Parametrizations in Control, Estimation and Filtering Problems: Accuracy Aspects

379 Pages·1993·15.59 MB·English

by  Michel Gevers PhD

Michel Gevers and Gang Li
Parametrizations in
Control, Estimation and
Filtering Problems:
Accuracy Aspects
With 60 Figures
Springer-Verlag
London Berlin Heidelberg New York
Paris Tokyo Hong Kong
Barcelona Budapest
Michel Gevers, PhD
Centre for Systems Engineering and Applied Mechanics (CESAME),
Universite Catholique de Louvain, Bitiment Euler,
B-1348 Louvain-Ia-Neuve, Belgium
Gang Li, PhD
School of Electrical and Electronic Engineering, Nanyang Technological
University, Nanyang Avenue, Singapore 2263
ISBN-13:978-1-4471-2041-4 e-ISBN-13:978-1-4471-2039-1
DOl: 10.1007/978-1-4471-2039-1
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
A catalog record for this book is available from the Library of Congress
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c Springer-Verlag London Limited 1993
Softcover reprint of the hardcover 1s t edition 1993
The publisher makes no representation, express or implied, with regard' to the
accuracy of the information contained in this book and cannot accept any legal
responsibility or liability for any errors or omissions that may be made.
Typesetting: Camera ready by authors
69/3830-543210 Printed on acid-free paper
Parametrizations in Control, Estimation and Filtering
Problems: Accuracy Aspects
Communications and Control Engineering Series
Editors: B.W. Dickinson·A. Fettweis·J.L. Massey·J.W. Modestino
E.D. Sontag· M. Thoma
CCES published titles include:
Sampled-Data Control Systems
J. Ackermann
Interactive System Identtfication
T. Bohlin
The Riccatti Equation
S. Bittanti, AJ. Laub and J.C. Willems (Eds.)
Nonlinear Control Systems
A.Isidori
Analysis and Design of Stream Ciphers
R.A. Rueppel
Sliding Modes in Control Optimization
V.1. Utkin
Fundamentals of Robotics
M. Vukobratovi6
To my parents
- Michel Gevers
To Qian-Rong
- Gang Li
Contents
Preface xi
1 Introduction 1
1.1 Motivation and general statement of objectives 1
1.2 A historical view and some motivating examples 4
1.3 Outline of the book. ... . . . . . . . . . . . . 17
2 Finite Word Length errors and computations 23
2.1 Introduction ........... ~ . 23
2.2 Representations of binary numbers . . . . . . 25
2.3 Overflow and quantization errors . . . . . . . 29
2.4 Arithmetic computations and roundoff errors 33
2.5 Dynamic range and scaling 39
2.6 Conclusions ................. . 43
3 Parametrizations in digital system design 45
3.1 Introduction.................. 45
3.2 State space realization set . . . . . . . . . . 49
3.3 Sensitivity measure of a state space realization. 50
3.4 Optimal realizations with respect to a sensitivity measure 55
3.5 Roundoff noise gain of state space realizations . . . . .. 60
3.6 Minimal roundoff noise gain realizations . . . . . . . .. 63
3.7 Relationship between sensitivity measure and roundoff
noise gain .......... . 66
3.8 Examples and simulations .. 68
3.9 Alternative approaches .. . 76
3.10 Conclusions ........ . 81
Appendix 3: Proof of Theorem 3.2 82
4 Frequency weighted optimal design 85
4.1 Introduction......................... 85
4.2 Minimization of a frequency weighted sensitivity measure 87
4.2.1 Weighted sensitivity measure of a realization .. 87
viii Contents
4.2.2 Optimal FWL realizations . . . . . . 90
4.2.3 Existence and uniqueness . . . . . . 91
4.3 Computation of the optimal realization set 95
4.4 Numerical example. . . . . . . . . . . . . . 99
4.5 Conclusions . . . . . . . . . . . . . . . . . . 103
Appendix 4.A: Proof of existence of a minimum. 104
Appendix 4.B: Computation of weighted Gramians 106
5 A new transfer function sensitivity measure 109
5.1 Introduction................... 109
5.2 Minimization of an L2 sensitivity measure . . 110
5.2.1 The L2 sensitivity measure of a realization 110
5.2.2 Optimal L2 sensitivity realizations . . . . . 111
5.2.3 Solution of the optimal realization problem 113
5.3 Relationship between L1/ L2 and L2 sensitivity measures 118
5.4 An example. 122
5.5 Conclusions............... 125
6 Pole and zero sensitivity minimization 127
6.1 Introduction............. 127
6.2 A pole-zero sensitivity measure . . . . 129
6.3 The eigenvalue sensitivity problem . . 131
6.4 Pole sensitivity minimization and normal matrices 139
6.5 Zero sensitivity measure . . . . . . . . . . . . . . . 142
6.6 Pole-zero sensitivity coordinate dependence . . . . 144
6.7 Optimal realizations for pole-zero sensitivity minimization146
6.8 Numerical example 151
6.9 Conclusions ......................... 156
7 A synthetic sensitivity - roundoff design 159
7.1 Introduction . . . . . . . . . . . . 159
7.2 A synthetic FWL noise gain . . . 160
7.3 Optimizing the Total Noise Gain 168
7.4 A numerical example . . . . . . . 175
............
7.5 Conclusions 179
Appendix 7: Existence of a constrained minimum . 180
8 Sparse optimal and suboptimal realizations 183
8.1 Introduction . . . . . . . . . 183
8.2 Sparse optimal realizations 186
Contents ix
8.2.1 Hessenberg optimal realizations . . 187
8.2.2 Schur optimal realizations . . . . . 190
8.2.3 Sparse block-balanced realizations 192
