Slotine • Li APPLIED NONLINEAR CONTROL

! i

APPLIEDNONLINEAR

CONTROL

Jean-Jacques E SlotineWeiping Li

AppliedNonlinearControl

JEAN-JACQUES E. SLOTINEMassachusetts Institute of Technology

WEIPING LIMassachusetts Institute of Technology'

Prentice HallEnglewood Cliffs, New Jersey 07632

Library of Congress Cataloging-in-Publication Data

Slotine, J.-J. E. (Jean-Jacques E.)Applied nonlinear control / Jean-Jacques E. Slotine, Weiping Li

p. cm.

Includes bibliographical references.

ISBN 0-13-040890-5

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

QA402.35.S56 1991 90-33365629.8'312-dc20 C1P

Editorial/production supervision andinterior design: JENNIFER WENZEL

Cover design: KAREN STEPHENSManufacturing Buyer: LORI BULWIN

= ^ = © 1991 by Prentice-Hall, Inc.^=&= A Division of Simon & Schuster

T k Englewood 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 14 13 12 1]

ISBN D-13-DHDfiTa-S

Prentice-Hall International (UK) Limited, LondonPrentice-Hall of Australia Pty. Limited, SydneyPrentice-Hall Canada Inc., TorontoPrentice-Hail 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

To Our Parents

Contents

Preface xi

1. Introduction 1

1.1 Why Nonlinear Control ? 1

1.2 Nonlinear System Behavior 4

1.3 An Overview of the Book 12

1.4 Notes and References 13

Part I: Nonlinear Systems Analysis 14Introduction to Part I 14

2. Phase Plane Analysis 17

2.1 Concepts of Phase Plane Analysis 182.1.1 Phase Portraits 182.1.2 Singular Points 202.1.3 Symmetry in Phase Plane Portraits 22

2.2 Constructing Phase Portraits 23

2.3 Determining Time from Phase Portraits 29

2.4 Phase Plane Analysis of Linear Systems 30

2.5 Phase Plane Analysis of Nonlinear Systems 32

2.6 Existence of Limit Cycles 36

2.7 Summary 38

2.8 Notes and References 38

2.9 Exercises 38

VI11

3. Fundamentals of Lyapunov Theory 40

3.1 Nonlinear Systems and Equilibrium Points 41

3.2 Concepts of Stability 47

3.3 Linearization and Local Stability 53

3.4 Lyapunov's Direct Method 573.4.1 Positive Definite Functions and Lyapunov Functions 583.4.2 Equilibrium Point Theorems 613.4.3 Invariant Set Theorems 68

3.5 System Analysis Based on Lyapunov's Direct Method 763.5.1 Lyapunov Analysis of Linear Time-Invariant Systems 773.5.2 Krasovskii's Method 833.5.3 The Variable Gradient Method 863.5.4 Physically Motivated Lyapunov Functions 883.5.5 Performance Analysis 91

3.6 Control Design Based on Lyapunov's Direct Method 94

3.7 Summary 95

3.8 Notes and References 96

3.9 Exercises 97

4. Advanced Stability Theory 100

4.1 Concepts of Stability for Non-Autonomous Systems 101

4.2 Lyapunov Analysis of Non-Autonomous Systems 105

4.2.1 Lyapunov's Direct Method for Non-Autonomous Systems 1054.2.2 Lyapunov Analysis of Linear Time-Varying Systems 114

4.2.3 The Linearization Method for Non-Autonomous Systems 116

4.3 * Instability Theorems 117

4.4 * Existence of Lyapunov Functions 1204.5 Lyapunov-Like Analysis Using Barbalat's Lemma 122

4.5.1 Asymptotic Properties of Functions and Their Derivatives 1224.5.2 Barbalat's Lemma 123

4.6 Positive Linear Systems 1264.6.1 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 1324.7.1 Block Combinations 1324.7.2 Passivity in Linear Systems 137