Table Of ContentAdaptive Filtering
Prediction and Control
Graham C. Goodwin
Director, ARC Center of Excellence for
Complex Dynamic Systems and Control
School of Electrical Engineering and Computer Science
University of New Castle
Callaghan, Australia
Kwai Sang Sin
Information Technology
The Hong Kong Jockey Club
Happy Valley, Hong Kong
DOVER PUBLICATIONS, INC.
MineoIa, New York
Copyriglzt
Copyright 0 1984 by Graham C. Goodwin and Kwai Sang Sin
All rights reserved.
Bibliogruplzicul Note
This Dover edition, first published in 2009, is an unabridged republication of the work
first published by Prentice-Hall, Englewood Cliffs, New Jersey, in 1984.
Library of Congress Cataloging-in-Publication Data
Goodwin, Graham C. (Graham Clifford), 1945-
Adaptive filtering prediction and control / Graham C. Goodwin, Kwai Sang Sin.
p. cm.
Originally published: Englewood Cliffs, N.J. : Prentice Hall, 1984.
Includes bibliographical references and index.
ISBN-I 3: 978-0-486-46932-4
ISBN-10: 0-486-46932-8
1. Discrete-time systems. 2. Filters (Mathematics). 3. Prediction theory. 4. Control
theory. I. Sin, Kwai Sang, 1952-
QA402 .G658 2009
003/ .83-dc22
2008044586
Manufactured in the llnited States of America
Dover Publications, Inc., 31 East 2nd Street, Mineold, N.Y. 11501
Preface
The object of this book is to present in a unified fashion the theory of adaptive
filtering, prediction, and control. The treatment is largely confined to linear discrete-
time systems, although natural extensions to nonlinear systems are also explored.
The emphasis on discrete-time systems reflects the growing importance of digital
computers in practical applications of this theory.
Adaptive techniques in filtering, prediction, and control have been extensively
studied for over a decade and numerous successful applications have been reported.
The development of the theory over the years has led to a much better understanding
of the performance of various adaptive algorithms. However, it was only recently
that a rigorous and comprehensive theory of convergence of adaptive algorithms
has emerged. This theory is very appealing from several points of view: it can be
applied in a unified manner to both deterministic and stochastic systems, yet it is
relatively simple and is easily understood with a minimum of background knowledge.
Perhaps more important, there is a close link between the convergence theory and
the performance of the algorithms in practice. This book summarizes the theoretical
and practical aspects of a large class of adaptive algorithms for potential users.
The philosophy of the presentation is that the relatively new material on
adaptive techniques is linked to standard results on design techniques applicable
when the system parameters are known. The book is aimed at two major groups of
readers : senior undergraduate students in engineering and applied mathematics, and
graduate students and research workers.
The book is divided into two parts: Part I deals with deterministic systems
and is suitable for both senior undergraduate and graduate students; Part I1 deals
with stochastic systems and is more suitable for graduate students. The two parts
xi
are further subdivided into chapters covering different design techniques for adaptive
filtering, prediction, and control. The book also contains appendices which sum-
marize the relevant background material, thus making the book substantially self-
contained. Throughout the book we have attempted to provide an adequate frame-
work that readers can build on when pursuing their own individual goals. Some
sections contain more difficult material or introduce ideas that are not heavily used
in subsequent work. These sections have been marked by an asterisk and may be
omitted on a first reading.
The book is an expanded version of lecture notes used for junior/graduate-level
courses at the University of Newcastle and the University of Houston and as the
basis of intensive courses on Adaptive Control at the University of Houston given
in December 1980 and the University of California, Los Angeles, in June 1982 and
June 1983. The book also includes the latest research results of the authors and
others in the field and should be suitable as background reading and as reference
for research workers.
We would like to pay tribute to a number of people who helped with the pre-
paration of this book. Our first thanks go to the large number of people who moti-
vated our interest in this topic by discussions, joint research, and correspondence.
We would particularly like to mention Karl Astrom, Peter Caines, Lennart Ljung,
and Peter Ramadge. In writing the book, we were also helped by useful discussions
with many others, including Brian Anderson, Howard Elliott, Tom Kailath, Bayliss
McInnis, Tino Mingori, John Moore, and many others.
It is a pleasure to thank Tony Cantoni, for his expert advice on numerous
practical questions, and Siew Wah Chan, who built the interface and organized
several adaptive control experiments which are discussed in the book. Special thanks
go to Lennart Ljung and Michel Gevers, who gave detailed and helpful comments
on the first draft. Several people made special contributions to particular sections
of the book. This includes Section 9.5.3, which was largely written by Michel Gevers
and Vincent Wertz; Appendix C, which was largely written by David Hill; and
Appendix E, which was initially prepared by Siew Wah Chan. The manuscript was
superbly typed by Maureen Byrnes with assistance from Dianne Piefke and Betty
Fewings, and the diagrams were expertly prepared by Phamie Sidell, Ann Pender,
and Wanda Lis. Finally, the first author would like to thank his wife, Rosslyn
Goodwin, and his children, Andrew and Sarah, for their generous support, under-
standing, and patience during the writing of the book.
