Linear Multivariable Control: A Geometric Approach

357 Pages·1974·9.102 MB·English

by  Dr. Walter Murray Wonham (auth.)

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continuaiion on page 345
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. Künzi
Systems Theory
101
W. Murray Wonham
Linear Multivariable Control
A Geometrie Approach
Springer-Verlag
Berlin Heidelberg GmbH 1974
Editorial Board
H. Albach . A. V. Balakrishnan . M. Beckmann (Managing Editor) . P. Dhrymes
J.
Green' W. Hildenbrand . W. Krelle . H. P. Künzi (Managing Editor) . K. Ritter
R. Sato . H. Schelbert . P. Schönfeld
Managing Editors
Prof. Dr. M. Beckmann Prof. Dr. H. P. Künzi
Brown University Universität Zürich
Providence, RI 02912/USA 8090 Zürich/Schweiz
Dr. Walter Murray Wonham
Department of Electrical
Engineering
University ofToronto
Toronto/Canada MSS lA4
Library of Congress Cataloging in Publication Data
Wonham, W M 1934-
Linear·mu1tivariab1e contro1.
(Lecture notes in economics and mathematical
systems ; 101 : Operations research)
Bibliography: p.
Inc1udes index.
1. Contro1 theory. 2. Algebras, Linear. I. Ti
t1e. II. Series: Lecture notes in economics and
mathematical systems 101. III. Series: Operations
research (Berlin).
QA~.3.W59 629.8'312 74-19470
AMS Subject Classifications (1970): 93B25
ISBN 978-3-662-22675-9 ISBN 978-3-662-22673-5 (eBook)
DOI 10.1007/978-3-662-22673-5
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned,
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Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the
publisher, the arnount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1974
Originally published by Springer-Verlag Berlin Heidelberg New York 1974
To Anne
PREFACE
In writing this monograph my objective is to present arecent, 'geometrie' approach to
the structural synthesis of multivariable control systems that are linear, time-invariant,
and of finite dynamic order. The book is addressed to graduate students specializing in
control, to engineering scientists engaged in control systems research and development,
and to mathematicians with some previous acquaintance with control problems.
The label 'geometrie' is applied for several reasons. First and obviously, the setting
is linear state space and the mathematics chiefly linear algebra in abstract (geometrie)
style. The basic ideas are the familiar system concepts of controllability and observability,
thought of as geometrie properties of distinguished state subspaces. Indeed, the geometry
was first brought in out of revulsion against the orgy of matrix manipulation which linear
control theory mainly consisted of, not so long ago. But secondlyand of greater interest,
the geometrie setting rather quickly suggested new methods of attacking synthesis which
have proved to be intuitive and economical; they are also easily reduced to matrix arith
metic as soonas you want to compute. The essence of the 'geometrie' approach is just this:
instead of looking directly for a feedback laW (say u = Fx) which would solve your synthesis
problem if a solution exists, first characterize solvability as a verifiable property of some
constructible state subspace, say J. Then, if all is weIl, you may calculate F from J
quite easily. When it works, the method converts what is usually an intractable nonlinear
problem in F, to a straightforward quasilinear one in J.
By this means the first reasonably complete structure theory has been given for two con
trol problems of longstanding interest: regulation, andnoninteraction. Of course, no claim
is made that themethodsemployedare the best, and I leave the reader to judge whetherone
sort of orgy has just been replaced by another.
The book is organized as follows. Chapter 0 is a quick review of linear algebra and se
lected rudiments of linear systems. It is assumed that the reader already has some working
knowledge in these areas. Chapters 1-3 cover mainly standard material on controllability
andobservability, although sometimes in a more 'geometrie' style than has beencustomary,
and at times with greater completeness than in the literature to date. The essentially new
concepts are (A, B)-invariant subspaces and (A, B)-controllability subspaces: these are in
troduced in Chapters 4 and 5, along with a few primitive applications by way of motivation
and illustration. The first major application - to tracking and regulation - is developed in
leisurely style through Chapters 6 - 8. In Chapters 6 and 7 purelyalgebraic conditions are
investigated, for output regulation alone and then for regulation along with internal stability •
Chapter 8 attacks the problem of qualitative insensitivity to small parameter variations.
