QUANTITATIVE FEEDBACK DESIGN OF LINEAR AND NONLINEAR CONTROL SYSTEMS

THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE

QUANTITATIVE FEEDBACK DESIGN OF LINEAR AND NONLINEAR CONTROL SYSTEMS

ODEDYANIV Faculty of Engineering, Tel Aviv University, Israel

Foreword by:

Isaac Horowitz

~.

" Springer Science+Business Media, LLC

Library <lf Congress Cataloging-iJt-Publicatiorl Data

Yaniv, Oded, Quantitative feedback design Df linear and nonlinear control

systems 1 Oded Yaniv ; foreword by Isaac Horowitz. p cm. -- (Kluwer international sedes in engineering and

computer science ; SECS 509) Includes bibliDgraphical references. ISBN 978-1-4419-5089-5 ISBN 978-1-4757-6331-7 (eBook) DOI 10.1007/978-1-4757-6331-7

1. feedback control systems. 1. Title. Il. Series. TJ216.Y36 1999 629. 8' 3 --dc2l

Copyright © 1999 by Springer Science+Business Media New York Originally pub1ished by Kluwer Academic Pub1ishers in 1999 Softcover reprint of the hardcover 1 st edition 1999

99-27956 CIP

AII rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanica1, photo­copying, recording, or otherwise, without ihe prîor writien permission of the publisher, Springer Science+Business Media, LLC.

Printed on acid-free paper,

NOTE TO READER

The software for all the examples, in the form of Matlab ™ script files, is available at http://www.eng.tau.ac.il/~yaniv.

FOR Michal and Yotam

Contents

Foreword

Preface Acknow ledgments Abbreviations Notation and Symbols

1. INTRODUCTION 1. Basic Components of Feedback Controlled Systems 2. Why Embed a Plant in a Feedback System? 3. The Design Process of Feedback Control Systems 4. Book Outline

Part I LINEAR SYSTEMS

2. BASICS OF SISO FEEDBACK CONTROLLED SYSTEMS 1. Introduction 2. Basic Frequency Domain Characteristics

2.1 Relative Stability, Cross-Over Frequency and Bandwidth 2.2 Conditionally Stable Systems 2.3 High Frequency Gain 2.4 Stability Analysis Using Nichols Charts

2.4.1 Continuous Systems 2.4.2 Discrete Time Systems 2.4.3 Some Remarks About the Nyquist Plot

3. Closed Loop Specifications 3.1 t-Domain Specification 3.2 w-Domain Specification 3.3 Translation of Specifications From t-Domain to w-

Domain 3.3.1 Model Based Technique 3.3.2 Krishnan and Cruickshanks' Technique

4. Performance Limitations of NMP or Unstable Systems 4.1 Stable Plants

4.1.1 Extension to Several RHP Zeros and/or Delay

xv

xvii

xix

xxi

1 1 2 4

9

15 15 15 16 17 18 20 21 22 25 25 26 27

29 30 33 34 35 37

x QUANTITATIVE FEEDBACK DESIGN

4.1.2 Comparison to Loop Transmissions Which Violate Assumption 2.1 40

4.2 Unstable Plants 42 4.2.1 Unstable Plants With a Single RHP Pole 43 4.2.2 An Example and Limitations 46 4.2.3 Extension to Several RHP Poles 47

5. Loop Shaping 49 6. Summary 55 7. Exercises 56 8. Notes and References 57

3. SYNTHESIS OF LTI CONTROLLERS FOR MISO LTI PLANTS 59 1. Introduction 59 2. One DOF System 59

2.1 Sensitivity Reduction Problem 61 2.2 Bound Calculations 62 2.3 Control Effort Problem 64 2.4 Examples 65

3. Two DOF Systems 73 3.1 Bound Calculations 75

3.1.1 Bound Calculations With The Aid of Plant Templates 75

3.1.2 Bound Calculations Using a Closed Form Algorithm 78

3.2 An Example 80 4. Extension to NMP Plants 81 5. Extension to Sampled Data Systems 85 6. Summary 88 7. Exercises 88 8. Notes and References 90

