = 0. Therefore, it is a good candidate for order reduction by approximation. Compute two second-order approximations to T, one that preserves the DC gain and one that truncates the lowest-energy state without changing the other states. Use balredOptions to specify the approximation methods, MatchDC and Truncate, respectively. matchopt truncopt Tmatch = Ttrunc = = balredOptions('StateElimMethod','MatchDC'); = balredOptions('StateElimMethod','Truncate'); balred(T,2,matchopt); balred(T,2,truncopt); Compare the frequency responses of the approximated models. bodeplot(T,Tmatch,Ttrunc) legend('Original','DC Match','Truncate') 6-32 Balanced Truncation Model Reduction The truncated model Ttrunc matches the original model well at high frequencies, but differs considerably at low frequency. Conversely, Tmatch yields a good match at low frequencies as expected, at the expense of high-frequency accuracy. You can also see the differences between the two methods by examining the time-domain response in different regimes. Compare the slow dynamics by looking at the step response of all three models with a long time horizon. stepplot(T,Tmatch,'r--',Ttrunc,1500) legend('Original','DC Match','Truncate') 6-33 6 Model Simplification As expected, on long time scales the DC-matched approximation Tmatch has a very similar response to the original model. Compare the fast transients in the step response. stepplot(T,Tmatch,'r',Ttrunc,'g--',0.5) legend('Original','DC Match','Truncate') 6-34 Balanced Truncation Model Reduction On short time scales, the truncated approximation Ttrunc provides a better match to the original model. Which approximation method you should use depends on which regime is most important for your application. Approximate Model with Unstable or Near-Unstable Pole This example shows how to compute a reduced-order approximation of a system when the system has unstable or near-unstable poles. When computing a reduced-order approximation, the balred command (or the Model Reducer app) does not eliminate unstable poles because doing so would fundamentally change the system dynamics. Instead, the software decomposes the model into stable and unstable parts and reduces the stable part of the model. 6-35 6 Model Simplification If your model has near-unstable poles, you might want to ensure that the reduced-order approximation preserves these dynamics. This example shows how to use the Offset option of balred to preserve poles that are close to the stable-unstable boundary. You can achieve the same result in the Model Reducer app, on the Balanced Truncation tab, under Options, using the Offset field, as shown: Load a model with unstable and near-unstable poles. load('reduce.mat','gasf35unst') gasf35unst is a 25-state SISO model with two unstable poles (Re(s) > 0). Examine the system poles to find the near-unstable poles. pzplot(gasf35unst) axis([-0.0015 0.0015 -0.0005 0.0005]) 6-36 Balanced Truncation Model Reduction The pole-zero plot shows several poles (marked by x) that fall in the left half-plane, but relatively close to the imaginary axis. These are the near-unstable poles. Two of these fall within 0.0005 of instability. Three more fall within 0.001 of instability. Examine a Hankel singular-value plot of the model. hsvplot(gasf35unst) 6-37 6 Model Simplification The plot shows the two unstable modes, but you cannot easily determine the energy contribution of the near-unstable poles. In your application, you might want to reduce the model without discarding those poles nearest to instability, even if they are of relatively low energy. Use the Offset option of balred to calculate a reduced-order system that preserves the two stable poles that are closest to the imaginary axis. The Offset option sets the boundary between poles that balred can discard, and poles that balred must preserve (treat as unstable). opts = balredOptions('Offset',0.0005); gasf_arr = balred(gasf35unst,[10 15],opts); Providing balred an array of target approximation orders [10 15] causes balred to return an array of approximated models. The array gasf_arr contains two models, a 6-38 Balanced Truncation Model Reduction 10th-order and a 15th-order approximation of gasf35unst. In both approximations, balred does not discard the two unstable poles or the two nearly-unstable poles. Compare the reduced-order approximations to the original model. bodeplot(gasf35unst,gasf_arr,'r--') The 15th order approximation is a good frequency-domain match to the original model. However, the 10th-order approximation shows changes in high-frequency dynamics, 6-39 6 Model Simplification which might be too large to be acceptable. The 15th-order approximation is likely a better choice. Frequency-Limited Balanced Truncation Focusing the energy-contribution calculation on a particular frequency region sometimes yields a good approximation to the dynamics of interest at a lower order than a reduction that takes all frequencies into account. For this example, reduce a high-order model with a focus on the dynamics in a particular interval. This example demonstrates frequency-limited balanced truncation at the command line, using options for the balred command. You can also perform frequency-limited balanced truncation in the Model Reducer app, on the Balanced Truncation tab, using the Select frequency range check box, as shown. Load a model and examine its frequency response. load(fullfile(matlabroot,'examples','control','build.mat'),'G') bodeplot(G) 6-40 Balanced Truncation Model Reduction G is a 48th-order model with several large peak regions around 5.2 rad/s, 13.5 rad/s, and 24.5 rad/s, and smaller peaks scattered across many frequencies. Examine the Hankel singular-value plot to see the energy contributions of the model's 48 states. hsvd(G) 6-41 6 Model Simplification The singular-value plot suggests that you can discard at least 20 states without significant impact on the overall system response. Suppose that for your application you are only interested in the dynamics near the second large peak, between 10 rad/s and 22 rad/s. Try a few reduced-model orders based on the Hankel singular value plot. Compare their frequency responses to the original model, especially in the region of that peak. G18 = balred(G,18); G10 = balred(G,10); bodeplot(G,G18,G10,logspace(0.5,1.5,100)); legend('Original','Order 18','Order 10'); 6-42 Balanced Truncation Model Reduction The 18th-order model is a good match to the dynamics in the region of interest. In the 10th order model, however, there is some degradation of the match. Focus the model reduction on the region of interest to obtain a good match with a lowerorder approximation. First, examine the state energy contributions in that frequency region only. Use hsvdOptions to specify the frequency interval for hsvd. hopt = hsvdOptions('FreqIntervals',[10,22]); hsvd(G,hopt) 6-43 6 Model Simplification Comparing this plot to the previous Hankel singular-value plot shows that in this frequency region, many fewer states contribute significantly to the dynamics than contribute to the overall dynamics. Try the same reduced-model orders again, this time choosing states to eliminate based only on their contribution to the frequency interval. Use balredOptions to specify the frequency interval for balred. bopt = balredOptions('StateElimMethod','Truncate','FreqIntervals',[10,22]); GLim18 = balred(G,18,bopt); GLim10 = balred(G,10,bopt); bodeplot(G,GLim18,GLim10,logspace(0.5,1.5,100)); legend('Original','Order 18','Order 10'); 6-44 See Also With the frequency-limited energy computation, a 10th-order approximation is as good in the region of interest as the 18th-order approximation computed without frequency limits. See Also Apps Model Reducer Functions balred | hsvplot 6-45 6 Model Simplification Related Examples 6-46 • “Mode-Selection Model Reduction” on page 6-57 • “Pole-Zero Simplification” on page 6-47 • “Model Reduction Basics” on page 6-2 Pole-Zero Simplification Pole-Zero Simplification Pole-zero simplification reduces the order of your model exactly by canceling pole-zero pairs or eliminating states that have no effect on the overall model response. Pole-zero pairs can be introduced, for example, when you construct closed-loop architectures. Normal small errors associated with numerical computation can convert such canceling pairs to near-canceling pairs. Removing these states preserves the model response characteristics while simplifying analysis and control design. Types of pole-zero simplification include: • Structural elimination — Eliminate states that are structurally disconnected from the inputs or outputs. Eliminating structurally disconnected states is a good first step in model reduction because the process does not involve any numerical computation. It also preserves the state structure of the remaining states. At the command line, perform structural elimination with sminreal. • Pole-zero cancellation or minimal realization — Eliminate canceling or near-canceling pole-zero pairs from transfer functions. Eliminate unobservable or uncontrollable states from state-space models. At the command line, perform this kind of simplification with minreal. In the Model Reducer app, the Pole-Zero Simplification method automatically eliminates structurally disconnected states and also performs pole-zero cancellation or minimal realization. Pole-Zero Simplification in the Model Reducer App Model Reducer provides an interactive tool for performing model reduction and examining and comparing the responses of the original and reduced-order models. To reduce a model by pole-zero simplification in Model Reducer: 1 Open the app and import a model to reduce. For instance, suppose that there is a model named build in the MATLAB workspace. The following command opens Model Reducer and imports the LTI model build. modelReducer(build) 2 In the Data Browser, select the model to reduce. Click Simplification. Pole-Zero 6-47 6 Model Simplification In the Pole-Zero Simplification tab, Model Reducer displays a plot of the frequency response of the original model and a reduced version of the model. The app also displays a pole-zero map of both models. 6-48 Pole-Zero Simplification The pole-zero map marks pole locations with x and zero locations with o. Note The frequency response is a Bode plot for SISO models, and a singular-value plot for MIMO models. 3 Optionally, change the tolerance with which Model Reducer identifies canceling pole-zero pairs. Model Reducer cancels pole-zero pairs that fall within the tolerance specified by the Simplification of pole-zero pairs value. In this case, no pole-zero 6-49 6 Model Simplification pairs are close enough together for Model Reducer to cancel them at the default tolerance of 1e-05. To cancel pairs that are a little further apart, move the slider to the right or enter a larger value in the text box. The blue x and o marks on the pole-zero map show the near-canceling pole-zero pairs in the original model that are eliminated from the simplified model. Poles and zeros remaining in the simplified model are marked with red x and o. 4 6-50 Try different simplification tolerances while observing the frequency response of the original and simplified model. Remove as many poles and zeros as you can while preserving the system behavior in the frequency region that is important for your Pole-Zero Simplification application. Optionally, examine absolute or relative error between the original and simplified model. Select the error-plot type using the buttons on the Pole-Zero Simplification tab. For more information about using the analysis plots, see “Visualize Reduced-Order Models in the Model Reducer App” on page 6-67. 5 When you have a simplified model that you want to store and analyze further, click . The new model appears in the Data Browser with a name that reflects the reduced model order. 6-51 6 Model Simplification After creating a reduced model in the Data Browser, you can continue changing the simplification parameters and create reduced models with different orders for analysis and comparison. You can now perform further analysis with the reduced model. For example: • Examine other responses of the reduced system, such as the step response or Nichols plot. To do so, use the tools on the Plots tab. See “Visualize Reduced-Order Models in the Model Reducer App” on page 6-67 for more information. • Export reduced models to the MATLAB workspace for further analysis or control design. On the Model Reducer tab, click Export. Generate MATLAB Code for Pole-Zero Simplification To create a MATLAB script you can use for further model-reduction tasks at the command line, click Create Reduced Model, and select Generate MATLAB Script. 6-52 Pole-Zero Simplification Model Reducer creates a script that uses the minreal command to perform model reduction with the parameters you have set on the Pole-Zero Simplification tab. The script opens in the MATLAB editor. Pole-Zero Cancellation at the Command Line To reduce the order of a model by pole-zero cancellation at the command line, use minreal. Create a model of the following system, where C is a PI controller, and G has a zero at rad/s. Such a low-frequency zero can arise from derivative action somewhere in the plant dynamics. For example, the plant may include a component that computes speed from position measurements. 6-53 6 Model Simplification G = zpk(3e-8,[-1,-3],1); C = pid(1,0.3); T = feedback(G*C,1) T = (s+0.3) (s-3e-08) ---------------------s (s+4.218) (s+0.7824) Continuous-time zero/pole/gain model. In the closed-loop model T, the integrator frequency zero of G. from C very nearly cancels the low- Force a cancellation of the integrator with the zero near the origin. Tred = minreal(T,1e-7) Tred = (s+0.3) -------------------(s+4.218) (s+0.7824) Continuous-time zero/pole/gain model. By default, minreal reduces transfer function order by canceling exact pole-zero pairs or near pole-zero pairs within sqrt(eps). Specifying 1e-7 as the second input causes minreal to eliminate pole-zero pairs within rad/s of each other. The reduced model Tred includes all the dynamics of the original closed-loop model T, except for the near-canceling zero-pole pair. Compare the frequency responses of the original and reduced systems. 6-54 Pole-Zero Simplification bode(T,Tred,'r--') legend('T','Tred') Because the canceled pole and zero do not match exactly, some extreme low-frequency dynamics evident in the original model are missing from Tred. In many applications, you can neglect such extreme low-frequency dynamics. When you increase the matching tolerance of minreal, make sure that you do not eliminate dynamic features that are relevant to your application. 6-55 6 Model Simplification See Also Apps Model Reducer Functions minreal | sminreal Related Examples 6-56 • “Balanced Truncation Model Reduction” on page 6-17 • “Mode-Selection Model Reduction” on page 6-57 • “Model Reduction Basics” on page 6-2 Mode-Selection Model Reduction Mode-Selection Model Reduction Model selection eliminates poles that fall outside a specific frequency range of interest. This method is useful when you want to focus your analysis on a particular subset of system dynamics. For instance, if you are working with a control system with bandwidth limited by actuator dynamics, you might discard higher-frequency dynamics in the plant. Eliminating dynamics outside the frequency range of interest reduces the numerical complexity of calculations with the model. There are two ways to compute a reducedorder model by mode selection: • At the command line, using the freqsep command. • In the Model Reducer, using the Mode Selection method. For more general information about model reduction, see “Model Reduction Basics” on page 6-2. Mode Selection in the Model Reducer App Model Reducer provides an interactive tool for performing model reduction and examining and comparing the responses of the original and reduced-order models. To approximate a model by mode selection in Model Reducer: 1 Open the app and import an LTI model to reduce. For instance, suppose that there is a model named Gms in the MATLAB workspace. The following command opens Model Reducer and imports the model. modelReducer(Gms) 2 In the Data Browser, select the model to reduce. Click Mode Selection. 6-57 6 Model Simplification In the Mode Selection tab, Model Reducer displays a plot of the frequency response of the original model and a reduced version of the model. The app also displays a pole-zero map of both models. 6-58 Mode-Selection Model Reduction The pole-zero map marks pole locations with x and zero locations with o. Note The frequency response is a Bode plot for SISO models, and a singular-value plot for MIMO models. 6-59 6 Model Simplification 3 Model Reducer eliminates poles that lie outside the shaded region. Change the shaded region to capture only the dynamics you want to preserve in the reduced model. There are two ways to do so. • On either the response plot or the pole-zero map, drag the boundaries of the shaded region or the shaded region itself. • On the Mode Selection tab, enter lower and upper cutoff frequencies. When you change the shaded regions or cutoff frequencies, Model Reducer automatically computes a new reduced-order model. All poles retained in the reduced 6-60 Mode-Selection Model Reduction model fall within the shaded region on the pole-zero map. The reduced model might contain zeros that fall outside the shaded region. 4 Optionally, examine absolute or relative error between the original and simplified model. Select the error-plot type using the buttons on the Mode Selection tab. For more information about using the analysis plots, see “Visualize Reduced-Order Models in the Model Reducer App” on page 6-67. 5 When you have one or more reduced models that you want to store and analyze further, click . The new model appears in the Data Browser. 6-61 6 Model Simplification After creating a reduced model in the Data Browser, you can continue adjusting the mode-selection region to create reduced models with different orders for analysis and comparison. You can now perform further analysis with the reduced model. For example: • Examine other responses of the reduced system, such as the step response or Nichols plot. To do so, use the tools on the Plots tab. See “Visualize Reduced-Order Models in the Model Reducer App” on page 6-67 for more information. • Export reduced models to the MATLAB workspace for further analysis or control design. On the Model Reducer tab, click Export. Generate MATLAB Code for Mode Selection To create a MATLAB script you can use for further model-reduction tasks at the command line, click Create Reduced Model, and select Generate MATLAB Script. 6-62 Mode-Selection Model Reduction Model Reducer creates a script that uses the freqsep command to perform model reduction with the parameters you have set on the Mode Selection tab. The script opens in the MATLAB editor. Mode Selection at the Command Line To reduce the order of a model by mode selection at the command line, use freqsep. This command separates a dynamic system model into slow and fast components around a specified frequency. For this example, load the model Gms and examine its frequency response. load modeselect Gms bodeplot(Gms) 6-63 6 Model Simplification Gms has two sets of resonances, one at relatively low frequency and the other at relatively high frequency. Suppose that you want to tune a controller for Gms, but the actuator in your system is limited to a bandwidth of about 3 rad/s, in between the two groups of resonances. To simplify calculation and tuning using Gms, you can use mode selection to eliminate the high-frequency dynamics. [Gms_s,Gms_f] = freqsep(Gms,30); freqsep decomposes Gms into slow and fast components such that Gms = Gms_s + Gms_f. All modes (poles) with natural frequency less than 30 are in Gms_s, and the higher-frequency poles are in Gms_f. bodeplot(Gms,Gms_s,Gms_f) legend('original','slow','fast') 6-64 Mode-Selection Model Reduction The slow component, Gms_s, contains only the lower-frequency resonances and matches the DC gain of the original model. Examine the orders of both models. order(Gms) ans = 18 order(Gms_s) ans = 10 When the high-frequency dynamics are unimportant for your application, you can use the 10th-order Gms_s instead of the original 18th-order model. If neglecting low-frequency dynamics is appropriate for your application, you can use Gms_f. To select modes that fall between a low-frequency and a high-frequency cutoff, use additional calls to freqsep. 6-65 6 Model Simplification See Also Model Reducer | freqsep Related Examples 6-66 • “Balanced Truncation Model Reduction” on page 6-17 • “Pole-Zero Simplification” on page 6-47 • “Model Reduction Basics” on page 6-2 Visualize Reduced-Order Models in the Model Reducer App Visualize Reduced-Order Models in the Model Reducer App The plotting tools in the Model Reducer app let you examine and compare time-domain and frequency-domain responses of the original model and the reduced models you create in the app. Use these tools to help verify that the reduced-order model you choose to work with preserves the system behaviors that are important for your application. For more general information about model reduction, see “Model Reduction Basics” on page 6-2. Error Plots By default, for any model reduction method, Model Reducer shows a frequency-response plot of both the original and reduced models. This plot is a Bode plot for SISO models, and a singular-value plot for MIMO models. To more closely examine the differences between an original model and a reduced model, you can use absolute error or relative error plots. On any model reduction tab, click Absolute error plot or Relative error plot to view these plots. 6-67 6 Model Simplification • Absolute error plot — Shows the singular values of G-Gr, where G is the original model and Gr is the current reduced model. • Relative error plot — Shows the singular values of (G-Gr)/G. This plot is useful when the model has very high or very low gain in the region that is important to your application. In such regions, absolute error can be misleading. For SISO models, the singular-value plot is the magnitude of the frequency response. Response Plots After you click to add one or more reduced models to the Data Browser, compare additional responses of the original and reduced models using the Plots tab. Create New Response Plot In the Data Browser, select one or more models to plot. (Ctrl-click to select multiple models.) Then, on the Plots tab, click the type of plot you want to create. Model Reducer creates the plot. 6-68 Visualize Reduced-Order Models in the Model Reducer App Add Model to Existing Plot In the Data Browser, select the model to add. Then, on the Plots tab, click the icon corresponding to the plot you want to update. Plots you have created appear on the left side of the plot gallery. 6-69 6 Model Simplification Model Reducer updates the plot with the new model. Tip To expand the gallery view, click . Plot Characteristics On any plot in Model Reducer: • To see response information and data values, click a line on the plot. 6-70 Visualize Reduced-Order Models in the Model Reducer App • To view system characteristics, right-click anywhere on the plot, as described in “Frequency-Domain Characteristics on Response Plots” on page 8-10. 6-71 6 Model Simplification Plot Tools Mouse over any plot to access plot tools at the upper right corner of the plot. 6-72 Visualize Reduced-Order Models in the Model Reducer App • • • and — Zoom in and zoom out. Click to activate, and drag the cursor over the region to zoom. The zoom icon turns dark when zoom is active. Right-click while zoom is active to access additional zoom options.Click the icon again to deactivate. — Pan. Click to activate, and drag the cursor across the plot area to pan. The pan icon turns dark when pan is active. Right-click while pan is active to access additional pan options. Click the icon again to deactivate. — Legend. By default, the plot legend is active. To toggle the legend off and on, click this icon. To move the legend, drag it to a new location on the plot. To change the way plots are tiled or sorted, use the options on the View tab. 6-73 6 Model Simplification See Also Model Reducer Related Examples 6-74 • “Balanced Truncation Model Reduction” on page 6-17 • “Mode-Selection Model Reduction” on page 6-57 • “Pole-Zero Simplification” on page 6-47 Linear Analysis 75 7 Time Domain Analysis • “Plotting System Responses” on page 7-2 • “Time-Domain Responses” on page 7-20 • “Time-Domain Response Data and Plots” on page 7-21 • “Time-Domain Characteristics on Response Plots” on page 7-24 • “Numeric Values of Time-Domain System Characteristics” on page 7-29 • “Time-Domain Responses of Discrete-Time Model” on page 7-31 • “Time-Domain Responses of MIMO Model” on page 7-34 • “Time-Domain Responses of Multiple Models” on page 7-36 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 • “Response from Initial Conditions” on page 7-45 • “Import LTI Model Objects into Simulink” on page 7-48 • “Analysis of Systems with Time Delays” on page 7-53 7 Time Domain Analysis Plotting System Responses This example shows how to plot the time and frequency responses of SISO and MIMO linear systems. Time Responses For illustration purposes, create the following third-order transfer function: sys = tf([8 18 32],[1 6 14 24]) sys = 8 s^2 + 18 s + 32 ----------------------s^3 + 6 s^2 + 14 s + 24 Continuous-time transfer function. You can plot the step and impulse responses of this system using the step and impulse commands: subplot(2,1,1) step(sys) subplot(2,1,2) impulse(sys) 7-2 Plotting System Responses You can also simulate the response to an arbitrary signal, for example, a sine wave, using the lsim command. The input signal appears in gray and the system's response in blue. clf t = 0:0.01:4; u = sin(10*t); lsim(sys,u,t) % u,t define the input signal 7-3 7 Time Domain Analysis You can use the plotting commands with continuous or discrete TF, SS, or ZPK models. For state-space models, you can also plot the unforced response from some given initial state, for example: A = [-0.8 3.6 -2.1;-3 -1.2 4.8;3 -4.3 -1.1]; B = [0; -1.1; -0.2]; C = [1.2 0 0.6]; D = -0.6; G = ss(A,B,C,D); x0 = [-1;0;2]; % initial state initial(G,x0) grid 7-4 Plotting System Responses Frequency Responses Frequency-domain analysis is key to understanding stability and performance properties of control systems. Bode plots, Nyquist plots, and Nichols chart are three standard ways to plot and analyze the frequency response of a linear system. You can create these plots using the bode, nichols, and nyquist commands. For example: sys = tf([8 18 32],[1 6 14 24]) sys = 8 s^2 + 18 s + 32 ----------------------- 7-5 7 Time Domain Analysis s^3 + 6 s^2 + 14 s + 24 Continuous-time transfer function. bode(sys) grid nyquist(sys) grid 7-6 Plotting System Responses nichols(sys) grid 7-7 7 Time Domain Analysis Pole/Zero Maps and Root Locus The poles and zeros of a system contain valuable information about its dynamics, stability, and limits of performance. For example, consider the feedback loop in Figure 1 where 7-8 Plotting System Responses Figure 1: SISO Feedback Loop. For the gain value k = 0.7, you can plot the closed-loop poles and zeros using pzmap: s = tf('s'); G = -(2*s+1)/(s^2+3*s+2); k = 0.7; T = feedback(G*k,1); pzmap(T) grid, axis([-2 0 -1 1]) 7-9 7 Time Domain Analysis The closed-loop poles (marked by blue x's) lie in the left half-plane so the feedback loop is stable for this choice of gain k. You can read the damping ratio of the closed-loop poles from this chart (see labels on the radial lines). Here the damping ratio is about 0.7, suggesting a well-damped closed-loop response as confirmed by: clf step(T) 7-10 Plotting System Responses To further understand how the loop gain k affects closed-loop stability, you can plot the locus of the closed-loop poles as a function of k: rlocus(G) grid 7-11 7 Time Domain Analysis Clicking where the locus intersects the y axis reveals that the closed-loop poles become unstable for k = 1.51. So the loop gain should remain smaller than 1.5 for closed-loop stability. 7-12 Plotting System Responses Response Characteristics Right-clicking on response plots gives access to a variety of options and annotations. In particular, the Characteristics menu lets you display standard metrics such as rise time and settling time for step responses, or peak gain and stability margins for frequency response plots. Using the example from the previous section, plot the closed-loop step response: step(T) 7-13 7 Time Domain Analysis Now, right-click on the plot to display the Peak Response and Settling Time Characteristics, and click on the blue dots to read the corresponding overshoot and settling time values: 7-14 Plotting System Responses Analyzing MIMO Systems All commands mentioned so far fully support multi-input multi-output (MIMO) systems. In the MIMO case, these commands produce arrays of plots. For example, the step response of the two-input, two-output system sys = rss(3,2,2); sys.A = [-0.5 -0.3 -0.2 ; 0 -1.3 -1.7; 0.4 1.7 -1.3]; is a 2-by-2 array of plots where each column shows the step response of a particular input channel: step(sys) 7-15 7 Time Domain Analysis If desired, you can group all four responses on a single plot by right-clicking on the plot and selecting the I/O Grouping -> All submenu. The resulting plot is shown below. 7-16 Plotting System Responses The following additional plots are useful for analyzing MIMO systems: • Singular value plot (sigma), which shows the principal gains of the frequency response • Pole/zero map for each I/O pair (iopzplot) For example, plot the peak gain of sys as a function of frequency: sigma(sys) grid 7-17 7 Time Domain Analysis Comparing Systems You can plot multiple systems at once using any of the response plot commands. You can assign a specific color, marker, or line style to each system for easy comparison. Using the feedback example above, plot the closed-loop step response for three values of the loop gain k in three different colors: k1 = 0.4; T1 = feedback(G*k1,1); k2 = 1; T2 = feedback(G*k2,1); step(T,'b',T1,'r',T2,'g') legend('k = 0.7','k = 0.4','k = 1') 7-18 See Also See Also bode | step More About • “Time-Domain Responses” on page 7-20 • “Frequency-Domain Responses” on page 8-2 7-19 7 Time Domain Analysis Time-Domain Responses When you perform time-domain analysis of a dynamic system model, you may want one or more of the following: • A plot of the system response as a function of time. • Numerical values of the system response in a data array. • Numerical values of characteristics of the system response such as peak response or settling time. Control System Toolbox time-domain analysis commands can obtain these results for any kind of dynamic system model (for example, continuous or discrete, SISO or MIMO, or arrays of models) except for frequency response data models. To obtain numerical data, use: • step,impulse,initial,lsim — System response data at a vector of time points. • stepinfo,lsiminfo — Numerical values of system response characteristics such as settling time and overshoot. To obtain response plots, use: • step,impulse,initial,lsim — Plot system response data, visualize response characteristics on plots, compare responses of multiple systems on a single plot. • stepplot,impulseplot,initialplot,lsimplot — Create system response plots with more plot-customization options. For details about plot customization, see “Plot Customization”. • Linear System Analyzer — App for plotting many types of system responses simultaneously, including both time-domain and frequency-domain responses See Also Related Examples 7-20 • “Time-Domain Response Data and Plots” on page 7-21 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 Time-Domain Response Data and Plots Time-Domain Response Data and Plots This example shows how to obtain step and impulse response data, as well as step and impulse response plots, from a dynamic system model. Create a transfer function model and plot its response to a step input at = 0. H = tf([8 18 32],[1 6 14 24]); step(H); When call step without output arguments, it plots the step response on the screen. Unless you specify a time range to plot, step automatically chooses a time range that illustrates the system dynamics. 7-21 7 Time Domain Analysis Calculate the step response data from = 0 (application of the step input) to = 8 s. [y,t] = step(H,8); When you call step with output arguments, the command returns the step response data y. The vector t contains corresponding time values. Plot the response of H to an impulse input applied at = 0. Plot the response with a grid. opts = timeoptions; opts.Grid = 'on'; impulseplot(H,opts) Use the timeoptions command to define options sets for customizing time-domain plots with commands like impulseplot and stepplot. 7-22 See Also Calculate 200 points of impulse response data from = 1 (one second after application of the impulse input) to = 3s. [y,t] = impulse(H,linspace(1,3,200)); As for step, you can omit the time vector to allow impulse to automatically select a time range. See Also impulse | impulseplot | step | stepplot | timeoptions Related Examples • “Time-Domain Characteristics on Response Plots” on page 7-24 • “Time-Domain Responses of Multiple Models” on page 7-36 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 More About • “Time-Domain Responses” on page 7-20 7-23 7 Time Domain Analysis Time-Domain Characteristics on Response Plots This example shows how to display system characteristics such as settling time and overshoot on step response plots. You can use similar procedures to display system characteristics on impulse response plots or initial value response plots, such as peak response or settling time. Create a transfer function model and plot its response to a step input at t = 0. H = tf([8 18 32],[1 6 14 24]); stepplot(H) Display the peak response on the plot. Right-click anywhere in the figure and select Characteristics > Peak Response from the menu. 7-24 Time-Domain Characteristics on Response Plots A marker appears on the plot indicating the peak response. Horizontal and vertical dotted lines indicate the time and amplitude of that response. 7-25 7 Time Domain Analysis Click the marker to view the value of the peak response and the overshoot in a datatip. 7-26 See Also You can use a similar procedure to select other characteristics such as settling time and rise time from the Characteristics menu and view the values. See Also impulse | lsiminfo | step | stepinfo Related Examples • “Numeric Values of Time-Domain System Characteristics” on page 7-29 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 7-27 7 Time Domain Analysis More About • 7-28 “Time-Domain Responses” on page 7-20 Numeric Values of Time-Domain System Characteristics Numeric Values of Time-Domain System Characteristics This example shows how to obtain numeric values of step response characteristics such as rise time, settling time, and overshoot using stepinfo. You can use similar techniques with lsiminfo to obtain characteristics of the system response to an arbitrary input or initial conditions. Create a dynamic system model and get numeric values of the system’s step response characteristics. H = tf([8 18 32],[1 6 14 24]); data = stepinfo(H) data = struct with fields: RiseTime: 0.2087 SettlingTime: 3.4972 SettlingMin: 1.1956 SettlingMax: 1.6871 Overshoot: 26.5302 Undershoot: 0 Peak: 1.6871 PeakTime: 0.5987 The output is a structure that contains values for several step response characteristics. To access these values or refer to them in other calculations, use dot notation. For example, data.Overshoot is the overshoot value. Calculate the time it takes the step response of H to settle within 0.5% of its final value. data = stepinfo(H,'SettlingTimeThreshold',0.005); t05 = data.SettlingTime t05 = 4.8896 By default, stepinfo defines the settling time as the time it takes for the output to settle within 0.02 (2%) of its final value. Specifying a more stringent 'SettlingTimeThreshold' of 0.005 results in a longer settling time. For more information about the options and the characteristics, see the stepinfo reference page. 7-29 7 Time Domain Analysis See Also lsiminfo | stepinfo Related Examples • “Time-Domain Characteristics on Response Plots” on page 7-24 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 More About • 7-30 “Time-Domain Responses” on page 7-20 Time-Domain Responses of Discrete-Time Model Time-Domain Responses of Discrete-Time Model This example shows how to obtain a step-response plot and step-response data for a discrete-time dynamic system model. Obtaining time-domain responses of discrete-time models is the same as for continuous-time models, except that the time sample points are limited by the sample time Ts of the model. You can use the techniques of this example with commands such as impulse, initial, impulseplot, and initialpot to obtain time-domain responses of discrete-time models. Create a discrete-time transfer function model and plot its response to a step input at 0. = H = tf([-0.06,0.4],[1,-1.6,0.78],0.1); step(H) 7-31 7 Time Domain Analysis For discrete-time models, step plots the response at multiples of the sample time, assuming a hold between samples. Compute the step response of H between 0.5 and 2.5 seconds. [y,t] = step(H,0.5:0.1:2.5); When you specify a time vector for the response of a discrete-time model, the time step must match the sample time Ts of the discrete-time model. The vector t contains the time points between 0.5 and 2.5 seconds, at multiples of the sample time of H, 0.1 s. The vector y contains the corresponding step response values. 7-32 See Also See Also impulse | impulseplot | initial | initialplot | step | stepplot Related Examples • “Time-Domain Responses of MIMO Model” on page 7-34 • “Time-Domain Responses of Multiple Models” on page 7-36 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 More About • “Time-Domain Responses” on page 7-20 7-33 7 Time Domain Analysis Time-Domain Responses of MIMO Model This example shows how to obtain impulse response data and plots for a multi-input, multi-output (MIMO) model using impulse. You can use the same techniques to obtain other types of time-domain responses of MIMO models. Create a MIMO model and plot its response to a t = 0 impulse at all inputs. H = rss(2,2,2); H.InputName = 'Control'; H.OutputName = 'Temperature'; impulse(H) 7-34 See Also impulse plots the response of each output to an impulse applied at each input. (Because rss generates a random state-space model, you might see different responses from those pictured.) The first column of plots shows the response of each output to an impulse applied at the first input, Control(1). The second column shows the response of each output to an impulse applied at the second input, Control(2). Calculate the impulse responses of all channels of H, and examine the size of the output. [y,t] = impulse(H); size(y) ans = 1×3 207 2 2 The first dimension of the data array y is the number of samples in the time vector t. The impulse command determines this number automatically if you do not supply a time vector. The remaining dimensions of y are the numbers of outputs and inputs in H. Thus, y(:,i,j) is the response at the i th output of H to an impulse applied at the j th input. See Also impulse | impulseplot | initial | initialplot | step | stepplot Related Examples • “Time-Domain Responses of Multiple Models” on page 7-36 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 More About • “Time-Domain Responses” on page 7-20 7-35 7 Time Domain Analysis Time-Domain Responses of Multiple Models This example shows how to compare the step responses of multiple models on a single plot using step. This example compares the step response of an uncontrolled plant to the closed-loop step response of the plant with two different PI controllers. You can use similar techniques with other response commands, such as impulse or initial, to obtain plots of responses of multiple models. For this example, obtain two models whose time responses you want to compare, and plot them on a single step plot. For instance, you can compare a third-order plant G, and the closed-loop response of G with a controller C1 having integral action. G = zpk([],[-5 -5 -10],100); C1 = pid(0,4.4); CL1 = feedback(G*C1,1); step(G,CL1); 7-36 Time-Domain Responses of Multiple Models When you provide multiple models to step as input arguments, the command displays the responses of both models on the same plot. If you do not specify a time range to plot, step attempts to choose a time range that illustrates the dynamics of all the models. Compare the step response of the closed-loop model with another controller. Specify plot colors and styles for each response. C2 = pid(2.9,7.1); CL2 = feedback(G*C2,1); step(G,'b--',CL1,'g-',CL2,'r-') 7-37 7 Time Domain Analysis You can specify custom plot color and style for each response in the plot. For example, 'g-' specifies a solid green line for response CL2. For additional plot customization options, use stepplot. See Also Linear System Analyzer | impulse | impulseplot | initial | initialplot | step | stepplot 7-38 See Also Related Examples • “Time-Domain Responses of MIMO Model” on page 7-34 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 More About • “Time-Domain Responses” on page 7-20 7-39 7 Time Domain Analysis Joint Time-Domain and Frequency-Domain Analysis This example shows how to compare multiple types of responses side by side, including both time-domain and frequency-domain responses, using the interactive Linear System Analyzer app. Obtain models whose responses you want to compare. For example, compare a third-order plant G, and the closed-loop responses of G with two different controllers, C1 and C2. G = zpk([],[-5 -5 -10],100); C1 = pid(0,4.4); T1 = feedback(G*C1,1); C2 = pid(2.9,7.1); T2 = feedback(G*C2,1); Open the Linear System Analyzer tool to examine the responses of the plant and the closed-loop systems. linearSystemAnalyzer(G,T1,T2) 7-40 Joint Time-Domain and Frequency-Domain Analysis By default, the Linear System Analyzer launches with a plot of the step response of the three systems. Click to add a legend to the plot. Add plots of the impulse responses to the Linear System Analyzer display. In the Linear System Analyzer, select Edit > Plot Configurations to open the Plot Configurations dialog box. 7-41 7 Time Domain Analysis Select the two-plot configuration. In the Response Type area, select Bode Magnitude for the second plot type. Click OK to add the Bode plots to the Linear System Analyzer display. Display the peak values of the Bode responses on the plot. Right-click anywhere in the Bode Magnitude plot and select Characteristics > Peak Response from the menu. 7-42 Joint Time-Domain and Frequency-Domain Analysis Markers appear on the plot indicating the peak response values. Horizontal and vertical dotted lines indicate the frequency and amplitude of those responses. Click on a marker to view the value of the peak response in a datatip. You can use a similar procedure to select other characteristics such as settling time and rise time from the Characteristics menu and view the values. You can also change the type of plot displayed in the Linear System Analyzer. For example, to change the first plot type to a plot of the impulse response, right-click anywhere in the plot. Select Plot Types > Impulse 7-43 7 Time Domain Analysis The displayed plot changes to show the impulse of the three systems. See Also Linear System Analyzer | impulse | impulseplot | initial | initialplot | step | stepplot Related Examples • “Time-Domain Responses of Multiple Models” on page 7-36 More About • 7-44 “Time-Domain Responses” on page 7-20 Response from Initial Conditions Response from Initial Conditions This example shows how to compute and plot the response of a state-space (ss) model to specified initial state values using initial. Load a state-space model. load ltiexamples sys_dc sys_dc.InputName = 'Volts'; sys_dc.OutputName = 'w'; sys_dc.StateName = {'Current','w'}; sys_dc sys_dc = A = Current w Current -4 0.75 w -0.03 -10 B = Current w Volts 2 0 C = w Current 0 w 1 D = w Volts 0 Continuous-time state-space model. This example uses the SISO, 2-state model sys_dc. This model represents a DC motor. The input is an applied voltage, and the output is the angular rate of the motor ω. The states of the model are the induced current (x1), and ω (x2). The model display in the command window shows the labeled input, output, and states. Plot the undriven evolution of the motor's angular rate from an initial state in which the induced current is 1.0 amp and the initial rotation rate is 5.0 rad/s. 7-45 7 Time Domain Analysis x0 = [1.0 5.0]; initial(sys_dc,x0) initial plots the time evolution from the specified initial condition on the screen. Unless you specify a time range to plot, initial automatically chooses a time range that illustrates the system dynamics. Calculate the time evolution of the output and the states of sys_dc from (application of the step input) to = 1 s. t = 0:0.01:1; [y,t,x] = initial(sys_dc,x0,t); 7-46 =0 See Also The vector y contains the output at each time step in t. The array x contains the state values at each time step. Therefore, in this example x is a 2-by-101 array. Each row of x contains the values of the two states of sys_dc at the corresponding time step. See Also impulse | initial | initialplot | step Related Examples • “Time-Domain Response Data and Plots” on page 7-21 • “Numeric Values of Time-Domain System Characteristics” on page 7-29 More About • “Time-Domain Responses” on page 7-20 7-47 7 Time Domain Analysis Import LTI Model Objects into Simulink Use the LTI System block to import linear system model objects into Simulink. You can simulate linear systems represented as LTI model objects, and incorporate such systems as elements of Simulink models of more complex systems. In the block parameters, set the LTI system variable parameter to the LTI model to import. For state-space models, set the Initial states parameter to a vector to specify non-zero initial states. Simulate LTI Model in Simulink The LTISystemBlockSimulation model shows how to use an LTI System block to simulate the response of a SISO transfer function to a step input. To specify a model for the LTI System block, set the LTI system variable block parameter to either: • The variable name of an LTI model in the MATLAB® workspace or model workspace, such as sys. • A MATLAB expression that evaluates to an LTI model, such as tf(1,[1 1]). For example, you can specify a state-space (ss), zero-pole-gain (zpk), or transfer function (tf) model. You can simulate SISO models or MIMO models, and continuous-time or discrete-time models. In LTISystemBlockSimulation model, the LTI system variable parameter is a MATLAB expression, tf(1,[1 2 5]), which creates a continuous-time SISO transfer function. If the specified system is a state-space (ss) model, then you can specify initial state values by setting the Initial states parameter. Simulate the model, and examine the result in the scope. 7-48 Import LTI Model Objects into Simulink 7-49 7 Time Domain Analysis This example simulates the system response to a step input at t = 2 s. Use the LTI System block to import an LTI model object anywhere in your Simulink model to simulate the linear system response to any input. Import MIMO LTI Model into Simulink This model shows how to use an LTI System block to represent a MIMO linear system in Simulink®. The LTI System block has one input and one output, even when you specify a MIMO model for the block. In that case, the block input and output become vector signals. For instance, the model LTISystemBlockMIMO uses an LTI system block to represent a MIMO plant in a control system. In this model, the LTI System specified in the block is Gm, a 2-output, 2-input transfer function model stored in the model workspace. A Mux block combines the two controller outputs into a vector signal for the LTI System block input. Similarly, a Demux block separates the vector output of the LTI System block into two scalar signals. Simulate the model, and examine the result in the scope. 7-50 Import LTI Model Objects into Simulink 7-51 7 Time Domain Analysis This example simulates a closed-loop system response to a t = 50 s step at the first input and a t = 150 s step at the second input. You can use the LTI system block anywhere you want to insert an LTI system into a Simulink model. See Also LTI System 7-52 Analysis of Systems with Time Delays Analysis of Systems with Time Delays You can use analysis commands such as step, bode, or margin to analyze systems with time delays. The software makes no approximations when performing such analysis. For example, consider the following control loop, where the plant is modeled as first-order plus dead time: You can model the closed-loop system from r to y with the following commands: s P C T = = = = tf('s'); 5*exp(-3.4*s)/(s+1); 0.1 * (1 + 1/(5*s)); feedback(P*C,1); T is a state-space model with an internal delay. For more information about models with internal delays, see “Closing Feedback Loops with Time Delays” on page 2-45. Plot the step response of T: stepplot(T) 7-53 7 Time Domain Analysis For more complicated interconnections, you can name the input and output signals of each block and use connect to automatically take care of the wiring. Suppose, for example, that you want to add feedforward to the control loop of the previous model. 7-54 Analysis of Systems with Time Delays You can derive the corresponding closed-loop model Tff by F = 0.3/(s+4); P.InputName = 'u'; P.OutputName = 'y'; C.InputName = 'e'; C.OutputName = 'uc'; F.InputName = 'r'; F.OutputName = 'uf'; Sum1 = sumblk('e','r','y','+-'); % e = r-y Sum2 = sumblk('u','uf','uc','++'); % u = uf+uc Tff = connect(P,C,F,Sum1,Sum2,'r','y'); and compare its response with the feedback only design. stepplot(T,Tff) legend('No feedforward','Feedforward') 7-55 7 Time Domain Analysis The state-space representation keeps track of the internal delays in both models. Considerations to Keep in Mind when Analyzing Systems with Internal Time Delays The time and frequency responses of delay systems can look odd and suspicious to those only familiar with delay-free LTI analysis. Time responses can behave chaotically, Bode plots can exhibit gain oscillations, etc. These are not software or numerical quirks but real features of such systems. Below are a few illustrations of these phenomena. Gain ripple: 7-56 Analysis of Systems with Time Delays s = tf('s'); G = exp(-5*s)/(s+1); T = feedback(G,.5); bodemag(T) Gain oscillations: G = 1 + 0.5 * exp(-3*s); bodemag(G) 7-57 7 Time Domain Analysis Jagged step response: G = exp(-s) * (0.8*s^2+s+2)/(s^2+s); T = feedback(G,1); stepplot(T) 7-58 Analysis of Systems with Time Delays Note the rearrivals (echoes) of the initial step function. Chaotic response: G = 1/(s+1) + exp(-4*s); T = feedback(1,G); stepplot(T,150) 7-59 7 Time Domain Analysis You can use Control System Toolbox tools to model and analyze these and other strangeappearing artifacts of internal delays. See Also Related Examples • 7-60 “Closing Feedback Loops with Time Delays” on page 2-45 See Also More About • “Time Delays in Linear Systems” on page 2-40 • “Internal Delays” on page 2-73 7-61 8 Frequency Domain Analysis • “Frequency-Domain Responses” on page 8-2 • “Frequency Response of a SISO System” on page 8-4 • “Frequency Response of a MIMO System” on page 8-6 • “Frequency-Domain Characteristics on Response Plots” on page 8-10 • “Numeric Values of Frequency-Domain Characteristics of SISO Model” on page 8-13 • “Pole and Zero Locations” on page 8-16 • “Assessing Gain and Phase Margins” on page 8-19 • “Analyzing Control Systems with Delays” on page 8-32 • “Analyzing the Response of an RLC Circuit” on page 8-50 8 Frequency Domain Analysis Frequency-Domain Responses When you perform frequency-domain analysis of a dynamic system model, you may want one or more of the following: • A plot of the system response as a function of frequency, or plots of pole and zero locations. • Numerical values of the system response in a data array. • Numerical values of characteristics of the system response such as stability margins, peak gains, or singular values. Control System Toolbox frequency-domain analysis commands can obtain these results for any kind of dynamic system model (for example, continuous or discrete, SISO or MIMO, or arrays of models). To obtain numerical data, use: • bode,freqresp,nichols,nyquist — System response data at a vector of frequency points. • margin,getPeakGain,getGainCrossover,sigma — Numerical values of system response characteristics such as gain margins, phase margins, and singular values. To obtain response plots, use: • bode,bodemag,nichols,nyquist — Plot system response data, visualize response characteristics on plots, compare responses of multiple systems on a single plot. • bodeplot,nicholsplot,nyquistplot,sigmaplot — Create system response plots with more plot-customization options. For details about plot customization, see “Plot Customization”. • Linear System Analyzer — App for plotting many types of system responses simultaneously, including both time-domain and frequency-domain responses To obtain pole-zero maps, use: • pzplot, iopzplot — Plot pole and zero locations in the complex plane. If you have a generalized state-space (genss) model of a control system, you can extract various transfer functions from it for analysis using frequency-domain and time-domain analysis commands. Extract responses from such models using getIOTransfer, getLoopTransfer, getSensitivity, and getCompSensitivity. 8-2 See Also See Also Related Examples • “Frequency Response of a SISO System” on page 8-4 • “Frequency Response of a MIMO System” on page 8-6 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 8-3 8 Frequency Domain Analysis Frequency Response of a SISO System This example shows how to plot the frequency response and obtain frequency response data for a single-input, single-output (SISO) dynamic system model. Create a transfer function model and plot its frequency response. H = tf([10,21],[1,1.4,26]); bode(H) When you can bode without output arguments, it plots the frequency response on the screen. Unless you specify a frequency range to plot, bode automatically chooses a frequency range based on the system dynamics. 8-4 See Also Calculate the frequency response between 1 and 13 rad/s. [mag,phase,w] = bode(H,{1,13}); When you call bode with output arguments, the command returns vectors mag and phase containing the magnitude and phase of the frequency response. The cell array input {1,13} tells bode to calculate the response at a grid of frequencies between 1 and 13 rad/s. bode returns the frequency points in the vector w. See Also bode | bodeoptions | bodeplot Related Examples • “Frequency Response of a MIMO System” on page 8-6 • “Numeric Values of Frequency-Domain Characteristics of SISO Model” on page 813 More About • “Frequency-Domain Responses” on page 8-2 8-5 8 Frequency Domain Analysis Frequency Response of a MIMO System This example shows how to examine the frequency response of a multi-input, multi-output (MIMO) system in two ways: by computing the frequency response, and by computing the singular values. Calculate the frequency response of a MIMO model and examine the size of the output. H = rss(2,2,2); H.InputName = 'Control'; H.OutputName = 'Temperature'; [mag,phase,w] = bode(H); size(mag) ans = 1×3 2 2 70 The first and second dimension of the data array mag are the number of outputs and inputs of H. The third dimension is the number of points in the frequency vector w. (The bode command determines this number automatically if you do not supply a frequency vector.) Thus, mag(i,j,:) is the frequency response from the j th input of H to the i th output, in absolute units. The phase data array phase takes the same form as mag. Plot the frequency response of each input/output pair in H. bode(H) 8-6 Frequency Response of a MIMO System bode plots the magnitude and the phase of the frequency response of each input/output pair in H. (Because rss generates a random state-space model, you might see different responses from those pictured.) The first column of plots shows the response from the first input, Control(1), to each output. The second column shows the response from the second input, Control(2), to each output. Plot the singular values of H as a function of frequency. sigma(H) 8-7 8 Frequency Domain Analysis sigma plots the singular values of the MIMO system H as a function of frequency. The maximum singular value at a particular frequency is the maximum gain of the system over all linear combinations of inputs at that frequency. Singular values can provide a better indication of the overall response, stability, and conditioning of a MIMO system than a channel-by-channel Bode plot. Calculate the singular values of H between 0.1 and 10 rad/s. [sv,w] = sigma(H,{0.1,10}); When you call sigma with output arguments, the command returns the singular values in the data array sv. The cell array input {0.1,10} tells sigma to calculate the singular values at a grid of frequencies between 0.1 and 10 rad/s. sigma returns these frequencies in the vector w. Each row of sv contains the singular values of H at the frequencies of w. 8-8 See Also See Also bode | bodeplot | sigma | sigmaplot Related Examples • “Numeric Values of Frequency-Domain Characteristics of SISO Model” on page 813 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 8-9 8 Frequency Domain Analysis Frequency-Domain Characteristics on Response Plots This example shows how to display system characteristics such as peak response on Bode response plots. You can use similar procedures to display system characteristics on other types of response plots. Create a transfer function model and plot its frequency response. H = tf([10,21],[1,1.4,26]); bodeplot(H) Display the peak response on the plot. Right-click anywhere in the figure and select Characteristics > Peak Response from the menu. 8-10 Frequency-Domain Characteristics on Response Plots A marker appears on the plot indicating the peak response. Horizontal and vertical dotted lines indicate the frequency and magnitude of that response. The other menu options add other system characteristics to the plot. 8-11 8 Frequency Domain Analysis Click the marker to view the magnitude and frequency of the peak response in a datatip. See Also Related Examples 8-12 • “Numeric Values of Frequency-Domain Characteristics of SISO Model” on page 813 • “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 • “Pole and Zero Locations” on page 8-16 Numeric Values of Frequency-Domain Characteristics of SISO Model Numeric Values of Frequency-Domain Characteristics of SISO Model This example shows how to obtain numeric values of several frequency-domain characteristics of a SISO dynamic system model, including the peak gain, dc gain, system bandwidth, and the frequencies at which the system gain crosses a specified frequency. Create a transfer function model and plot its frequency response. H = tf([10,21],[1,1.4,26]); bodeplot(H) 8-13 8 Frequency Domain Analysis Plotting the frequency response gives a rough idea of the frequency-domain characteristics of the system. H includes a pronounced resonant peak, and rolls off at 20 dB/decade at high frequency. It is often desirable to obtain specific numeric values for such characteristics. Calculate the peak gain and the frequency of the resonance. [gpeak,fpeak] = getPeakGain(H); gpeak_dB = mag2db(gpeak) gpeak_dB = 17.7579 getPeakGain returns both the peak location fpeak and the peak gain gpeak in absolute units. Using mag2db to convert gpeak to decibels shows that the gain peaks at almost 18 dB. Find the band within which the system gain exceeds 0 dB, or 1 in absolute units. wc = getGainCrossover(H,1) wc = 2×1 1.2582 12.1843 getGainCrossover returns a vector of frequencies at which the system response crosses the specified gain. The resulting wc vector shows that the system gain exceeds 0 dB between about 1.3 and 12.2 rad/s. Find the dc gain of H. The Bode response plot shows that the gain of H tends toward a finite value as the frequency approaches zero. The dcgain command finds this value in absolute units. k = dcgain(H); Find the frequency at which the response of H rolls off to –10 dB relative to its dc value. fb = bandwidth(H,-10); bandwidth returns the first frequency at which the system response drops below the dc gain by the specified value in dB. 8-14 See Also See Also bandwidth | getGainCrossover | getPeakGain Related Examples • “Pole and Zero Locations” on page 8-16 More About • “Frequency-Domain Responses” on page 8-2 8-15 8 Frequency Domain Analysis Pole and Zero Locations This example shows how to examine the pole and zero locations of dynamic systems both graphically using pzplot and numerically using pole and zero. Examining the pole and zero locations can be useful for tasks such as stability analysis or identifying near-canceling pole-zero pairs for model simplification. This example compares two closed-loop systems that have the same plant and different controllers. Create dynamic system models representing the two closed-loop systems. G = zpk([],[-5 -5 -10],100); C1 = pid(2.9,7.1); CL1 = feedback(G*C1,1); C2 = pid(29,7.1); CL2 = feedback(G*C2,1); The controller C2 has a much higher proportional gain. Otherwise, the two closed-loop systems CL1 and CL2 are the same. Graphically examine the pole and zero locations of CL1 and CL2. pzplot(CL1,CL2) grid 8-16 Pole and Zero Locations pzplot plots pole and zero locations on the complex plane as x and o marks, respectively. When you provide multiple models, pzplot plots the poles and zeros of each model in a different color. Here, there poles and zeros of CL1 are blue, and those of CL2 are green. The plot shows that all poles of CL1 are in the left half-plane, and therefore CL1 is stable. From the radial grid markings on the plot, you can read that the damping of the oscillating (complex) poles is approximately 0.45. The plot also shows that CL2 contains poles in the right half-plane and is therefore unstable. Compute numerical values of the pole and zero locations of CL2. z = zero(CL2); p = pole(CL2); 8-17 8 Frequency Domain Analysis zero and pole return column vectors containing the zero and pole locations of the system. See Also pole | pzplot | zero Related Examples • “Numeric Values of Frequency-Domain Characteristics of SISO Model” on page 8-13 More About • 8-18 “Frequency-Domain Responses” on page 8-2 Assessing Gain and Phase Margins Assessing Gain and Phase Margins This example shows how to examine the effect of stability margins on closed-loop response characteristics of a control system. Stability of a Feedback Loop Stability generally means that all internal signals remain bounded. This is a standard requirement for control systems to avoid loss of control and damage to equipment. For linear feedback systems, stability can be assessed by looking at the poles of the closedloop transfer function. Consider for example the SISO feedback loop: Figure 1: SISO Feedback Loop. For a unit loop gain k, you can compute the closed-loop transfer function T using: G = tf([.5 1.3],[1 1.2 T = feedback(G,1); 1.6 0]); To obtain the poles of T, type pole(T) ans = -0.2305 + 1.3062i -0.2305 - 1.3062i -0.7389 + 0.0000i The feedback loop for k=1 is stable since all poles have negative real parts. 8-19 8 Frequency Domain Analysis How Stable is Stable? Checking the closed-loop poles gives us a binary assessment of stability. In practice, it is more useful to know how robust (or fragile) stability is. One indication of robustness is how much the loop gain can change before stability is lost. You can use the root locus plot to estimate the range of k values for which the loop is stable: rlocus(G) Clicking on the point where the locus intersects the y axis reveals that this feedback loop is stable for 8-20 Assessing Gain and Phase Margins This range shows that with k=1, the loop gain can increase 270% before you lose stability. Gain and Phase Margins Changes in the loop gain are only one aspect of robust stability. In general, imperfect plant modeling means that both gain and phase are not known exactly. Because modeling errors are most damaging near the gain crossover frequency (frequency where open-loop gain is 0dB), it also matters how much phase variation can be tolerated at this frequency. The phase margin measures how much phase variation is needed at the gain crossover frequency to lose stability. Similarly, the gain margin measures what relative gain variation is needed at the gain crossover frequency to lose stability. Together, these two numbers give an estimate of the "safety margin" for closed-loop stability. The smaller the stability margins, the more fragile stability is. You can display the gain and phase margins on a Bode plot as follows. First create the plot: bode(G), grid 8-21 8 Frequency Domain Analysis Then, right-click on the plot and select the Characteristics -> Minimum Stability Margins submenu. Finally, click on the blue dot markers. The resulting plot is shown below: 8-22 Assessing Gain and Phase Margins This indicates a gain margin of about 9 dB and a phase margin of about 45 degrees. The corresponding closed-loop step response exhibits about 20% overshoot and some oscillations. step(T), title('Closed-loop response for k=1') 8-23 8 Frequency Domain Analysis If we increase the gain to k=2, the stability margins are reduced to [Gm,Pm] = margin(2*G); GmdB = 20*log10(Gm) % gain margin in dB Pm % phase margin in degrees GmdB = 2.7471 Pm = 8-24 Assessing Gain and Phase Margins 8.6328 and the closed-loop response has poorly damped oscillations, a sign of near instability. step(feedback(2*G,1)), title('Closed-loop response for k=2') Systems with Multiple Gain or Phase Crossings Some systems have multiple gain crossover or phase crossover frequencies, which leads to multiple gain or phase margin values. For example, consider the feedback loop 8-25 8 Frequency Domain Analysis Figure 2: Feedback Loop with Multiple Phase Crossovers The closed-loop response for k=1 is stable: G = tf(20,[1 7]) * tf([1 3.2 7.2],[1 -1.2 0.8]) * tf([1 -8 400],[1 33 700]); T = feedback(G,1); step(T), title('Closed-loop response for k=1') 8-26 Assessing Gain and Phase Margins To assess how robustly stable this loop is, plot its Bode response: bode(G), grid 8-27 8 Frequency Domain Analysis Then, right-click on the plot and select the Characteristics -> All Stability Margins submenu to show all the crossover frequencies and associated stability margins. The resulting plot is shown below. 8-28 Assessing Gain and Phase Margins Note that there are two 180 deg phase crossings with corresponding gain margins of -9.35dB and +10.6dB. Negative gain margins indicate that stability is lost by decreasing the gain, while positive gain margins indicate that stability is lost by increasing the gain. This is confirmed by plotting the closed-loop step response for a plus/minus 6dB gain variation about k=1: k1 = 2; T1 = feedback(G*k1,1); k2 = 1/2; T2 = feedback(G*k2,1); step(T,'b',T1,'r',T2,'g',12), legend('k = 1','k = 2','k = 0.5') 8-29 8 Frequency Domain Analysis The plot shows increased oscillations for both smaller and larger gain values. You can use the command allmargin to compute all stability margins. Note that gain margins are expressed as gain ratios, not dB. Use mag2db to convert the values to dB. m = allmargin(G) GainMargins_dB = mag2db(m.GainMargin) m = struct with fields: 8-30 See Also GainMargin: GMFrequency: PhaseMargin: PMFrequency: DelayMargin: DMFrequency: Stable: [0.3408 3.3920] [1.9421 16.4807] 68.1178 7.0762 0.1680 7.0762 1 GainMargins_dB = -9.3510 10.6091 See Also margin | pole Related Examples • “Pole and Zero Locations” on page 8-16 8-31 8 Frequency Domain Analysis Analyzing Control Systems with Delays This example shows how to use Control System Toolbox™ to analyze and design control systems with delays. Control of Processes with Delays Many processes involve dead times, also referred to as transport delays or time lags. Controlling such processes is challenging because delays cause linear phase shifts that limit the control bandwidth and affect closed-loop stability. Using the state-space representation, you can create accurate open- or closed-loop models of control systems with delays and analyze their stability and performance without approximation. The state-space (SS) object automatically keeps track of "internal" delays when combining models, see the "Specifying Time Delays" tutorial for more details. Example: PI Control Loop with Dead Time Consider the standard setpoint tracking loop: where the process model P has a 2.6 second dead time and the compensator C is a PI controller: You can specify these two transfer functions as s = tf('s'); P = exp(-2.6*s)*(s+3)/(s^2+0.3*s+1); C = 0.06 * (1 + 1/s); To analyze the closed-loop response, construct a model T of the closed-loop transfer from ysp to y. Because there is a delay in this feedback loop, you must convert P and C to state space and use the state-space representation for analysis: 8-32 Analyzing Control Systems with Delays T = feedback(P*C,1) T = A = x1 x2 x3 x1 -0.36 1 0 x2 -1.24 0 1 x3 -0.18 0 0 B = x1 x2 x3 u1 0.5 0 0 C = y1 x1 0.12 x2 0.48 x3 0.36 D = y1 u1 0 (values computed with all internal delays set to zero) Internal delays (seconds): 2.6 Continuous-time state-space model. The result is a third-order model with an internal delay of 2.6 seconds. Internally, the state-space object T tracks how the delay is coupled with the remaining dynamics. This structural information is not visible to users, and the display above only gives the A,B,C,D values when the delay is set to zero. Use the STEP command to plot the closed-loop step response from ysp to y: step(T) 8-33 8 Frequency Domain Analysis The closed-loop oscillations are due to a weak gain margin as seen from the open-loop response P*C: margin(P*C) 8-34 Analyzing Control Systems with Delays There is also a resonance in the closed-loop frequency response: bode(T) grid, title('Closed-loop frequency response') 8-35 8 Frequency Domain Analysis To improve the design, you can try to notch out the resonance near 1 rad/s: notch = tf([1 0.2 1],[1 .8 1]); C = 0.05 * (1 + 1/s); Tnotch = feedback(P*C*notch,1); step(Tnotch), grid 8-36 Analyzing Control Systems with Delays Pade Approximation of Time Delays Many control design algorithms cannot handle time delays directly. A common workaround consists of replacing delays by their Pade approximations (all-pass filters). Because this approximation is only valid at low frequencies, it is important to compare the true and approximate responses to choose the right approximation order and check the approximation validity. Use the PADE command to compute Pade approximations of LTI models with delays. For the PI control example above, you can compare the exact closed-loop response T with the response obtained for a first-order Pade approximation of the delay: 8-37 8 Frequency Domain Analysis T1 = pade(T,1); step(T,'b',T1,'r',100) grid, legend('Exact','First-Order Pade') The approximation error is fairly large. To get a better approximation, try a second-order Pade approximation of the delay: T2 = pade(T,2); step(T,'b',T2,'r',100) grid, legend('Exact','Second-Order Pade') 8-38 Analyzing Control Systems with Delays The responses now match closely except for the non-minimum phase artifact introduced by the Pade approximation. Sensitivity Analysis Delays are rarely known accurately, so it is often important to understand how sensitive a control system is to the delay value. Such sensitivity analysis is easily performed using LTI arrays and the InternalDelay property. For example, to analyze the sensitivity of the notched PI control above, create 5 models with delay values ranging from 2.0 to 3.0: tau = linspace(2,3,5); Tsens = repsys(Tnotch,[1 1 5]); % 5 delay values % 5 copies of Tnotch 8-39 8 Frequency Domain Analysis for j=1:5 Tsens(:,:,j).InternalDelay = tau(j); end % jth delay value -> jth model Then use STEP to create an envelope plot: step(Tsens) grid, title('Closed-loop response for 5 delay values between 2.0 and 3.0') This plot shows that uncertainty on the delay value has little effect on closed-loop characteristics. Note that while you can change the values of internal delays, you cannot change how many there are because this is part of the model structure. To eliminate some internal delays, set their value to zero or use PADE with order zero: 8-40 Analyzing Control Systems with Delays Tnotch0 = Tnotch; Tnotch0.InternalDelay = 0; bode(Tnotch,'b',Tnotch0,'r',{1e-2,3}) grid, legend('Delay = 2.6','No delay','Location','SouthWest') Discretization You can use C2D to discretize continuous-time delay systems. Available methods include zero-order hold (ZOH), first-order hold (FOH), and Tustin. For models with internal delays, the ZOH discretization is not always "exact," i.e., the continuous and discretized step responses may not match: 8-41 8 Frequency Domain Analysis Td = c2d(T,1); step(T,'b',Td,'r') grid, legend('Continuous','ZOH Discretization') Warning: Discretization is only approximate due to internal delays. Use faster sampling rate if discretization error is large. To correct such discretization gaps, reduce the sampling period until the continuous and discrete responses match closely: Td = c2d(T,0.05); step(T,'b',Td,'r') grid, legend('Continuous','ZOH Discretization') 8-42 Analyzing Control Systems with Delays Warning: Discretization is only approximate due to internal delays. Use faster sampling rate if discretization error is large. Note that internal delays remain internal in the discretized model and do not inflate the model order: order(Td) Td.InternalDelay ans = 3 8-43 8 Frequency Domain Analysis ans = 52 Some Unique Features of Delay Systems The time and frequency responses of delay systems can look bizarre and suspicious to those only familiar with delay-free LTI analysis. Time responses can behave chaotically, Bode plots can exhibit gain oscillations, etc. These are not software quirks but real features of such systems. Below are a few illustrations of these phenomena Gain ripples: G = exp(-5*s)/(s+1); T = feedback(G,.5); bodemag(T) 8-44 Analyzing Control Systems with Delays Gain oscillations: G = 1 + 0.5 * exp(-3*s); bodemag(G) 8-45 8 Frequency Domain Analysis Jagged step response (note the "echoes" of the initial step): G = exp(-s) * (0.8*s^2+s+2)/(s^2+s); T = feedback(G,1); step(T) 8-46 Analyzing Control Systems with Delays Chaotic response: G = 1/(s+1) + exp(-4*s); T = feedback(1,G); step(T) 8-47 8 Frequency Domain Analysis See Also margin | pade Related Examples • “Analyzing the Response of an RLC Circuit” on page 8-50 More About • 8-48 “Time Delays in Linear Systems” on page 2-40 See Also • “Time-Delay Approximation” on page 2-48 8-49 8 Frequency Domain Analysis Analyzing the Response of an RLC Circuit This example shows how to analyze the time and frequency responses of common RLC circuits as a function of their physical parameters using Control System Toolbox™ functions. Bandpass RLC Network The following figure shows the parallel form of a bandpass RLC circuit: Figure 1: Bandpass RLC Network. The transfer function from input to output voltage is: 8-50 Analyzing the Response of an RLC Circuit The product LC controls the bandpass frequency while RC controls how narrow the passing band is. To build a bandpass filter tuned to the frequency 1 rad/s, set L=C=1 and use R to tune the filter band. Analyzing the Frequency Response of the Circuit The Bode plot is a convenient tool for investigating the bandpass characteristics of the RLC network. Use tf to specify the circuit's transfer function for the values %|R=L=C=1|: R = 1; L = 1; C = 1; G = tf([1/(R*C) 0],[1 1/(R*C) 1/(L*C)]) G = s ----------s^2 + s + 1 Continuous-time transfer function. Next, use bode to plot the frequency response of the circuit: bode(G), grid 8-51 8 Frequency Domain Analysis As expected, the RLC filter has maximum gain at the frequency 1 rad/s. However, the attenuation is only -10dB half a decade away from this frequency. To get a narrower passing band, try increasing values of R as follows: R1 = 5; G1 = tf([1/(R1*C) 0],[1 1/(R1*C) 1/(L*C)]); R2 = 20; G2 = tf([1/(R2*C) 0],[1 1/(R2*C) 1/(L*C)]); bode(G,'b',G1,'r',G2,'g'), grid legend('R = 1','R = 5','R = 20') 8-52 Analyzing the Response of an RLC Circuit The resistor value R=20 gives a filter narrowly tuned around the target frequency of 1 rad/s. Analyzing the Time Response of the Circuit We can confirm the attenuation properties of the circuit G2 (R=20) by simulating how this filter transforms sine waves with frequency 0.9, 1, and 1.1 rad/s: t = 0:0.05:250; opt = timeoptions; opt.Title.FontWeight = 'Bold'; subplot(311), lsim(G2,sin(t),t,opt), title('w = 1') subplot(312), lsim(G2,sin(0.9*t),t,opt), title('w = 0.9') subplot(313), lsim(G2,sin(1.1*t),t,opt), title('w = 1.1') 8-53 8 Frequency Domain Analysis The waves at 0.9 and 1.1 rad/s are considerably attenuated. The wave at 1 rad/s comes out unchanged once the transients have died off. The long transient results from the poorly damped poles of the filters, which unfortunately are required for a narrow passing band: damp(pole(G2)) Pole -2.50e-02 + 1.00e+00i -2.50e-02 - 1.00e+00i 8-54 Damping 2.50e-02 2.50e-02 Frequency (rad/TimeUnit) 1.00e+00 1.00e+00 Time Constant (TimeUnit) 4.00e+01 4.00e+01 Analyzing the Response of an RLC Circuit Interactive GUI To analyze other standard circuit configurations such as low-pass and high-pass RLC networks, click on the link below to launch an interactive GUI. In this GUI, you can change the R,L,C parameters and see the effect on the time and frequency responses in real time. Open the RLC Circuit GUI rlc_gui 8-55 8 Frequency Domain Analysis See Also bodeplot | lsim | stepplot Related Examples • 8-56 “Joint Time-Domain and Frequency-Domain Analysis” on page 7-40 9 Sensitivity Analysis • “Model Array with Single Parameter Variation” on page 9-2 • “Model Array with Variations in Two Parameters” on page 9-6 • “Study Parameter Variation by Sampling Tunable Model” on page 9-9 • “Sensitivity of Control System to Time Delays” on page 9-12 9 Sensitivity Analysis Model Array with Single Parameter Variation This example shows how to create a one-dimensional array of transfer functions using the stack command. One parameter of the transfer function varies from model to model in the array. You can use such an array to investigate the effect of parameter variation on your model, such as for sensitivity analysis. Create an array of transfer functions representing the following low-pass filter at three values of the roll-off frequency, a. Create transfer function models representing the filter with roll-off frequency at a = 3, 5, and 7. F1 = tf(3,[1 3]); F2 = tf(5,[1 5]); F3 = tf(7,[1 7]); Use the stack command to build an array. Farray = stack(1,F1,F2,F3); The first argument to stack specifies the array dimension along which stack builds an array. The remaining arguments specify the models to arrange along that dimension. Thus, Farray is a 3-by-1 array of transfer functions. Concatenating models with MATLAB® array concatenation commands, instead of with stack, creates multi-input, multi-output (MIMO) models rather than model arrays. For example: G = [F1;F2;F3]; creates a one-input, three-output transfer function model, not a 3-by-1 array. When working with a model array that represents parameter variations, You can associate the corresponding parameter value with each entry in the array. Set the SamplingGrid property to a data structure that contains the name of the parameter and the sampled parameter values corresponding with each model in the array. This assignment helps you keep track of which model corresponds to which parameter value. 