Table Of ContentStable Adaptive
Systems
Kumpati S. Narendra
Anuradha M. Annaswamy
STABLE ADAPTIVE SYSTEMS
S.
KUMPATI NARENDRA
Center for Systems Science
Yale University
M.
ANURADHA ANNASWAMY
Department of Mechanical Engineering
Massachusetts Institute of Technology
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright
Copyright© 1989 by Kumpati S. Narendra and Anuradha M. Annaswamy
All rights reserved.
Bibliographical Note
This Dover edition, first published in 2005, is an unabridged republication of
the work originally published in 1989 by Prentice-Hall, Inc., Englewood Cliffs,
New Jersey, as part of the Information and System Sciences series. An errata
list (pages 495-496) has been added to the present edition.
Library of Congress Cataloging-in-Publication Data
Narendra, Kumpati S.
Stable adaptive systems I Kumpati S. Narendra, Anuradha M.
Annaswamy.
p.cm.
Originally published: Englewood Cliffs, N.J. : Prentice Hall, cl 989.
Includes bibliographical references and index.
ISBN-13: 978-0-486-44226-6 (pbk.)
ISBN-10: 0-486-44226-8 (pbk.)
1. Adaptive control systems. I. Annaswamy, Anuradha M., 1956---- II.
Title.
TJ217.N37 2005
629.8'36-dc22
2004065741
Manufactured in the United States by Courier Corporation
44226803
www.doverpublications.com
To the memory of my mentor, Philippe LeCorbeiller,
and
to my graduate students
K.S.N.
To my grandfather, Pandit Narayan Jayaram Sharma
A.M.A.
Contents
Preface xv
1 Introduction 1
1.1 Introduction 1
1.2 Systems Theory 2
1.3 Adaptive Systems 5
1.3.1 What Is an Adaptive System?, 9
1.3.2 Parameter Adaptive and Structurally Adaptive Systems, 12
1.4 Direct and Indirect Control 13
1.4.1 Parameter Perturbation Method, 14
1.4 .2 Sensitivity Method, 18
1.5 Model Reference Adaptive Systems and Self-Tuning
Regulators 21
1.5.1 Identification Model, 22
1.5.2 Reference Model, 22
1.5.3 Reference Model with Inputs, 24
1.5.4 Model Reference Adaptive Control, 25
1.5.5 Self-Tuning Regulators, 28
1.6 Stable Adaptive Systems 28
1.6.1 Error Models, 30
1. 7 Applications 31
1.8 Summary 32
vii
viii Contents
1.9 Scope of the Book 34
Problems 35
References 39
2 Stability Theory 42
2.1 Introduction 42
2.2 Notation 44
2.3 Linear Systems 45
2.3.1 Linear Time-Invariant Systems, 46
2.3.2 Almost Time-Invariant Systems, 47
2.4 Lyapunov Stability 50
2.4.1 Definitions, 50
2.4.2 Lyapunov's Direct Method, 53
2.4.3 V(x, t) > 0, V(x, t) ~ 0, 56
2.4.4 LT/ Systems and Lyapunov Stability, 60
2.4.5 Linear Time-Varying Systems, 61
2.5 Positive Real Functions 63
2.5.1 Positive Real Transfer Matrices, 65
2.6 Kalman-Yakubovich Lemma 65
2.7 Input-Output Stability 67
2.8 Stability of Adaptive Systems 71
2.8.1 1wo Differential Equations That Arise in Adaptive
Systems, 71
2.8.2 Growth Rates of Signals in Dynamical Systems, 77
2.8.3 Stability in Terms of a Parameter Vector 8, 86
2.9 Other Stability Concepts 88
2.9.J Total Stability, 89
2.9.2 BIBO Stability, 92
2.10 Summary 94
Problems 94
References 96
3 Simple Adaptive Systems 99
3.1 Introduction 99
3.2 Algebraic Systems 100
3.2.1 Identification, JOO
3.3 Dynamical Systems 102
3.3.1 The Identification Problem: Scalar Case, 102
3.3.2 The Control Problem: Scalar Case (Direct Control), 109
3.3.3 The Control Problem: Scalar Case (Indirect Control), 117
Contents Ix
3.3.4 Indirect Control: Dynamic Adjustment of Control
Parameters, 121
3.3.5 Examples, 123
3.4 State Variables Accessible 126
3.4.1 The Identification Problem: Vector Case, 126
3.4.2 The Control Problem, 128
3.5 Summary 136
Problems 136
References 139
4 Adaptive Observers 140
4.1 Introduction 140
4.2 The Luenberger Observer 141
4.3 Adaptive Observers 144
4.3.1 Statement of the Problem, 144
43.2 Minimal Realization of Adaptive Observers, 145
4.3.3 Nonminimal Realization of Adaptive Observers, 151
4.3.4 Modified Adaptive Observer, 158
4.4 Design of Adaptive Observers 162
4.4.1 Representation 1,163
4.4.2 Representation 2,165
4.4.3 Hybrid Algorithms, 166
4.4.4 Time-Varying Adaptive Gains, 169
4.4.5 Integral Algorithm, 170
4.4.6 Multiple Models, 171
4.4.7 Adaptive Algorithms: An Overview, 173
4.5 Summary 177
Problems 177
References 179
5 The Control Problem 181
5.1 Introduction 181