4. SYNTHESIS OF LTI CONTROLLERS FOR MIMO LTI PLANTS 93 1. Synthesis of One DOF Feedback Systems 94

1.1 2 x 2 Plants and Disturbances at the Plant's Inputs 95 1.2 2 x 2 Plants and Disturbances at the Plant's Outputs 113 1.3 n x n Plants and Disturbances at the Plants' Inputs 125 1.4 n x n Plants and Disturbances at the Plants' Outputs 134 1.5 Design Improvements by Iteration 135 1.6 Shortcuts in Low Frequency Bound Calculations 137

2. Synthesis of Two DOF Feedback Systems 137 2.1 2 x 2 Plants 139 2.2 n x n Plants 148 2.3 Model Matching Specifications 157 2.4 Shortcuts in Low Frequency Bound Calculations 158

3. Synthesis for Margins at the Plants' Outputs 160 3.1 2 x 2 Plants and Diagonal Controllers 163 3.2 n x n Plants and Diagonal Controllers 170

Contents xi

4. Synthesis for Margins at the Plants' Inputs 180 5. Synthesis of Non-diagonal Controllers for n x m Plants 186 6. Synthesis for Minimum Phase Diagonal Elements 194

6.1 2 x 2 Plants 196 6.2 n x n Plants 201

7. Synthesis for the General Control Problem Using LFT Notation 205 7.1 Some Special Cases 206 7.2 Statement of the Problem 208 7.3 Development of the Design Equations 209

7.3.1 Special Cases 210 7.4 A Design Procedure For The Stated Problems 211

8. Sensitivity Reduction Limitations and Tradeoffs in NMP Feedback

9. 10. 11. 12. 13. 14. 15. 16.

Systems 212 8.1 SISO Plants 214 8.2 MIMO Plants 217 8.3 Sensitivity Reduction Limitations For a Single Row of

S 217 8.4 Sensitivity Reduction Limitations For Several Rows of

S 222 8.5 Necessary Conditions 8.6 A Design Example Exercises Appendix A Appendix B Appendix C AppendixD Appendix E Summary Notes and References

225 227 230 237 238 239 239 239 240 241

Part II NONLINEAR SYSTEMS

5. SYNTHESIS OF LTI CONTROLLERS FOR NONLINEAR SISO PLANTS 247

1. Synthesis for Tracking Specifications 247 1.1 The Schauder Technique 250 1.2 G.ui~eli~es for the Choice of PN,y, dN,y and Theoretical

Limitatlons 253 1.2.1 How to choose the pairs PN,y, dN,y 253 1.2.2 Some Theoretical Limitations 256

1.3 The Homotopic Invariance Technique 258 1.4 An Example 261

1.4.1 Implementation of the Design Procedure -A Four-Step Process 261

2. Synthesis for Zeroing the Plant Output 264 2.1 The Schauder Technique 267

xu QUANTITATIVE FEEDBACK DESIGN

3. 4. 5.

2.2 G.ui~eli!1es for the Choice of PN,y, dN,y and Theoretical LImItatIOns 2.2.1 How to choose the pairs PN,y, dN,y 2.2.2 Some Theoretical Limitations

2.3 The Homotopic Invariance Technique 2.4 An Example

2.4.1 Implementation of the Design Procedure - a Two-Step Process

Appendix A Summary Notes and References

270 270 274 275 276

276 278 280 280

6. SYNTHESIS OF LTV CONTROLLERS FOR NONLINEAR SISO PLANTS 281

1. Statement of the Problem 281 2. The Design Procedure 283

2.1 Guidelines for the Choice of PN,y, dN,y and the Time Slices 284

3. An Example 286 3.1 Statement of the Problem 286 3.2 Single Time Slice Design 288 3.3 The Two Consecutive Time Slices Design 290