9-2 Model Array with Single Parameter Variation Farray.SamplingGrid = struct('alpha',[3 5 7]); Farray Farray(:,:,1,1) [alpha=3] = 3 ----s + 3 Farray(:,:,2,1) [alpha=5] = 5 ----s + 5 Farray(:,:,3,1) [alpha=7] = 7 ----s + 7 3x1 array of continuous-time transfer functions. The parameter values in Farray.SamplingGrid are displayed along with the each transfer function in the array. Plot the frequency response of the array to examine the effect of parameter variation on the filter behavior. bodeplot(Farray) 9-3 9 Sensitivity Analysis When you use analysis commands such as bodeplot on a model array, the resulting plot shows the response of each model in the array. Therefore, you can see the range of responses that results from the parameter variation. See Also stack More About • 9-4 “Model Arrays” on page 2-100 See Also • “Select Models from Array” on page 2-103 9-5 9 Sensitivity Analysis Model Array with Variations in Two Parameters This example shows how to create a two-dimensional (2-D) array of transfer functions using for loops. One parameter of the transfer function varies in each dimension of the array. You can use the technique of this example to create higher-dimensional arrays with variations of more parameters. Such arrays are useful for studying the effects of multipleparameter variations on system response. The second-order single-input, single-output (SISO) transfer function depends on two parameters: the damping ratio, and , and the natural frequency, . If both vary, you obtain multiple transfer functions of the form: where and represent different measurements or sampled values of the variable parameters. You can collect all of these transfer functions in a single variable to create a two-dimensional model array. Preallocate memory for the model array. Preallocating memory is an optional step that can enhance computation efficiency. To preallocate, create a model array of the required size and initialize its entries to zero. H = tf(zeros(1,1,3,3)); In this example, there are three values for each parameter in the transfer function H. Therefore, this command creates a 3-by-3 array of single-input, single-output (SISO) zero transfer functions. Create arrays containing the parameter values. zeta = [0.66,0.71,0.75]; w = [1.0,1.2,1.5]; 9-6 Model Array with Variations in Two Parameters Build the array by looping through all combinations of parameter values. for i = 1:length(zeta) for j = 1:length(w) H(:,:,i,j) = tf(w(j)^2,[1 2*zeta(i)*w(j) w(j)^2]); end end H is a 3-by-3 array of transfer functions. a single column of H. The parameter varies as you move from model to model along varies as you move along a single row. Plot the step response of H to see how the parameter variation affects the step response. stepplot(H) 9-7 9 Sensitivity Analysis You can set the SamplingGrid property of the model array to help keep track of which set of parameter values corresponds to which entry in the array. To do so, create a grid of parameter values that matches the dimensions of the array. Then, assign these values to H.SamplingGrid with the parameter names. [zetagrid,wgrid] = ndgrid(zeta,w); H.SamplingGrid = struct('zeta',zetagrid,'w',wgrid); When you display H, the parameter values in H.SamplingGrid are displayed along with the each transfer function in the array. See Also ndgrid More About 9-8 • “Model Arrays” on page 2-100 • “Study Parameter Variation by Sampling Tunable Model” on page 9-9 Study Parameter Variation by Sampling Tunable Model Study Parameter Variation by Sampling Tunable Model This example shows how to sample a parametric model of a second-order filter across a grid of parameter values using sampleBlock. Consider the second-order filter represented by: Sample this filter at varying values of the damping constant and the natural frequency . Create a parametric model of the filter by using tunable elements for and . wn = realp('wn',3); zeta = realp('zeta',0.8); F = tf(wn^2,[1 2*zeta*wn wn^2]) F = Generalized continuous-time state-space model with 1 outputs, 1 inputs, 2 states, and wn: Scalar parameter, 5 occurrences. zeta: Scalar parameter, 1 occurrences. Type "ss(F)" to see the current value, "get(F)" to see all properties, and "F.Blocks" t F is a genss model with two tunable Control Design Blocks, the realp blocks wn and zeta. The blocks wn and zeta have initial values of 3 and 0.8, respectively. Sample F over a 2-by-3 grid of (wn, zeta) values. wnvals = [3;5]; zetavals = [0.6 0.8 1.0]; Fsample = sampleBlock(F,'wn',wnvals,'zeta',zetavals); Here, sampleBlock samples the model independently over the two values and three values. Thus, Fsample is a 2-by-3 array of state-space models. Each entry in the array is a state-space model that represents F evaluated at the corresponding (wn, zeta) pair. For example, Fsample(:,:,2,3) has wn = 5 and zeta = 1.0. Set the SamplingGrid property of the model array to help keep track of which set of parameter values corresponds to which entry in the array. To do so, create a grid of 9-9 9 Sensitivity Analysis parameter values that matches the dimensions of the array. Then, assign these values to Fsample.SamplingGrid in a structure with the parameter names. [wngrid,zetagrid] = ndgrid(wnvals,zetavals); Fsample.SamplingGrid = struct('wn',wngrid,'zeta',zetagrid); The ndgrid command produces the full 2-by-3 grid of (wn, zeta) combinations. When you display Fsample in the command window, the parameter values in Fsample.SamplingGrid are displayed along with the each transfer function in the array. The parameter information is also available in response plots. For instance, examine the step response of Fsample. stepplot(Fsample) 9-10 See Also The step response plots show the variation in the natural frequency and damping constant across the six models in the array. When you click on one of the responses in the plot, the datatip includes the corresponding wn and zeta values as specified in Fsample.SamplingGrid. See Also sampleBlock More About • “Models with Tunable Coefficients” on page 1-19 9-11 9 Sensitivity Analysis Sensitivity of Control System to Time Delays This example shows how to examine the sensitivity of a closed-loop control system to time delays within the system. Time delays are rarely known accurately, so it is often important to understand how sensitive a control system is to the delay value. Such sensitivity analysis is easily performed using LTI arrays and the InternalDelay property. For example, consider the notched PI control system developed in "PI Control Loop with Dead Time" from the example "Analyzing Control Systems with Delays." The following commands create an LTI model of that closed-loop system, a third-order plant with an input delay, a PI controller and a notch filter. s = tf('s'); G = exp(-2.6*s)*(s+3)/(s^2+0.3*s+1); C = 0.06 * (1 + 1/s); T = feedback(ss(G*C),1); notch = tf([1 0.2 1],[1 .8 1]); C = 0.05 * (1 + 1/s); Tnotch = feedback(ss(G*C*notch),1); Examine the internal delay of the closed-loop system Tnotch. Tnotch.InternalDelay ans = 2.6000 The 2.6-second input delay of the plant G becomes an internal delay of 2.6 s in the closedloop system. To examine the sensitivity of the responses of Tnotch to variations in this delay, create an array of copies of Tnotch. Then, vary the internal delay across the array. Tsens = repsys(Tnotch,[1 1 5]); tau = linspace(2,3,5); for j = 1:5; Tsens(:,:,j).InternalDelay = tau(j); end The array Tsens contains five models with internal delays that range from 2.0 to 3.0. Examine the step responses of these models. stepplot(Tsens) 9-12 See Also The plot shows that uncertainty on the delay value has a small effect on closed-loop characteristics. See Also More About • “Time Delays in Linear Systems” on page 2-40 9-13 10 Passivity and Conic Sectors • “About Passivity and Passivity Indices” on page 10-2 • “About Sector Bounds and Sector Indices” on page 10-9 • “Passivity Indices” on page 10-19 • “Parallel Interconnection of Passive Systems” on page 10-24 • “Series Interconnection of Passive Systems” on page 10-27 • “Feedback Interconnection of Passive Systems” on page 10-31 10 Passivity and Conic Sectors About Passivity and Passivity Indices Passive control is often part of the safety requirements in applications such as process control, tele-operation, human-machine interfaces, and system networks. A system is passive if it cannot produce energy on its own, and can only dissipate the energy that is stored in it initially. More generally, an I/O map is passive if, on average, increasing the output y requires increasing the input u. For example, a PID controller is passive because the control signal (the output) moves in the same direction as the error signal (the input). But a PID controller with delay is not passive, because the control signal can move in the opposite direction from the error, a potential cause of instability. Most physical systems are passive. The Passivity Theorem holds that the negativefeedback interconnection of two strictly passive systems is passive and stable. As a result, it can be desirable to enforce passivity of the controller for a passive system, or to passivate the operator of a passive system, such as the driver of a car. In practice, passivity can easily be destroyed by the phase lags introduced by sensors, actuators, and communication delays. These problems have led to extension of the Passivity Theorem that consider excesses or shortages of passivity, frequency-dependent measures of passivity, and a mix of passivity and small-gain properties. Passive Systems A linear system is passive if all input/output trajectories satisfy: where denotes the transpose of . For physical systems, the integral typically represents the energy going into the system,. Thus passive systems are systems that only consume or dissipate energy. As a result, passive systems are intrinsically stable. In the frequency domain, passivity is equivalent to the "positive real" condition: 10-2 About Passivity and Passivity Indices For SISO systems, this is saying that Nyquist plot lies in the right-half plane. at all frequencies, so the entire nyquist(tf([1 3 5],[5 6 1])) Nyquist plot of passive system Passive systems have the following important properties for control purposes: • The inverse of a passive system is passive. • The parallel interconnection of passive systems is passive (see “Parallel Interconnection of Passive Systems” on page 10-24). 10-3 10 Passivity and Conic Sectors • The feedback interconnection of passive systems is passive (see “Feedback Interconnection of Passive Systems” on page 10-31). When controlling a passive system with unknown or variable characteristics, it is therefore desirable to use a passive feedback law to guarantee closed-loop stability. This task can be rendered difficult given that delays and significant phase lag destroy passivity. Directional Passivity Indices For stability, knowing whether a system is passive or not does not tell the full story. It is often desirable to know by how much it is passive or fails to be passive. In addition, a shortage of passivity in the plant can be compensated by an excess of passivity in the controller, and vice versa. It is therefore important to measure the excess or shortage of passivity, and this is where passivity indices come into play. There are different types of indices with different applications. One class of indices measure the excess or shortage of passivity in a particular direction of the input/output space. For example, the input passivity index is defined as the largest for all trajectories and such that: . The system G is input strictly passive (ISP) , and has a shortage of passivity when . The input passivity index is also when called the input feedforward passivity (IFP) index because it corresponds to the minimum static feedforward action needed to make the system passive. In the frequency domain, the input passivity index is characterized by: 10-4 About Passivity and Passivity Indices where denotes the smallest eigenvalue. In the SISO case, leftmost point on the Nyquist curve. Similarly, the output passivity index is defined as the largest for all trajectories and is the abscissa of the such that: . The system G is output strictly passive (OSP) , and has a shortage of passivity when . The output passivity index is when also called the output feedback passivity (OFP) index because it corresponds to the minimum static feedback action needed to make the system passive. In the frequency domain, the output passivity index of a minimum-phase system given by: In the SISO case, is the abscissa of the leftmost point on the Nyquist curve of Combining these two notions leads to the I/O passivity index, which is the largest that: is . such 10-5 10 Passivity and Conic Sectors A system with the direction is very strictly passive. More generally, we can define the index in as the largest such that: The input, output, and I/O passivity indices all correspond to special choices of and are collectively referred to as directional passivity indices. You can use getPassiveIndex to compute any of these indices for linear systems in either parametric or FRD form. You can also use passiveplot to plot the input, output, or I/O passivity indices as a function of frequency. This plot provides insight into which frequency bands have weaker or stronger passivity. There are many results quantifying how the input and output passivity indices propagate through parallel, series, or feedback interconnections. There are also results quantifying the excess of input or output passivity needed to compensate a given shortage of passivity in a feedback loop. For details, see: • “Parallel Interconnection of Passive Systems” on page 10-24 • “Series Interconnection of Passive Systems” on page 10-27 • “Feedback Interconnection of Passive Systems” on page 10-31 Relative Passivity Index The positive real condition for passivity: is equivalent to the small gain condition: 10-6 About Passivity and Passivity Indices We can therefore use the peak gain of Specifically, let Then that is passive if and only if as a measure of passivity. , and is finite if and only if indicates a shortage of passivity. Note is minimum phase. We refer to as the relative passivity index, or R-index. In the time domain, the R-index is the smallest that: for all trajectories and . When such is minimum phase, you can use passiveplot to plot the principal gains of . This plot is entirely analogous to the singular value plot (see sigma), and shows how the degree of passivity changes with frequency and direction. The following result is analogous to the Small Gain Theorem for feedback loops. It gives a simple condition on R-indices for compensating a shortage of passivity in one system by an excess of passivity in the other. Small-R Theorem: Let and and , respectively. If and be two linear systems with passivity R-indices , then the negative feedback interconnection of is stable. 10-7 10 Passivity and Conic Sectors See Also getPassiveIndex | isPassive | passiveplot Related Examples 10-8 • “Passivity Indices” on page 10-19 • “Parallel Interconnection of Passive Systems” on page 10-24 • “Series Interconnection of Passive Systems” on page 10-27 • “Feedback Interconnection of Passive Systems” on page 10-31 • “About Sector Bounds and Sector Indices” on page 10-9 About Sector Bounds and Sector Indices About Sector Bounds and Sector Indices Conic Sectors In its simplest form, a conic sector is the 2-D region delimited by two lines, and . The shaded region is characterized by the inequality generally, any such sector can be parameterized as: where is a 2x2 symmetric indefinite matrix ( . More has one positive and one negative the sector matrix. This concept generalizes to higher dimensions. eigenvalue). We call In an N-dimensional space, a conic sector is a set: 10-9 10 Passivity and Conic Sectors where is again a symmetric indefinite matrix. Sector Bounds Sector bounds are constraints on the behavior of a system. Gain constraints and passivity constraints are special cases of sector bounds. If for all nonzero input trajectories the output trajectory of a linear system then the output trajectories of , satisfies: lie in the conic sector with matrix . Selecting different matrices imposes different conditions on the system's response. For example, consider trajectories and the following values: These values correspond to the sector bound: This sector bound is equivalent to the passivity condition for : In other words, passivity is a particular sector bound on the system defined by: 10-10 About Sector Bounds and Sector Indices Frequency-Domain Condition Because the time-domain condition must hold for all , deriving an equivalent frequency-domain bound takes a little care and is not always possible. Let the following: be (any) decomposition of the indefinite matrix When condition: into its positive and negative parts. is square and minimum phase (has no unstable zeros), the time-domain is equivalent to the frequency-domain condition: It is therefore enough to check the sector inequality for real frequencies. Using the decomposition of Note that , this is also equivalent to: is square when has as many negative eigenvalues as input channels in . If this condition is not met, it is no longer enough (in general) to just look at real frequencies. Note also that if sector bound to hold. is square, then it must be minimum phase for the This frequency-domain characterization is the basis for sectorplot. Specifically, sectorplot plots the singular values of as a function of frequency. The sector bound is satisfied if and only if the largest singular value stays below 1. Moreover, the plot contains useful information about the frequency bands where the sector bound is satisfied or violated, and the degree to which it is satisfied or violated. 10-11 10 Passivity and Conic Sectors For instance, examine the sector plot of a 2-output, 2-input system for a particular sector. rng(4); H = rss(3,4,2); Q = [-5.12 2.16 2.16 -1.22 -2.04 -0.28 2.17 -1.11 sectorplot(H,Q) -2.04 -0.28 -3.35 0.00 2.17 -1.11 0.00 0.18]; The plot shows that the largest singular value of exceeds 1 below about 0.5 rad/s and in a narrow band around 3 rad/s. Therefore, H does not satisfy the sector bound represented by Q. 10-12 About Sector Bounds and Sector Indices Relative Sector Index We can extend the notion of relative passivity index to arbitrary sectors. Let LTI system, and let: be an orthogonal decomposition of into its positive and negative parts, as is readily obtained from the Schur decomposition of Because increasing , or R-index, is makes : more negative, the inequality is usually large enough. However, there are cases when it can never be satisfied, in satisfied for which case the R-index is only of . The relative sector index such that for all output trajectories defined as the smallest be an . Clearly, the original sector bound is satisfied if and . To understand the geometrical interpretation of the R-index, consider the family of cones with matrix . In 2D, the cone slant angle (see diagram below). More generally, sector with matrix , an R-index value is proportional to before some output trajectory of Similarly, a value means that we must increase measure of how well the response of by . Thus, given a conic means that we can reduce the cone) by a factor to include all output trajectories of is related to (narrow leaves the conic sector. (widen the cone) by a factor . This clearly makes the R-index a relative fits in a particular conic sector. 10-13 10 Passivity and Conic Sectors In the diagram, and When is square and minimum phase, the R-index can also be characterized in the frequency domain as the smallest Using elementary algebra, this leads to: 10-14 such that: About Sector Bounds and Sector Indices In other words, the R-index is the peak gain of the (stable) transfer function , and the singular values of can be seen as the "principal" R-indices at each frequency. This also explains why plotting the R-index vs. frequency looks like a singular value plot (see sectorplot). There is a complete analogy between relative sector index and system gain. Note, however, that this analogy only holds when is square and minimum phase. Directional Sector Index Similarly, we can extend the notion of directional passivity index to arbitrary sectors. Given a conic sector with matrix the largest , and a direction such that for all output trajectories The directional passivity index for a system , the directional sector index is : corresponds to: The directional sector index measures by how much we need to deform the sector in the direction to make it fit tightly around the output trajectories of is satisfied if and only if the directional index is positive. . The sector bound Common Sectors There are many ways to specify sector bounds. Next we review commonly encountered expressions and give the corresponding system and sector matrix form used by getSectorIndex and sectorplot: for the standard For simplicity, these descriptions use the notation: 10-15 10 Passivity and Conic Sectors and omit the requirement. Passivity Passivity is a sector bound with: Gain constraint The gain constraint is a sector bound with: Ratio of distances Consider the "interior" constraint, where are scalars and . This is a sector bound with: The underlying conic sector is symmetric with respect to constraint, is a sector bound with: 10-16 . Similarly, the "exterior" About Sector Bounds and Sector Indices Double inequality When dealing with static nonlinearities, it is common to consider conic sectors of the form where is the nonlinearity output. While this relationship is not a sector bound per se, it clearly implies: along all I/O trajectories and for all bound with: . This condition in turn is equivalent to a sector Product form Generalized sector bounds of the form: correspond to: As before, the static sector bound: 10-17 10 Passivity and Conic Sectors implies the integral sector bound above. QSR dissipative A system is QSR-dissipative if it satisfies: This is a sector bound with: See Also getSectorCrossover | getSectorIndex | sectorplot Related Examples • 10-18 “About Passivity and Passivity Indices” on page 10-2 Passivity Indices Passivity Indices This example shows how to compute various measures of passivity for linear timeinvariant systems. Passive Systems A linear system G(s) is passive when all I/O trajectories where denotes the transpose of satisfy . To measure "how passive" a system is, we use passivity indices. • The input passivity index is defined as the largest such that The system G is "input strictly passive" (ISP) when . is also called the "input feedforward passivity" (IFP) index and corresponds to the minimum feedforward action needed to make the system passive. • The output passivity index is defined as the largest such that 10-19 10 Passivity and Conic Sectors The system G is "output strictly passive" (OSP) when . is also called the "output feedback passivity" (OFP) index and corresponds to the minimum feedback action needed to make the system passive. • The I/O passivity index is defined as the largest The system is "very strictly passive" (VSP) if such that . Circuit Example as Consider the following example. We take the current as the input and the voltage the output. Based on Kirchhoff's current and voltage law, we obtain the transfer function for Let , , and . R = 2; L = 1; C = 0.1; s = tf('s'); G = (L*s+R)*(R*s+1/C)/(L*s^2 + 2*R*s+1/C); Use isPassive to check whether 10-20 is passive. Passivity Indices PF = isPassive(G) PF = logical 1 Since PF = true, indices of is passive. Use getPassiveIndex to compute the passivity . % Input passivity index nu = getPassiveIndex(G,'in') nu = 2 % Output passivity index rho = getPassiveIndex(G,'out') rho = 0.2857 % I/O passivity index tau = getPassiveIndex(G,'io') tau = 0.2642 Since , the system is very strictly passive. Frequency-Domain Characterization A linear system is passive if and only if it is "positive real": The smallest eigenvalue of the left-hand-side is related to the input passivity index where denotes the smallest eigenvalue. Similarly, when output passivity index is given by: : is minimum-phase, the 10-21 10 Passivity and Conic Sectors Verify this for the circuit example. Plot the Nyquist plot of the circuit transfer function. nyquist(G) The entire Nyquist plot lies in the right-half plane so is positive real. The leftmost so the input passivity index is , the point on the Nyquist curve is same value we obtained earlier. Similarly, the leftmost point on the Nyquist curve for gives the output passivity index value Relative Passivity Index It can be shown that the "positive real" condition 10-22 . See Also is equivalent to the small gain condition The relative passivity index (R-index) is the peak gain over frequency of when is minimum phase, and otherwise: In the time domain, the R-index is the smallest such that The system is passive if and only if , and the smaller is, the more passive the system is. Use getPassiveIndex to compute the R-index for the circuit example. R = getPassiveIndex(G) R = 0.5556 The resulting value indicates that the circuit is a very passive system. See Also getPassiveIndex | isPassive Related Examples • “About Passivity and Passivity Indices” on page 10-2 • “Parallel Interconnection of Passive Systems” on page 10-24 • “Series Interconnection of Passive Systems” on page 10-27 • “Feedback Interconnection of Passive Systems” on page 10-31 10-23 10 Passivity and Conic Sectors Parallel Interconnection of Passive Systems This example illustrates the properties of a parallel interconnection of passive systems. Parallel Interconnection of Passive Systems Consider an interconnection of two subsystems interconnected system maps the input and to the output in parallel. The . If both systems and are passive, then the interconnected system to be passive. Take for example Both systems are passive. G1 = tf([0.1,1],[1,2]); isPassive(G1) ans = logical 1 G2 = tf([1,2,1],[1,3,10]); isPassive(G2) ans = logical 1 10-24 is guaranteed Parallel Interconnection of Passive Systems We can therefore expect their parallel interconnection to be passive, as confirmed by H = parallel(G1,G2); isPassive(H) ans = logical 1 Passivity Indices for Parallel Interconnection There is a relationship between the passivity indices of indices of the interconnected system for and , and let and . Let and the passivity denote the input passivity indices denote the output passivity indices. If all these indices are nonnegative, then the input passivity index parallel interconnection and and and the output passivity index for the satisfy In other words, we can infer some minimum level of input and output passivity for the parallel connection from the input and output passivity indices of and . For details, see the paper by Yu, H., "Passivity and dissipativity as design and analysis tools for networked control systems," Chapter 2, PhD Thesis, University of Notre Dame, 2012. Verify the lower bound for the input passivity index . % Input passivity index for G1 nu1 = getPassiveIndex(G1,'input'); % Input passivity index for G2 nu2 = getPassiveIndex(G2,'input'); % Input passivity index for H nu = getPassiveIndex(H,'input') nu = 0.3777 % Lower bound nu1+nu2 ans = 0.1474 10-25 10 Passivity and Conic Sectors Similarly, verify the lower bound for the output passivity index of % Output passivity index for G1 rho1 = getPassiveIndex(G1,'output'); % Output passivity index for G2 rho2 = getPassiveIndex(G2,'output'); % Output passivity index for H rho = getPassiveIndex(H,'output') rho = 0.6450 % Lower bound rho1*rho2/(rho1+rho2) ans = 0.2098 See Also getPassiveIndex | isPassive Related Examples 10-26 • “About Passivity and Passivity Indices” on page 10-2 • “Series Interconnection of Passive Systems” on page 10-27 • “Feedback Interconnection of Passive Systems” on page 10-31 . Series Interconnection of Passive Systems Series Interconnection of Passive Systems This example illustrates the properties of a series interconnection of passive systems. Series Interconnection of Passive Systems Consider an interconnection of two subsystems system is given by the mapping from input and to output in series. The interconnected . In contrast with parallel and feedback interconnections, passivity of the subsystems and does not guarantee passivity for the interconnected system . Take for example Both systems are passive as confirmed by G1 = tf([5 3 1],[1,2,1]); isPassive(G1) ans = logical 1 G2 = tf([1,1,5,.1],[1,2,3,4]); isPassive(G2) ans = logical 1 However the series interconnection of and is not passive: 10-27 10 Passivity and Conic Sectors H = G2*G1; isPassive(H) ans = logical 0 This is confirmed by verifying that the Nyquist plot of nyquist(H) 10-28 is not positive real. Series Interconnection of Passive Systems Passivity Indices for Series Interconnection While the series interconnection of passive systems is not passive in general, there is a relationship between the passivity indices of . Let and and and and the passivity indices of denote the input passivity indices for and , and let denote the output passivity indices. If all these indices are positive, then the input passivity index satisfy and the output passivity index for the series interconnection In other words, the shortage of passivity at the inputs or outputs of is no worse than the right-hand-side expressions. For details, see the paper by Arcak, M. and Sontag, E.D., "Diagonal stability of a class of cyclic systems and its connection with the secant criterion," Automatica, Vol 42, No. 9, 2006, pp. 1531-1537. Verify these lower bounds for the example above. % Output passivity index for G1 rho1 = getPassiveIndex(G1,'output'); % Output passivity index for G2 rho2 = getPassiveIndex(G2,'output'); % Input passivity index for H=G2*G1 nu = getPassiveIndex(H,'input') nu = -1.2875 % Lower bound -0.125/(rho1*rho2) ans = -2.4119 Similarly, verify the lower bound for the output passivity index of . % Input passivity index for G1 nu1 = getPassiveIndex(G1,'input'); % Input passivity index for G2 nu2 = getPassiveIndex(G2,'input'); % Output passivity index for H=G2*G1 rho = getPassiveIndex(H,'output') 10-29 10 Passivity and Conic Sectors rho = -0.6966 % Lower bound -0.125/(nu1*nu2) ans = -5.9420 See Also getPassiveIndex | isPassive Related Examples 10-30 • “About Passivity and Passivity Indices” on page 10-2 • “Parallel Interconnection of Passive Systems” on page 10-24 • “Feedback Interconnection of Passive Systems” on page 10-31 Feedback Interconnection of Passive Systems Feedback Interconnection of Passive Systems This example illustrates the properties of a feedback interconnection of passive systems. Feedback Interconnection of Passive Systems Consider an interconnection of two subsystems interconnected system maps the input and to the output in feedback. The . If both systems and are passive, then the interconnected system to be passive. Take for example is guaranteed Both systems are passive as confirmed by G1 = tf([1,1,1],[1,1,4]); isPassive(G1) ans = logical 1 G2 = tf([1,2],[1,5]); isPassive(G2) ans = logical 1 10-31 10 Passivity and Conic Sectors The interconnected system is therefore passive. H = feedback(G1,G2); isPassive(H) ans = logical 1 This is confirmed by verifying that the Nyquist plot of nyquist(H) 10-32 is positive real. Feedback Interconnection of Passive Systems Passivity Indices for Feedback Interconnection There is a relationship between the passivity indices of indices of the interconnected system for and , and let and . Let and the passivity denote the input passivity indices denote the output passivity indices. If all these indices are positive, then the input passivity index feedback interconnection and and and the output passivity index for the satisfy In other words, we can infer some minimum level of input and output passivity for the closed-loop system from the input and output passivity indices of and . For details, see the paper by Zhu, F. and Xia, M and Antsaklis, P.J., "Passivity analysis and passivation of feedback systems using passivity indices," American Control Conference , 2014, pp. 1833-1838. Verify the lower bound for the input passivity index . % Input passivity index for G1 nu1 = getPassiveIndex(G1,'input'); % Output passivity index for G2 rho2 = getPassiveIndex(G2,'output'); % Input passivity index for H nu = getPassiveIndex(H,'input') nu = 0.1293 % Lower bound nu1*rho2/(nu1+rho2) ans = 2.4923e-06 Similarly, verify the lower bound for the output passivity index of . % Output passivity index for G1 rho1 = getPassiveIndex(G1,'output'); % Input passivity index for G2 nu2 = getPassiveIndex(G2,'input'); % Output passivity index for H rho = getPassiveIndex(H,'output') 10-33 10 Passivity and Conic Sectors rho = 0.4485 % Lower bound rho1+nu2 ans = 0.4000 See Also getPassiveIndex | isPassive Related Examples 10-34 • “About Passivity and Passivity Indices” on page 10-2 • “Parallel Interconnection of Passive Systems” on page 10-24 • “Series Interconnection of Passive Systems” on page 10-27 • “Passive Control with Communication Delays” on page 17-44 Control Design 35 11 PID Controller Design • “PID Controller Design at the Command Line” on page 11-2 • “Designing Cascade Control System with PI Controllers” on page 11-9 • “Tune 2-DOF PID Controller (Command Line)” on page 11-15 • “Tune 2-DOF PID Controller (PID Tuner)” on page 11-21 • “PID Controller Types for Tuning” on page 11-31 • “PID Controller Tuning in Simulink” on page 11-37 • “Design PID Controller Using Estimated Frequency Response” on page 11-46 • “Design Family of PID Controllers for Multiple Operating Points” on page 11-56 • “Design PID Controller Using Simulated I/O Data” on page 11-65 11 PID Controller Design PID Controller Design at the Command Line This example shows how to design a PID controller for the plant given by: As a first pass, create a model of the plant and design a simple PI controller for it. sys = zpk([],[-1 -1 -1],1); [C_pi,info] = pidtune(sys,'PI') C_pi = 1 Kp + Ki * --s with Kp = 1.14, Ki = 0.454 Continuous-time PI controller in parallel form. info = struct with fields: Stable: 1 CrossoverFrequency: 0.5205 PhaseMargin: 60.0000 C_pi is a pid controller object that represents a PI controller. The fields of info show that the tuning algorithm chooses an open-loop crossover frequency of about 0.52 rad/s. Examine the closed-loop step response (reference tracking) of the controlled system. T_pi = feedback(C_pi*sys, 1); step(T_pi) 11-2 PID Controller Design at the Command Line To improve the response time, you can set a higher target crossover frequency than the result that pidtune automatically selects, 0.52. Increase the crossover frequency to 1.0. [C_pi_fast,info] = pidtune(sys,'PI',1.0) C_pi_fast = 1 Kp + Ki * --s with Kp = 2.83, Ki = 0.0495 11-3 11 PID Controller Design Continuous-time PI controller in parallel form. info = struct with fields: Stable: 1 CrossoverFrequency: 1 PhaseMargin: 43.9973 The new controller achieves the higher crossover frequency, but at the cost of a reduced phase margin. Compare the closed-loop step response with the two controllers. T_pi_fast = feedback(C_pi_fast*sys,1); step(T_pi,T_pi_fast) axis([0 30 0 1.4]) legend('PI','PI,fast') 11-4 PID Controller Design at the Command Line This reduction in performance results because the PI controller does not have enough degrees of freedom to achieve a good phase margin at a crossover frequency of 1.0 rad/s. Adding a derivative action improves the response. Design a PIDF controller for Gc with the target crossover frequency of 1.0 rad/s. [C_pidf_fast,info] = pidtune(sys,'PIDF',1.0) C_pidf_fast = 1 s Kp + Ki * --- + Kd * -------s Tf*s+1 11-5 11 PID Controller Design with Kp = 2.72, Ki = 0.985, Kd = 1.72, Tf = 0.00875 Continuous-time PIDF controller in parallel form. info = struct with fields: Stable: 1 CrossoverFrequency: 1 PhaseMargin: 60.0000 The fields of info show that the derivative action in the controller allows the tuning algorithm to design a more aggressive controller that achieves the target crossover frequency with a good phase margin. Compare the closed-loop step response and disturbance rejection for the fast PI and PIDF controllers. T_pidf_fast = feedback(C_pidf_fast*sys,1); step(T_pi_fast, T_pidf_fast); axis([0 30 0 1.4]); legend('PI,fast','PIDF,fast'); 11-6 PID Controller Design at the Command Line You can compare the input (load) disturbance rejection of the controlled system with the fast PI and PIDF controllers. To do so, plot the response of the closed-loop transfer function from the plant input to the plant output. S_pi_fast = feedback(sys,C_pi_fast); S_pidf_fast = feedback(sys,C_pidf_fast); step(S_pi_fast,S_pidf_fast); axis([0 50 0 0.4]); legend('PI,fast','PIDF,fast'); 11-7 11 PID Controller Design This plot shows that the PIDF controller also provides faster disturbance rejection. See Also pid | pidtune More About 11-8 • “Choosing a PID Controller Design Tool” • “Designing Cascade Control System with PI Controllers” on page 11-9 • “PID Controller Design for Fast Reference Tracking” Designing Cascade Control System with PI Controllers Designing Cascade Control System with PI Controllers This example shows how to design a cascade control loop with two PI controllers using the pidtune command. Introduction to Cascade Control Cascade control is mainly used to achieve fast rejection of disturbance before it propagates to the other parts of the plant. The simplest cascade control system involves two control loops (inner and outer) as shown in the block diagram below. Controller C1 in the outer loop is the primary controller that regulates the primary controlled variable y1 by setting the set-point of the inner loop. Controller C2 in the inner loop is the secondary controller that rejects disturbance d2 locally before it propagates to P1. For a cascade control system to function properly, the inner loop must respond much faster than the outer loop. In this example, you will design a single loop control system with a PI controller and a cascade control system with two PI controllers. The responses of the two control systems are compared for both reference tracking and disturbance rejection. Plant In this example, the inner loop plant P2 is The outer loop plant P1 is 11-9 11 PID Controller Design P2 = zpk([],-2,3); P1 = zpk([],[-1 -1 -1],10); Designing a Single Loop Control System with a PI Controller Use pidtune command to design a PI controller in standard form for the whole plant model P = P1 * P2. The desired open loop bandwidth is 0.2 rad/s, which roughly corresponds to the response time of 10 seconds. % P % C % C C The plant model is P = P1*P2 = P1*P2; Use a PID or PIDSTD object to define the desired controller structure = pidstd(1,1); Tune PI controller for target bandwidth is 0.2 rad/s = pidtune(P,C,0.2); C = 1 1 Kp * (1 + ---- * ---) Ti s with Kp = 0.0119, Ti = 0.849 Continuous-time PI controller in standard form 11-10 Designing Cascade Control System with PI Controllers Designing a Cascade Control System with Two PI Controllers The best practice is to design the inner loop controller C2 first and then design the outer loop controller C1 with the inner loop closed. In this example, the inner loop bandwidth is selected as 2 rad/s, which is ten times higher than the desired outer loop bandwidth. In order to have an effective cascade control system, it is essential that the inner loop responds much faster than the outer loop. Tune inner-loop controller C2 with open-loop bandwidth at 2 rad/s. C2 = pidtune(P2,pidstd(1,1),2); C2 C2 = 1 1 Kp * (1 + ---- * ---) Ti s with Kp = 0.244, Ti = 0.134 Continuous-time PI controller in standard form Tune outer-loop controller C1 with the same bandwidth as the single loop system. % Inner loop system when the control loop is closed first clsys = feedback(P2*C2,1); % Plant seen by the outer loop controller C1 is clsys*P1 C1 = pidtune(clsys*P1,pidstd(1,1),0.2); C1 C1 = 1 1 Kp * (1 + ---- * ---) Ti s with Kp = 0.015, Ti = 0.716 Continuous-time PI controller in standard form Performance Comparison First, plot the step reference tracking responses for both control systems. 11-11 11 PID Controller Design % single loop system for reference tracking sys1 = feedback(P*C,1); sys1.Name = 'Single Loop'; % cascade system for reference tracking sys2 = feedback(clsys*P1*C1,1); sys2.Name = 'Cascade'; % plot step response figure; step(sys1,'r',sys2,'b') legend('show','location','southeast') title('Reference Tracking') Secondly, plot the step disturbance rejection responses of d2 for both control systems. 11-12 Designing Cascade Control System with PI Controllers % single loop system for rejecting d2 sysd1 = feedback(P1,P2*C); sysd1.Name = 'Single Loop'; % cascade system for rejecting d2 sysd2 = P1/(1+P2*C2+P2*P1*C1*C2); sysd2.Name = 'Cascade'; % plot step response figure; step(sysd1,'r',sysd2,'b') legend('show') title('Disturbance Rejection') 11-13 11 PID Controller Design From the two response plots you can conclude that the cascade control system performs much better in rejecting disturbance d2 while the set-point tracking performances are almost identical. See Also pidstd | pidtune More About 11-14 • “Choosing a PID Controller Design Tool” • “PID Controller Design at the Command Line” on page 11-2 • “Tune PID Controller to Favor Reference Tracking or Disturbance Rejection (Command Line)” Tune 2-DOF PID Controller (Command Line) Tune 2-DOF PID Controller (Command Line) This example shows how to design a two-degree-of-freedom (2-DOF) PID controller at the command line. The example also compares the 2-DOF controller performance to the performance achieved with a 1-DOF PID controller. 2-DOF PID controllers include setpoint weighting on the proportional and derivative terms. Compared to a 1-DOF PID controller, a 2-DOF PID controller can achieve better disturbance rejection without significant increase of overshoot in setpoint tracking. A typical control architecture using a 2-DOF PID controller is shown in the following diagram. For this example, design a 2-DOF controller for the plant given by: Suppose that your target bandwidth for the system is 1.5 rad/s. wc = 1.5; G = tf(1,[1 0.5 0.1]); C2 = pidtune(G,'PID2',wc) C2 = 1 u = Kp (b*r-y) + Ki --- (r-y) + Kd*s (c*r-y) s 11-15 11 PID Controller Design with Kp = 1.26, Ki = 0.255, Kd = 1.38, b = 0.665, c = 0 Continuous-time 2-DOF PID controller in parallel form. Using the type 'PID2' causes pidtune to generate a 2-DOF controller, represented as a pid2 object. The display confirms this result. pidtune tunes all controller coefficients, including the setpoint weights b and c, to balance performance and robustness. To compute the closed-loop response, note that a 2-DOF PID controller is a 2-input, 1output dynamic system. You can resolve the controller into two channels, one for the reference signal and one for the feedback signal, as shown in the diagram. (See “Continuous-Time 2-DOF PID Controller Representations” on page 2-16 for more information.) Decompose the controller into the components Cr and Cy, and use them to compute the closed-loop response from r to y. C2tf Cr = Cy = T2 = = tf(C2); C2tf(1); C2tf(2); Cr*feedback(G,Cy,+1); To examine the disturbance-rejection performance, compute the transfer function from d to y. S2 = feedback(G,Cy,+1); 11-16 Tune 2-DOF PID Controller (Command Line) For comparison, design a 1-DOF PID controller with the same bandwidth and compute the corresponding transfer functions. Then compare the step responses. C1 = pidtune(G,'PID',wc); T1 = feedback(G*C1,1); S1 = feedback(G,C1); subplot(2,1,1) stepplot(T1,T2) title('Reference Tracking') subplot(2,1,2) stepplot(S1,S2) title('Disturbance Rejection') legend('1-DOF','2-DOF') 11-17 11 PID Controller Design The plots show that adding the second degree of freedom eliminates the overshoot in the reference-tracking response without any cost to disturbance rejection. You can improve disturbance rejection too using the DesignFocus option. This option causes pidtune to favor disturbance rejection over setpoint tracking. opt = pidtuneOptions('DesignFocus','disturbance-rejection'); C2dr = pidtune(G,'PID2',wc,opt) C2dr = 1 u = Kp (b*r-y) + Ki --- (r-y) + Kd*s (c*r-y) s with Kp = 1.72, Ki = 0.593, Kd = 1.25, b = 0, c = 0 Continuous-time 2-DOF PID controller in parallel form. With the default balanced design focus, pidtune selects a b value between 0 and 1. For this plant, when you change design focus to favor disturbance rejection, pidtune sets b = 0 and c = 0. Thus, pidtune automatically generates an I-PD controller to optimize for disturbance rejection. (Explicitly specifying an I-PD controller without setting the design focus yields a similar controller.) Compare the closed-loop responses using all three controllers. C2dr_tf = tf(C2dr); Cdr_r = C2dr_tf(1); Cdr_y = C2dr_tf(2); T2dr = Cdr_r*feedback(G,Cdr_y,+1); S2dr = feedback(G,Cdr_y,+1); subplot(2,1,1) stepplot(T1,T2,T2dr) title('Reference Tracking') subplot(2,1,2) stepplot(S1,S2,S2dr); title('Disturbance Rejection') legend('1-DOF','2-DOF','2-DOF rejection focus') 11-18 Tune 2-DOF PID Controller (Command Line) The plots show that the disturbance rejection is further improved compared to the balanced 2-DOF controller. This improvement comes with some sacrifice of referencetracking performance, which is slightly slower. However, the reference-tracking response still has no overshoot. Thus, using 2-DOF control can improve disturbance rejection without sacrificing as much reference tracking performance as 1-DOF control. These effects on system performance depend strongly on the properties of your plant. For some plants and some control 11-19 11 PID Controller Design bandwidths, using 2-DOF control or changing the design focus has less or no impact on the tuned result. See Also pid2 | pidtune More About 11-20 • “Designing PID Controllers with PID Tuner” • “Two-Degree-of-Freedom PID Controllers” on page 2-16 • “Tune 2-DOF PID Controller (PID Tuner)” on page 11-21 • “Analyze Design in PID Tuner” Tune 2-DOF PID Controller (PID Tuner) Tune 2-DOF PID Controller (PID Tuner) This example shows how to design a two-degree-of-freedom (2-DOF) PID controller using PID Tuner. The example also compares the 2-DOF controller performance to the performance achieved with a 1-DOF PID controller. In this example, you represent the plant as an LTI model on page 1-13. For information about using PID Tuner to tune a PID Controller (2DOF) block in a Simulink model, see “Design Two-Degree-of-Freedom PID Controllers” (Simulink Control Design). 2-DOF PID controllers include setpoint weighting on the proportional and derivative terms. Compared to a 1-DOF PID controller, a 2-DOF PID controller can achieve better disturbance rejection without significant increase of overshoot in setpoint tracking. A typical control architecture using a 2-DOF PID controller is shown in the following diagram. For this example, first design a 1-DOF controller for the plant given by: G ( s) = 1 2 . s + 0 .5 s + 0 .1 G = tf(1,[1 0.5 0.1]); pidTuner(G,'PID') 11-21 11 PID Controller Design Suppose for this example that your application requires a faster response than the PID Tuner initial design. In the text box next to the Response Time slider, enter 2. 11-22 Tune 2-DOF PID Controller (PID Tuner) The resulting response is fast, but has a considerable amount of overshoot. Design a 2DOF controller to improve the overshoot. First, set the 1-DOF controller as the baseline controller for comparison. Click the Export arrow and select Save as Baseline. 11-23 11 PID Controller Design Design the 2-DOF controller. In the Type menu, select PID2. 11-24 Tune 2-DOF PID Controller (PID Tuner) PID Tuner generates a 2-DOF controller with the same target response time. The controller parameters displayed at the bottom right show that PID Tuner tunes all 11-25 11 PID Controller Design controller coefficients, including the setpoint weights b and c, to balance performance and robustness. Compare the 2-DOF controller performance (solid line) with the performance of the 1-DOF controller that you stored as the baseline (dotted line). Adding the second degree of freedom eliminates the overshoot in the reference tracking response. Next, add a step response plot to compare the disturbance rejection performance of the two controllers. Select Add Plot > Input Disturbance Rejection. 11-26 Tune 2-DOF PID Controller (PID Tuner) PID Tuner tiles the disturbance-rejection plot side by side with the reference-tracking plot. 11-27 11 PID Controller Design The disturbance-rejection performance is identical with both controllers. Thus, using a 2DOF controller eliminates reference-tracking overshoot without any cost to disturbance rejection. You can improve disturbance rejection too by changing the PID Tuner design focus. First, click the Export arrow and select Save as Baseline again to set the 2-DOF controller as the baseline for comparison. Change the PID Tuner design focus to favor reference tracking without changing the response time or the transient-behavior coefficient. To do so, click the Focus menu, select Input disturbance rejection. 11-28 Options, and in Tune 2-DOF PID Controller (PID Tuner) PID Tuner automatically retunes the controller coefficients with a focus on disturbancerejection performance. 11-29 11 PID Controller Design With the default balanced design focus, PID Tuner selects a b value between 0 and 1. For this plant, when you change design focus to favor disturbance rejection, PID Tuner sets b = 0 and c = 0. Thus, PID Tuner automatically generates an I-PD controller to optimize for disturbance rejection. (Explicitly specifying an I-PD controller without setting the design focus yields a similar controller.) The response plots show that with the change in design focus, the disturbance rejection is further improved compared to the balanced 2-DOF controller. This improvement comes with some sacrifice of reference-tracking performance, which is slightly slower. However, the reference-tracking response still has no overshoot. Thus, using 2-DOF control can improve disturbance rejection without sacrificing as much reference tracking performance as 1-DOF control. These effects on system performance depend strongly on the properties of your plant and the speed of your controller. For some plants and some control bandwidths, using 2-DOF control or changing the design focus has less or no impact on the tuned result. See Also pidTuner More About 11-30 • “Designing PID Controllers with PID Tuner” • “Two-Degree-of-Freedom PID Controllers” on page 2-16 • “Tune 2-DOF PID Controller (Command Line)” on page 11-15 • “Analyze Design in PID Tuner” PID Controller Types for Tuning PID Controller Types for Tuning PID Tuner and the pidtune command can tune many PID and 2-DOF PID controller types. The term controller type refers to which terms are present in the controller action. For example, a PI controller has only a proportional and an integral term, while a PIDF controller contains proportional, integrator, and filtered derivative terms. This topic summarizes the types of PID controllers available for tuning with PID Tuner and pidtune. Specifying PID Controller Type To select the controller type, use one of these methods: • For command-line tuning, provide the type argument to the pidtune command. For example, C = pidtune(G,'PI') tunes a PI controller for plant G. • For tuning in PID Tuner: • Provide the type argument to the pidTuner command when you open PID Tuner. For example, pidTuner(G,'PIDF2') opens PID Tuner with an initial design that is a 2-DOF PID controller with a filter on the derivative term. • Provide the baseline-controller Cbase argument to the pidTuner command when you open PID Tuner. PID Tuner designs a controller of the same type as Cbase. For example, suppose C0 is a pid controller object that has proportional and derivative action only (PD controller). Then, pidTuner(G,C0) opens PID Tuner with an initial design that is a PD controller. • In PID Tuner, use the Type menu to change controller types. 11-31 11 PID Controller Design 11-32 PID Controller Types for Tuning 1-DOF Controllers The following table summarizes the available 1-DOF PID controller types and provides representative controller formulas for parallel form. The standard-form and discrete-time formulas are analogous. Type Controller Actions Continuous-Time Controller Formula (parallel form) Discrete-Time Controller Formula (parallel form, ForwardEuler integration method) P Proportional only Kp Kp I Integral only PI Proportional and integral PD Proportional and derivative PDF PID PIDF Proportional and derivative with first-order filter on derivative term Proportional, integral, and derivative Proportional, integral, and derivative with first-order filter on derivative term Ki s Kp + Ki Ki s K p + Kd s Ts z -1 K p + Ki Ts z -1 K p + Kd z-1 Ts K ds Tf s + 1 K p + Kd Kp + Ki + Kd s s K p + Ki Ts z -1 + Kd z -1 Ts Kp + Ki Kd s + s Tf s + 1 K p + Ki Ts + Kd z -1 Kp + 1 Tf + Ts z -1 1 Tf + 2-DOF Controllers PID Tuner can automatically design 2-DOF PID controller types with free setpoint weights. The following table summarizes the 2-DOF controller types in PID Tuner. The standard-form and discrete-time formulas are analogous. For more information about 211-33 Ts z -1 11 PID Controller Design DOF PID controllers generally, see “Two-Degree-of-Freedom PID Controllers” on page 216. Type Controller Actions PI2 2-DOF proportional and integral PD2 2-DOF proportional and derivative PDF2 2-DOF proportional and derivative with first-order filter on derivative term u = K p ( br - y ) + K d 2-DOF proportional, integral, and derivative u = K p ( br - y ) + Ki T ( r - y) +uK=dKs (pcr( br- y-)y ) + Ki s ( r - y) + z -1 s u = K p ( br - y ) + Ki T s y )i s ( r - y) + ( r - y) +uK=dK p ( br -( ycr) +- K s Tf s + 1 z -1 PID2 PIDF2 2-DOF proportional, integral, and derivative with first-order filter on derivative term Continuous-Time Controller Formula (parallel form) u = K p ( br - y ) + Discrete-Time Controller Formula (parallel form, ForwardEuler integration method) Ki T ( r - y) u = K p ( br - y ) + Ki s ( r - y) s z -1 z -1 u = K p ( br - y ) + K d s ( cr - y) u = K p ( br - y ) + K d ( cr - y) Ts s u ( cr= -Kyp)( br - y ) + K d Tf s + 1 1 T Tf + s z -1 2-DOF Controllers with Fixed Setpoint Weights With PID control, step changes in the reference signal can cause spikes in the control signal contributed by the proportional and derivative terms. By fixing the setpoint weights of a 2-DOF controller, you can mitigate the influence on the control signal exerted by changes in the reference signal. For example, consider the relationship between the inputs r (setpoint) and y (feedback) and the output u (control signal) of a continuous-time 2-DOF PID controller. u = K p ( br - y ) + 11-34 Ki ( r - y) + K d s ( cr - y) s ( cr See Also If you set b = 0 and c = 0, then changes in the setpoint r do not feed through directly to either the proportional or the derivative terms in u. The b = 0, c = 0 controller is called an I-PD type controller. I-PD controllers are also useful for improving disturbance rejection. Use PID Tuner to design the fixed-setpoint-weight controller types summarized in the following table. The standard-form and discrete-time formulas are analogous. Type Controller Actions Continuous-Time Discrete-Time Controller Formula Controller Formula (parallel form) (parallel form, ForwardEuler integration method) I-PD 2-DOF PID with b = 0, c = 0 u = -K p y + Ki T - K p y + K i s ( r - y) - K ( r - y ) -uK=d sy z -1 s I-PDF 2-DOF PIDF with b = 0, c = 0 u = -K p y + Ki T s ( r - y ) -uK=d - K p y + yK i s ( r - y) - K s Tf s + 1 z -1 ID-P 2-DOF PID with b = 0, c = 1 u = -K p y + Ki T ( r - y ) +uK=ds-(Kr p-yy+) K i s ( r - y) + K z -1 s IDF-P 2-DOF PIDF with b = 0, c = 1 u = -K p y + Ki T s ( r - y ) +uK=d - K p y + (Kr -i y)s ( r - y) + K s Tf s + 1 z -1 PI-D 2-DOF PID with b = 1, c = 0 u = K p (r - y) + Ki T ( r - uy)=-KKpd(syr - y ) + Ki s ( r - y) z-1 s PI-DF 2-DOF PIDF with b = 1, c = 0 u = K p (r - y) + Ki T s ( r - uy)=-KKpd( r - y ) + yKi s ( r - y) s Tf s + 1 z-1 See Also pidTuner | pidtune 11-35 11 PID Controller Design More About 11-36 • “Designing PID Controllers with PID Tuner” • “Proportional-Integral-Derivative (PID) Controllers” on page 2-13 • “Two-Degree-of-Freedom PID Controllers” on page 2-16 • “PID Controller Design at the Command Line” on page 11-2 • “PID Controller Design for Fast Reference Tracking” • “Tune 2-DOF PID Controller (Command Line)” on page 11-15 • “Tune 2-DOF PID Controller (PID Tuner)” on page 11-21 PID Controller Tuning in Simulink PID Controller Tuning in Simulink This example shows how to automatically tune a PID Controller block using PID Tuner. Introduction of the PID Tuner PID Tuner provides a fast and widely applicable single-loop PID tuning method for the Simulink® PID Controller blocks. With this method, you can tune PID controller parameters to achieve a robust design with the desired response time. A typical design workflow with the PID Tuner involves the following tasks: (1) Launch the PID Tuner. When launching, the software automatically computes a linear plant model from the Simulink model and designs an initial controller. (2) Tune the controller in the PID Tuner by manually adjusting design criteria in two design modes. The tuner computes PID parameters that robustly stabilize the system. (3) Export the parameters of the designed controller back to the PID Controller block and verify controller performance in Simulink. Open the Model Open the engine speed control model with PID Controller block and take a few moments to explore it. open_system('scdspeedctrlpidblock') 11-37 11 PID Controller Design Design Overview In this example, you design a PI controller in an engine speed control loop. The goal of the design is to track the reference signal from a Simulink step block scdspeedctrlpidblock/Speed Reference. The design requirement are: • Settling time under 5 seconds • Zero steady-state error to the step reference input. In this example, you stabilize the feedback loop and achieve good reference tracking performance by designing the PI controller scdspeedctrl/PID Controller in the PID Tuner. Open PID Tuner To launch the PID Tuner, double-click the PID Controller block to open its block dialog. In the Main tab, click Tune. Initial PID Design When the PID Tuner launches, the software computes a linearized plant model seen by the controller. The software automatically identifies the plant input and output, and uses the current operating point for the linearization. The plant can have any order and can have time delays. The PID Tuner computes an initial PI controller to achieve a reasonable tradeoff between performance and robustness. By default, step reference tracking performance displays in the plot. The following figure shows the PID Tuner dialog with the initial design: 11-38 PID Controller Tuning in Simulink Display PID Parameters Click Show parameters to view controller parameters P and I, and a set of performance and robustness measurements. In this example, the initial PI controller design gives a settling time of 2 seconds, which meets the requirement. 11-39 11 PID Controller Design Adjust PID Design in PID Tuner The overshoot of the reference tracking response is about 7.5 percent. Since we still have some room before reaching the settling time limit, you could reduce the overshoot by increasing the response time. Move the response time slider to the left to increase the closed loop response time. Notice that when you adjust response time, the response plot and the controller parameters and performance measurements update. The following figure shows an adjusted PID design with an overshoot of zero and a settling time of 4 seconds. The designed controller effectively becomes an integral-only controller. 11-40 PID Controller Tuning in Simulink 11-41 11 PID Controller Design Complete PID Design with Performance Trade-Off In order to achieve zero overshoot while reducing the settling time below 2 seconds, you need to take advantage of both sliders. You need to make control response faster to reduce the settling time and increase the robustness to reduce the overshoot. For example, you can reduce the response time from 3.4 to 1.5 seconds and increase robustness from 0.6 to 0.72. The following figure shows the closed-loop response with these settings: 11-42 PID Controller Tuning in Simulink 11-43 11 PID Controller Design Write Tuned Parameters to PID Controller Block After you are happy with the controller performance on the linear plant model, you can test the design on the nonlinear model. To do this, click Update Block in the PID Tuner. This action writes the parameters back to the PID Controller block in the Simulink model. The following figure shows the updated PID Controller block dialog: 11-44 PID Controller Tuning in Simulink Completed Design The following figure shows the response of the closed-loop system: The response shows that the new controller meets all the design requirements. You can also use the Control System Designer to design the PID Controller block, when the PID Controller block belongs to a multi-loop design task. See the example “Single Loop Feedback/Prefilter Compensator Design” (Simulink Control Design). bdclose('scdspeedctrlpidblock') 11-45 11 PID Controller Design Design PID Controller Using Estimated Frequency Response This example shows how to design a PI controller using a frequency response estimated from a Simulink model. This is an alternative PID design workflow when the linearized plant model is invalid for PID design (for example, when the plant model has zero gain). Open the Model Open the engine control model and take a few moments to explore it. mdl = 'scdenginectrlpidblock'; open_system(mdl) The PID loop includes a PI controller in parallel form that manipulates the throttle angle to control the engine speed. The PI controller has default gains that makes the closed loop system oscillate. We want to design the controller using the PID Tuner that is launched from the PID block dialog. open_system([mdl '/Engine Speed (rpm)']) sim(mdl) 11-46 Design PID Controller Using Estimated Frequency Response Close the scope. close_system([mdl '/Engine Speed (rpm)']) PID Tuner Obtaining a Plant Model with Zero Gain From Linearization In this example, the plant seen by the PID block is from throttle angle to engine speed. Linearization input and output points are already defined at the PID block output and the engine speed measurement respectively. Linearization at the initial operating point gives a plant model with zero gain. To verify the zero linearization, first obtain the linearization input and output points from the model. io = getlinio(mdl); Then, linearize the plant at its initial operating point. linsys = linearize(mdl,io) linsys = 11-47 11 PID Controller Design D = EngineSpeed Throttle Ang 0 Static gain. The reason for obtaining zero gain is that there is a triggered subsystem (Compression) in the linearization path and the analytical block-by-block linearization does not support event-based subsystems. Since PID Tuner uses the same approach to obtain a linear plant model, *PID Tuner also obtains a plant model with zero gain and rejects it during the launching process. To launch the PID Tuner, open the PID block dialog, and click Tune. An information dialog opens and indicates that the plant model linearized at the initial operating point has zero gain and cannot be used to design a PID controller. 11-48 Design PID Controller Using Estimated Frequency Response An alternative way to obtain a linear plant model is to directly estimate the frequency response data from the Simulink model, create an frd system in the MATLAB workspace, and import it back to PID Tuner to continue PID design. Obtain Estimated Frequency Response Data Using Sinestream Signals The sinestream input signal is the most reliable input signal for estimating an accurate frequency response of a Simulink model using the frestimate function. For more information on how to use frestimate, see “Frequency Response Estimation Using Simulation-Based Techniques” (Simulink Control Design). 11-49 11 PID Controller Design In this example, create a sine stream that sweeps frequency from 0.1 to 10 rad/sec with an amplitude of 1e-3. You can inspect the estimation results using the bode plot. Construct the sinestream signal. in = frest.Sinestream('Frequency',logspace(-1,1,50),'Amplitude',1e-3); Estimate the frequency response. This process can take a few minutes. sys = frestimate(mdl,io,in); Display the estimated frequency response. bode(sys) 11-50 Design PID Controller Using Estimated Frequency Response Design PI Controller sys is an frd system that represents the plant frequency response at the initial operating point. To use it in PID Tuner, we need to import it after PID Tuner is launched. Click Plant, and select Import. 11-51 11 PID Controller Design Click Importing an LTI system, and in the list, select sys. Then, click "OK" to import the frd system into PID Tuner. The automated design returns a stabilizing controller. Click Add Plot, and select Open-Loop Bode plot. The plot shows reasonable gain and phase margin. Click Show Parameters to see the gain and phase margin values. Time domain response plots are not available for frd plant models. 11-52 Design PID Controller Using Estimated Frequency Response 11-53 11 PID Controller Design To update the PID block P and I gains, click Update Block. Simulate Closed-Loop Performance in Simulink Model Simulation in Simulink shows that the new PI controller provides good performance when controlling the nonlinear model. 11-54 Design PID Controller Using Estimated Frequency Response Close the model. bdclose(mdl) 11-55 11 PID Controller Design Design Family of PID Controllers for Multiple Operating Points This example shows how to design an array of PID controllers for a nonlinear plant in Simulink that operates over a wide range of operating points. Open Plant Model The plant is a continuous stirred tank reactor (CSTR) that operates over a wide range of operating points. A single PID controller can effectively use the coolant temperature to regulate the output concentration around a small operating range that the PID controller is designed for. However, since the plant is a strongly nonlinear system, control performance degrades if operating point changes significantly. The closed-loop system can even become unstable. Open the CSTR plant model. mdl = 'scdcstrctrlplant'; open_system(mdl) 11-56 Design Family of PID Controllers for Multiple Operating Points For more information on this system, see [1]. Introduction to Gain Scheduling A common approach to solve the nonlinear control problem is using gain scheduling with linear controllers. Generally speaking, designing a gain scheduling control system takes four steps: 1 Obtain a plant model for each operating region. The usual practice is to linearize the plant at several equilibrium operating points. 11-57 11 PID Controller Design 2 Design a family of linear controllers, such as PID controllers, for the plant models obtained in the previous step. 3 Implement a scheduling mechanism such that the controller coefficients, such as PID gains, are changed based on the values of the scheduling variables. Smooth (bumpless) transfer between controllers is required to minimize disturbance to plant operation. 4 Assess control performance with simulation. For more information on gain scheduling, see [2]. This example focuses on designing a family of PID controllers for the nonlinear CSTR plant. Obtain Linear Plant Models for Multiple Operating Points The output concentration C is used to identify different operating regions. The CSTR plant can operate at any conversion rate between a low conversion rate (C = 9) and a high conversion rate (C = 2). In this example, divide the operating range into eight regions represented by C = 2 through 9. Specify the operating regions. C = [2 3 4 5 6 7 8 9]; Create an array of default operating point specifications. op = operspec(mdl,numel(C)); Initialize the operating point specifications by specifying that the output concentration is a known value, and specifying the output concentration value. for ct = 1:numel(C) op(ct).Outputs.Known = true; op(ct).Outputs.y = C(ct); end Compute the equilibrium operating points corresponding to the values of C. opoint = findop(mdl,op,findopOptions('DisplayReport','off')); Linearize the plant at these operating points. Plants = linearize(mdl,opoint); 11-58 Design Family of PID Controllers for Multiple Operating Points Since the CSTR plant is nonlinear, the linear models display different characteristics. For example, plant models with high and low conversion rates are stable, while the others are not. isstable(Plants,'elem')' ans = 1x8 logical array 1 1 0 0 0 0 1 1 Design PID Controllers for the Plant Models To design multiple PID controllers in batch, use the pidtune function. The following command generates an array of PID controllers in parallel form. The desired open-loop crossover frequency is at 1 rad/sec and the phase margin is the default value of 60 degrees. Controllers = pidtune(Plants,'pidf',pidtuneOptions('Crossover',1)); Display the controller for C = 4. Controllers(:,:,4) ans = 1 s Kp + Ki * --- + Kd * -------s Tf*s+1 with Kp = -12.4, Ki = -1.74, Kd = -16, Tf = 0.00875 Continuous-time PIDF controller in parallel form. To analyze the closed-loop responses for step setpoint tracking, first construct the closedloop systems. clsys = feedback(Plants*Controllers,1); Plot closed-loop responses 11-59 11 PID Controller Design figure hold on for ct = 1:length(C) % Select a system from the LTI array sys = clsys(:,:,ct); sys.Name = ['C=',num2str(C(ct))]; sys.InputName = 'Reference'; % Plot step response stepplot(sys,20); end legend('show','location','southeast') 11-60 Design Family of PID Controllers for Multiple Operating Points All the closed loops are stable, but the overshoots of the loops with unstable plants (C = 4, through 7) are too large. To improve the results, increase the target open-loop bandwidth to 10 rad/sec. Design updated controllers for the unstable plant models. Controllers = pidtune(Plants,'pidf',10); Display the controller for C = 4. Controllers(:,:,4) ans = 1 s Kp + Ki * --- + Kd * -------s Tf*s+1 with Kp = -283, Ki = -151, Kd = -128, Tf = 0.0183 Continuous-time PIDF controller in parallel form. Construct the closed-loop systems, and plot the closed-loop step responses for the new controllers. clsys = feedback(Plants*Controllers,1); figure hold on for ct = 1:length(C) % Select a system from the LTI array sys = clsys(:,:,ct); set(sys,'Name',['C=',num2str(C(ct))],'InputName','Reference'); % Plot step response stepplot(sys,20); end legend('show','location','southeast') 11-61 11 PID Controller Design All the closed-loop responses are now satisfactory. For comparison, examine the response when you use the same controller at all operating points. Create another set of closedloop systems, where each one uses the C = 2 controller, and plot their responses. clsys_flat = feedback(Plants*Controllers(:,:,1),1); figure stepplot(clsys,clsys_flat,20) legend('C-dependent Controllers','Single Controller') 11-62 Design Family of PID Controllers for Multiple Operating Points The array of PID controllers designed separately for each concentration gives considerably better performance than a single controller. However, the closed-loop responses shown above are computed based on linear approximations of the full nonlinear system. To validate the design, implement the scheduling mechanism in your model using the PID Controller block. Close the model. bdclose(mdl) 11-63 11 PID Controller Design References [1] Seborg, D. E., T. F. Edgar, and D. A. Mellichamp. Process Dynamics and Control, 2nd Ed., Wiley, 2004, pp. 34-36. [2] Rugh, W. J., and J. S. Shamma. "Research on gain scheduling." Automatica, Issue 36, 2000, pp. 1401-1425. 11-64 Design PID Controller Using Simulated I/O Data Design PID Controller Using Simulated I/O Data This example shows how to tune a PID controller for plants that cannot be linearized. You use PID Tuner to identify a plant for your model. Then tune the PID controller using the identified plant. This example uses a buck converter model that requires Simscape™ Electrical™ software. Buck Converter Model Buck converters convert DC to DC. This model uses a switching power supply to convert a 30V DC supply into a regulated DC supply. The converter is modeled using MOSFETs rather than ideal switches to ensure that device on-resistances are correctly represented. The converter response from reference voltage to measured voltage includes the MOSFET switches. PID design requires a linear model of the system from the reference voltage to the measured voltage. However, because of the switches, automated linearization results in a zero system. In this example, using PID Tuner, you identify a linear model of the system using simulation instead of linearization. For more information on the buck converter model, see the Simscape Electrical example elec_switching_power_supply. open_system('scdbuckconverter') sim('scdbuckconverter') 11-65 11 PID Controller Design The model is configured with a reference voltage that switches from 15 to 25 Volts at 0.004 seconds and a load current that is active from 0.0025 to 0.005 seconds. The controller is initialized with default gains and results in overshoot and slow settling time. open_system('scdbuckconverter/Scope 1') open_system('scdbuckconverter/Scope 2') 11-66 Design PID Controller Using Simulated I/O Data 11-67 11 PID Controller Design Simulate Model to Generate I/O Data To open the PID Tuner, in the Feedback controller subsystem, open the PID Controller block dialog, and click Tune. PID Tuner indicates that the model cannot be linearized and returned a zero system. 11-68 Design PID Controller Using Simulated I/O Data PID Tuner provides several alternatives when linearization fails. In the Plant drop-down list, you can select one of the following methods: • Import - Import a linear model from the MATLAB workspace. • Re-linearize Closed Loop - Linearize the model at different simulation snapshot times. • Identify New Plant - Identify a plant model using measured data. For this example, click Identify New Plant to open the Plant Identification tool. 11-69 11 PID Controller Design To open a tool that simulates the model to collect data for plant identification, on the Plant Identification tab, click Get I/O Data > Simulate Data. 11-70 Design PID Controller Using Simulated I/O Data On the Simulate I/O Data tab, you simulate the plant seen by the controller. The software temporarily: • Removes the PID Controller block from the model. • Injects a signal where the output of the PID block used to be. • Measures the resulting signal where the input to the PID block used to be. This data describes the response of the plant seen by the controller. The PID Tuner uses this response data to estimate a linear plant model. Configure the input signal as a step input with the following properties: 11-71 11 PID Controller Design • Sample Time ( ) = 5e-6 - Controller sample rate. • Offset ( ) = 0.51 - Output offset value that puts the converter in a state where the output voltage is near 15V and gives the operating point around which to tune the controller. • Onset Time ( ) = 0.003 - Delay to allow sufficient time for the converter to reach the 15V steady state before applying the step change. • Step Amplitude ( ) = 0.4 - Step size of the controller output (plant input) to apply to the model. This value is added to the offset value so that the actual plant input steps from 0.51 to 0.91. Note that the controller output (plant input) is limited to the range [0.01 0.95]. 11-72 Design PID Controller Using Simulated I/O Data Select Show Input Response, Show Offset Response, and Show Identification Data. Then, click the Run Simulation. The Plant Identification plot is updated. 11-73 11 PID Controller Design The red curve is the offset response. The offset response is the plant response to a constant input of . The response shows that the model has some transients with a constant input, in particular: • The [0 0.001] second range where the converter reaches the 15V steady state. Recall that this signal is the control error signal and hence drops to zero as steady state is reached. • The [0.0025 0.004] second range where the converter reacts to the current load being applied while the reference voltage is maintained at 15V. • The 0.004 second point where the reference voltage signal is changed from 15V to 25V resulting in a larger control error signal. 11-74 Design PID Controller Using Simulated I/O Data • The [0.005 0.006] second range where the converter reacts to the current load being removed. The blue curve shows the complete plant response that contains the contributions from the initial transients (significant for times < 0.001 seconds), the response to the cyclic current load (time durations 0.0025 to 0.005 seconds), reference voltage change (at 0.004 seconds), and response to the step test signal (applied at time 0.003 seconds). In contrast, the red curve is the response to only the initial transients, reference voltage step, and cyclic current load. The green curve is the data that will be used for plant identification. This curve is the change in response due to the step test signal, which is the difference between the blue (input response) and red (offset response) curves taking into account the negative feedback sign. To use the measured data to identify a plant model, click Apply. Then, to return to plant identification, click Close. Plant Identification PID Tuner identifies a plant model using the data generated by simulating the model. You tune the identified plant parameters so that the identified plant response, when provided the measured input, matches the measured output. 11-75 11 PID Controller Design You can manually adjust the estimated model. Click and drag the plant curve and pole location (X) to adjust the identified plant response so that it matches the identification data as closely as possible. 11-76 Design PID Controller Using Simulated I/O Data To tune the identified plant using automated identification, click Auto Estimate. The automated tuning response is not much better than the interactive tuning. The identified plant and identification data do not match well. Change the plant structure to get a better match: • In the Structure drop-down list, select Underdamped pair. • Click and drag the 2nd order envelope to match the identified data as closely as possible (almost critically damped). • Click Auto Estimate to fine tune the plant model. 11-77 11 PID Controller Design 11-78 Design PID Controller Using Simulated I/O Data To designate the identified model as the current plant for controller tuning, Click Apply. PID Tuner then automatically tunes a controller for the identified plant and updates the Reference Tracking step plot. Controller Tuning The PID Tuner automatically tunes a PID controller for the identified plant. The tuned controller response has about 5% overshoot and a settling time of around 6 seconds. Click the Reference Tracking step plot to make it the current figure. 11-79 11 PID Controller Design The controller output is the duty cycle for the PWM system and must be limited to [0.01 0.95]. To confirm that the controller output satisfies these bounds, create a controller effort plot. On the PID Tuner tab, in the Add Plot drop-down list, under Step, click Controller effort. Move the newly created Controller effort plot to the second plot group. 11-80 Design PID Controller Using Simulated I/O Data In the Controller effort plot, the tuned response (solid line) shows a large control effort required at the start of the simulation. To achieve a settling time of about 4 seconds and overshoot of 9%, adjust the Response Time and Transient Behavior sliders. These adjustments reduce the maximum control effort to the acceptable range. 11-81 11 PID Controller Design To update the Simulink block with the tuned controller values, click Update Block. To confirm the PID controller performance, simulate the Simulink model. 11-82 Design PID Controller Using Simulated I/O Data bdclose('scdbuckconverter') 11-83 12 Classical Control Design • “Choosing a Control Design Approach” on page 12-2 • “Control System Designer Tuning Methods” on page 12-4 • “Design Requirements” on page 12-9 • “Feedback Control Architectures” on page 12-21 • “Design Multiloop Control System” on page 12-24 • “Multimodel Control Design” on page 12-35 • “Bode Diagram Design” on page 12-49 • “Root Locus Design” on page 12-64 • “Nichols Plot Design” on page 12-80 • “Edit Compensator Dynamics” on page 12-93 • “Design Compensator Using Automated Tuning Methods” on page 12-100 • “Analyze Designs Using Response Plots” on page 12-117 • “Compare Performance of Multiple Designs” on page 12-127 • “Design Hard-Disk Read/Write Head Controller” on page 12-133 • “Design Compensator for Plant Model with Time Delays” on page 12-147 • “Design Compensator for Systems Represented by Frequency Response Data” on page 12-155 • “Design Internal Model Controller for Chemical Reactor Plant” on page 12-161 • “Design LQG Tracker Using Control System Designer” on page 12-176 • “Export Design to MATLAB Workspace” on page 12-188 • “Generate Simulink Model for Control Architecture” on page 12-191 • “Tune Simulink Blocks Using Compensator Editor” on page 12-193 • “Single Loop Feedback/Prefilter Compensator Design” on page 12-201 • “Cascaded Multiloop Feedback Design” on page 12-211 • “Reference Tracking of DC Motor with Parameter Variations” on page 12-222 12 Classical Control Design Choosing a Control Design Approach Control System Toolbox provides several approaches to tuning control systems. Use the following table to determine which approach best supports what you want to do. PID Tuning Classical Control Design Architecture PID loops with unit Control systems feedback (1-DOF and having SISO 2-DOF) controllers in common single-loop or multiloop configurations (see “Feedback Control Architectures” on page 12-21) Any architecture, including any number of SISO or MIMO feedback loops Control Design Approach Automatically tune • Graphically tune PID gains to balance poles and zeros performance and on design plots, robustness such as Bode, root locus, and Nichols Automatically tune controller parameters to meet design requirements you specify, such as setpoint tracking, stability margins, disturbance rejection, and loop shaping (see “Tuning Goals”) • Automatically tune compensators using response optimization (Simulink Design Optimization™), LQG synthesis, or IMC tuning Analysis of Control Time and frequency System responses for Performance reference tracking and disturbance rejection 12-2 Multiloop, Multiobjective Tuning Any combination of system responses Any combination of system responses See Also PID Tuning Interface Classical Control Design • Graphical tuning Graphical tuning using PID Tuner using Control (see “Designing System Designer PID Controllers with PID Tuner”) • Programmatic tuning using pidtune (see “PID Controller Design at the Command Line” on page 11-2) Multiloop, Multiobjective Tuning • Graphical tuning using Control System Tuner • Programmatic tuning using systune (see “Programmatic Tuning”) See Also More About • “PID Controller Tuning” • “Classical Control Design” • “Tuning with Control System Tuner” • “Programmatic Tuning” 12-3 12 Classical Control Design Control System Designer Tuning Methods Using Control System Designer, you can tune compensators using various graphical and automated tuning methods. Graphical Tuning Methods Use graphical tuning methods to interactively add, modify, and remove controller poles, zeros, and gains. Tuning Method Description Useful For Bode Editor Tune your compensator to achieve Adjusting open-loop bandwidth a specific open-loop frequency and designing to gain and phase response (loop shaping). margin specifications. Closed-Loop Bode Editor Tune your prefilter to improve closed-loop system response. Root Locus Editor Tune your compensator to produce Designing to time-domain design closed-loop pole locations that specifications, such as maximum satisfy your design specifications. overshoot and settling time. Nichols Editor Tune your compensator to achieve Adjusting open-loop bandwidth a specific open-loop response (loop and designing to gain and phase shaping), combining gain and margin specifications. phase information on a Nichols plot. Improving reference tracking, input disturbance rejection, and noise rejection. When using graphical tuning, you can modify the compensator either directly from the editor plots or using the compensator editor. A common design approach is to roughly tune your compensator using the editor plots, and then use the compensator editor to fine-tune the compensator parameters. For more information, see “Edit Compensator Dynamics” on page 12-93 The graphical tuning methods are not mutually exclusive. For example, you can tune your compensator using both the Bode editor and root locus editor simultaneously. This option is useful when designing to both time-domain and frequency-domain specifications. For examples of graphical tuning, see the following: 12-4 Control System Designer Tuning Methods • “Bode Diagram Design” on page 12-49 • “Root Locus Design” on page 12-64 • “Nichols Plot Design” on page 12-80 Automated Tuning Methods Use automated tuning methods to automatically tune compensators based on your design specifications. Tuning Method Description Requirements and Limitations PID Tuning Automatically tune PID gains to balance performance and robustness or tune controllers using classical PID tuning formulas. Classical PID tuning formulas require a stable or integrating effective plant. Optimization Based Tuning Optimize compensator parameters Requires Simulink Design using design requirements Optimization software. specified in graphical tuning and Tunes the parameters of a analysis plots. previously defined controller structure. LQG Synthesis Design a full-order stabilizing Maximum controller order feedback controller as a lineardepends on the effective plant quadratic-Gaussian (LQG) tracker. dynamics. Loop Shaping Find a full-order stabilizing feedback controller with a specified open-loop bandwidth or shape. Requires Robust Control Toolbox software. Maximum controller order depends on the effective plant dynamics. 