Nonlinear Systems

766 Pages·2001·36.546 MB·768\766

by  Hassan K. Khalil

Nonlinear Systems
Third Edition

HASSAN K. KHALIL
Department of Electrical and Computer Engineering
Michigan State University
PRENTICE HALL
Upper Saddle River, New Jersey 07458
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Prentice © 2002, 1996by Prentice Hall
| Hall. Prentice-Hall, Inc.
ee (Upper Saddle River, NJ 07458
All rights reserved. No part ofthis book may be reproduced in any form or by any means,
without permission in writing from the publisher.
The author and publisher ofthis book have used their best efforts in preparing this book. These
efforts include the development, research, and testing ofthe theories and programs to determine
their effectiveness. The author and publisher make no warranty of any kind, expressed or
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Printed in the United States of America
10098765 4
ISBN Q-13-0b73849-7
Pearson Education Ltd., London
Pearson Education Australia Pty. Ltd., Sydney
Pearson Education Singapore, Pte. Ltd.
Pearson Education North Asia Ltd., Hong Kong
Pearson Education Canada, Inc., Toronto
Pearson Educacion de Mexico, S.A. de C.V.
Pearson Education—Japan, Tokyo
Pearson Education Malaysia, Pte. Ltd.
Pearson Education, Upper Saddle River, New Jersey
To my mentor,
Petar V. Kokotovié
and my family
Amina, Mohammad, Omar, Yousuf, and Suzanne