3.3.1 Design on the First Time Slice 290 3.3.2 Design on the Second Time Slice 292

3.4 Comparisons and Discussion 293 4. Summary 294

7. SYNTHESIS OF LTI CONTROLLERS FOR NONLINEAR MIMO PLANTS 295

1. Synthesis for Tracking Specifications 295 1.1 The Schauder Technique 299 1.2 Guidelines for Choosing of PN,y, dN,y and Remarks 305 1.3 The Homotopic Invariance Technique 308 1.4 An Example 310

2. Synthesis for Zeroing the Plant Outputs 314 2.1 The Schauder Technique 317 2.2 Guidelines for Choosing PN,y, dN,y and Remarks 319 2.3 The Homotopic Invariance Technique 322 2.4 An example 323

3. Synthesis of the MIMO LTI Problem 326 3.1 2 x 2 plants 328

3.1.1 A design Procedure With Less Stringent High Frequency Conditions 331

3.1.2 A Third Design Procedure 332 3.2 n x n Plants 332

3.2.1 A Design Procedure With Less Stringent High Frequency Conditions 334

3.2.2 A Third Design Procedure 336

Contents Xlll

4. Rational TF Approximations From Input Output Data 337 4.1 SISO Systems 338 4.2 MIMO systems 339

5. Summary 341 6. Notes and References 341

8. SYNTHESIS OF LTV CONTROLLERS FOR NONLINEAR MIMO PLANTS 343

1. Statement of the Problem 343 2. The Design Procedure 344

2.1 Guidelines for the Choice of PN,y, dN,y and Time Slices 346

3. An Example 4. Notes

Index

347 354 367

Foreword

This book is a very welcome, valuable addition to the Feedback Control literature in general, and especially to its practical, Engineering part. There are so few books devoted to genuine, practical Feedback Control design. There is a great need for many more such books, in order to hopefully overcome the huge gap between much of Engineering Control Academia and Industry, if only for appreciation of the tremendous power of the simplest kind of feedback compensation, when it is properly understood and quantitatively formulated.

It is also very important that such books be written by active designers, whose elbows have been deep in detailed designs and their practical implementation. We are satiated with superficial review books. It is natural that such a variety of research-oriented practical designers will present a variety of approaches, insights - each his own special techniques and short cuts. This is a boon to the genuine student and design engineer, enhancing his perspective and design tools.

There are very few self-contained books dealing with the theory and practice of Quantitative Feedback Theory. This book is one of the only two that are available. Control engineers will find in this book many-detailed examples, which can be easily applied to many industrial applications, from Robotics to Flight control to Ecology.

This book presents feedback synthesis for single-input single-output and multi-input multi-output linear time invariant and nonlinear plants, based on the QFT method. It includes design details and graphs which do not appear in the literature. These will enable engineers and researchers understand QFT in great depth. Many examples are presented, to help the serious reader to understand and apply the extremely powerful design techniques.

Professor Oded Yaniv has extensive, highly successful practical design ex­perience. He has solved practical, extremely challenging problems which were given up as hopeless. In addition he is a leading researcher in this highly im­portant, but academia - mutilated subject. The Control community is indebted

XVI QUANTITATIVE FEEDBACK DESIGN

to him for this sorely needed book. It is hoped that they will take advantage of this gift.

Isaac Horowitz Dec. 1998

Preface

This book presents practical techniques for designing CONTROL systems for linear and nonlinear plants based on "Quantitative Feedback Theory" (QFT), whose origins can be traced back to I. Horowitz (1959). The QFT is a frequency domain method based on two observations: (i) feedback is needed to achieve a desired plant output response in the presence of plant uncertainty and/or unknown disturbances; and (U) a controller which produces less control effort is preferred, that is, a controller whose bandwidth is smaller is preferable. The QFT is quantitative in the sense that it synthesizes a controller for the exact amount of plant uncertainty, disturbances and required specifications. One additional very attractive property of the QFT technique is the fact that "narrow" bandwidth controllers result as compared to designs which assume special structures for plant uncertainties, disturbances and specifications such as norm-bounded uncertainties etc.