GRAHAMC. GOODWIN
New South Wales, Australia KWAIS ANGS IN
xii Preface
PREFACE xi
I INTRODUCTION TO ADAPTIVE TECHNIQUES 1
1.1 Filtering, 1
1.2 Prediction, 2
1.3 Control, 2
PART I: DETERMINISTIC SYSTEMS
2 MODELS FOR DETERMINISTIC DYNAMICAL SYSTEMS 7
2.1 Introduction, 7
2.2 State-Space Models, 8
2.2.1 General, 8
2.2.2 Controllable State-Space Models, 8
2.2.3 Observable State-Space Models, 14
2.2.4 Minimal State-Space Models, 18
2.3 Difference Operator Representations, 20
2.3.1 General, 20
V
2.3.2 Right Difference Operator Representations, 21
2.3.3 Lejl Difference Operator Representations, 26
2.3.4 Deterministic Autoregressive Moving-Average Models, 32
2.3.5 Irreducible Diflerence Operator Representations, 33
2.4 Models for Bilinear Systems, 36
3 PARAMETER ESTIMATION FOR DETERMINISTIC SYSTEMS 47
3.1 Introduction, 47
3.2 On-Line Estimation Schemes, 49
3.3 Equation Error Methods for Deterministic Systems, 50
3.4 Parameter Convergence, 68
3.4.1 The Orthogonalized Projection Algorithm, 68
3.4.2 The Least-Squares Algorithm, 69
3.4.3 The Projection Algorithm, 69
3.4.4 Persistent Excitation, 72
3.5 Output Error Methods, 82
3.6 Parameter Estimation with Bounded Noise, 88
3.7 Constrained Parameter Estimation, 91
3.8 Parameter Estimation for Multi-output Systems, 94
3.9 Concluding Remarks, 98
4 DETERMINISTIC ADAPTIVE PREDICTION 106
4.1 Introduction, 106
4.2 Predictor Structures, 106
4.2.1 Prediction with Known Models, 107
4.2.2 Restricted Complexity Predictors, 110
4.3 Adaptive Prediction, 110
4.3.1 Direct Adaptive Prediction, 111
4.3.2 Indirect Adaptive Prediction, 113
4.4 Concluding Remarks, 11 5
5 CONTROL OF LINEAR DETERMINISTIC SYSTEMS 118
5.1 Introduction, 118
5.2 Minimum Prediction Error Controllers, 120
5.2.1 One-Step-Ahead Control (The SISO Case), I20
5.2.2 Model Reference Control (The SISO Case), 129
5.2.3 One-Step-Ahead Design for Multi-input Multi-output Systems, 132
5.2.4 Robustness Considerations, 143
vi Contents
5.3 Closed-Loop Pole Assignment, 146
5.3.1 Introduction, 146
5.3.2 The Pole Assignment Algorithm (Difference Operator Formulation), I47
5.3.3 Rapprochement with State- Variable Feedback, 148
5.3.4 Rapprochement with Minimum Prediction Error Control, 154
5.3.5 The Internal Model Principle, 155
5.3.6 Some Design Considerations, 157
5.4 An Illustrative Example, 163
6 ADAPTIVE CONTROL OF LINEAR DETERMINISTIC SYSTEMS 178
6.1 Introduction, 178
6.2 The Key Technical Lemma, 181
6.3 Minimum Prediction Error Adaptive Controllers (Direct Approach), 182
6.3.1 One-Step-Ahead Adaptive Control (The SISO Case), 182
6.3.2 Model Reference Adaptive Control, I99
6.3.3 One-Step-Ahead Adaptive Controllers
for Multi-input Multi-output Systems, 202
6.4 Minimum Prediction Error Adaptive Controllers (Indirect Approach), 207
6.5 Adaptive Algorithms for Closed-Loop Pole Assignment, 209
6.6 Adaptive Control of Nonlinear Systems, 218
6.7 Adaptive Control of Time-Varying Systems, 224
6.8 Some Implementation Considerations, 228
PART 11: STOCHASTIC SYSTEMS
7 OPTIMAL FILTERING AND PREDICTION 245
7.1 Introduction, 245
7.2 Stochastic State-Space Models, 246
7.3 Linear Optimal Filtering and Prediction, 248
7.3.I The Kalman Filter, 248
7.3.2 Fixed-Lag Smoothing, 256
7.3.3 Fixed-Point Smoothing, 259
7.3.4 Optimal Prediction, 261
7.4 Filtering and Prediction Using Stochastic ARMA Models, 262
7.4.1 The Stochastic ARMA Model, 262
7.4.2 Optimal Filters and Predictors in ARMA Form, 267
7.5 Restricted Complexity Filters and Predictors, 275
7.5.1 General Filters, 275
7.5.2 Whitening Filters, 278
7.5.3 Levinson Predictors, 280