Contents
Preface xiii
1 Introduction R
1.1 Nonlinear Models and Nonlinear Phenomena r
1.2 Examples. ..........-2.-2.-.22-- l
O
1.2.1 Pendulum Equation .........
l
O
1.2.2 Tunnel-Diode Circuit ........ m
O
1.2.3 Mass-Spring System ........
1.2.4 Negative-Resistance Oscillator .. .
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1.2.5 Artificial Neural Network ...... 14
1.2.6 Adaptive Control .......... 16
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1.2.7 Common Nonlinearities ....... 18
1.3 Exercises . 2... 2... ee 24
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2 Second-Order Systems 30
2.1 Qualitative Behavior of Linear Systems . . . 37
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2.2 Multiple Equilibria. 2... 2... 2. 2. 46
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2.3 Qualitative Behavior Near Equilibrium Points 51
2.4 Limit Cycles .. 2... .....0.....0.. 54
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2.0 Numerical Construction of Phase Portraits . 59
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2.6 Existence of Periodic Orbits. ........ 61
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2.7 Bifurcation... 2.2... ..0.. 0-0-0200 - 69
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2.8 Exercises... 2... 2. ee ee ee ee 76
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3 Fundamental Properties 87
3.1 Existence and Uniqueness .......... 88
3.2 Continuous Dependence onInitial Conditions
and Parameters ............... 95
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3.3 Differentiability of Solutions and Sensitivity
Equations .............-.+-.. 99
3.4 Comparison Principle ............ 102
3.9 Exercises... 2... ee ee ee eee 105
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vii
vill CONTENTS
Lyapunov Stability 111
4.1 Autonomous Systems ........-00 02 eee 112
4.2 The Invariance Principle... .............2.2-2.2-02005% 126
4.3 Linear Systems and Linearization ..............-..-000- 133
4.4 Comparison Functions... ..........2. 2.0000 eee eee 144
4.5 Nonautonomous Systems ............- 2.05002 ee eee 147
4.6 Linear Time-Varying Systems and Linearization ............. 156
4.7 Converse Theorems .........0 000 2 eee ee ee 162
4.8 Boundedness and Ultimate Boundedness ................. 168
4.9 Input-to-State Stability ....................-.22040. 174
4.10 Exercises... 2... ee 181
Input-Output Stability 195
5.1 £ Stability... 2.2... ee ee ee 195
5.2 £ Stability of State Models... 2... ........-.-...--.000- 201
5.3 Lo Gain 2... ee 209
5.4 Feedback Systems: The Small-Gain Theorem .............. 217
5.5 Exercises... ee ee 222
Passivity 227
6.1 Memoryless Functions... .......2.. 000. ee ee ee ee 228
6.2 State Models... 2... 2... ee ee 233
6.3 Positive Real Transfer Functions ..................26. 237
6.4 £9 and Lyapunov Stability ....................0.... 241
6.5 Feedback Systems: Passivity Theorems... ..........--.08- 245
6.6 Exercises. 2... ee 259
Frequency Domain Analysis of Feedback Systems 263
7.1 Absolute Stability 2... ......0..0.2....2.....2...000. 264
7.1.1 Circle Criterion 2... 0.0... ee ee ee ee 265
7.1.2 Popov Criterion... 2... 2.2.2.0... 0000002 eee eee 275
7.2 The Describing Function Method .............2..-22-008.- 280
7.3 Exercises... ee 296
Advanced Stability Analysis 303
8.1 The Center Manifold Theorem .......-.........22-0008- 303
8.2 Region of Attraction... 2... 2.20.02... 22002. eee ee eee 312
8.3 Invariance-like Theorems .........0. 00. ee eee ee ee ee 322
8.4 Stability of Periodic Solutions... ......-...2-..2...002..0. 329
8.5 Exercises. 2... ee 304
CONTENTS
9 Stability of Perturbed Systems 339
9.1 Vanishing Perturbation ...........2. 2000 eee eee ene 340
9.2 Nonvanishing Perturbation 346
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9.3 Comparison Method... 2... ....0..0 00. eee ee es 300
9.4 Continuity of Solutions on the Infinite Interval 309
9.5 Interconnected Systems 308
9.6 Slowly Varying Systems .. 2... 2.2.02. eee ee 365
9.7 Exercises . 6... ee 372
10 Perturbation Theory and Averaging 381
10.1 The Perturbation Method... ...............2.220004 382
10.2 Perturbation on the Infinite Interval ................... 393
10.3 Periodic Perturbation of Autonomous Systems 397
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10.4 Averaging 2.2... 2. ee 402
10.5 Weakly Nonlinear Second-Order Oscillators 411
10.6 General Averaging 413
10.7 Exercises 419
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11 Singular Perturbations
11.1 The Standard Singular Perturbation Model... 2... ......2.2...
11.2 Time-Scale Properties of the Standard Model
11.3 Singular Perturbation on the Infinite Interval
11.4 Slow and Fast Manifolds
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11.5 Stability Analysis 2... 2.2.2... 0.202.202 eee ee ee
11.6 Exercises . 2... ee
12 Feedback Control
12.1 Control Problems . 2... 0.0.0.0... 000 eee eee ee
12.2 Stabilization via Linearization... 2... 2. ee
12.3 Integral Control . 2 2. ee eee
12.4 Integral Control via Linearization
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12.5 Gain Scheduling .. 2... 2. ee ee eee
12.6 Exercises 2. 2 2
13 Feedback Linearization
13.1 Motivation... 2... ee ee
13.2 Input-Output Linearization... 2... ee ee
13.3 Full-State Linearization
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13.4 State Feedback Control
13.4.1 Stabilization .......0...0.. 0.000000 ee eee
13.4.2 Tracking... 2... 2. ee ee ee
13.5 Exercises
x CONTENTS
14 Nonlinear Design Tools 551
14.1 Sliding Mode Control .. 2... 2... 2 eee 552
14.1.1 Motivating Example ..............-.....2-4. 552
14.1.2 Stabilization .. 2... . ee 563
14.1.3 Tracking... 2... ee ee 572
14.1.4 Regulation via Integral Control... .............. 575
14.2 Lyapunov Redesign .. 2... 2.2.2.2... ee ee ee ee 579
14.2.1 Stabilization... 2... 2... ee ee 579
14.2.2 Nonlinear Damping ..................+2005 588
14.3 Backstepping .. 2... 2... 2. ee ee 589
14.4 Passivity-Based Control ............02... 2020200202 ee 604
14.5 High-Gain Observers... 2... ee ee 610
14.5.1 Motivating Example ......... So ee 612
14.5.2 Stabilization .. 2... 2... ee ee 619
14.5.3 Regulation via Integral Control... .............. 623
14.6 Exercises 2. 2. ee 625
A Mathematical Review 647
B Contraction Mapping 653
C Proofs 657
C.1 Proof of Theorems 3.1 and3.2 2. ...2..02..00. 0.0.02. 000005 657
C.2 Proof of Lemma 3.4......0.0.0.0. 2.000002 eee ee ee 659
C.3 Proof of Lemma 4.1......0.0...0. 0.0.2.0 00020808 0G 661
C.4 Proof of Lemma 4.3.......0.0.0 2.0.00 002 eee eee 662
C.5 Proof of Lemma 4.4......0...0 0.0.00. eee ee ee ee 662
C.6 Proof of Lemma 4.5......0...0.0.0 0.0.00. ee eee ee eee 663
C.7 Proof of Theorem 4.16 ......0... 020.0000 0. eee eee 665
C.8 Proof of Theorem 4.17 2... 2.0.0.0... 0.0000 eee ee ee 669
C.9 Proof of Theorem 4.18 2... ....0.02. 2.0.00. 002 eee eee 675
C.10 Proof of Theorem 5.4 .....0.0.0.020 0.000 eee ee ee ee 676
C.11 Proof of Lemma6.l......0..02020.0.00..00.02. 0.2.2. 0084 677
C.12 Proof of Lemma 6.2.....0.0.0.0.0 0.0000 eee ee ee 680
C.13 Proof of Lemma 7.1......0.0..0.. 0.00 00. ee eee ee 684
C.14 Proof of Theorem 7.4 ....0..0.0. 0.0.00. 00 eee ee ee 688
C.15 Proof of Theorems 8.1 and 8.3 ......0..0202.... 2.000208. 690
C.16 Proof of Lemma 8.1 ......0.0.0.00. 0... 0000 eee eee en 699
C.17 Proof of Theorem 11.1 ....0..0.002... 0.2.00. 0 02. ee 700
C.18 Proof of Theorem 11.2 ....0..0.00. 2.2002 eee ee 706
C.19 Proof of Theorem 12.1 .....0.0..00.0 0... 000080 eee 708
C.20 Proof of Theorem 12.2
C.21 Proof of Theorem 13.1
C.22 Proof of Theorem 13.2
x1
CONTENTS
C.23 Proof of Theorem 14.6 . 2... 2. ee 713
719
Note and References
724
Bibliography
740
Symbols
TAQ
Index