APPLIEDNONLINEAR

CONTROL

Jean-Jacques E SlotineWeipingLi

AppliedNonlinearControl

JEAN-JACQUES E. SLOTlNEMassachusetts Institute of Technology

WEIPING LIMassachusetts Institute of Technologv

Prentice HallEnglewood Cliffs, New Jersey 07632

Library of Congress Cataloging·in·Publication Data

Siotine, I.-I. E. (Jean-Jacques E.)Applied nonlinear control/Jean-Jacques E. Siotine, Weiping Li

p. em.

Includes bibliographical references.

ISBN 0-13-040890-5

1, Nonlinear control theory. I. Li, Weiping.

QA402.35.S56 1991629.8'312-dc20

Editorial/production supervision andinterior design: JENNIFER WENZEL

Cover design: KAREN STEPHENSManufacturing Buyer: LORI BULW[N

© 1991 by Prentice-Hall, Inc.A Division of Simon & SchusterEnglewood Cliffs, New Jersey 07632

All rights reserved. No part of this book may bereproduced, in any form or by any means,without permission in writing from the publisher.

Printed in the United States of America

20 19 18 17 16 15

II. Title.

90-33365CIP

ISBN 0-13-040890-5

Prentice-Hall International (UK) Limited, LondonPrentice-Hall of Australia Pty. Limited, SydneyPrentice-Hall Canada Inc., TorontoPrentice-Hall Hispanoamericana, S.A., MexicoPrentice-Hall of India Private Limited, New DelhiPrentice-Hall of Japan, Inc.. TokyoSimon & Schuster Asia Pte. Ltd., SingaporeEditora Prentice-Hall do Brasil, Ltda., Rio de Janeiro

Contents

Preface

1. Introduction

1.1 Why Nonlinear Control?

1.2 Nonlinear System Behavior

1.3 An Overview of the Book

1.4 Notes and References

Part I: Nonlinear Systems AnalysisIntroduction to Part I 14

2. Phase Plane Analysis

2.1 Concepts of Phase Plane Analysis2.1.1 Phase Portraits 182.1.2 Singular Points 202.1.3 Symmetly in Phase Plane Portraits 22

2.2 Constructing Phase Portraits

2.3 Determining Time from Phase Portraits

2.4 Phase Plane Analysis of Linear Systems

2.5 Phase Plane Analysis of Nonlinear Systems

2.6 Existence of Limit Cycles

2.7 Summary

2.8 Notes and References

2.9 Exercises

xi

1

1

4

12

13

14

17

18

23

29

30

32

36

38

38

38

vii

viii

3. Fundamentals of Lyapunov Theory 40

3.1 Nonlinear Systems and Equilibrium Points

3.2 Concepts of Stability

3.3 Linearization and Local Stability

3.4 Lyapunov's Direct Method

3.4.1 Positive Definite Functions and Lyapunov Functions3.4.2 Equilibrium Point Theorems 613.4.3 Invariant Set Theorems 68

3.5 System Analysis Based on Lyapunov's Direct Method

3.5.1 Lyapunov Analysis ofLinear Time-Invariant Systems3.5.2 Krasovskii's Method 833.5.3 The Variable Gradient Method 863.5.4 Physically Motivated Lyapunov Functions 883.5.5 Pelformance Analysis 91

3.6 Control Design Based on Lyapunov's Direct Method

3.7 Summary

3.8 Notes and References

3.9 Exercises

58

77

41

47

53

57

76

94

95

9697

100

101105

lOS

]]6

117

120122

122

126

4.1 Concepts of Stability for Non-Autonomous Systems

4.2 Lyapunov Analysis of Non-Autonomous Systems

4.2.1 Lyapullov'S Direct Methodfor Non-Autonomous Systems4.2.2 Lyapullov Analysis ofLinear Time-Varying Systems 1144.2.3 * The Linearization Methodfor Non-Autono~ousSystems

4.3 * Instability Theorems

4.4 * Existence of Lyapunov Functions4.5 Lyapunov-Like Analysis Using Barbalat's Lemma

4.5.1 Asymptotic Properties ofFunctions and Their Derivatives4.5.2 Barbalat's Lemma 123

4.6 Positive Linear Systems

4.6.1 PR and SPR Transfer Functions 1264.6.2 The Kalman-Yakubovich Lemma 1304.6.3 Positive Real Transfer Matrices 131

4.7 The Passivity Formalism 132

4.7.1 Block Combinations 1324.7.2 Passivity in Linear Systems 137

4. Advanced Stability Theory

ix

40 4.8 * Absolute Stability 1424.9 * Establishing Boundedness of Signals 147

41 4.10 * Existence and Unicity of Solutions 15147 4.11 Summary 15353 4.12 Notes and References 15357 4.13 Exercises 154