This book is about feedback synthesis for single-input single-output (SISO), multi-input single-output (MISO) and mUlti-input multi-output (MIMO) linear time invariant and nonlinear plants, by the QFT method. It includes design details which do not appear in the literature, and thus can help engineers and researchers understand QFT in greater depth. Many examples are presented, so that the reader can study them and acquire the experience and understanding of the design techniques involved. In particular each MIMO technique is first explained for the simple two-input two-output case, and then extended to the general multi-input multi-output case.

Local linearization is most often used to design feedback around nonlinear plants. The QFT techniques are suitable for feedback design around the locally linearized model because the design takes into account the exact uncertainty without inducing over-design. However this technique may not be suitable for systems whose operating point changes rapidly. One design method which overcomes this difficulty and is presented in this book is the global linearization approach which is based on functional analysis theory. The reader who is

XVlll QUANTITATIVE FEEDBACK DESIGN

unfamiliar with this branch of mathematics may skip the proofs, since a good understanding of Linear Systems Theory is really all that is required in order to understand the material presented in this book.

This book can be used as a text in any course on control system design at both the graduate and undergraduate levels. The prerequisites are a course in clas­sical linear systems, minimal understanding of sampled data systems and the z-transform (only for section 5.), and an introductory course in c1assicallinear control theory. Some functional analysis is needed for the nonlinear techniques but these can be explained and applied without understanding the underlying mathematical theory. Engineers are highly encouraged to use this book to ap­ply QFT to industrial applications; the techniques are not only quantitative but also offer the following two important properties for practical feedback design (i) insight into tradeoffs among various considerations such as: solution com­plexity, amount of uncertainty, specifications, amount of scheduling, sampling rate, and cost of feedback; and (ii) no need for a model in state-space form or any other form, since the QFT techniques can be applied to the measured plant responses at a dense set of frequencies. Property (ii) enables us to extend QFT to many nonlinear plants. Researchers in control systems are encouraged to use this book in order to gain insight into synthesis of feedback systems based on the QFT methodology, compare their design results with that of QFT, and assess and improve their design results where applicable.

All of the examples where implemented using Matlab™ version 5.3, the script files can be found at the following Web site: http://www.eng.tau.ac.il/yaniv.

ODED YANIV

Acknowledgments

It is a pleasure for me to acknowledge several individuals who have helped me throughout the years. The first is Prof. Isaac Horowitz who was my Ph.D supervisor and as such taught me QFT. Prof. C.H. Houpis hosted me sev­eral times at the Air-force institute of technology (AFIT), Dayton Ohio, and encouraged me to develop QFT software and the extension of the Golubev identification algorithm to MIMO systems. Prof. Yossi Chait and I have been working together for the last 10 years and along with Craig Borghesani we developed the QFT toolbox. Dr. Yahali Theodor has shared his extensive prac­tical experience with me in the form of many long discussions and I've enjoyed numerous stimulating discussions with my colleague Dr. Per-Olof Gutmann. Dr. Marcel Sidi and I cooperated on the topic of bandwidth limitations of non minimum-phase SISO systems. Special appreciation is due to Igor Chipovet­sky for preparation of many of the LTI MIMO examples, to Ronen Boneh for preparation of the SISO LTV example and to Haim Weiss from Rafael, Israel, for his contribution to section 1.3. To Arik Dickman whose excellent insight into industrial control problems he shared with me during our cooperative years in the control industry. My final word of acknowledgment goes to Aron Pila of IMI (Israel Military Industries - Advanced Systems Division) who did an outstanding job of reviewing the manuscript, improving its presentation and generating some of the plots and figures. His contribution is deeply appreciated.