12-5 12 Classical Control Design Tuning Method Description Requirements and Limitations Internal Model Control (IMC) Tuning Obtain a full-order stabilizing Assumes that your control system feedback controller using the IMC uses an IMC architecture that design method. contains a predictive model of your plant dynamics. Maximum controller order depends on the effective plant dynamics. A common design approach is to generate an initial compensator using PID tuning, LQG synthesis, loop shaping, or IMC tuning. You can then improve the compensator performance using either optimization-based tuning or graphical tuning. For more information on automated tuning methods, see “Design Compensator Using Automated Tuning Methods” on page 12-100. Effective Plant for Tuning An effective plant is the system controlled by a compensator that contains all elements of the open loop in your model other than the compensator you are tuning. The following diagrams show examples of effective plants: Knowing the properties of the effective plant seen by your compensator can help you understand which tuning methods work for your system. For example, some automated Ÿ tuning methods apply only to compensators whose open loops ( L = C P ) have stable 12-6 Control System Designer Tuning Methods Ÿ effective plants ( P ). Also, for tuning methods such as IMC and loop shaping, the maximum controller order depends on the dynamics of the effective plant. Select a Tuning Method To select a tuning method, in Control System Designer, click Tuning Methods. 12-7 12 Classical Control Design See Also Control System Designer Related Examples 12-8 • “Bode Diagram Design” on page 12-49 • “Root Locus Design” on page 12-64 • “Nichols Plot Design” on page 12-80 • “Design Compensator Using Automated Tuning Methods” on page 12-100 Design Requirements Design Requirements This topic describes time-domain and frequency-domain design requirements available in Control System Designer. Each requirement defines an exclusion region, indicated by a yellow shaded area. To satisfy a requirement, a response plot must remain outside of the associated exclusion region. 12-9 12 Classical Control Design If you have Simulink Design Optimization software installed, you can use response optimization techniques to find a compensator that meets your specified design requirements. For examples of optimization-based control design using design requirements, see “Optimize LTI System to Meet Frequency-Domain Requirements” (Simulink Design Optimization) and “Design Optimization-Based PID Controller for Linearized Simulink Model (GUI)” (Simulink Design Optimization). For other Control System Designer tuning methods, you can use the specified design requirements as visual guidelines during the tuning process. Add Design Requirements You can add design requirements either directly to existing plots or, when using optimization-based tuning, from the Response Optimization dialog box. Add Requirements to Existing Plots You can add design requirements directly to existing: • Bode, root locus, and Nichols editor plots. • Analysis plots: • Root locus plots and pole/zero maps • Bode diagrams • Nichols plots • Step and impulse responses To add a design requirement to a plot, in Control System Designer, right-click the plot, and select Design Requirements > New. 12-10 Design Requirements In the New Design Requirement dialog box, in the Design requirement type drop-down list, select the type of requirement to add. You can select any valid requirement for the associated plot type. In the Design requirement parameters section, configure the requirement properties. Parameters are dependent on the type of requirement you select. To create the specified requirement and add it to the plot, click OK. Add Requirements from Response Optimization Dialog Box When using optimization-based tuning, you can add design requirements from the Response Optimization dialog box. To do so, on the Design Requirements tab, click Add new design requirement. 12-11 12 Classical Control Design In the New Design Requirement dialog box, select a Design requirement type from the drop-down list. In the Requirement for response drop-down list, specify the response to which to apply the design requirement. You can select any response in Data Browser. In the Design requirement parameters section, configure the requirement properties. Parameters are dependent on the type of requirement you select. To create the specified design requirement, click OK. In the Response Optimization dialog box, on the Design Requirements tab, the new requirement is added to the table. The app also adds the design requirement to a corresponding editor or analysis plot. The plot type used depends on the selected design requirement type. If the requirement is for a Bode, root locus, or Nichols plot and: • A corresponding editor plot is open, the requirement is added to that plot. • Only a corresponding analysis plot is open, the requirement is added to that plot. • No corresponding plot is open, the requirement is added to a new Editor plot. Otherwise, if the requirement is for a different plot type, the requirement is added to an appropriate analysis plot. For example, a Step requirement bound is added to a new step analysis plot. 12-12 Design Requirements Edit Design Requirements To edit an existing requirement, in Control System Designer, right-click the corresponding plot, and select Design Requirements > Edit. In the Edit Design Requirement dialog box, in the Design requirement drop-down list, select a design requirement to edit. You can select any existing design requirement from the current plot. In the Design requirement parameters section, specify the requirement properties. Parameters are dependent on the type of requirement you select. When you change a parameter, the app automatically updates the requirement display in the associated plot. You can also interactively adjust design requirements by dragging the edges or vertices of the shaded exclusion region in the associated plot. 12-13 12 Classical Control Design Root Locus and Pole-Zero Plot Requirements Settling Time Specifying a settling time for a continuous-time system adds a vertical boundary line to the root locus or pole-zero plot. This line represents pole locations associated with the specified settling time. This boundary is exact for a second-order system with no zeros. For higher order systems, the boundary is an approximation based on second-order dominant systems. To satisfy this requirement, your system poles must be to the left of the boundary line. For a discrete-time system, the design requirement boundary is a curved line centered on the origin. In this case, your system poles must be within the boundary line to satisfy the requirement. Percent Overshoot Specifying percent overshoot for a continuous-time system adds two rays to the plot that start at the origin. These rays are the locus of poles associated with the specified overshoot value. In the discrete-time case, the design requirement adds two curves originating at (1,0) and meeting on the real axis in the left-hand plane. 12-14 Design Requirements Note The percent overshoot (p.o.) design requirement can be expressed in terms of the damping ratio, ζ: Ê pz p.o. = 100 exp Á Á 1- z 2 Ë ˆ ˜ ˜ ¯ Damping Ratio Specifying a damping ratio for a continuous-time system adds two rays to the plot that start at the origin. These rays are the locus of poles associated with the specified overshoot value. This boundary is exact for a second-order system and, for higher order systems, is an approximation based on second-order dominant systems. To meet this requirement, your system poles must be to the left of the boundary lines. For discrete-time systems, the design requirement adds two curves originating at (1,0) and meeting on the real axis in the left-hand plane. In this case, your system poles must be within the boundary curves. Natural Frequency Specifying a natural frequency bound adds a semicircle to the plot that is centered around the origin. The radius of the semicircle equals the natural frequency. If you specify a natural frequency lower bound, the system poles must remain outside this semicircle. If you specify a natural frequency upper bound, the system poles must remain within this semicircle. Region Constraint To specify a region constraint, define two or more vertices of a piece-wise linear boundary line. For each vertex, specify Real and Imaginary components. This requirement adds a shaded exclusion region on one side of the boundary line. To switch the exclusion region to the opposite side of the boundary, in the response plot, right-click the requirement, and select Flip. To satisfy this requirement, your system poles must be outside of the exclusion region. 12-15 12 Classical Control Design Open-Loop and Closed-Loop Bode Diagram Requirements Upper Gain Limit You can specify upper gain limits for both open-loop and closed-loop Bode responses. A gain limit consists of one or more line segments. For the start and end points of each segment, specify a frequency, Freq, and magnitude, Mag. You can also specify the slope of the line segment in dB/decade. When you change the slope, the magnitude for the end point updates. If you are using optimization-based tuning, you can assign a tuning Weight to each segment to indicate their relative importance. In the Type drop-down list you can select whether to constrain the magnitude to be above or below the specified boundary. Lower Gain Limit You can specify lower gain limits in the same way as upper gain limits. Gain and Phase Margin You can specify a lower bound for the gain margin, the phase margin, or both. The specified bounds appear in text on the Bode magnitude plot. Note Gain and phase margin requirements are only applicable to open-loop Bode diagrams. 12-16 Design Requirements Open-Loop Nichols Plot Requirements Phase Margin Specify a minimum phase margin as a positive value. Graphically, Control System Designer displays this requirement as a region of exclusion along the 0 dB open-loop gain axis. Gain Margin Specify a minimum gain margin value. Graphically, Control System Designer displays this requirement as a region of exclusion along the -180 degree open-loop phase axis. Closed-Loop Peak Gain Specify a minimum closed-loop peak gain value. The specified dB value can be positive or negative. The design requirement follows the curves of the Nichols plot grid. As a best practice, have the grid on when using a closed-loop peak gain requirement. Gain-Phase Design Requirement To specify a gain-phase design requirement, define two or more vertices of a piece-wise linear boundary line. For each vertex, specify Open-Loop phase and Open-Loop gain values. This requirement adds a shaded exclusion region on one side of the boundary line. To switch the exclusion region to the opposite side of the boundary, in the Nichols plot, right-click the requirement, and select Flip. Display Location When editing a phase margin, gain margin, or closed-loop peak gain requirement, you can specify the display location as -180 ± k360 degrees, where k is an integer value. 12-17 12 Classical Control Design If you enter an invalid location, the closest valid location is selected. While displayed graphically at only one location, these requirements apply regardless of actual phase; that is, they are applied for all values of k. Step and Impulse Response Requirements Upper Time Response Bound You can specify upper time response bounds for both step and impulse responses. A time-response bound consists of one or more line segments. For the start and end points of each segment, specify a Time and Amplitude value. You can also specify the slope of the line segment. When you change the slope, the amplitude for the end point updates. 12-18 Design Requirements If you are using optimization-based tuning, you can assign a tuning Weight to each segment to indicate its relative importance. In the Type drop-down list, you can select whether to constrain the response to be above or below the specified boundary. Lower Time Response Bound You can specify lower time response bounds for both step and impulse responses in the same way as upper gain limits. Step Response Bound For a step response plot, you can also specify a step response bound design requirement. To define a step response bound requirement, specify the following step response parameters: • Final value — Final steady-state value • Rise time — Time required to reach the specified percentage, % Rise, of the Final value • Settling time — Time at which the response enters and stays within the settling percentage, % Settling, of the Final value • % Overshoot — Maximum percentage overshoot above the Final value • % Undershoot — Maximum percentage undershoot below the Initial value In Control System Designer, step response plots always use an Initial value and a Step time of 0 12-19 12 Classical Control Design See Also More About 12-20 • “Optimize LTI System to Meet Frequency-Domain Requirements” (Simulink Design Optimization) • “Design Optimization-Based PID Controller for Linearized Simulink Model (GUI)” (Simulink Design Optimization) Feedback Control Architectures Feedback Control Architectures When you open Control System Designer from MATLAB, you can select one of six possible control architecture configurations. Configura Block Diagram tion Features 1 • Single feedback loop • Compensator (C) and plant (G) in forward path • Sensor dynamics (H) in feedback path • Prefilter F 2 • Single feedback loop • Compensator (C) and sensor dynamics (H) in feedback path • Prefilter F 3 • Compensator (C) and plant (G) in forward path • Feedforward prefilter F for input disturbance attenuation • Sensor dynamics (H) in feedback path 12-21 12 Classical Control Design Configura Block Diagram tion Features 4 • Nested multiloop architecture • Outer loop with compensator (C1) in forward path • Inner loop with compensator (C2) in feedback path • Sensor dynamics (H) in feedback path 5 • Standard Internal Model Control (IMC) architecture • Compensator (C) in forward path • Plant G1 and plant predictive model G2 • Disturbance model Gd • Prefilter F 6 • Cascaded multiloop architecture with the inner loop in the forward path of the outer loop. • Compensator (C1, C2) in the forward path and sensor dynamics (H1, H2) in the feedback path (both loops) • Prefilter F If your control application does not match one of the supported control architectures, you can use block diagram algebra to convert your system to match an architecture. For an example of such an application, see “Design Multiloop Control System” on page 12-24. Note If you are unable to match your application to one of the supported control architectures, consider using the Control System Tuner app to design your control system. 12-22 See Also See Also Control System Designer | sisoinit More About • “Design Multiloop Control System” on page 12-24 12-23 12 Classical Control Design Design Multiloop Control System In many applications, a single-loop control system is not feasible due to your plant design or design requirements. If you have a design with an inner and outer loop, you can use Control System Designer to design compensators for both loops. The typical workflow is to tune the compensator for the inner loop first, by isolating the inner loop from the rest of the control system. Once the inner loop is satisfactorily tuned, tune the outer loop to achieve your desired closed-loop response. System Model For this example develop a position control system for a DC motor. A single-loop angular velocity controller is designed in “Bode Diagram Design” on page 12-49. To design an angular position controller, add an outer loop that contains an integrator. Define a state-space plant model, as described in “SISO Example: The DC Motor”. % Define the motor parameters R = 2.0 L = 0.5 Km = .015 Kb = .015 Kf = 0.2 J = 0.02 % Create the state-space model A = [-R/L -Kb/L; Km/J -Kf/J] 12-24 Design Multiloop Control System B = [1/L; 0]; C = [0 1]; D = [0]; sys_dc = ss(A,B,C,D); Design Objectives The design objective is to minimize the closed-loop step response settling time, while maintaining an inner-loop phase margin of at least 65 degrees with maximum bandwidth: • Minimal closed-loop step response settling time. • Inner-loop phase margin of at least 65 degrees. • Maximum inner-loop bandwidth. Match System To Control Architecture Control System Designer has six possible control architectures from which you can choose. For more information on these architectures, see “Feedback Control Architectures” on page 12-21. For this example use Configuration 4, which has an inner and outer control loop. Currently, the control system structure does not match Configuration 4. However, using block diagram algebra, you can modify the system model by adding: • An integrator to the motor output to get the angular displacement. • A differentiator to the inner-loop feedback path. 12-25 12 Classical Control Design At the MATLAB command line, add the integrator to the motor plant model. plant = sys_dc*tf(1,[1,0]); Create an initial model of the inner-loop compensator that contains the feedback differentiator. Cdiff = tf('s'); Define Control Architecture Open Control System Designer. controlSystemDesigner In Control System Designer, on the Control System tab, click Edit Architecture. In the Edit Architecture dialog box, under Select Control Architecture, click the fourth architecture. 12-26 Design Multiloop Control System Import the plant and controller models from the MATLAB workspace. In the Blocks tab, for: • Controller C2, specify a Value of Cdiff. • Plant G, specify a Value of plant. Click OK. The app updates the control architecture and imports the specified models for the motor plant and the inner-loop controller. In Control System Designer, the following plots open: 12-27 12 Classical Control Design • Bode Editor for LoopTransfer_C1 — Open-loop Bode Editor for the outer loop • Root Locus Editor for LoopTransfer_C1 — Open-loop Root Locus Editor for the outer loop • Bode Editor for LoopTransfer_C2 — Open-loop Bode Editor for the inner loop • Root Locus Editor for LoopTransfer_C2 — Open-loop root Locus Editor for the inner loop • IOTransfer_r2y: step — Overall closed-loop step response from input r to output y For this example, close the Bode Editor for LoopTransfer_C1 and Root Locus Editor for LoopTransfer_C2 plots. Since the inner loop is tuned first, configure the plots to view just the inner-loop Bode editor plot. On the View tab, click Single, and click Bode Editor for LoopTransfer_C2. Isolate Inner Loop To isolate the inner loop from the rest of the control system architecture, add a loop opening to the open-loop response of the inner loop. In the Data Browser, right-click LoopTransfer_C2, and select Open Selection. To add a loop opening at the output of outer-loop compensator, C1, in the Open-Loop Transfer Function dialog box, click uC1. 12-28 Add loop opening location to list. Then, select Design Multiloop Control System Click OK. The app adds a loop opening at the selected location. This opening removes the effect of the outer control loop on the open-loop transfer function of the inner loop. The Bode Editor response plot updates to reflect the new open-loop transfer function. 12-29 12 Classical Control Design Tune Inner Loop To increase the bandwidth of the inner loop, increase the gain of compensator C2. In the Bode Editor plot, drag the magnitude response upward until the phase margin is 65 degrees. This corresponds to a compensator gain of 107. Increasing the gain further reduces the phase margin below 65 degrees. 12-30 Design Multiloop Control System Alternatively, you can adjust the gain value using the compensator editor. For more information, see “Edit Compensator Dynamics” on page 12-93. Tune Outer Loop With the inner loop tuned, you can now tune the outer loop to reduce the closed-loop settling time. 12-31 12 Classical Control Design In Control System Designer, on the View tab, select Left/Right. Arrange the plots to display the Root Locus for LoopTransfer_C1 and IOTransfer_r2y_step plots simultaneously. To view the current settling time, right-click in the step response plot and select Characteristics > Settling Time. The current closed-loop settling time is greater than 500 seconds. In the Root Locus Editor, increase the gain of compensator C1. As the gain increases, the complex pole pair moves toward a slower time constant and the real pole moves 12-32 See Also toward a faster time constant. A gain of 600 produces a good compromise between rise time and settling time. With a closed-loop settling time below 0.8 seconds and an inner-loop phase margin of 65 degrees, the design satisfies the design requirements. See Also Control System Designer 12-33 12 Classical Control Design More About • 12-34 “Feedback Control Architectures” on page 12-21 Multimodel Control Design Multimodel Control Design Typically, the dynamics of a system are not known exactly and may vary. For example, system dynamics can vary because of: • Parameter value variations caused by manufacturing tolerances — For example, the resistance value of a resistor is typically within a range about the nominal value, 5 Ω +/– 5%. • Operating conditions — For example, aircraft dynamics change based on altitude and speed. Any controller you design for such a system must satisfy the design requirements for all potential system dynamics. Control Design Overview To design a controller for a system with varying dynamics: 1 Sample the variations. 2 Create an LTI model for each sample. 3 Create an array of sampled LTI models. 4 Design a controller for a nominal representative model from the array. 5 Analyze the controller design for all models in the array. 6 If the controller design does not satisfy the requirements for all the models, specify a different nominal model and redesign the controller. Model Arrays In Control System Designer, you can specify multiple models for any plant or sensor in the current control architecture using an array of LTI models (see “Model Arrays” on page 2-100). If you specify model arrays for more than one plant or sensor, the lengths of the arrays must match. Create Model Arrays To create arrays for multimodel control design, you can: • Create multiple LTI models using the tf, ss, zpk, or frd commands. 12-35 12 Classical Control Design % Specify model parameters. m = 3; b = 0.5; k = 8:1:10; T = 0.1:.05:.2; % Create an array of LTI models. for ct = 1:length(k); G(:,:,ct) = tf(1,[m,b,k(ct)]); end • Create an array of LTI models using the stack command. % Create individual LTI models. G1 = tf(1, [1 1 8]); G2 = tf(1, [1 1 9]); G3 = tf(1, [1 1 10]); % Combine models in an array. G = stack(1,G1,G2,G3); • Perform batch linearizations at multiple operating points. Then export the computed LTI models to create an array of LTI models. See the example “Reference Tracking of DC Motor with Parameter Variations” (Simulink Control Design). (Requires Simulink Control Design software) • Sample an uncertain state-space (uss) model using usample. • Compute a uss model from a Simulink model. Then use usubs or usample to create an array of LTI models. See “Obtain Uncertain State-Space Model from Simulink Model” (Robust Control Toolbox). (Requires Robust Control Toolbox software) • Specify a core Simulink block to linearize to a uss or ufrd model. See “Specify Uncertain Linearization for Core or Custom Simulink Blocks” (Robust Control Toolbox). (Requires Simulink Control Design and Robust Control Toolbox software) Import Model Arrays to Control System Designer To import models as arrays, you can pass them as input arguments when opening Control System Designer from the MATLAB command line. For more information, see Control System Designer. You can also import model arrays into Control System Designer when configuring the control architecture. In the Edit Architecture dialog box: • In the Value text box, specify the name of an LTI model from the MATLAB workspace. • To import block data from the MATLAB workspace or from a MAT-file in your current working directory, click 12-36 . Multimodel Control Design Nominal Model What Is a Nominal Model? The nominal model is a representative model in the array of LTI models that you use to design the controller in Control System Designer. Use the editor and analysis plots to visualize and analyze the effect of the controller on the remaining plants in the array. You can select any model in the array as your nominal model. For example, you can choose a model that: • Represents the expected nominal operating point of your system. 12-37 12 Classical Control Design • Is an average of the models in the array. • Represents a worst-case plant. • Lies closest to the stability point. Tip You can plot and analyze the open-loop dynamics of the system on a Bode plot to determine which model to choose as nominal. Specify Nominal Model To select a nominal model from the array of LTI models, in Control System Designer, click Multimodel Configuration. Then, in the Multimodel Configuration dialog box, select a Nominal model index. The default index is 1. For each plant or sensor that is defined as a model array, the app selects the model at the specified index as the nominal model. Otherwise, the app uses scalar expansion to apply the single LTI model for all model indices. For example, for the following control architecture: if G and H are both three-element arrays and the nominal model index is 2, the software uses the second element in both the arrays to compute the nominal model: Nominal Model r y 2 2 The nominal response from r to y is: 12-38 Multimodel Control Design T= CG2 1 + CG2 H2 The app also computes and plots the responses showing the effect of C on the remaining pairs of plant and sensor models — G1H1 and G3H3. If only G is an array of LTI models, and the specified nominal model is 2, then the control architecture for nominal response is: Nominal Model r y 2 In this case, the nominal response from r to y is: T= CG2 1 + CG2 H The app also computes and plots the responses showing the effect of C on the remaining pairs of plant and sensor model — G1H and G3H. Frequency Grid The frequency response of a system is computed at a series of frequency values, called a frequency grid. By default, Control System Designer computes a logarithmically equally spaced grid based on the dynamic range of each model in the array. Specify a custom frequency grid when: • The automatic grid has more points than you require. To improve computational efficiency, specify a less dense grid spacing. • The automatic grid is not sufficiently dense within a particular frequency range. For example, if the response does not capture the resonant peak dynamics of an underdamped system, specify a more dense grid around the corner frequency. 12-39 12 Classical Control Design • You are only interested in the response within specific frequency ranges. To improve computational efficiency, specify a grid that covers only the frequency ranges of interest. For more information on specifying logarithmically spaced vectors, see logspace. Note Modifying the frequency grid does not affect the frequency response computation for the nominal model. The app always uses the Auto select option to compute the nominal model frequency response. Design Controller for Multiple Plant Models This example shows how to design a compensator for a set of plant models using Control System Designer. 1 Create Array of Plant Models Create an array of LTI plant models using the stack command. % Create an array of LTI models to model plant (G) variations. G1 = tf(1,[1 1 8]); G2 = tf(1,[1 1 9]); G3 = tf(1,[1 1 10]); G = stack(1,G1,G2,G3); 2 Create Array of Sensor Models Similarly, create an array of sensor models. H1 = tf(1,[1/0.1,1]); H2 = tf(1,[1/0.15,1]); H3 = tf(1,[1/0.2,1]); H = stack(1,H1,H2,H3); 3 Open Control System Designer Open Control System Designer, and import the plant and sensor model arrays. controlSystemDesigner(G,1,H) 12-40 Multimodel Control Design 4 The app opens and imports the plant and sensor model arrays. Configure Analysis Plot To view the closed-loop step response in a larger plot, in Control System Designer, on the View tab, click Single. Then, click IOTransfer_r2y: step. 12-41 12 Classical Control Design By default the step response shows only the nominal response. To display the individual responses for the other model indices, right-click the plot area, and select Multimodel Configuration > Individual Responses. 12-42 Multimodel Control Design Note To view an envelope of all model responses, right-click the plot area, and select Multimodel Configuration > Bounds The plot updates to display the responses for the other models. 5 Select Nominal Model On the Control System tab, click Multimodel Configuration. 12-43 12 Classical Control Design In the Multimodel Configuration dialog box, specify a Nominal Model Index of 2. Click Close. 12-44 Multimodel Control Design 6 The selected nominal model corresponds to the average system response. Design Compensator To design a compensator using the nominal model, you can use any of the supported “Control System Designer Tuning Methods” on page 12-4. 12-45 12 Classical Control Design For this example, use the Compensator Editor to manually specify the compensator dynamics. Add an integrator to the compensator and set the compensator gain to 0.4. For more information, see “Edit Compensator Dynamics” on page 12-93. 7 Analyze Results The tuned controller produces a step response with minimal overshoot for the nominal models and a worst-case overshoot less than 10%. 12-46 See Also See Also Control System Designer Related Examples • “Model Arrays” on page 2-100 12-47 12 Classical Control Design • 12-48 “Control System Designer Tuning Methods” on page 12-4 Bode Diagram Design Bode Diagram Design Bode diagram design is an interactive graphical method of modifying a compensator to achieve a specific open-loop response (loop shaping). To interactively shape the open-loop response using Control System Designer , use the Bode Editor. In the editor, you can adjust the open-loop bandwidth and design to gain and phase margin specifications. To adjust the loop shape, you can add poles and zeros to your compensator and adjust their values directly in the Bode Editor, or you can use the Compensator Editor. For more information, see “Edit Compensator Dynamics” on page 12-93. For information on all of the tuning methods available in Control System Designer, see “Control System Designer Tuning Methods” on page 12-4. Tune Compensator For DC Motor Using Bode Diagram Graphical Tuning This example shows how to design a compensator for a DC motor using Bode diagram graphical tuning techniques. Plant Model and Requirements The transfer function of the DC motor plant, as described in “SISO Example: The DC Motor”, is: G= 1.5 2 s + 14 s + 40 .02 For this example, the design requirements are: • Rise time of less than 0.5 seconds • Steady-state error of less than 5% • Overshoot of less than 10% • Gain margin greater than 20 dB • Phase margin greater than 40 degrees 12-49 12 Classical Control Design Open Control System Designer At the MATLAB command line, create a transfer function model of the plant, and open Control System Designer in the Bode Editor configuration. G = tf(1.5,[1 14 40.02]); controlSystemDesigner('bode',G); The app opens and imports G as the plant model for the default control architecture, Configuration 1. In the app, the following response plots open: • Open-loop Bode Editor for the LoopTransfer_C response. This response is the open-loop transfer function GC, where C is the compensator and G is the plant. • Step Response for the IOTransfer_r2y response. This response is the input-output transfer function for the overall closed-loop system. Tip To open the open-loop Bode Editor when Control System Designer is already open, on the Control System tab, in the Tuning Methods drop-down list, select Bode Editor. In the Select Response to Edit dialog box, select an existing response to plot, or create a New Open-Loop Response. To view the open-loop frequency response and closed-loop step response simultaneously, on the Views tab, click Left/Right. The app displays the Bode Editor and Step Response plots side-by-side. 12-50 Bode Diagram Design Adjust Bandwidth Since the design requires a rise time less than 0.5 seconds, set the open-loop DC crossover frequency to about 3 rad/s. To a first-order approximation, this crossover frequency corresponds to a time constant of 0.33 seconds. To make the crossover easier to see, turn on the plot grid. Right-click the Bode Editor plot area, and select Grid. The app adds a grid to the Bode response plots. To adjust the crossover frequency increase the compensator gain. In the Bode Editor plot, in the Magnitude response plot, drag the response upward. Doing so increases the gain of the compensator. 12-51 12 Classical Control Design As you drag the magnitude plot, the app computes the compensator gain and updates the response plots. Drag the magnitude response upward until the crossover frequency is about 3 rad/s. 12-52 Bode Diagram Design View Step Response Characteristics To add the rise time to the Step Response plot, right-click the plot area, and select Characteristics > Rise Time. To view the rise time, move the cursor over the rise time indicator. 12-53 12 Classical Control Design The rise time is around 0.23 seconds, which satisfies the design requirements. Similarly, to add the peak response to the Step Response plot, right-click the plot area, and select Characteristics > Peak Response. 12-54 Bode Diagram Design The peak overshoot is around 3.5%. Add Integrator To Compensator To meet the 5% steady-state error requirement, eliminate steady-state error from the closed-loop step response by adding an integrator to your compensator. In the Bode Editor right-click in the plot area, and select Add Pole/Zero > Integrator. 12-55 12 Classical Control Design Adding an integrator produces zero steady-state error. However, changing the compensator dynamics also changes the crossover frequency, increasing the rise time. To reduce the rise time, increase the crossover frequency to around 3 rad/s. Adjust Compensator Gain To return the crossover frequency to around 3 rad/s, increase the compensator gain further. Right-click the Bode Editor plot area, and select Edit Compensator. In the Compensator Editor dialog box, in the Compensator section, specify a gain of 99, and press Enter. 12-56 Bode Diagram Design The response plots update automatically. The rise time is around 0.4 seconds, which satisfies the design requirements. However, the peak overshoot is around 32%. A compensator consisting of a gain and an integrator is not sufficient to meet the design requirements. Therefore, the compensator requires additional dynamics. Add Lead Network to Compensator In the Bode Editor, review the gain margin and phase margin for the current compensator design. The design requires a gain margin greater than 20 dB and phase 12-57 12 Classical Control Design margin greater than 40 degrees. The current design does not meet either of these requirements. To increase the stability margins, add a lead network to the compensator. In the Bode Editor, right-click and select Add Pole/Zero > Lead. To specify the location of the lead network pole, click on the magnitude response. The app adds a real pole (red X) and real zero (red O) to the compensator and to the Bode Editor plot. In the Bode Editor, drag the pole and zero to change their locations. As you drag them, the app updates the pole/zero values and updates the response plots. To decrease the magnitude of a pole or zero, drag it towards the left. Since the pole and zero are on the negative real axis, dragging them to the left moves them closer to the origin in the complex plane. 12-58 Bode Diagram Design Tip As you drag a pole or zero, the app displays the new value in the status bar, on the right side. As an initial estimate, drag the zero to a location around -7 and the pole to a location around -11. 12-59 12 Classical Control Design The phase margin meets the design requirements; however, the gain margin is still too low. Edit Lead Network Pole and Zero To improve the controller performance, tune the lead network parameters. In the Compensator Editor dialog box, in the Dynamics section, click the Lead row. 12-60 Bode Diagram Design In the Edit Selected Dynamics section, in the Real Zero text box, specify a location of -4.3, and press Enter. This value is near the slowest (left-most) pole of the DC motor plant. In the Real Pole text box, specify a value of -28, and press Enter. When you modify a lead network parameters, the Compensator and response plots update automatically. In the app, in the Bode Editor, the gain margin of 20.5 just meets the design requirement. 12-61 12 Classical Control Design To add robustness to the system, in the Compensator Editor dialog box, decrease the compensator gain to 84.5, and press Enter. The gain margin increases to 21.8, and the response plots update. In Control System Designer, in the response plots, compare the system performance to the design requirements. The system performance characteristics are: • Rise time is 0.445 seconds. • Steady-state error is zero. 12-62 See Also • Overshoot is 3.39%. • Gain margin is 21.8 dB. • Phase margin is 65.6 degrees. The system response meets all of the design requirements. See Also Control System Designer | bodeplot More About • “Edit Compensator Dynamics” on page 12-93 • “Control System Designer Tuning Methods” on page 12-4 • “Root Locus Design” on page 12-64 • “Nichols Plot Design” on page 12-80 12-63 12 Classical Control Design Root Locus Design Root locus design is a common control system design technique in which you edit the compensator gain, poles, and zeros in the root locus diagram. As the open-loop gain, k, of a control system varies over a continuous range of values, the root locus diagram shows the trajectories of the closed-loop poles of the feedback system. For example, in the following tracking system: P(s) is the plant, H(s) is the sensor dynamics, and k is an adjustable scalar gain The closed-loop poles are the roots of q ( s) = 1 + kP ( s) H ( s ) The root locus technique consists of plotting the closed-loop pole trajectories in the complex plane as k varies. You can use this plot to identify the gain value associated with a desired set of closed-loop poles. Tune Electrohydraulic Servomechanism Using Root Locus Graphical Tuning This example shows how to design a compensator for an electrohydraulic servomechanism using root locus graphical tuning techniques. Plant Model A simple version of an electrohydraulic servomechanism model consists of • A push-pull amplifier (a pair of electromagnets) 12-64 Root Locus Design • A sliding spool in a vessel of high-pressure hydraulic fluid • Valve openings in the vessel to allow for fluid to flow • A central chamber with a piston-driven ram to deliver force to a load • A symmetrical fluid return vessel The force on the spool is proportional to the current in the electromagnet coil. As the spool moves, the valve opens, allowing the high-pressure hydraulic fluid to flow through the chamber. The moving fluid forces the piston to move in the opposite direction of the spool. For more information on this model, including the derivation of a linearized model, see [1]. You can use the input voltage to the electromagnet to control the ram position. When measurements of the ram position are available, you can use feedback for the ram position control, as shown in the following, where Gservo represents the servomechanism: 12-65 12 Classical Control Design Design Requirements For this example, tune the compensator, C(s) to meet the following closed-loop step response requirements: • The 2% settling time is less than 0.05 seconds. • The maximum overshoot is less than 5%. Open Control System Designer At the MATLAB command line, load a linearized model of the servomechanism, and open Control System Designer in the root locus editor configuration. load ltiexamples Gservo controlSystemDesigner('rlocus',Gservo); The app opens and imports Gservo as the plant model for the default control architecture, Configuration 1. In Control System Designer, a Root Locus Editor plot and input-output Step Response open. To view the open-loop frequency response and closed-loop step response simultaneously, on the Views tab, click Left/Right. 12-66 Root Locus Design The app displays Bode Editor and Step Response plots side-by-side. 12-67 12 Classical Control Design In the closed-loop step response plot, the rise time is around two seconds, which does not satisfy the design requirements. To make the root locus diagram easier to read, zoom in. In the Root Locus Editor, rightclick the plot area and select Properties. In the Property Editor dialog box, on the Limits tab, specify Real Axis and Imaginary Axis limits from -500 to 500. 12-68 Root Locus Design Click Close. Increase Compensator Gain To create a faster response, increase the compensator gain. In the Root Locus Editor, right-click the plot area and select Edit Compensator. In the Compensator Editor dialog box, specify a gain of 20. 12-69 12 Classical Control Design In the Root Locus Editor plot, the closed-loop pole locations move to reflect the new gain value. Also, the Step Response plot updates. 12-70 Root Locus Design The closed-loop response does not satisfy the settling time requirement and exhibits unwanted ringing. Increasing the gain makes the system underdamped and further increases lead to instability. Therefore, to meet the design requirements, you must specify additional compensator dynamics. For more information on adding and editing compensator dynamics, see “Edit Compensator Dynamics” on page 12-93. 12-71 12 Classical Control Design Add Poles to Compensator To add a complex pole pair to the compensator, in the Root Locus Editor, right-click the plot area and select Add Pole/Zero > Complex Pole. Click the plot area where you want to add one of the complex poles. The app adds the complex pole pair to the root locus plot as red X’s, and updates the step response plot. In the Root Locus Editor, drag the new poles to locations near –140 ± 260i. As you drag one pole, the other pole updates automatically. 12-72 Root Locus Design Tip As you drag a pole or zero, the app displays the new value in the status bar, on the right side. 12-73 12 Classical Control Design Add Zeros to Compensator To add a complex zero pair to your compensator, in the Compensator Editor dialog box, right-click the Dynamics table, and select Add Pole/Zero > Complex Zero 12-74 Root Locus Design The app adds a pair of complex zeros at –1 ± i to your compensator In the Dynamics table, click the Complex Zero row. Then in the Edit Selected Dynamics section, specify a Real Part of -170 and an Imaginary Part of 430. 12-75 12 Classical Control Design The compensator and response plots automatically update to reflect the new zero locations. 12-76 Root Locus Design In the Step Response plot, the settling time is around 0.1 seconds, which does not satisfy the design requirements. Adjust Pole and Zero Locations The compensator design process can involve some trial and error. Adjust the compensator gain, pole locations, and zero locations until you meet the design criteria. One possible compensator design that satisfies the design requirements is: 12-77 12 Classical Control Design • Compensator gain of 10 • Complex poles at –110 ± 140i • Complex zeros at –70 ± 270i In the Compensator Editor dialog box, configure your compensator using these values. In the Step Response plot, the settling time is around 0.05 seconds. To verify the exact settling time, right-click the Step Response plot area and select Characteristics > Settling Time. A settling time indicator appears on the response plot. 12-78 See Also To view the settling time, move the cursor over the settling time indicator. The settling time is about 0.043 seconds, which satisfies the design requirements. References [1] Clark, R. N. Control System Dynamics, Cambridge University Press, 1996. See Also Control System Designer | rlocusplot More About • “Edit Compensator Dynamics” on page 12-93 • “Control System Designer Tuning Methods” on page 12-4 • “Bode Diagram Design” on page 12-49 • “Nichols Plot Design” on page 12-80 12-79 12 Classical Control Design Nichols Plot Design Nichols plot design is an interactive graphical method of modifying a compensator to achieve a specific open-loop response (loop shaping). Unlike “Bode Diagram Design” on page 12-49, Nichols plot design uses Nichols plots to view the open-loop frequency response. Nichols plots combine gain and phase information into a single plot, which is useful when you are designing to gain and phase margin specifications. You can also use the Nichols plot grid lines to estimate the closed-loop response (see ngrid). For more information on Nichols plots, see nicholsplot. Tune Compensator For DC Motor Using Nichols Plot Graphical Design This example shows how to design a compensator for a DC motor using Nichols plot graphical tuning techniques. Plant Model and Requirements The transfer function of the DC motor plant, as described in “SISO Example: The DC Motor”, is: G= 1.5 2 s + 14 s + 40 .02 For this example, the design requirements are: • Rise time of less than 0.5 seconds • Steady-state error of less than 5% • Overshoot of less than 10% • Gain margin greater than 20 dB • Phase margin greater than 40 degrees Open Control System Designer At the MATLAB command line, create a transfer function model of the plant, and open Control System Designer in the Nichols Editor configuration. G = tf(1.5,[1 14 40.02]); controlSystemDesigner('nichols',G); 12-80 Nichols Plot Design The app opens and imports G as the plant model for the default control architecture, Configuration 1. In the app, the following response plots open: • Open-loop Nichols Editor for the LoopTransfer_C response. This response is the open-loop transfer function GC, where C is the compensator and G is the plant. • Step Response for the IOTransfer_r2y response. This response is the input-output transfer function for the overall closed-loop system. Tip To open the open-loop Nichols Editor when Control System Designer is already open, on the Control System tab, in the Tuning Methods drop-down list, select Nichols Editor. In the Select Response to Edit dialog box, select an existing response to plot, or create a New Open-Loop Response. To view the open-loop frequency response and closed-loop step response simultaneously, on the Views tab, click Left/Right. The app displays the Nichols Editor and Step Response plots side-by-side. Adjust Bandwidth Since the design requires a rise time less than 0.5 seconds, set the open-loop DC crossover frequency to about 3 rad/s. To a first-order approximation, this crossover frequency corresponds to a time constant of 0.33 seconds. 12-81 12 Classical Control Design To adjust the crossover frequency increase the compensator gain. In the Nichols Editor, drag the response upward. Doing so increases the gain of the compensator. As you drag the Nichols plot, the app computes the compensator gain and updates the response plots. Drag the Nichols plot upward until the crossover frequency is about 3 rad/s. 12-82 Nichols Plot Design View Step Response Characteristics To add the rise time to the Step Response plot, right-click the plot area, and select Characteristics > Rise Time. To view the rise time, move the cursor over the rise time indicator. 12-83 12 Classical Control Design The rise time is around 0.23 seconds, which satisfies the design requirements. Similarly, to add the peak response to the Step Response plot, right-click the plot area, and select Characteristics > Peak Response. 12-84 Nichols Plot Design The peak overshoot is around 3.5%. Add Integrator To Compensator To meet the 5% steady-state error requirement, eliminate steady-state error from the closed-loop step response by adding an integrator to your compensator. In the Nichols Editor right-click in the plot area, and select Add Pole/Zero > Integrator. 12-85 12 Classical Control Design Adding an integrator produces zero steady-state error. However, changing the compensator dynamics also changes the crossover frequency, increasing the rise time. To reduce the rise time, increase the crossover frequency to around 3 rad/s. Adjust Compensator Gain To return the crossover frequency to around 3 rad/s, increase the compensator gain further. Right-click the Nichols Editor plot area, and select Edit Compensator. In the Compensator Editor dialog box, in the Compensator section, specify a gain of 99, and press Enter. 12-86 Nichols Plot Design The response plots update automatically. The rise time is around 0.4 seconds, which satisfies the design requirements. However, the peak overshoot is around 32%. A compensator consisting of a gain and an integrator is not sufficient to meet the design requirements. Therefore, the compensator requires additional dynamics. Add Lead Network to Compensator In the Nichols Editor, review the gain margin and phase margin for the current compensator design. The design requires a gain margin greater than 20 dB and phase 12-87 12 Classical Control Design margin greater than 40 degrees. The current design does not meet either of these requirements. To increase the stability margins, add a lead network to the compensator. In the Nichols Editor, right-click and select Add Pole/Zero > Lead. To specify the location of the lead network pole, click on the magnitude response. The app adds a real pole (red X) and real zero (red O) to the compensator and to the Nichols Editor plot. In the Nichols Editor, drag the pole and zero to change their locations. As you drag them, the app updates the pole/zero values and updates the response plots. To decrease the magnitude of a pole or zero, drag it towards the left. Since the pole and zero are on the negative real axis, dragging them to the left moves them closer to the origin in the complex plane. Tip As you drag a pole or zero, the app displays the new value in the status bar, on the right side. 12-88 Nichols Plot Design As an initial estimate, drag the zero to a location around -7 and the pole to a location around -11. The phase margin meets the design requirements; however, the gain margin is still too low. Edit Lead Network Pole and Zero To improve the controller performance, tune the lead network parameters. In the Compensator Editor dialog box, in the Dynamics section, click the Lead row. 12-89 12 Classical Control Design In the Edit Selected Dynamics section, in the Real Zero text box, specify a location of -4.3, and press Enter. This value is near the slowest (left-most) pole of the DC motor plant. In the Real Pole text box, specify a value of -28, and press Enter. When you modify a lead network parameters, the Compensator and response plots update automatically. In the app, in the Nichols Editor, the gain margin of 20.5 just meets the design requirement. 12-90 Nichols Plot Design To add robustness to the system, in the Compensator Editor dialog box, decrease the compensator gain to 84.5, and press Enter. The gain margin increases to 21.8, and the response plots update. In Control System Designer, in the response plots, compare the system performance to the design requirements. The system performance characteristics are: • Rise time is 0.445 seconds. • Steady-state error is zero. 12-91 12 Classical Control Design • Overshoot is 3.39%. • Gain margin is 21.8 dB. • Phase margin is 65.6 degrees. The system response meets all of the design requirements. See Also Control System Designer | nicholsplot More About 12-92 • “Edit Compensator Dynamics” on page 12-93 • “Control System Designer Tuning Methods” on page 12-4 • “Bode Diagram Design” on page 12-49 • “Root Locus Design” on page 12-64 Edit Compensator Dynamics Edit Compensator Dynamics Using Control System Designer, you can manually edit compensator dynamics to achieve your design goals. In particular, you can adjust the compensator gain, and you can add the following compensator dynamics: • Real and complex poles, including integrators • Real and complex zeros, including differentiators • Lead and lag networks • Notch filters You can add dynamics and modify compensator parameters using the Compensator Editor or using the graphical Bode Editor, Root Locus Editor, or Nichols Editor plots. Compensator Editor To open the Compensator Editor dialog box, in Control System Designer, in an editor plot area, right-click and select Edit Compensator. Alternatively, in the Data Browser, in the Controllers section, right-click the compensator you want to edit and click Open Selection. 12-93 12 Classical Control Design The Compensator Editor displays the transfer function for the currently selected compensator. You can select a different compensator to edit using the drop-down list. By default, the compensator transfer function displays in the time constant format. You can select a different format by changing the corresponding Control System Designer preference. In Control System Designer, on the Control System tab, click Preferences. In the Control System Designer Preferences dialog box, on the Options tab, select a Compensator Format. To add poles and zeros to your compensator, in the Compensator Editor, right-click in the Dynamics table and, under Add Pole/Zero, select the type of pole/zero you want to add. 12-94 Edit Compensator Dynamics The app adds a pole or zero of the selected type with default parameters. To edit a pole or zero, in the Dynamics table, click on the pole/zero type you want to edit. Then, in the Edit Selected Dynamics section, in the text boxes, specify the pole and zero locations. To delete poles and zeros, in the Dynamics table, click on the pole/zero type you want to delete. Then, right-click and select Delete Pole/Zero. 12-95 12 Classical Control Design Graphical Compensator Editing You can also add and adjust poles and zeros directly from Bode Editor, Root Locus Editor, or Nichols Editor plots. Use this method to roughly place poles and zeros in the correct area before fine-tuning their locations using the Compensator Editor. To add poles and zeros directly from an editor plot, right-click the plot area and, under Add Pole/Zero, select the type of pole/zero you want to add. In the editor plot, the app displays the editable compensator poles and zeros as red X’s and O’s respectively. In the editor plots, you can drag poles and zeros to adjust their locations. As you drag a pole or zero, the app displays the new value in the status bar, on the right side. To delete a pole or zero, right-click the plot area and select Delete Pole/Zero. Then, in the editor plot, click the pole or zero you want to delete. 12-96 Edit Compensator Dynamics Poles and Zeros You can add the following poles and zeros to your compensator: • Real pole/zero — Specify the pole/zero location on the real axis • Complex poles/zeros — Specify complex conjugate pairs by: • Setting the real and imaginary parts directly. • Setting the natural frequency, ωn, and damping ratio, ξ. • Integrator — Add a pole at the origin to eliminate steady-state error for step inputs and DC inputs. • Differentiator — Add a zero at the origin. Lead and Lag Networks You can add lead networks, lag networks, and combination lead-lag networks to your compensator. Network Type Description Use This To Lead One pole and one zero on the negative real axis, with the zero having a lower natural frequency • Increase stability margins One pole and one zero on the negative real axis, with the pole having a lower natural frequency • Reduce high-frequency gain A combination of a lead network and a lag network Combine the effects of lead and lag networks Lag Lead-Lag • Increase system bandwidth • Reduce rise time • Increase phase margin • Improve steady-state accuracy To add a lead-lag network, add separate lead and lag networks. To configure a lead or lag network for your compensator, use one of the following options: • Specify the pole and zero locations. Placing the pole and zero further apart increases the amount of phase angle change. 12-97 12 Classical Control Design • Specify the maximum amount of phase angle change and the frequency at which this change occurs. The app automatically computes the pole and zero locations. When graphically changing pole and zero locations for a lead or lag compensator, in the editor plot, you can drag the pole and zeros independently. Notch Filters If you know that your system has disturbances at a particular frequency, you can add a notch filter to attenuate the gain of the system at that frequency. The notch filter transfer function is: s2 + 2x1wn s + wn2 s2 + 2x2wn s + wn2 where • ωn is the natural frequency of the notch. • The ratio ξ2/ξ1 sets the depth of the notch. To configure a notch filter for your compensator, in the Compensator Editor dialog box, you can specify the: • Natural Frequency — Attenuated frequency • Notch Depth and Notch Width • Damping for the complex poles and zeros of the transfer function. When graphically editing a notch filter, in the Bode Editor, you can drag the bottom of the notch to adjust ωn and the notch depth. To adjust the width of the notch without changing ωn or the notch depth, you can drag the edges of the notch. 12-98 See Also See Also Control System Designer More About • “Bode Diagram Design” on page 12-49 • “Root Locus Design” on page 12-64 • “Nichols Plot Design” on page 12-80 12-99 12 Classical Control Design Design Compensator Using Automated Tuning Methods This example shows how to tune a compensator using automated tuning methods in Control System Designer. Select Tuning Method To select an automated tuning method, in Control System Designer, click Tuning Methods. 12-100 Design Compensator Using Automated Tuning Methods Select one of the following tuning methods: • PID Tuning — Tune PID gains to balance performance and robustness or use classical tuning formulas. • Optimization Based Tuning — Optimize compensator parameters using design requirements implemented in graphical tuning and analysis plots (requires Simulink Design Optimization software). • LQG Synthesis — Design a full-order stabilizing feedback controller as a linearquadratic-Gaussian (LQG) tracker. • Loop Shaping — Find a full-order stabilizing feedback controller with a specified open-loop bandwidth or shape (requires Robust Control Toolbox software). • Internal Model Control (IMC) Tuning — Obtain a full-order stabilizing feedback controller using the IMC design method. Select Compensator and Loop to Tune In the dialog box for your selected tuning method, in the Compensator section, select the compensator and loop to tune. • Compensator — Select a compensator to tune from the drop-down list. The app displays the current compensator transfer function. • Select Loop to Tune — Select an existing open-loop transfer function to tune from the drop-down list. You can select any open-loop transfer function from the Data Browser that includes the selected compensator in series • Add New Loop — Create a new loop to tune. In the Open-Loop Transfer Function dialog box, select signals and loop openings to configure the loop transfer function. 12-101 12 Classical Control Design Note For optimization-based tuning, you do not specify the compensator and loop to tune in this way. Instead, you define the compensator structure and select compensator and prefilter parameters to optimize. For more information, see “Select Tunable Compensator Elements” (Simulink Design Optimization). PID Tuning Using Control System Designer, you can automatically tune any of the following PID controller types: • P — Proportional-only control • I — Integral-only control • PI — Proportional-integral control • PD — Proportional-derivative control • PDF — Proportional-derivative control with a low-pass filter on the derivative term • PID — Proportional-integral-derivative control • PIDF — Proportional-integral-derivative control with a low-pass filter on the derivative term To open the PID Tuning dialog box, in Control System Designer, click Tuning Methods, and select PID Tuning. 12-102 Design Compensator Using Automated Tuning Methods Robust Response Time The robust response time algorithm automatically tunes PID parameters to balance performance and robustness. Using the robust response time method, you can: • Tune all parameters for any type of PID controller. • Design for plants that are stable, unstable, or integrating. To tune your compensator using this method: 1 In the PID Tuning dialog box, in the Specifications section, in the Tuning method drop-down list, select Robust response time. 12-103 12 Classical Control Design 2 Select a Controller type. If you choose PD or PID, check Design with first order derivative filter to design a PDF or PIDF controller, respectively. Tip Adding derivative action to the controller gives the algorithm more freedom to achieve both adequate phase margin and faster response time. 3 In the Design mode drop-down list, select one of the following: • Time — Specify controller performance using time-domain parameters. 12-104 Design Compensator Using Automated Tuning Methods • Response Time — Specify a faster or slower controller response time. To modify the response time by a factor of ten, use the left or right arrows. • Transient Behavior — Specify the controller transient behavior. You can make the controller more aggressive at disturbance rejection or more robust against plant uncertainty. • Frequency — Specify controller performance using frequency-domain parameters. 12-105 12 Classical Control Design • Bandwidth — Specify the closed-loop bandwidth of the control system. To produce a faster response time, increase the bandwidth. To modify the bandwidth by a factor of ten, use the left or right arrows. • Phase Margin — Specify a target phase margin for the system. To reduce overshoot and create a more robust controller, increase the phase margin. 4 To apply the specified controller design to the selected compensator, click Update Compensator. Note If you previously specified the controller structure manually or using a different automated tuning method, that structure is lost when you click Update Compensator. 5 By default, the app automatically computes controller parameters for balanced performance and robustness. To revert to these default parameters at any time, click Reset Parameters. Classical Design Formulas You can use classical PID design formulas to tune P, PI, PID, and PIDF controllers. These design formulas: • Require a stable or integrating plant. For more information about the effective plant seen by the compensator, see “Effective Plant for Tuning” on page 12-6. 12-106 Design Compensator Using Automated Tuning Methods • Cannot tune the derivative filter. If you select a PIDF controller, the classical design methods set the filter time constant to Td/10, where Td is the tuned derivative time. To tune your compensator using a classical method: 1 In the PID Tuning dialog box, in the Specifications section, in the Tuning method drop-down list, select Classical design formulas. 2 Select a Controller type. Tip Adding derivative action to the compensator gives the algorithm more freedom to achieve both adequate phase margin and faster response time. 3 In the Formula drop-down list, select a classical design formula. • Approximate MIGO frequency response — Compute controller parameters using closed-loop, frequency-domain, approximate M-constrained integral gain optimization (see [1]). 12-107 12 Classical Control Design • Approximate MIGO step response — Compute controller parameters using open-loop, time-domain, approximate M-constrained integral gain optimization (see [1]). • Chien-Hrones-Reswick — Approximate the plant as a first-order model with a time delay, and compute PID parameters using a Chien-Hrones-Reswick lookup table for 0% overshoot and disturbance rejection (see [2]). • Skogestad IMC — Approximate the plant as a first-order model with a time delay, and compute PID parameters using Skogestad design rules (see [3]). Note This method is different from selecting “Internal Model Control Tuning” on page 12-113 as the full-order compensator tuning method. • Ziegler-Nichols frequency response — Compute controller parameters from a Ziegler-Nichols lookup table, based on the ultimate gain and frequency of the system (see [2]). • Ziegler-Nichols step response — Approximate the plant as a first-order model with a time delay, and compute PID parameters using the Ziegler-Nichols design method (see [2]). 4 Apply the specified controller design to the selected compensator. Click Update Compensator. Note If you previously specified the controller structure manually or using a different automated tuning method, that structure is lost when you click Update Compensator. Optimization-Based Tuning Optimization-based tuning is available only if you have Simulink Design Optimization software installed. You can use this method to design control systems for LTI models by optimizing controller parameters. Note Optimization-based tuning only changes the values of controller parameters and not the controller structure itself. For information on adding or removing compensator elements, see “Edit Compensator Dynamics” on page 12-93. To design a controller using optimization-based tuning: 12-108 Design Compensator Using Automated Tuning Methods 1 Define the structure of the compensators you want to tune. Typically, you design an initial controller either manually or using a different automated tuning method. 2 Open the Response Optimization dialog box. In Control System Designer, click Tuning Methods, and select Optimization-Based Tuning. 3 Select compensator parameters to optimize. On the Compensators tab, in the Optimize column, select the compensator elements to tune. You can optimize elements for any compensator listed in the Data Browser. To select all elements for a given compensator, select the check box that corresponds to that compensator. Any elements that you do not select in the Optimize column remain at their current values during optimization. 4 For each compensator element, specify: • Initial guess — Starting point for the optimization algorithm. To use the current element Value as the Initial guess, click a row in the table, and click Use Value as Initial guess. • Minimum and Maximum bounds on the element value. The optimization constrains the search results to the specified range. • Typical value scaling factor for normalizing the compensator elements. 12-109 12 Classical Control Design 5 On the Design Requirements tab, in the Optimize column, select the design requirements to satisfy during optimization. Each design requirement is associated with a plot of a specific response in the Data Browser. To select all requirements for a given response, select the corresponding check box. For information on adding and editing design requirements, see “Design Requirements” on page 12-9. 6 (optional) Configure optimization options. On the Optimization tab, click Optimization options. 7 Click Start Optimization. For examples of optimization-based tuning, see “Optimize LTI System to Meet FrequencyDomain Requirements” (Simulink Design Optimization) and “Design Optimization-Based PID Controller for Linearized Simulink Model (GUI)” (Simulink Design Optimization). LQG Synthesis Linear-quadratic-Gaussian (LQG) control is a technique for designing optimal dynamic regulators and setpoint trackers. This technique allows you to trade off performance and control effort, and to take into account process disturbances and measurement noise. 12-110 Design Compensator Using Automated Tuning Methods LQG synthesis generates a full-order feedback controller that guarantees closed-loop stability. The designed controller contains an integrator, which guarantees zero steadystate error for plants without a free differentiator. To design an LQG controller: 1 Open the LQG Synthesis dialog box. In Control System Designer, click Tuning Methods, and select LQG Synthesis. 2 Specify the transient behavior of the controller using the Controller response slider. You can make the controller more aggressive at disturbance rejection or more robust against plant uncertainty. If you believe your model is accurate and that the manipulated variable has a large enough range, an aggressive controller is preferable. 3 Specify an estimate of the level of output measurement noise for your application using the Measurement noise slider. To produce a more robust controller, specify a larger noise estimate. 4 Specify your controller order preference using the Desired controller order slider. The maximum controller order is dependent on the effective plant dynamics. 5 Apply the specified controller design to the selected compensator. Click Update Compensator. 12-111 12 Classical Control Design Note If you previously specified the controller structure manually or using a different automated tuning method, that structure is lost when you click Update Compensator. For an example of LQG synthesis using Control System Designer, see “Design LQG Tracker Using Control System Designer” on page 12-176. Loop Shaping If you have Robust Control Toolbox software installed, you can use loop shaping to design SISO compensators in Control System Designer. Loop shaping generates a stabilizing feedback controller to match, as closely as possible, a target loop shape. You can specify this loop shape as a bandwidth or an open-loop frequency response. To design a controller using loop shaping: 1 Open the Loop Shaping dialog box. In Control System Designer, click Tuning Methods, and select Loop Shaping. 2 Select one of the following tuning preferences: • Target bandwidth — Specify a Target open-loop bandwidth, wb , to produce a wb . s • Target loop shape — Specify the Target open-loop shape as a tf, ss, or zpk object. To limit the frequencies over which to match the target loop shape, specify the Frequency range for loop shaping as a two-element row vector. loop shape of the specified bandwidth over an integrator, 3 Specify your controller order preference using the Desired controller order slider. The maximum controller order is dependent on the effective plant dynamics. 4 Apply the specified controller design to the selected compensator. Click Update Compensator. Note If you previously specified the controller structure manually or using a different automated tuning method, that structure is lost when you click Update Compensator. 12-112 Design Compensator Using Automated Tuning Methods Internal Model Control Tuning Internal model control (IMC) uses a predictive model of the plant dynamics to compute control actions. IMC design generates a full-order feedback controller that guarantees closed-loop stability when there is no model error. The tuned compensator also contains an integrator, which guarantees zero steady-state offset for plants without a free differentiator. You can use this tuning method for both stable and unstable plants. To design an IMC controller: 1 Select and configure the IMC control architecture. In Control System Designer, click Edit Architecture. In the Edit Architecture dialog box, select the fifth control architecture, and import the plant model, G1, predictive model, G2, and disturbance model Gd. 12-113 12 Classical Control Design Click OK. 2 12-114 Open the Internal Model Control (IMC) Tuning dialog box. In Control System Designer, click Tuning Methods, and select Internal Model Control (IMC) Tuning. Design Compensator Using Automated Tuning Methods 3 Specify a Dominant closed-loop time constant. The default value is 5% of the open-loop settling time. In general, increasing this value slows down the closed-loop system and makes it more robust. 4 Specify your controller order preference using the Desired controller order slider. The maximum controller order is dependent on the effective plant dynamics. 5 Apply the specified controller design to the selected compensator. Click Update Compensator. Note If you previously specified the controller structure manually or using a different automated tuning method, that structure is lost when you click Update Compensator. For an example of IMC tuning, see “Design Internal Model Controller for Chemical Reactor Plant” on page 12-161. 12-115 12 Classical Control Design References [1] Åström, K. J. and Hägglund, T. “Replacing the Ziegler-Nichols Tuning Rules.” Chapter 7 in Advanced PID Control, Research Triangle Park, NC: Instrumentation, Systems, and Automation Society, 2006, pp. 233–270. [2] Åström, K. J. and Hägglund, T. “Ziegler-Nichols' and Related Methods.” Section 6.2 in Advanced PID Control, Research Triangle Park, NC: Instrumentation, Systems, and Automation Society, 2006, pp. 167–176. [3] Skogestad, S., “Simple analytic rules for model reduction and PID controller tuning.” Journal of Process Control, Vol. 13, No. 4, 2003, pp. 291–309. See Also Control System Designer More About 12-116 • “Control System Designer Tuning Methods” on page 12-4 • “Design Internal Model Controller for Chemical Reactor Plant” on page 12-161 • “Design LQG Tracker Using Control System Designer” on page 12-176 Analyze Designs Using Response Plots Analyze Designs Using Response Plots This example shows how to analyze your control system designs using the plotting tools in Control System Designer. There are two types of Control System Designer plots: • Analysis plots — Use these plots to visualize your system performance and display response characteristics. • Editor plots — Use these plots to visualize your system performance and interactively tune your compensator dynamics using graphical tuning methods. Analysis Plots Use analysis plots to visualize your system performance and display response characteristics. You can also use these plots to compare multiple control system designs. For more information, see “Compare Performance of Multiple Designs” on page 12-127. To create a new analysis plot, in Control System Designer, on the Control System tab, click New Plot, and select the type of plot to add. 12-117 12 Classical Control Design In the new plot dialog box, specify an existing or new response to plot. 12-118 Analyze Designs Using Response Plots Note Using analysis plots, you can compare the performance of multiple designs stored in the Data Browser. For more information, see “Compare Performance of Multiple Designs” on page 12-127. Plot Existing Response To plot an existing response, in the Select Response to Plot drop-down list, select an existing response from the Data Browser. The details for the selected response are displayed in the text box. To plot the selected response, click Plot. Plot New Response To plot a new response, specify the following: • Select Response to Plot — Select the type of response to create. • New Input-Output Transfer Response — Create a transfer function response for specified input signals, output signals, and loop openings. • New Open-Loop Response — Create an open-loop transfer function response at a specified location with specified loop openings. • New Sensitivity Transfer Response — Create a sensitivity response at a specified location with specified loop openings. • Response Name — Enter a name in the text box. • Signal selection boxes — Specify signals as inputs, outputs, or loop openings by . If you open Control System Designer from: clicking • MATLAB — Select a signal using the Architecture block diagram for reference. • Simulink — Select an existing signal from the current control system architecture, or add a signal from the Simulink model. Use , , and to reorder and delete signals. To add the specified response to the Data Browser and create the selected plot, click Plot. 12-119 12 Classical Control Design Editor Plots Use editor plots to visualize your system performance and interactively tune your compensator dynamics using graphical tuning methods. To create a new editor plot, in Control System Designer, on the Control System tab, click Tuning Methods, and select one of the Graphical Tuning methods. For examples of graphical tuning using editor plots, see: • “Bode Diagram Design” on page 12-49 • “Root Locus Design” on page 12-64 • “Nichols Plot Design” on page 12-80 For more information on interactively editing compensator dynamics, see “Edit Compensator Dynamics” on page 12-93. 12-120 Analyze Designs Using Response Plots Plot Characteristics On any analysis plot in Control System Designer: • To see response information and data values, click a line on the plot. • To view system characteristics, right-click anywhere on the plot, as described in “Frequency-Domain Characteristics on Response Plots” on page 8-10. 12-121 12 Classical Control Design Plot Tools Mouse over any analysis plot to access plot tools at the upper right corner of the plot. 12-122 Analyze Designs Using Response Plots • • and — Zoom in and zoom out. Click to activate, and drag the cursor over the region to zoom. The zoom icon turns dark when zoom is active. Right-click while zoom is active to access additional zoom options. Click the icon again to deactivate. — Pan. Click to activate, and drag the cursor across the plot area to pan. The pan icon turns dark when pan is active. Right-click while pan is active to access additional pan options. Click the icon again to deactivate. 12-123 12 Classical Control Design • — Legend. By default, the plot legend is inactive. To toggle the legend on and off, click this icon. To move the legend, drag it to a new location on the plot. To change the way plots are tiled or sorted, use the options on the View tab. Design Requirements You can add graphical representations of design requirements to any editor or analysis plots. These requirements define shaded exclusion regions in the plot area. 12-124 Analyze Designs Using Response Plots Use these regions as guidelines when analyzing and tuning your compensator designs. To meet a design requirement, your response plots must remain outside of the corresponding shaded area. To add design requirements to a plot, right-click anywhere on the plot and select Design Requirements > New. 12-125 12 Classical Control Design In the New Design Requirement dialog box, specify the Design requirement type, and define the Design requirement parameters. Each type of design requirement has a different set of parameters to configure. For more information on adding design requirements to analysis and editor plots, see “Design Requirements” on page 12-9. See Also More About 12-126 • “Control System Designer Tuning Methods” on page 12-4 • “Compare Performance of Multiple Designs” on page 12-127 • “Design Requirements” on page 12-9 Compare Performance of Multiple Designs Compare Performance of Multiple Designs This example shows how to compare the performance of two different control system designs. Such comparison is useful, for example, to see the effects of different tuning methods or compensator structures. Store First Design In this example, the first design is the compensator tuned graphically in “Bode Diagram Design” on page 12-49. After tuning the compensator with this first tuning method, store the design in Control System Designer. On the Control System tab, in the Designs section, click appears in the Data Browser in the Designs area. Store. The stored design 12-127 12 Classical Control Design The stored design contains the tuned values of the controller and filter blocks. The app does not store the values of any fixed blocks. To rename the stored design, in the Data Browser, double-click the design, and specify a new name. Compute New Design On the Control System tab, tune the compensator using a different tuning method. Under Tuning Methods, select PID Tuning. 