Adaptive Systems: An Introduction

358 Pages·1996·8.61 MB·English

by  Iven Mareels, Jan Willem Polderman (auth.)

Systems & Control: Foundations & Applications
Founding Editor
Christopher I. Byrnes, Washington University
To my dad
Iven
To the memory of my father
Jan Willem
Iven Mareels
Jan Willem Polderman
Adaptive Systems
An Introduction
Springer Science + Business Media, LLC
I ven Mareels Jan Willem Polderman
Department of Engineering Department of Applied Mathematics
Faculty of Engineering & University of Twente
Information Technology 7500 AE Enschede
Australian National University The Netherlands
ACT 2000, Australia
Library of Congress Cataloging-in-Publication Data
Mareels,lven, 1959-
Adaptive systems : An introduction / by Iven Mareels and Jan
Willem Polderman.
p. cm. -- (Systems & control)
Includes bibliographical references and index.
ISBN 978-1-4612-6414-9 ISBN 978-0-8176-8142-5 (eBook)
DOI 10.1007/978-0-8176-8142-5
1. Adaptive control systems. 1. Polderman, lan Willem, 1956-
II. Title. III. Series.
TJ217. M362 1996
629.8' 36--dc20 96-24283
CIP
Printed on acid-free paper ~®
H{J2J
© 1996 Springer Science+Business Media New York
Originally published by Birkhauser Boston in 1996
Softcover reprint of the hardcover 1s t edition 1996
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ISBN 978-1-4612-6414-9
Typeset by the Authors in 0TEX.
987 6 543 2 1
Contents
Preface xiii
Acknowledgements xvi
1 Adaptive Systems 1
1.1 Introduction.........
1.2 Adaptive systems: examples
1.2.1 Adaptive control . . I
1.2.2 Adaptive signal processing . 3
1.2.3 Adaptive systems versus classical techniques 3
1.3 General structure of adaptive control systems 4
1.3.1 Introduction. . . . . 4
1.3.2 The general structure 4
1.3.3 The error signal . . . 8
1.3.4 The tuner ..... . 9
1.3.5 Certainty equivalence. 11
1.3.6 Design and analysis. 12
1.4 Illustrating the concepts . . . 13
1.4.1 The MIT rule for adaptive control: feedforward case 13
1.4.2 The MIT rule for adaptive control: feedback problem . 16
] .4.3 An adaptive pole placement scheme 19
1.4.4 A universal controller 21
1.4.5 Echo cancelling. 22
1.5 Summary of chapter. 25
1.6 Notes and references 25
1. 7 Exercises ..... . 25
VI Contents
2 Systems And Their Representations 27
2.1 Introduction 27
2.2 Notation .. 28
2.3 The behavior 28
2.4 Latent variables 31
2.5 Equivalent representations 32
2.6 Controllability. 33
2.7 Observability 35
2.8 Stability .... 37
2.9 Elimination of Latent variables 39
2.10 TheringlR[~,Cll ..... 43
2.11 An example ........ 47
2.12 A word about the notation 48
2.13 Summary of chapter. 49
2.14 Notes and references 49
3 Adaptive systems : principles of identification 50
3.1 Introduction ........... . 50
3.2 Object of interest and model class 51
3.2.1 Object of interest . . . . . 51
3.2.2 Model class ....... . 53
3.3 Identification criterion and algorithms 58
3.3.1 Least squares identification. . 58
3.3.2 Recursive Least Squares (RLS) 59
3.3.3 Projection algorithm . . . . . . 63
3.3.3.1 Basic projection algorithm 63
3.3.3.2 Normalized Least Mean Square (NLMS) 64
3.3.3.3 Projection with dead zone ...... . 65
3.3.3.4 Least Mean Square Algorithm (LMS) . 66
3.4 Data model assumptions .. 67
3.4.1 Stable data filter .. 67
3.4.2 Data in model class. 67
3.4.3 Information content of data. 69
3.4.4 Data do not fit model class . 70
3.5 Analysis of identification algorithms 71
3.5.1 Properties of recursive least squares 71
Contents vii
3.5.1.1 Consistency for RLS. . . . . . . . . . . 74
3.5.1.2 Consistency with model errors for RLS . 76
3.5.2 Properties of the NLMS algorithm . . . . . . . . . 78
3.5.2.1 With NLMS the equation error converges. 78
3.5.2.2 Consistency for NLMS ... . . . . . . . 80
3.5.2.3 Consistency with model errors for NLMS 82
3.5.3 Projection algorithm with dead zone . 84
3.5.4 Tracking properties . . . . . . . . . . 86
3.5.4.1 NLMS algorithm can track 87
3.5.4.2 RLS algorithm cannot track 88
3.5.5 Incorporating prior knowledge in algorithms 91
3.6 Persistency of excitation 91
3.7 Summary of chapter. 96
3.8 Notes and references 96
3.9 Exercises ..... . 97
4 Adaptive Pole Assignment 103
4.1 Introduction. 103
4.2 Preliminaries 105
4.3 The system and its representations 107
4.4 Equilibrium analysis ...... . 109
4.4.1 The error model. . . . . . 110
4.4.2 How much can be learned,
and how much must be learned? 110
4.5 An algorithm for adaptive pole assignment. 114
4.5.1 The adaptive system .... 114
4.6 Analysis of the algorithm . . . . . . 117
4.6.1 N onminimal representation. 118
4.6.2 Minimal representation . . . 120
4.7 Filtered signals . . . . . . . . . . . 124
4.7.1 Filter representation ofi/o systems . 124