The result is a simplified, 'generic' version of the general algebraic setup, leading finally
to a structurally stable synthesis, as required in any practical implementation. A similar
plan is followed in treating the second main topic, noninteracting control: first the algebraic
VI
development, in Cbapters 9 and 10, then generic solvability in Cbapter 11. No description
is attempted of structurally stable synthesis of noninteracting controllers, as this is seen
to require adaptive control, at a level of complexity beyond the domain of strict1y linear
structures; but its feasibility in principle should be clear. The two closing Chapters 12
and 13 deal with quadratic optimization. While not strongly dependent on the preceding geo
metrie ideas the presentation, via dynamic programming, is perhaps a little more complete
tban wbat is available in this style in current textbooks. In any event the topic is standard
in most courses on linear contro!.
The framework throughout is state space, only casual use being made of frequency do
main representations and procedures. It would be a highly worthwhile project to link the
'geometrie approach' with some of the re cent synthesis techniques based on transfer ma
trices. Again for the future, intriguing possibilities exist for the use of geometrie methods
in exploring other major problems of multivariable system structure: for instance, the
contrasting philosophies of hierarchical and decentralized contro!. I hope the book may be
seminal in these respects.
A word on pedagogy. The main text is devoted to the theoretical development. To mini
mize clutter, nearly all routine numerical examples have been placed among the exercises
at the end of each chapter. With these as guide the reader should easily learn to translate
the relatively abstract language of the theory, with its stress on the qualitative and geo
metrie, into the computational language of everyday matrix arithmetic. While the book is
not primarily a design manual, the computational procedures sketched out have all been
programmed in APL and successfully run on systems of (modest) dynamic order 10 to 15.
But much worthwhile and interesting work can and should be done on numerical aspects which
are here entirely ignored.
More than half this book is based on published research coauthored with several col
leagues and graduate students, and it is a pleasure to re-affirm my considerable debt to
them: Steve Morse, Boyd Pearson, Ellis Fabian, Bruce Francis and Omar Sebakhy. In
addition I owe much to conversations with Ted Davison, Mike Sain, Harold Smith, Jakov
Snyders, Shi-Ho Wang, Ming Chan, Witold Gesing, Jan Van den Kieboom and Joe Yuan.
Finally, thanks are due to Professor A. V. Balakrishnan for his editorial encouragement
to publish this work in the Springer-Verlag 'Lecture Notes' series; and to Mrs. Rita de
Clercq Zubli for her expert preparation of the typeseript.
Toronto W.M. Wonham
June, 1974
CONTENTS
CHAPTER O. MATHEMATICAL PRELIMINARIES 1
0.1 Notation •..••.••..•••.•••....•.•••..•.... 1
0.2 Linear Spaces •••••••••••••••••••••••.••••• 1
0.3 Subspaces •••••••••••••••••••••••••••••••• 2
0.4 Maps and Matrice s . . . • . . . . • • . • . • . • . . . . • • . . • . 5
0.5 Factor Spaces . . . . . . . . . . . . . . . . • . . . . . . . . . . • . 8
0.6 Commutative Diagrams ••••••••••••••••••••••• 10
0.7 Invariant Subspaces. Induced Maps •••••••••••••• 11
0.8 Characteristic Polynomial. Spectrum ••••••••••••• 12
0.9 Po lynomial Rings • • • • • • • • • • • • • • • • • • • • • • • • • • • 13
0.10 Rational Canonical Structure • • • • • • • • • • • • • • • • • • • 14
0.11 Jordan Decomposition. • • . • • • • • • • • • • • • • • • • • . • • 17
0.12 Dual Spaces. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 21
0.13 Inner Product Spaces • • • • • • • • • • • • • • • • • • • • • • • . 23
0.14 Hermitian and Symmetrie Maps ••••••••••••••••• 24
0.15 Well-Posedness and Genericity •••.•••••••...••. 25