12-128 Compare Performance of Multiple Designs To design a controller with the default Robust response time specifications, in the PID Tuning dialog box, click Update Compensator. Compare New Design with Stored Design Update all plots to reflect both the new design and the stored design. On the Control System tab, click Compare. In the Compare Designs dialog box, the current design is checked by default. To compare a design with the current design, check the corresponding box. All analysis plots update to reflect the checked designs. The blue trace corresponds to the current design. Refer to the plot legend to identify the responses for other designs. 12-129 12 Classical Control Design To compare a stored design with the current design, the sample times of the current design and stored design must be the same. To modify the sample time of the current design to match that of a stored design, on the Control System tab, select Edit Architecture > Sample Time Conversion. Then, in the Sample Time Conversion dialog box, specify the sample time and a rate conversion methods for each block in the architecture. 12-130 Compare Performance of Multiple Designs Restore Previously Saved Design Under some conditions, it is useful to restore a previously stored design. For example, when designing a compensator for a Simulink model, you can write the current compensator values to the model (see “Update Simulink Model and Validate Design” (Simulink Control Design)). To test a stored compensator in your model, first restore the stored design as the current design. To do so, in Control System Designer, click you want to make current. Retrieve. Select the stored design that As with design comparison, to retrieve a stored design, the sample times of the current design and stored design must be the same. 12-131 12 Classical Control Design Note The retrieved design overwrites the current design. If necessary, store the current design before retrieving a previously stored design. See Also More About 12-132 • “Analyze Designs Using Response Plots” on page 12-117 • “Control System Designer Tuning Methods” on page 12-4 Design Hard-Disk Read/Write Head Controller Design Hard-Disk Read/Write Head Controller This example shows how to design a computer hard-disk read/write head position controller using classical control design methods. Create Read/Write Head Model Using Newton's laws, model the read/write head using the following differential equation: Here, • is the inertia of the head assembly. • is the viscous damping coefficient of the bearings. • is the return spring constant. • is the motor torque constant. • is the angular position of the head. • is the input current. Taking the Laplace transform, the transfer function from to is 12-133 12 Classical Control Design Specify the physical constants of the model, such that: • • • • J = 0.01; C = 0.004; K = 10; Ki = 0.05; Define the transfer function using these constants. num = Ki; den = [J C K]; H = tf(num,den) H = 0.05 ----------------------0.01 s^2 + 0.004 s + 10 Continuous-time transfer function. Discretize the Model To design a digital controller that provides accurate positioning of the read/write head, first discretize the continuous-time plant. Specify the sample time. Ts = 0.005; 12-134 Design Hard-Disk Read/Write Head Controller Discretize the model. Since the controller will have a digital-to-analog converter (with a zero-order hold) connected to its input, use the c2d command with the 'zoh' discretization method. Hd = c2d(H,Ts,'zoh') Hd = 6.233e-05 z + 6.229e-05 ----------------------z^2 - 1.973 z + 0.998 Sample time: 0.005 seconds Discrete-time transfer function. Compare the Bode plots of the continuous-time and discrete-time models. bodeplot(H,'-',Hd,'--') legend('Continuous-time','Discrete-time') 12-135 12 Classical Control Design To analyze the discretized system, plot its step response. stepplot(Hd) 12-136 Design Hard-Disk Read/Write Head Controller The step response has significant oscillation, which is most likely due to light damping. Check the damping for the open-loop poles of the system. damp(Hd) Pole 9.87e-01 + 1.57e-01i 9.87e-01 - 1.57e-01i Magnitude 9.99e-01 9.99e-01 Damping 6.32e-03 6.32e-03 Frequency (rad/seconds) Time Constant (seconds) 3.16e+01 3.16e+01 5.00e+00 5.00e+00 As expected, the poles have light equivalent damping and are near the unit circle. Therefore, you must design a compensator that increases the damping in the system. 12-137 12 Classical Control Design Add a Compensator Gain The simplest compensator is a gain factor with no poles or zeros. Try to select an appropriate feedback gain using the root locus technique. The root locus plots the closedloop pole trajectories as a function of the feedback gain. rlocus(Hd) The poles quickly leave the unit circle and go unstable. Therefore, you must introduce some lead to the system. Add a Lead Network Define a lead compensator with a zero at 12-138 and a pole at . Design Hard-Disk Read/Write Head Controller D = zpk(0.85,0,1,Ts); The corresponding open-loop model is the series connection of the compensator and plant. oloop = Hd*D oloop = 6.2328e-05 (z+0.9993) (z-0.85) -----------------------------z (z^2 - 1.973z + 0.998) Sample time: 0.005 seconds Discrete-time zero/pole/gain model. To see how the lead compensator affects the open-loop frequency response, compare the Bode plots of Hd and oloop. bodeplot(Hd,'--',oloop,'-') legend('Plant','Plant plus lead compensator') 12-139 12 Classical Control Design The compensator adds lead to the system, which shifts the phase response upward in the frequency range . Examine the behavior of the closed-loop system poles using a root locus plot. Set the limits of both the x-axis and y-axis from -1 to 1. rlocus(oloop) zgrid xlim([-1 1]) ylim([-1 1]) 12-140 Design Hard-Disk Read/Write Head Controller The closed-loop poles now remain within the unit circle for some time. To create a data marker for the plot, click the root locus curve. Find the point on the curve where the damping is greatest by dragging the marker. The maximum damping of 0.782 corresponds to a feedback gain of 4.07e+03. 12-141 12 Classical Control Design Analyze Design To analyze this design, first define the closed-loop system, which consists of the open-loop system with a feedback gain of 4.07e+03. 12-142 Design Hard-Disk Read/Write Head Controller k = 4.07e+03; cloop = feedback(oloop,k); Plot the closed-loop step response. stepplot(cloop) 12-143 12 Classical Control Design This response depends on your closed-loop setpoint. The one shown here is relatively fast and settles in about 0.06 seconds. Therefore, the closed-loop disk drive system has a seek time of 0.06 seconds. While this seek time is relatively slow by modern standards, you also started with a lightly-damped system. It is good practice to examine the robustness of your design. To do so, compute the gain and phase margins for your system. First, form the unity feedback open-loop system by connecting the compensator, plant, and feedback gain in series. olk = k*oloop; Next, compute the margins for this open-loop model. [Gm,Pm,Wcg,Wcp] = margin(olk) 12-144 Design Hard-Disk Read/Write Head Controller Gm = 3.8360 Pm = 43.3068 Wcg = 296.7978 Wcp = 105.4680 This command returns the gain margin, Gm, the phase margin Pm, and their respective cross-over frequencies, Wcg and Wcp. Convert the gain margin to dB. 20*log10(Gm) ans = 11.6775 You can also display the margins graphically. margin(olk) 12-145 12 Classical Control Design This design is robust and can tolerate an 11-dB gain increase or a 40-degree phase lag in the open-loop system without going unstable. By continuing this design process, you may be able to find a compensator that stabilizes the open-loop system and reduces the seek time further. See Also bodeplot | feedback | margin | rlocus 12-146 Design Compensator for Plant Model with Time Delays Design Compensator for Plant Model with Time Delays This example shows how to design a compensator for a plant with time delays using Control System Designer. Analysis and Design of Feedback Systems with Time Delays When working with time delay systems it is advantageous to work with analysis and design tools that directly support time delays so that performance and stability can be evaluated exactly. However, many control design techniques and algorithms cannot directly handle time delays. A common workaround consists of replacing delays by their Pade approximations (all-pass filters). Because this approximation is only valid at low frequencies, it is important to choose the right approximation order and check the approximation validity. Control System Designer provides a variety of design and analysis tools. Some of these tools support time delays exactly while others support time delays indirectly through approximations. Use these tools to design compensators for your control system and visualize the compromises made when using approximations. Plant Model For this example, which uses a unity feedback configuration, the plant model has a time delay: Create the plant model. G = tf(1,[1,1],'InputDelay',0.5); Tools that Support Time Delays In the app, the following tools support time delays directly: 12-147 12 Classical Control Design • Bode and Nichols Editors • Time Response Plots • Frequency Response Plots Open Control System Designer, importing the plant model and using a Bode editor configuration. controlSystemDesigner({'bode'},G) 12-148 Design Compensator for Plant Model with Time Delays The phase response of the Bode plot shows the roll-off effect from the exact representation of the delay. The beginning of the step response shows an exact representation of the 0.5 second delay. Open a Nyquist plot of the open-loop response. In the Data Browser, right-click LoopTransfer_C, and select Plot > nyquist. The Nyquist response wrapping around the origin in a spiral fashion is the result of the exact representation of the time delay. 12-149 12 Classical Control Design Tools that Approximate Time Delays In the app, the following tools approximate time delays: • Root Locus Editor • Pole/Zero Plots • Many of the automated tuning methods When using approximations, the results are not exact and depend on the validity of the approximation. Each tool in Control System Designer provides a warning pane to indicate when time-delays are approximated. Open a root locus editor plot for the open-loop response. Click Tuning Methods, and select Root Locus Editor. In the Select Response to Edit dialog box, click Plot. 12-150 Design Compensator for Plant Model with Time Delays The Root Locus Editor shows a notification that the plot is using a time delay approximation. This notification can be minimized by clicking on the arrow icon to the left. 12-151 12 Classical Control Design Change Approximation Settings To change the approximation settings, click the hyperlink in the notification. In the Control System Designer Preferences dialog box, on the Time Delays tab, specify a Pade order of 4. Alternatively, you can set the bandwidth over which you want the approximation to be accurate. 12-152 Design Compensator for Plant Model with Time Delays The higher-order Pade approximation adds poles and zeros to the root locus plot. 12-153 12 Classical Control Design See Also Control System Designer 12-154 Design Compensator for Systems Represented by Frequency Response Data Design Compensator for Systems Represented by Frequency Response Data This example shows how to design a compensator for a plant model defined by frequency response data (FRD) using Control System Designer. Acquire Frequency Response Data (FRD) Plant Model Non-parametric representations of plant models, such as frequency response data, are often used for analysis and control design. These FRD models are typically obtained from: 1) Signal analyzer hardware that performs frequency domain measurements on systems. 2) Non-parametric estimation techniques using the systems time response data. You can use the following products to estimate FRD models: Simulink® Control Design™: • Function: frestimate • Example: “Frequency Response Estimation Using Simulation-Based Techniques” (Simulink Control Design). Signal Processing Toolbox™: • Function: tfestimate. System Identification Toolbox™: • Functions: etfe, spa, spafdr FRD Model and Design Requirements In this example, design an engine speed controller that actuates the engine throttle angle: 12-155 12 Classical Control Design The frequency response of the engine is already estimated. Load and view the data. load FRDPlantDemoData.mat AnalyzerData AnalyzerData = struct with fields: Response: [594x1 double] Frequency: [594x1 double] FrequencyUnits: 'rad/s' Create an FRD model object: FRDPlant = frd(AnalyzerData.Response,AnalyzerData.Frequency,... 'Unit',AnalyzerData.FrequencyUnits); The design requirements are: • Zero steady-state error for step reference speed changes • Phase margin greater than 60 degrees • Gain margin greater than 20 dB. Design Compensator Open Control System Designer. controlSystemDesigner({'bode','nichols'},FRDPlant)| The Control System Designer opens with both Bode and Nichols open-loop editors. 12-156 Design Compensator for Systems Represented by Frequency Response Data You can design the compensator by shaping the open-loop frequency response in either the Bode editor or Nichols editor. In these editors, interactively modify the gain, poles, and zeros of the compensator. To satisfy the tracking requirement of zero steady-state error, add an integrator to the compensator. Right-click the Bode editor plot area, and select Add Pole/Zero > Integrator. To meet the gain and phase margin requirements, add a zero to the compensator. Rightclick the Bode editor plot area, and select Add Pole/Zero > Real Zero. Modify the 12-157 12 Classical Control Design location of the zero and the gain of the compensator until you satisfy the margin requirements. One possible design that satisfies the design requirements is: This compensator design, which is a PI controller, achieves a 20.7 dB gain margin and a 70.8 degree phase margin. 12-158 Design Compensator for Systems Represented by Frequency Response Data Export the designed compensator to the workspace. Click Export. Validate the Design Validate the controller performance by simulating the engine response using a nonlinear model in Simulink®. For this example, the validation simulation results are in EngineStepResponse. Plot the response of the engine to a reference speed change from 2000 to 2500 RPM: plot(EngineStepResponse.Time,EngineStepResponse.Speed) title('Engine Step Response') xlabel('Time (s)') ylabel('Engine Speed (RPM)') 12-159 12 Classical Control Design The response shows zero steady-state error and well-behaved transients with the following metrics. stepinfo(EngineStepResponse.Speed,EngineStepResponse.Time) ans = struct with RiseTime: SettlingTime: SettlingMin: SettlingMax: Overshoot: Undershoot: Peak: PeakTime: fields: 1.1048 1.7194 2.4501e+03 2.5078e+03 0.3127 0 2.5078e+03 2.3853 See Also Control System Designer 12-160 Design Internal Model Controller for Chemical Reactor Plant Design Internal Model Controller for Chemical Reactor Plant This example shows how to design a compensator in an IMC structure for series chemical reactors, using Control System Designer. Model-based control systems are often used to track setpoints and reject load disturbances in process control applications. Plant Model The plant for this example is a chemical reactor system, comprised of two well-mixed tanks. The reactors are isothermal and the reaction in each reactor is first order on component A: 12-161 12 Classical Control Design Material balance is applied to the system to generate a dynamic model of the system. The tank levels are assumed to stay constant because of the overflow nozzle and hence there is no level control involved. For details about this plant, see Example 3.3 in Chapter 3 of "Process Control: Design Processes and Control Systems for Dynamic Performance" by Thomas E. Marlin. The following differential equations describe the component balances: At steady state, the material balances are: where , , and are steady-state values. Substitute, the following design specifications and reactor parameters: 12-162 Design Internal Model Controller for Chemical Reactor Plant The resulting steady-state concentrations in the two reactors are: where For this example, design a controller to maintain the outlet concentration of reactant from , in the presence of any disturbance in feed concentration, the second reactor, The manipulated variable is the molar flowrate of the reactant, F, entering the first reactor. . Linear Plant Models In this control design problem, the plant model is and the disturbance model is This chemical process can be represented using the following block diagram: 12-163 12 Classical Control Design where 12-164 Design Internal Model Controller for Chemical Reactor Plant Based on the block diagram, obtain the plant and disturbance models as follows: Create the plant model at the command line: s = tf('s'); G1 = (13.3259*s+3.2239)/(8.2677*s+1)^2; G2 = G1; Gd = 0.4480/(8.2677*s+1)^2; G1 is the real plant used in controller evaluation. G2 is an approximation of the real plant and it is used as the predictive model in the IMC structure. G2 = G1 means that there is no model mismatch. Gd is the disturbance model. Define IMC Structure in Control System Designer Open Control System Designer. controlSystemDesigner 12-165 12 Classical Control Design Select the IMC control architecture. In Control System Designer, click Edit Architecture. In the Edit Architecture dialog box, select Configuration 5. 12-166 Design Internal Model Controller for Chemical Reactor Plant Load the system data. For G1, G2, and Gd, specify a model Value. 12-167 12 Classical Control Design Tune Compensator Plot the open-loop step response of G1. step(G1) 12-168 Design Internal Model Controller for Chemical Reactor Plant Right-click the plot and select Characteristics > Rise Time submenu. Click the blue rise time marker. 12-169 12 Classical Control Design The rise time is about 25 seconds and we want to tune the IMC compensator to achieve a faster closed-loop response time. To tune the IMC compensator, in Control System Designer, click Tuning Methods, and select Internal Model Control (IMC) Tuning. Select a Dominant closed-loop time constant of 2 and a Desired controller order of 2. 12-170 Design Internal Model Controller for Chemical Reactor Plant To view the closed-loop step response, in Control System Designer, double-click the IOTransfer_r2y:step plot tab. 12-171 12 Classical Control Design Control Performance with Model Mismatch When designing the controller, we assumed G1 was equal to G2. In practice, they are often different, and the controller needs to be robust enough to track setpoints and reject disturbances. Create model mismatches between G1 and G2 and examine the control performance at the MATLAB command line in the presence of both setpoint change and load disturbance. Export the IMC Compensator to the MATLAB workspace. Click Export. In the Export Model dialog box, select compensator model C. 12-172 Design Internal Model Controller for Chemical Reactor Plant Click Export. Convert the IMC structure to a classic feedback control structure with the controller in the feedforward path and unit feedback. C = zpk([-0.121 -0.121],[-0.242, -0.466],2.39); C_new = feedback(C,G2,+1) C_new = 2.39 (s+0.121)^4 --------------------------------------------(s-0.0001594) (s+0.121) (s+0.1213) (s+0.2419) Continuous-time zero/pole/gain model. Define the following plant models: • No Model Mismatch: G1p = (13.3259*s+3.2239)/(8.2677*s+1)^2; 12-173 12 Classical Control Design • G1 time constant changed by 5%: G1t = (13.3259*s+3.2239)/(8.7*s+1)^2; • G1 gain is increased by 3 times: G1g = 3*(13.3259*s+3.2239)/(8.2677*s+1)^2; Evaluate the setpoint tracking performance. step(feedback(G1p*C_new,1),feedback(G1t*C_new,1),feedback(G1g*C_new,1)) legend('No Model Mismatch','Mismatch in Time Constant','Mismatch in Gain') Evaluate the disturbance rejection performance. 12-174 See Also step(Gd*feedback(1,G1p*C_new),Gd*feedback(1,G1t*C_new),Gd*feedback(1,G1g*C_new)) legend('No Model Mismatch','Mismatch in Time Constant','Mismatch in Gain') The controller is fairly robust to uncertainties in the plant parameters. See Also Control System Designer More About • “Design Compensator Using Automated Tuning Methods” on page 12-100 12-175 12 Classical Control Design Design LQG Tracker Using Control System Designer This example shows how to use LQG synthesis to design a feedback controller for a disk drive read/write head using Control System Designer. For details about the system and model, see Chapter 14 of "Digital Control of Dynamic Systems," by Franklin, Powell, and Workman. Disk Drive Model Below is a picture of the system to be modeled. 12-176 Design LQG Tracker Using Control System Designer The model input is the current driving the voice coil motor, and the output is the position error signal (PES, in % of track width). To learn more about the 10th order model, see “Digital Servo Control of a Hard-Disk Drive”. The plant includes a small time delay. For the purpose of this example, ignore this delay. load diskdemo Gr = tf(1e6,[1 12.5 0]); Gf1 = tf(w1*[a1 b1*w1],[1 Gf2 = tf(w2*[a2 b2*w2],[1 Gf3 = tf(w3*[a3 b3*w3],[1 Gf4 = tf(w4*[a4 b4*w4],[1 G = (ss(Gf1)+Gf2+Gf3+Gf4) 2*z1*w1 2*z2*w2 2*z3*w3 2*z4*w4 * Gr; w1^2]); % first resonance w2^2]); % second resonance w3^2]); % third resonance w4^2]); % fourth resonance % convert to state space for accuracy Design Overview In this example, design a full-ordered LQG tracker, which places the read/write head at the correct position. Tune the LQG tracker to achieve specific performance requirements and reduce the controller order as much as possible. For example, turn the LQG tracker into a PI controller format. Open Control System Designer Open Control System Designer, importing the plant model. controlSystemDesigner(G) By default, Control System Designer displays the step response of the closed-loop system along with Bode and root locus graphical editors for the open-loop response. Maximize the step response. Double-click the IOTransfer_r2y: step plot tab. Details about how to use the Control System Designer are described in “Getting Started with the Control System Designer”. 12-177 12 Classical Control Design The default unity gain compensator produces a stable closed-loop system with large oscillations. Design a Full-Order LQG Tracker Click Tuning Methods, and select LQG Synthesis. 12-178 Design LQG Tracker Using Control System Designer In the LQG Synthesis dialog box, in the Specifications section, set requirements on the controller performance: • Controller response - Specify the controller transient behavior. You can make the controller more aggressive at disturbance rejection or more robust against plant uncertainty. If you believe your model is accurate and that the manipulated variable has a large enough range, an aggressive controller is preferable. • Measurement noise - Specify an estimate of the level of output measurement noise for your application. To produce a more robust controller, specify a larger noise estimate. 12-179 12 Classical Control Design • Desired controller order - Specify your controller order preference. Use default slider settings as the initial controller design. Click Update Compensator. The new Compensator is displayed, and the step response updates. 12-180 Design LQG Tracker Using Control System Designer To design a more aggressive controller, move the Controller response slider to the far left. The more aggressive controller reduces the overshoot by 50% and reduces the settling time by 70%. 12-181 12 Classical Control Design Design a Reduced-Order LQG Tracker To create a PI controller, reset the Controller response slider to the middle default value, and set the Desired controller order to 1. 12-182 Design LQG Tracker Using Control System Designer Click Update Compensator. 12-183 12 Classical Control Design This controller produces a heavily oscillating closed-loop system. To make the controller less aggressive, move the Controller response slider to the right. 12-184 Design LQG Tracker Using Control System Designer Click Update Compensator. 12-185 12 Classical Control Design The step response shows that the PI controller design provides a good starting point for optimization-based design. For more information, see “Getting Started with the Control System Designer”. See Also Control System Designer 12-186 See Also More About • “Design Compensator Using Automated Tuning Methods” on page 12-100 12-187 12 Classical Control Design Export Design to MATLAB Workspace After designing your controller in Control System Designer, you can export your design to the MATLAB Workspace for further analysis or design. To export your design: 12-188 1 In Control System Designer, on the Control System tab, under Export, click Export tuned blocks. 2 In the Export Model dialog box, in the Select Design drop-down list, choose the design that you want to export. You can select either the Current Design or one of the stored designs from the Data Browser. See Also 3 In the Export models to MATLAB Workspace table, in the Export column, select the models you want to export. For all designs, you can export the controller and prefilter models. Also, for the Current Design, you can export the fixed block models and any responses from the Data Browser. For more information on the prefilter, controller, and fixed blocks in each control architecture, see “Feedback Control Architectures” on page 12-21. 4 In the Export as column, you can specify an alternate name for the exported model. Exporting a model with the same name as an existing variable in the MATLAB Workspace overwrites the variable. 5 To save the selected models to the MATLAB Workspace, click Export. See Also Apps Control System Designer 12-189 12 Classical Control Design More About 12-190 • “Feedback Control Architectures” on page 12-21 • “Generate Simulink Model for Control Architecture” on page 12-191 Generate Simulink Model for Control Architecture Generate Simulink Model for Control Architecture After designing your controllers in Control System Designer, to simulate your system, you can automatically generate a Simulink model for your control architecture. To do so, on the Control System tab, under Export, click Create Simulink model. The app exports the controllers and fixed blocks for the current design to the MATLAB Workspace and generates a Simulink model that matches the current control architecture. For more information on the controllers and fixed blocks in each control architecture, see “Feedback Control Architectures” on page 12-21. For example, if you design a control system using configuration 1, Control System Designer exports C, F, G, and H to the MATLAB Workspace and generates the following Simulink model. 12-191 12 Classical Control Design In the generated model, the Input block is a Signal Generator. Using this block, you simulate your model with different input waveforms, such as sine waves or random signals. To generate a step response, replace the Input block with a Step block. To generate a Simulink model for a stored design, first make that design current. On the Control System tab, under Retrieve, select the design for which you want to generate a model. See Also Control System Designer More About 12-192 • “Feedback Control Architectures” on page 12-21 • “Export Design to MATLAB Workspace” on page 12-188 Tune Simulink Blocks Using Compensator Editor Tune Simulink Blocks Using Compensator Editor This example shows how to tune Simulink® blocks using the Compensator Editor dialog box in Control System Designer. Open the Model This example uses a model of a speed control system for a sparking ignition engine. The initial compensator has been designed in a fashion similar to the method shown in “Single Loop Feedback/Prefilter Compensator Design” (Simulink Control Design). Open and explore the engine speed control model. open_system('scdspeedctrl') Introduction This example uses the Compensator Editor to tune Simulink blocks. When tuning a block in Simulink using Control System Designer, you can tune the block parameters directly or you can tune a zero-pole-gain representation of the block. For example, in the speed control example there is a PID controller with filtered derivative scdspeedctrl/PID Controller: 12-193 12 Classical Control Design 12-194 Tune Simulink Blocks Using Compensator Editor This block implements the traditional PID with filtered derivative as: In this block P, I, D, and N are the parameters that are available for tuning. Another approach is to reformulate the block transfer function to use zero-pole-gain format: This formulation of poles, zeros, and gains allows for direct graphical tuning on design plots such as Bode, root locus, and Nichols plots. Additionally, Control System Designer allows for both representations to be tuned using the compensator editor. The tuning of both representations is available for all supported blocks in Simulink Control Design™. For more information, see “What Blocks Are Tunable?” (Simulink Control Design). Open Control System Designer In this example, to tune the compensators in this feedback system, open a preconfigured Control System Designer session by double clicking the subsystem in the lower lefthand corner of the model. Compensator Editor Dialog Box You can view the representations of the PID compensator using the Compensator Editor dialog box. To open the Compensator Editor, in the Data Browser, in the Controllers and Fixed Blocks section, double-click scdspeedctrl_PID_Controller. In the Compensator Editor dialog box, in the Compensator section, you can view and edit any of the compensators in your system. 12-195 12 Classical Control Design On the Pole/Zero tab, you can add, delete, and edit compensator poles and zeros. Since the PID with filtered derivative is fixed in structure, the number of poles and zeros is limited to having up to two zeros, one pole, and an integrator at s = 0. On the Parameter tab, you can independently tune the P, I, D, and N parameters. 12-196 Tune Simulink Blocks Using Compensator Editor Enter new parameters values in the Value column. To interactively tune the parameters, use the sliders. You can change the slider limits using the Min Value and Max Value columns. When you change parameter values, any associated editor and analysis plots automatically update. Complete Design The design requirements in “Single Loop Feedback/Prefilter Compensator Design” (Simulink Control Design) can be met with the following controller parameters: 12-197 12 Classical Control Design • scdspeedctrl/PID Controller: P = 0.0012191 I = 0.0030038 • scdspeedctrl/Reference Filter: Numerator = 10 Denominator = [1 10] In the Compensator Editor dialog box, specify these parameters. Then, in Control System Designer, view the closed-loop responses. 12-198 Tune Simulink Blocks Using Compensator Editor 12-199 12 Classical Control Design Update Simulink Model To write the compensator parameters back to the Simulink model, click Update Blocks. You can then test your design on the nonlinear model. bdclose('scdspeedctrl') 12-200 Single Loop Feedback/Prefilter Compensator Design Single Loop Feedback/Prefilter Compensator Design This example shows how to tune multiple compensators (feedback and prefilter) to control a single loop using Control System Designer. Open the Model Open the engine speed control model and take a few moments to explore it. open_system('scdspeedctrl') Design Overview This example introduces the process of designing a single-loop control system with both feedback and prefilter compensators. The goal of the design is to: • Track the reference signal from a Simulink step block scdspeedctrl/Speed Reference. The design requirement is to have a settling time of under 5 seconds and zero steady-state error to the step reference input. • Reject an unmeasured output disturbance specified in the subsystem scdspeedctrl/ External Disturbance. The design requirement is to reduce the peak deviation to 190 RPM and to have zero steady-state error for a step disturbance input. In this example, the stabilization of the feedback loop and the rejection of the output disturbance are achieved by designing the PID compensator scdspeedctrl/PID Controller. The prefilter scdspeedctrl/Reference Filter is used to tune the response of the feedback system to changes in the reference tracking. 12-201 12 Classical Control Design Open Control System Designer This example uses Control System Designer to tune the compensators in the feedback system. To open the Control System Designer • Launch a pre-configured Control System Designer session by double-clicking the subsystem in the lower left corner of the model. • Configure Control System Designer using the following procedure. Start a New Design To open Control System Designer, in the Simulink model window, select Analysis > Control Design > Control System Designer. The Edit Architecture dialog box opens when the Control System Designer launches. 12-202 Single Loop Feedback/Prefilter Compensator Design In the Edit Architecture dialog box, on the Blocks tab, click Add Blocks, and select the following blocks to tune: • scdspeedctrl/Reference Filter 12-203 12 Classical Control Design • scdspeedctrl/PID Controller On the Signals tab, the analysis points defined in the Simulink model are automatically added as Locations. • Input: scdspeedctrl/Speed Reference output port 1 • Input scdspeedctrl/External Disturbance/Step Disturbance output port 1 • Output scdspeedctrl/Speed Output output port 1 12-204 Single Loop Feedback/Prefilter Compensator Design On the Linearization Options tab, in the Operating Point drop-down list, select Model Initial Condition. Create new plots to view the step responses while tuning the controllers. In Control System Designer, click New Plot, and select New Step. In the Select Response to Plot drop-down menu, select New Input-Output Transfer Response. Configure the response as follows: 12-205 12 Classical Control Design To view the response, click Plot. Similarly, create a step response plot to show the disturbance rejection. In the New Step to plot dialog box, configure the response as follows: 12-206 Single Loop Feedback/Prefilter Compensator Design Tune Compensators Control System Designer contains several methods tuning a control system: • Manually tune the parameters of each compensator using the compensator editor. For more information, see “Tune Simulink Blocks Using Compensator Editor” (Simulink Control Design). • Graphically tune the compensator poles, zeros, and gains using open/closed-loop Bode, root locus, or Nichols editor plots. Click Tuning Methods, and select an editor under Graphical Tuning. • Optimize compensator parameters using both time-domain and frequency-domain design requirements (requires Simulink Design Optimization™ software). Click 12-207 12 Classical Control Design Tuning Methods, and select Optimization based tuning. For more information, see “Enforcing Time and Frequency Requirements on a Single-Loop Controller Design” (Simulink Design Optimization). • Compute initial compensator parameters using automated tuning based on parameters such as closed-loop time constants. Click Tuning Methods, and select either PID Tuning, Internal Model Control (IMC) Tuning, Loop Shaping (requires Robust Control Toolbox™ software), or LQG Synthesis. Completed Design The following compensator parameters satisfy the design requirements: • scdspeedctrl/PID Controller has parameters: P = 0.0012191 I = 0.0030038 • scdspeedctrl/Reference Filter: Numerator = 10 Denominator = [1 10] The responses of the closed-loop system are shown below: 12-208 Single Loop Feedback/Prefilter Compensator Design 12-209 12 Classical Control Design Update Simulink Model To write the compensator parameters back to the Simulink model, click Update Blocks. You can then test your design on the nonlinear model. bdclose('scdspeedctrl') See Also Control System Designer 12-210 Cascaded Multiloop Feedback Design Cascaded Multiloop Feedback Design This example shows how to tune two cascaded feedback loops in Simulink® Control Design™ using Control System Designer. This example designs controllers for two cascaded feedback loops in an airframe model such that the acceleration component (az) tracks reference signals with a maximum rise time of 0.5 seconds. The feedback loop structure in this example uses the body rate (q) as an inner feedback loop and the acceleration (az) as an outer feedback loop. Open the airframe model. open_system('scdairframectrl') The two feedback controllers are: • scdairframectrl/q Control - A discrete-time integrator and a gain block stabilize the inner loop. 12-211 12 Classical Control Design open_system('scdairframectrl/q Control') • scdairframectrl/az Control - A discrete-time integrator, a discrete transfer function, and a gain block stabilize the outer loop. open_system('scdairframectrl/az Control') Decoupling Loops in Multiloop Systems The typical design procedure for cascaded feedback systems is to first design the inner loop and then the outer loop. In Control System Designer, it is possible to design both loops simultaneously; by default, when designing a multi-loop feedback system the coupling effects between loops are taken into account. However, when designing two feedback loops simultaneously, it can be necessary decouple the feedback loops; that is, remove the effect of the outer loop when tuning the inner loop. In this example, you design the inner feedback loop (q) with the effect of the outer loop (az) removed. Configure Control System Designer To design a controller using Control System Designer, you must: • Select the controller blocks that you want to tune. • Create the open-loop and closed-loop responses that you want to view. For this example, you can: • Launch a preconfigured Control System Designer session by double-clicking the subsystem in the lower left corner of the model. • Configure the Control System Designer using the following procedure. 12-212 Cascaded Multiloop Feedback Design To open Control System Designer, in the Simulink model window, select Analysis > Control Design > Control System Designer. In the Edit Architecture dialog box, on the Blocks tab, click Add Blocks. In the Select Blocks to Tune dialog box, select the following blocks, and click OK. • scdairframectrl/q Control/q Gain • scdairframectrl/az Control/az Gain • scdairframectrl/az Control/az DTF On the Signals tab, the analysis points defined in the Simulink model are automatically added as Locations. • Input: scdairframectrl/Step az - Output port 1 • Output: scdairframectrl/Airframe Model - Output port 1 To use the selected blocks and signals, click OK. In the Data Browser, the Responses section contains the following open-loop responses, which Control System Designer automatically recognizes as potential feedback loops for open-loop design. • Output port 1 of scdairframectrl/az Control/az DTF • Output port 1 of scdairframectrl/az Control/az Gain 12-213 12 Classical Control Design • Output port 1 of scdairframectrl/q Control/q Gain Open graphical Bode editors for each of the following responses. In Control System Designer, select Tuning Methods > Bode Editor. Then, in the Select Response to Edit drop-down list, select the corresponding open-loop responses, and click Plot. • Open Loop at outport 1 of scdairframectrl/az Control/az DTF • Open Loop at outport 1 of scdairframectrl/q Control/q Gain To view the closed-loop response of the feedback system, create a step plot for a new input-output transfer function response. Select New Plot > New Step. Then, in the New 12-214 Cascaded Multiloop Feedback Design Step to plot dialog box, in the Select Response to Plot drop-down list, select New Input-Output Transfer Response. Add scdairframectrl/Step az/1 as an input signal and scdairframectrl/ Airframe Model/1 as an output signal. Click Plot. Remove Effect of Outer Feedback Loop In the outer-loop bode editor plot, Bode Editor for LoopTransfer_scdairframectrl_az_Control_az_DTF, increase the gain of the feedback loop by dragging the magnitude response upward. The inner-loop bode editor plot, Bode 12-215 12 Classical Control Design Editor for LoopTransfer_scdairframectrl_q_Control_q_Gain, also changes. This change is a result of the coupling between the feedback loops. A more systematic approach is to first design the inner feedback loop, with the outer loop open. To remove the effect of the outer loop when designing the inner loop, add a loop opening to the open-loop response of the inner loop. In the Data Browser, in the Responses area, right-click the inner loop response, and select Open Selection. 12-216 Cascaded Multiloop Feedback Design In the Open-Loop Transfer Function dialog box, specify scdairframectrl/az Control/az DTF/1 as the loop opening. Click OK. 12-217 12 Classical Control Design In the outer-loop Bode editor plot, increase the gain by dragging the magnitude response. Since the loops are decoupled, the inner-loop Bode editor plot does not change. You can now complete the design of the inner loop without the effect of the outer loop and simultaneously design the outer loop while taking the effect of the inner loop into account. Tune Compensators Control System Designer contains several methods tuning a control system: 12-218 Cascaded Multiloop Feedback Design • Manually tune the parameters of each compensator using the compensator editor. For more information, see “Tune Simulink Blocks Using Compensator Editor” (Simulink Control Design). • Graphically tune the compensator poles, zeros, and gains using open/closed-loop Bode, root locus, or Nichols editor plots. Click Tuning Methods, and select an editor under Graphical Tuning. • Optimize compensator parameters using both time-domain and frequency-domain design requirements (requires Simulink Design Optimization™ software). Click Tuning Methods, and select Optimization based tuning. For more information, see “Enforcing Time and Frequency Requirements on a Single-Loop Controller Design” (Simulink Design Optimization). • Compute initial compensator parameters using automated tuning based on parameters such as closed-loop time constants. Click Tuning Methods, and select either PID Tuning, Internal Model Control (IMC) Tuning, Loop Shaping (requires Robust Control Toolbox™ software), or LQG Synthesis. Complete Design The following compensator parameters satisfy the design requirements: • scdairframectrl/q Control/q Gain: K_q = 2.7717622 • scdairframectrl/az Control/az Gain: K_az = 0.00027507 • scdairframectrl/az Control/az DTF: Numerator = [100.109745 -99.109745] Denominator = [1 -0.88893] The response of the closed-loop system is shown below: 12-219 12 Classical Control Design 12-220 See Also Update Simulink Model To write the compensator parameters back to the Simulink model, click Update Blocks. You can then test your design on the nonlinear model. bdclose('scdairframectrl') See Also Control System Designer More About • “Design Multiloop Control System” on page 12-24 12-221 12 Classical Control Design Reference Tracking of DC Motor with Parameter Variations This example shows how to generate an array of LTI models that represent the plant variations of a control system from a Simulink model. This array of models is used in Control System Designer for control design. DC Motor Model In armature-controlled DC motors, the applied voltage Va controls the angular velocity of the shaft. A simplified model of the DC motor is shown below. Open the Simulink model for the DC motor. 12-222 Reference Tracking of DC Motor with Parameter Variations mdl = 'scdDCMotor'; open_system(mdl) Perform Batch Linearization The goal of the controller is to provide tracking to step changes in reference angular velocity. For this example, the physical constants for the motor are: • R = 2.0 +/- 10% Ohms • L = 0.5 Henrys • Km = 0.1 Torque constant • Kb = 0.1 Back emf constant • Kf = 0.2 Nms • J = 0.02 +/- .01 kg m^2/s^2 Note that parameters R and J are specified as a range of values. To design a controller which will work for all physical parameter values, create a representative set of plants by sampling these values. For parameters R and J, use their nominal, minimum, and maximum values. R = [2,1.8,2.2]; J = [.02,.03,.01]; To create an LTI array of plant models, batch linearize the DC motor plant. For each combination of the sample values of R and J, linearize the Simulink model. To do so, specify a linearization input point at the output of the controller block and a linearization output point with a loop opening at the output of the load block as shown in the model. 12-223 12 Classical Control Design Get the linearization analysis points specified in the model. io = getlinio(mdl); Vary the plant parameters R and J. [R_grid,J_grid] = ndgrid(R,J); params(1).Name = 'R'; params(1).Value = R_grid; params(2).Name = 'J'; params(2).Value = J_grid; Linearize the model for each parameter value combination. sys = linearize(mdl,io,params); Open Control System Designer Open Control System Designer, and import the array of plant models. using the following command. controlSystemDesigner(sys) Using Control System Designer, you can design a controller for the nominal plant model while simultaneously visualizing the effect on the other plant models as shown below. 12-224 Reference Tracking of DC Motor with Parameter Variations The root locus editor displays the root locus for the nominal model and the closed-loop pole locations associated with the other plant models. The Bode editor displays both the nominal model response and the responses of the other plant models. The step responses show that reference tracking is not achieved for any of the plant models. 12-225 12 Classical Control Design Design Controller Using the tools in Control System Designer, design the following compensator for reference tracking. The resulting design is shown below. The closed-loop step response shows that the goal of reference tracking is achieved with zero steady-state error for all models defined in the plant set. However, if a zero percent overshoot requirement is necessary, not all responses would satisfy this requirement. 12-226 Reference Tracking of DC Motor with Parameter Variations Export Design and Validate in Simulink Model To export the designed controller to the MATLAB workspace, click Export. In the Export Model dialog box, select C, and click Export. Write the controller parameters to the Simulink model. [Cnum,Cden] = tfdata(C,'v'); hws = get_param(mdl, 'modelworkspace'); assignin(hws,'Cnum',Cnum) assignin(hws,'Cden',Cden) 12-227 12 Classical Control Design More Information For more information on using the multimodel features of Control System Designer, see “Multimodel Control Design” on page 12-35. bdclose('scdDCMotor') 12-228 13 State-Space Control Design • “Extended and Unscented Kalman Filter Algorithms for Online State Estimation” on page 13-2 • “Generate Code for Online State Estimation in MATLAB” on page 13-11 • “Validate Online State Estimation at the Command Line” on page 13-15 • “Validate Online State Estimation in Simulink” on page 13-18 • “Troubleshoot Online State Estimation” on page 13-22 • “Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter” on page 13-24 • “Estimate States of Nonlinear System with Multiple, Multirate Sensors” on page 13-45 • “Regulate Pressure in Drum Boiler” on page 13-59 13 State-Space Control Design Extended and Unscented Kalman Filter Algorithms for Online State Estimation You can use discrete-time extended and unscented Kalman filter algorithms for online state estimation of discrete-time nonlinear systems. If you have a system with severe nonlinearities, the unscented Kalman filter algorithm may give better estimation results. You can perform the state estimation in Simulink and at the command line. To perform the state estimation, you first create the nonlinear state transition function and measurement function for your system. At the command line, you use the functions to construct the extendedKalmanFilter or unscentedKalmanFilter object for desired algorithm, and specify whether the process and measurement noise terms in the functions are additive or non-additive. After you create the object, you use the predict and correct commands to estimate the states using real-time data. For information about the order in which to execute these commands, see the predict and correct reference pages. In Simulink, you specify these function in the Extended Kalman Filter and Unscented Kalman Filter blocks. You also specify whether the process and measurement noise terms in the functions are additive or non-additive. In the blocks, the software decides the order in which prediction and correction of state estimates is done. Extended Kalman Filter Algorithm The extendedKalmanFilter command and Extended Kalman Filter block implement the first-order discrete-time Kalman filter algorithm. Assume that the state transition and measurement equations for a discrete-time nonlinear system have non-additive process and measurement noise terms with zero mean and covariance matrices Q and R, respectively: x[ k + 1] = f ( x[ k ], w[ k],us[ k ]) y[ k ] = h( x[ k], v[ k],um [ k]) w[ k] ~ (0, Q[ k ]) v[ k] ~ ( 0, R[ k]) Here f is a nonlinear state transition function that describes the evolution of states x from one time step to the next. The nonlinear measurement function h relates x to the measurements y at time step k. These functions can also have additional input arguments 13-2 Extended and Unscented Kalman Filter Algorithms for Online State Estimation that are denoted by us and um. The process and measurement noise are w and v, respectively. You provide Q and R. In the block, the software decides the order of prediction and correction of state estimates. At the command line, you decide the order. For information about the order in which to execute these commands, see the predict and correct reference pages. Assuming that you implement the correct command before predict, the software implements the algorithm as follows: 1 Initialize the filter object with initial values of the state, x[0], and state estimation error covariance matrix, P. xˆ [0 | -1 ] = E( x[ 0]) P[ 0 | -1] = E( x[0 ] - xˆ[ 0 | -1])( x[0 ] - xˆ [0 | -1])T Here x̂ is the state estimate and xˆ [ ka | kb ] denotes the state estimate at time step ka using measurements at time steps 0,1,...,kb. So xˆ [0 | -1 ] is the best guess of the state value before you make any measurements. You specify this value when you construct the filter. 2 For time steps k = 0,1,2,3,..., perform the following: a Compute the Jacobian of the measurement function, and update the state and state estimation error covariance using the measured data, y[k]. At the command line, the correct command performs this update. C[ k] = ∂h ∂x x[ k|k-1] S[ k] = ∂h ∂v x[ k|k-1] The software calculates these Jacobian matrices numerically unless you specify the analytical Jacobian. K[ k ] = P[ k |k - 1 ]C[ k]T (C[ k ]P[ k | k - 1]C[ k]T + S[ k] R[ k]S[ k]T ) -1 x[ k| k ] = x[ k | k - 1] + K[ k ]( y[ k ] - h( x[ k| k - 1 ], 0, um [ k ]) P[ k | k] = P[ k| k - 1] - K[ k ] C[ k] P[ k | k - 1 ] 13-3 13 State-Space Control Design Here K is the Kalman gain. b Compute the Jacobian of the state transition function, and predict the state and state estimation error covariance at the next time step. In the software, the predict command performs this prediction. A[ k ] = G[ k] = ∂f ∂x x[ k|k] ∂f ∂ w x[ k|k] The software calculates these Jacobian matrices numerically unless you specify the analytical Jacobian. This numerical computation may increase processing time and numerical inaccuracy of the state estimation. P[ k + 1| k] = A[ k ]P[ k | k] A[ k ]T + G[ k ]Q[ k ]G[ k]T x[ k + 1| k ] = f ( x[ k| k ], 0, us[ k ]) These values are used by the correct command in the next time step. The Extended Kalman Filter block supports multiple measurement functions. These measurements can have different sample times as long as their sample time is an integer multiple of the state transition sample time. In this case, a separate correction step is performed corresponding to measurements from each measurement function. The algorithm steps described previously assume that you have non-additive noise terms in the state transition and measurement functions. If you have additive noise terms in the functions, the changes in the algorithm are: • If the process noise w is additive, that is the state transition equation has the form x[ k] = f ( x[ k - 1], us[ k - 1 ]) + w[ k - 1 ] , then the Jacobian matrix G[k] is an identity matrix. • If the measurement noise v is additive, that is the measurement equation has the form y[ k ] = h( x[ k], um [ k]) + v[ k ] , then the Jacobian matrix S[k] is an identity matrix. Additive noise terms in the state and transition functions reduce the processing time. The first-order extended Kalman filter uses linear approximations to nonlinear state transition and measurement functions. As a result, the algorithm may not be reliable if 13-4 Extended and Unscented Kalman Filter Algorithms for Online State Estimation the nonlinearities in your system are severe. The unscented Kalman filter algorithm may yield better results in this case. Unscented Kalman Filter Algorithm The unscented Kalman filter algorithm and Unscented Kalman Filter block use the unscented transformation to capture the propagation of the statistical properties of state estimates through nonlinear functions. The algorithm first generates a set of state values called sigma points. These sigma points capture the mean and covariance of the state estimates. The algorithm uses each of the sigma points as an input to the state transition and measurement functions to get a new set of transformed state points. The mean and covariance of the transformed points is then used to obtain state estimates and state estimation error covariance. Assume that the state transition and measurement equations for an M-state discrete-time nonlinear system have additive process and measurement noise terms with zero mean and covariances Q and R, respectively: x[ k + 1] = f ( x[ k ], us[ k]) + w[ k] y[ k ] = h( x[ k], um [ k]) + v[ k ] w[ k] ~ (0, Q[ k ]) v[ k] ~ ( 0, R[ k]) You provide the initial values of Q and R in the ProcessNoise and MeasurementNoise properties of the unscented Kalman filter object. In the block, the software decides the order of prediction and correction of state estimates. At the command line, you decide the order. For information about the order in which to execute these commands, see the predict and correct reference pages. Assuming that you implement the correct command before predict, the software implements the algorithm as follows: 1 Initialize the filter object with initial values of the state, x[0], and state estimation error covariance, P. xˆ [0 | -1 ] = E( x[ 0]) P[ 0 | -1] = E( x[0 ] - xˆ[ 0 | -1])( x[0 ] - xˆ [0 | -1])T 13-5 13 State-Space Control Design Here x̂ is the state estimate and xˆ [ ka | kb ] denotes the state estimate at time step ka using measurements at time steps 0,1,...,kb. So xˆ [0 | -1 ] is the best guess of the state value before you make any measurements. You specify this value when you construct the filter. 2 For each time step k, update the state and state estimation error covariance using the measured data, y[k]. In the software, the correct command performs this update. a Choose the sigma points xˆ ( i) [ k |k - 1] at time step k. xˆ (0) [ k | k - 1] = xˆ[ k |k - 1] xˆ ( i) [ k |k - 1] ( i) Dx = D x( M +i ) = = xˆ [ k| k - 1 ] + Dx( i) ( )i - ( cP[ k| k - 1 ] ) i cP[ k| k - 1 ] i = 1,..., 2 M i = 1,..., M i = 1,..., M Where c = a 2 ( M + k ) is a scaling factor based on number of states M, and the parameters α and κ. For more information about the parameters, see “Effect of Alpha, Beta, and Kappa Parameters” on page 13-9. root of cP such that b cP ( cP ) 13-6 = cP and ( cP )i is the ith column of cP . Use the nonlinear measurement function to compute the predicted measurements for each of the sigma points. yˆ ( i) [ k | k - 1] = h( xˆ ( i) [ k | k - 1] , um [k]) c T cP is the matrix square i = 0,1,..., 2 M Combine the predicted measurements to obtain the predicted measurement at time k. Extended and Unscented Kalman Filter Algorithms for Online State Estimation 2M yˆ[ k ] = Â WM(i) yˆ(i) [k| k - 1] i =0 ( 0) WM =1i WM = d M 2 a (M + k ) 1 2a 2 ( M + k ) i = 1, 2,..., 2 M Estimate the covariance of the predicted measurement. Add R[k] to account for the additive measurement noise. 2M Py = Â Wc(i) ( y(i)[k|k - 1] - y[k])( y(i)[k| k - 1] - y[k])T + R[k] i =0 Wc( 0) = (2 - a 2 + b ) Wci 2 M 2 a (M +k) = 1 / (2a ( M + k )) i = 1, 2,..., 2M For information about β parameter, see “Effect of Alpha, Beta, and Kappa Parameters” on page 13-9. e Estimate the cross-covariance between xˆ [ k| k - 1 ] and yˆ[ k ] . Pxy = 1 2M Â (x(i)[k| k - 1] - x[k| k - 1])( y(i)[k| k - 1] - y[k])T 2a 2 ( m + k ) i =1 The summation starts from i = 1 because xˆ (0) [ k | k - 1] - xˆ [ k| k - 1] = 0 . f Obtain the estimated state and state estimation error covariance at time step k. K = Pxy Py-1 xˆ [ k| k ] = xˆ[ k | k - 1] + K ( y[ k ] - yˆ [ k]) P[ k | k] = P[ k| k - 1] - KPy K T k Here K is the Kalman gain. 13-7 13 State-Space Control Design 3 Predict the state and state estimation error covariance at the next time step. In the software, the predict command performs this prediction. a Choose the sigma points xˆ ( i) [ k |k ] at time step k. xˆ (0) [ k | k] = xˆ [ k| k ] xˆ ( i) [ k |k ] = xˆ[ k | k] + Dx(i ) D x( i) ( = D x( M +i ) = b )i - ( cP[ k| k ] ) i cP[ k| k ] i = 1,..., 2 M i = 1,..., M i = 1,..., M Use the nonlinear state transition function to compute the predicted states for each of the sigma points. xˆ ( i) [ k + 1 | k] = f ( xˆ ( i) [ k | k] ,us[k]) c Combine the predicted states to obtain the predicted states at time k+1. These values are used by the correct command in the next time step. 2M Â WM(i) xˆ(i)[k + 1| k] xˆ [ k + 1| k ] = i=0 ( 0) WM =1- i WM = d M 2 a (M + k ) 1 2a 2 ( M + k ) i = 1, 2,..., 2 M Compute the covariance of the predicted state. Add Q[k] to account for the additive process noise. These values are used by the correct command in the next time step. 2M Â Wc(i) (x(i)[k + 1| k] - x[k + 1| k])(x(i)[k + 1| k] - x[k + 1| k])T + Q[k] P[ k + 1| k] = i =0 Wc( 0) Wci 13-8 = (2 - a 2 + b ) 2 M 2 a (M +k) = 1 / (2a ( M + k )) i = 1, 2,..., 2 M Extended and Unscented Kalman Filter Algorithms for Online State Estimation The Unscented Kalman Filter block supports multiple measurement functions. These measurements can have different sample times as long as their sample time is an integer multiple of the state transition sample time. In this case, a separate correction step is performed corresponding to measurements from each measurement function. The previous algorithm is implemented assuming additive noise terms in the state transition and measurement equations. If the noise terms are non-additive, the main changes to the algorithm are: • The correct command generates 2*(M+V)+1 sigma points using P[k|k-1] and R[k], where V is the number of elements in measurement noise v[k]. The R[k] term is no longer added in the algorithm step 2(d) because the extra sigma points capture the impact of measurement noise on Py. • The predict command generates 2*(M+W)+1 sigma points using P[k|k] and Q[k], where W is the number of elements in process noise w[k]. The Q[k] term is no longer added in the algorithm step 3(d) because the extra sigma points capture the impact of process noise on P[k+1|k]. Effect of Alpha, Beta, and Kappa Parameters To compute the state and its statistical properties at the next time step, the unscented Kalman filter algorithm generates a set of state values distributed around the mean state value. The algorithm uses each sigma points as an input to the state transition and measurement functions to get a new set of transformed state points. The mean and covariance of the transformed points is then used to obtain state estimates and state estimation error covariance. The spread of the sigma points around the mean state value is controlled by two parameters α and κ. A third parameter, β, impacts the weights of the transformed points during state and measurement covariance calculations. • α — Determines the spread of the sigma points around the mean state value. It is usually a small positive value. The spread of sigma points is proportional to α. Smaller values correspond to sigma points closer to the mean state. • κ — A second scaling parameter that is usually set to 0. Smaller values correspond to sigma points closer to the mean state. The spread is proportional to the square-root of κ. • β — Incorporates prior knowledge of the distribution of the state. For Gaussian distributions, β = 2 is optimal. 13-9 13 State-Space Control Design You specify these parameters in the Alpha, Kappa, and Beta properties of the unscented Kalman filter. If you know the distribution of state and state covariance, you can adjust these parameters to capture the transformation of higher-order moments of the distribution. The algorithm can track only a single peak in the probability distribution of the state. If there are multiple peaks in the state distribution of your system, you can adjust these parameters so that the sigma points stay around a single peak. For example, choose a small Alpha to generate sigma points close to the mean state value. References [1] Simon, D. Optimal State Estimation: Kalman, H Infinity, and Nonlinear Approaches. John Wiley and Sons Inc., 2006. See Also Functions extendedKalmanFilter | unscentedKalmanFilter Blocks Extended Kalman Filter | Unscented Kalman Filter 13-10 Generate Code for Online State Estimation in MATLAB Generate Code for Online State Estimation in MATLAB You can generate C/C++ code from MATLAB code that uses extendedKalmanFilter, unscentedKalmanFilter and particleFilter objects for online state estimation. C/C++ code is generated using the codegen command from MATLAB Coder software. Use the generated code to deploy online estimation algorithms to an embedded target. You can also deploy online estimation code by creating a standalone application using MATLAB Compiler™ software. To generate C/C++ code for online state estimation: 1 Create a function to declare your filter object as persistent, and initialize the object. You define the object as persistent to maintain the object states between calls. function [CorrectedX] = ukfcodegen(output) % Declare object as persistent. persistent obj; if isempty(obj) % Initialize the object. obj = unscentedKalmanFilter(@vdpStateFcn,@vdpMeasurementFcn,[2;0]); obj.MeasurementNoise = 0.01; end % Estimate the states. CorrectedX = correct(obj,output); predict(obj); end The function creates an unscented Kalman filter object for online state estimation of a van der Pol oscillator with two states and one output. You use the previously written and saved state transition and measurement functions, vdpStateFcn.m and vdpMeasurementFcn.m, and specify the initial state values for the two states as [2;0]. Here output is the measured output data. Save the ukfcodegen.m function on the MATLAB path. Alternatively, you can specify the full path name for this function. In the ukfcodegen.m function, the persistent object is initialized with condition if isempty(obj) to ensure that the object is initialized only once, when the function is called the first time. Subsequent calls to the function only execute the predict and correct commands to update the state estimates. During initialization, you specify the nontunable properties of the object, such as StateTransitionFcn (specified in ukfcodegen.m as vdpStateFcn.m) and MeasurementFcn (specified in ukfcodegen.m as vdpMeasurementFcn.m). After that, you can specify only the 13-11 13 State-Space Control Design tunable properties. For more information, see “Tunable and Nontunable Object Properties” on page 13-13. 2 In the state transition and measurement functions you must use only commands that are supported for code generation. For a list of these commands, see “Functions and Objects Supported for C/C++ Code Generation — Category List” (MATLAB Coder). Include the compilation directive %#codegen in these functions to indicate that you intend to generate code for the function. Adding this directive instructs the MATLAB Code Analyzer to help you diagnose and fix violations that would result in errors during code generation. For an example, type vdpStateFcn.m at the command line. Generate C/C++ code and MEX-files using the codegen command from MATLAB Coder software. codegen ukfcodegen -args {1} The syntax -args {1} specifies an example of an argument to your function. The argument sets the dimensions and data types of the function argument output as a double-precision scalar. Note If you want a filter with single-precision floating-point variables, you must specify the initial value of the states as single-precision during object construction. obj = unscentedKalmanFilter(@vdpStateFcn,@vdpMeasurementFcn,single([2;0])) Then to generate code, use the following syntax. codegen ukfcodegen -args {{single(1)} 3 Use the generated code. • Use the generated C/C++ code to deploy online state estimation to an embedded target. • Use the generated MEX-file for testing the compiled C/C++ code in MATLAB. The generated MEX-file is also useful for accelerating simulations of state estimation algorithms in MATLAB. Load the estimation data. Suppose that your output data is stored in the measured_data.mat file. load measured_data.mat output Estimate the states by calling the generated MEX-file. 13-12 See Also for i = 1:numel(output) XCorrected = ukfcodegen_mex(output(i)); end This example generates C/C++ code for compiling a MEX-file. To generate code for other targets, see codegen in the MATLAB Coder documentation. Tunable and Nontunable Object Properties Property Type Extended Kalman Filter Object Unscented Kalman Particle Filter Filter Object Object Tunable properties that you can specify multiple times either during object construction, or afterward using dot notation State, StateCovariance, ProcessNoise, and MeasurementNoise State, Particles and StateCovariance, Weights ProcessNoise, MeasurementNoise , Alpha, Beta, and Kappa Nontunable properties that you can specify only once, either during object construction, or afterward using dot notation, but before using the predict or correct commands StateTransitionF StateTransitionF StateTransitionF cn, cn and cn, MeasurementFcn, MeasurementFcn MeasurementLikel StateTransitionJ ihoodFcn, acobianFcn, and StateEstimationM MeasurementJacob ethod, ianFcn StateOrientation , ResamplingPolicy and ResamplingMethod Nontunable properties that you must specify during object construction HasAdditiveProce ssNoise and HasAdditiveMeasu rementNoise HasAdditiveProce ssNoise and HasAdditiveMeasu rementNoise See Also extendedKalmanFilter | particleFilter | unscentedKalmanFilter 13-13 13 State-Space Control Design More About 13-14 • “Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter” on page 13-24 • “Validate Online State Estimation at the Command Line” on page 13-15 • “Troubleshoot Online State Estimation” on page 13-22 Validate Online State Estimation at the Command Line Validate Online State Estimation at the Command Line After you use the extendedKalmanFilter, unscentedKalmanFilter or particleFilter commands for online state estimation of a nonlinear system, validate the estimation before deploying the code in your application. If the validation indicates low confidence in the estimation, then see “Troubleshoot Online State Estimation” on page 13-22 for next steps. After you have validated the online estimation results, you can generate C/C++ code or a standalone application using MATLAB Coder or MATLAB Compiler software. To validate the performance of your filter, perform state estimation using measured or simulated output data from different scenarios. • Obtain output data from your system at different operating conditions and input values — To ensure that estimation works well under all operating conditions of interest. For example, suppose that you want to track the position and velocity of a vehicle using noisy position measurements. Measure the data at different vehicle speeds and slow and sharp maneuvers. • For each operating condition of interest, obtain multiple sets of experimental or simulated data with different noise realizations — To ensure different noise values do not deteriorate estimation performance. For each of these scenarios, test the filter performance by examining the output estimation error and state estimation error. For an example about performing and validating online state estimation, see “Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter” on page 13-24. Examine Output Estimation Error The output estimation error is the difference between the measured output, y, and the estimated output, yEstimated. You can obtain the estimated output at each time step by using the measurement function of the system. For example, if vdpMeasurementFcn.m is the measurement function for your nonlinear system, and you are performing state estimation using an extended Kalman filter object, obj, you can compute the estimated output using the current state estimates as: yEstimated = vdpMeasurementFcn(obj.State); estimationError = y-yEstimated; 13-15 13 State-Space Control Design Here obj.State is the state value xˆ [ k| k - 1 ] after you estimate the states using the predict command. xˆ [ k| k - 1 ] is the predicted state estimate for time k, estimated using measured output until a previous time k-1. The estimation errors (residuals) must have the following characteristics: • Small magnitude — Small errors relative to the size of the outputs increase confidence in the estimated values. • Zero mean • Low autocorrelation, except at zero time lag — To compute the autocorrelation, you can use the xcorr command from Signal Processing Toolbox™ software. Examine State Estimation Error for Simulated Data When you simulate the output data of your nonlinear system and use that data for state estimation, you know the true state values. You can compute the errors between estimated and true state values and analyze the errors. The estimated state value at any time step is the value stored in obj.State after you estimate the states using the predict or correct command. The state estimation errors must satisfy the following characteristics: • Small magnitude • Zero mean • Low autocorrelation, except at zero time lag You can also compute the covariance of the state estimation error and compare it to the state estimation error covariance stored in the StateCovariance property of the filter. Similar values increase confidence in the performance of the filter. See Also extendedKalmanFilter | particleFilter | unscentedKalmanFilter More About • 13-16 “Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter” on page 13-24 See Also • “Extended and Unscented Kalman Filter Algorithms for Online State Estimation” on page 13-2 • “Troubleshoot Online State Estimation” on page 13-22 • “Generate Code for Online State Estimation in MATLAB” on page 13-11 13-17 13 State-Space Control Design Validate Online State Estimation in Simulink After you use the Extended Kalman Filter, Unscented Kalman Filter or Particle Filter blocks for online state estimation of a nonlinear system, validate the estimation before deploying the code in your application. If the validation indicates low confidence in the estimation, then see “Troubleshoot Online State Estimation” on page 13-22 for next steps. After you have validated the online estimation results, you can generate C/C++ code for the blocks using Simulink Coder software. To validate the performance of your filter, perform state estimation using measured or simulated output data from these scenarios. • Obtain output data from your system at different operating conditions and input values — To ensure that estimation works well under all operating conditions of interest. For example, suppose that you want to track the position and velocity of a vehicle using noisy position measurements. Measure the data at different vehicle speeds and slow and sharp maneuvers. • For each operating condition of interest, obtain multiple sets of experimental or simulated data with different noise realizations — To ensure that different noise values do not deteriorate estimation performance. For each of these scenarios, test the filter performance by examining the residuals and state estimation error. Examine Residuals The residual, or output estimation error, is the difference between the measured system output yMeasured[k], and the estimated system output yPredicted[k|k-1] at time step k. Here, yPredicted[k|k-1] is the estimated output at time step k, which is predicted using output measurements until time step k-1. The blocks do not explicitly output yPredicted[k|k-1], however you can compute the output using the estimated state values and your state transition and measurement functions. For an example, see “Compute Residuals and State Estimation Errors” on page 13-19. The residuals must have the following characteristics: • Small magnitude — Small errors relative to the size of the outputs increase confidence in the estimated values. 13-18 Validate Online State Estimation in Simulink • Zero mean • Low autocorrelation, except at zero time lag — To compute the autocorrelation, you can use the Autocorrelation block from DSP System Toolbox™ software. Examine State Estimation Error for Simulated Data When you simulate the output data of your nonlinear system and use that data for state estimation, you know the true state values. You can compute the errors between estimated and true state values and analyze the errors. The estimated state value at any time step is output at the xhat port of the blocks. The state estimation errors must satisfy the following characteristics: • Small magnitude • Zero mean • Low autocorrelation, except at zero time lag You can also compute the covariance of the state estimation error, and compare it to the state estimation error covariance that is output by the blocks in the P port of the blocks. Similar values increase confidence in the performance of the filter. Compute Residuals and State Estimation Errors This example shows how to estimate the states of a discrete-time Van der Pol oscillator and compute state estimation errors and residuals for validating the estimation. The residuals are the output estimation errors, that is, they are the difference between the measured and estimated outputs. In the Simulink™ model, the Van der Pol Oscillator block implements the oscillator with nonlinearity parameter, mu, equal to 1. The oscillator has two states. A noisy measurement of the first state x1 is available. The model uses the Unscented Kalman Filter block to estimate the states of the oscillator. Since the block requires discrete-time inputs, the Rate Transition block samples x1 to give the discretized output measurement yMeasured[k] at time step k. The Unscented Kalman Filter block outputs the estimated state values xhat[k|k] at time step k, using yMeasured until time k. The filter block uses the previously written and saved state transition and measurement functions, vdpStateFcn.m and vdpMeasurementFcn.m. For information about these functions, see “Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter” on page 13-24. 13-19 13 State-Space Control Design To validate the state estimation, the model computes the residuals in the Generate Residual block. In addition, since the true state values are known, the model also computes the state estimation errors. To compute the residuals, the Generate Residual block first computes the estimated output yPredicted[k|k-1] using the estimated states and state transition and measurement functions. Here, yPredicted[k|k-1] is the estimated output at time step k, predicted using output measurements until time step k-1. The block then computes the residual at time step k as yMeasured[k] - yPredicted[k|k-1]. Examine the residuals and state estimation errors, and ensure that they have a small magnitude, zero mean, and low autocorrelation. In this example, the Unscented Kalman Filter block outputs xhat[k|k] because the Use the current measurements to improve state estimates parameter of the block is 13-20 See Also selected. If you clear this parameter, the block instead outputs xhat[k|k-1], the predicted state value at time step k, using yMeasured until time k-1. In this case, compute yPredicted[k|k-1] = MeasurementFcn(xhat[k|k-1]), where MeasurementFcn is the measurement function for your system. See Also Extended Kalman Filter | Kalman Filter | Particle Filter | Unscented Kalman Filter More About • “Extended and Unscented Kalman Filter Algorithms for Online State Estimation” on page 13-2 • “Troubleshoot Online State Estimation” on page 13-22 13-21 13 State-Space Control Design Troubleshoot Online State Estimation After you perform state estimation of a nonlinear system using linear, extended, or unscented Kalman filter or particle filter algorithms, you validate the estimation before deploying the code in your application. If the validation indicates low confidence in the estimation, check the following filter properties that you specified: • Initial state and state covariance values — If you find that the measured and estimated outputs of your system are diverging at the beginning of state estimation, check the initial values that you specified. • State transition and measurement functions — Verify that the functions you specify are a good representation of the nonlinear system. If the true system is continuous-time, to implement the algorithms, you discretize the state transition and measurement equations and use the discretized versions. If the state estimation results are not satisfactory, consider decreasing the sample time used for discretization. Alternatively, try a different discretization method. For an example of how to discretize a continuoustime state transition function, type edit vdpStateFcn.m at the command line. Also see, “Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter” on page 13-24. • Process and measurement noise covariance values — If the difference between estimated and measured outputs of your system is large, try specifying different values for the process and measurement noise covariance values. • Choice of algorithm — If you are using the extended Kalman filter algorithm, you can try the unscented Kalman filter, or the particle filter algorithm instead. The unscented Kalman filter and particle filter may capture the nonlinearities in the system better. To troubleshoot state estimation, you can create multiple versions of the filter with different properties, perform state estimation, and choose the filter that gives the best validation results. At the command line, if you want to copy an existing filter object and then modify properties of the copied object, use the clone command. Do not create additional objects using syntax obj2 = obj. Any changes made to the properties of the new object created in this way (obj2) also change the properties of the original object (obj). See Also Functions extendedKalmanFilter | particleFilter | unscentedKalmanFilter 13-22 See Also Blocks Extended Kalman Filter | Particle Filter | Unscented Kalman Filter More About • “Validate Online State Estimation at the Command Line” on page 13-15 • “Generate Code for Online State Estimation in MATLAB” on page 13-11 • “Validate Online State Estimation in Simulink” on page 13-18 13-23 13 State-Space Control Design Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter This example shows how to use the unscented Kalman filter and particle filter algorithms for nonlinear state estimation for the van der Pol oscillator. This example also uses the Signal Processing Toolbox™. Introduction Control System Toolbox™ offers three commands for nonlinear state estimation: • extendedKalmanFilter: First-order, discrete-time extended Kalman filter • unscentedKalmanFilter: Discrete-time unscented Kalman filter • particleFilter: Discrete-time particle filter A typical workflow for using these commands is as follows: 1 Model your plant and sensor behavior. 2 Construct and configure the extendedKalmanFilter, unscentedKalmanFilter or particleFilter object. 3 Perform state estimation by using the predict and correct commands with the object. 