.~

5. Describing Function Analysis 15776

5.1 Describing Function Fundamentals 1585.1.1 An Example ofDescribing Function Analysis 1585.1.2 Applications Domain 1625.1.3 Basic Assumptions 1645.1.4 Basic Definitions 165

94 5.1.5 Computing Describing Functions 16795 5.2 Common Nonlinearities In Control Systems 16996 5.3 Describing Functions of Common Nonlinearities 17297 5.4 Describing Function Analysis of Nonlinear Systems 179

5.4.1 The Nyquist Criterion and Its Extension 1805.4.2 Existence ofLimit Cycles 182

100 5.4.3 Stability ofLimit Cycles 1845.4.4 Reliability ofDescribing Function Analysis 186

101 5.5 Summary187

105 5.6 Notes and References 188

1055.7 Exercises 188

l-t116

117 Part II: Nonlinear Control Systems Design 191

120Introduction to Part II 191

122122 6. Feedback Linearization 207

1266.1 Intuitive Concepts 208

6.1.1 Feedback Linearization And The Canonical Form 2086.1.2 Input-State Linearization 2136.1.3 Input-Output Linearization 216

1326.2 Mathematical Tools 2296.3 Input-State Linearization of SISO Systems 236

x

6.4 Input-Output Linearization of SISO Systems

6.5 * Multi-Input Systems6.6 Summary

6.7 Notes and References

6.8 Exercises

246

266

270

271

271

7. Sliding Control

7.1 Sliding Surfaces7.1.1 A Notational Simplification 2787.1.2 * Filippo v,s Construction of the Equivalent Dynamics7.1.3 Pe/fect Peiformance - At a Price 2857.1.4 Direct 1mplementations ofSwitching Control Laws

7.2 Continuous Approximations of Switching Control Laws

7.3 The Modeling/Perfonnance Trade-Offs

7.4 * Multi-Input Systems7.5 Summary

7.6 Notes and References

7.7 Exercises

276

277

283

289

290

301

303

306

307

307

8. Adaptive Control

8.1 Basic Concepts in Adaptive Control8.1.1 Why Adaptive Control? 3128.1.2 What 1s Adaptive Control? 3158.1.3 How To Design Adaptive Controllers? 323

8.2 Adaptive Control of First-Order Systems

8.3 Adaptive Control of Linear Systems With Full State Feedback

8.4 Adaptive Control of Linear Systems With Output Feedback8.4.1 Linear Systems With Relative Degree One 3408.4.2 Linear Systems With Higher Relative Degree 346

8.5 Adaptive Control of Nonlinear Systems

8.6 Robustness of Adaptive Control Systems

8.7 * On-Line Parameter Estimation8.7.1 Linear Parametrization Model 3598.7.2 Prediction-Error-Based Estimation Methods 3648.7.3 The Gradient Estimator 3648.7.4 The Standard Least-Squares Estimator 370

311

312

326

335

339

350

353

358

xi

246 8.7.5 Least-Squares With Exponential Forgetting 374266 8.7.6 Bounded-Gain Forgetting 376

2708.7.7 Concluding Remarks and Implementation Issues 381

2718.8 Composite Adaptation 382

2718.9 Summary 3888.10 Notes and References 3898.11 Exercises 389

276

2779. Control of Multi-Input Physical Systems 392

283 9.1 Robotics as a Prototype 3939.1.1 Position Control 394

'9 9.1.2 Trajectory Control 397

290 9.2 Adaptive Robot Trajectory Control 403301 9.2.1 The Basic Algorithm 404

303 9.2 .2 * Composite Adaptive Trajectory Control411

306 9.3 Putting Physics in Control416

3079.3.1 High-Frequency Unmodeled Dynamics 4169.3.2 Conservative and Dissipative Dynamics 418

307 9.3.3 Robotics as a Metaphor 4199.4 Spacecraft Control 422

9.4.1 The Spacecraft Model 422311 9.4.2 Attitude Control 425

9.5 Summary 432312 9.6 Notes and References 433

9.7 Exercises 433

326 BIBLIOGRAPHY 437

335 INDEX 459

339

350353

358