4.7.2 Application to adaptive pole assignment. 128
4.8 Modification of the projection algorithm 133
4.9 Summary of chapter. 135
4.10 Notes and references 135
4.11 Exercises . . . . . . 136
Vlll Contents
5 Direct Adaptive Model Reference Control 139
5.1 Introduction ...... . 139
5.2 Basic problem definition 140
5.3 Model reference control: nonadaptive solution. 142
5.4 Error model construction 147
5.5 Equilibrium analysis .. 152
5.6 Adaptive algorithm ... 155
5.6.1 Adaptive model reference control algorithm. 155
5.7 Analysis of the adaptive system 156
5.7.1 Stability of the adaptive system 157
5.7.2 Parameter convergence? ... 162
5.8 Adaptive model reference control with
disturbance rejection . . . . . . . . . 164
5.8.1 The Internal Model Principle. 164
5.8.2 Model reference control with disturbance rejection 167
5.8.3 Adaptive model reference control with
known disturbance rejection . . . . . . . . . . . . 168
5.8.4 Adaptive model reference and disturbance rejection control 169
5.9 Summary of chapter. 169
5.10 Notes and references 170
5.11 Exercises .... 171
6 Universal Controllers 172
6.1 Introduction... 172
6.2 Existence of solutions. 174
6.3 The first order case . . 174
6.3.1 Sign b known. 175
6.3.2 The Nussbaum controller: sign b unknown 178
6.3.3 The Willems & Byrnes controller: sign b unknown 183
6.4 Higher order systems . . . . . . . . . . . . . . . . . . . 185
6.4.1 High gain feedback . . . . . . . . . . . . . . . 186
6.4.2 Willems-Byrnes controller: sign of qn-I known 189
6.4.3 Willems-Byrnes controller: sign qn-I unknown 190
6.5 Martensson's algorithm ....... . 193
6.5.1 The adaptive control problem 194
6.5.2 The main result 195
6.5.3 Dense curves . 197
Contents ix
6.504 A dense curve based on an enumeration of QN 198
6.6 Summary of chapter. 198
6.7 Notes and references 199
6.8 Exercises . . . . . . 199
7 The pole/zero cancellation problem 204
7.1 Introduction....................... . 204
7.2 The pole/zero cancellation problem in adaptive control . 205
7.3 Combining direct and indirect adaptive control .... . 207
7.3.1 The first order case . . . . . . . . . . . . . . . . 207
7.3.1.1 Problem statement and reparametrization . . 207
7.3 .1.2 Equilibrium analysis . . 208
7.3.1.3 Adaptive algorithm . . 209
7.3.2 The higher order case. . . . . . . 210
7.3.2.1 Problem statement and reparametrization . . 210
7.3.2.2 Equilibrium analysis. . 212
7.3.2.3 Adaptive algorithm . 218
704 Adaptive Excitation ......... . . 219
704.1 The first order case . . . . . . .220
704.1.1 Problem statement. .220
704.1.2 Adaptive algorithm .220
704.2 The higher order case ..... .223
704.2.1 Problem statement. .223
704.2.2 Adaptive algorithm .223
7.5 A more fundamental viewpoint .... .225
7.5.1 The connection with tunability . .226
7.5.2 Alternative parametrizations .227
7.6 Conclusions ..... .228
7.7 Summary of chapter. .228
7.8 Notes and references .228
7.9 Exercises ..... . .230
8 Averaging Analysis For Adaptive Systems 232
8.1 Introduction...... .232
8.2 Averaging ............... . .233
8.2.1 An illustration ........ . . 235
8.2.2 Some notation and preliminaries .239
x Contents
8.2.3 Finite horizon averaging result ....... . · 241
8.2.4 Infinite horizon result. . . . . . . . . . . . . .244
8.3 Transforming an adaptive system into standard form · 251
8.4 Averaging approximation ........... . · 258
8.5 Application: the MIT rule for adaptive control. .260
8.5.1 System description ..... .260
8.5.2 Frozen system for MIT rule · 261
8.5.3 Averaging for MIT rule ... · 261
8.5.4 Interpretation of averaged system . 263
8.5.4.1 Case I: Reference model equals plant Zm == Zp . 263
8.5.4.2 Case II: Constant reference signal. . 263
8.5.4.3 Case III: General problem. . . . 264
8.5.4.4 How slow is slow adaptation? . .266
8.6 Application: echo cancellation in telephony .267
8.6.1 Echo cancellation .......... . .267
8.6.2 System description and assumptions .268
8.6.3 Analysis ......... . · 271
8.6.3.1 The frozen system 271
8.6.3.2 The averaged update equation . 272
8.6.3.3 Analysis of the averaged equation. .274
8.6.3.4 DEC system behavior .277
8.6.3.5 General observations .279
8.7 Summary of chapter. .280
8.8 Notes and references · 281
8.9 Exercises ..... . .282
9 Dynamics of adaptive systems: A case study 286
9.1 Introduction........... · 286
9.2 The example .......... . · 287
9.3 Global analysis and bifurcations .289
9.4 Adaptive system behavior: ideal case · 291
9.5 Adaptive system behavior: undermodelled case .294
9.5.1 Parameter range ...... . .296
9.5.2 Equilibria .......... . .296
9.5.3 Beyond period 1 bifurcations . .298
o ....... .
9.5.4 Summary d =1= .299