4 Analyze results to gain confidence in filter performance 5 Deploy the filter on your hardware. You can generate code for these filters using MATLAB Coder™. This example first uses the unscentedKalmanFilter command to demonstrate this workflow. Then it demonstrates the use of particleFilter. Plant Modeling and Discretization The unscented Kalman filter (UKF) algorithm requires a function that describes the evolution of states from one time step to the next. This is typically called the state transition function. unscentedKalmanFilter supports the following two function forms: 1 2 13-24 Additive process noise: Non-additive process noise: Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter Here f(..) is the state transition function, x is the state, w is the process noise. u is optional and represents additional inputs to f, for instance system inputs or parameters. u can be specified as zero or more function arguments. Additive noise means that the state and process noise is related linearly. If the relationship is nonlinear, use the second form. When you create the unscentedKalmanFilter object, you specify f(..) and also whether the process noise is additive or non-additive. The system in this example is the van der Pol oscillator with mu=1. This 2-state system is described with the following set of nonlinear ordinary differential equations (ODE): Denote this equation as , where . The process noise w does not appear in the system model. You can assume it is additive for simplicity. unscentedKalmanFilter requires a discrete-time state transition function, but the plant model is continuous-time. You can use a discrete-time approximation to the continuoustime model. Euler discretization is one common approximation method. Assume that your sample time is , and denote the continuous-time dynamics you have as discretization approximates the derivative operator as discrete-time state-transition function is: . Euler . The resulting The accuracy of this approximation depends on the sample time . Smaller values provide better approximations. Alternatively, you can use a different discretization method. For instance, higher order Runge-Kutta family of methods provide a higher accuracy at the expense of more computational cost, given a fixed sample time . Create this state-transition function and save it in a file named vdpStateFcn.m. Use the sample time . You provide this function to the unscentedKalmanFilter during object construction. 13-25 13 State-Space Control Design addpath(fullfile(matlabroot,'examples','control_featured','main')) % add example data type vdpStateFcn function x = vdpStateFcn(x) % vdpStateFcn Discrete-time approximation to van der Pol ODEs for mu = 1. % Sample time is 0.05s. % % Example state transition function for discrete-time nonlinear state % estimators. % % xk1 = vdpStateFcn(xk) % % Inputs: % xk - States x[k] % % Outputs: % xk1 - Propagated states x[k+1] % % See also extendedKalmanFilter, unscentedKalmanFilter % Copyright 2016 The MathWorks, Inc. %#codegen % The tag %#codegen must be included if you wish to generate code with % MATLAB Coder. % Euler integration of continuous-time dynamics x'=f(x) with sample time dt dt = 0.05; % [s] Sample time x = x + vdpStateFcnContinuous(x)*dt; end function dxdt = vdpStateFcnContinuous(x) %vdpStateFcnContinuous Evaluate the van der Pol ODEs for mu = 1 dxdt = [x(2); (1-x(1)^2)*x(2)-x(1)]; end Sensor Modeling unscentedKalmanFilter also needs a function that describes how the model states are related to sensor measurements. unscentedKalmanFilter supports the following two function forms: 1 13-26 Additive measurement noise: Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter 2 Non-additive measurement noise: h(..) is the measurement function, v is the measurement noise. u is optional and represents additional inputs to h, for instance system inputs or parameters. u can be specified as zero or more function arguments. You can add additional system inputs following the u term. These inputs can be different than the inputs in the state transition function. For this example, assume you have measurements of the first state percentage error: within some This falls into the category of non-additive measurement noise because the measurement noise is not simply added to a function of states. You want to estimate both from the noisy measurements. and Create this state transition function and save it in a file named vdpMeasurementNonAdditiveNoiseFcn.m. You provide this function to the unscentedKalmanFilter during object construction. type vdpMeasurementNonAdditiveNoiseFcn function yk = vdpMeasurementNonAdditiveNoiseFcn(xk,vk) % vdpMeasurementNonAdditiveNoiseFcn Example measurement function for discrete % time nonlinear state estimators with non-additive measurement noise. % % yk = vdpNonAdditiveMeasurementFcn(xk,vk) % % Inputs: % xk - x[k], states at time k % vk - v[k], measurement noise at time k % % Outputs: % yk - y[k], measurements at time k % % The measurement is the first state with multiplicative noise % % See also extendedKalmanFilter, unscentedKalmanFilter % Copyright 2016 The MathWorks, Inc. 13-27 13 State-Space Control Design %#codegen % The tag %#codegen must be included if you wish to generate code with % MATLAB Coder. yk = xk(1)*(1+vk); end Unscented Kalman Filter Construction Construct the filter by providing function handles to the state transition and measurement functions, followed by your initial state guess. The state transition model has additive noise. This is the default setting in the filter, hence you do not need to specify it. The measurement model has non-additive noise, which you must specify through setting the HasAdditiveMeasurementNoise property of the object as false. This must be done during object construction. If your application has non-additive process noise in the state transition function, specify the HasAdditiveProcessNoise property as false. % Your initial state guess at time k, utilizing measurements up to time k-1: xhat[k|k-1 initialStateGuess = [2;0]; % xhat[k|k-1] % Construct the filter ukf = unscentedKalmanFilter(... @vdpStateFcn,... % State transition function @vdpMeasurementNonAdditiveNoiseFcn,... % Measurement function initialStateGuess,... 'HasAdditiveMeasurementNoise',false); Provide your knowledge of the measurement noise covariance R = 0.2; % Variance of the measurement noise v[k] ukf.MeasurementNoise = R; ProcessNoise property stores the process noise covariance. It is set to account for model inaccuracies and the effect of unknown disturbances on the plant. We have the true model in this example, but discretization introduces errors. This example did not include any disturbances for simplicity. Set it to a diagonal matrix with less noise on the first state, and more on the second state to reflect the knowledge that the second state is more impacted by modeling errors. ukf.ProcessNoise = diag([0.02 0.1]); 13-28 Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter Estimation Using predict and correct Commands In your application, the measurement data arriving from your hardware in real-time are processed by the filters as they arrive. This operation is demonstrated in this example by generating a set of measurement data first, and then feeding it to the filter one step at a time. Simulate the van der Pol oscillator for 5 seconds with the filter sample time 0.05 [s] to generate the true states of the system. T = 0.05; % [s] Filter sample time timeVector = 0:T:5; [~,xTrue]=ode45(@vdp1,timeVector,[2;0]); Generate the measurements assuming that a sensor measures the first state, with a standard deviation of 45% error in each measurement. rng(1); % Fix the random number generator for reproducible results yTrue = xTrue(:,1); yMeas = yTrue .* (1+sqrt(R)*randn(size(yTrue))); % sqrt(R): Standard deviation of noise Pre-allocate space for data that you will analyze later Nsteps = numel(yMeas); % Number of time steps xCorrectedUKF = zeros(Nsteps,2); % Corrected state estimates PCorrected = zeros(Nsteps,2,2); % Corrected state estimation error covariances e = zeros(Nsteps,1); % Residuals (or innovations) Perform online estimation of the states x using the correct and predict commands. Provide generated data to the filter one time step at a time. for k=1:Nsteps % Let k denote the current time. % % Residuals (or innovations): Measured output - Predicted output e(k) = yMeas(k) - vdpMeasurementFcn(ukf.State); % ukf.State is x[k|k-1] at this poi % Incorporate the measurements at time k into the state estimates by % using the "correct" command. This updates the State and StateCovariance % properties of the filter to contain x[k|k] and P[k|k]. These values % are also produced as the output of the "correct" command. [xCorrectedUKF(k,:), PCorrected(k,:,:)] = correct(ukf,yMeas(k)); % Predict the states at next time step, k+1. This updates the State and % StateCovariance properties of the filter to contain x[k+1|k] and % P[k+1|k]. These will be utilized by the filter at the next time step. 13-29 13 State-Space Control Design predict(ukf); end Unscented Kalman Filter Results and Validation Plot the true and estimated states over time. Also plot the measured value of the first state. figure(); subplot(2,1,1); plot(timeVector,xTrue(:,1),timeVector,xCorrectedUKF(:,1),timeVector,yMeas(:)); legend('True','UKF estimate','Measured') ylim([-2.6 2.6]); ylabel('x_1'); subplot(2,1,2); plot(timeVector,xTrue(:,2),timeVector,xCorrectedUKF(:,2)); ylim([-3 1.5]); xlabel('Time [s]'); ylabel('x_2'); 13-30 Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter The top plot shows the true, estimated, and the measured value of the first state. The filter utilizes the system model and noise covariance information to produce an improved estimate over the measurements. The bottom plot shows the second state. The filter is is successful in producing a good estimate. The validation of unscented and extended Kalman filter performance is typically done using extensive Monte Carlo simulations. These simulations should test variations of process and measurement noise realizations, plant operating under various conditions, initial state and state covariance guesses. The key signal of interest used for validating the state estimation is the residuals (or innovations). In this example, you perform residual analysis for a single simulation. Plot the residuals. 13-31 13 State-Space Control Design figure(); plot(timeVector, e); xlabel('Time [s]'); ylabel('Residual (or innovation)'); The residuals should have: 1 Small magnitude 2 Zero mean 3 No autocorrelation, except at zero lag The mean value of the residuals is: mean(e) 13-32 Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter ans = -0.0012 This is small relative to the magnitude of the residuals. The autocorrelation of the residuals can be calculated with the xcorr function in the Signal Processing Toolbox. [xe,xeLags] = xcorr(e,'coeff'); % 'coeff': normalize by the value at zero lag % Only plot non-negative lags idx = xeLags>=0; figure(); plot(xeLags(idx),xe(idx)); xlabel('Lags'); ylabel('Normalized correlation'); title('Autocorrelation of residuals (innovation)'); 13-33 13 State-Space Control Design The correlation is small for all lags except 0. The mean correlation is close to zero, and the correlation does not show any significant non-random variations. These characteristics increase confidence in filter performance. In reality the true states are never available. However, when performing simulations, you have access to real states and can look at the errors between estimated and true states. These errors must satisfy similar criteria to the residual: 1 Small magnitude 2 Variance within filter error covariance estimate 3 Zero mean 4 Uncorrelated. First, look at the error and the estimate. uncertainty bounds from the filter error covariance eStates = xTrue-xCorrectedUKF; figure(); subplot(2,1,1); plot(timeVector,eStates(:,1),... % Error for timeVector, sqrt(PCorrected(:,1,1)),'r', ... % 1-sigma timeVector, -sqrt(PCorrected(:,1,1)),'r'); % 1-sigma xlabel('Time [s]'); ylabel('Error for state 1'); title('State estimation errors'); subplot(2,1,2); plot(timeVector,eStates(:,2),... % Error for timeVector,sqrt(PCorrected(:,2,2)),'r', ... % 1-sigma timeVector,-sqrt(PCorrected(:,2,2)),'r'); % 1-sigma xlabel('Time [s]'); ylabel('Error for state 2'); legend('State estimate','1-sigma uncertainty bound',... 'Location','Best'); 13-34 the first state upper-bound lower-bound the second state upper-bound lower-bound Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter The error bound for state 1 approaches 0 at t=2.15 seconds because of the sensor model (MeasurementFcn). It has the form . At t=2.15 seconds the true and estimated states are near zero, which implies the measurement error in absolute terms is also near zero. This is reflected in the state estimation error covariance of the filter. Calculate what percentage of the points are beyond the 1-sigma uncertainty bound. distanceFromBound1 = abs(eStates(:,1))-sqrt(PCorrected(:,1,1)); percentageExceeded1 = nnz(distanceFromBound1>0) / numel(eStates(:,1)); distanceFromBound2 = abs(eStates(:,2))-sqrt(PCorrected(:,2,2)); percentageExceeded2 = nnz(distanceFromBound2>0) / numel(eStates(:,2)); [percentageExceeded1 percentageExceeded2] 13-35 13 State-Space Control Design ans = 1×2 0.1386 0 The first state estimation errors exceed the uncertainty bound approximately 14% of the time steps. Less than 30% of the errors exceeding the 1-sigma uncertainty bound implies good estimation. This criterion is satisfied for both states. The 0% percentage for the second state means that the filter is conservative: most likely the combined process and measurement noises are too high. Likely a better performance can be obtained by tuning these parameters. Calculate the mean autocorrelation of state estimation errors. Also plot the autocorrelation. mean(eStates) ans = 1×2 -0.0103 0.0200 [xeStates1,xeStatesLags1] = xcorr(eStates(:,1),'coeff'); % 'coeff': normalize by the va [xeStates2,xeStatesLags2] = xcorr(eStates(:,2),'coeff'); % 'coeff' % Only plot non-negative lags idx = xeStatesLags1>=0; figure(); subplot(2,1,1); plot(xeStatesLags1(idx),xeStates1(idx)); xlabel('Lags'); ylabel('For state 1'); title('Normalized autocorrelation of state estimation error'); subplot(2,1,2); plot(xeStatesLags2(idx),xeStates2(idx)); xlabel('Lags'); ylabel('For state 2'); 13-36 Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter The mean value of the errors is small relative to the value of the states. The autocorrelation of state estimation errors shows little non-random variations for small lag values, but these are much smaller than the normalized peak value 1. Combined with the fact that the estimated states are accurate, this behavior of the residuals can be considered as satisfactory results. Particle Filter Construction Unscented and extended Kalman filters aim to track the mean and covariance of the posterior distribution of the state estimates by different approximation methods. These methods may not be sufficient if the nonlinearities in the system are severe. In addition, for some applications, just tracking the mean and covariance of the posterior distribution of the state estimates may not be sufficient. Particle filter can address these problems by 13-37 13 State-Space Control Design tracking the evolution of many state hypotheses (particles) over time, at the expense of higher computational cost. The computational cost and estimation accuracy increases with the number of particles. The particleFilter command in Control System Toolbox implements a discrete-time particle filter algorithm. This section walks you through constructing a particleFilter for the same van der Pol oscillator used earlier in this example, and highlights the similarities and differences with the unscented Kalman filter. The state transition function you provide to particleFilter must perform two tasks. One, sampling the process noise from any distribution appropriate for your system. Two, calculating the time propagation of all particles (state hypotheses) to the next step, including the effect of process noise you calculated in step one. type vdpParticleFilterStateFcn function particles = vdpParticleFilterStateFcn(particles) % vdpParticleFilterStateFcn Example state transition function for particle % filter % % Discrete-time approximation to van der Pol ODEs for mu = 1. % Sample time is 0.05s. % % predictedParticles = vdpParticleFilterStateFcn(particles) % % Inputs: % particles - Particles at current time. Matrix with dimensions % [NumberOfStates NumberOfParticles] matrix % % Outputs: % predictedParticles - Predicted particles for the next time step % % See also particleFilter % Copyright 2017 The MathWorks, Inc. %#codegen % The tag %#codegen must be included if you wish to generate code with % MATLAB Coder. [numberOfStates, numberOfParticles] = size(particles); % Time-propagate each particle 13-38 Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter % % Euler integration of continuous-time dynamics x'=f(x) with sample time dt dt = 0.05; % [s] Sample time for kk=1:numberOfParticles particles(:,kk) = particles(:,kk) + vdpStateFcnContinuous(particles(:,kk))*dt; end % Add Gaussian noise with variance 0.025 on each state variable processNoise = 0.025*eye(numberOfStates); particles = particles + processNoise * randn(size(particles)); end function dxdt = vdpStateFcnContinuous(x) %vdpStateFcnContinuous Evaluate the van der Pol ODEs for mu = 1 dxdt = [x(2); (1-x(1)^2)*x(2)-x(1)]; end There are differences between the state transition function you supply to unscentedKalmanFilter and particleFilter. The state transition function you used for unscented Kalman filter just described propagation of one state hypothesis to the next time step, instead of a set of hypotheses. In addition, the process noise distribution was defined in ProcessNoise property of the unscentedKalmanFilter, just by its covariance. Particle filter can consider arbitrary distributions that may require more statistical properties to be defined. This arbitrary distribution and its parameters are fully defined in the state transition function you provide to the particleFilter. The measurement likelihood function you provide to particleFilter must also perform two tasks. One, calculating measurement hypotheses from particles. Two, calculating the likelihood of each particle from the sensor measurement and the hypotheses calculated in step one. type vdpExamplePFMeasurementLikelihoodFcn function likelihood = vdpExamplePFMeasurementLikelihoodFcn(particles,measurement) % vdpExamplePFMeasurementLikelihoodFcn Example measurement likelihood function % % The measurement is the first state. % % likelihood = vdpParticleFilterMeasurementLikelihoodFcn(particles, measurement) % % Inputs: % particles - NumberOfStates-by-NumberOfParticles matrix that holds % the particles % 13-39 13 State-Space Control Design % Outputs: % likelihood - A vector with NumberOfParticles elements whose n-th % element is the likelihood of the n-th particle % % See also extendedKalmanFilter, unscentedKalmanFilter % Copyright 2017 The MathWorks, Inc. %#codegen % The tag %#codegen must be included if you wish to generate code with % MATLAB Coder. % Validate the sensor measurement numberOfMeasurements = 1; % Expected number of measurements validateattributes(measurement, {'double'}, {'vector', 'numel', numberOfMeasurements}, 'vdpExamplePFMeasurementLikelihoodFcn', 'measurement'); % The measurement is first state. Get all measurement hypotheses from particles predictedMeasurement = particles(1,:); % Assume the ratio of the error between predicted and actual measurements % follow a Gaussian distribution with zero mean, variance 0.2 mu = 0; % mean sigma = 0.2 * eye(numberOfMeasurements); % variance % Use multivariate Gaussian probability density function, calculate % likelihood of each particle numParticles = size(particles,2); likelihood = zeros(numParticles,1); C = det(2*pi*sigma) ^ (-0.5); for kk=1:numParticles errorRatio = (predictedMeasurement(kk)-measurement)/predictedMeasurement(kk); v = errorRatio-mu; likelihood(kk) = C * exp(-0.5 * (v' / sigma * v) ); end end Now construct the filter, and initialize it with 1000 particles around the mean [2; 0] with 0.01 covariance. The covariance is small because you have high confidence in your guess [2; 0]. pf = particleFilter(@vdpParticleFilterStateFcn,@vdpExamplePFMeasurementLikelihoodFcn); initialize(pf, 1000, [2;0], 0.01*eye(2)); 13-40 Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter Optionally, pick the state estimation method. This is set by the StateEstimationMethod property of particleFilter, which can take the value 'mean' (default) or 'maxweight'. When StateEstimationMethod is 'mean', the object extracts a weighted mean of the particles from the Particles and Weights properties as the state estimate. 'maxweight' corresponds to choosing the particle (state hypothesis) as the state estimate. Alternatively, you can access Particles and Weights properties of the object and extract your state estimate via an arbitrary method of your choice. pf.StateEstimationMethod ans = 'mean' particleFilter lets you specify various resampling options via its ResamplingPolicy and ResamplingMethod properties. This example uses the default settings in the filter. See the particleFilter documentation for further details on resampling. pf.ResamplingMethod ans = 'multinomial' pf.ResamplingPolicy ans = particleResamplingPolicy with properties: TriggerMethod: 'ratio' SamplingInterval: 1 MinEffectiveParticleRatio: 0.5000 Start the estimation loop. This represents measurements arriving over time, step by step. % Estimate xCorrectedPF = zeros(size(xTrue)); for k=1:size(xTrue,1) % Use measurement y[k] to correct the particles for time k xCorrectedPF(k,:) = correct(pf,yMeas(k)); % Filter updates and stores Particles[k|k % The result is x[k|k]: Estimate of states at time k, utilizing % measurements up to time k. This estimate is the mean of all particles % because StateEstimationMethod was 'mean'. % % Now, predict particles at next time step. These are utilized in the % next correct command 13-41 13 State-Space Control Design predict(pf); % Filter updates and stores Particles[k+1|k] end Plot the state estimates from particle filter: figure(); subplot(2,1,1); plot(timeVector,xTrue(:,1),timeVector,xCorrectedPF(:,1),timeVector,yMeas(:)); legend('True','Particlte filter estimate','Measured') ylim([-2.6 2.6]); ylabel('x_1'); subplot(2,1,2); plot(timeVector,xTrue(:,2),timeVector,xCorrectedPF(:,2)); ylim([-3 1.5]); xlabel('Time [s]'); ylabel('x_2'); 13-42 Nonlinear State Estimation Using Unscented Kalman Filter and Particle Filter The top plot shows the true value, particle filter estimate, and the measured value of the first state. The filter utilizes the system model and noise information to produce an improved estimate over the measurements. The bottom plot shows the second state. The filter is successful in producing a good estimate. The validation of the particle filter performance involves performing statistical tests on residuals, similar to those that were performed earlier in this example for unscented Kalman filter results. Summary This example has shown the steps of constructing and using an unscented Kalman filter and a particle filter for state estimation of a nonlinear system. You estimated states of a 13-43 13 State-Space Control Design van der Pol oscillator from noisy measurements, and validated the estimation performance. rmpath(fullfile(matlabroot,'examples','control_featured','main')) % remove example data See Also extendedKalmanFilter | particleFilter | unscentedKalmanFilter More About 13-44 • “Extended and Unscented Kalman Filter Algorithms for Online State Estimation” on page 13-2 • “Validate Online State Estimation at the Command Line” on page 13-15 • “Troubleshoot Online State Estimation” on page 13-22 • “Generate Code for Online State Estimation in MATLAB” on page 13-11 Estimate States of Nonlinear System with Multiple, Multirate Sensors Estimate States of Nonlinear System with Multiple, Multirate Sensors This example shows how to perform nonlinear state estimation in Simulink™ for a system with multiple sensors operating at different sample rates. The Extended Kalman Filter block in Control System Toolbox™ is used to estimate the position and velocity of an object using GPS and radar measurements. Introduction The toolbox has three Simulink blocks for nonlinear state estimation: • Extended Kalman Filter: Implements the first-order discrete-time extended Kalman filter algorithm. • Unscented Kalman Filter: Implements the discrete-time unscented Kalman filter algorithm. • Particle Filter: Implements a discrete-time particle filter algorithm. These blocks support state estimation using multiple sensors operating at different sample rates. A typical workflow for using these blocks is as follows: 1 Model your plant and sensor behavior using MATLAB or Simulink functions. 2 Configure the parameters of the block. 3 Simulate the filter and analyze results to gain confidence in filter performance. 4 Deploy the filter on your hardware. You can generate code for these filters using Simulink Coder™ software. This example uses the Extended Kalman Filter block to demonstrate the first two steps of this workflow. The last two steps are briefly discussed in the Next Steps section. The goal in this example is to estimate the states of an object using noisy measurements provided by a radar and a GPS sensor. The states of the object are its position and velocity, which are denoted as xTrue in the Simulink model. If you are interested in the Particle Filter block, please see the example "Parameter and State Estimation in Simulink Using Particle Filter Block". addpath(fullfile(matlabroot,'examples','ident_featured','main')) % add example data open_system('multirateEKFExample'); 13-45 13 State-Space Control Design Plant Modeling The extended Kalman filter (EKF) algorithm requires a state transition function that describes the evolution of states from one time step to the next. The block supports the following two function forms: • Additive process noise: • Nonadditive process noise: Here f(..) is the state transition function, x is the state, and w is the process noise. u is optional, and represents additional inputs to f, for instance system inputs or parameters. Additive noise means that the next state and process noise linearly. If the relationship is nonlinear, use the nonadditive form. are related The function f(...) can be a MATLAB Function that comply with the restrictions of MATLAB Coder™, or a Simulink Function block. After you create f(...), you specify the function name and whether the process noise is additive or nonadditive in the Extended Kalman Filter block. In this example, you are tracking the north and east positions and velocities of an object on a 2-dimensional plane. The estimated quantities are: 13-46 Estimate States of Nonlinear System with Multiple, Multirate Sensors Here is the discrete-time index. The state transition equation used is of the nonadditive form The filter assumes that variance where walk: , where is the state vector, and is the process noise. is a zero-mean, independent random variable with known . The A and G matrices are: is the sample time. The third row of A and G model the east velocity as a random . In reality, position is a continuous-time variable and is the integral of velocity over time . The first row of A and G represent a discrete approximation to this kinematic relationship: . The second and fourth rows of A and G represent the same relationship between the north velocity and position. This state transition model is linear, but the radar measurement model is nonlinear. This nonlinearity necessitates the use of a nonlinear state estimator such as the extended Kalman filter. In this example you implement the state transition function using a Simulink Function block. To do so, • Add a Simulink Function block to your model from the Simulink/User-Defined Functions library • Click on the name shown on the Simulink Function block. Edit the function name, and add or remove input and output arguments, as necessary. In this example the name for the state transition function is stateTransitionFcn. It has one output argument (xNext) and two input arguments (x, w). 13-47 13 State-Space Control Design • Though it is not required in this example, you can use any signals from the rest of your Simulink model in the Simulink Function. To do so, add Inport blocks from the Simulink/Sources library. Note that these are different than the ArgIn and ArgOut blocks that are set through the signature of your function (xNext = stateTransitionFcn(x, w)). • In the Simulink Function block, construct your function utilizing Simulink blocks. • Set the dimensions for the input and output arguments x, w, and xNext in the Signal Attributes tab of the ArgIn and ArgOut blocks. The data type and port dimensions must be consistent with the information you provide in the Extended Kalman Filter block. Analytical Jacobian of the state transition function is also implemented in this example. Specifying the Jacobian is optional. However, this reduces the computational burden, and in most cases increases the state estimation accuracy. Implement the Jacobian function as a Simulink function because the state transition function is a Simulink function. open_system('multirateEKFExample/Simulink Function - State Transition Jacobian'); 13-48 Estimate States of Nonlinear System with Multiple, Multirate Sensors Sensor modeling - Radar The Extended Kalman Filter block also needs a measurement function that describes how the states are related to measurements. The following two function forms are supported: • Additive measurement noise: • Nonadditive measurement noise: Here h(..) is the measurement function, and v is the measurement noise. u is optional, and represents additional inputs to h, for instance system inputs or parameters. These inputs can differ from the inputs in the state transition function. In this example a radar located at the origin measures the range and angle of the object at 20 Hz. Assume that both of the measurements have about 5% noise. This can be modeled by the following measurement equation: Here and are the measurement noise terms, each with variance 0.05^2. That is, most of the measurements have errors less than 5%. The measurement noise is and are not simply added to the measurements, but nonadditive because instead they depend on the states x. In this example, the radar measurement equation is implemented using a Simulink Function block. open_system('multirateEKFExample/Simulink Function - Radar Measurements'); 13-49 13 State-Space Control Design Sensor modeling - GPS A GPS measures the east and north positions of the object at 1 Hz. Hence, the measurement equation for the GPS sensor is: Here and are measurement noise terms with the covariance matrix [10^2 0; 0 10^2]. That is, the measurements are accurate up to approximately 10 meters, and the errors are uncorrelated. The measurement noise is additive because the noise terms affect the measurements linearly. Create this function, and save it in a file named gpsMeasurementFcn.m. When the measurement noise is additive, you must not specify the noise terms in the function. You provide this function name and measurement noise covariance in the Extended Kalman Filter block. type gpsMeasurementFcn function y = gpsMeasurementFcn(x) % gpsMeasurementFcn GPS measurement function for state estimation % % Assume the states x are: % [EastPosition; NorthPosition; EastVelocity; NorthVelocity] 13-50 Estimate States of Nonlinear System with Multiple, Multirate Sensors %#codegen % The %#codegen tag above is needed is you would like to use MATLAB Coder to % generate C or C++ code for your filter y = x([1 2]); % Position states are measured end Filter Construction Configure the Extended Kalman Filter block to perform the estimation. You specify the state transition and measurement function names, initial state and state error covariance, and process and measurement noise characteristics. In the System Model tab of the block dialog, specify the following parameters: State Transition 1 Specify the state transition function, stateTransitionFcn, in Function. Since you have the Jacobian of this function, select Jacobian, and specify the Jacobian function, stateTransitionJacobianFcn. 2 Select Nonadditive in the Process Noise drop-down list because you explicitly stated how the process noise impacts the states in your function. 3 Specify the process noise covariance as [0.2 0; 0 0.2]. As explained in the Plant Modeling section of this example, process noise terms define the random walk of the velocities in each direction. The diagonal terms approximately capture how much the velocities can change over one sample time of the state transition function. The offdiagonal terms are set to zero, which is a naive assumption that velocity variations in the north and east directions are uncorrelated. Initialization 1 Specify your best initial state estimate in Initial state. In this example, specify [100; 100; 0; 0]. 2 Specify your confidence in your state estimate guess in Initial covariance. In this example, specify 10. The software interprets this value as the true state values are of your initial estimate. You can specify a separate value for likely to be within each state by setting Initial covariance as a vector. You can specify crosscorrelations in this uncertainty by specifying it as a matrix. 13-51 13 State-Space Control Design Since there are two sensors, click Add Measurement to specify a second measurement function. Measurement 1 1 Specify the name of your measurement function, radarMeasurementFcn, in Function. 2 Select Nonadditive in the Measurement Noise drop-down list because you explicitly stated how the process noise impacts the measurements in your function. 3 Specify the measurement noise covariance as [0.05^2 0; 0 0.05^2] per the discussion in the Sensor Modeling - Radar section. Measurement 2 13-52 1 Specify the name of your measurement function, gpsMeasurementFcn, in Function. 2 This sensor model has additive noise. Therefore, specify the GPS measurement noise as Additive in the Measurement Noise drop-down list. 3 Specify the measurement noise covariance as [100 0; 0 100]. Estimate States of Nonlinear System with Multiple, Multirate Sensors 13-53 13 State-Space Control Design In the Multirate tab, since the two sensors are operating at different sample rates, perform the following configuration: 13-54 1 Select Enable multirate operation. 2 Specify the state transition sample time. The state transition sample time must be the smallest, and all measurement sample times must be an integer multiple of the state transition sample time. Specify State Transition sample time as 0.05, the sample time of the fastest measurement. Though not required in this example, it is possible to have a smaller sample time for state transition than all measurements. This means there will be some sample times without any measurements. For these sample times the filter generates state predictions using the state transition function. 3 Specify the Measurement 1 sample time (Radar) as 0.05 seconds and Measurement 2 (GPS) as 1 seconds. Estimate States of Nonlinear System with Multiple, Multirate Sensors Simulation and Results Test the performance of the Extended Kalman filter by simulating a scenario where the object travels in a square pattern with the following maneuvers: • At t = 0, the object starts at • It heads north at until t = 20 seconds. 13-55 13 State-Space Control Design • It heads east at between t = 20 and t = 45 seconds. • It heads south at between t = 45 and t = 85 seconds. • It heads west at between t = 85 and t = 185 seconds. Generate the true state values corresponding to this motion: Ts = 0.05; % [s] Sample rate for the true states [t, xTrue] = generateTrueStates(Ts); % Generate position and velocity profile over 0-18 Simulate the model. For instance, look at the actual and estimated velocities in the east direction: sim('multirateEKFExample'); open_system('multirateEKFExample/Scope - East Velocity'); 13-56 Estimate States of Nonlinear System with Multiple, Multirate Sensors The plot shows the true velocity in the east direction, and its extended Kalman filter estimates. The filter successfully tracks the changes in velocity. The multirate nature of the filter is most apparent in the time range t = 20 to 30 seconds. The filter makes large corrections every second (GPS sample rate), while the corrections due to radar measurements are visible every 0.05 seconds. 13-57 13 State-Space Control Design Next Steps 1 Validate the state estimation: The validation of unscented and extended Kalman filter performance is typically done using extensive Monte Carlo simulations. For more information, see “Validate Online State Estimation in Simulink” on page 13-18. 2 Generate code: The Unscented and Extended Kalman Filter blocks support C and C+ + code generation using Simulink Coder™ software. The functions you provide to these blocks must comply with the restrictions of MATLAB Coder™ software (if you are using MATLAB functions to model your system) and Simulink Coder software (if you are using Simulink Function blocks to model your system). Summary This example has shown how to use the Extended Kalman Filter block in System Identification Toolbox. You estimated position and velocity of an object from two different sensors operating at different sampling rates. close_system('multirateEKFExample', 0); See Also Extended Kalman Filter | Particle Filter | Unscented Kalman Filter More About 13-58 • “Extended and Unscented Kalman Filter Algorithms for Online State Estimation” on page 13-2 • “Validate Online State Estimation in Simulink” on page 13-18 • “Troubleshoot Online State Estimation” on page 13-22 Regulate Pressure in Drum Boiler Regulate Pressure in Drum Boiler This example shows how to use Simulink® Control Design™ software, using a drum boiler as an example application. Using the operating point search function, the example illustrates model linearization as well as subsequent state observer and LQR design. In this drum-boiler model, the control problem is to regulate boiler pressure in the face of random heat fluctuations from the furnace by adjusting the feed water flow rate and the nominal heat applied. For this example, 95% of the random heat fluctuations are less than 50% of the nominal heating value, which is not unusual for a furnace-fired boiler. Open the Model Open the Simulink model. mdl = 'Boiler_Demo'; open_system(mdl) When you open boiler control model the software initializes the controller sizes. u0 and y0 are set after the operating point computation and are therefore initially set to zero. The observer and regulator are computed during the controller design step and are also initially set to zero. 13-59 13 State-Space Control Design Find Nominal Operating Point and Linearize Model The model initial state values are defined in the Simulink model. Using these state values, find the steady-state operating point using the findop function. Create an operating point specification where the state values are known. opspec = operspec(mdl); opspec.States(1).Known = 1; opspec.States(2).Known = 1; opspec.States(3).Known = [1;1]; Adjust the operating point specification to indicate that the inputs must be computed and that they are lower-bounded. opspec.Inputs(1).Known = [0;0]; opspec.Inputs(1).Min = [0;0]; % Inputs unknown % Input minimum value Add an output specification to the operating point specification; this is necessary to ensure that the output operating point is computed during the solution process. opspec = addoutputspec(opspec,[mdl '/Boiler'],1); opspec.Outputs(1).Known = 0; % Outputs unknown opspec.Outputs(1).Min = 0; % Output minimum value Compute the operating point, and generate an operating point search report. [opSS,opReport] = findop(mdl,opspec); Operating point search report: --------------------------------Operating point search report for the Model Boiler_Demo. (Time-Varying Components Evaluated at time t=0) Operating point specifications were successfully met. States: ---------(1.) Boiler_Demo/Boiler/Steam volume x: 5.6 dx: 7.85e-13 (0) (2.) Boiler_Demo/Boiler/Temperature x: 180 dx: -5.93e-14 (0) (3.) Boiler_Demo/Observer/Internal x: 0 dx: 0 (0) 13-60 Regulate Pressure in Drum Boiler x: 0 dx: Inputs: ---------(1.) Boiler_Demo/Input u: 2.41e+05 u: 100 [0 Inf] [0 Inf] Outputs: ---------(1.) Boiler_Demo/Boiler y: 1e+03 [0 Inf] 0 (0) Before linearizing the model around this point, specify the input and output signals for the linear model. First, specify the input points for linearization. Boiler_io(1) = linio([mdl '/Sum'],1,'input'); Boiler_io(2) = linio([mdl '/Demux'],2,'input'); Next, specify the open-loop output points for linearization. Boiler_io(3) = linio([mdl '/Boiler'],1,'openoutput'); setlinio(mdl,Boiler_io); Find a linear model around the chosen operating point. Lin_Boiler = linearize(mdl,opSS,Boiler_io); Finally, using the minreal function, make sure that the model is a minimum realization. Lin_Boiler = minreal(Lin_Boiler); 1 state removed. Design Regulator and State Observer Using this linear model, design an LQR regulator and Kalman filter state observer. First, find the controller offsets to make sure that the controller is operating around the chosen linearization point by retrieving the computed operating point. u0 = opReport.Inputs.u; y0 = opReport.Outputs.y; Now, design the regulator using the lqry function. Tight regulation of the output is required while input variation should be limited. 13-61 13 State-Space Control Design Q = diag(1e8); % Output regulation R = diag([1e2,1e6]); % Input limitation [K,S,E] = lqry(Lin_Boiler,Q,R); Design the Kalman state observer using the kalman function. For this example, the main noise source is process noise. The noise enters the system only through one input, hence the form of G and H. [A,B,C,D] = ssdata(Lin_Boiler); G = B(:,1); H = 0; QN = 1e4; RN = 1e-1; NN = 0; [Kobsv,L,P] = kalman(ss(A,[B G],C,[D H]),QN,RN); Simulate Model Simulate the model for the designed controller. sim(mdl) Plot the process input and output signals. The following figure shows the feed water actuation signal in kg/s. plot(FeedWater.time/60,FeedWater.signals.values) title('Feedwater flow rate [kg/s]'); ylabel('Flow [kg/s]') xlabel('time [min]') grid on 13-62 Regulate Pressure in Drum Boiler The following plot shows the heat actuation signal in kJ. plot(Heat.time/60,Heat.signals.values/1000) title('Applied heat [kJ]'); ylabel('Heat [kJ]') xlabel('time [min]') grid on 13-63 13 State-Space Control Design The next figure shows the heat disturbance in kJ. The disturbance varies by as much as 50% of the nominal heat value. plot(HeatDist.time/60,HeatDist.signals.values/1000) title('Heat disturbance [kJ]'); ylabel('Heat [kJ]') xlabel('time [min]') grid on 13-64 Regulate Pressure in Drum Boiler The figure below shows the corresponding drum pressure in kPa. The pressure varies by about 1% of the nominal value even though the disturbance is relatively large. plot(DrumPressure.time/60,DrumPressure.signals.values) title('Drum pressure [kPa]'); ylabel('Pressure [kPa]') xlabel('time [min]') grid on bdclose(mdl) 13-65 13 State-Space Control Design 13-66 Control System Tuning 67 14 Control System Tuning • “Automated Tuning Overview” on page 14-3 • “Choosing an Automated Tuning Approach” on page 14-5 • “Automated Tuning Workflow” on page 14-7 • “Specify Control Architecture in Control System Tuner” on page 14-9 • “Open Control System Tuner for Tuning Simulink Model” on page 14-13 • “Specify Operating Points for Tuning in Control System Tuner” on page 14-15 • “Specify Blocks to Tune in Control System Tuner” on page 14-24 • “View and Change Block Parameterization in Control System Tuner” on page 14-26 • “Setup for Tuning Control System Modeled in MATLAB” on page 14-35 • “How Tuned Simulink Blocks Are Parameterized” on page 14-36 • “Specify Goals for Interactive Tuning” on page 14-39 • “Quick Loop Tuning of Feedback Loops in Control System Tuner” on page 14-48 • “Quick Loop Tuning” on page 14-58 • “Step Tracking Goal” on page 14-62 • “Step Rejection Goal” on page 14-68 • “Transient Goal” on page 14-74 • “LQR/LQG Goal” on page 14-81 • “Gain Goal” on page 14-86 • “Variance Goal” on page 14-92 • “Reference Tracking Goal” on page 14-97 • “Overshoot Goal” on page 14-104 • “Disturbance Rejection Goal” on page 14-109 • “Sensitivity Goal” on page 14-114 • “Weighted Gain Goal” on page 14-119 • “Weighted Variance Goal” on page 14-124 • “Minimum Loop Gain Goal” on page 14-129 14 Control System Tuning • “Maximum Loop Gain Goal” on page 14-135 • “Loop Shape Goal” on page 14-141 • “Margins Goal” on page 14-148 • “Passivity Goal” on page 14-153 • “Conic Sector Goal” on page 14-158 • “Weighted Passivity Goal” on page 14-165 • “Poles Goal” on page 14-171 • “Controller Poles Goal” on page 14-176 • “Manage Tuning Goals” on page 14-179 • “Generate MATLAB Code from Control System Tuner for Command-Line Tuning” on page 14-181 • “Interpret Numeric Tuning Results” on page 14-184 • “Visualize Tuning Goals” on page 14-189 • “Create Response Plots in Control System Tuner” on page 14-198 • “Examine Tuned Controller Parameters in Control System Tuner” on page 14-205 • “Compare Performance of Multiple Tuned Controllers” on page 14-207 • “Create and Configure slTuner Interface to Simulink Model” on page 14-212 • “Stability Margins in Control System Tuning” on page 14-218 • “Tune Control System at the Command Line” on page 14-223 • “Speed Up Tuning with Parallel Computing Toolbox Software” on page 14-225 • “Validate Tuned Control System” on page 14-227 • “Extract Responses from Tuned MATLAB Model at the Command Line” on page 14-232 14-2 Automated Tuning Overview Automated Tuning Overview The control system tuning tools such as systune and Control System Tuner automatically tune control systems from high-level tuning goals you specify, such as reference tracking, disturbance rejection, and stability margins. The software jointly tunes all the free parameters of your control system regardless of control system architecture or the number of feedback loops it contains. For example, the model of the following illustration represents a multiloop control system for a helicopter. This control system includes a number of fixed elements, such as the helicopter model itself and the roll-off filters. The inner control loop provides static output feedback for decoupling. The outer loop includes PI controllers for setpoint tracking. The tuning tools jointly optimize the gains in the SOF and PI blocks to meet setpoint tracking, stability margin, and other requirements that you specify. These tools allow you to specify any control structure and designate which blocks in your system are tunable. Control systems are tuned to meet your specific performance and robustness goals subject to feasibility constraints such as actuator limits, sensor accuracy, computing power, or energy consumption. The library of tuning goals lets you capture these objectives in a form suitable for fast automated tuning. This library includes standard control objectives such as reference tracking, disturbance rejection, loop shapes, closed14-3 14 Control System Tuning loop damping, and stability margins. Using these tools, you can perform multi-objective tuning of control systems having any structure. See Also Control System Designer | systune More About 14-4 • “Choosing an Automated Tuning Approach” on page 14-5 • “Automated Tuning Workflow” on page 14-7 Choosing an Automated Tuning Approach Choosing an Automated Tuning Approach You can tune control systems at the MATLAB command line or using the Control System Tuner App. Control System Tuner provides an interactive graphical interface for specifying your tuning goals and validating the performance of the tuned control system. Use Control System Tuner to tune control systems consisting of any number of feedback loops, with tunable components having any structure (such as PID, gain block, or statespace). You can represent your control architecture in MATLAB as a tunable generalized 14-5 14 Control System Tuning state-space (genss) model. If you have Simulink Control Design software, you can tune a control system represented by a Simulink model. Use the graphical interface to configure your tuning goals, examine response plots, and validate your controller design. The systune command can perform all the same tuning tasks as Control System Tuner. Tuning at the command line allows you to write scripts for repeated tuning tasks. systune also provides advanced techniques such as tuning a controller for multiple plants, or designing gain-scheduled controllers. To use the command-line tuning tools, you can represent your control architecture in MATLAB as a tunable generalized statespace (genss) model. If you have Simulink Control Design software, you can tune a control system represented by a Simulink model using an slTuner interface. Use the TuningGoal requirement objects to configure your tuning goals. Analysis commands such as getIOTransfer and viewGoal let you examine and validate the performance of your tuned system. See Also Control System Designer | systune More About • 14-6 “Automated Tuning Workflow” on page 14-7 Automated Tuning Workflow Automated Tuning Workflow Whether you are tuning a control system at the command line or using Control System Tuner, the basic workflow includes the following steps: 1 Define your control architecture, by building a model of your control system from fixed-value blocks and blocks with tunable parameters. You can do so in one of several ways: • Create a Simulink model of your control system. (Tuning a Simulink model requires Simulink Control Design software.) • Use a predefined control architecture available in Control System Tuner. • At the command line, build a tunable genss model of your control system out of numeric LTI models and tunable control design blocks. For more information, see “Specify Control Architecture in Control System Tuner” on page 14-9. 2 Set up your model for tuning. • In Control System Tuner, identify which blocks of the model you want to tune. See Model Setup for Control System Tuner. • If tuning a Simulink model at the command line, create and configure the slTuner interface to the model. See Setup for Tuning Simulink Models at the Command Line. 3 Specify your tuning goals. Use the library of tuning goals to capture requirements such as reference tracking, disturbance rejection, stability margins, and more. • In Control System Tuner, use the graphical interface to specify tuning goals. See Tuning Goals (Control System Tuner). • At the command-line, use the TuningGoal requirement objects to specify your tuning goals. See Tuning Goals (programmatic tuning). 4 Tune the model. Use the systune command or Control System Tuner to optimize the tunable parameters of your control system to best meet your specified tuning goals. Then, analyze the tuned system responses and validate the design. Whether at the command line or in Control System Tuner, you can plot system responses to examine any aspects of system performance you need to validate your design. • For tuning and validating in Control System Tuner, see Tuning, Analysis, and Validation (Control System Tuner). 14-7 14 Control System Tuning • For tuning at the command line, see Tuning, Analysis, and Validation (programmatic tuning). 14-8 Specify Control Architecture in Control System Tuner Specify Control Architecture in Control System Tuner About Control Architecture Control System Tuner lets you tune a control system having any architecture. Control system architecture defines how your controllers interact with the system under control. The architecture comprises the tunable control elements of your system, additional filter and sensor components, the system under control, and the interconnections among all these elements. For example, a common control system architecture is the single-loop feedback configuration of the following illustration: G is the plant model, and H the sensor dynamics. These are usually the fixed components of the control system. The prefilter F and feedback controller C are the tunable elements. Because control systems are so conveniently expressed in this block diagram form, these elements are referred to as fixed blocks and tunable blocks. Control System Tuner gives you several ways to define your control system architecture: • Use the predefined feedback structure of the illustration. • Model any control system architecture in MATLAB by building a generalized statespace (genss) model from fixed LTI components and tunable control design blocks. • Model your control system in Simulink and specify the blocks to tune in Control System Tuner (requires Simulink Control Design software). Predefined Feedback Architecture If your control system has the single-loop feedback configuration of the following illustration, use the predefined feedback structure built into Control System Tuner. 14-9 14 Control System Tuning For example, suppose you have a DC motor for which you want to tune a PID controller. The response of the motor is modeled as G(s) = 1/(s + 1)2. Create a fixed LTI model representing the plant, and a tunable PID controller model. Gmot = zpk([],[-1,-1],1); Cmot = tunablePID('Cmot','PID'); Open Control System Tuner. controlSystemTuner Control System Tuner opens, set to tune this default architecture. Next, specify the values of the blocks in the architecture. Click configuration dialog box. to open the Standard feedback Enter the values for C and G that you created. Control System Tuner reads these values from the MATLAB workspace. Click OK. The default value for the sensor dynamics is a fixed unity-gain transfer function. The default value for the filter F is a tunable gain block. You can now select blocks to tune, create tuning goals, and tune the control system. 14-10 Specify Control Architecture in Control System Tuner Arbitrary Feedback Control Architecture If your control architecture does not match Control System Tuner’s predefined control architecture, you can create a generalized state-space (genss) model with tunable components representing your controller elements. For example, suppose you want to tune the cascaded control system of the following illustration, that includes two tunable PID controllers. . r + PID - C1 u1 + - PI C2 u2 x2 G2 y2 G1 y1 x1 Create tunable control design blocks for the controllers, and fixed LTI models for the plant components, G1 and G2. Also include optional loop-opening locations x1 and x2. These locations indicate where you can open loops or inject signals for the purpose of specifying requirements for tuning the system. G2 = zpk([],-2,3); G1 = zpk([],[-1 -1 -1],10); C20 = tunablePID('C2','pi'); C10 = tunablePID('C1','pid'); X1 = AnalysisPoint('X1'); X2 = AnalysisPoint('X2'); Connect these components to build a model of the entire closed-loop control system. InnerLoop = feedback(X2*G2*C20,1); CL0 = feedback(G1*InnerLoop*C10,X1); CL0.InputName = 'r'; CL0.OutputName = 'y'; CL0 is a tunable genss model. Specifying names for the input and output channels allows you to identify them when you specify tuning requirements for the system. 14-11 14 Control System Tuning Open Control System Tuner to tune this model. controlSystemTuner(CL0) You can now select blocks to tune, create tuning goals, and tune the control system. Control System Architecture in Simulink If you have Simulink Control Design software, you can model an arbitrary control system architecture in a Simulink model and tune the model in Control System Tuner. See “Open Control System Tuner for Tuning Simulink Model” on page 14-13. See Also More About 14-12 • “Building Tunable Models” on page 17-10 • “Specify Blocks to Tune in Control System Tuner” on page 14-24 • “Specify Goals for Interactive Tuning” on page 14-39 Open Control System Tuner for Tuning Simulink Model Open Control System Tuner for Tuning Simulink Model To open Control System Tuner for tuning a Simulink model, open the model. In the Simulink Editor, select Analysis > Control Design > Control System Tuner. Each instance of Control System Tuner is linked to the Simulink model from which it is opened. The title bar of the Control System Tuner window reflects the name of the associated Simulink model. 14-13 14 Control System Tuning Command-Line Equivalents At the MATLAB command line, use the controlSystemTuner command to open Control System Tuner for tuning a Simulink model. For example, the following command opens Control System Tuner for the model rct_helico.slx. controlSystemTuner('rct_helico') If SLT0 is an slTuner interface to the Simulink model, the following command opens Control System Tuner using the information in the interface, such as blocks to tune and analysis points. controlSystemTuner(SLT0) See Also Related Examples • “Specify Operating Points for Tuning in Control System Tuner” on page 14-15 • “Specify Blocks to Tune in Control System Tuner” on page 14-24 More About • 14-14 “Automated Tuning Workflow” on page 14-7 Specify Operating Points for Tuning in Control System Tuner Specify Operating Points for Tuning in Control System Tuner About Operating Points in Control System Tuner When you use Control System Tuner with a Simulink model, the software computes system responses and tunes controller parameters for a linearization of the model. That linearization can depend on model operating conditions. By default, Control System Tuner linearizes at the operating point specified in the model, which comprises the initial state values in the model (the model initial conditions). You can specify one or more alternate operating points for tuning the model. Control System Tuner lets you compute two types of alternate operating points: • Simulation snapshot time. Control System Tuner simulates the model for the amount of time you specify, and linearizes using the state values at that time. Simulation snapshot linearization is useful, for instance, when you know your model reaches an equilibrium state after a certain simulation time. • Steady-state operating point. Control System Tuner finds a steady-state operating point at which some specified condition is met (trimming). For example, if your model represents an automobile motor, you can compute an operating point at which the motor operates steadily at 2000 rpm. For more information on finding steady-state operating points, see “About Operating Points” (Simulink Control Design) and “Compute Steady-State Operating Points” (Simulink Control Design). Linearize at Simulation Snapshot Times This example shows how to compute linearizations at one or more simulation snapshot times. In the Control System tab, in the Operating Point menu, select Linearize At. 14-15 14 Control System Tuning In the Enter snapshot times to linearize dialog box, specify one or more simulation snapshot times. Click OK. 14-16 Specify Operating Points for Tuning in Control System Tuner When you are ready to analyze system responses or tune your model, Control System Tuner computes linearizations at the specified snapshot times. If you enter multiple snapshot times, Control System Tuner computes an array of linearized models, and displays analysis plots that reflect the multiple linearizations in the array. In this case, Control System Tuner also takes into account all linearizations when tuning parameters. This helps to ensure that your tuned controller meets your design requirements at a variety of operating conditions. Compute Operating Points at Simulation Snapshot Times This example shows how to compute operating points at one or more simulation snapshot times. Doing so stores the operating point within Control System Tuner. When you later want to analyze or tune the model at a stored operating point, you can select the stored operating point from the Operating Point menu. In the Control System tab, in the Operating Point menu, select Take simulation snapshot. 14-17 14 Control System Tuning In the Enter snapshot times to linearize dialog box, in the Simulation snapshot times field, enter one or more simulation snapshot times. Enter multiple snapshot times as a vector. 14-18 Specify Operating Points for Tuning in Control System Tuner Click Take Snapshots. Control System Tuner simulates the model and computes the snapshot operating points. Compute additional snapshot operating points if desired. Enter additional snapshot times and click Take Snapshots. Close the dialog box when you are done. When you are ready to analyze responses or tune your model, select the operating point at which you want to linearize the model. In the Control System tab, in the Operating Point menu, stored operating points are available. 14-19 14 Control System Tuning 14-20 Specify Operating Points for Tuning in Control System Tuner If you entered a vector of snapshot times, all the resulting operating points are stored together in an operating-point vector. You can use this vector to tune a control system at several operating points simultaneously. Compute Steady-State Operating Points This example shows how to compute a steady-state operating point with specified conditions. Doing so stores the operating point within Control System Tuner. When you later want to analyze or tune the model at a stored operating point, you can select the stored operating point from the Operating Point menu. In the Control System tab, in the Operating Point menu, select Trim model. 14-21 14 Control System Tuning In the Trim the model dialog box, enter the specifications for the steady-state state values at which you want to find an operating point. For an example showing how to use the Trim the model dialog box to specify the conditions for a steady-state operating point search, see “Compute Operating Points from Specifications Using Linear Analysis Tool” (Simulink Control Design). When you have entered your state specifications, click Start trimming. Control System Tuner finds an operating point that meets the state specifications and stores it. When you are ready to analyze responses or tune your model, select the operating point at which you want to linearize the model. In the Control System tab, in the Operating Point menu, stored operating points are available. 14-22 See Also See Also Related Examples • “Specify Blocks to Tune in Control System Tuner” on page 14-24 • “Robust Tuning Approaches” (Robust Control Toolbox) 14-23 14 Control System Tuning Specify Blocks to Tune in Control System Tuner To select which blocks of your Simulink model to tune in Control System Tuner: 1 In the Tuning tab, click 2 Click Add Blocks. Control System Tuner analyzes your model to find blocks that can be tuned. 3 In the Select Blocks to Tune dialog box, use the nodes in the left panel to navigate through your model structure to the subsystem that contains blocks you want to tune. Check Tune? for the blocks you want to tune. The parameters of blocks you do not check remain constant when you tune the model. Select Blocks. The Select tuned Blocks dialog opens. Tip To find a block in your model, select the block in the Block Name list and click Highlight Selected Block. 4 14-24 Click OK. The Select tuned blocks dialog box now reflects the blocks you added. See Also To import the current value of a block from your model into the current design in Control System Tuner, select the block in the Blocks list and click Sync from Model. Doing so is useful when you have tuned a block in Control System Tuner, but wish to restore that block to its original value. To store the current design before restoring a block value, in the Control System tab, click Store. See Also Related Examples • “View and Change Block Parameterization in Control System Tuner” on page 14-26 More About • “How Tuned Simulink Blocks Are Parameterized” on page 14-36 14-25 14 Control System Tuning View and Change Block Parameterization in Control System Tuner Control System Tuner parameterizes every block that you designate for tuning. • When you tune a Simulink model, Control System Tuner automatically assigns a default parameterization to tunable blocks in the model. The default parameterization depends on the type of block. For example, a PID Controller block configured for PI structure is parameterized by proportional gain and integral gain as follows: 1 u = K p + Ki . s Kp and Ki are the tunable parameters whose values are optimized to satisfy your specified tuning goals. • When you tune a predefined control architecture or a MATLAB (generalized statespace) model, you define the parameterization of each tunable block when you create it at the MATLAB command line. For example, you can use tunablePID to create a tunable PID block. Control System Tuner lets you view and change the parameterization of any block to be tuned. Changing the parameterization can include changing the structure or current parameter values. You can also designate individual block parameters fixed (non-tunable) or limit their tuning range. View Block Parameterization To access the parameterization of a block that you have designated as a tuned block, in the Data Browser, in the Tuned Blocks area, double-click the name of a block. The Tuned Block Editor dialog box opens, displaying the current block parameterization. 14-26 View and Change Block Parameterization in Control System Tuner 14-27 14 Control System Tuning The fields of the Tuned Block Editor display the type of parameterization, such as PID, State-Space, or Gain. For more specific information about the fields, click . Note To find a tuned block in the Simulink model, right-click the block name in the Data Browser and select Highlight. Fix Parameter Values or Limit Tuning Range You can change the current value of a parameter, fix its current value (make the parameter nontunable), or limit the parameter’s tuning range. To change a current parameter value, type a new value in its text box. Alternatively, click to use a variable editor to change the current value. If you attempt to enter an invalid value, the parameter returns to its previous value. Click 14-28 to access and edit additional properties of each parameter. View and Change Block Parameterization in Control System Tuner • Minimum — Minimum value that the parameter can take when the control system is tuned. • Maximum — Maximum value that the parameter can take when the control system is tuned. • Free — When the value is true, Control System Toolbox tunes the parameter. To fix the value of the parameter, set Free to false. For array-valued parameters, you can set these properties independently for each entry in the array. For example, for a vector-valued gain of length 3, enter [1 10 100] to set the current value of the three gains to 1, 10, and 100 respectively. Alternatively, click use a variable editor to specify such values. to 14-29 14 Control System Tuning For vector or matrix-valued parameters, you can use the Free parameter to constrain the structure of the parameter. For example, to restrict a matrix-valued parameter to be a diagonal matrix, set the current values of the off-diagonal elements to 0, and set the corresponding entries in Free to false. Custom Parameterization When tuning a control system represented by a Simulink model or by a “Predefined Feedback Architecture” on page 14-9, you can specify a custom parameterization for any tuned block using a generalized state-space (genss) model. To do so, create and configure a genss model in the MATLAB workspace that has the desired parameterization, initial parameter values, and parameter properties. In the Change parameterization dialog box, select Custom. In the Parameterization area, the variable name of the genss model. For example, suppose you want to specify a tunable low-pass filter, F = a/(s +a), where a is the tunable parameter. First, at the MATLAB command line, create a tunable genss model that represents the low-pass filter structure. a = realp('a',1); F = tf(a,[1 a]); F = Generalized continuous-time state-space model with 1 outputs, 1 inputs, 1 states, and the following blocks: a: Scalar parameter, 2 occurrences. Type "ss(F)" to see the current value, "get(F)" to see all properties, and "F.Blocks" to interact with the blocks. Then, in the Tuned Block Editor, enter F in the Parameterization area. 14-30 View and Change Block Parameterization in Control System Tuner When you specify a custom parameterization for a Simulink block, you might not be able to write the tuned block value back to the Simulink model. When writing values to Simulink blocks, Control System Tuner skips blocks that cannot represent the tuned value in a straightforward and lossless manner. For example, if you reparameterize a PID Controller Simulink block as a third-order state-space model, Control System Tuner will not write the tuned value back to the block. Block Rate Conversion When Control System Tuner writes tuned parameters back to the Simulink model, each tuned block value is automatically converted from the sample time used for tuning, to the sample time of the Simulink block. When the two sample times differ, the Tuned Block Editor contains additional rate conversion options that specify how this resampling operation is performed for the corresponding block. 14-31 14 Control System Tuning By default, Control System Tuner performs linearization and tuning in continuous time (sample time = 0). You can specify discrete-time linearization and tuning and change the sample time. To do so, on the Control System tab, click Linearization Options. Sample time for tuning reflects the sample time specified in the Linearization Options dialog box. The remaining rate conversion options depend on the parameterized block. 14-32 View and Change Block Parameterization in Control System Tuner Rate Conversion for Parameterized PID Blocks For parameterization of continuous-time PID Controller and PID Controller (2-DOF) blocks, you can independently specify the rate-conversion methods as discretization formulas for the integrator and derivative filter. Each has the following options: • Trapezoidal (default) — Integrator or derivative filter discretized as (Ts/2)*(z +1)/(z-1), where Ts is the target sample time. • Forward Euler — Ts/(z-1). • Backward Euler — Ts*z/(z-1). For more information about PID discretization formulas, see “Discrete-Time ProportionalIntegral-Derivative (PID) Controllers” on page 2-24. For discrete-time PID Controller and PID Controller (2-DOF) blocks, you set the integrator and derivative filter methods in the block dialog box. You cannot change them in the Tuned Block Editor. Rate Conversion for Other Parameterized Blocks For blocks other than PID Controller blocks, the following rate-conversion methods are available: • Zero-order hold — Zero-order hold on the inputs. For most dynamic blocks this is the default rate-conversion method. • Tustin — Bilinear (Tustin) approximation. • Tustin with prewarping — Tustin approximation with better matching between the original and rate-converted dynamics at the prewarp frequency. Enter the frequency in the Prewarping frequency field. • First-order hold — Linear interpolation of inputs. • Matched (SISO only) — Zero-pole matching equivalents. For more detailed information about these rate-conversion methods, see “ContinuousDiscrete Conversion Methods” on page 5-26. Blocks with Fixed Rate Conversion Methods For the following blocks, you cannot set the rate-conversion method in the Tuned Block Editor. 14-33 14 Control System Tuning • Discrete-time PID Controller and PID Controller (2-DOF) block. Set the integrator and derivative filter methods in the block dialog box. • Gain block, because it is static. • Transfer Fcn Real Zero block. This block can only be tuned at the sample time specified in the block. • Block that has been discretized using the Model Discretizer. Sample time for this block is specified in the Model Discretizer itself. See Also Related Examples • “Specify Blocks to Tune in Control System Tuner” on page 14-24 More About • 14-34 “How Tuned Simulink Blocks Are Parameterized” on page 14-36 Setup for Tuning Control System Modeled in MATLAB Setup for Tuning Control System Modeled in MATLAB To model your control architecture in MATLAB for tuning in Control System Tuner, construct a tunable model of the control system that identifies and parameterizes its tunable elements. You do so by combining numeric LTI models of the fixed elements with parametric models of the tunable elements. The result is a tunable generalized statespace genss model. Building a tunable genss model for Control System Tuner is the same as building such a model for tuning at the command line. For information about building such models, “Setup for Tuning MATLAB Models”. When you have a tunable genss model of your control system, use the controlSystemTuner command to open Control System Tuner. For example, if T0 is the genss model, the following command opens Control System Tuner for tuning T0: controlSystemTuner(T0) See Also Related Examples • “Specify Goals for Interactive Tuning” on page 14-39 14-35 14 Control System Tuning How Tuned Simulink Blocks Are Parameterized Blocks With Predefined Parameterization When you tune a Simulink model, either with Control System Tuner or at the command line through an slTuner interface, the software automatically assigns a predefined parameterization to certain Simulink blocks. For example, for a PID Controller block set to the PI controller type, the software automatically assigns the parameterization Kp + Ki/s, where Kp and Ki are the tunable parameters. For blocks that have a predefined parameterization, you can write tuned values back to the Simulink model for validating the tuned controller. Blocks that have a predefined parameterization include the following: Simulink Library Blocks with Predefined Parameterization Math Operations Gain Continuous • State-Space • Transfer Fcn • Zero-Pole • PID Controller • PID Controller (2DOF) Discrete • Discrete State-Space • Discrete Transfer Fcn • Discrete Zero-Pole • Discrete Filter • Discrete PID Controller • Discrete PID Controller (2DOF) Lookup Tables • 1-D Lookup Table • 2-D Lookup Table • n-D Lookup Table Control System Toolbox 14-36 LTI System How Tuned Simulink Blocks Are Parameterized Simulink Library Blocks with Predefined Parameterization Discretizing (Model Discretizer Blocks) • Discretized State-Space • Discretized Transfer Fcn • Discretized Zero-Pole • Discretized LTI System • Discretized Transfer Fcn (with initial states) Simulink Extras/Additional Linear State-Space (with initial outputs) Scalar Expansion The following tunable blocks support scalar expansion: • Discrete Filter • Gain • 1-D Lookup Table, 2-D Lookup Table, n-D Lookup Table • PID Controller, PID Controller (2DOF) Scalar expansion means that the block parameters can be scalar values even when the input and output signals are vectors. For example, you can use a Gain block to implement y = k*u with scalar k and vector u and y. To do so, you set the Multiplication mode of the block to Element-wise(K.*u), and set the gain value to the scalar k. When a tunable block uses scalar expansion, its default parameterization uses tunable scalars. For example, in the y = k*u Gain block, the software parameterizes the scalar k as a tunable real scalar (realp of size [1 1]). If instead you want to tune different gain values for each channel, replace the scalar gain k by a N-by-1 gain vector in the block dialog, where N is the number of channels, the length of the vectors u and y. The software then parameterizes the gain as a realp of size [N 1]. Blocks Without Predefined Parameterization You can specify blocks for tuning that do not have a predefined parameterization. When you do so, the software assigns a state-space parameterization to such blocks based upon the block linearization. For blocks that do not have a predefined parameterization, the software cannot write tuned values back to the block, because there is no clear mapping 14-37 14 Control System Tuning between the tuned parameters and the block. To validate a tuned control system that contains such blocks, you can specify a block linearization in your model using the value of the tuned parameterization. (See “Specify Linear System for Block Linearization Using MATLAB Expression” (Simulink Control Design) for more information about specifying block linearization.) View and Change Block Parameterization You can view and edit the current parameterization of every block you designate for tuning. • In Control System Tuner, see “View and Change Block Parameterization in Control System Tuner” on page 14-26. • At the command line, use getBlockParam to view the current block parameterization. Use setBlockParam to change the block parameterization. 14-38 Specify Goals for Interactive Tuning Specify Goals for Interactive Tuning This example shows how to specify your tuning goals for automated tuning in Control System Tuner. Use the New Goal menu to create a tuning goal such as a tracking requirement, disturbance rejection specification, or minimum stability margins. Then, when you are ready to tune your control system, use Manage Goals to designate which goals to enforce. This example creates tuning goals for tuning the sample model rct_helico. Choose Tuning Goal Type In Control System Tuner, in the Tuning tab, click New Goal. Select the type of goal you want to create. A tuning goal dialog box opens in which you can provide the detailed specifications of your goal. For example, select Tracking of step commands to make a particular step response of your control system match a desired response. 14-39 14 Control System Tuning Choose Signal Locations for Evaluating Tuning Goal Specify the signal locations in your control system at which the tuning goal is evaluated. For example, the step response goal specifies that a step signal applied at a particular input location yields a desired response at a particular output location. Use the Step Response Selection section of the dialog box to specify these input and output locations. 14-40 Specify Goals for Interactive Tuning (Other tuning goal types, such as loop-shape or stability margins, require you to specify only one location for evaluation. The procedure for specifying the location is the same as illustrated here.) Under Specify step-response inputs, click input locations appears. Add signal to list. A list of available If the signal you want to designate as a step-response input is in the list, click the signal to add it to the step-response inputs. If the signal you want to designate does not appear, and you are tuning a Simulink model, click Select signal from model. In the Select signals dialog box, build a list of the signals you want. To do so, click signals in the Simulink model editor. The signals that you click appear in the Select signals dialog box. Click one signal to create a SISO tuning goal, and click multiple signals to create a MIMO tuning goal. Click Add signal(s). The Select signals dialog box closes, returning you to the new tuning-goal specification dialog box. 14-41 14 Control System Tuning The signals you selected now appear in the list of step-response inputs in the tuning goal dialog box. Similarly, specify the locations at which the step response is measured to the stepresponse outputs list. For example, the following configuration constrains the response to a step input applied at theta-ref and measured at theta in the Simulink model rct_helico. 