Abbreviations Notation and Symbols

Abbreviations

DOF - Degree-Of-Freedom LHP - Left-Half-Plane LFT - Linear Fractional Transformation LTI - Linear-Time-Invariant LTV - Linear-Time-Varying MIMO - Multi-Input-Multi-Output MTF - Matrix Transfer Function MP - Minimum-Phase NL - Non-Linear NMP - Non-Minimum-Phase QFT - Quantitative Feedback Theory RHP - Right-Half-Plane SISO - SingJe-Input-Single-Output SIMO - Single-Input-Multi-Output MISO - Multi-Input-Single-Output TF - Transfer Function ZOH - Zero-Order-Hold

Notation

• Capital or lower case non-bold italic letters denote SISO TF's, for example P, G, F,g, f.

• Signals and their Laplace transforms are denoted by the same italic Jetter. In order to avoid confusion, 's', 'jw' or 't' are added as arguments whenever necessary, for example r, r(t), r(s), r(jw).

• The capital letter 'P' , in all its forms, is reserved for the plant: P, P k> P, Pk> PN,y, P N,y·

• For LTI plants, in order to avoid confusion, 's', 'jw' or 't' are added as arguments whenever necessary, for example P(s), P(jw), P(s), P(jw).

• Bold capital letters stand for matrices; the same non-bold lower-case letters stand for its entries: M = [mij].

XXll QUANTITATIVE FEEDBACK DESIGN

• Bold lower-case letters stand for vectors; the same non-bold lower-case letters stand for its entries: v = [Vi], v = [VI, ... , Vn], v = [VI, ... , Vn]T.

• MIMO plant inverses p-I have entries: p-I = [nij].

• Sets are denoted by parenthesis { }, for example {P} is a set of plants P.

• Matrix and vector inequalities, for example A ~ B, means element by element inequality.

• The notation PN,y (P N,y), means that the plant, P (P), is a function of N and y (N and y).

• The notation dN,y (dN,y), means that the disturbance, d (d), is a function of Nand y (N and y).

• diag(gl' ... , gn) denotes a square n x n matrix (or MTF) whose diagonal ii element is gi and all other elements are zeros.

Symbols

H2 - The Hardy space of all complex valued functions F (s) which are analytic in the open right half plane and which satisfy the condition

[suP~>a 2~ I~oo F(~ + jw)dw] < 00

Hoo - The Hardy space of all complex-valued functions F(s) which are analytic and bounded in the open right half plane, Re s > O. Bounded means that there is a real number b such that I F(s) 1< b, Re s > O. The least such bound b is the Hoo norm of F denoted by II F 1100 . Equivalently II F 1100= sup{1 F(s) I: Re s > o}. By the maximum Modulus Theorem, the open right half plane can be replaced by the imaginary axis and thus II F 1100= sUPw{1 F(jw) I: w E R}.

L2 - The Hilbert space of all measurable functions f : R+ -+ R with the property that Iaoo II f (t) 112 dt < 00, and where the inner product is defined as the square of the norm, that is < f, f >= Iaoo II f(t) 112 dt, while II . II signifies the Euclidean norm.

L~ x n _ The set of k x n matrices whose elements are in L 2 .

L} - The Hilbert space of all measurable functions f : R+ -+ R with the

{ f(t) 0 < t < T . property that h E L2 where h = 0 t >-T - for all fillIte T.

RHocF The subset of Hoo consisting of real-rational functions. FE RHoo if and only if F is proper <I F(oo) 1 is finite) and stable i.e. F has no poles in the closed right half plane, Re s 2: o.

RkxnHoo - The set of k x n matrices whose elements are in RHoo.

RH2 - The subset of H2 consisting of real-rational functions. F E RH2 if and only if F is stable and strictly proper.

RkxnH2 - The set of k x n matrices whose elements are in RH2 •

R - The set of real numbers.

R+ - The set of non negative real numbers.

Rkxn- The set of k x n matrices whose elements are in R.

XXlll