Linear Time Varying Systems and Sampled-data Systems

370 Pages·2001·3.85 MB·English

by  Akira Ichikawa

Lecture Notes
in Control and Information Sciences 265
Editors: M. Thoma • M. Morari
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Series Advisory Board
A. Bensoussan • M.]. Grimble • P. Kokotovic • A.B. Kurzhanski •
H. Kwakernaak • ].L. Massey
Authors
Akira Ichikawa, PhD
Department of Electrical and Electronic Engineering, Shizuoka University,
Hamamatsu 432 8561, napaJ
Hitoshi Katayama, PhD
Department of Electro-Mechanical Engineering,
Osaka Electro-Communication University, Neyagawa 572 8530, Japan
ISBN 1-85233-439-8 Springer-Verlag London Berlin Heidelberg
British Library Cataloguing in Publication Data
A catalog record for this book is available from the British Library
Library of Congress Cataloging-in-Publication Data
Ichikawa, Akira, 1947-
Linear time varying systems and sampled-data systems / Akira Ichikawa and Hitoshi Katayama.
p. cm. -- (Lecture notes in control and information sciences ; 265)
Includes bibliographical references and index.
ISBN 1-85233-439-8 (alk. paper)
.1 Adaptive control systems. 2. Linear systems. I. Katayama, Hitoshi, 1965- II. Title
III. Series.
712173 .I16 2001
629.8'36--dc21 00-069842
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Preface
The system theory for linear time-invariant systems is now mature and offers
a wide range of system theoretic concepts, design methods and solutions to
optimal or suboptimal control problems including the design of state feed-
back controllers and observers, optimal quadratic regulators, Kalman filters,
coprime factorization and Youla-parametrization of stabilizing controllers, H2
control, differential games, ~oH control and robust control. One of the most
important recent developments is, without doubt, ~oH control. Since the be-
ginning of the linear systems theory many researchers have made constant
efforts to extend the theory to time-varying systems and sampled-data sys-
tems as well as to infinite dimensional systems. Although there are many
excellent books on the systems theory of linear time-invariant systems, there
are not many books covering recent developments for time-varying systems.
In this monograph we consider linear optimal regulators, 2H control, differen-
tial games, ~oH control and filtering, and develop the theory for time-varying
systems and jump systems. Jump systems arise when impulse controls are
involved. As is well-known sampled-data systems can be written as jump
systems with constant coefficients which are regarded as periodic systems
with period equal to the sampling period. One of our main motivations for
writing this monograph is to develop the 2H and ~oH theory of sampled-data
systems from the jump system point of view. The jump system is a natural
state-space representation of sampled-data systems and original signals and
parameters are maintained in the new system. The 2H and ooH problems
for jump systems can be treated in a unified manner as for time-invariant
systems. Moreover, they can be directly extended to more general cases of
delayed observation, first-order hold and infinite dimensional systems. Jump
systems are also useful to design stabilizing controllers for certain nonlinear
systems. Since lump systems with constant coefficients are periodic systems
and hence time-varying systems, it is useful to develop the system theory
for time-varying systems. Extension of the system theory to time-varying
systems seems routine, but there are some inherent features of time-varying
systems. For example, frequency domain arguments cannot be extended and
the state-space approach is needed. Some arguments for time-invariant sys-
tems may not have easy extensions to time-varying systems. The ooH theory
based on X and Y Riccati equations is such an example as we see in Chapter
2. Hence the systems theory for time-varying systems itself is important and
vi
interesting and gives some new points of view "1o new insights into the system
theory of time-invariant systems.
In Chapter 2 we consider continuous-time systems and consider stabil-
ity, quadratic control, differential games, ~oH control, H~ filtering and H2
control. In ~oH control and filtering we allow for initial uncertainty in the
system and develop the general theory of this case. We give examples and
computer simulations for most of main results. Chapter 3 is concerned with
discrete-time systems and discusses the same topics as in Chapter 2. Chapter
4 introduces the jump system which contains both continuous- and discrete-
time features and discusses the same problems as in earlier chapters. Chapter
5 covers a special case of jump systems which arises from the sampled-data
systems with zero-order hold and applies the main results of Chapter 4 to
them. Finally in Chapter 6 we discuss further developments in the theory of
jump systems. We first give an extension to infinite dimensions and as an
example we consider H2 and ~oH control for sampled-data systems with first-
order hold. We also introduce sampled-data fuzzy systems which can express
certain nonlinear sampled-data systems and show how to design stabilizing