14-42 Specify Goals for Interactive Tuning Tip To highlight any selected signal in the Simulink model, click from the input or output list, click reorder them using and . To remove a signal . When you have selected multiple signals, you can . Specify Loop Openings Most tuning goals can be enforced with loops open at one or more locations in the control system. Click tuning goal. Add loop opening location to list to specify such locations for the 14-43 14 Control System Tuning Define Other Specifications of the Tuning Goal The tuning goal dialog box prompts you to specify other details about the tuning goal. For example, to create a step response requirement, you provide details of the desired step response in the Desired Response area of the Step Response Goal dialog box. Some tuning goals have additional options in an Options section of the dialog box. For information about the fields for specifying a particular tuning goal, click tuning goal dialog box. in the Store the Tuning Goal for Tuning When you have finished specifying the tuning goal, click OK in the tuning goal dialog box. The new tuning goal appears in the Tuning Goals section of the Data Browser. A new figure opens displaying a graphical representation of the tuning goal. When you tune your control system, you can refer to this figure to evaluate graphically how closely the tuned system satisfies the tuning goal. 14-44 Specify Goals for Interactive Tuning Tip To edit the specifications of the tuning goal, double-click the tuning goal in the Data Browser. Activate the Tuning Goal for Tuning When you have saved your tuning goal, click goals. New Goal to create additional tuning When you are ready to tune your control system, click Manage Goals to select which tuning goals are active for tuning. In the Manage Tuning Goals dialog box, 14-45 14 Control System Tuning Active is checked by default for any new goals. Uncheck Active for any tuning goal that you do not want enforced. You can also designate one or more tuning goals as Hard goals. Control System Tuner attempts to satisfy hard requirements, and comes as close as possible to satisfying remaining (soft) requirements subject to the hard constraints. By default, new goals are designated soft goals. Check Hard for any goal to designate it a hard goal. For example, if you tune with the following configuration, Control System Tuner optimizes StepRespGoal1, subject to MarginsGoal1. The tuning goal PolesGoal1 is ignored. Deactivating tuning goals or designating some goals as soft requirements can be useful when investigating the tradeoffs between different tuning requirements. For example, if you do not obtain satisfactory performance with all your tuning goals active and hard, you might try another design in which less crucial goals are designated as soft or deactivated entirely. See Also Related Examples • 14-46 “Manage Tuning Goals” on page 14-179 See Also • “Quick Loop Tuning of Feedback Loops in Control System Tuner” on page 14-48 • “Create Response Plots in Control System Tuner” on page 14-198 14-47 14 Control System Tuning Quick Loop Tuning of Feedback Loops in Control System Tuner This example shows how to tune a Simulink model of a control system to meet a specified bandwidth and specified stability margins in Control System Tuner, without explicitly creating tuning goals that capture these requirements. You can use a similar approach for quick loop tuning of control systems modeled in MATLAB. This example demonstrates how the Quick Loop Tuning option of Control System Tuner generates tuning goals from your crossover frequency and gain and phase margin specifications. This option lets you quickly set up SISO or MIMO feedback loops for tuning using a loop-shaping approach. The example also shows how to add further tuning requirements to the control system after using the Quick Loop Tuning option. Quick Loop Tuning is the Control System Tuner equivalent of the looptune command. Set up the Model for Tuning Open the Simulink model. open_system('rct_distillation') This model represents a distillation column, captured in the two-input, two-output plant G. The tunable elements are the decoupling gain matrix DM, and the two PI controllers, PI_L and PI_V. (For more information about this model, see “Decoupling Controller for a Distillation Column” on page 15-20.) 14-48 Quick Loop Tuning of Feedback Loops in Control System Tuner Suppose your goal is to tune the MIMO feedback loop between r and y to a bandwidth between 0.1 and 0.5 rad/s. Suppose you also require a gain margin of 7 dB and a phase margin of 45 degrees. You can use the Quick Loop Tuning option to quickly configure Control System Tuner for these goals. In the Simulink model editor, open Control System Tuner by selecting Analysis > Control Design > Control System Tuner. Designate the blocks you want to tune. In the Tuning tab of Control System Tuner, click Select Blocks. In the Select tuned blocks dialog box, click Add blocks. Then, select DM, PI_L, and PI_V for tuning. (For more information about selecting tuned blocks, see “Specify Blocks to Tune in Control System Tuner” on page 14-24.) The model is now ready to tune to the target bandwidth and stability margins. Specify the Goals for Quick Loop Tuning In the Tuning tab, select New Goal > Quick Loop Tuning. For Quick Loop Tuning, you need to identify the actuator signals and sensor signals that separate the plant portion of the control system from the controller, which for the purpose of Quick Loop Tuning is the rest of the control system. The actuator signals are the controller outputs that drive the plant, or the plant inputs. The sensor signals are the measurements of plant output that feed back into the controller. In this control system, 14-49 14 Control System Tuning the actuator signals are represented by the vector signal u, and the sensor signals by the vector signal y. In the Quick Loop Tuning dialog box, under Specify actuator signals (controls), add the actuator signal, u. Similarly, under Specify sensor signals (measurements), add the sensor signal, y (For more information about specifying signals for tuning, see “Specify Goals for Interactive Tuning” on page 14-39.) Under Desired Goals, in the Target gain crossover region field, enter the target bandwidth range, [0.1 0.5]. Enter the desired gain margin and phase margin in the corresponding fields. 14-50 Quick Loop Tuning of Feedback Loops in Control System Tuner 14-51 14 Control System Tuning Click OK. Control System Tuner automatically generates tuning goals that capture the desired goals you entered in the dialog box. Examine the Automatically-Created Tuning Goals In this example, Control System Tuner creates a Loop Shape Goal and a Margins Goal. If you had changed the pole-location settings in the Quick Loop Tuning dialog box, a Poles goal would also have been created. 14-52 Quick Loop Tuning of Feedback Loops in Control System Tuner Click Manage Goals to examine the automatically-created goals. By default, the goals are active and designated as soft tuning goals. You can double-click the tuning goals to examine their parameters, which are automatically computed and populated. You can also examine the graphical representations of the tuning goals. In the Tuning tab, examine the LoopTuning1_LoopShapeGoal plot. 14-53 14 Control System Tuning The target crossover range is expressed as a Loop Shape goal with an integrator openloop gain profile. The shaded areas of the graph show that the permitted crossover range is [0.1 0.5] rad/s, as you specified in the Quick Loop Tuning dialog box. Similarly, your margin requirements are captured in the LoopTuning1_MarginsGoal plot. Tune the Model Tune to tune the model to meet the automatically-created tuning goals. In the Click tuning goal plots, you can see that the requirements are satisfied. 14-54 Quick Loop Tuning of Feedback Loops in Control System Tuner To create additional plots for examining other system responses, see “Create Response Plots in Control System Tuner” on page 14-198. Change Design Requirements If you want to change your design requirements after using Quick Loop Tuning, you can edit the automatically-created tuning goals and tune the model again. You can also create additional tuning goals. For example, add a requirement that limits the response to a disturbance applied at the plant inputs. Limit the response to a step command at dL and dV at the outputs, y, to be well damped, to settle in less than 20 seconds, and not exceed 4 in amplitude. Select New Goal > Rejection of step disturbances and enter appropriate values in the Step Rejection Goal dialog box. (For more information about creating tuning goals, see “Specify Goals for Interactive Tuning” on page 14-39.) 14-55 14 Control System Tuning 14-56 See Also You can now retune the model to meet all these tuning goals. See Also looptune (for slTuner) Related Examples • “Specify Operating Points for Tuning in Control System Tuner” on page 14-15 • “Manage Tuning Goals” on page 14-179 • “Setup for Tuning Control System Modeled in MATLAB” on page 14-35 • “Stability Margins in Control System Tuning” on page 14-218 14-57 14 Control System Tuning Quick Loop Tuning Purpose Tune SISO or MIMO feedback loops using a loop-shaping approach in Control System Tuner. Description Quick Loop Tuning lets you tune your system to meet open-loop gain crossover and stability margin requirements without explicitly creating tuning goals that capture these requirements. You specify the feedback loop whose open-loop gain you want to shape by designating the actuator signals (controls) and sensor signals (measurements) that form the loop. Actuator signals are the signals that drive the plant. The sensor signals are the plant outputs that feed back into the controller. You enter the target loop bandwidth and desired gain and phase margins. You can also specify constraints on pole locations of the tuned system, to eliminate fast dynamics. Control System Tuner automatically creates Tuning Goals that capture your specifications and ensure integral action at frequencies below the target loop bandwidth. Creation In the Tuning tab of Control System Tuner, select New Goal > Quick Loop Tuning to specify loop-shaping requirements. Command-Line Equivalent When tuning control systems at the command line, use looptune (for slTuner) or looptune for tuning feedback loops using a loop-shaping approach. Feedback Loop Selection Use this section of the dialog box to specify input, output, and loop-opening locations for evaluating the tuning goal. • Specify actuator signals (controls) 14-58 Quick Loop Tuning Designate one or more signals in your model as actuator signals. These are the input signals that drive the plant. To tune a SISO feedback loop, select a single-valued input signal. To tune MIMO loop, select multiple signals or a vector-valued signal. • Specify sensor signals (measurements) Designate one or more signals in your model as sensor signals. These are the plant outputs that provide feedback into the controller. To tune a SISO feedback loop, select a single-valued input signal. To tune MIMO loop, select multiple signals or a vectorvalued signal. • Compute the response with the following loops open Designate additional locations at which to open feedback loops for the purpose of tuning the loop defined by the control and measurement signals. Quick Loop Tuning tunes the open-loop response of the loop defined by the control and measurement signals. If you want your specifications for that loop to apply with other feedback loops in the system opened, specify loop-opening locations in this section of the dialog box. For example, if you are tuning a cascaded-loop control system with an inner loop and an outer loop, you might want to tune the inner loop with the outer loop open. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can and . For more information on how to specify signal locations reorder them using for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Desired Goals Use this section of the dialog box to specify desired characteristics of the tuned system. Control System Tuner converts these into Loop Shape, Margin, and Poles goals. • Target gain crossover region Specify a frequency range in which the open-loop gain should cross 0 dB. Specify the frequency range as a row vector of the form [min,max], expressed in frequency units of your model. Alternatively, if you specify a single target frequency, wc, the target range is taken as [wc/10^0.1,wc*10^0.1], or wc ± 0.1 decade. 14-59 14 Control System Tuning • Gain margin (db) Specify the desired gain margin in decibels. For MIMO control system, the gain margin is the multiloop disk margin. For information about multiloop disk margins, see “Stability Analysis Using Disk Margins” (Robust Control Toolbox). • Phase margin (degrees) Specify the desired phase margin in degrees. For MIMO control system, the phase margin is the multiloop disk margin. For information about multiloop disk margins, see “Stability Analysis Using Disk Margins” (Robust Control Toolbox). • Keep poles inside the following region Specify minimum decay rate and maximum natural frequency for the closed-loop poles of the tuned system. While the other Quick Loop Tuning options specify characteristics of the open-loop response, these specifications apply to the closed-loop dynamics. The minimum decay rate you enter constrains the closed-loop pole locations to: • Re(s) < -mindecay, for continuous-time systems. • log(|z|) < -mindecay*Ts, for discrete-time systems with sample time Ts. The maximum frequency you enter constrains the closed-loop poles to satisfy |s| < maxfreq for continuous time, or |log(z)| < maxfreq*Ts for discrete-time systems with sample time Ts. This constraint prevents fast dynamics in the closed-loop system. Options Use this section of the dialog box to specify additional characteristics. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. 14-60 See Also For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms Control System Tuner uses looptuneSetup (for slTuner) or looptuneSetup to convert Quick Loop Tuning specifications into tuning goals. See Also Related Examples • “Quick Loop Tuning of Feedback Loops in Control System Tuner” on page 14-48 • “Specify Goals for Interactive Tuning” on page 14-39 • “Visualize Tuning Goals” on page 14-189 • “Manage Tuning Goals” on page 14-179 • “Stability Margins in Control System Tuning” on page 14-218 14-61 14 Control System Tuning Step Tracking Goal Purpose Make the step response from specified inputs to specified outputs closely match a target response, when using Control System Tuner. Description Step Tracking Goal constrains the step response between the specified signal locations to match the step response of a stable reference system. The constraint is satisfied when the relative difference between the tuned and target responses falls within the tolerance you specify. You can use this goal to constrain a SISO or MIMO response of your control system. You can specify the reference system for the target step response in terms of first-order system characteristics (time constant) or second-order system characteristics (natural frequency and percent overshoot). Alternatively, you can specify a custom reference system as a numeric LTI model. 14-62 Step Tracking Goal Creation In the Tuning tab of Control System Tuner, select New Goal > Tracking of step commands to create a Step Tracking Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.StepTracking to specify a step response goal. Step Response Selection Use this section of the dialog box to specify input, output, and loop-opening locations for evaluating the tuning goal. • Specify step-response inputs Select one or more signal locations in your model at which to apply the step input. To constrain a SISO response, select a single-valued input signal. For example, to constrain the step response from a location named 'u' to a location named 'y', click Add signal to list and select 'u'. To constrain a MIMO response, select multiple signals or a vector-valued signal. • Specify step-response outputs Select one or more signal locations in your model at which to measure the response to the step input. To constrain a SISO response, select a single-valued output signal. For example, to constrain the step response from a location named 'u' to a location named 'y', click Add signal to list and select 'y'. To constrain a MIMO response, select multiple signals or a vector-valued signal. For MIMO systems, the number of outputs must equal the number of outputs. • Compute step response with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. 14-63 14 Control System Tuning Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Desired Response Use this section of the dialog box to specify the shape of the desired step response. • First-order characteristics Specify the desired step response (the reference model Href) as a first-order response with time constant τ: H ref = 1 /t . s +1 / t Enter the desired value for τ in the Time Constant text box. Specify τ in the time units of your model. • Second-order characteristics Specify the desired step response as a second-order response with time constant τ, and natural frequency 1/τ. Enter the desired value for τ in the Time Constant text box. Specify τ in the time units of your model. Enter the target overshoot percentage in the Overshoot text box. The second-order reference system has the form: H ref = (1 / t ) 2 . 2 s2 + 2 ( z / t ) s + (1 / t ) The damping constant ζ is related to the overshoot percentage by ζ = cos(atan2(pi,-log(overshoot/100))). 14-64 Step Tracking Goal • Custom reference model Specify the reference system for the desired step response as a dynamic system model, such as a tf, zpk, or ss model. Enter the name of the reference model in the MATLAB workspace in the LTI model to match text field. Alternatively, enter a command to create a suitable reference model, such as tf(1,[1 1.414 1]). The reference model must be stable and must have DC gain of 1 (zero steady-state error). The model can be continuous or discrete. If the model is discrete, it can include time delays which are treated as poles at z = 0. The reference model can be MIMO, provided that it is square and that its DC singular value (sigma) is 1. Then number of inputs and outputs of the reference model must match the dimensions of the inputs and outputs specified for the step response goal. For best results, the reference model should also include intrinsic system characteristics such as non-minimum-phase zeros (undershoot). If your selected inputs and outputs define a MIMO system and you apply a SISO reference system, the software attempts to match the diagonal channels of the MIMO system. In that case, cross-couplings tend to be minimized. Options Use this section of the dialog box to specify additional characteristics of the step response goal. • Keep % mismatch below Specify the relative matching error between the actual (tuned) step response and the target step response. Increase this value to loosen the matching tolerance. The relative matching error, erel, is defined as: erel = y ( t ) - yref ( t ) 1 - yref ( t ) 2. 2 y(t) – yref(t) is the response mismatch, and 1 – yref(t) is the step-tracking error of the target model. ◊ 2 denotes the signal energy (2-norm). 14-65 14 Control System Tuning • Adjust for step amplitude For a MIMO tuning goal, when the choice of units results in a mix of small and large signals in different channels of the response, this option allows you to specify the relative amplitude of each entry in the vector-valued step input. This information is used to scale the off-diagonal terms in the transfer function from reference to tracking error. This scaling ensures that cross-couplings are measured relative to the amplitude of each reference signal. For example, suppose that tuning goal is that outputs 'y1' and 'y2' track reference signals 'r1'and 'r2'. Suppose further that you require the outputs to track the references with less than 10% cross-coupling. If r1 and r2 have comparable amplitudes, then it is sufficient to keep the gains from r1 to y2 and r2 and y1 below 0.1. However, if r1 is 100 times larger than r2, the gain from r1 to y2 must be less than 0.001 to ensure that r1 changes y2 by less than 10% of the r2 target. To ensure this result, set Adjust for step amplitude to Yes. Then, enter [100,1] in the Amplitudes of step commands text box. This tells Control System Tuner to take into account that the first reference signal is 100 times greater than the second reference signal. The default value, No , means no scaling is applied. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. 14-66 See Also For Step Response Goal, f(x) is given by: 1 T ( s, x) - H ref ( s ) s 2. f (x) = 1 erel Href ( s) - I s 2 ( ) ( ) T(s,x) is the closed-loop transfer function between the specified inputs and outputs, evaluated with parameter values x. Href(s) is the reference model. erel is the relative error (see “Options” on page 14-65). ◊ 2 denotes the H2 norm (see norm). This tuning goal also imposes an implicit stability constraint on the closed-loop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 14-67 14 Control System Tuning Step Rejection Goal Purpose Set a minimum standard for rejecting step disturbances, when using Control System Tuner. Description Use Step Rejection Goal to specify how a step disturbance injected at a specified location in your control system affects the signal at a specified output location. You can specify the desired response in time-domain terms of peak value, settling time, and damping ratio. Control System Tuner attempts to make the actual rejection at least as good as the desired response. Alternatively, you can specify the response as a stable reference model having DC-gain. In that case, the tuning goal is to reject the disturbance as well as or better than the reference model. To specify disturbance rejection in terms of a frequency-domain attenuation profile, use Disturbance Rejection Goal. 14-68 Step Rejection Goal When you create a tuning goal in Control System Tuner, a tuning-goal plot is generated. The dotted line shows the target step response you specify. The solid line is the current corresponding response of your system. Creation In the Tuning tab of Control System Tuner, select New Goal > Rejection of step disturbance to create a Step Rejection Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.StepRejection to specify a step response goal. Step Disturbance Response Selection Use this section of the dialog box to specify input, output, and loop-opening locations for evaluating the tuning goal. • Specify step disturbance inputs Select one or more signal locations in your model at which to apply the input. To constrain a SISO response, select a single-valued input signal. For example, to constrain the step-disturbance response from a location named 'u' to a location named 'y', click Add signal to list and select 'u'. To constrain a MIMO response, select multiple signals or a vector-valued signal. • Specify step response outputs Select one or more signal locations in your model at which to measure the response to the step disturbance. To constrain a SISO response, select a single-valued output signal. For example, to constrain the transient response from a location named 'u' to Add signal to list and select 'y'. To constrain a a location named 'y', click MIMO response, select multiple signals or a vector-valued signal. For MIMO systems, the number of outputs must equal the number of outputs. • Compute the response with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you 14-69 14 Control System Tuning identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Desired Response to Step Disturbance Use this section of the dialog box to specify the shape of the desired response to the step disturbance. Control System Tuner attempts to make the actual response at least as good as the desired response. • Response Characteristics Specify the desired response in terms of time-domain characteristics. Enter the maximum amplitude, maximum settling time, and minimum damping constant in the text boxes. • Reference Model Specify the desired response in terms of a reference model. Enter the name of the reference model in the MATLAB workspace in the Reference Model text field. Alternatively, enter a command to create a suitable reference model, such as tf([1 0],[1 1.414 1]). The reference model must be stable and must have zero DC gain. The model can be continuous or discrete. If the model is discrete, it can include time delays which are treated as poles at z = 0. For best results, the reference model and the open-loop response from the disturbance to the output should have similar gains at the frequency where the reference model gain peaks. 14-70 Step Rejection Goal Options Use this section of the dialog box to specify additional characteristics of the step rejection goal. • Adjust for amplitude of input signals and Adjust for amplitude of output signals For a MIMO tuning goal, when the choice of units results in a mix of small and large signals in different channels of the response, this option allows you to specify the relative amplitude of each entry in the vector-valued signals. This information is used to scale the off-diagonal terms in the transfer function from the tuning goal inputs to outputs. This scaling ensures that cross-couplings are measured relative to the amplitude of each reference signal. When these options are set to No, the closed-loop transfer function being constrained is not scaled for relative signal amplitudes. When the choice of units results in a mix of small and large signals, using an unscaled transfer function can lead to poor tuning results. Set the option to Yes to provide the relative amplitudes of the input signals and output signals of your transfer function. For example, suppose the tuning goal constrains a 2-input, 2-output transfer function. Suppose further that second input signal to the transfer function tends to be about 100 times greater than the first signal. In that case, select Yes and enter [1,100] in the Amplitudes of input signals text box. Adjusting signal amplitude causes the tuning goal to be evaluated on the scaled transfer function Do–1T(s)Di, where T(s) is the unscaled transfer function. Do and Di are diagonal matrices with the Amplitudes of output signals and Amplitudes of input signals values on the diagonal, respectively. The default value, No, means no scaling is applied. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth 14-71 14 Control System Tuning models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms Evaluating Tuning Goals When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning requirement is a hard constraint. Step Rejection Goal aims to keep the gain from disturbance to output below the gain of the reference model. The scalar value of the requirement f(x) is given by: f ( x ) = WF ( s ) Tdy ( s, x ) • , or its discrete-time equivalent. Here, Tdy(s,x) is the closed-loop transfer function of the constrained response, and ◊ • denotes the H∞ norm (see norm). WF is a frequency weighting function derived from the step-rejection profile you specify in the tuning goal. The gain of WF roughly matches the inverse of the reference model for gain values within 60 dB of the peak gain. For numerical reasons, the weighting function levels off outside this range, unless you specify a reference model that changes slope outside this range. This adjustment is called regularization. Because poles of WF close to s = 0 or s = Inf might lead to poor numeric conditioning for tuning, it is not recommended to specify reference models with very low-frequency or very high-frequency dynamics.For more information about regularization and its effects, see “Visualize Tuning Goals” on page 14189. Implicit Constraints This tuning goal also imposes an implicit stability constraint on the closed-loop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly 14-72 See Also constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 14-73 14 Control System Tuning Transient Goal Purpose Shape how the closed-loop system responds to a specific input signal when using Control System Tuner. Use a reference model to specify the desired transient response. Description Transient Goal constrains the transient response from specified input locations to specified output locations. This requirement specifies that the transient response closely match the response of a reference model. The constraint is satisfied when the relative difference between the tuned and target responses falls within the tolerance you specify. You can constrain the response to an impulse, step, or ramp input signal. You can also constrain the response to an input signal that is given by the impulse response of an input filter you specify. 14-74 Transient Goal Creation In the Tuning tab of Control System Tuner, select New Goal > Transient response matching to create a Transient Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.Transient to specify a step response goal. Response Selection Use this section of the dialog box to specify input, output, and loop-opening locations for evaluating the tuning goal. • Specify response inputs Select one or more signal locations in your model at which to apply the input. To constrain a SISO response, select a single-valued input signal. For example, to constrain the transient response from a location named 'u' to a location named 'y', Add signal to list and select 'u'. To constrain a MIMO response, select click multiple signals or a vector-valued signal. • Specify response outputs Select one or more signal locations in your model at which to measure the transient response. To constrain a SISO response, select a single-valued output signal. For example, to constrain the transient response from a location named 'u' to a location named 'y', click Add signal to list and select 'y'. To constrain a MIMO response, select multiple signals or a vector-valued signal. For MIMO systems, the number of outputs must equal the number of outputs. • Compute the response with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. 14-75 14 Control System Tuning Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Initial Signal Selection Select the input signal shape for the transient response you want to constrain in Control System Tuner. • Impulse — Constrain the response to a unit impulse. • Step — Constrain the response to a unit step. Using Step is equivalent to using a Step Tracking Goal. • Ramp — Constrain the response to a unit ramp, u = t. • Other — Constrain the response to a custom input signal. Specify the custom input signal by entering a transfer function (tf or zpkmodel) in the Use impulse response of filter field. The custom input signal is the response of this transfer function to a unit impulse. This transfer function represents the Laplace transform of the desired custom input signal. For example, to constrain the transient response to a unit-amplitude sine wave of frequency w, enter tf(w,[1,0,w^2]). This transfer function is the Laplace transform of sin(wt). The transfer function you enter must be continuous, and can have no poles in the open right-half plane. The series connection of this transfer function with the reference system for the desired transient response must have no feedthrough term. Desired Transient Response Specify the reference system for the desired transient response as a dynamic system model, such as a tf, zpk, or ss model. The Transient Goal constrains the system response to closely match the response of this system to the input signal you specify in Initial Signal Selection. Enter the name of the reference model in the MATLAB workspace in the Reference Model field. Alternatively, enter a command to create a suitable reference model, such as 14-76 Transient Goal tf(1,[1 1.414 1]). The reference model must be stable, and the series connection of the reference model with the input shaping filter must have no feedthrough term. Options Use this section of the dialog box to specify additional characteristics of the transient response goal. • Keep % mismatch below Specify the relative matching error between the actual (tuned) transient response and the target response. Increase this value to loosen the matching tolerance. The relative matching error, erel, is defined as: gap = y ( t ) - yref ( t ) yref ( tr) ( t) 2 . 2 y(t) – yref(t) is the response mismatch, and 1 – yref(tr)(t) is the transient portion of yref (deviation from steady-state value or trajectory). ◊ 2 denotes the signal energy (2norm). The gap can be understood as the ratio of the root-mean-square (RMS) of the mismatch to the RMS of the reference transient. • Adjust for amplitude of input signals and Adjust for amplitude of output signals For a MIMO tuning goal, when the choice of units results in a mix of small and large signals in different channels of the response, this option allows you to specify the relative amplitude of each entry in the vector-valued signals. This information is used to scale the off-diagonal terms in the transfer function from the tuning goal inputs to outputs. This scaling ensures that cross-couplings are measured relative to the amplitude of each reference signal. When these options are set to No, the closed-loop transfer function being constrained is not scaled for relative signal amplitudes. When the choice of units results in a mix of small and large signals, using an unscaled transfer function can lead to poor tuning results. Set the option to Yes to provide the relative amplitudes of the input signals and output signals of your transfer function. For example, suppose the tuning goal constrains a 2-input, 2-output transfer function. Suppose further that second input signal to the transfer function tends to be about 14-77 14 Control System Tuning 100 times greater than the first signal. In that case, select Yes and enter [1,100] in the Amplitudes of input signals text box. Adjusting signal amplitude causes the tuning goal to be evaluated on the scaled transfer function Do–1T(s)Di, where T(s) is the unscaled transfer function. Do and Di are diagonal matrices with the Amplitudes of output signals and Amplitudes of input signals values on the diagonal, respectively. The default value, No, means no scaling is applied. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Tips • When you use this requirement to tune a control system, Control System Tuner attempts to enforce zero feedthrough (D = 0) on the transfer that the requirement constrains. Zero feedthrough is imposed because the H2 norm, and therefore the value of the tuning goal (see “Algorithms” on page 14-79), is infinite for continuous-time systems with nonzero feedthrough. Control System Tuner enforces zero feedthrough by fixing to zero all tunable parameters that contribute to the feedthrough term. Control System Tuner returns an error when fixing these tunable parameters is insufficient to enforce zero feedthrough. In such cases, you must modify the requirement or the control structure, or manually fix some tunable parameters of your system to values that eliminate the feedthrough term. When the constrained transfer function has several tunable blocks in series, the software’s approach of zeroing all parameters that contribute to the overall feedthrough might be conservative. In that case, it is sufficient to zero the feedthrough 14-78 Transient Goal term of one of the blocks. If you want to control which block has feedthrough fixed to zero, you can manually fix the feedthrough of the tuned block of your choice. To fix parameters of tunable blocks to specified values, see “View and Change Block Parameterization in Control System Tuner” on page 14-26. • This tuning goal also imposes an implicit stability constraint on the closed-loop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning requirement is a hard constraint. For Transient Goal, f(x) is based upon the relative gap between the tuned response and the target response: gap = y ( t ) - yref ( t ) yref ( tr) ( t) 2 . 2 y(t) – yref(t) is the response mismatch, and 1 – yref(tr)(t) is the transient portion of yref (deviation from steady-state value or trajectory). ◊ 2 denotes the signal energy (2-norm). The gap can be understood as the ratio of the root-mean-square (RMS) of the mismatch to the RMS of the reference transient. 14-79 14 Control System Tuning See Also Related Examples 14-80 • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 LQR/LQG Goal LQR/LQG Goal Purpose Minimize or limit Linear-Quadratic-Gaussian (LQG) cost in response to white-noise inputs, when using Control System Tuner. Description LQR/LQG Goal specifies a tuning requirement for quantifying control performance as an LQG cost. It is applicable to any control structure, not just the classical observer structure of optimal LQG control. The LQG cost is given by: J = E(z(t)′ QZ z(t)). z(t) is the system response to a white noise input vector w(t). The covariance of w(t) is given by: E(w(t)w(t)′) = QW. The vector w(t) typically consists of external inputs to the system such as noise, disturbances, or command. The vector z(t) includes all the system variables that characterize performance, such as control signals, system states, and outputs. E(x) denotes the expected value of the stochastic variable x. The cost function J can also be written as an average over time: Ê1 J = lim E Á T Æ• Ë T ˆ Ú0 z (t )’ QZ z (t ) dt ˜¯ . T Creation In the Tuning tab of Control System Tuner, select New Goal > LQR/LQG objective to create an LQR/LQG Goal. 14-81 14 Control System Tuning Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.LQG to specify an LQR/LQG goal. Signal Selection Use this section of the dialog box to specify noise input locations and performance output locations. Also specify any locations at which to open loops for evaluating the tuning goal. • Specify noise inputs (w) Select one or more signal locations in your model as noise inputs. To constrain a SISO response, select a single-valued input signal. For example, to constrain the LQG cost for a noise input 'u' and performance output 'y', click Add signal to list and select 'u'. To constrain the LQG cost for a MIMO response, select multiple signals or a vector-valued signal. • Specify performance outputs (z) Select one or more signal locations in your model as performance outputs. To constrain a SISO response, select a single-valued output signal. For example, to constrain the LQG cost for a noise input 'u' and performance output 'y', click Add signal to list and select 'y'. To constrain the LQG cost for a MIMO response, select multiple signals or a vector-valued signal. • Evaluate LQG objective with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can and . For more information on how to specify signal locations reorder them using for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. 14-82 LQR/LQG Goal LQG Objective Use this section of the dialog box to specify the noise covariance and performance weights for the LQG goal. • Performance weight Qz Performance weights, specified as a scalar or a matrix. Use a scalar value to specify a multiple of the identity matrix. Otherwise specify a symmetric nonnegative definite matrix. Use a diagonal matrix to independently scale or penalize the contribution of each variable in z. The performance weights contribute to the cost function according to: J = E(z(t)′ Qz z(t)). When you use the LQG goal as a hard goal, the software tries to drive the cost function J < 1. When you use it as a soft goal, the cost function J is minimized subject to any hard goals and its value is contributed to the overall objective function. Therefore, select Qz values to properly scale the cost function so that driving it below 1 or minimizing it yields the performance you require. • Noise Covariance Qw Covariance of the white noise input vector w(t), specified as a scalar or a matrix. Use a scalar value to specify a multiple of the identity matrix. Otherwise specify a symmetric nonnegative definite matrix with as many rows as there are entries in the vector w(t). A diagonal matrix means the entries of w(t) are uncorrelated. The covariance of w(t is given by: E(w(t)w(t)′) = QW. When you are tuning a control system in discrete time, the LQG goal assumes: E(w[k]w[k]′) = QW/Ts. Ts is the model sample time. This assumption ensures consistent results with tuning in the continuous-time domain. In this assumption, w[k] is discrete-time noise obtained by sampling continuous white noise w(t) with covariance QW. If in your system w[k] is a truly discrete process with known covariance QWd, use the value Ts*QWd for the QW value. 14-83 14 Control System Tuning Options Use this section of the dialog box to specify additional characteristics of the LQG goal. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Tips When you use this requirement to tune a control system, Control System Tuner attempts to enforce zero feedthrough (D = 0) on the transfer that the requirement constrains. Zero feedthrough is imposed because the H2 norm, and therefore the value of the tuning goal, is infinite for continuous-time systems with nonzero feedthrough. Control System Tuner enforces zero feedthrough by fixing to zero all tunable parameters that contribute to the feedthrough term. Control System Tuner returns an error when fixing these tunable parameters is insufficient to enforce zero feedthrough. In such cases, you must modify the requirement or the control structure, or manually fix some tunable parameters of your system to values that eliminate the feedthrough term. When the constrained transfer function has several tunable blocks in series, the software’s approach of zeroing all parameters that contribute to the overall feedthrough might be conservative. In that case, it is sufficient to zero the feedthrough term of one of the blocks. If you want to control which block has feedthrough fixed to zero, you can manually fix the feedthrough of the tuned block of your choice. To fix parameters of tunable blocks to specified values, see “View and Change Block Parameterization in Control System Tuner” on page 14-26. 14-84 See Also Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For LQR/LQG Goal, f(x) is given by the cost function J: J = E(z(t)′ Qz z(t)). When you use the LQG requirement as a hard goal, the software tries to drive the cost function J < 1. When you use it as a soft goal, the cost function J is minimized subject to any hard goals and its value is contributed to the overall objective function. Therefore, select Qz values to properly scale the cost function so that driving it below 1 or minimizing it yields the performance you require. See Also Related Examples • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 14-85 14 Control System Tuning Gain Goal Purpose Limit gain of a specified input/output transfer function, when using Control System Tuner. Description Gain Goal limits the gain from specified inputs to specified outputs. If you specify multiple inputs and outputs, Gain Goal limits the largest singular value of the transfer matrix. (See sigma for more information about singular values.) You can specify a constant maximum gain at all frequencies. Alternatively, you can specify a frequency-dependent gain profile. Use Gain Goal, for example, to enforce a custom roll-off rate in a particular frequency band. To do so, specify a maximum gain profile in that band. You can also use Gain Goal to enforce disturbance rejection across a particular input/output pair by constraining the gain to be less than 1. 14-86 Gain Goal When you create a tuning goal in Control System Tuner, a tuning-goal plot is generated. The dotted line shows the gain profile you specify. The shaded area on the plot represents the region in the frequency domain where the gain goal is not satisfied. By default, Gain Goal constrains a closed-loop gain. To constrain a gain computed with one or more loops open, specify loop-opening locations in the I/O Transfer Selection section of the dialog box. Creation In the Tuning tab of Control System Tuner, select New Goal > Gain limits to create a Gain Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.Gain to specify a maximum gain goal. I/O Transfer Selection Use this section of the dialog box to specify the inputs and outputs of the transfer function that the tuning goal constrains. Also specify any locations at which to open loops for evaluating the tuning goal. • Specify input signals Select one or more signal locations in your model as inputs to the transfer function that the tuning goal constrains. To constrain a SISO response, select a single-valued input signal. For example, to constrain the gain from a location named 'u' to a Add signal to list and select 'u'. To constrain the location named 'y', click largest singular value of a MIMO response, select multiple signals or a vector-valued signal. • Specify output signals Select one or more signal locations in your model as outputs of the transfer function that the tuning goal constrains. To constrain a SISO response, select a single-valued output signal. For example, to constrain the gain from a location named 'u' to a location named 'y', click Add signal to list and select 'y'. To constrain the largest singular value of a MIMO response, select multiple signals or a vector-valued signal. 14-87 14 Control System Tuning • Compute input/output gain with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Options Use this section of the dialog box to specify additional characteristics of the gain goal. • Limit gain to Enter the maximum gain in the text box. You can specify a scalar value or a frequencydependent gain profile. To specify a frequency-dependent gain profile, enter a SISO numeric LTI model. For example, you can specify a smooth transfer function (tf, zpk, or ss model). Alternatively, you can sketch a piecewise maximum gain using an frd model. When you do so, the software automatically maps the profile to a smooth transfer function that approximates the desired minimum disturbance rejection. For example, to specify a gain profile that rolls off at –40dB/decade in the frequency band from 8 to 800 rad/s, enter frd([0.8 8 800],[10 1 1e-4]). You must specify a SISO transfer function. If you specify multiple input signals or output signals, the gain profile applies to all I/O pairs between these signals. If you are tuning in discrete time, you can specify the maximum gain profile as a discrete-time model with the same sampling time as you use for tuning. If you specify the gain profile in continuous time, the tuning software discretizes it. Specifying the gain profile in discrete time gives you more control over the gain profile near the Nyquist frequency. 14-88 Gain Goal • Stabilize I/O transfer By default, the tuning goal imposes a stability requirement on the closed-loop transfer function from the specified inputs to outputs, in addition to the gain constraint. If stability is not required or cannot be achieved, select No to remove the stability requirement. For example, if the gain constraint applies to an unstable open-loop transfer function, select No. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. • Adjust for signal amplitude When this option is set to No, the closed-loop transfer function being constrained is not scaled for relative signal amplitudes. When the choice of units results in a mix of small and large signals, using an unscaled transfer function can lead to poor tuning results. Set the option to Yes to provide the relative amplitudes of the input signals and output signals of your transfer function. For example, suppose the tuning goal constrains a 2-input, 2-output transfer function. Suppose further that second input signal to the transfer function tends to be about 100 times greater than the first signal. In that case, select Yes and enter [1,100] in the Amplitude of input signals text box. Adjusting signal amplitude causes the tuning goal to be evaluated on the scaled transfer function Do–1T(s)Di, where T(s) is the unscaled transfer function. Do and Di are diagonal matrices with the Amplitude of output signals and Amplitude of input signals values on the diagonal, respectively. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. 14-89 14 Control System Tuning For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms Evaluating Tuning Goals When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Gain Goal, f(x) is given by: f ( x ) = WF ( s ) Do-1 T ( s, x ) Di , • or its discrete-time equivalent. Here, T(s,x) is the closed-loop transfer function between the specified inputs and outputs, evaluated with parameter values x. Do and Di are the scaling matrices described in “Options” on page 14-88. ◊ getPeakGain). • denotes the H∞ norm (see The frequency weighting function WF is the regularized gain profile, derived from the maximum gain profile you specify. The gain of WF roughly matches the inverse of the gain profile you specify, inside the frequency band you set in the Enforce goal in frequency range field of the tuning goal. WF is always stable and proper. Because poles of WF(s) close to s = 0 or s = Inf might lead to poor numeric conditioning for tuning, it is not recommended to specify maximum gain profiles with very low-frequency or very highfrequency dynamics. For more information about regularization and its effects, see “Visualize Tuning Goals” on page 14-189. Implicit Constraints This tuning goal also imposes an implicit stability constraint on the closed-loop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the 14-90 See Also default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 14-91 14 Control System Tuning Variance Goal Purpose Limit white-noise impact on specified output signals, when using Control System Tuner. Description Variance Goal imposes a noise attenuation constraint that limits the impact on specified output signals of white noise applied at specified inputs. The noise attenuation is measured by the ratio of the noise variance to the output variance. For stochastic inputs with a nonuniform spectrum (colored noise), use “Weighted Variance Goal” on page 14-124 instead. Creation In the Tuning tab of Control System Tuner, select New Goal > Signal variance attenuation to create a Variance Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.Variance to specify a constraint on noise amplification. I/O Transfer Selection Use this section of the dialog box to specify noise input locations and response outputs. Also specify any locations at which to open loops for evaluating the tuning goal. • Specify stochastic inputs Select one or more signal locations in your model as noise inputs. To constrain a SISO response, select a single-valued input signal. For example, to constrain the gain from a location named 'u' to a location named 'y', click Add signal to list and select 'u'. To constrain the noise amplification of a MIMO response, select multiple signals or a vector-valued signal. • Specify stochastic outputs 14-92 Variance Goal Select one or more signal locations in your model as outputs for computing response to the noise inputs. To constrain a SISO response, select a single-valued output signal. For example, to constrain the gain from a location named 'u' to a location named 'y', click Add signal to list and select 'y'. To constrain the noise amplification of a MIMO response, select multiple signals or a vector-valued signal. • Compute output variance with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Options Use this section of the dialog box to specify additional characteristics of the variance goal. • Attenuate input variance by a factor Enter the desired noise attenuation from the specified inputs to outputs. This value specifies the maximum ratio of noise variance to output variance. When you tune a control system in discrete time, this requirement assumes that the physical plant and noise process are continuous, and interprets the desired noise attenuation as a bound on the continuous-time H2 norm. This ensures that continuoustime and discrete-time tuning give consistent results. If the plant and noise processes are truly discrete, and you want to bound the discrete-time H2 norm instead, multiple the desired attenuation value by tuning. Ts . Ts is the sample time of the model you are 14-93 14 Control System Tuning • Adjust for signal amplitude When this option is set to No, the closed-loop transfer function being constrained is not scaled for relative signal amplitudes. When the choice of units results in a mix of small and large signals, using an unscaled transfer function can lead to poor tuning results. Set the option to Yes to provide the relative amplitudes of the input signals and output signals of your transfer function. For example, suppose the tuning goal constrains a 2-input, 2-output transfer function. Suppose further that second input signal to the transfer function tends to be about 100 times greater than the first signal. In that case, select Yes and enter [1,100] in the Amplitude of input signals text box. Adjusting signal amplitude causes the tuning goal to be evaluated on the scaled transfer function Do–1T(s)Di, where T(s) is the unscaled transfer function. Do and Di are diagonal matrices with the Amplitude of output signals and Amplitude of input signals values on the diagonal, respectively. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Tips • When you use this requirement to tune a control system, Control System Tuner attempts to enforce zero feedthrough (D = 0) on the transfer that the requirement constrains. Zero feedthrough is imposed because the H2 norm, and therefore the value of the tuning goal (see “Algorithms” on page 14-95), is infinite for continuous-time systems with nonzero feedthrough. Control System Tuner enforces zero feedthrough by fixing to zero all tunable parameters that contribute to the feedthrough term. Control System Tuner returns an 14-94 Variance Goal error when fixing these tunable parameters is insufficient to enforce zero feedthrough. In such cases, you must modify the requirement or the control structure, or manually fix some tunable parameters of your system to values that eliminate the feedthrough term. When the constrained transfer function has several tunable blocks in series, the software’s approach of zeroing all parameters that contribute to the overall feedthrough might be conservative. In that case, it is sufficient to zero the feedthrough term of one of the blocks. If you want to control which block has feedthrough fixed to zero, you can manually fix the feedthrough of the tuned block of your choice. To fix parameters of tunable blocks to specified values, see “View and Change Block Parameterization in Control System Tuner” on page 14-26. • This tuning goal also imposes an implicit stability constraint on the closed-loop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Variance Goal, f(x) is given by: f ( x ) = Attenuation ◊ T ( s, x) . 2 T(s,x) is the closed-loop transfer function from Input to Output. ◊ norm (see norm). 2 denotes the H2 For tuning discrete-time control systems, f(x) is given by: 14-95 14 Control System Tuning f (x) = Attenuation Ts T ( z, x) . 2 Ts is the sample time of the discrete-time transfer function T(z,x). See Also Related Examples 14-96 • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 Reference Tracking Goal Reference Tracking Goal Purpose Make specified outputs track reference inputs with prescribed performance and fidelity, when using Control System Tuner. Limit cross-coupling in MIMO systems. Description Reference Tracking Goal constrains tracking between the specified signal locations. The constraint is satisfied when the maximum relative tracking error falls below the value you specify at all frequencies. The relative error is the gain from reference input to tracking error as a function of frequency. You can specify the maximum error profile directly as a function of frequency. Alternatively, you can specify the tracking goal a target DC error, peak error, and response time. These parameters are converted to the following transfer function that describes the maximum frequency-domain tracking error: MaxError = ( PeakError ) s + wc ( DCError ) . s + wc Here, ωc is 2/(response time). The following plot illustrates these relationships for an example set of values. 14-97 14 Control System Tuning When you create a tuning goal in Control System Tuner, a tuning-goal plot is generated. The dotted line shows the error profile you specify. The shaded area on the plot represents the region in the frequency domain where the tuning goal is not satisfied. Creation In the Tuning tab of Control System Tuner, select New Goal > Reference Tracking to create a Reference Tracking Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.Tracking to specify a tracking goal. Response Selection Use this section of the dialog box to specify input, output, and loop-opening locations for evaluating the tuning goal. • Specify reference inputs Select one or more signal locations in your model as reference signals. To constrain a SISO response, select a single-valued reference signal. For example, to constrain the 14-98 Reference Tracking Goal response from a location named 'u' to a location named 'y', click Add signal to list and select 'u'. To constrain a MIMO response, select multiple signals or a vectorvalued signal. • Specify reference-tracking outputs Select one or more signal locations in your model as reference-tracking outputs. To constrain a SISO response, select a single-valued output signal. For example, to constrain the step response from a location named 'u' to a location named 'y', click Add signal to list and select 'y'. To constrain a MIMO response, select multiple signals or a vector-valued signal. For MIMO systems, the number of outputs must equal the number of outputs. • Evaluate tracking performance with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Tracking Performance Use this section of the dialog box to specify frequency-domain constraints on the tracking error. Response time, DC error, and peak error Select this option to specify the tracking error in terms of response time, percent steadystate error, and peak error across all frequencies. These parameters are converted to the following transfer function that describes the maximum frequency-domain tracking error: 14-99 14 Control System Tuning MaxError = ( PeakError ) s + wc ( DCError ) . s + wc When you select this option, enter the following parameters in the text boxes: • Response Time — Enter the target response time. The tracking bandwidth is given by ωc = 2/Response Time. Express the target response time in the time units of your model. • Steady-state error (%) — Enter the maximum steady-state fractional tracking error, expressed in percent. For MIMO tracking goals, this steady-state error applies to all I/O pairs. The steady-state error is the DC error expressed as a percentage, DCError/ 100. • Peak error across frequency (%) — Enter the maximum fractional tracking error across all frequencies, expressed in percent. Maximum error as a function of frequency Select this option to specify the maximum tracking error profile as a function of frequency. Enter a SISO numeric LTI model in the text box. For example, you can specify a smooth transfer function (tf, zpk, or ss model). Alternatively, you can sketch a piecewise error profile using an frd model. When you do so, the software automatically maps the error profile to a smooth transfer function that approximates the desired error profile. For example, to specify a maximum error of 0.01 below about 1 rad/s, gradually rising to a peak error of 1 at 100 rad/s, enter frd([0.01 0.01 1],[0 1 100]). For MIMO tracking goals, this error profile applies to all I/O pairs. If you are tuning in discrete time, you can specify the maximum error profile as a discrete-time model with the same sampling time as you use for tuning. If you specify the attenuation profile in continuous time, the tuning software discretizes it. Specifying the error profile in discrete time gives you more control over the profile near the Nyquist frequency. Options Use this section of the dialog box to specify additional characteristics of the tracking goal. 14-100 Reference Tracking Goal • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. • Adjust for step amplitude For a MIMO tuning goal, when the choice of units results in a mix of small and large signals in different channels of the response, this option allows you to specify the relative amplitude of each entry in the vector-valued step input. This information is used to scale the off-diagonal terms in the transfer function from reference to tracking error. This scaling ensures that cross-couplings are measured relative to the amplitude of each reference signal. For example, suppose that tuning goal is that outputs 'y1' and 'y2' track reference signals 'r1'and 'r2'. Suppose further that you require the outputs to track the references with less than 10% cross-coupling. If r1 and r2 have comparable amplitudes, then it is sufficient to keep the gains from r1 to y2 and r2 and y1 below 0.1. However, if r1 is 100 times larger than r2, the gain from r1 to y2 must be less than 0.001 to ensure that r1 changes y2 by less than 10% of the r2 target. To ensure this result, set Adjust for step amplitude to Yes. Then, enter [100,1] in the Amplitudes of step commands text box. This tells Control System Tuner to take into account that the first reference signal is 100 times greater than the second reference signal. The default value, No , means no scaling is applied. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). 14-101 14 Control System Tuning Algorithms Evaluating Tuning Goals When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Tracking Goal, f(x) is given by: f ( x ) = WF ( s ) ( T ( s, x ) - I ) • , or its discrete-time equivalent. Here, T(s,x) is the closed-loop transfer function between the specified inputs and outputs, and ◊ • denotes the H∞ norm (see getPeakGain). WF is a frequency weighting function derived from the error profile you specify in the tuning goal. The gain of WF roughly matches the inverse of the error profile for gain values between –20 dB and 60 dB. For numerical reasons, the weighting function levels off outside this range, unless you specify a reference model that changes slope outside this range. This adjustment is called regularization. Because poles of WF close to s = 0 or s = Inf might lead to poor numeric conditioning of the systune optimization problem, it is not recommended to specify error profiles with very low-frequency or very high-frequency dynamics. For more information about regularization and its effects, see “Visualize Tuning Goals” on page 14-189. Implicit Constraints This tuning goal also imposes an implicit stability constraint on the closed-loop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. 14-102 See Also See Also Related Examples • “Specify Goals for Interactive Tuning” on page 14-39 • “Visualize Tuning Goals” on page 14-189 • “Manage Tuning Goals” on page 14-179 14-103 14 Control System Tuning Overshoot Goal Purpose Limit overshoot in the step response from specified inputs to specified outputs, when using Control System Tuner. Description Overshoot Goal limits the overshoot in the step response between the specified signal locations. The constraint is satisfied when the overshoot in the tuned response is less than the target overshoot The software maps the maximum overshoot to a peak gain constraint, assuming secondorder system characteristics. Therefore, for tuning higher-order systems, the overshoot constraint is only approximate. In addition, the Overshoot Goal cannot reliably reduce the overshoot below 5%. When you create a tuning goal in Control System Tuner, a tuning-goal plot is generated. The shaded area on the plot represents the region in the frequency domain where the tuning goal is not satisfied. 14-104 Overshoot Goal Creation In the Tuning tab of Control System Tuner, select New Goal > Maximum overshoot to create an Overshoot Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.Overshoot to specify a step response goal. Response Selection Use this section of the dialog box to specify input, output, and loop-opening locations for evaluating the tuning goal. • Specify step-response inputs Select one or more signal locations in your model at which to apply the step input. To constrain a SISO response, select a single-valued input signal. For example, to constrain the step response from a location named 'u' to a location named 'y', click Add signal to list and select 'u'. To constrain a MIMO response, select multiple signals or a vector-valued signal. • Specify step-response outputs Select one or more signal locations in your model at which to measure the response to the step input. To constrain a SISO response, select a single-valued output signal. For example, to constrain the step response from a location named 'u' to a location named 'y', click Add signal to list and select 'y'. To constrain a MIMO response, select multiple signals or a vector-valued signal. For MIMO systems, the number of outputs must equal the number of outputs. • Evaluate overshoot with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. 14-105 14 Control System Tuning Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Options Use this section of the dialog box to specify additional characteristics of the overshoot goal. • Limit % overshoot to Enter the maximum percent overshoot. Overshoot Goal cannot reliably reduce the overshoot below 5% • Adjust for step amplitude For a MIMO tuning goal, when the choice of units results in a mix of small and large signals in different channels of the response, this option allows you to specify the relative amplitude of each entry in the vector-valued step input. This information is used to scale the off-diagonal terms in the transfer function from reference to tracking error. This scaling ensures that cross-couplings are measured relative to the amplitude of each reference signal. For example, suppose that tuning goal is that outputs 'y1' and 'y2' track reference signals 'r1'and 'r2'. Suppose further that you require the outputs to track the references with less than 10% cross-coupling. If r1 and r2 have comparable amplitudes, then it is sufficient to keep the gains from r1 to y2 and r2 and y1 below 0.1. However, if r1 is 100 times larger than r2, the gain from r1 to y2 must be less than 0.001 to ensure that r1 changes y2 by less than 10% of the r2 target. To ensure this result, set Adjust for step amplitude to Yes. Then, enter [100,1] in the Amplitudes of step commands text box. This tells Control System Tuner to take into account that the first reference signal is 100 times greater than the second reference signal. The default value, No , means no scaling is applied. • Apply goal to 14-106 Overshoot Goal Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Overshoot Goal, f(x) reflects the relative satisfaction or violation of the goal. The percent deviation from f(x) = 1 roughly corresponds to the percent deviation from the specified overshoot target. For example, f(x) = 1.2 means the actual overshoot exceeds the target by roughly 20%, and f(x) = 0.8 means the actual overshoot is about 20% less than the target. Overshoot Goal uses T • as a proxy for the overshoot, based on second-order model characteristics. Here, T is the closed-loop transfer function that the requirement constrains. The overshoot is tuned in the range from 5% ( T • = 1) to 100% ( T Overshoot Goal is ineffective at forcing the overshoot below 5%. • ). This tuning goal also imposes an implicit stability constraint on the closed-loop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. 14-107 14 Control System Tuning See Also Related Examples 14-108 • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 Disturbance Rejection Goal Disturbance Rejection Goal Purpose Attenuate disturbances at particular locations and in particular frequency bands, when using Control System Tuner. Description Disturbance Rejection Goal specifies the minimum attenuation of a disturbance injected at a specified location in a control system. When you use this tuning goal, the software attempts to tune the system so that the attenuation of a disturbance at the specified location exceeds the minimum attenuation factor you specify. This attenuation factor is the ratio between the open- and closed-loop sensitivities to the disturbance, and is a function of frequency. The following diagram illustrates how the attenuation factor is calculated. Suppose you specify a location in your control system, y, which is the output of a block A. In that case, the software calculates the closed-loop sensitivity at out to a signal injected at in. The software also calculates the sensitivity with the control loop opened at the location z. A y z To specify a Disturbance Rejection Goal, you specify one or more locations at which to attenuate disturbance. You also provide the frequency-dependent minimum attenuation factor as a numeric LTI model. You can achieve disturbance attenuation only inside the control bandwidth. The loop gain must be larger than one for the disturbance to be attenuated (attenuation factor > 1). 14-109 14 Control System Tuning When you create a tuning goal in Control System Tuner, a tuning-goal plot is generated. The dotted line shows the gain profile you specify. The shaded area on the plot represents the region in the frequency domain where the tuning goal is not satisfied. The solid line is the current corresponding response of your system. If you prefer to specify sensitivity to disturbance at a location, rather than disturbance attenuation, you can use “Sensitivity Goal” on page 14-114. Creation In the Tuning tab of Control System Tuner, select New Goal > Disturbance rejection to create a Disturbance Rejection Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.Rejection to specify a disturbance rejection goal. Disturbance Scenario Use this section of the dialog box to specify the signal locations at which to inject the disturbance. You can also specify loop-opening locations for evaluating the tuning goal. 14-110 Disturbance Rejection Goal • Inject disturbances at the following locations Select one or more signal locations in your model at which to measure the disturbance attenuation. To constrain a SISO response, select a single-valued location. For example, to attenuate disturbance at a location named 'y', click Add signal to list and select 'y'. To constrain a MIMO response, select multiple signals or a vectorvalued signal. • Evaluate disturbance rejection with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Rejection Performance Specify the minimum disturbance attenuation as a function of frequency. Enter a SISO numeric LTI model whose magnitude represents the desired attenuation profile as a function of frequency. For example, you can specify a smooth transfer function (tf, zpk, or ss model). Alternatively, you can sketch a piecewise minimum disturbance rejection using an frd model. When you do so, the software automatically maps the profile to a smooth transfer function that approximates the desired minimum disturbance rejection. For example, to specify an attenuation factor of 100 (40 dB) below 1 rad/s, that gradually drops to 1 (0 dB) past 10 rad/s, enter frd([100 100 1 1],[0 1 10 100]). If you are tuning in discrete time, you can specify the attenuation profile as a discretetime model with the same sampling time as you use for tuning. If you specify the attenuation profile in continuous time, the tuning software discretizes it. Specifying the attenuation profile in discrete time gives you more control over the profile near the Nyquist frequency. 14-111 14 Control System Tuning Options Use this section of the dialog box to specify additional characteristics of the disturbance rejection goal. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. Regardless of the limits you enter, a disturbance rejection goal can only be enforced within the control bandwidth. • Equalize cross-channel effects For multiloop or MIMO disturbance rejection requirements, the feedback channels are automatically rescaled to equalize the off-diagonal (loop interaction) terms in the openloop transfer function. Select Off to disable such scaling and shape the unscaled open-loop response. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms Evaluating Tuning Goals When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. 14-112 See Also The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Disturbance Rejection Goal, f(x) is given by: f ( x ) = max WS ( jw ) S ( jw, x ) wŒ W • , or its discrete-time equivalent. Here, S(jω,x) is the closed-loop sensitivity function measured at the disturbance location. Ω is the frequency interval over which the requirement is enforced, specified in the Enforce goal in frequency range field. WS is a frequency weighting function derived from the attenuation profile you specify. The gains of WS and the specified profile roughly match for gain values ranging from –20 dB to 60 dB. For numerical reasons, the weighting function levels off outside this range, unless the specified gain profile changes slope outside this range. This adjustment is called regularization. Because poles of WS close to s = 0 or s = Inf might lead to poor numeric conditioning for tuning, it is not recommended to specify loop shapes with very lowfrequency or very high-frequency dynamics.For more information about regularization and its effects, see “Visualize Tuning Goals” on page 14-189. Implicit Constraints This tuning goal imposes an implicit stability constraint on the closed-loop sensitivity function measured at the specified, evaluated with loops opened at the specified loopopening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 14-113 14 Control System Tuning Sensitivity Goal Purpose Limit sensitivity of feedback loops to disturbances, when using Control System Tuner. Description Sensitivity Goal limits the sensitivity of a feedback loop to disturbances. You specify the maximum sensitivity as a function of frequency. Constrain the sensitivity to be smaller than one at frequencies where you need good disturbance rejection. To specify a Sensitivity Goal, you specify one or more locations at which to limit sensitivity. You also provide the frequency-dependent maximum sensitivity as a numeric LTI model whose magnitude represents the desired sensitivity as a function of frequency. When you create a tuning goal in Control System Tuner, a tuning-goal plot is generated. The dotted line shows the gain profile you specify. The shaded area on the plot represents the region in the frequency domain where the tuning goal is not satisfied. 14-114 Sensitivity Goal If you prefer to specify disturbance attenuation at a particular location, rather than sensitivity to disturbance, you can use “Disturbance Rejection Goal” on page 14-109. Creation In the Tuning tab of Control System Tuner, select New Goal > Sensitivity of feedback loops to create a Sensitivity Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.Sensitivity to specify a disturbance rejection goal. Sensitivity Evaluation Use this section of the dialog box to specify the signal locations at which to compute the sensitivity to disturbance. You can also specify loop-opening locations for evaluating the tuning goal. • Measure sensitivity at the following locations Select one or more signal locations in your model at which to measure the sensitivity to disturbance. To constrain a SISO response, select a single-valued location. For example, to limit sensitivity at a location named 'y', click Add signal to list and select 'y'. To constrain a MIMO response, select multiple signals or a vector-valued signal. • Evaluate disturbance rejection with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can and . For more information on how to specify signal locations reorder them using for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. 14-115 14 Control System Tuning Sensitivity Bound Specify the maximum sensitivity as a function of frequency. Enter a SISO numeric LTI model whose magnitude represents the desired sensitivity bound as a function of frequency. For example, you can specify a smooth transfer function (tf, zpk, or ss model). Alternatively, you can sketch a piecewise maximum sensitivity using an frd model. When you do so, the software automatically maps the profile to a smooth transfer function that approximates the desired sensitivity. For example, to specify a sensitivity that rolls up at 20 dB per decade and levels off at unity above 1 rad/s, enter frd([0.01 1 1],[0.001 0.1 100]). If you are tuning in discrete time, you can specify the maximum sensitivity profile as a discrete-time model with the same sampling time as you use for tuning. If you specify the sensitivity profile in continuous time, the tuning software discretizes it. Specifying the profile in discrete time gives you more control over the profile near the Nyquist frequency. Options Use this section of the dialog box to specify additional characteristics of the sensitivity goal. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. • Equalize cross-channel effects For multiloop or MIMO sensitivity requirements, the feedback channels are automatically rescaled to equalize the off-diagonal (loop interaction) terms in the openloop transfer function. Select Off to disable such scaling and shape the unscaled open-loop response. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or block14-116 Sensitivity Goal parameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms Evaluating Tuning Goals When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Sensitivity Goal, f(x) is given by: f ( x ) = WS ( s) S ( s, x ) • , or its discrete-time equivalent. Here, S(s,x) is the closed-loop sensitivity function measured at the location specified in the tuning goal. ◊ • denotes the H∞ norm (see norm). WS is a frequency weighting function derived from the sensitivity profile you specify. The gain of WS roughly matches the inverse of the specified profile for gain values ranging from –20 dB to 60 dB. For numerical reasons, the weighting function levels off outside this range, unless the specified gain profile changes slope outside this range. This adjustment is called regularization. Because poles of WS close to s = 0 or s = Inf might lead to poor numeric conditioning for tuning, it is not recommended to specify sensitivity profiles with very low-frequency or very high-frequency dynamics.For more information about regularization and its effects, see “Visualize Tuning Goals” on page 14-189. Implicit Constraint This tuning goal imposes an implicit stability constraint on the closed-loop sensitivity function measured at the specified, evaluated with loops opened at the specified loopopening locations. The dynamics affected by this implicit constraint are the stabilized 14-117 14 Control System Tuning dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples 14-118 • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 Weighted Gain Goal Weighted Gain Goal Purpose Frequency-weighted gain limit for tuning with Control System Tuner. Description Weighted Gain Goal limits the gain of the frequency-weighted transfer function WL(s)H(s)WR(s), where H(s) is the transfer function between inputs and outputs you specify. WL(s) and WR(s) are weighting functions that you can use to emphasize particular frequency bands. Weighted Gain Goal constrains the peak gain of WL(s)H(s)WR(s) to values less than 1. If H(s) is a MIMO transfer function, Weighted Gain Goal constrains the largest singular value of H(s). By default, Weighted Gain Goal constrains a closed-loop gain. To constrain a gain computed with one or more loops open, specify loop-opening locations in the I/O Transfer Selection section of the dialog box. Creation In the Tuning tab of Control System Tuner, select New Goal > Frequency-weighted gain limit to create a Weighted Gain Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.WeightedGain to specify a weighted gain goal. I/O Transfer Selection Use this section of the dialog box to specify the inputs and outputs of the transfer function that the tuning goal constrains. Also specify any locations at which to open loops for evaluating the tuning goal. • Specify input signals Select one or more signal locations in your model as inputs to the transfer function that the tuning goal constrains. To constrain a SISO response, select a single-valued input signal. For example, to constrain the gain from a location named 'u' to a 14-119 14 Control System Tuning location named 'y', click Add signal to list and select 'u'. To constrain the largest singular value of a MIMO response, select multiple signals or a vector-valued signal. • Specify output signals Select one or more signal locations in your model as outputs of the transfer function that the tuning goal constrains. To constrain a SISO response, select a single-valued output signal. For example, to constrain the gain from a location named 'u' to a Add signal to list and select 'y'. To constrain the location named 'y', click largest singular value of a MIMO response, select multiple signals or a vector-valued signal. • Compute input/output gain with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Weights Use the Left weight WL and Right weight WR text boxes to specify the frequencyweighting functions for the tuning goal. The tuning goal ensures that the gain H(s) from the specified input to output satisfies the inequality: ||WL(s)H(s)WR(s)||∞ < 1. WL provides the weighting for the output channels of H(s), and WR provides the weighting for the input channels. You can specify scalar weights or frequency-dependent weighting. To specify a frequency-dependent weighting, use a numeric LTI model whose 14-120 Weighted Gain Goal magnitude represents the desired weighting function. For example, enter tf(1,[1 0.01]) to specify a high weight at low frequencies that rolls off above 0.01 rad/s. If the tuning goal constrains a MIMO transfer function, scalar or SISO weighting functions automatically expand to any input or output dimension. You can specify different weights for each channel by specifying matrices or MIMO weighting functions. The dimensions H(s) must be commensurate with the dimensions of WL and WR. For example, if the constrained transfer function has two inputs, you can specify diag([1 10]) as WR. If you are tuning in discrete time, you can specify the weighting functions as discrete-time models with the same sampling time as you use for tuning. If you specify the weighting functions in continuous time, the tuning software discretizes them. Specifying the weighting functions in discrete time gives you more control over the weighting functions near the Nyquist frequency. Options Use this section of the dialog box to specify additional characteristics of the weighted gain goal. • Stabilize I/O transfer By default, the tuning goal imposes a stability requirement on the closed-loop transfer function from the specified inputs to outputs, in addition to the gain constraint. If stability is not required or cannot be achieved, select No to remove the stability requirement. For example, if the gain constraint applies to an unstable open-loop transfer function, select No. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To 14-121 14 Control System Tuning enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Weighted Gain Goal, f(x) is given by: f ( x ) = WL H ( s, x ) WR • . H(s,x) is the closed-loop transfer function between the specified inputs and outputs, evaluated with parameter values x. Here, ◊ • denotes the H∞ norm (see getPeakGain). This tuning goal also imposes an implicit stability constraint on the weighted closed-loop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples • 14-122 “Specify Goals for Interactive Tuning” on page 14-39 See Also • “Visualize Tuning Goals” on page 14-189 • “Manage Tuning Goals” on page 14-179 14-123 14 Control System Tuning Weighted Variance Goal Purpose Frequency-weighted limit on noise impact on specified output signals for tuning with Control System Tuner. Description Weighted Variance Goal limits the noise impact on the outputs of the frequency-weighted transfer function WL(s)H(s)WR(s), where H(s) is the transfer function between inputs and outputs you specify. WL(s) and WR(s) are weighting functions you can use to model a noise spectrum or emphasize particular frequency bands. Thus, you can use Weighted Variance Goal to tune the system response to stochastic inputs with a nonuniform spectrum such as colored noise or wind gusts. Weighted Variance minimizes the response to noise at the inputs by minimizing the H2 norm of the frequency-weighted transfer function. The H2 norm measures: • The total energy of the impulse response, for deterministic inputs to the transfer function. • The square root of the output variance for a unit-variance white-noise input, for stochastic inputs to the transfer function. Equivalently, the H2 norm measures the rootmean-square of the output for such input. Creation In the Tuning tab of Control System Tuner, select New Goal > Frequency-weighted variance attenuation to create a Weighted Variance Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.WeightedVariance to specify a weighted gain goal. I/O Transfer Selection Use this section of the dialog box to specify noise input locations and response outputs. Also specify any locations at which to open loops for evaluating the tuning goal. 14-124 Weighted Variance Goal • Specify stochastic inputs Select one or more signal locations in your model as noise inputs. To constrain a SISO response, select a single-valued input signal. For example, to constrain the gain from a location named 'u' to a location named 'y', click Add signal to list and select 'u'. To constrain the noise amplification of a MIMO response, select multiple signals or a vector-valued signal. • Specify stochastic outputs Select one or more signal locations in your model as outputs for computing response to the noise inputs. To constrain a SISO response, select a single-valued output signal. For example, to constrain the gain from a location named 'u' to a location named 'y', click Add signal to list and select 'y'. To constrain the noise amplification of a MIMO response, select multiple signals or a vector-valued signal. • Compute output variance with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Weights Use the Left weight WL and Right weight WR text boxes to specify the frequencyweighting functions for the tuning goal. WL provides the weighting for the output channels of H(s), and WR provides the weighting for the input channels. 14-125 14 Control System Tuning You can specify scalar weights or frequency-dependent weighting. To specify a frequencydependent weighting, use a numeric LTI model whose magnitude represents the desired weighting as a function of frequency. For example, enter tf(1,[1 0.01]) to specify a high weight at low frequencies that rolls off above 0.01 rad/s. To limit the response to a nonuniform noise distribution, enter as WR an LTI model whose magnitude represents the noise spectrum. If the tuning goal constrains a MIMO transfer function, scalar or SISO weighting functions automatically expand to any input or output dimension. You can specify different weights for each channel by specifying MIMO weighting functions. The dimensions H(s) must be commensurate with the dimensions of WL and WR. For example, if the constrained transfer function has two inputs, you can specify diag([1 10]) as WR. If you are tuning in discrete time, you can specify the weighting functions as discrete-time models with the same sampling time as you use for tuning. If you specify the weighting functions in continuous time, the tuning software discretizes them. Specifying the weighting functions in discrete time gives you more control over the weighting functions near the Nyquist frequency. Options Use this section of the dialog box to specify additional characteristics of the weighted variance goal. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). 14-126 Weighted Variance Goal Tips • When you use this requirement to tune a control system, Control System Tuner attempts to enforce zero feedthrough (D = 0) on the transfer that the requirement constrains. Zero feedthrough is imposed because the H2 norm, and therefore the value of the tuning goal (see “Algorithms” on page 14-127), is infinite for continuous-time systems with nonzero feedthrough. Control System Tuner enforces zero feedthrough by fixing to zero all tunable parameters that contribute to the feedthrough term. Control System Tuner returns an error when fixing these tunable parameters is insufficient to enforce zero feedthrough. In such cases, you must modify the requirement or the control structure, or manually fix some tunable parameters of your system to values that eliminate the feedthrough term. When the constrained transfer function has several tunable blocks in series, the software’s approach of zeroing all parameters that contribute to the overall feedthrough might be conservative. In that case, it is sufficient to zero the feedthrough term of one of the blocks. If you want to control which block has feedthrough fixed to zero, you can manually fix the feedthrough of the tuned block of your choice. To fix parameters of tunable blocks to specified values, see “View and Change Block Parameterization in Control System Tuner” on page 14-26. • This tuning goal also imposes an implicit stability constraint on the weighted closedloop transfer function between the specified inputs to outputs, evaluated with loops opened at the specified loop-opening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Weighted Variance Goal, f(x) is given by: 14-127 14 Control System Tuning f ( x ) = WL H ( s, x ) WR . 2 H(s,x) is the closed-loop transfer function between the specified inputs and outputs, evaluated with parameter values x. ◊ 2 denotes the H2 norm (see norm). For tuning discrete-time control systems, f(x) is given by: f (x) = 1 WL ( z ) H ( z, x ) WR ( z ) . 2 Ts Ts is the sample time of the discrete-time transfer function H(z,x). See Also Related Examples 14-128 • “Specify Goals for Interactive Tuning” on page 14-39 • “Visualize Tuning Goals” on page 14-189 • “Manage Tuning Goals” on page 14-179 Minimum Loop Gain Goal Minimum Loop Gain Goal Purpose Boost gain of feedback loops at low frequency when using Control System Tuner. Description Minimum Loop Gain Goal enforces a minimum loop gain in a particular frequency band. This tuning goal is useful, for example, for improving disturbance rejection at a particular location. Minimum Loop Gain Goal imposes a minimum gain on the open-loop frequency response (L) at a specified location in your control system. You specify the minimum open-loop gain as a function of frequency (a minimum gain profile). For MIMO feedback loops, the specified gain profile is interpreted as a lower bound on the smallest singular value of L. When you tune a control system, the minimum gain profile is converted to a minimum gain constraint on the inverse of the sensitivity function, inv(S) = (I + L). The following figure shows a typical specified minimum gain profile (dashed line) and a resulting tuned loop gain, L (blue line). The green region represents gain profile values that are forbidden by this requirement. The figure shows that when L is much larger than 1, imposing a minimum gain on inv(S) is a good proxy for a minimum open-loop gain. 14-129 14 Control System Tuning Minimum Loop Gain Goal is a constraint on the open-loop gain of the specified control loop. Thus, the loop gain is computed with the loop open at the specified location. To compute the gain with loop openings at other points in the control system, use the Compute response with the following loops open option in the Open-Loop Response Selection section of the dialog box. Minimum Loop Gain Goal and Maximum Loop Gain Goal specify only low-gain or highgain constraints in certain frequency bands. When you use these requirements, the software determines the best loop shape near crossover. When the loop shape near crossover is simple or well understood (such as integral action), you can use “Loop Shape Goal” on page 14-141 to specify that target loop shape. 14-130 Minimum Loop Gain Goal Creation In the Tuning tab of Control System Tuner, select New Goal > Minimum gain for open-loop response to create a Minimum Gain Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.MinLoopGain to specify a minimum loop gain goal. Open-Loop Response Selection Use this section of the dialog box to specify the signal locations at which to compute the open-loop gain. You can also specify additional loop-opening locations for evaluating the tuning goal. • Shape open-loop response at the following locations Select one or more signal locations in your model at which to compute and constrain the open-loop gain. To constrain a SISO response, select a single-valued location. For Add signal example, to constrain the open-loop gain at a location named 'y', click to list and select 'y'. To constrain a MIMO response, select multiple signals or a vector-valued signal. • Compute response with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can and . For more information on how to specify signal locations reorder them using for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. 14-131 14 Control System Tuning Desired Loop Gain Use this section of the dialog box to specify the target minimum loop gain. • Pure integrator K/s Check to specify a pure integrator shape for the target minimum loop gain. The software chooses the integrator constant, K, based on the values you specify for a target minimum gain and frequency. For example, to specify an integral gain profile with crossover frequency 10 rad/s, enter 1 in the Choose K to keep gain above text box. Then, enter 10 in the at the frequency text box. The software chooses the integrator constant such that the minimum loop gain is 1 at 10 rad/s. • Other gain profile Check to specify the minimum gain profile as a function of frequency. Enter a SISO numeric LTI model whose magnitude represents the desired gain profile. For example, you can specify a smooth transfer function (tf, zpk, or ss model). Alternatively, you can sketch a piecewise target loop gain using an frd model. When you do so, the software automatically maps the profile to a smooth transfer function that approximates the desired minimum loop gain. For example, to specify minimum gain of 100 (40 dB) below 0.1 rad/s, rolling off at a rate of –20 dB/dec at higher frequencies, enter frd([100 100 10],[0 1e-1 1]). If you are tuning in discrete time, you can specify the minimum gain profile as a discrete-time model with the same sampling time as you use for tuning. If you specify the gain profile in continuous time, the tuning software discretizes it. Specifying the profile in discrete time gives you more control over the profile near the Nyquist frequency. Options Use this section of the dialog box to specify additional characteristics of the minimum loop gain goal. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. 14-132 Minimum Loop Gain Goal • Stabilize closed loop system By default, the tuning goal imposes a stability requirement on the closed-loop transfer function from the specified inputs to outputs, in addition to the gain constraint. If stability is not required or cannot be achieved, select No to remove the stability requirement. For example, if the gain constraint applies to an unstable open-loop transfer function, select No. • Equalize loop interactions For multi-loop or MIMO loop gain constraints, the feedback channels are automatically rescaled to equalize the off-diagonal (loop interaction) terms in the open-loop transfer function. Select Off to disable such scaling and shape the unscaled open-loop response. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms Evaluating Tuning Goals When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Minimum Loop Gain Goal, f(x) is given by: ( f ( x ) = WS D -1 SD ) •. 14-133 14 Control System Tuning D is a diagonal scaling (for MIMO loops). S is the sensitivity function at Location. WS is a frequency-weighting function derived from the minimum loop gain profile you specify. The gain of this function roughly matches the specified loop gain for values ranging from –20 dB to 60 dB. For numerical reasons, the weighting function levels off outside this range, unless the specified gain profile changes slope outside this range. This adjustment is called regularization. Because poles of WS close to s = 0 or s = Inf might lead to poor numeric conditioning for tuning, it is not recommended to specify gain profiles with very low-frequency or very high-frequency dynamics. For more information about regularization and its effects, see “Visualize Tuning Goals” on page 14-189. Although S is a closed-loop transfer function, driving f(x) < 1 is equivalent to enforcing a lower bound on the open-loop transfer function, L, in a frequency band where the gain of L is greater than 1. To see why, note that S = 1/(1 + L). For SISO loops, when |L| >> 1, |S | ≈ 1/|L|. Therefore, enforcing the open-loop minimum gain requirement, |L| > |WS|, is roughly equivalent to enforcing |WsS| < 1. For MIMO loops, similar reasoning applies, with ||S|| ≈ 1/σmin(L), where σmin is the smallest singular value. Implicit Constraints This tuning goal imposes an implicit stability constraint on the closed-loop sensitivity function measured at the specified, evaluated with loops opened at the specified loopopening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples 14-134 • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 Maximum Loop Gain Goal Maximum Loop Gain Goal Purpose Suppress gain of feedback loops at high frequency when using Control System Tuner. Description Maximum Loop Gain Goal enforces a maximum loop gain in a particular frequency band. This tuning goal is useful, for example, for increasing system robustness to unmodeled dynamics. Maximum Loop Gain Goal imposes a maximum gain on the open-loop frequency response (L) at a specified location in your control system. You specify the maximum open-loop gain as a function of frequency (a maximum gain profile). For MIMO feedback loops, the specified gain profile is interpreted as an upper bound on the largest singular value of L. When you tune a control system, the maximum gain profile is converted to a maximum gain constraint on the complementary sensitivity function, T) = L/(I + L). The following figure shows a typical specified maximum gain profile (dashed line) and a resulting tuned loop gain, L (blue line). The shaded region represents gain profile values that are forbidden by this requirement. The figure shows that when L is much smaller than 1, imposing a maximum gain on T is a good proxy for a maximum open-loop gain. 14-135 14 Control System Tuning Maximum Loop Gain Goal is a constraint on the open-loop gain of the specified control loop. Thus, the loop gain is computed with the loop open at the specified location. To compute the gain with loop openings at other points in the control system, use the Compute response with the following loops open option in the Open-Loop Response Selection section of the dialog box. Maximum Loop Gain Goal and Minimum Loop Gain Goal specify only high-gain or lowgain constraints in certain frequency bands. When you use these requirements, the software determines the best loop shape near crossover. When the loop shape near crossover is simple or well understood (such as integral action), you can use “Loop Shape Goal” on page 14-141 to specify that target loop shape. 14-136 Maximum Loop Gain Goal Creation In the Tuning tab of Control System Tuner, select New Goal > Maximum gain for open-loop response to create a Maximum Gain Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.MaxLoopGain to specify a maximum loop gain goal. Open-Loop Response Selection Use this section of the dialog box to specify the signal locations at which to compute the open-loop gain. You can also specify additional loop-opening locations for evaluating the tuning goal. • Shape open-loop response at the following locations Select one or more signal locations in your model at which to compute and constrain the open-loop gain. To constrain a SISO response, select a single-valued location. For Add signal example, to constrain the open-loop gain at a location named 'y', click to list and select 'y'. To constrain a MIMO response, select multiple signals or a vector-valued signal. • Compute response with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can and . For more information on how to specify signal locations reorder them using for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. 14-137 14 Control System Tuning Desired Loop Gain Use this section of the dialog box to specify the target maximum loop gain. • Pure integrator K/s Check to specify a pure integrator shape for the target maximum loop gain. The software chooses the integrator constant, K, based on the values you specify for a target maximum gain and frequency. For example, to specify an integral gain profile with crossover frequency 10 rad/s, enter 1 in the Choose K to keep gain below text box. Then, enter 10 in the at the frequency text box. The software chooses the integrator constant such that the maximum loop gain is 1 at 10 rad/s. • Other gain profile Check to specify the maximum gain profile as a function of frequency. Enter a SISO numeric LTI model whose magnitude represents the desired gain profile. For example, you can specify a smooth transfer function (tf, zpk, or ss model). Alternatively, you can sketch a piecewise target loop gain using an frd model. When you do so, the software automatically maps the profile to a smooth transfer function that approximates the desired maximum loop gain. For example, to specify maximum gain of 100 (40 dB) below 0.1 rad/s, rolling off at a rate of –20 dB/dec at higher frequencies, enter frd([100 100 10],[0 1e-1 1]). If you are tuning in discrete time, you can specify the maximum gain profile as a discrete-time model with the same sampling time as you use for tuning. If you specify the gain profile in continuous time, the tuning software discretizes it. Specifying the profile in discrete time gives you more control over the profile near the Nyquist frequency. Options Use this section of the dialog box to specify additional characteristics of the maximum loop gain goal. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. 14-138 Maximum Loop Gain Goal • Stabilize closed loop system By default, the tuning goal imposes a stability requirement on the closed-loop transfer function from the specified inputs to outputs, in addition to the gain constraint. If stability is not required or cannot be achieved, select No to remove the stability requirement. For example, if the gain constraint applies to an unstable open-loop transfer function, select No. • Equalize loop interactions For multi-loop or MIMO loop gain constraints, the feedback channels are automatically rescaled to equalize the off-diagonal (loop interaction) terms in the open-loop transfer function. Select Off to disable such scaling and shape the unscaled open-loop response. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms Evaluating Tuning Goals When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Maximum Loop Gain Goal, f(x) is given by: ( f ( x ) = WT D-1TD ) •. 14-139 14 Control System Tuning Here, D is a diagonal scaling (for MIMO loops). T is the complementary sensitivity function at the specified location. WT is a frequency-weighting function derived from the maximum loop gain profile you specify. The gain of this function roughly matches the inverse of the specified loop gain for values ranging from –60 dB to 20 dB. For numerical reasons, the weighting function levels off outside this range, unless the specified gain profile changes slope outside this range. This adjustment is called regularization. Because poles of WT close to s = 0 or s = Inf might lead to poor numeric conditioning for tuning, it is not recommended to specify gain profiles with very low-frequency or very highfrequency dynamics. For more information about regularization and its effects, see “Visualize Tuning Goals” on page 14-189. Although T is a closed-loop transfer function, driving f(x) < 1 is equivalent to enforcing an upper bound on the open-loop transfer, L, in a frequency band where the gain of L is less than one. To see why, note that T = L/(I + L). For SISO loops, when |L| << 1, |T| ≈ |L|. Therefore, enforcing the open-loop maximum gain requirement, |L| < 1/|WT|, is roughly equivalent to enforcing |WTT| < 1. For MIMO loops, similar reasoning applies, with ||T|| ≈ σmax(L), where σmax is the largest singular value. Implicit Constraints This tuning goal imposes an implicit stability constraint on the closed-loop sensitivity function measured at the specified, evaluated with loops opened at the specified loopopening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples 14-140 • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 Loop Shape Goal Loop Shape Goal Purpose Shape open-loop response of feedback loops when using Control System Tuner. Description Loop Shape Goal specifies a target gain profile (gain as a function of frequency) of an open-loop response. Loop Shape Goal constrains the open-loop, point-to-point response (L) at a specified location in your control system. When you tune a control system, the target open-loop gain profile is converted into constraints on the inverse sensitivity function inv(S) = (I + L) and the complementary sensitivity function T = 1–S. These constraints are illustrated for a representative tuned system in the following figure. 14-141 14 Control System Tuning Where L is much greater than 1, a minimum gain constraint on inv(S) (green shaded region) is equivalent to a minimum gain constraint on L. Similarly, where L is much smaller than 1, a maximum gain constraint on T (red shaded region) is equivalent to a maximum gain constraint on L. The gap between these two constraints is twice the crossover tolerance, which specifies the frequency band where the loop gain can cross 0 dB. For multi-input, multi-output (MIMO) control systems, values in the gain profile greater than 1 are interpreted as minimum performance requirements. Such values are lower 14-142 Loop Shape Goal bounds on the smallest singular value of the open-loop response. Gain profile values less than one are interpreted as minimum roll-off requirements, which are upper bounds on the largest singular value of the open-loop response. For more information about singular values, see sigma. Use Loop Shape Goal when the loop shape near crossover is simple or well understood (such as integral action). To specify only high gain or low gain constraints in certain frequency bands, use “Minimum Loop Gain Goal” on page 14-129 or “Maximum Loop Gain Goal” on page 14-135. When you do so, the software determines the best loop shape near crossover. Creation In the Tuning tab of Control System Tuner, select New Goal > Target shape for openloop response to create a Loop Shape Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.LoopShape to specify a loop-shape goal. Open-Loop Response Selection Use this section of the dialog box to specify the signal locations at which to compute the open-loop gain. You can also specify additional loop-opening locations for evaluating the tuning goal. • Shape open-loop response at the following locations Select one or more signal locations in your model at which to compute and constrain the open-loop gain. To constrain a SISO response, select a single-valued location. For example, to constrain the open-loop gain at a location named 'y', click Add signal to list and select 'y'. To constrain a MIMO response, select multiple signals or a vector-valued signal. • Compute response with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. 14-143 14 Control System Tuning Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Desired Loop Shape Use this section of the dialog box to specify the target loop shape. • Pure integrator wc/s Check to specify a pure integrator and crossover frequency for the target loop shape. For example, to specify an integral gain profile with crossover frequency 10 rad/s, enter 10 in the Crossover frequency wc text box. • Other gain profile Check to specify the target loop shape as a function of frequency. Enter a SISO numeric LTI model whose magnitude represents the desired gain profile. For example, you can specify a smooth transfer function (tf, zpk, or ss model). Alternatively, you can sketch a piecewise target loop shape using an frd model. When you do so, the software automatically maps the profile to a smooth transfer function that approximates the desired loop shape. For example, to specify a target loop shape of 100 (40 dB) below 0.1 rad/s, rolling off at a rate of –20 dB/decade at higher frequencies, enter frd([100 100 10],[0 1e-1 1]). If you are tuning in discrete time, you can specify the loop shape as a discrete-time model with the same sample time that you are using for tuning. If you specify the loop shape in continuous time, the tuning software discretizes it. Specifying the loop shape in discrete time gives you more control over the loop shape near the Nyquist frequency. Options Use this section of the dialog box to specify additional characteristics of the loop shape goal. • Enforce loop shape within 14-144 Loop Shape Goal Specify the tolerance in the location of the crossover frequency, in decades. For example, to allow gain crossovers within half a decade on either side of the target crossover frequency, enter 0.5. Increase the crossover tolerance to increase the ability of the tuning algorithm to enforce the target loop shape for all loops in a MIMO control system. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. • Stabilize closed loop system By default, the tuning goal imposes a stability requirement on the closed-loop transfer function from the specified inputs to outputs, in addition to the gain constraint. If stability is not required or cannot be achieved, select No to remove the stability requirement. For example, if the gain constraint applies to an unstable open-loop transfer function, select No. • Equalize loop interactions For multi-loop or MIMO loop gain constraints, the feedback channels are automatically rescaled to equalize the off-diagonal (loop interaction) terms in the open-loop transfer function. Select Off to disable such scaling and shape the unscaled open-loop response. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). 14-145 14 Control System Tuning Algorithms Evaluating Tuning Goals When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Loop Shape Goal, f(x) is given by: f (x) = WS S WT T . • S = D–1[I – L(s,x)]–1D is the scaled sensitivity function. L(s,x) is the open-loop response being shaped. D is an automatically-computed loop scaling factor. (If Equalize loop interactions is set to Off, then D = I.) T = S – I is the complementary sensitivity function. WS and WT are frequency weighting functions derived from the specified loop shape. The gains of these functions roughly match your specified loop shape and its inverse, respectively, for values ranging from –20 dB to 60 dB. For numerical reasons, the weighting functions level off outside this range, unless the specified gain profile changes slope outside this range. Because poles of WS or WT close to s = 0 or s = Inf might lead to poor numeric conditioning for tuning, it is not recommended to specify loop shapes with very low-frequency or very high-frequency dynamics. For more information about regularization and its effects, see “Visualize Tuning Goals” on page 14-189. Implicit Constraints This tuning goal imposes an implicit stability constraint on the closed-loop sensitivity function measured at the specified, evaluated with loops opened at the specified loopopening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the 14-146 See Also default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 14-147 14 Control System Tuning Margins Goal Purpose Enforce specified gain and phase margins when using Control System Tuner. Description Margins Goal enforces specified gain and phase margins on a SISO or MIMO feedback loop. For MIMO feedback loops, the gain and phase margins are based on the notion of disk margins, which guarantee stability for concurrent gain and phase variations in all feedback channels. For information about multiloop disk margins, see “Stability Analysis Using Disk Margins” (Robust Control Toolbox). In Control System Tuner, the shaded area on the plot represents the region in the frequency domain where the margins goal is not met. For more information about interpreting this plot, see “Stability Margins in Control System Tuning” on page 14-218. 14-148 Margins Goal Creation In the Tuning tab of Control System Tuner, select New Goal > Minimum stability margins to create a Margins Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.Margins to specify a stability margin goal. Feedback Loop Selection Use this section of the dialog box to specify the signal locations at which to measure stability margins. You can also specify additional loop-opening locations for evaluating the tuning goal. • Measure stability margins at the following locations Select one or more signal locations in your model at which to compute and constrain the stability margins. To constrain a SISO loop, select a single-valued location. For Add example, to constrain the stability margins at a location named 'y', click signal to list and select 'y'. To constrain a MIMO loop, select multiple signals or a vector-valued signal. • Measure stability margins with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can and . For more information on how to specify signal locations reorder them using for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. 14-149 14 Control System Tuning Desired Margins Use this section of the dialog box to specify the minimum gain and phase margins for the feedback loop. • Gain margin (dB) Enter the required minimum gain margin for the feedback loop as a scalar value expressed in dB. • Phase margin (degrees) Enter the required minimum phase margin for the feedback loop as a scalar value expressed in degrees. For MIMO feedback loops, the gain and phase margins are based on the notion of disk margins, which guarantee stability for concurrent gain and phase variations in all feedback channels. For information about multiloop disk margins, see “Stability Analysis Using Disk Margins” (Robust Control Toolbox). Options Use this section of the dialog box to specify additional characteristics of the stability margin goal. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. For best results with stability margin requirements, pick a frequency band extending about one decade on each side of the gain crossover frequencies. • D scaling order This value controls the order (number of states) of the scalings involved in computing MIMO stability margins. Static scalings (scaling order 0) are used by default. Increasing the order may improve results at the expense of increased computations. If the stability margin plot shows a large gap between the optimized and actual margins, 14-150 Margins Goal consider increasing the scaling order. See “Stability Margins in Control System Tuning” on page 14-218. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Margins Goal, f(x) is given by: f ( x ) = 2a S - a I • . S = D–1[I – L(s,x)]–1D is the scaled sensitivity function. L(s,x) is the open-loop response being shaped. D is an automatically-computed loop scaling factor. α is a scalar parameter computed from the specified gain and phase margin. This tuning goal imposes an implicit stability constraint on the closed-loop sensitivity function measured at the specified, evaluated with loops opened at the specified loopopening locations. The dynamics affected by this implicit constraint are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly 14-151 14 Control System Tuning constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples 14-152 • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 • “Stability Margins in Control System Tuning” on page 14-218 Passivity Goal Passivity Goal Purpose Enforce passivity of specific input/output map when using Control System Tuner. Description Passivity Goal enforces passivity of the response of the transfer function between the specified signal locations. A system is passive if all its I/O trajectories (u(t),y(t)) satisfy: T T Ú0 y (t ) u ( t) dt > 0, for all T > 0. Equivalently, a system is passive if its frequency response is positive real, which means that for all ω > 0, G ( jw ) + G ( jw ) H >0 Passivity Goal creates a constraint that enforces: T T T T T T Ú0 y (t ) u ( t) dt > n Ú0 u (t ) u (t ) dt + r Ú0 y( t) y (t ) dt, for all T > 0. To enforce the overall passivity condition, set the minimum input passivity index (ν) and the minimum output passivity index (ρ) to zero. To enforce an excess of passivity at the inputs or outputs, set ν or ρ to a positive value. To permit a shortage of passivity, set ν or ρ to a negative value. See “About Passivity and Passivity Indices” on page 10-2 for more information about these indices. 14-153 14 Control System Tuning In Control System Tuner, the shaded area on the plot represents the region in the frequency domain in which the tuning goal is not met. The plot shows the value of the index described in “Algorithms” on page 14-156. Creation In the Tuning tab of Control System Tuner, select New Goal > Passivity Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.Passivity to specify a passivity constraint. I/O Transfer Selection Use this section of the dialog box to specify the inputs and outputs of the transfer function that the tuning goal constrains. Also specify any locations at which to open loops for evaluating the tuning goal. • Specify input signals Select one or more signal locations in your model as inputs to the transfer function that the tuning goal constrains. To constrain a SISO response, select a single-valued 14-154 Passivity Goal input signal. For example, to constrain the gain from a location named 'u' to a Add signal to list and select 'u'. To constrain the location named 'y', click passivity of a MIMO response, select multiple signals or a vector-valued signal. • Specify output signals Select one or more signal locations in your model as outputs of the transfer function that the tuning goal constrains. To constrain a SISO response, select a single-valued output signal. For example, to constrain the gain from a location named 'u' to a Add signal to list and select 'y'. To constrain the location named 'y', click passivity of a MIMO response, select multiple signals or a vector-valued signal. • Compute input/output gain with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Options Use this section of the dialog box to specify additional characteristics of the passivity goal. • Minimum input passivity index Enter the target value of ν in the text box. To enforce an excess of passivity at the specified inputs, set ν > 0. To permit a shortage of passivity, set ν < 0. By default, the passivity goal enforces ν = 0, passive at the inputs with no required excess of passivity. • Minimum output passivity index 14-155 14 Control System Tuning Enter the target value of ρ in the text box. To enforce an excess of passivity at the specified outputs, set ρ > 0. To permit a shortage of passivity, set ρ < 0. By default, the passivity goal enforces ρ = 0, passive at the outputs with no required excess of passivity. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Passivity Goal, for a closed-loop transfer function G(s,x) from the specified inputs to the specified outputs, f(x) is given by: f ( x) = 14-156 R , Rmax = 106. 1 + R Rmax See Also R is the relative sector index (see getSectorIndex) of [G(s,x); I], for the sector represented by: Ê 2r Q=Á Ë -I -I ˆ ˜, 2n ¯ where ρ is the minimum output passivity index and ν is the minimum input passivity index specified in the dialog box. Rmax is fixed at 106, included to avoid numeric errors for very large R. This tuning goal imposes an implicit minimum-phase constraint on the transfer function G + I. The transmission zeros of G + I are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 • “Passive Control of Water Tank Level” on page 18-251 • “About Passivity and Passivity Indices” on page 10-2 14-157 14 Control System Tuning Conic Sector Goal Purpose Enforce sector bound on specific input/output map when using Control System Tuner. Description Conic Sector Goal creates a constraint that restricts the output trajectories of a system. If for all nonzero input trajectories u(t), the output trajectory z(t) = (Hu)(t) of a linear system H satisfies: T T Ú0 z (t ) Q z (t ) dt < 0, for all T ≥ 0, then the output trajectories of H lie in the conic sector described by the symmetric indefinite matrix Q. Selecting different Q matrices imposes different conditions on the system response. When you create a Conic Sector Goal, you specify the input signals, output signals, and the sector geometry. 14-158 Conic Sector Goal In Control System Tuner, the shaded area on the plot represents the region in the frequency domain in which the tuning goal is not met. The plot shows the value of the Rindex described in “About Sector Bounds and Sector Indices” on page 10-9. Creation In the Tuning tab of Control System Tuner, select New Goal > Conic Sector Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.ConicSector to specify a step response goal. I/O Transfer Selection Use this section of the dialog box to specify the inputs and outputs of the transfer function that the tuning goal constrains. Also specify any locations at which to open loops for evaluating the tuning goal. • Specify input signals Select one or more signal locations in your model as inputs to the transfer function that the tuning goal constrains. To constrain a SISO response, select a single-valued input signal. For example, to constrain the gain from a location named 'u' to a location named 'y', click Add signal to list and select 'u'. To constrain the passivity of a MIMO response, select multiple signals or a vector-valued signal. • Specify output signals Select one or more signal locations in your model as outputs of the transfer function that the tuning goal constrains. To constrain a SISO response, select a single-valued output signal. For example, to constrain the gain from a location named 'u' to a location named 'y', click Add signal to list and select 'y'. To constrain the passivity of a MIMO response, select multiple signals or a vector-valued signal. • Compute input/output gain with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. 14-159 14 Control System Tuning Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Options Specify additional characteristics of the conic sector goal using this section of the dialog box. • Conic Sector Matrix Enter the sector geometry Q, specified as: • A matrix, for constant sector geometry. Q is a symmetric square matrix that is ny on a side, where ny is the number of output signals you specify for the goal. The matrix Q must be indefinite to describe a well-defined conic sector. An indefinite matrix has both positive and negative eigenvalues. In particular, Q must have as many negative eigenvalues as there are input signals specified for the tuning goal (the size of the vector input signal u(t)). • An LTI model, for frequency-dependent sector geometry. Q satisfies Q(s)’ = Q(–s). In other words, Q(s) evaluates to a Hermitian matrix at each frequency. For more information, see “About Sector Bounds and Sector Indices” on page 10-9. • Regularization Regularization parameter, specified as a real nonnegative scalar value. Regularization keeps the evaluation of the tuning goal numerically tractable when other tuning goals tend to make the sector bound ill-conditioned at some frequencies. When this condition occurs, set Regularization to a small (but not negligible) fraction of the typical norm of the feedthrough term in H. For example, if you anticipate the norm of the feedthrough term of H to be of order 1 during tuning, try setting Regularization to 0.001. For more information about the conditions that require regularization, see the Regularization property of TuningGoal.ConicSector. • Enforce goal in frequency range 14-160 Conic Sector Goal Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Tips Constraining Input and Output Trajectories to Conic Sector Consider the following control system. r e + C u G y − Suppose that the signal u is marked as an analysis point in the model you are tuning. Suppose also that G is the closed-loop transfer function from u to y. A common application is to create a tuning goal that constrains all the I/O trajectories {u(t),y(t)} of G to satisfy: (t )ˆ Ê y (t )ˆ ÁÁ ˜˜ Q ÁÁ ˜˜ dt < 0, Ë u ( t) ¯ Ë u ( t) ¯ TÊy Ú0 T 14-161 14 Control System Tuning for all T ≥ 0. Constraining the I/O trajectories of G is equivalent to restricting the output trajectories z(t) of the system H = [G;I] to the sector defined by: T T Ú0 z (t ) Q z (t ) dt < 0. (See “About Sector Bounds and Sector Indices” on page 10-9 for more details about this equivalence.) To specify a constraint of this type using Conic Sector Goal, specify u as the input signal, and specify y and u as output signals. When you specify u as both input and output, Conic Sector Goal sets the corresponding transfer function to the identity. Therefore, the transfer function that the goal constrains is H = [G;I] as intended. This treatment is specific to Conic Sector Goal. For other tuning goals, when the same signal appears in both inputs and outputs, the resulting transfer function is zero in the absence of feedback loops, or the complementary sensitivity at that location otherwise. This result occurs because when the software processes analysis points, it assumes that the input is injected after the output. See “Mark Signals of Interest for Control System Analysis and Design” on page 2-89 for more information about how analysis points work. Algorithms Let Q = W1W1T - W2W2T be an indefinite factorization of Q, where W1T W2 = 0 . If W2T H ( s ) is square and minimum phase, then the time-domain sector bound T T Ú0 z (t ) Q z (t ) dt < 0, is equivalent to the frequency-domain sector condition, H ( - jw ) QH ( jw ) < 0 for all frequencies. Conic Sector Goal uses this equivalence to convert the time-domain characterization into a frequency-domain condition that Control System Tuner can handle in the same way it handles gain constraints. To secure this equivalence, Conic Sector Goal 14-162 See Also also makes W2T H ( s ) minimum phase by making all its zeros stable. The transmission zeros affected by this minimum-phase condition are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. For sector bounds, the R-index plays the same role as the peak gain does for gain constraints (see “About Sector Bounds and Sector Indices” on page 10-9). The condition H ( - jw ) QH ( jw ) < 0 is satisfied at all frequencies if and only if the R-index is less than one. The plot that Control System Tuner displays for Conic Sector Goal shows the R-index value as a function of frequency (see sectorplot). When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x), where x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Conic Sector Goal, for a closed-loop transfer function H(s,x) from the specified inputs to the specified outputs, f(x) is given by: f ( x) = R , Rmax = 106. 1 + R Rmax R is the relative sector index (see getSectorIndex) of H(s,x), for the sector represented by Q. See Also Related Examples • “About Sector Bounds and Sector Indices” on page 10-9 • “Specify Goals for Interactive Tuning” on page 14-39 14-163 14 Control System Tuning 14-164 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 Weighted Passivity Goal Weighted Passivity Goal Purpose Enforce passivity of a frequency-weighted transfer function when tuning in Control System Tuner. Description Weighted Passivity Goal enforces the passivity of H(s) = WL(s)T(s)WR(s), where T(s) is the transfer function from specified inputs to outputs. WL(s) and WR(s) are frequency weights used to emphasize particular frequency bands. A system is passive if all its I/O trajectories (u(t),y(t)) satisfy: T T Ú0 y (t ) u ( t) dt > 0, for all T > 0. Weighted Passivity Goal creates a constraint that enforces: T T T T T T Ú0 y (t ) u ( t) dt > n Ú0 u (t ) u (t ) dt + r Ú0 y( t) y (t ) dt, for the trajectories of the weighted transfer function H(s), for all T > 0. To enforce the overall passivity condition, set the minimum input passivity index (ν) and the minimum output passivity index (ρ) to zero. To enforce an excess of passivity at the inputs or outputs of the weighted transfer function, set ν or ρ to a positive value. To permit a shortage of passivity, set ν or ρ to a negative value. See getPassiveIndex for more information about these indices. 14-165 14 Control System Tuning In Control System Tuner, the shaded area on the plot represents the region in the frequency domain in which the tuning goal is not met. The plot shows the value of the index described in “Algorithms” on page 14-169. Creation In the Tuning tab of Control System Tuner, select New Goal > Weighted Passivity Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.WeightedPassivity to specify a step response goal. I/O Transfer Selection Use this section of the dialog box to specify the inputs and outputs of the transfer function that the tuning goal constrains. Also specify any locations at which to open loops for evaluating the tuning goal. • Specify input signals 14-166 Weighted Passivity Goal Select one or more signal locations in your model as inputs to the transfer function that the tuning goal constrains. To constrain a SISO response, select a single-valued input signal. For example, to constrain the gain from a location named 'u' to a location named 'y', click Add signal to list and select 'u'. To constrain the passivity of a MIMO response, select multiple signals or a vector-valued signal. • Specify output signals Select one or more signal locations in your model as outputs of the transfer function that the tuning goal constrains. To constrain a SISO response, select a single-valued output signal. For example, to constrain the gain from a location named 'u' to a Add signal to list and select 'y'. To constrain the location named 'y', click passivity of a MIMO response, select multiple signals or a vector-valued signal. • Compute input/output gain with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Weights Use the Left weight WL and Right weight WR text boxes to specify the frequencyweighting functions for the tuning goal. H(s) = WL(s)T(s)WR(s), where T(s) is the transfer function from specified inputs to outputs. WL provides the weighting for the output channels of H(s), and WR provides the weighting for the input channels. You can specify scalar weights or frequency-dependent weighting. To specify a frequency-dependent weighting, use a numeric LTI model whose magnitude represents the desired weighting function. For example, enter tf(1,[1 0.01]) to specify a high weight at low frequencies that rolls off above 0.01 rad/s. 14-167 14 Control System Tuning If the tuning goal constrains a MIMO transfer function, scalar or SISO weighting functions automatically expand to any input or output dimension. You can specify different weights for each channel by specifying matrices or MIMO weighting functions. The dimensions H(s) must be commensurate with the dimensions of WL and WR. For example, if the constrained transfer function has two inputs, you can specify diag([1 10]) as WR. If you are tuning in discrete time, you can specify the weighting functions as discrete-time models with the same sampling time as you use for tuning. If you specify the weighting functions in continuous time, the tuning software discretizes them. Specifying the weighting functions in discrete time gives you more control over the weighting functions near the Nyquist frequency. Options Use this section of the dialog box to specify additional characteristics of the step response goal. • Minimum input passivity index Enter the target value of ν in the text box. To enforce an excess of passivity at the specified inputs, set ν > 0. To permit a shortage of passivity, set ν < 0. By default, the passivity goal enforces ν = 0, passive at the inputs with no required excess of passivity. • Minimum output passivity index Enter the target value of ρ in the text box. To enforce an excess of passivity at the specified outputs, set ρ > 0. To permit a shortage of passivity, set ρ < 0. By default, the passivity goal enforces ρ = 0, passive at the outputs with no required excess of passivity. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or block14-168 Weighted Passivity Goal parameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Weighted Passivity Goal, for a closed-loop transfer function T(s,x) from the specified inputs to the specified outputs, and the weighted transfer function H(s,x) = WL(s)T(s,x)WR(s), f(x) is given by: f ( x) = R , Rmax = 106. 1 + R Rmax R is the relative sector index (see getSectorIndex) of [H(s,x); I], for the sector represented by: Ê 2r Q=Á Ë -I -I ˆ ˜, 2n ¯ where ρ is the minimum output passivity index and ν is the minimum input passivity index specified in the dialog box. Rmax is fixed at 106, included to avoid numeric errors for very large R. This tuning goal imposes an implicit minimum-phase constraint on the weighted transfer function H + I. The transmission zeros of H + I are the stabilized dynamics for this tuning goal. The Minimum decay rate and Maximum natural frequency tuning options control the lower and upper bounds on these implicitly constrained dynamics. If the 14-169 14 Control System Tuning optimization fails to meet the default bounds, or if the default bounds conflict with other requirements, on the Tuning tab, use Tuning Options to change the defaults. See Also Related Examples 14-170 • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 • “About Passivity and Passivity Indices” on page 10-2 Poles Goal Poles Goal Purpose Constrain the dynamics of the closed-loop system, specified feedback loops, or specified open-loop configurations, when using Control System Tuner. Description Poles Goal constrains the dynamics of your entire control system or of specified feedback loops of your control system. Constraining the dynamics of a feedback loop means constraining the dynamics of the sensitivity function measured at a specified location in the control system. Using Poles Goal, you can specify finite minimum decay rate or minimum damping for the poles in the control system or specified loop. You can specify a maximum natural frequency for these poles, to eliminate fast dynamics in the tuned control system. 14-171 14 Control System Tuning In Control System Tuner, the shaded area on the plot represents the region in the frequency domain where the pole location constraints are not met. To constrain dynamics or ensure stability of a single tunable component of the control system, use “Controller Poles Goal” on page 14-176. Creation In the Tuning tab of Control System Tuner, select New Goal > Constraint on closedloop dynamics to create a Poles Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.Poles to specify a disturbance rejection goal. Feedback Configuration Use this section of the dialog box to specify the portion of the control system for which you want to constrain dynamics. You can also specify loop-opening locations for evaluating the tuning goal. • Entire system Select this option to constrain the locations of closed-loop poles of the control system. • Specific feedback loop(s) Select this option to specify one or more feedback loops to constrain. Specify a feedback loop by selecting a signal location in your control system. Poles Goal constrains the dynamics of the sensitivity function measured at that location. (See getSensitivity for information about sensitivity functions.) To constrain the dynamics of a SISO loop, select a single-valued location. For example, to constrain the dynamics of the sensitivity function measured at a location named 'y', click Add signal to list and select 'y'. To constrain the dynamics of a MIMO loop, select multiple signals or a vector-valued signal. • Compute poles with the following loops open Select one or more signal locations in your model at which to open a feedback loop for the purpose of evaluating this tuning goal. The tuning goal is evaluated against the 14-172 Poles Goal open-loop configuration created by opening feedback loops at the locations you identify. For example, to evaluate the tuning goal with an opening at a location named 'x', click Add signal to list and select 'x'. Tip To highlight any selected signal in the Simulink model, click from the input or output list, click . To remove a signal . When you have selected multiple signals, you can reorder them using and . For more information on how to specify signal locations for a tuning goal, see “Specify Goals for Interactive Tuning” on page 14-39. Pole Location Use this section of the dialog box to specify the limits on pole locations. • Minimum decay rate Enter the target minimum decay rate for the system poles. Closed-loop system poles that depend on the tunable parameters are constrained to satisfy Re(s) < MinDecay for continuous-time systems, or log(|z|) < -MinDecay*Ts for discretetime systems with sample time Ts. This constraint helps ensure stable dynamics in the tuned system. Enter 0 to impose no constraint on the decay rate. • Minimum damping Enter the target minimum damping of closed-loop poles of tuned system, as a value between 0 and 1. Closed-loop system poles that depend on the tunable parameters are constrained to satisfy Re(s) < -MinDamping*|s|. In discrete time, the damping ratio is computed using s = log(z)/Ts. Enter 0 to impose no constraint on the damping ratio. • Maximum natural frequency Enter the target maximum natural frequency of poles of tuned system, in the units of the control system model you are tuning. When you tune the control system using this requirement, closed-loop system poles that depend on the tunable parameters are constrained to satisfy |s| < MaxFrequency for continuous-time systems, or | 14-173 14 Control System Tuning log(z)| < MaxFrequency*Ts for discrete-time systems with sample time Ts. This constraint prevents fast dynamics in the control system. Enter Inf to impose no constraint on the natural frequency. Options Use this section of the dialog box to specify additional characteristics of the poles goal. • Enforce goal in frequency range Limit the enforcement of the tuning goal to a particular frequency band. Specify the frequency band as a row vector of the form [min,max], expressed in frequency units of your model. For example, to create a tuning goal that applies only between 1 and 100 rad/s, enter [1,100]. By default, the tuning goal applies at all frequencies for continuous time, and up to the Nyquist frequency for discrete time. The Poles Goal applies only to poles with natural frequency within the range you specify. • Apply goal to Use this option when tuning multiple models at once, such as an array of models obtained by linearizing a Simulink model at different operating points or blockparameter values. By default, active tuning goals are enforced for all models. To enforce a tuning requirement for a subset of models in an array, select Only Models. Then, enter the array indices of the models for which the goal is enforced. For example, suppose you want to apply the tuning goal to the second, third, and fourth models in a model array. To restrict enforcement of the requirement, enter 2:4 in the Only Models text box. For more information about tuning for multiple models, see “Robust Tuning Approaches” (Robust Control Toolbox). Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. 14-174 See Also For Poles Goal, f(x) reflects the relative satisfaction or violation of the goal. For example, if your Poles Goal constrains the closed-loop poles of a feedback loop to a minimum damping of ζ = 0.5, then: • f(x) = 1 means the smallest damping among the constrained poles is ζ = 0.5 exactly. • f(x) = 1.1 means the smallest damping ζ = 0.5/1.1 = 0.45, roughly 10% less than the target. • f(x) = 0.9 means the smallest damping ζ = 0.5/0.9 = 0.55, roughly 10% better than the target. See Also Related Examples • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 14-175 14 Control System Tuning Controller Poles Goal Purpose Constrain the dynamics of a specified tunable block in the tuned control system, when using Control System Tuner. Description Controller Poles Goal constrains the dynamics of a tunable block in your control system model. Controller Poles Goal can impose a stability constraint on the specified block. You can also specify a finite minimum decay rate, a minimum damping rate, or a maximum natural frequency for the poles of the block. These constraints allow you to eliminate fast dynamics and control ringing in the response of the tunable block. In Control System Tuner, the shaded area on the plot represents the region in the frequency domain where the pole location constraints are not met. The constraint applies to all poles in the block except fixed integrators, such as the I term of a PID controller. 14-176 Controller Poles Goal To constrain dynamics or ensure stability of an entire control system or a feedback loop in the control system, use “Poles Goal” on page 14-171. Creation In the Tuning tab of Control System Tuner, select New Goal > Constraint on controller dynamics to create a Controller Poles Goal. Command-Line Equivalent When tuning control systems at the command line, use TuningGoal.ControllerPoles to specify a controller poles goal. Constrain Dynamics of Tuned Block From the drop-down menu, select the tuned block in your control system to which to apply the Controller Poles Goal. If the block you want to constrain is not in the list, add it to the Tuned Blocks list. In Control System Tuner, in the Tuning tab, click Select Blocks. For more information about adding tuned blocks, see “Specify Blocks to Tune in Control System Tuner” on page 14-24. Keep Poles Inside the Following Region Use this section of the dialog box to specify the limits on pole locations. • Minimum decay rate Enter the desired minimum decay rate for the poles of the tunable block. Poles of the block are constrained to satisfy Re(s) < -MinDecay for continuous-time blocks, or log(|z|) < -MinDecay*Ts for discrete-time blocks with sample time Ts. Specify a nonnegative value to ensure that the block is stable. If you specify a negative value, the tuned block can include unstable poles. • Minimum damping Enter the desired minimum damping ratio of poles of the tunable block, as a value between 0 and 1. Poles of the block that depend on the tunable parameters are constrained to satisfy Re(s) < -MinDamping*|s|. In discrete time, the damping ratio is computed using s=log(z)/Ts. 14-177 14 Control System Tuning • Maximum natural frequency Enter the target maximum natural frequency of poles of the tunable block, in the units of the control system model you are tuning. Poles of the block are constrained to satisfy |s| < MaxFrequency for continuous-time blocks, or |log(z)| < MaxFrequency*Ts for discrete-time blocks with sample time Ts. This constraint prevents fast dynamics in the tunable block. Algorithms When you tune a control system, the software converts each tuning goal into a normalized scalar value f(x). Here, x is the vector of free (tunable) parameters in the control system. The software then adjusts the parameter values to minimize f(x) or to drive f(x) below 1 if the tuning goal is a hard constraint. For Controller Poles Goal, f(x) reflects the relative satisfaction or violation of the goal. For example, if your Controller Poles Goal constrains the pole of a tuned block to a minimum damping of ζ = 0.5, then: • f(x) = 1 means the damping of the pole is ζ = 0.5 exactly. • f(x) = 1.1 means the damping is ζ = 0.5/1.1 = 0.45, roughly 10% less than the target. • f(x) = 0.9 means the damping is ζ = 0.5/0.9 = 0.55, roughly 10% better than the target. See Also Related Examples 14-178 • “Specify Goals for Interactive Tuning” on page 14-39 • “Manage Tuning Goals” on page 14-179 • “Visualize Tuning Goals” on page 14-189 Manage Tuning Goals Manage Tuning Goals Control System Tuner lets you designate one or more tuning goals as hard goals. This designation gives you a way to differentiate must-have goals from nice-to-have goals. Control System Tuner attempts to satisfy hard requirements by driving their associated cost functions below 1. Subject to that constraint, the software comes as close as possible to satisfying remaining (soft) requirements. For best results, make sure you can obtain a reasonable design with all goals treated as soft goals before attempting to enforce any goal as a hard constraint. By default, new goals are designated soft goals. In the Tuning tab, click Manage Goals to open the Manage tuning goals dialog box. Check Hard for any goal to designate it a hard goal. You can also designate any tuning goal as inactive for tuning. In this case the software ignores the tuning goal entirely. Use this dialog box to select which tuning goals are active when you tune the control system. Active is checked by default for any new goals. Uncheck Active for any design goal that you do not want enforced. For example, if you tune with the following configuration, Control System Tuner optimizes StepRespGoal1, subject to MarginsGoal1. The tuning goal PolesGoal1 is ignored. 14-179 14 Control System Tuning All tuning goals you have created in the Control System Tuner session are listed in the dialog box. To edit an existing tuning goal, select it in the list and click Edit. To delete a tuning goal from the list, select it and click Remove. To add more tuning goals to the list, in Control System Tuner, in the Tuning tab, click New Goal. For more information about creating tuning goals, see “Specify Goals for Interactive Tuning” on page 14-39. 14-180 Generate MATLAB Code from Control System Tuner for Command-Line Tuning Generate MATLAB Code from Control System Tuner for Command-Line Tuning You can generate a MATLAB script in Control System Tuner for tuning a control system at the command line. Generated scripts are useful when you want to programmatically reproduce a result you obtained interactively. A generated MATLAB script also enables you to programmatically perform multiple tuning operations with variations in tuning goals, system parameters, or model conditions such as operating point. Tip You can also save a Control System Tuner session to reproduce within Control System Tuner. To do so, in the Control System tab, click Save Session. To generate a MATLAB script in Control System Tuner, in the Tuning tab, click Tune . Select Script with current values. The MATLAB Editor displays the generated script, which script reproduces programmatically the current tuning configuration of Control System Tuner. 14-181 14 Control System Tuning For example, suppose you generate a MATLAB script after completing all steps in the example “Control of a Linear Electric Actuator Using Control System Tuner” on page 18109. The generated script computes the operating point used for tuning, designates the blocks to tune, creates the tuning goals, and performs other operations to reproduce the result at the command line. The first section of the script creates the slTuner interface to the Simulinkmodel (rct_linact in this example). The slTuner interface stores a linearization of the model and parameterizations of the blocks to tune. %% Create system data with slTuner interface TunedBlocks = {'rct_linact/Current Controller/Current PID'; ... 'rct_linact/Speed Controller/Speed PID'}; AnalysisPoints = {'rct_linact/Speed Demand (rpm)/1'; ... 'rct_linact/Current Sensor/1'; ... 'rct_linact/Hall Effect Sensor/1'; ... 'rct_linact/Speed Controller/Speed PID/1'; ... 'rct_linact/Current Controller/Current PID/1'}; OperatingPoints = 0.5; % Specify the custom options Options = slTunerOptions('AreParamsTunable',false); % Create the slTuner object CL0 = slTuner('rct_linact',TunedBlocks,AnalysisPoints,OperatingPoints,Options); The slTuner interface also specifies the operating point at which the model is linearized, and marks as analysis points all the signal locations required to specify the tuning goals for the example. (See “Create and Configure slTuner Interface to Simulink Model” on page 14-212.) If you are tuning a control system modeled in MATLAB instead of Simulink, the first section of the script constructs a genss model that has equivalent dynamics and parameterization to the genss model of the control system that you specified Control System Tuner. Next, the script creates the three tuning goals specified in the example. The script uses TuningGoal objects to capture these tuning goals. For instance, the script uses TuningGoal.Tracking to capture the Tracking Goal of the example. %% Create tuning goal to follow reference commands with prescribed performance % Inputs and outputs Inputs = {'rct_linact/Speed Demand (rpm)/1'}; Outputs = {'rct_linact/Hall Effect Sensor/1[rpm]'}; % Tuning goal specifications ResponseTime = 0.1; % Approximately reciprocal of tracking bandwidth DCError = 0.001; % Maximum steady-state error PeakError = 1; % Peak error across frequency % Create tuning goal for tracking TR = TuningGoal.Tracking(Inputs,Outputs,ResponseTime,DCError,PeakError); TR.Name = 'TR'; % Tuning goal name 14-182 See Also After creating the tuning goals, the script sets any algorithm options you had set in Control System Tuner. The script also designates tuning goals as soft or hard goals, according to the configuration of tuning goals in Control System Tuner. (See “Manage Tuning Goals” on page 14-179.) %% Create option set for systune command Options = systuneOptions(); %% Set soft and hard goals SoftGoals = [ TR ; ... MG1 ; ... MG2 ]; HardGoals = []; In this example, all the goals are designated as soft goals when the script is generated. Therefore, HardGoals is empty. Finally, the script tunes the control system by calling systune on the slTuner interface using the tuning goals and options. %% Tune the parameters with soft and hard goals [CL1,fSoft,gHard,Info] = systune(CL0,SoftGoals,HardGoals,Options); The script also includes an optional call to viewGoal, which displays graphical representations of the tuning goals to aid you in interpreting and validating the tuning results. Uncomment this line of code to generate the plots. %% View tuning results % viewGoal([SoftGoals;HardGoals],CL1); You can add calls to functions such getIOTransfer to make the script generate additional analysis plots. See Also Related Examples • “Create and Configure slTuner Interface to Simulink Model” on page 14-212 • “Tune Control System at the Command Line” on page 14-223 • “Validate Tuned Control System” on page 14-227 14-183 14 Control System Tuning Interpret Numeric Tuning Results When you tune a control system with systune or Control System Tuner, the software provides reports that give you an overview of how well the tuned control system meets your design requirements. Interpreting these reports requires understanding how the tuning algorithm optimizes the system to satisfy your tuning goals. (The software also provides visualizations of the tuning goals and system responses to help you see where and by how much your requirements are not satisfied. For information about using these plots, see “Visualize Tuning Goals” on page 14-189.) Tuning-Goal Scalar Values The tuning software converts each tuning goal into a normalized scalar value which it then constrains (hard goals) or minimizes (soft goals). Let fi(x) and gj(x) denote the scalar values of the soft and hard goals, respectively. Here, x is the vector of tunable parameters in the control system to tune. The tuning algorithm solves the minimization problem: Minimize max fi ( x ) subject to max g j ( x ) < 1 , for xmin < x < xmax . i j xmin and xmax are the minimum and maximum values of the free parameters of the control system. (For information about the specific functions used to evaluate each type of requirement, see the reference pages for each tuning goal.) When you use both soft and hard tuning goals, the software solves the optimization as a sequence of subproblems of the form: min max ( a f ( x ) , g ( x ) ) . x The software adjusts the multiplier α so that the solution of the subproblems converges to the solution of the original constrained optimization problem. The tuning software reports the final scalar values for each tuning goal. When the final value of fi(x) or gj(x) is less than 1, the corresponding tuning goal is satisfied. Values greater than 1 indicate that the tuning goal is not satisfied for at least some conditions. For instance, a tuning goal that describes a frequency-domain constraint might be satisfied at some frequencies and not at others. The closer the value is to 1, the closer the tuning goal is to being satisfied. Thus these values give you an overview of how successfully the tuned system meets your requirements. 14-184 Interpret Numeric Tuning Results The form in which the software presents the optimized tuning-goal values depends on whether you are tuning with Control System Tuner or at the command line. Tuning Results at the Command Line The systune command returns the control system model or slTuner interface with the tuned parameter values. systune also returns the best achieved values of each fi(x) and gj(x) as the vector-valued output arguments fSoft and gHard, respectively. See the systune reference page for more information. (To obtain the final tuning goal values on their own, use evalGoal.) By default, systune displays the best achieved final values of the tuning goals in the command window. For instance, in the example “PID Tuning for Setpoint Tracking vs. Disturbance Rejection” on page 18-13, systune is called with one soft requirement, R1, and two hard requirements R2 and R3. T1 = systune(T0,R1,[R2 R3]); Final: Soft = 1.12, Hard = 0.99988, Iterations = 143 This display indicates that the largest optimized value of the hard tuning goals is less than 1, so both hard goals are satisfied. The soft goal value is slightly greater than one, indicating that the soft goal is nearly satisfied. You can use tuning-goal plots to see in what regimes and by how much the tuning goals are violated. (See “Visualize Tuning Goals” on page 14-189.) You can obtain additional information about the optimization progress and values using the info output of systune. To make systune display additional information during tuning, use systuneOptions. Tuning Results in Control System Tuner In Control System Tuner, when you click , the app compiles a Tuning Report summarizing the best achieved values of fi(x) and gj(x). To view the tuning report immediately after tuning a control system, click Tuning Report at the bottom-right corner of Control System Tuner. 14-185 14 Control System Tuning The tuning report displays the final fi(x) and gj(x) values obtained by the algorithm. The Hard Goals area shows the minimized gi(x) values and indicates which are satisfied. The Soft Goals area highlights the largest of the minimized fi(x) values as Worst Value, and lists the values for all the requirements. In this example, the hard goal is satisfied, while the soft goals are nearly satisfied. As in the command-line case, you can use tuninggoal plots to see where and by how much tuning goals are violated. (See “Visualize Tuning Goals” on page 14-189.) 14-186 Interpret Numeric Tuning Results Tip You can view a report from the most recent tuning run at any time. In the Tuning tab, click Tune , and select Tuning Report. Improve Tuning Results If the tuning results do not adequately meet your design requirements, adjust your set of tuning goals to improve the results. For example: • Designate tuning goals that are must-have requirements as hard goals. Or, relax tuning goals that are not absolute requirements by designating them as soft goals. • Limit the frequency range in which frequency-domain goals are enforced. • In Control System Tuner, use the Enforce goal in frequency range field of the tuning goal dialog box. • At the command line, use the Focus property of the TuningGoal object. If the tuning results do satisfy your design requirements, you can validate the tuned control system as described in “Validate Tuned Control System” on page 14-227. 14-187 14 Control System Tuning See Also evalGoal | systune | systune (for slTuner) | viewGoal Related Examples 14-188 • “Visualize Tuning Goals” on page 14-189 • “Validate Tuned Control System” on page 14-227 Visualize Tuning Goals Visualize Tuning Goals When you tune a control system with systune or Control System Tuner, use tuning-goal plots to visualize your design requirements against the tuned control system responses. Tuning-goal plots show graphically where and by how much tuning goals are satisfied or violated. This visualization lets you examine how close your control system is to ideal performance. It can also help you identify problems with tuning and provide clues on how to improve your design. Tuning-Goal Plots How you obtain tuning-goal plots depends on your work environment. • At the command line, use viewGoal. • In Control System Tuner, each tuning goal that you create generates a tuning-goal plot. When you tune the control system, these plots update to reflect the tuned design. The form of the tuning-goal plot depends on the specific tuning goal you use. Time-Domain Goals For time-domain tuning goals, the tuning-goal plot is a time-domain plot of the relevant system response. The following plot, adapted from the example “MIMO Control of Diesel Engine” on page 18-184, shows a typical tuning-goal plot for a time-domain disturbancerejection goal. The dashed lines represent the worst acceptable step response specified in the tuning goal. The solid line shows the corresponding response of the tuned system. 14-189 14 Control System Tuning Frequency-Domain Goals The plots for frequency-domain tuning goals show the target response and the tuned response in the frequency domain. The following plot, adapted from the example “FixedStructure Autopilot for a Passenger Jet” on page 18-226, shows a plot for a gain goal (TuningGoal.Gain at the command line). This tuning goal limits the gain between a specified input and output to a frequency-dependent profile. In the plot, the dashed line shows the gain profile specified in the tuning goal. If the tuned system response (solid line) enters the shaded region, the tuning goal is violated. In this case, the tuning goal is satisfied at all frequencies. 14-190 Visualize Tuning Goals Margin Goals For information about interpreting tuning-goal plots for stability-margin goals, see “Stability Margins in Control System Tuning” on page 14-218. Difference Between Dashed Line and Shaded Region With some frequency-domain tuning goals, there might be a difference between the gain profile you specify in the tuning goal, and the profile the software uses for tuning. In this case, the shaded region of the plot reflects the profile that the software uses for tuning. The gain profile you specify and the gain profile used for tuning might differ if: 14-191 14 Control System Tuning • You tune a control system in discrete time, but specify the gain profile in continuous time. • The software modifies the asymptotes of the specified gain profile to improve numeric stability. Continuous-Time Gain Profile for Discrete-Time Tuning When you tune a discrete-time control system, you can specify frequency-dependent tuning goals using discrete-time or continuous-time transfer functions. If you use a continuous-time transfer function, the tuning algorithm discretizes the transfer function before tuning. For instance, suppose that you specify a tuning goal as follows. W = zpk([],[0 -150 -150],1125000); Req = TuningGoal.MaxLoopGain('Xloc',W); Suppose further that you use the tuning goal with systune to tune a discrete-time genss model or slTuner interface. CL is the resulting tuned control system. To examine the result, generate a tuning-goal plot. viewGoal(Req,CL) 14-192 Visualize Tuning Goals The plot shows W, the continuous-time maximum loop gain that you specified, as a dashed line. The shaded region shows the discretized version of W that systune uses for tuning. The discretized maximum loop gain cuts off at the Nyquist frequency corresponding to the sample time of CL. Near that cutoff, the shaded region diverges from the dashed line. The plot highlights that sometimes it is preferable to specify tuning goals for discretetime tuning using discrete-time gain profiles. In particular, specifying a discrete-time profile gives you more control over the behavior of the gain profile near the Nyquist frequency. 14-193 14 Control System Tuning Modifications for Numeric Stability When you use a tuning goal with a frequency-dependent specification, the tuning algorithm uses a frequency-weighting function to compute the normalized value of the tuning goal. This weighting function is derived from the gain profile that you specify. For numeric tractability, weighting functions must be stable and proper. For numeric stability, their dynamics must be in the same frequency range as the control system dynamics. For these reasons, the software might adjust the specified gain profile to eliminate undesirable low-frequency or high-frequency dynamics or asymptotes. The process of modifying the tuning goal for better numeric conditioning is called regularization. For example, consider the following tracking goal. R1 = TuningGoal.Tracking('r','y',tf([1 0 0],[1 2 1])); viewGoal(R1) 14-194 Visualize Tuning Goals Here the control bandwidth is about 1 rad/s and the gain profile has two zeros at s = 0, which become unstable poles in the weighting function (see TuningGoal.Tracking for details). The regularization moves these zeros to about 0.01 rad/s, and the maximum tracking error levels off at about 10–3 (0.1%). If you need better tracking accuracy, you can explicitly specify the cutoff frequency in the error profile. R2 = TuningGoal.Tracking('r','y',tf([1 0 5e-8],[1 2 1])); viewGoal(R2) set(gca,'Ylim',[1e-4,10]) 14-195 14 Control System Tuning However, for numeric safety, the regularized weighting function always levels off at very low and very high frequencies, regardless of the specified gain profile. Access the Regularized Functions When you are working at the command line, you can obtain the regularized gain profile using the getWeight or getWeights commands. For details, see the reference pages for the individual tuning goals for which the tuning algorithm performs regularization: • TuningGoal.Gain • TuningGoal.LoopShape • TuningGoal.MaxLoopGain 14-196 See Also • TuningGoal.MinLoopGain • TuningGoal.Rejection • TuningGoal.Sensitivity • TuningGoal.StepRejection • TuningGoal.Tracking In Control System Tuner, you cannot view the regularized weighting functions directly. Instead, use the tuning-goal commands to generate an equivalent tuning goal, and use getWeight or getWeights to access the regularized functions. Improve Tuning Results If the tuning results do not adequately meet your design requirements, adjust your set of tuning goals to improve the results. For example: • Designate tuning goals that are must-have requirements as hard goals. Or, relax tuning goals that are not absolute requirements by designating them as soft goals. • Limit the frequency range in which frequency-domain goals are enforced. • In Control System Tuner, use the Enforce goal in frequency range field of the tuning goal dialog box. • At the command line, use the Focus property of the TuningGoal object. If the tuning results do satisfy your design requirements, you can validate the tuned control system as described in “Validate Tuned Control System” on page 14-227. See Also viewGoal Related Examples • “Interpret Numeric Tuning Results” on page 14-184 • “Create Response Plots in Control System Tuner” on page 14-198 • “Validate Tuned Control System” on page 14-227 14-197 14 Control System Tuning Create Response Plots in Control System Tuner This example shows how to create response plots for analyzing system performance in Control System Tuner. Control System Tuner can generate many types of response plots in the time and frequency domains. You can view responses of SISO or MIMO transfer functions between inputs and outputs at any location in your model. When you tune your control system, Control System Tuner updates the response plots to reflect the tuned design. Use response plots to validate the performance of the tuned control system. This example creates response plots for analyzing the sample model rct_helico. Choose Response Plot Type In Control System Tuner, in the Control System tab, click of plot you want to create. New Plot. Select the type A new plot dialog box opens in which you specify the inputs and outputs of the portion of your control system whose response you want to plot. For example, select New step to create a step response plot from specified inputs to specified outputs of your system. 