0.16 Linear Systems • • • • • • • • • • • • • • • • • • • • • • • • • • • • 27
0.17 Transfer Matrices. Signal Flow Graphs ••••••••••• 29
0.18 Rouche' s Theorem . • • • • • • • • • • • • • • • • • • • • • • • • • 31
0.19 Exercises •••••••••••••••••••••••••••••••• 31
0.20 Notes and References .•••••••••••••••••.••••• 34
CHAPTER 1. INTRODUCTION TO CONTROLLABILITY • . • • • • • • • •• 35
1.1 Reachability.............................. 35
1. 2 Controllability • • • • • • • • • • • • • • • • • • • • • • . • • • • •. 36
1.3 Single-Input Systems .••••••••••••••••••••••• 38
1.4 Multi-Input Systems. • • • • • • • • • • • • • • • • • • • • • • •• 39
1.5 Controllability is Generic . . • • • • • • • • • • • • • • • • • •• 43
1. 6 Exercise s. • • • • • • • • • • • • • • • • • • • • • • • • • • . • • •. 43
1. 7 Notes and References . • • • • • • • • • • • • • • • • • • • • • •. 45
CHAPTER 2. CONTROLLABILITY, FEEDBACK AND POLE
ASSIGNMENT • . • • • • • • • • • • • • • • • • • • • • • • • • • • •• 46
2. 1 Controllability and Feedback • • • • • • • • • • • • • • • • • •• 46
2.2 Pole Assignment • . • • • • • • • • • • • • • • • • • • • • • • • •. 48
2.3 Incomplete Controllability and Pole Shifting • • • • • • • •• 50
2. 4 stabilizability............................. 52
2. 5 Exercises................................ 53
2.6 Notes and References . • . • • • • • • • • • • • • • • • • • • • •• 54
CHAPTER 3. OBSERVABILITY AND DYNAMIC OBSERVERS . • . • . •• 55
3.1 Observability.............................. 55
3.2 Unobservable Subspace • • • • • • • • • • • • • • • • • • • • • •• 57
3.3 Full Order Dynamic Observer • • • • • • • • • • • • • • • • •. 58
3.4 Minimal Order Dynamic Observer ••••••••••••••. 60
3. 5 Observers and Pole Shifting. • • • • • • • • • • • • • • • • • •• 63
3. 6 Detectability.............................. 66
VIII
CHAPTER 3. OBSERVABILITY AND DYNAMIC OBSERVERS (cont'd)
3. 7 Detectors and Pole Shifting • • • • • • • • • • • • • • • • • • •• 67
3.8 Pole Shifting by Dynamic Compensation • • • • • • • • • • •• 72
3.9 Observer for a Single Linear Functional ••••••••••• 80
3.10 Preservation of Observability and Detectability • • • . • .• 81
3.11 Exercises................................ 83
3.12 Notes and References • • • • • • • • • • • • • • • • • • • • • • •• 88
CHAPTER 4. DISTURBANCE DECOUPLING AND OUTPUT
STABILIZATION • • • • • • • • • • • • • • • • • • • • • • • • • • .• 90
4.1 Disturbance Decoupling Problem (DDP) • • • • • • • • • • •• 90
4.2 (A, B)-Invariant Subspaces • . • • • • • • • • • • • • • • • • • •• 91
4.3 Solution of DDP • • • • • • • • • • • • • • • • • • • • • • • • • • •• 94
4.4 Output stabilization Problem (OSP) • • • • • • • • • . • • • •• 95
4.5 Exercises................................ 100
4. 6 Notes and References • • • • • • • • • • • • • • • • • • • . • . •• 104
CHAPTER 5. CONTROLLABILITY SUBSPACES ••••.•.•••••.•••• 105
5.1 Controllability Subspaces •.•.••..••••••.••..••. 105
5.2 Spectral Assignability • • • • • • • • • • • • • • • • • • • • • • •. 108
5.3 Controllability Subspace Algorithm • • . • • • . • • • • • • •• 110
5.4 Supremal Controllability Subspace. • • • • • • • . . . • • • .• 112
5.5 Disturbance Decoupling with stability ••.••••••.... 117
5.6 Controllability Indices •.••.••••••••••••••••.• 121
5.7 Exercises................................ 127
5. 8 Notes and References • . . • • . • • • • • • • • • • • • • • • • •• 132
CHAPTER 6. TRACKING AND REGULATION I: OUTPUT
STABILIZATION . • • • • • • • • • • • • • • • • • • • • • • . • . •• 133
6.1 Restricted Regulator Problem (RRP) • • • • • • • • • • • • .• 134
6.2 Solvability of RRP • • • • . . • • • • • . . • • • • • • . • • . . •• 136
6.3 Extended Regulator Problem (ERP) .••••••..•.•••• 143
6.4 Example................................. 147
6.5 Concluding Remark ••.•••••••••••.•••••.•••• 150
6.6 Exercises................................ 150