8.3 Theoretical versus actual sensitivity measure 193
8.3.1 Pole sensitivity of Schur realizations 194
8.3.2 A numerical example . . . . . . . . . . 197
8.4 Sparse quasi-optimal realizations . . . . . . . 201
8.4.1 Constrained similarity transformations. 202
8.4.2 Extensions of the Bomar and Hung algorithm. 209
8.5 Sparse suboptimal realizations 211
8.6 Conclusion ............... 214
9 Parametrizations in control problems 217
9.1 Introduction.................... 217
9.2 Implementation of a pole placement controller. 223
9.2.1 The ideal pole placement controller. . . 223
9.2.2 Finite precision aspects in a closed loop compen-
sator: problem formulation ........... 225
9.2.3 Sensitivity analysis and optimal structures. .. 227
9.2.4 Roundoff noise study and optimal realizations. 231
9.2.5 Design example . . . . . . . 235
9.3 FWL LQG controller design. . . . . . . . . . . . . 244
9.3.1 The 'ideal' LQG controller. . . . . . . . . . 245
9.3.2 Roundoff noise study of an LQG controller 248
9.3.3 Optimal implementations of FWL LQG controllers250
9.3.4 Optimal FWL LQG controllers . . . . . . . 253
9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . 255
Appendix 9: Sensitivity functions of closed loop system 257
10 Synthetic FWL compensator design 261
10.1 Introduction ............. . 261
10.2 State space description of a compensator. 263
10.3 Analysis of FWL effects of a compensator 268
10.4 Optimal FWL compensator realizations 276
10.5 A design example . 279
10.6 Conclusion ..... 0' . . . . . • . . . . 285
11 Parametrizations in the Delta operator 289
11.1 Introduction and motivation .. . 289
11.2 Delta operator parametrizations .... . . 293
x Contents
11.3 Sensitivity of delta parametrizations 296
11.3.1 Sensitivity measure. . . 297
11.3.2 Optimal realization set. 299
11.3.3 Numerical example . 302
11.4 Roundoff noise analysis .... 305
11.4.1 Introduction ...... 305
11.4.2 The roundoff noise gain of shift and delta opera-
tor realizations . . . . . . . . . . . . . . . . .. 305
11.4.3 Minimization of the roundoff noise gain .. .. 308
11.4.4 New conditions for superiority of o-realizations 310
11.4.5 Numerical examples . . 314
11.5 Conclusions . . . . . . . . . . . 315
Appendix 11: Proof of Theorem 11.5 317
12 Generalized transfer function parametrizations for
adaptive estimation 319
12.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . 319
12.2 Parameter estimation and the information matrix. . 321
12.3 Connection between parametrization and data filtering . 327
12.4 Optimal and suboptimal design choices. . . . . . . . . . 332
12.4.1 Manipulating the information matrix with data
information . . . . . . 333
12.4.2 A robustness property . . . . . . 334
12.5 ,-operator parametrizations . . . . . . . 335
12.5.1 A transfer function in ,-operator 336
12.5.2 Computing the information matrix 337
12.5.3 Analysis of the information matrix 341
12.5.4 Comparison with o-operator parametrizations. 346
12.6 Applications in estimation and adaptive filtering 348
12.7 Conclusions . . . . . . . . . . . . . . . . . . . . . 356
References 359
Index 369
Preface
This book is all about finite wordlength errors in digital filters, con
trollers and estimators, and how to minimize the deleterious effects of
these errors on the performance of these devices. This does by no means
imply that all about finite wordlength errors in filters, controllers and
estimators is to be found in this book.
We first ventured into the world of finite wordlength effects in 1987
when Gang Li began his PhD thesis in this area. Our more experienced
readers might well say 'This shows', but we believe that the extent of
our new contributions largely offsets our relative inexperience about
the subject that might surface here and there in the book. Our naive
view on the subject of finite wordlength errors in 1987 could probably
be summarized as follows:
• numerical errors due to finite wordlength encoding and roundoff
are something that one has to live with, and there is probably
not much that can be done about them except to increase the
wordlength by improvements on the hardware;
• these errors are as old as finite arithmetic and numerical analysis
and they must therefore be well understood by now;
• thus, if something can be done to minimize their effects, it must
have been analysed and put into practice a long time ago.
It is almost fair to say that we were wrong on all counts. Indeed,
whereas numerical analysis has been in a fairly advanced state of de
velopment for some years, its practical implications in terms of filter
design has only begun to be understood in the late seventies. The first
results came out in 1976-77 and gave rise to the now celebrated MRH
structures (for Mullis, Roberts and Hwang, the inventors of digital filter
realizations that minimize the roundoff noise gain). More surprisingly,
the first hard results on the design of digital filters minimizing a certain
sensitivity measure of the transfer function with respect to coefficient
errors date back only to 1986. Our own first effort in this area was
to address the corresponding feedback controller design problem, with