14-198 Create Response Plots in Control System Tuner Specify Transfer Function Choose which transfer function associated with the specified inputs and outputs you want to analyze. For most response plots types, the Select Response to Plot menu lets you choose one of the following transfer functions: • New Input-Output Transfer Response — Transfer function between specified inputs and outputs, computed with loops open at any additionally specified loopopening locations. • New Sensitivity Transfer Response — Sensitivity function computed at the specified location and with loops open at any specified loop-opening locations. • New Open-Loop Response — Open loop point-to-point transfer function computed at the specified location and with loops open at any additionally specified loop-opening locations. • Entire System Response — For Pole/Zero maps and I/O Pole/Zero maps only. Plot the pole and zero locations for the entire closed-loop control system. • Response of Tuned Block — For Pole/Zero maps and I/O Pole/Zero maps only. Plot the pole and zero locations of tuned blocks. Name the Response Type a name for the response in the Response Name text box. Once you have specified signal locations defining the response, Control System Tuner stores the response under this name. When you create additional new response plots, the response appears by this name in Select Response to Plot menu. Choose Signal Locations for Evaluating System Response Specify the signal locations in your control system at which to evaluate the selected response. For example, the step response plot displays the response of the system at one 14-199 14 Control System Tuning or more output locations to a unit step applied at one or more input locations. Use the Specify input signals and Specify output signals sections of the dialog box to specify these locations. (Other tuning goal types, such as loop-shape or stability margins, require you to specify only one location for evaluation. The procedure for specifying the location is the same as illustrated here.) Under Specify input signals, click locations appears. Add signal to list. A list of available input If the signal you want to designate as a step-response input is in the list, click the signal to add it to the step-response inputs. If the signal you want to designate does not appear, and you are tuning a Simulink model, click Select signal from model. In the Select signals dialog box, build a list of the signals you want. To do so, click signals in the Simulink model editor. The signals that you click appear in the Select signals dialog box. Click one signal to create a SISO response, and click multiple signals to create a MIMO response. Click Add signal(s). The Select signals dialog box closes. 14-200 Create Response Plots in Control System Tuner The signal or signals you selected now appear in the list of step-response inputs in the new-plot dialog box. Similarly, specify the locations at which the step response is measured to the stepresponse outputs list. For example, the following configuration plots the MIMO response to a step input applied at theta-ref and phi-ref and measured at theta and phi in the Simulink model rct_helico. 14-201 14 Control System Tuning Tip To highlight any selected signal in the Simulink model, click from the input or output list, click reorder them using and . To remove a signal . When you have selected multiple signals, you can . Specify Loop Openings You can evaluate most system responses with loops open at one or more locations in the control system. Click the response. Add loop opening location to list to specify such locations for Store and Plot the Response When you have finished specifying the response, click Plot in the new plot dialog box. The new response appears in the Responses section of the Data Browser. A new figure opens 14-202 Create Response Plots in Control System Tuner displaying the response plot. When you tune your control system, you can refer to this figure to evaluate the performance of the tuned system. Tip To edit the specifications of the response, double-click the response in the Data Browser. Any plots using that response update to reflect the edited response. View response characteristics such as rise-times or peak values by right-clicking on the plot. Other options for managing and organizing multiple plots are available in the View tab. 14-203 14 Control System Tuning See Also Related Examples 14-204 • “Compare Performance of Multiple Tuned Controllers” on page 14-207 • “Examine Tuned Controller Parameters in Control System Tuner” on page 14-205 • “Visualize Tuning Goals” on page 14-189 Examine Tuned Controller Parameters in Control System Tuner Examine Tuned Controller Parameters in Control System Tuner After you tune your control system, Control System Tuner gives you two ways to view the current values of the tuned block parameters: • In the Data Browser, in the Tuned Blocks area, select the block whose parameters you want to view. A text summary of the block and its current parameter values appears in the Data Browser in the Data Preview area. • In the Data Browser, in the Tuned Blocks area, double-click the block whose parameters you want to view. The Tuned Block Editor opens, displaying the current values of the parameters. For array-valued parameters, click editor displaying values in the array. to open a variable 14-205 14 Control System Tuning Note To find a tuned block in the Simulink model, right-click the block name in the Data Browser and select Highlight. See Also Related Examples • 14-206 “View and Change Block Parameterization in Control System Tuner” on page 14-26 Compare Performance of Multiple Tuned Controllers Compare Performance of Multiple Tuned Controllers Control System Tuner lets you compare the performance of a control system tuned with two different sets of tuning goals. Such comparison is useful, for example, to see the effect on performance of changing a tuning goal from hard goal to soft goal. Comparing performance is also useful to see the effect of adding an additional tuning goal when an initial design fails to satisfy all your performance requirements either in the linearized system or when validated against a full nonlinear model. This example compares tuning results for the sample model rct_linact. Store First Design After tuning a control system with a first set of design requirements, store the design in Control System Tuner. In the Control System tab, click Browser in the Designs area. Store. The stored design appears in the Data 14-207 14 Control System Tuning Change the name of the stored design, if desired, by right-clicking on the data browser entry. 14-208 Compare Performance of Multiple Tuned Controllers Compute New Design In the Tuning tab, make any desired changes to the tuning goals for the second design. For example, add new tuning goals or edit existing tuning goals to change specifications. Or, in Manage Goals, change which tuning goals are active and which are designated hard constraints or soft requirements. When you are ready, retune the control system with the new set of tuning goals. Click Tune. Control System Tuner updates the current design (the current set of controller parameters) with the new tuned design. All existing plots, which by default show the current design, are updated to reflect the new current design. Compare New Design with Stored Design Update all plots to reflect both the new design and the stored design. In the Control System tab, click Compare. The Compare Designs dialog box opens. In the Compare Designs dialog box, the current design is checked by default. Check the box for the design you want to compare to the current design. All response plots and 14-209 14 Control System Tuning tuning goal plots update to reflect the checked designs. The solid trace corresponds to the current design. Other designs are identified by name in the plot legend. Use the same procedure save and compare as many designs as you need. Restore Previously Saved Design Under some conditions, it is useful to restore the tuned parameter values from a Update Blocks previously saved design as the current design. For example, clicking writes the current parameter values to the Simulink model. If you decide to test a stored controller design in your full nonlinear model, you must first restore those stored values as the current design. 14-210 See Also To do so, click design. Retrieve. Select the stored design that you want to make the current See Also Related Examples • “Create Response Plots in Control System Tuner” on page 14-198 14-211 14 Control System Tuning Create and Configure slTuner Interface to Simulink Model This example shows how to create and configure an slTuner interface for a Simulink® model. The slTuner interface parameterizes blocks in your model that you designate as tunable and allows you to tune them using systune. The slTuner interface generates a linearization of your Simulink model, and also allows you to extract linearized system responses for analysis and validation of the tuned control system. For this example, create and configure an slTuner interface for tuning the Simulink model rct_helico, a multiloop controller for a rotorcraft. Open the model. open_system('rct_helico'); The control system consists of two feedback loops. The inner loop (static output feedback) provides stability augmentation and decoupling. The outer loop (PI controllers) provides the desired setpoint tracking performance. Suppose that you want to tune this model to meet the following control objectives: 14-212 Create and Configure slTuner Interface to Simulink Model • Track setpoint changes in theta, phi, and r with zero steady-state error, specified rise times, minimal overshoot, and minimal cross-coupling. • Limit the control bandwidth to guard against neglected high-frequency rotor dynamics and measurement noise. • Provide strong multivariable gain and phase margins (robustness to simultaneous gain/phase variations at the plant inputs and outputs). The systune command can jointly tune the controller blocks SOF and the PI controllers to meet these design requirements. The slTuner interface sets up this tuning task. Create the slTuner interface. ST0 = slTuner('rct_helico',{'PI1','PI2','PI3','SOF'}); This command initializes the slTuner interface with the three PI controllers and the SOF block designated as tunable. Each tunable block is automatically parameterized according to its type and initialized with its value in the Simulink model. To configure the slTuner interface, designate as analysis points any signal locations of relevance to your design requirements. First, add the outputs and reference inputs for the tracking requirements. addPoint(ST0,{'theta-ref','theta','phi-ref','phi','r-ref','r'}); When you create a TuningGoal.Tracking object that captures the tracking requirement, this object references the same signals. Configure the slTuner interface for the stability margin requirements. Designate as analysis points the plant inputs and outputs (control and measurement signals) where the stability margins are measured. addPoint(ST0,{'u','y'}); Display a summary of the slTuner interface configuration in the command window. ST0 slTuner tuning interface for "rct_helico": 4 Tuned blocks: (Read-only TunedBlocks property) -------------------------Block 1: rct_helico/PI1 14-213 14 Control System Tuning Block 2: rct_helico/PI2 Block 3: rct_helico/PI3 Block 4: rct_helico/SOF 8 Analysis points: -------------------------Point 1: Port 1 of rct_helico/theta-ref Point 2: Signal "theta", located at port 1 of rct_helico/Demux1 Point 3: Port 1 of rct_helico/phi-ref Point 4: Signal "phi", located at port 2 of rct_helico/Demux1 Point 5: Port 1 of rct_helico/r-ref Point 6: Signal "r", located at port 3 of rct_helico/Demux1 Point 7: Signal "u", located at port 1 of rct_helico/Mux3 Point 8: Signal "y", located at port 1 of rct_helico/Helicopter No permanent openings. Use the addOpening command to add new permanent openings. Properties with dot notation get/set access: Parameters : [] OperatingPoints : [] (model initial condition will be used.) BlockSubstitutions : [] Options : [1x1 linearize.SlTunerOptions] Ts : 0 In the command window, click on any highlighted signal to see its location in the Simulink model. In addition to specifying design requirements, you can use analysis points for extracting system responses. For example, extract and plot the step responses between the reference signals and 'theta', 'phi', and 'r'. T0 = getIOTransfer(ST0,{'theta-ref','phi-ref','r-ref'},{'theta','phi','r'}); stepplot(T0,1) 14-214 Create and Configure slTuner Interface to Simulink Model All the step responses are unstable, including the cross-couplings, because this model has not yet been tuned. After you tune the model, you can similarly use the designated analysis points to extract system responses for validating the tuned system. If you want to examine system responses at locations that are not needed to specify design requirements, add these locations to the slTuner interface as well. For example, plot the sensitivity function measured at the output of the block roll-off 2. addPoint(ST0,'dc') dcS0 = getSensitivity(ST0,'dc'); bodeplot(dcS0) 14-215 14 Control System Tuning Suppose you want to change the parameterization of tunable blocks in the slTuner interface. For example, suppose that after tuning the model, you want to test whether changing from PI to PID controllers yields improved results. Change the parameterization of the three PI controllers to PID controllers. PID0 PID1 PID2 PID3 = = = = pid(0,0.001,0.001,.01); tunablePID('C1',PID0); tunablePID('C2',PID0); tunablePID('C3',PID0); % initial value for PID controllers setBlockParam(ST0,'PI1',PID1,'PI2',PID2,'PI3',PID3); 14-216 See Also After you configure the slTuner interface to your Simulink model, you can create tuning goals and tune the model using systune or looptune. See Also addBlock | addPoint | getIOTransfer | getSensitivity | setBlockParam | slTuner Related Examples • “Mark Signals of Interest for Control System Analysis and Design” on page 2-89 • “Multiloop Control of a Helicopter” on page 18-217 • “Control of a Linear Electric Actuator” on page 18-99 14-217 14 Control System Tuning Stability Margins in Control System Tuning When you visualize stability-margin goals in Control System Tuner or with viewGoal, the margins are displayed as a function of frequency. For instance, the following plot shows a typical result of tuning a control system with systune or Control System Tuner when you use a tuning goal that constrains stability margins.. . You obtain this plot in one of the following ways: • Tuning in Control System Tuner using a “Margins Goal” on page 14-148 or “Quick Loop Tuning” on page 14-58. • Tuning at the command line using systune with TuningGoal.Margins. If S is the control system model or slTuner interface, and Req is a TuningGoal.Margins goal, obtain the stability-margin plot by entering: 14-218 Stability Margins in Control System Tuning viewGoal(Req,S) The plot shows that how much gain or phase variation a system can tolerate without going unstable can depend on the frequency of the perturbation. Gain and Phase Margins For SISO systems, the gain and phase margins at a frequency ω indicate how much the gain or phase of the open-loop response L(jω) can change without loss of stability. For example, a gain margin of 5dB at 2 rad/s indicates that closed-loop stability is maintained when the loop gain increases or decreases by as much as 5dB at this frequency. Gain and phase margins typically vary across frequencies. For MIMO systems, gain and phase margins are interpreted as follows: • Gain margin: Stability is preserved when the gain increases or decreases by up to the gain margin value in each channel of the feedback loop. • Phase margin: Stability is preserved when the phase increases or decreases by up to the phase margin value in each channel of the feedback loop. In MIMO systems, the gain or phase can change in all channels at once, and by a different amount in each channel. The Margins Goal and TuningGoal.Margins rely on the notion of disk margin for MIMO systems. (See “Algorithm” on page 14-222.) Like SISO stability margins, gain and phase margins in MIMO systems typically vary across frequency. Combined Gain and Phase Variations To assess robustness to changes in both gain and phase, use the following chart. 14-219 14 Control System Tuning For example, if the gain margin plot in Control System Tuner indicates a 10 dB margin at a particular frequency, then trace the contour starting at (Gain,Phase) = (10,0) to see how a given amount of phase variation reduces the allowable gain variation at that frequency. For example, if the phase can vary by 30 degrees than the gain can only vary by about 8.4 dB (red mark). Interpreting the Gain and Phase Margin Plot The stability-margin plot for Margins Goal or TuningGoal.Margins shows in shaded yellow the region where the target margins are not met. The plot displays the current gain and phase margins (computed using the current values of the tunable parameters in the control system) as a blue trace. 14-220 Stability Margins in Control System Tuning These gain and phase margin curves are obtained using an exact calculation involving μanalysis. For computational efficiency, however, the tuning algorithm uses an approximate calculation that can yield smaller margins in parts of the frequency range. To see the lower bound used by the tuner, right-click on the plot, and select Systems > Tuned Lower Bound. If there is a significant gap between the true margins and the tuner approximation, try increasing the D-scaling order. The default order is zero (static scaling). For tuning in Control System Tuner, set the D-scaling order in the Margins Goal dialog box. For command-line tuning, set this value using the ScalingOrder property of TuningGoal.Margins. 14-221 14 Control System Tuning Algorithm The gain and phase margin values are both derived from the disk margin. The disk margin measures the radius of a circular exclusion region centered near the critical point. (See “Stability Analysis Using Disk Margins” (Robust Control Toolbox).) This radius is a decreasing function of the scaled norm: min D diagonal D -1 ( I - L ( jw ) ) ( I + L ( jw ) ) -1 D . 2 Unlike the traditional gain and phase margins, the disk margins and associated gain and phase margins guarantee that the open-loop response L(jω) stays at a safe distance from the critical point at all frequencies. See Also TuningGoal.Margins | diskmargin More About 14-222 • “Loop Shape and Stability Margin Specifications” on page 18-42 • “Margins Goal” on page 14-148 • “Stability Analysis Using Disk Margins” (Robust Control Toolbox) Tune Control System at the Command Line Tune Control System at the Command Line After specifying your tuning goals using TuningGoal objects (see “Tuning Goals”), use systune to tune the parameters of your model. The systune command lets you designate one or more design goals as hard goals. This designation gives you a way to differentiate must-have goals from nice-to-have tuning goals.systune attempts to satisfy hard requirements by driving their associated cost functions below 1. Subject to that constraint, the software comes as close as possible to satisfying remaining (soft) requirements. For best results, make sure you can obtain a reasonable design with all goals treated as soft goals before attempting to enforce any goal as a hard constraint. Organize your TuningGoal objects into a vector of soft requirements and a vector of hard requirements. For example, suppose you have created a tracking requirement, a rejection requirement, and stability margin requirements at the plant inputs and outputs. The following commands tune the control system represented by T0, treating the stability margins as hard goals, the tracking and rejection requirements as soft goals. (T0 is either a genss model or an slTuner interface previously configured for tuning.) SoftReqs = [Rtrack,Rreject]; HardReqs = [RmargIn,RmargOut]; [T,fSoft,gHard] = systune(T0,SoftReqs,HardReqs); systune converts each tuning requirement into a normalized scalar value, f for the soft constraints and g for the hard constraints. The command adjusts the tunable parameters of T0 to minimize the f values, subject to the constraint that each g < 1. systune returns the vectors fSoft and gHard that contain the final normalized values for each tuning goal in SoftReqs and HardReqs. Use systuneOptions to configure additional options for the systune algorithm, such as the number of independent optimization runs, convergence tolerance, and output display options. See Also systune | systune (for slTuner) | systuneOptions 14-223 14 Control System Tuning More About • 14-224 “Interpret Numeric Tuning Results” on page 14-184 Speed Up Tuning with Parallel Computing Toolbox Software Speed Up Tuning with Parallel Computing Toolbox Software If you have the Parallel Computing Toolbox software installed, you can speed up the tuning of fixed-structure control systems. When you run multiple randomized optimization starts with systune, looptune, or hinfstruct, parallel computing speeds up tuning by distributing the optimization runs among workers. To distribute randomized optimization runs among workers: If Automatically create a parallel pool is not selected in your Parallel Computing Toolbox preferences (Parallel Computing Toolbox), manually start a parallel pool using parpool. For example: parpool; If Automatically create a parallel pool is selected in your preferences, you do not need to manually start a pool. Create a systuneOptions, looptuneOptions, or hinfstructOptions set that specifies multiple random starts. For example, the following options set specifies 20 random restarts to run in parallel for tuning with looptune: options = systuneOptions('RandomStart',20,'UseParallel',true); Setting UseParallel to true enables parallel processing by distributing the randomized starts among available workers in the parallel pool. Use the options set when you call the tuning command. For example, if you have already created a tunable control system model, CL0, and tunable controller, and tuning requirement vectors SoftReqs and HardReqs, the following command uses parallel computing to tune the control system of CL0 with systune. [CL,fSoft,gHard,info] = systune(CL0,SoftReq,Hardreq,options); To learn more about configuring a parallel pool, see the Parallel Computing Toolbox documentation. See Also parpool 14-225 14 Control System Tuning Related Examples • “Using Parallel Computing to Accelerate Tuning” on page 18-94 More About • 14-226 “Specify Your Parallel Preferences” (Parallel Computing Toolbox) Validate Tuned Control System Validate Tuned Control System When you tune a control system using systune or Control System Tuner, you must validate the results of tuning. The tuning results provide numeric and graphical indications of how well your tuning goals are satisfied. (See “Interpret Numeric Tuning Results” on page 14-184 and “Visualize Tuning Goals” on page 14-189.) Often, you want to examine other system responses using the tuned controller parameters. If you are tuning a Simulink model, you must also validate the tuned controller against the full nonlinear system. At the command line and in Control System Tuner, there are several tools to help you validate the tuned control system. Extract and Plot System Responses In addition to the system responses corresponding to your tuning goals (see “Visualize Tuning Goals” on page 14-189), you can evaluate the tuned system performance by plotting other system responses. For instance, evaluate reference tracking or overshoot performance by plotting the step response of transfer function from the reference input to the controlled output. Or, evaluate stability margins by examining an open-loop transfer function. You can extract any transfer function you need for analysis from the tuned model of your control system. Extract System Responses at the Command Line The tuning tools include analysis functions that let you extract responses from your tuned control system. For generalized state-space (genss) models, use: • getIOTransfer • getLoopTransfer • getSensitivity • getCompSensitivity For an slTuner interface, use: • getIOTransfer (for slTuner) • getLoopTransfer (for slTuner) • getSensitivity (for slTuner) 14-227 14 Control System Tuning • getCompSensitivity (for slTuner) In either case, the extracted responses are represented by state-space (ss) models. You can analyze these models using commands such as step, bode, sigma, or margin. For instance, suppose that you are tuning the control system of the example “Multiloop Control of a Helicopter” on page 18-217. You have created an slTuner interface ST0 for the Simulink model. You have also specified tuning goals TrackReq, MarginReq1, MarginReq2, and PoleReq. You tune the control system using systune. AllReqs = [TrackReq,MarginReq1,MarginReq2,PoleReq]; ST1 = systune(ST0,AllReqs); Final: Soft = 1.11, Hard = -Inf, Iterations = 128 Suppose also that ST0 has analysis points that include signals named theta-ref, theta, phi-ref, and phi. Use getIOTransfer to extract the tuned transfer functions from theta-ref and phi-ref to theta and phi. T1 = getIOTransfer(ST1,{'theta-ref','phi-ref'},{'theta','phi'}); step(T1,5) 14-228 Validate Tuned Control System The step plot shows that the extracted transfer function is the 2-input, 2-output response from the specified reference inputs to the specified outputs. For an example that shows how to extract responses from a tuned genss model, see “Extract Responses from Tuned MATLAB Model at the Command Line” on page 14-232. For additional examples, see “Validating Results” on page 18-54. System Responses in Control System Tuner For information about extracting and plotting system responses in Control System Tuner, see “Create Response Plots in Control System Tuner” on page 14-198. 14-229 14 Control System Tuning Validate Design in Simulink Model When you tune a Simulink model, the software evaluates tuning goals for a linearization of the model. Similarly, analysis commands such as getIOTransfer extract linearized system responses. Therefore, you must validate the tuned controller parameters by simulating the full nonlinear model with the tuned controller parameters, even if the tuned linear system meets all your design requirements. To do so, write the tuned parameter values to the model. Tip If you tune the Simulink model at an operating point other than the model initial condition, initialize the model at the same operating point before validating the tuned controller parameters. See “Simulate Simulink Model at Specific Operating Point” (Simulink Control Design). Write Parameters at the Command Line To write tuned block values from a tuned slTuner interface to the corresponding Simulink model, use the writeBlockValue command. For example, suppose ST1 is the tuned slTuner interface returned by systune. The following command writes the tuned parameters from ST1 to the associated Simulink model. writeBlockValue(ST1) Simulate the Simulink model to evaluate system performance with the tuned parameter values. Write Parameters in Control System Tuner To write tuned block parameters to a Simulink model, in the Control System tab, click Update Blocks. 14-230 See Also Control System Tuner transfers the current values of the tuned block parameters to the corresponding blocks in the Simulink model. Simulate the model to evaluate system performance using the tuned parameter values. To update Simulink model with parameter values from a previous design stored in Control System Tuner, click Retrieve and select the stored design that you want to make the current design. Then click Update Blocks. See Also Related Examples • “Extract Responses from Tuned MATLAB Model at the Command Line” on page 14232 • “Create Response Plots in Control System Tuner” on page 14-198 • “Visualize Tuning Goals” on page 14-189 14-231 14 Control System Tuning Extract Responses from Tuned MATLAB Model at the Command Line This example shows how to analyze responses of a tuned control system by using getIOTransfer to compute responses between various inputs and outputs of a closedloop model of the system. You can obtain other responses using similar functions such as getLoopTransfer and getSensitivity. Consider the following control system. r + - C1 + - C2 d2 d1 G2 G1 y X2 X1 Suppose you have used systune to tune a genss model of this control system. The result is a genss model, T, which contains tunable blocks representing the controller elements C1 and C2. The tuned model also contains AnalysisPoint blocks that represent the analysis-point locations, X1 and X2. Analyze the tuned system performance by examining various system responses extracted from T. For example, examine the response at the output, y, to a disturbance injected at the point d1. H1 = getIOTransfer(T,'X1','y'); H1 represents the closed-loop response of the control system to a disturbance injected at the implicit input associated with the AnalysisPoint block X1, which is the location of d1: 14-232 See Also H1 is a genss model that includes the tunable blocks of T. H1 allows you to validate the disturbance response of your tuned system. For example, you can use analysis commands such as bodeplot or stepplot to analyze H1. You can also use getValue to obtain the current value of H1, in which all the tunable blocks are evaluated to their current numeric values. Similarly, examine the response at the output to a disturbance injected at the point d2. H2 = getIOTransfer(T,'X2','y'); You can also generate a two-input, one-output model representing the response of the control system to simultaneous disturbances at both d1 and d2. To do so, provide getIOTransfer with a cell array that specifies the multiple input locations. H = getIOTransfer(T,{'X1','X2'},'y'); See Also AnalysisPoint | getCompSensitivity | getIOTransfer | getLoopTransfer | getSensitivity Related Examples • “Interpret Numeric Tuning Results” on page 14-184 14-233 15 Loop-Shaping Design • “Structure of Control System for Tuning With looptune” on page 15-2 • “Set Up Your Control System for Tuning with looptune” on page 15-4 • “Tune MIMO Control System for Specified Bandwidth” on page 15-6 • “Tuning Feedback Loops with LOOPTUNE” on page 15-13 • “Decoupling Controller for a Distillation Column” on page 15-20 • “Tuning of a Digital Motion Control System” on page 15-32 15 Loop-Shaping Design Structure of Control System for Tuning With looptune looptune tunes the feedback loop illustrated below to meet default requirements or requirements that you specify. G u y C C represents the controller and G represents the plant. The sensor outputsy (measurement signals) and actuator outputs u (control signals) define the boundary between plant and controller. The controller is the portion of your control system whose inputs are measurements, and whose outputs are control signals. Conversely, the plant is the remainder—the portion of your control system that receives control signals as inputs, and produces measurements as outputs. For example, in the control system of the following illustration, the controller C receives the measurement y, and the reference signal r. The controller produces the controls qL and qV as outputs. C r e + - pL D PIL pV PIV qL G y qV The controller C has a fixed internal structure. C includes a gain matrix D , the PI controllers PI_L and PI_V, and a summing junction. The looptune command tunes free 15-2 Structure of Control System for Tuning With looptune parameters of C such as the gains in D and the proportional and integral gains of PI_L and PI_V. You can also use looptune to co-tune free parameters in both C and G. 15-3 15 Loop-Shaping Design Set Up Your Control System for Tuning with looptune Set Up Your Control System for looptunein MATLAB To set up your control system in MATLAB for tuning with looptune: 1 Parameterize the tunable elements of your controller. You can use predefined structures such as tunablePID, tunableGain, and tunableTF. Or, you can create your own structure from elementary tunable parameters (realp). 2 Use model interconnection commands such as series and connect to build a tunable genss model representing the controller C0. 3 Create a Numeric LTI model on page 1-13 representing the plant G. For co-tuning the plant and controller, represent the plant as a tunable genss model. Set Up Your Control System for looptune in Simulink To set up your control system in Simulink for tuning with systune (requires Simulink Control Design software): 1 Use slTuner to create an interface to the Simulink model of your control system. When you create the interface, you specify which blocks to tune in your model. 2 Use addPoint to specify the control and measurement signals that define the boundaries between plant and controller. Use addOpening to mark optional loopopening or signal injection sites for specifying and assessing open-loop requirements. The slTuner interface automatically linearizes your Simulink model. The slTuner interface also automatically parametrizes the blocks that you specify as tunable blocks. For more information about this linearization, see the slTuner reference page and “How Tuned Simulink Blocks Are Parameterized” on page 14-36. See Also Related Examples 15-4 • “Tune MIMO Control System for Specified Bandwidth” on page 15-6 • “Tuning Feedback Loops with LOOPTUNE” on page 15-13 See Also More About • “Structure of Control System for Tuning With looptune” on page 15-2 15-5 15 Loop-Shaping Design Tune MIMO Control System for Specified Bandwidth This example shows how to tune the following control system to achieve a loop crossover frequency between 0.1 and 1 rad/s, using looptune. The plant, G, is a two-input, two-output model (y is a two-element vector signal). For this example, the transfer function of G is given by: This sample plant is based on the distillation column described in more detail in the example “Decoupling Controller for a Distillation Column” on page 15-20. To tune this control system, you first create a numeric model of the plant. Then you create tunable models of the controller elements and interconnect them to build a controller model. Then you use looptune to tune the free parameters of the controller model. Finally, examine the performance of the tuned system to confirm that the tuned controller yields desirable performance. Create a model of the plant. s = tf('s'); G = 1/(75*s+1)*[87.8 -86.4; 108.2 -109.6]; G.InputName = {'qL','qV'}; G.OutputName = 'y'; When you tune the control system, looptune uses the channel names G.InputName and G.OutputName to interconnect the plant and controller. Therefore, assign these channel names to match the illustration. When you set G.OutputName = 'y', the G.OutputName is automatically expanded to {'y(1)';'y(2)'}. This expansion occurs because G is a two-output system. Represent the components of the controller. 15-6 Tune MIMO Control System for Specified Bandwidth D = tunableGain('Decoupler',eye(2)); D.InputName = 'e'; D.OutputName = {'pL','pV'}; PI_L = tunablePID('PI_L','pi'); PI_L.InputName = 'pL'; PI_L.OutputName = 'qL'; PI_V = tunablePID('PI_V','pi'); PI_V.InputName = 'pV'; PI_V.OutputName = 'qV'; sum1 = sumblk('e = r - y',2); The control system includes several tunable control elements. PI_L and PI_V are tunable PI controllers. These elements represented by tunablePID models. The fixed control structure also includes a decoupling gain matrix D, represented by a tunable tunableGain model. When the control system is tuned, D ensures that each output of G tracks the corresponding reference signal r with minimal crosstalk. Assigning InputName and OutputName values to these control elements allows you to interconnect them to create a tunable model of the entire controller C as shown. When you tune the control system, looptune uses these channel names to interconnect C and G. The controller C also includes the summing junction sum1. This a two-channel summing junction, because r and y are vector-valued signals of dimension 2. Connect the controller components. C0 = connect(PI_L,PI_V,D,sum1,{'r','y'},{'qL','qV'}); C0 is a tunable genss model that represents the entire controller structure. C0 stores the tunable controller parameters and contains the initial values of those parameters. Tune the control system. The inputs to looptune are G and C0, the plant and initial controller models that you created. The input wc = [0.1,1] sets the target range for the loop bandwidth. This 15-7 15 Loop-Shaping Design input specifies that the crossover frequency of each loop in the tuned system fall between 0.1 and 1 rad/min. wc = [0.1,1]; [G,C,gam,Info] = looptune(G,C0,wc); Final: Peak gain = 0.973, Iterations = 19 Achieved target gain value TargetGain=1. The displayed Peak Gain = 0.949 indicates that looptune has found parameter values that achieve the target loop bandwidth. looptune displays the final peak gain value of the optimization run, which is also the output gam. If gam is less than 1, all tuning requirements are satisfied. A value greater than 1 indicates failure to meet some requirement. If gam exceeds 1, you can increase the target bandwidth range or relax another tuning requirement. looptune also returns the tuned controller model C. This model is the tuned version of C0. It contains the PI coefficients and the decoupling matrix gain values that yield the optimized peak gain value. Display the tuned controller parameters. showTunable(C) Decoupler = D = y1 y2 u1 2.139 -1.583 u2 -1.303 1.295 Name: Decoupler Static gain. ----------------------------------PI_L = 1 Kp + Ki * --s with Kp = 1.5, Ki = 0.0427 Name: PI_L Continuous-time PI controller in parallel form. 15-8 Tune MIMO Control System for Specified Bandwidth ----------------------------------PI_V = 1 Kp + Ki * --s with Kp = -1.74, Ki = -0.0532 Name: PI_V Continuous-time PI controller in parallel form. Check the time-domain response for the control system with the tuned coefficients. To produce a plot, construct a closed-loop model of the tuned control system. Plot the step response from reference to output. T = connect(G,C,'r','y'); step(T) 15-9 15 Loop-Shaping Design The decoupling matrix in the controller permits each channel of the two-channel output signal y to track the corresponding channel of the reference signal r, with minimal crosstalk. From the plot, you can how well this requirement is achieved when you tune the control system for bandwidth alone. If the crosstalk still exceeds your design requirements, you can use a TuningGoal.Gain requirement object to impose further restrictions on tuning. Examine the frequency-domain response of the tuned result as an alternative method for validating the tuned controller. figure('Position',[100,100,520,1000]) loopview(G,C,Info) 15-10 Tune MIMO Control System for Specified Bandwidth 15-11 15 Loop-Shaping Design The first plot shows that the open-loop gain crossovers fall within the specified interval [0.1,1]. This plot also includes the maximum and tuned values of the sensitivity function and complementary sensitivity . The second and third plots show that the MIMO stability margins of the tuned system (blue curve) do not exceed the upper limit (yellow curve). See Also Related Examples • “Decoupling Controller for a Distillation Column” on page 15-20 More About • 15-12 “Structure of Control System for Tuning With looptune” on page 15-2 Tuning Feedback Loops with LOOPTUNE Tuning Feedback Loops with LOOPTUNE This example shows the basic workflow of tuning feedback loops with the looptune command. looptune is similar to systune and meant to facilitate loop shaping design by automatically generating the tuning requirements. Engine Speed Control This example uses a simple engine speed control application as illustration. The control system consists of a single PID loop and the PID controller gains must be tuned to adequately respond to step changes in the desired speed. Specifically, we want the response to settle in less than 5 seconds with little or no overshoot. Figure 1: Engine Speed Control Loop We use the following fourth-order model of the engine dynamics. load rctExamples Engine bode(Engine), grid 15-13 15 Loop-Shaping Design Specifying the Tunable Elements We need to tune the four PID gains to achieve the desired performance. Use the tunablePID class to parameterize the PID controller. PID0 = tunablePID('SpeedController','pid') PID0 = Parametric continuous-time PID controller "SpeedController" with formula: 1 s Kp + Ki * --- + Kd * -------s Tf*s+1 15-14 Tuning Feedback Loops with LOOPTUNE and tunable parameters Kp, Ki, Kd, Tf. Type "pid(PID0)" to see the current value and "get(PID0)" to see all properties. Building a Tunable Model of the Feedback Loop looptune tunes the generic SISO or MIMO feedback loop of Figure 2. This feedback loop models the interaction between the plant and the controller. Note that this is a positive feedback interconnection. Figure 2: Generic Feedback Loop For the speed control loop, the plant of the PID and the prefilter is the engine model and the controller consists . Figure 3: Feedback Loop for Engine Speed Control To use looptune, create models for and in Figure 3. Assign names to the inputs and outputs of each model to specify the feedback paths between plant and controller. 15-15 15 Loop-Shaping Design Note that the controller measurement "speed". F = tf(10,[1 10]); has two inputs: the speed reference "r" and the speed % prefilter G = Engine; G.InputName = 'throttle'; G.OutputName = 'speed'; C0 = PID0 * [F , -1]; C0.InputName = {'r','speed'}; C0.OutputName = 'throttle'; Here C0 is a generalized state-space model (genss) that depends on the tunable PID block PID0. Tuning the Controller Parameters You can now use looptune to tune the PID gains subject to a simple control bandwidth requirement. To achieve the 5-second settling time, the gain crossover frequency of the open-loop response should be approximately 1 rad/s. Given this basic requirement, looptune automatically shapes the open-loop response to provide integral action, highfrequency roll-off, and adequate stability margins. Note that you could specify additional requirements to further constrain the design, see "Decoupling Controller for a Distillation Column" for an example. wc = 1; % target gain crossover frequency [~,C,~,Info] = looptune(G,C0,wc); Final: Peak gain = 0.999, Iterations = 6 Achieved target gain value TargetGain=1. The final value is less than 1, indicating that the desired bandwidth was achieved with adequate roll-off and stability margins. looptune returns the tuned controller C. Use getBlockValue to retrieve the tuned value of the PID block. PIDT = getBlockValue(C,'SpeedController') PIDT = 1 s Kp + Ki * --- + Kd * -------s Tf*s+1 15-16 Tuning Feedback Loops with LOOPTUNE with Kp = 0.000638, Ki = 0.00254, Kd = -0.000272, Tf = 0.999 Name: SpeedController Continuous-time PIDF controller in parallel form. Validating Results Use loopview to validate the design and visualize the loop shaping requirements implicitly enforced by looptune. clf, loopview(G,C,Info) 15-17 15 Loop-Shaping Design Next plot the closed-loop response to a step command in engine speed. The tuned response satisfies our requirements. T = connect(G,C,'r','speed'); clf, step(T) See Also looptune 15-18 % closed-loop transfer from r to speed See Also More About • “Decoupling Controller for a Distillation Column” on page 15-20 15-19 15 Loop-Shaping Design Decoupling Controller for a Distillation Column This example shows how to use looptune to decouple the two main feedback loops in a distillation column. Distillation Column Model This example uses a simple model of the distillation column shown below. Figure 1: Distillation Column In the so-called LV configuration, the controlled variables are the concentrations yD and yB of the chemicals D (tops) and B (bottoms), and the manipulated variables are the reflux L and boilup V. This process exhibits strong coupling and large variations in steady-state gain for some combinations of L and V. For more details, see Skogestad and Postlethwaite, Multivariable Feedback Control. The plant is modeled as a first-order transfer function with inputs L,V and outputs yD,yB: 15-20 Decoupling Controller for a Distillation Column The unit of time is minutes (all plots are in minutes, not seconds). s = tf('s'); G = [87.8 -86.4 ; 108.2 -109.6]/(75*s+1); G.InputName = {'L','V'}; G.OutputName = {'yD','yB'}; Control Architecture The control objectives are as follows: • Independent control of the tops and bottoms concentrations by ensuring that a change in the tops setpoint Dsp has little impact on the bottoms concentration B and vice versa • Response time of about 4 minutes with less than 15% overshoot • Fast rejection of input disturbances affecting the effective reflux L and boilup V To achieve these objectives we use the control architecture shown below. This architecture consists of a static decoupling matrix DM in series with two PI controllers for the reflux L and boilup V. open_system('rct_distillation') 15-21 15 Loop-Shaping Design Controller Tuning in Simulink with LOOPTUNE The looptune command provides a quick way to tune MIMO feedback loops. When the control system is modeled in Simulink, you just specify the tuned blocks, the control and measurement signals, and the desired bandwidth, and looptune automatically sets up the problem and tunes the controller parameters. looptune shapes the open-loop response to provide integral action, roll-off, and adequate MIMO stability margins. Use the slTuner interface to specify the tuned blocks, the controller I/Os, and signals of interest for closed-loop validation. ST0 = slTuner('rct_distillation',{'PI_L','PI_V','DM'}); % Signals of interest addPoint(ST0,{'r','dL','dV','L','V','y'}) Set the control bandwidth by specifying the gain crossover frequency for the open-loop response. For a response time of 4 minutes, the crossover frequency should be approximately 2/4 = 0.5 rad/min. wc = 0.5; Use TuningGoal objects to specify the remaining control objectives. The response to a step command should have less than 15% overshoot. The response to a step disturbance at the plant input should be well damped, settle in less than 20 minutes, and not exceed 4 in amplitude. OS = TuningGoal.Overshoot('r','y',15); DR = TuningGoal.StepRejection({'dL','dV'},'y',4,20); Next use looptune to tune the controller blocks PI_L, PI_V, and DM subject to the disturbance rejection requirement. Controls = {'L','V'}; Measurements = 'y'; [ST,gam,Info] = looptune(ST0,Controls,Measurements,wc,OS,DR); Final: Peak gain = 0.996, Iterations = 70 Achieved target gain value TargetGain=1. The final value is near 1 which indicates that all requirements were met. Use loopview to check the resulting design. The responses should stay outside the shaded areas. 15-22 Decoupling Controller for a Distillation Column figure('Position',[0,0,1000,1200]) loopview(ST,Info) 15-23 15 Loop-Shaping Design Use getIOTransfer to access and plot the closed-loop responses from reference and disturbance to the tops and bottoms concentrations. The tuned responses show a good compromise between tracking and disturbance rejection. figure Ttrack = getIOTransfer(ST,'r','y'); step(Ttrack,40), grid, title('Setpoint tracking') Treject = getIOTransfer(ST,{'dV','dL'},'y'); step(Treject,40), grid, title('Disturbance rejection') 15-24 Decoupling Controller for a Distillation Column Comparing the open- and closed-loop disturbance rejection characteristics in the frequency domain shows a clear improvement inside the control bandwidth. clf, sigma(G,Treject), grid title('Principal gains from input disturbances to outputs') legend('Open-loop','Closed-loop') 15-25 15 Loop-Shaping Design Adding Constraints on the Tuned Variables Inspection of the controller obtained above shows that the second PI controller has negative gains. getBlockValue(ST,'PI_V') ans = 1 Kp + Ki * --s 15-26 Decoupling Controller for a Distillation Column with Kp = -3.31, Ki = -0.439 Name: PI_V Continuous-time PI controller in parallel form. This is due to the negative signs in the second input channels of the plant . In addition, the tunable elements are over-parameterized because multiplying DM by two and dividing the PI gains by two does not change the overall controller. To address these issues, fix the (1,1) entry of DM to 1 and the (2,2) entry to -1. DM = getBlockParam(ST0,'DM'); DM.Gain.Value = diag([1 -1]); DM.Gain.Free = [false true;true false]; setBlockParam(ST0,'DM',DM) Re-tune the controller for the reduced set of tunable parameters. [ST,gam,Info] = looptune(ST0,Controls,Measurements,wc,OS,DR); Final: Peak gain = 0.995, Iterations = 66 Achieved target gain value TargetGain=1. The step responses look similar but the values of DM and the PI gains are more suitable for implementation. figure('Position',[0,0,700,350]) subplot(121) Ttrack = getIOTransfer(ST,'r','y'); step(Ttrack,40), grid, title('Setpoint tracking') subplot(122) Treject = getIOTransfer(ST,{'dV','dL'},'y'); step(Treject,40), grid, title('Disturbance rejection') 15-27 15 Loop-Shaping Design showTunable(ST) Block 1: rct_distillation/PI_L = 1 Kp + Ki * --s with Kp = 17.2, Ki = 2.3 Name: PI_L Continuous-time PI controller in parallel form. ----------------------------------Block 2: rct_distillation/PI_V = 1 Kp + Ki * --s 15-28 Decoupling Controller for a Distillation Column with Kp = 13.8, Ki = 1.85 Name: PI_V Continuous-time PI controller in parallel form. ----------------------------------Block 3: rct_distillation/DM = D = y1 y2 u1 1 1.225 u2 -0.7913 -1 Name: DM Static gain. Equivalent Workflow in MATLAB If you do not have a Simulink model of the control system, you can use LTI objects and Control Design blocks to create a MATLAB representation of the following block diagram. Figure 2: Block Diagram of Control System First parameterize the tunable elements using Control Design blocks. Use the tunableGain object to parameterize DM and fix DM(1,1)=1 and DM(2,2)=-1. This creates a 2x2 static gain with the off-diagonal entries as tunable parameters. DM = tunableGain('Decoupler',diag([1 -1])); DM.Gain.Free = [false true;true false]; 15-29 15 Loop-Shaping Design Similarly, use the tunablePID object to parameterize the two PI controllers: PI_L = tunablePID('PI_L','pi'); PI_V = tunablePID('PI_V','pi'); Next construct a model C0 of the controller in Figure 2. C0 = blkdiag(PI_L,PI_V) * DM * [eye(2) -eye(2)]; % Note: I/O names should be consistent with those of G C0.InputName = {'Dsp','Bsp','yD','yB'}; C0.OutputName = {'L','V'}; Now tune the controller parameters with looptune as done previously. % Crossover frequency wc = 0.5; % Overshoot and disturbance rejection requirements OS = TuningGoal.Overshoot({'Dsp','Bsp'},{'yD','yB'},15); DR = TuningGoal.StepRejection({'L','V'},{'yD','yB'},4,20); % Tune controller gains [~,C] = looptune(G,C0,wc,OS,DR); Final: Peak gain = 1, Iterations = 70 Achieved target gain value TargetGain=1. To validate the design, close the loop with the tuned compensator C and simulate the step responses for setpoint tracking and disturbance rejection. Tcl = connect(G,C,{'Dsp','Bsp','L','V'},{'yD','yB'}); figure('Position',[0,0,700,350]) subplot(121) Ttrack = Tcl(:,[1 2]); step(Ttrack,40), grid, title('Setpoint tracking') subplot(122) Treject = Tcl(:,[3 4]); Treject.InputName = {'dL','dV'}; step(Treject,40), grid, title('Disturbance rejection') 15-30 See Also The results are similar to those obtained in Simulink. See Also looptune | looptune (slTuner) More About • “Tuning of a Digital Motion Control System” on page 15-32 15-31 15 Loop-Shaping Design Tuning of a Digital Motion Control System This example shows how to use Control System Toolbox™ to tune a digital motion control system. Motion Control System The motion system under consideration is shown below. Figure 1: Digital motion control hardware This device could be part of some production machine and is intended to move some load (a gripper, a tool, a nozzle, or anything else that you can imagine) from one angular position to another and back again. This task is part of the "production cycle" that has to be completed to create each product or batch of products. 15-32 Tuning of a Digital Motion Control System The digital controller must be tuned to maximize the production speed of the machine without compromising accuracy and product quality. To do this, we first model the control system in Simulink using a 4th-order model of the inertia and flexible shaft: open_system('rct_dmc') The "Tunable Digital Controller" consists of a gain in series with a lead/lag controller. 15-33 15 Loop-Shaping Design Figure 2: Digital controller Tuning is complicated by the presence of a flexible mode near 350 rad/s in the plant: G = linearize('rct_dmc','rct_dmc/Plant Model'); bode(G,{10,1e4}), grid 15-34 Tuning of a Digital Motion Control System Compensator Tuning We are seeking a 0.5 second response time to a step command in angular position with minimum overshoot. This corresponds to a target bandwidth of approximately 5 rad/s. The looptune command offers a convenient way to tune fixed-structure compensators like the one in this application. To use looptune, first instantiate the slTuner interface to automatically acquire the control structure from Simulink. Note that the signals of interest are already marked as Linear Analysis Points in the Simulink model. ST0 = slTuner('rct_dmc',{'Gain','Leadlag'}); Next use looptune to tune the compensator parameters for the target gain crossover frequency of 5 rad/s: 15-35 15 Loop-Shaping Design Measurement = 'Measured Position'; % controller input Control = 'Leadlag'; % controller output ST1 = looptune(ST0,Control,Measurement,5); Final: Peak gain = 0.948, Iterations = 26 Achieved target gain value TargetGain=1. A final value below or near 1 indicates success. Inspect the tuned values of the gain and lead/lag filter: showTunable(ST1) Block 1: rct_dmc/Tunable Digital Controller/Gain = D = y1 u1 1.556e-05 Name: Gain Static gain. ----------------------------------Block 2: rct_dmc/Tunable Digital Controller/Leadlag = 4.968 s + 8.291 --------------s + 15 Name: Leadlag Continuous-time transfer function. Design Validation To validate the design, use the slTuner interface to quickly access the closed-loop transfer functions of interest and compare the responses before and after tuning. T0 = getIOTransfer(ST0,'Reference','Measured Position'); T1 = getIOTransfer(ST1,'Reference','Measured Position'); step(T0,T1), grid legend('Original','Tuned') 15-36 Tuning of a Digital Motion Control System The tuned response has significantly less overshoot and satisfies the response time requirement. However these simulations are obtained using a continuous-time lead/lag compensator (looptune operates in continuous time) so we need to further validate the design in Simulink using a digital implementation of the lead/lag compensator. Use writeBlockValue to apply the tuned values to the Simulink model and automatically discretize the lead/lag compensator to the rate specified in Simulink. writeBlockValue(ST1) You can now simulate the response of the continuous-time plant with the digital controller: sim('rct_dmc'); t = yout.time; % angular position logged in "yout" variable 15-37 15 Loop-Shaping Design y = yout.signals.values; step(T1), hold, plot(t,y,'r--') legend('Continuous','Hybrid (Simulink)') Current plot held The simulations closely match and the coefficients of the digital lead/lag can be read from the "Leadlag" block in Simulink. Tuning an Additional Notch Filter Next try to increase the control bandwidth from 5 to 50 rad/s. Because of the plant resonance near 350 rad/s, the lead/lag compensator is no longer sufficient to get 15-38 Tuning of a Digital Motion Control System adequate stability margins and small overshoot. One remedy is to add a notch filter as shown in Figure 3. Figure 3: Digital Controller with Notch Filter To tune this modified control architecture, create an slTuner instance with the three tunable blocks. ST0 = slTuner('rct_dmcNotch',{'Gain','Leadlag','Notch'}); By default the "Notch" block is parameterized as any second-order transfer function. To retain the notch structure specify the coefficients as real parameters and create a parametric model N of the transfer function shown above: wn = realp('wn',300); zeta1 = realp('zeta1',1); zeta2 = realp('zeta2',1); 15-39 15 Loop-Shaping Design zeta1.Minimum = 0; zeta1.Maximum = 1; % 0 <= zeta1 <= 1 zeta2.Minimum = 0; zeta2.Maximum = 1; % 0 <= zeta2 <= 1 N = tf([1 2*zeta1*wn wn^2],[1 2*zeta2*wn wn^2]); % tunable notch filter Then associate this parametric notch model with the "Notch" block in the Simulink model. Because the control system is tuned in the continuous time, you can use a continuoustime parameterization of the notch filter even though the "Notch" block itself is discrete. setBlockParam(ST0,'Notch',N); Next use looptune to jointly tune the "Gain", "Leadlag", and "Notch" blocks with a 50 rad/s target crossover frequency. To eliminate residual oscillations from the plant resonance, specify a target loop shape with a -40 dB/decade roll-off past 50 rad/s. % Specify target loop shape with a few frequency points Freqs = [5 50 500]; Gains = [10 1 0.01]; TLS = TuningGoal.LoopShape('Notch',frd(Gains,Freqs)); Measurement = 'Measured Position'; % controller input Control = 'Notch'; % controller output ST2 = looptune(ST0,Control,Measurement,TLS); Final: Peak gain = 1.05, Iterations = 61 The final gain is close to 1, indicating that all requirements are met. Compare the closedloop step response with the previous designs. T2 = getIOTransfer(ST2,'Reference','Measured Position'); clf step(T0,T1,T2,1.5), grid legend('Original','Lead/lag','Lead/lag + notch') 15-40 Tuning of a Digital Motion Control System To verify that the notch filter performs as expected, evaluate the total compensator C and the open-loop response L and compare the Bode responses of G, C, L: % Get tuned block values (in the order blocks are listed in ST2.TunedBlocks) [g,LL,N] = getBlockValue(ST2,'Gain','Leadlag','Notch'); C = N * LL * g; L = getLoopTransfer(ST2,'Notch',-1); bode(G,C,L,{1e1,1e3}), grid legend('G','C','L') 15-41 15 Loop-Shaping Design This Bode plot confirms that the plant resonance has been correctly "notched out." Discretizing the Notch Filter Again use writeBlockValue to discretize the tuned lead/lag and notch filters and write their values back to Simulink. Compare the MATLAB and Simulink responses: writeBlockValue(ST2) sim('rct_dmcNotch'); t = yout.time; y = yout.signals.values; step(T2), hold, plot(t,y,'r--') legend('Continuous','Hybrid (Simulink)') 15-42 Tuning of a Digital Motion Control System Current plot held The Simulink response exhibits small residual oscillations. The notch filter discretization is the likely culprit because the notch frequency is close to the Nyquist frequency pi/ 0.002=1570 rad/s. By default the notch is discretized using the ZOH method. Compare this with the Tustin method prewarped at the notch frequency: wn = damp(N); Ts = 0.002; % natural frequency of the notch filter % sample time of discrete notch filter Nd1 = c2d(N,Ts,'zoh'); Nd2 = c2d(N,Ts,'tustin',c2dOptions('PrewarpFrequency',wn(1))); clf, bode(N,Nd1,Nd2) 15-43 15 Loop-Shaping Design legend('Continuous','Discretized with ZOH','Discretized with Tustin',... 'Location','NorthWest') The ZOH method has significant distortion and prewarped Tustin should be used instead. To do this, specify the desired rate conversion method for the notch filter block: setBlockRateConversion(ST2,'Notch','tustin',wn(1)) writeBlockValue(ST2) writeBlockValue now uses Tustin prewarped at the notch frequency to discretize the notch filter and write it back to Simulink. Verify that this gets rid of the oscillations. sim('rct_dmcNotch'); t = yout.time; 15-44 Tuning of a Digital Motion Control System y = yout.signals.values; step(T2), hold, plot(t,y,'r--') legend('Continuous','Hybrid (Simulink)') Current plot held Direct Discrete-Time Tuning Alternatively, you can tune the controller directly in discrete time to avoid discretization issues with the notch filter. To do this, specify that the Simulink model should be linearized and tuned at the controller sample time of 0.002 seconds: ST0 = slTuner('rct_dmcNotch',{'Gain','Leadlag','Notch'}); ST0.Ts = 0.002; 15-45 15 Loop-Shaping Design To prevent high-gain control and saturations, add a requirement that limits the gain from reference to control signal (output of Notch block). GL = TuningGoal.Gain('Reference','Notch',0.01); Now retune the controller at the specified sampling rate and verify the tuned open- and closed-loop responses. ST2 = looptune(ST0,Control,Measurement,TLS,GL); % Closed-loop responses T2 = getIOTransfer(ST2,'Reference','Measured Position'); clf step(T0,T1,T2,1.5), grid legend('Original','Lead/lag','Lead/lag + notch') Final: Peak gain = 1.04, Iterations = 33 15-46 Tuning of a Digital Motion Control System % Open-loop responses [g,LL,N] = getBlockValue(ST2,'Gain','Leadlag','Notch'); C = N * LL * g; L = getLoopTransfer(ST2,'Notch',-1); bode(G,C,L,{1e1,2e3}), grid legend('G','C','L') 15-47 15 Loop-Shaping Design The results are similar to those obtained when tuning the controller in continuous time. Now validate the digital controller against the continuous-time plant in Simulink. writeBlockValue(ST2) sim('rct_dmcNotch'); t = yout.time; y = yout.signals.values; step(T2), hold, plot(t,y,'r--') legend('Discrete','Hybrid (Simulink)') Current plot held 15-48 See Also This time, the hybrid response closely matches its discrete-time approximation and no further adjustment of the notch filter is required. See Also looptune (slTuner) More About • “Tuning Feedback Loops with LOOPTUNE” on page 15-13 15-49 16 Gain-Scheduled Controllers 16 Gain-Scheduled Controllers Gain Scheduling Basics Gain scheduling is an approach to control of nonlinear systems using a family of linear controllers, each providing satisfactory control for a different operating point of the system. Gain-scheduled control is typically implemented using a controller whose gains are automatically adjusted as a function of scheduling variables that describe the current operating point. Such variables can include time, external operating conditions, or system states such as orientation or velocity. Gain-scheduled control systems are often designed by choosing a small set of operating points, the design points, and designing a suitable linear controller for each point. In operation, the system switches or interpolates between these controllers according to the current values of the scheduling variables. Gain scheduling is most suitable when the scheduling variables are external parameters that vary slowly compared to the control bandwidth, such as the ambient temperature of a chemical reaction or the speed of a cruising aircraft. Gain scheduling is most challenging when the scheduling variables depend on fast-varying states of the system. Because local linear performance near operating points is no guarantee of global performance in nonlinear systems, extensive simulation-based validation is required. See [1] for an overview of gain scheduling and its challenges. To design a gain-scheduled control system, you need: • An operating range, defined as a set of ranges within which the values of relevant system parameters remain during operation. For instance, if your system is a cruising aircraft, then the operating range might be an incidence angle between –20° and 20° and airspeed in the range 200-250 m/s. • Some measurable variables that indicate where in the operating range the system is at a given time. These signals are the scheduling variables. For the aircraft system, the scheduling variables might be the incidence angle and the airspeed. • A gain schedule, which comprises the formulas or data tables that return the appropriate controller gains for given values of the scheduling variables. For the aircraft system, the gain schedule gives appropriate controller gains for any combination of incidence angle and airspeed within the operating range. Gain Scheduling in Simulink Control System Toolbox provides blocks that help you model gain-scheduled control systems in Simulink. These blocks let you implement common control-system elements 16-2 See Also with variable parameters. For instance, the Varying PID Controller block accepts PID gains as inputs. In your model, you use blocks such as n-D Lookup Table or MATLAB Function blocks to implement the gain schedule. For more information and examples, see “Model Gain-Scheduled Control Systems in Simulink” on page 16-4. Tune Gain Schedules If you have Simulink Control Design, you can use systune to tune gain schedules to achieve a control system that meets performance objectives across the entire operating range. For more information, see “Tune Gain Schedules in Simulink” on page 16-15. References [1] Rugh, W.J., and J.S. Shamma, “Research on Gain Scheduling”, Automatica, 36 (2000), pp. 1401-1425. See Also More About • “Model Gain-Scheduled Control Systems in Simulink” on page 16-4 • “Tune Gain Schedules in Simulink” on page 16-15 16-3 16 Gain-Scheduled Controllers Model Gain-Scheduled Control Systems in Simulink In Simulink, you can model gain-scheduled control systems in which controller gains or coefficients depend on scheduling variables such as time, operating conditions, or model parameters. The library of linear parameter-varying blocks in Control System Toolbox lets you implement common control-system elements with variable gains. Use blocks such as lookup tables or MATLAB Function blocks to implement the gain schedule, which gives the dependence of these gains on the scheduling variables. To model a gain-scheduled control system in Simulink: 1 Identify the scheduling variables and the signals that represent them in your model. For instance, if your system is a cruising aircraft, then the scheduling variables might be the incidence angle and the airspeed of the aircraft. 2 Use a lookup table block or a MATLAB Function block to implement a gain or coefficient that depends on the scheduling variables. If you do not have lookup table values or MATLAB expressions for gain schedules that meet your performance requirements, you can use systune to tune them. See “Tune Gain Schedules in Simulink” on page 16-15. 3 Replace ordinary control elements with gain-scheduled elements. For instance, instead of a fixed-coefficient PID controller, use a Varying PID Controller block, in which the gain schedules determine the PID gains. 4 Add scheduling logic and safeguards to your model as needed. Model Scheduled Gains A gain schedule converts the current values of the scheduling variables into controller gains. There are several ways to implement a gain schedule in Simulink. Available blocks for implementing lookup tables include: • Lookup tables — A lookup table is a list of breakpoints and corresponding gain values. When the scheduling variables fall between breakpoints, the lookup table interpolates between the corresponding gains. Use the following blocks to implement gain schedules as lookup tables. • 1-D Lookup Table, 2-D Lookup Table, n-D Lookup Table — For a scalar gain that depends on one, two, or more scheduling variables. • Matrix Interpolation — For a matrix-valued gain that depends on one, two, or three scheduling variables. (This block is in the Simulink Extras library.) 16-4 Model Gain-Scheduled Control Systems in Simulink • MATLAB Function block — When you have a functional expression relating the gains to the scheduling variables, use a MATLAB Function block. If the expression is a smooth function, using a MATLAB function can result in smoother gain variations than a lookup table. Also, if you use a code-generation product such as Simulink Coderto implement the controller in hardware, a MATLAB function can result in a more memory-efficient implementation than a lookup table. If you have Simulink Control Design, you can use systune to tune gain schedules implement as either lookup tables or MATLAB functions. See “Tune Gain Schedules in Simulink” on page 16-15. Scheduled Gain in Controller As an example, The model rct_CSTR includes a PI controller and a lead compensator in which the controller gains are implemented as lookup tables using 1-D Lookup Table blocks. Open that model and examine the controllers. open_system(fullfile(matlabroot,'examples','controls_id','rct_CSTR.slx')) 16-5 16 Gain-Scheduled Controllers Both the Concentration controller and Temperature controller blocks take the CSTR plant output, Cr, as an input. This value is both the controlled variable of the system and the scheduling variable on which the controller action depends. Double-click the Concentration controller block. 16-6 Model Gain-Scheduled Control Systems in Simulink This block is a PI controller in which the proportional gain Kp and integrator gain Ki are determined by feeding the scheduling parameter Cr into a 1-D Lookup Table block. Similarly, the Temperature controller block contains three gains implemented as lookup tables. Gain-Scheduled Equivalents for Commonly Used Control Elements Use the Linear Parameter Varying block library of Control System Toolbox to implement common control elements with variable parameters or coefficients. These blocks provide common elements in which the gains or parameters are available as external inputs. The following table lists some applications of these blocks. 16-7 16 Gain-Scheduled Controllers Block Application • Varying Lowpass Filter Use these blocks to implement a Butterworth lowpass filter in which the cutoff frequency varies with scheduling variables. • Discrete Varying Lowpass • Varying Notch Filter • Discrete Varying Notch • Varying PID Controller • Discrete Varying PID • Varying 2DOF PID • Discrete Varying 2DOF PID • Varying Transfer Function • Discrete Varying Transfer Function • Varying State Space • Discrete Varying State Space • Varying Observer Form • Discrete Varying Observer Form Use these blocks to implement a notch filter in which the notch frequency, width, and depth vary with scheduling variables. These blocks are preconfigured versions of the PID Controller and PID Controller (2DOF) blocks. Use them to implement PID controllers in which the PID gains vary with scheduling variables. Use these blocks to implement a transfer function of any order in which the polynomial coefficients of the numerator and denominator vary with scheduling variables. Use these blocks to implement a statespace controller in which the A, B, C, and D matrices vary with the scheduling variables. Use these blocks to implement a gainscheduled observer-form state-space controller, such as an LQG controller. In such a controller, the A, B, C, D matrices and the state-feedback and state-observer gain matrices vary with the scheduling variables. Gain-Scheduled Notch Filter For example, the subsystem in the following illustration uses a Varying Notch Filter block to implement a filter whose notch frequency varies as a function of two scheduling variables. The relationship between the notch frequency and the scheduling variables is implemented in a MATLAB function. 16-8 Model Gain-Scheduled Control Systems in Simulink Gain-Scheduled PI Controller As another example, the following subsystem is a gain-scheduled discrete-time PI controller in which both the proportional and integral gains depend on the same scheduling variable. This controller uses 1-D Lookup Table blocks to implement the gain schedules. 16-9 16 Gain-Scheduled Controllers Matrix-Valued Gain Schedules You can also implement matrix-valued gain schedules Simulink. A matrix-valued gain schedule takes one or more scheduling variables and returns a matrix rather than a scalar value. For instance, suppose that you want to implement a time-varying LQG controller of the form: dxe = Axe + Bu + L ( y - Cxe - Du) u = - Kxe , where, in general, the state-space matrices A, B, C, and D, the state-feedback matrix K, and the observer-gain matrix L all vary with time. In this case, time is the scheduling variable, and the gain schedule determines the values of the matrices at a given time. In your Simulink model, you can implement matrix-valued gain schedules using: • MATLAB Function block — Specify a MATLAB function that takes scheduling variables and returns matrix values. • Matrix Interpolation block — Specify a lookup table to associate a matrix value with each scheduling-variable breakpoint. Between breakpoints, the block interpolates the matrix elements. (This block is in the Simulink Extras library.) For the LQG controller, use either MATLAB Function blocks or Matrix Interpolation blocks to implement the time-varying matrices as inputs to a Varying Observer Form block. For example: 16-10 Model Gain-Scheduled Control Systems in Simulink In this implementation, the time-varying matrices are each implemented as a MATLAB Function block in which the associated function takes the simulation time and returns a matrix of appropriate dimensions. If you have Simulink Control Design, you can tune matrix-valued gain schedules implemented as either MATLAB Function blocks or as Matrix Interpolation blocks. However, to tune a Matrix Interpolation block, you must set Simulate using to Interpreted execution. See the Matrix Interpolation block reference page for information about simulation modes. 16-11 16 Gain-Scheduled Controllers Custom Gain-Scheduled Control Structures You can also use the scheduled gains to build your own control elements. For example, the model rct_CSTR includes a gain-scheduled lead compensator with three coefficients that depend on the scheduling variable, CR. To see how this compensator is implemented, open the model and examine the Temperature controller subsystem. Here, the overall gain Kt, the zero location a, and the pole location b are each implemented as a 1-D lookup table that takes the scheduling variable as input. The lookup tables feed directly into product blocks. 16-12 Model Gain-Scheduled Control Systems in Simulink Tunability of Gain Schedules For a lookup table or MATLAB Function block that implements a gain schedule to be tunable with systune, it must ultimately feed into either: • A block in the Linear Parameter Varying block library. • A Product block that applies the gain to a given signal. For instance, if the Product block takes as inputs a scheduled gain g(α) and a signal u(t), then the output signal of the block is y(t) = g(α)u(t). There can be one or more of the following blocks between the lookup table or MATLAB Function block and the Product block or parameter-varying block: • Gain • Bias • Blocks that are equivalent to a unit gain in the linear domain, including: • Transport Delay, Variable Transport Delay • Saturate, Deadzone • Rate Limiter, Rate Transition • Quantizer, Memory, Zero-Order Hold • MinMax • Data Type Conversion • Signal Specification • Switch blocks, including: • Switch • Multiport Switch • Manual Switch Inserting such blocks can be useful, for example, to constrain the gain value to a certain range, or to specify how often the gain schedule is updated. 16-13 16 Gain-Scheduled Controllers See Also Related Examples 16-14 • “Tune Gain Schedules in Simulink” on page 16-15 • “Gain-Scheduled Control of a Chemical Reactor” on page 16-52 Tune Gain Schedules in Simulink Tune Gain Schedules in Simulink Typically, gain-scheduled controllers are fixed single-loop or multiloop control structures in which controller gains vary with operating condition. A gain schedule converts the scheduling variables that describe the current operating condition into appropriate controller gains. In Simulink, you can implement gain schedules using lookup tables or MATLAB functions. (See “Model Gain-Scheduled Control Systems in Simulink” on page 16-4.) If you have Simulink Control Design, you can use systune to tune these gain schedules so that the full nonlinear system meets your design requirements. Tuning gain schedules amounts to identifying appropriate values for lookup-table data or finding the right function to embed in a MATLAB Function block. For systune, you parameterize the gain schedules as functions of the scheduling variables with tunable coefficients. Workflow for Tuning Gain Schedules The general workflow for tuning gain-scheduled control systems is: 1 Select a set of design points that adequately covers the operating range over which you are tuning. A design point is a set of scheduling-variable values that describe a particular operating condition. The set of design points can be a regular grid of values or a scattered set. Typically, you start with a few design points. If the performance that your tuned system achieves at the design points is not maintained between design points, add more design points and retune. 2 Obtain a collection of linear models describing the linearized plant dynamics at the selected design points. Ways to obtain the array of linear models include: • Linearize a Simulink model at each operating condition represented in the grid of design points. For example, if each design point corresponds to a steady-state operating condition, you can trim the plant at each design point and linearize at the resulting operating point. Or, if your scheduling variable is time, you can linearize at a series of simulation snapshots. • Sample an LPV model of the plant at the design points. For more information, see “Plant Models for Gain-Scheduled Controller Tuning” on page 16-18. 3 Create an slTuner interface for tuning the Simulink. When you do so, you substitute the array of linear models for the plant, so that the slTuner interface contains a set 16-15 16 Gain-Scheduled Controllers of closed-loop tunable models corresponding to each design point. For more information, see “Multiple Design Points in slTuner Interface” on page 16-26. 4 Model the gain schedules as parametric gain surfaces. A parametric gain surface is a basis-function expansion with tunable coefficients. For a vector σ of scheduling variables, such expansion is of the form: K (s ) = K0 + K1 F1 ( n (s ) ) + … + K M FM ( n ( s ) ) . n(σ) is a normalization function. For tuning with systune, you use tunableSurface to represent the parametric gain surface K(σ). In the slTuner interface you create for tuning, use setBlockParam to associate the resulting gain surface with the block that represents the gain schedule. systune tunes the coefficients K0,...,KM over all the design points. For more information, see “Parameterize Gain Schedules” on page 16-32. 5 Specify your tuning goals using TuningGoal objects. You can specify tuning goals that apply at all design points or at a subset of design points. You can also specify tuning goals that vary from design point to design point. For example, you might define a minimum gain margin that becomes increasingly stringent as a particular scheduling variable increases in magnitude. For information about specifying tuning goals that vary with design point, see “Change Requirements with Operating Condition” on page 16-42. For information about specifying tuning goals generally, see “Tuning Goals”. 6 Use systune to tune the control system. systune tunes the set of parameters, K0,...,KM, against all plant models in the design grid simultaneously (multimodel tuning). 7 Validate the tuning results. You can examine the tuned gain surfaces and validate the performance of the linearized system at each design point. However, local linear performance does not guarantee global performance in nonlinear systems. Therefore, it is important to perform simulation-based validation using the tuned gain schedules. For more information, see “Validate Gain-Scheduled Control Systems” on page 1646. 16-16 See Also See Also More About • “Model Gain-Scheduled Control Systems in Simulink” on page 16-4 • “Gain-Scheduled Control of a Chemical Reactor” on page 16-52 • “Tuning of Gain-Scheduled Three-Loop Autopilot” on page 16-70 16-17 16 Gain-Scheduled Controllers Plant Models for Gain-Scheduled Controller Tuning Gain scheduling is a control approach for controlling a nonlinear plant. To tune a gainscheduled control system, you need a collection of linear models that approximate the nonlinear dynamics near selected design points. Generally, the dynamics of the plant are described by nonlinear differential equations of the form: x& = f ( x, u,s ) y = g ( x, u,s ) . Here, x is the state vector, u is the plant input, and y is the plant output. These nonlinear differential equations can be known explicitly for a particular system. More commonly, they are specified implicitly, such as by a Simulink model. You can convert these nonlinear dynamics into a family of linear models that describe the local behavior of the plant around a family of operating points (x(σ),u(σ)), parameterized by the scheduling variables, σ. Deviations from the nominal operating condition are defined as: d x = x - x (s ) , d u = u - u ( s ) . These deviations are governed, to first order, by linear parameter-varying dynamics: d& x = A ( s ) d x + B( s ) d u, ∂f ( x (s ) ,u (s )) ∂x ∂g C (s ) = ( x (s ), u (s )) ∂x A (s ) = d y = C (s ) d x + D (s ) d u, ∂f ( x (s ), u (s )) ∂u ∂g D(s ) = ( x (s ), u (s )). ∂u B (s ) = This continuum of linear approximations to the nonlinear dynamics is called a linear parameter-varying (LPV) model: dx = A (s ) x + B ( s ) u dt y = C (s ) x + D ( s ) u. 16-18 Plant Models for G