6.7 Notes and References • • • . • • • • • . • • • • • • • • • • • • .• 151
CHAPTER 7. TRACKING AND REGULATION 11: INTERNAL
STABILIZATION • • • • • • • • • • • • • • • • • • . • • . . • • . .• 152
7.1 Solvability of RPIS: General Considerations ..•••...• 153
7.2 Constructive Solution of RPIS: 71 = 0 • • • . • • • . • • • • •• 156
7.3 Constructive Solution of RPIS: 71 Arbitrary •• • • • • • •• 161
7.4 Application: Regulation against step Disturbances • • • •• 166
7. 5 Application: static Decoupling . • • • • • • • • • • • • • • • •. 167
7.6 Example 1: RPIS Unsolvable • • • • • • • • • • • • • • • . • •• 168
7.7 Example 2: Servo-Regulator................... 170
7.8 Exercises • • • • • • • • • • • • • • • • • • • • • • • • . • • . • • •• 175
7.9 Notes and References • • • • . . • • • • • . • • • • • • . • • • •• 183
CHAPTER 8. TRACKING AND REGULATION III: STRUCTURALLY
STABLE SYNTHESIS . • • • • • • • • • • • • • . • • • • • . • • •• 184
8.1 Preliminaries............................. 184
8.2 Example 1: structural Stability ••••••••••••••••• 185
8.3 Well-Posedness and Genericity. • • • • • • • • • • • • • • • •. 187
IX
CHAPI'ER 8. TRACKING AND REGULATION III: STRUCTURALLY
STABLE SYNTHESIS (cont'd)
8.4 Synthesis, Case I: C = D, 'nD = 0 •••••••••••••••• 192
8.5 Synthesis, Cas~!I: KerC C KerD, 'nD Minimal •••••• 205
8.6 Synthesis, Case !II: KerC C KerD, Dual Observer •••• 210
8.7 Example 2: Ill-Posed RPIS •••••••••••••••••••• 214
8.8 Example 3: Well-Posed RPIS. Strong Synthesis •••••• 217
8.9 On Practical Synthesis •.••••••••••••••••••••• 219
8.10 The Internal Model Principle ••••••••••••••••••. 222
8.11 Exercises •••••••••••••••••••••••••••••••• 223
8.12 Notes and References •••••••••••••••••••••••• 226
CHAPTER 9. NONINTERACTING CONTROL I: BASIC PRINCIPLES •• 227
9.1 Decoupling: Systems Formulation •••••.••••••••• 227
9.2 Restricted Decoupling Problem (RDP) ••••••••••••• 229
9.3 Solution of RDP: Outputs Complete ••••••••••••.•• 231
9.4 Extended Decoupling Problem (EDP) •••••••••••••• 233
9.5 Solution of EDP •••••••••••••••••••••••••••• 235
9.6 Naive Extension •••••••••••••••••••.•••••••. 240
9.7 Example ••••••.•••••••••••••••••••••••••• 242
9.8 Partial Decoupling • . • • • • • • • • • • • • • • • • • • • • • • • . 243
9.9 Exercises •••••••••••••••••••••••••••••••• 245
9.10 Notes and References •••••••••••••••••••••••• 246
CHAPTER 10. NONINTERACTING CONTROL!I: EFFICIENT
COMPENSATION • • • • • • • • • • • • • • • • • • . • • . • • • • •• 248
10.1 The Radical •••••••••••••••••••••••••••••. 248
10.2 Efficient Extension •••••••••••.•••••••••••••• 253
10.3 Efficient Decoupling ••••••••••••••••••••••••• 258
10.4 Minimal Order Compensation: d(ß) = 2 •..••••.•••• 263
10.5 Minimal Order Compensation: d( ß) = k •••••••••••• 269
10.6 Exercises ••••••••••••••••••.••••.•••••••• 274
10.7 Notes and References •••••••••••••••••••••••. 275
CHAPTER 11. NONINTERACTING CONTROL III: GENERIC
SOLVABILITY ...•.......•..•..•........••.• 277
11.1 Generic Solvability of EDP ••••••••••••.•••••.. 277
11. 2 State Space Extension Bounds . • • • • • . • • . • • • • • • • • . 284
11.3 Significance of Generic Solvability ••••••••••••••• 289
11. 4 Exercises................................ 290
11. 5 Notes and References • • • • • • • • • • • • • • • • • • • • • • • • 290
CHAPTER 12. QUADRATIC OPTIMIZATION I: EXISTENCE AND
UNIQUENESS .••••••••••••••••••••••••••••• 291
12.1 Quadratic Optimization ••••••••.•.•••••••••••• 291
12.2 Dynamic Programming: Heuristics •••••••••••••• 292
12.3 Dynamic Programming: Rigor.................. 294
12.4 Matrix Quadratic Equation. • • • • • • • • • . • • • . • • • • • • 298
12.5 Exercises................................ 302
12.6 Notes and References •••••••••••••••••••••••• 304
CHAPTER 13. QUADRATIC OPTIMIZATION II: DYNAMIC RESPONSE. 306
13.1 Dynamic Response: Generalities .••••••••••••••• 306
13.2 Example 1: First-Order System ••••••••••••••••• 306
13.3 Example 2: Second-Order System ••••••••••••••. 307