Contents vii
7.6 Lattice Filters and Predictors, 283
7.6.1 Lattice Filters, 283
7.6.2 Lattice Predictors, 290
7.7 The Extended Kalman Filter, 293
8 PARAMETER EST1M ATlON
FOR STOCHASTIC DYNAMIC SYSTEMS 301
8.1 Introduction, 301
8.2 Off-Line Prediction Error Algorithms, 303
8.3 Sequential Prediction Error Methods, 308
8.3.1 General Systems, 309
8.3.2 Linear Systems, 313
8.4 Algorithms Based on Pseudo Linear Regressions, 319
8.5 Convergence Analysis of Sequential Algorithms, 323
8.5.1 The Stochastic Gradient Algorithm, 323
8.5.2 The Least-Squares Form of the Pseudo Linear Regression Algorithm, 329
8.5.3 The Stochastic Key Technical Lemma, 333
8.5.4 The ODE Approach to the Analysis of Sequential Algorithms, 335
8.6 Parameter Convergence, 339
8.6.1 The Ordinary Least-Squares Algorithm, 339
8.6.2 The Pseudo Linear Regression Algorithm, 341
8.7 Concluding Remarks, 346
9 ADAPTIVE FILTERING AND PREDICTION 360
9.1 Introduction, 360
9.2 Adaptive Optimal State Estimation, 362
9.2.1 The Extended Kalman Filter Approach, 362
9.2.2 The Prediction Error Approach, 366
9.2.3 Self-Tuning Fixed-Lag Smoothers, 368
9.3 Adaptive Optimal Prediction, 371
9.3.1 Indirect Adaptive Prediction, 373
9.3.2 Direct Adaptive Prediction, 375
9.4 Restricted Complexity Adaptive Filters, 379
9.4.1 Adaptive Deconvolution, 379
9.4.2 Adaptive Noise Canceling, 382
9.5 Adaptive Lattice Filters, 389
9.5.1 The Bootstrap Method, 389
9.5.2 The Prediction Error Method, 391
9.5.3 The Exact Least-Squares Method, 393
viii Contents
10 CONTROL OF STOCHASTIC SYSTEMS 407
10.1 Introduction, 407
10.2 The Application of Deterministic Design Methods, 408
10.3 Stochastic Minimum Prediction Error Controllers, 410
10.3.1 Minimum Variance Control (The SISO Case), 411
10.3.2 Model Reference Stochastic Control, 420
10.3.3 Control of Multi-input Multi-output Stochastic Systems, 422
10.4 The Linear Quadratic Gaussian Optimal Control Problem, 426
10.4.1 The Separation Principle, 427
10.4.2 The Tracking Problem, 430
10.4.3 Rapprochement with Minimum Variance Control, 431
11 ADAPTIVE CONTROL OF STOCHASTIC SYSTEMS 436
11.1 Introduction, 436
11.2 Concepts of Dual Control and Certainty Equivalence Control, 438
11.3 Stochastic Minimum Prediction Error Adaptive Controllers, 442
11.3.1 Adaptive Minimum Variance Control, 442
11.3.2 Stochastic Model Reference Adaptive Control, 456
11.3.3 Multi-input Multi-output Systems, 457
11.3.4 Convergence Analysis, 458
11.4 Adaptive Pole Placement and Adaptive Optimal Controllers, 463
11.5 Concluding Remarks, 464
APPENDICES
A A BRIEF REVIEW OF SOME RESULTS
FROM SYSTEMS THEORY 472
A.l State-Space Models, 472
A.2 Notes on 2-Transforms, 477
A.3 Properties of Polynomial Matrices, 479
A.4 Relatively Prime Polynomials, 48 1
A.5 Properties of Generalized Eigenvalues and Eigenvectors, 483
B A SUMMARY OF SOME STABILITY RESULTS 484
B.l Definitions, 484
B.2 Lyapunov Theorems, 485
B.3 Linear Time-Invariant Systems, 486
Contents ix
C PASSIVE SYSTEMS THEORY 489
C.l Introduction, 489
C.2 Preliminaries, 489
C.3 Frequency-Domain Properties, 492
C.4 Storage Functions, 494
D PROBABILITY THEORY AND STOCHASTIC PROCESSES 496
D.l Probability Theory, 496
D.2 Random Variables, 497
D.3 Independence and Conditional Probabilities, 497
D.4 Stochastic Processes, 498
D.5 Convergence, 499
D.6 Gaussian Stochastic Processes, 503
D.7 Minimum Variance Estimators, 504
D.8 Maximum Likelihood Estimation, 505
E MATRIX RlCCATl EQUATIONS 508
E.l Introduction, 508
E.2 The Algebraic Riccati Equation, 509
E.3 The Riccati Difference Equation, 513
REFERENCES 516
INDEX 535
X Contents