output feedback controllers using jump systems.
Chapter 2 is an introduction to time-varying continuous-time systems
while Chapter 3 is an introduction to discrete-time systems and either of them
can be read independently of the rest of the monograph. To read Chapter 4
the materials in Chapters 2 and 3 will be very helpful. To read Section 6.1
elements of functional analysis are necessary.
Co~e~s
1 Introduction 1
1.1 Continuous-time Systems and Discrete-time Systems ..... 2
1.2 Jump Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Sampled-data Systems . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Infinite Dimensional Systems and Sampled-data
Fuzzy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Continuous-time Systems 7
2.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Lyapunov Equations . . . . . . . . . . . . . . . . . . . 7
2.1.2 Performance Measures of Stable Systems ....... 16
2.1.3 Quadratic Control . . . . . . . . . . . . . . . . . . . . 19
2.1.4 Disturbance Attenuation Problems ........... 27
2.2 H~o Control and Differential Games .............. 36
2.2.1 Finite Horizon Problems . . . . . . . . . . . . . . . . . 38
2.2.2 The Infinite Horizon Problem .............. 45
2.3 Hoo Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.2 Full Information Problem ................ 56
2.3.3 Disturbance Feedforward Problem ........... 67
2.3.4 Backward Systems . . . . . . . . . . . . . . . . . . . . 69
2.3.5 Proofs of Main Results . . . . . . . . . . . . . . . . . . 73
2.4 Hoo Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.5 H2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.5.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . 84
2.5.2 Proofs of Main Results . . . . . . . . . . . . . . . . . . 86
2.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 94
3 Discrete-time Systems 95
3.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.1.1 Lyapunov Equations . . . . . . . . . . . . . . . . . . . 95
viii Contents
3.1.2 Performance Measures of Stable Systems ....... 102
3.1.3 Quadratic Control . . . . . . . . . . . . . . . . . . . . 106
3.1.4 Disturbance Attenuation Problems ........... 113
3.2 H~ Control and Quadratic Games ............... 122
3.2.1 Finite Horizon Problems . . . . . . . . . . . . . . . . . 123
3.2.2 The Infinite Horizon Problem .............. 132
3.3 Hc¢ Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.3.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . 139
3.3.2 Full Information Problem ................ 146
3.3.3 Disturbance Feedforward Problem ........... 154
3.3.4 Backward Systems . . . . . . . . . . . . . . . . . . . . 156
3.3.5 Proofs of Main Results . . . . . . . . . . . . . . . . . . 159
3.4 H~ Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
3.5 H2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.5.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . 171
3.5.2 Proofs of Main Results . . . . . . . . . . . . . . . . . . 173
3.6 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 182
4 Jump Systems 183
4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
4.1.1 Lyapunov Equations . . . . . . . . . . . . . . . . . . . 183
4.1.2 Performance Measures of Stable Systems ....... 192
4.1.3 Quadratic Control . . . . . . . . . . . . . . . . . . . . 198
4.1.4 Disturbance Attenuation Problems ........... 207
4.2 H~ Control ............................ 223
4.2.1 Main Results ....................... 223
4.2.2 H~ Riccati Equations .................. 229
4.2.3 Backward Systems .................... 251
4.2.4 Proofs of Main Results .................. 257
4.2.5 The General Case .................... 264
4.3 H~ Filtering ........................... 268
4.4 H2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
4.4.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . 277
4.4.2 Proofs of Main Results . . . . . . . . . . . . . . . . . . 278
4.5 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 289
5 Sampled-data Systems 291
5.1 Jump System Approach . . . . . . . . . . . . . . . . . . . . . 291
5.1.1 Transformation to Jump Systems ............ 291
5.1.2 Comments on the Sampling Period ........... 293
5.1.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 295
5.1.4 Quadratic Control . . . . . . . . . . . . . . . . . . . . 298
5.2 H~ Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
5.2.1 Finite Horizon Problems . . . . . . . . . . . . . . . . . 301
5.2.2 The Infinite Horizon Problem .............. 308
Contents ix
5.3 H2 Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
5.4 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 319
6 Further Developments 321
6.1 Jump Systems in Infinite Dimensions .............. 321
6.1.1 Ha Control . . . . . . . . . . . . . . . . . . . . . . . . 322
6.1.2 H2 Control . . . . . . . . . . . . . . . . . . . . . . . . 324
6.1.3 Sampled-data Systems with First-order Hold ..... 328
6.2 Sampled-data Fuzzy Systems .................. 333
6.2.1 Sampled-data Fuzzy Systems .............. 333
6.2.2 The Case with Premise Variable y ........... 338
6.2.3 The Case with Premise Variable x ........... 339
6.3 Notes and References . . . . . . . . . . . . . . . . . . . . . . . 346
A Basic Results of Functional Analysis 349
References 353
Index 359