applied nonlinear control

APPLIED

NONLINEAR

CONTROL

Jean-Jacques E Slotine

WeipingLi

Applied

Nonlinear

Control

JEAN-JACQUES E. SLOTlNE

Massachusetts Institute of Technology

WEIPING LI

Massachusetts Institute of Technologv

Prentice Hall

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 1991

629.8'312-dc20

Editorial/production supervision and interior design: JENNIFER WENZEL Cover design: KAREN STEPHENS Manufacturing Buyer: LORI BULW[N

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

All rights reserved. No part of this book may be reproduced, 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. 90-33365

CIP

ISBN

0-13-040890-5

Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall Canada Inc., Toronto

Prentice-Hall Hispanoamericana, S.A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc .. Tokyo

Simon&Schuster Asia Pte. Ltd., Singapore Editora 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 Analysis

Introduction to Part I 14

2. Phase Plane Analysis

2.1 Concepts of Phase Plane Analysis

2.1.1 Phase Portraits 18

2.1.2 Singular Points 20

2.1.3 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 Functions 3.4.2 Equilibrium Point Theorems 61

3.4.3 Invariant Set Theorems 68

3.5 System Analysis Based on Lyapunov's Direct Method

3.5.1 Lyapunov Analysis of Linear Time-Invariant Systems

3.5.2 Krasovskii's Method 83

3.5.3 The Variable Gradient Method 86

3.5.4 Physically Motivated Lyapunov Functions 88 3.5.5 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

96

97

100

101

105 lOS ]]6 117

120

122

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 Systems

4.2.2 Lyapullov Analysis ofLinear Time- Varying Systems 114

4.2.3 *The Linearization MethodforNon-Autono~ousSystems

4.3

*

Instability Theorems

4.4

*

Existence of Lyapunov Functions

4.5 Lyapunov-Like Analysis Using Barbalat's Lemma

4.5.1 Asymptotic Properties of Functions and Their Derivatives

4.5.2 Barbalat's Lemma 123

4.6 Positive Linear Systems

4.6.1 PR and SPR Transfer Functions 126 4.6.2 The Kalman-Yakubovich Lemma 130 4.6.3 Positive Real Transfer Matrices 131

4.7 The Passivity Formalism 132

4.7.1 Block Combinations 132 4.7.2 Passivity in Linear Systems 137

ix

40

4.8

*

Absolute Stability

142

4.9

*

Establishing Boundedness of Signals

147

41

4.10

*

Existence and Unicity of Solutions

151

47

4.11

Summary

₁₅₃

53

4.12

Notes and References

153

57

4.13

Exercises

154

.~

5.

Describing Function Analysis

157

76

5.1

Describing Function Fundamentals

158

5.1.1 An Example of Describing Function Analysis 158 5.1.2 Applications Domain 162

5.1.3 Basic Assumptions 164 5.1.4 Basic Definitions 165

94

5.1.5 Computing Describing Functions 167

95

_5.2

Common Nonlinearities In Control Systems

169

96

_5.3

Describing Functions of Common Nonlinearities

172

97

_5.4

Describing Function Analysis of Nonlinear Systems

179

5.4.1 The Nyquist Criterion and Its Extension 180 5.4.2 Existence of Limit Cycles 182

100

5.4.3 Stability of Limit Cycles 184

5.4.4 Reliability of Describing Function Analysis 186

101

5.5

Summary

187

105

5.6

Notes and References

188

105

5.7

Exercises

188

l-t

116

117

Part II: Nonlinear Control Systems Design

191

120

Introduction to Part II 191

122

122

_{6. Feedback Linearization}

₂₀₇

126

6.1

Intuitive Concepts

208

6.1.1 Feedback Linearization And The Canonical Form 208 6.1.2 Input-State Linearization 213

6.1.3 Input-Output Linearization 216

132

6.2

Mathematical Tools

229

6.3

Input-State Linearization of SISO Systems

236

x

6.4 Input-Output Linearization of SISO Systems 6.5

*

Multi-Input Systems

6.6 Summary

6.7 Notes and References 6.8 Exercises 246 266 270 271 271

7.

Sliding Control

7.1 Sliding Surfaces 7.1.1 A Notational Simplification 278

7.1.2

*

Filippo v,s Construction of the Equivalent Dynamics 7.1.3 Pe/fect Peiformance - At a Price 285

7.1.4 Direct 1mplementations ofSwitching Control Laws

7.2 Continuous Approximations of Switching Control Laws 7.3 The Modeling/Perfonnance Trade-Offs

7.4

*

Multi-Input Systems 7.5 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 Control

8.1.1 Why Adaptive Control? 312 8.1.2 What1sAdaptive Control? 315

8.1.3 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 Feedback

8.4.1 Linear Systems With Relative Degree One 340 8.4.2 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 Estimation

8.7.1 Linear Parametrization Model 359

8.7.2 Prediction-Error-Based Estimation Methods 364 8.7.3 The Gradient Estimator 364

8.7.4 The Standard Least-Squares Estimator 370

311

312 326 335 339 350 353 358

xi

246

_{8.7.5 Least-Squares With Exponential Forgetting 374}

266

8.7.6 Bounded-Gain Forgetting 376

270

8.7.7 Concluding Remarks and Implementation Issues 381

271

8.8

Composite Adaptation

382

271

8.9

Summary

388

8.10

Notes and References

389

8.11

Exercises

389

276

277

9.

Control of Multi-Input Physical Systems

392

283

9.1

Robotics as a Prototype

393

9.1.1 Position Control 394

'9 9.1.2 Trajectory Control 397

290

9.2

Adaptive Robot Trajectory Control

403

301

9.2.1 The Basic Algorithm 404

303

9.2 .2 *Composite Adaptive Trajectory Control 411

306

9.3

Putting Physics in Control

416

307

_{9.3.2 Conservative and Dissipative Dynamics}9.3.1 High-Frequency Unmodeled Dynamics 416₄₁₈

307

9.3.3 Robotics as a Metaphor 419

9.4

Spacecraft Control

422

9.4.1 The Spacecraft Model 422

311

9.4.2 Attitude Control 425

9.5

Summary

432

312

9.6

Notes and References

433

9.7

Exercises

433

326

BIBLIOGRAPHY

437

335

INDEX

459

339

350

353

358