5.2 Statement of the Problem 182
5.3 Adaptive Control of Plants with Relative Degree
Unity 184
5.3.1 Case (i) kp Unknown, 186
5.3.2 Case (ii) Unknown Zeros, 187
5.3.3 Case (iii) Unknown Poles, 190
=
5.3.4 General Case (n* 1), 192
Contents
X
5.4 Adaptive Control of Plants with Relative Degree
?: 2 199
5.4.1 Bezout Identity, 199
5.4.2 n* ?: 2: Algebraic Part, 202
=
5.4.3 n* 2: Analytic Part, 203
5.4.4 n* ?: 2: Analytic Part, 204
5.4.5 kp Known, 206
5.4.6 kp Unknown, 209
5.5 Proof of Global Stability of the Adaptive System 212
5.5.1 kp Known, 213
5.5.2 kp Unknown, 218
5.6 Comments on the Control Problem 222
5.7 Some Remarks on the Controller Structure 224
5.8 Stable Adaptive Co~trol Using the Indirect
Approach 226
=
5.8.1 n* 1, 227
z
5.8.2 n* 2, 230
5.9 Combined Direct and Indirect Approach 231
5.10 Summary 233
Problems 234
References 236
6 Persistent Excitation 238
6.1 Introduction 238
6.2 Persistent Excitation in Adaptive Systems 239
6.3 Definitions 246
6.3.1 Examples, 248
6.4 Properties of Persistently Exciting Functions 249
6.4.1 Algebraic Transformations, 249
6.4.2 Dynamic Transformations, 251
6.4.3 Almost Time-Invariant Systems, 256
6.4.4 Persistent Excitation of Unbounded Signals, 257
6.5 Applications to Adaptive Systems 258
6.5.1 Parameter Convergence in Adaptive Observers, 258
6.5.2 Parameter Convergence in Adaptive Control, 260
6.5.3 Design Considerations, 263
6.6 Persistent Excitation and Uniform Asymptotic
Stability 266
6.7 Summary 268
Problems 269
References 271
Contents xi
7 Error Models 273
7.1 Introduction 273
7.2 Error Model 1 274
7.3 Error Model 2 278
7.4 Error Model 3 279
7.5 Error Model 4 282
7.6 Further Comments on Error Models 1-4 286
7.7 Use of Error Models in Adaptive Systems 287
Problems 288
References 290
8 Robust Adaptive Control 291
8.1 Introduction 291
8.2 Adaptive Observers in the Presence
of Disturbances 293
8.2.1 Identifier, 293
8.2.2 Adaptive Observers, 294
8.2.3 Error Model, 295
8.3 Adaptive Control of a First-Order Plant in the Presence
of Bounded Disturbances 297
8.3.1 Exact Cancellation of Disturbance, 297
8.3 .2 Bounded Disturbances: Statement of the Problem, 301
8.3 .3 Category 1: Modifications of the Adaptive Law, 306
8.3.4 Category 2: Persistent Excitation of the Reference
Input, 312
8.3.5 Performance of the Adaptive System, 316
8.4 Adaptive Control of an nth Order Plant 318
8.4.1 Robustness without Persistent Excitation; 322
8.4.2 Robustness with Persistent Excitation, 327
8.4.3 Comments on Adaptive Algorithms for Bounded
Disturbances, 330
8.5 Hybrid Adaptive Control 331
8.5.1 Hybrid Error Models, 331
8.5.2 Hybrid Adaptive Control, 334
8.5.3 Hybrid Adaptive Control in the Presence of Bounded
Disturbances, 336
8.6 Adaptive Control of Time-Varying Plants 337
8.6.1 Adaptive Control of a First-Order Plant, 338
8.6.2 Time-Varying Plants of Higher Order, 342
8.7 Reduced-Order Controllers 345
Contents
xii
8.7.1 Instability with Reduced Order Controllers:
An Example, 346
8.7.2 Statement of the Problem, 347
8.7.3 Small Parameter µ in P, 348
8.7.4 Transfer Function W m(s) , 350
8.8 Summary 353
Probl~ms 354
References 356
9 . The Control Problem: Relaxation of Assumptions 359
9.1 Introduction 359
9.2 Assumption (i): Sign of kp 360
9.3 Assumption (iii): Upper Bound on the Order
of the Plant 366
=
9.3.l Regulation of Plant with Relative Degree n* l, 366
=
9.3.2 n* 2, 368
9.3.3 Arbitrary n*, 369
9.3.4 Universal Controller, 373
9.4 Assumption (iv): Zeros of Wp(s) in o::- 374
9.4.1 Direct Control,.375
9.4.2 Indirect Control, 378
9 .5 Command Generator Tracking Method 383
9.5.1 Statement of the Problem, 383
9.6 Summary 387
Problems 388
References 389
1O Multivariable Adaptive Systems 392
10.1 Introduction 392
10.2 Mathematical Preiiminaries 393
10.3 Multivariable Adaptive Identification 399
I 0.3.1 Statement of the Problem, 399
10.3.2 The Adaptive Observer: Minimal Realization, 400
10.3.3 The Adaptive Observer: Nonminima/ Realization, 401
10.4 Adaptive Control Problem 402
10.5 Adaptive Control of Two Input-Two Output
Plants 406
10.6 Dynamic Precompensator for a General Multivariable
Plant 408
10.6.J Design of the Adaptive Controller, 410
