Table Of ContentSystems & Control: Foundations & Applications
Founding Editor
Christopher I. Byrnes, Washington University
Mare A. Peters
Pablo A. Iglesias
Minimum Entropy Control
for Time-Varying Systems
Springer Science+Business Media, LLC
Marc A. Peters Pablo A. Iglesias
Dept. of Electrical Dept. of Electrical
& Computer Engineering & Computer Engineering
The Johns Hopkins University The Johns Hopkins University
Baltimore, MD 21218 Baltimore, MD 21218
Printed on acid-free paper
© 1997 Springer Science+Business Media New York
Origina11y published by Birkhăuser Boston in 1997
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ISBN 978-1-4612-7366-0 ISBN 978-1-4612-1982-8 (eBook)
DOI 10.1007/978-1-4612-1982-8
Typeset by the Authors in TEX.
987 6 543 2 1
Contents
Preface ix
1 Introduction 1
1.1 Optimal control problems 2
1.2 Minimum entropy control 6
1.3 The maximum entropy principle 8
1.4 Extensions to time-varying systems. 12
1.5 Organization of the book ..... . 13
2 Preliminaries 15
2.1 Discrete-time time-varying systems 16
2.2 State-space realizations 21
2.3 Time-reverse systems .. 24
3 Induced Operator Norms 26
3.1 Characterizations of the induced norm 27
3.2 Time-varying hybrid systems . . . . . 30
3.2.1 Sampled continuous-time systems. 31
3.2.2 Continuous-time systems with piecewise constant inputs 37
3.2.3 Hybrid feedback systems . 41
3.3 Computational issues . 44
4 Discrete-Time Entropy 46
4.1 Entropy of a discrete-time time-varying system 47
4.2 Properties..................... 49
4.2.1 Equivalence with the entropy integral . 52
4.2.2 Entropy in terms of a state-space realization 53
v
vi Contents
4.3 Entropy and information theory. 54
4.4 Entropy of an anti-causal system 57
4.5 Entropy and the W-transform . 61
4.6 Entropy of a non-linear system . 64
5 Connections With Related Optimal Control Problems 69
5.1 Relationship with 1-loo control. 70
5.2 Relationship with 1-l2 control 73
5.3 Average cost functions . 79
5.3.1 Average 1-l2 cost ... 79
5.3.2 Average entropy . . . 82
5.4 Time-varying risk-sensitive control 85
5.5 Problems defined on a finite horizon 91
6 Minimum Entropy Control 94
6.1 Problem statement 95
6.2 Basic results ....... . 96
6.3 Full information .... . 100
6.3.1 Characterizing all closed-loop systems 100
6.3.2 FI minimum entropy controller 104
6.4 Full control . . . . . . . . . . . . . . . . . . . 105
6.5 Disturbance feedforward ........... . 110
6.5.1 Characterizing all closed-loop systems 111
6.5.2 DF minimum entropy controller 114
6.6 Output estimation 116
6.7 Output feedback . 118
6.8 Stability concepts. 123
7 Continuous-Time Entropy 128
7.1 Classes of systems considered . . . . . . . . . . . . 129
7.2 Entropy of a continuous-time time-varying system 130
7.3 Properties ...................... . 131
7.3.1 Equivalence with the entropy integral .. . 136
7.3.2 Entropy in terms of a state-space realization 137
7.3.3 Relationship with discrete-time entropy .. 139
7.4 Connections with related optimal control problems 143
7.4.1 Relationship with 1-loo control ..... . 143
7.4.2 Relationship with 1-1.2 control ..... . 146
7.4.3 Relationship with risk-sensitive control. 149
Contents vii
7.5 Minimum entropy control 150
A Proof of Theorem 6.5 154
B Proof of Theorem 7.21 167
Bibliography 172
Notation 180
Index 183
Preface
Minimum entropy control has been studied extensively for linear time-
invariant systems, both in the continuous-time and discrete-time cases.
Controllers that satisfy a closed-loop minimum entropy condition are known
to have considerable advantages over other optimal controllers. While guar-
anteeing an 1leX) norm bound, the entropy is an upper bound for the H2
norm of the system, and thus minimum entropy controllers provide a degree
of performance sacrificed by other Hoc controllers.
These advantages make it desirable to extend the theory of minimum
entropy control to other settings. This is not straightforward, since the
notion of entropy is defined in terms of the system's transfer function,
which other systems may not admit.
In this book we provide a time-domain theory of the entropy criterion.
For linear time-invariant systems, this time-domain notion of entropy is
equivalent to the usual frequency domain criterion. Moreover, this time-
domain notion of entropy enables us to define a suitable entropy for other
classes of systems, including the class of linear time-varying systems. Fur-
thermore, by working with this time-domain definition of the entropy we
are able to gain new interpretations of the advantages of minimum entropy
control. In particular we consider the connections between the time-varying
minimum entropy control problem and the time-varying analogues to the
H2, Hoc and risk-sensitive control problem.
The majority of the work presented here arose as part of the first au-
thor's doctoral dissertation in the Department of Electrical and Computer
Engineering at The Johns Hopkins University. We are grateful to our col-
leagues in the department, particularly to Jack Rugh, who has always been
there to answer our questions.
The second author would also like to thank Professor Harry Dym. It
IX
x Preface
was during a visit to Rehovot in May 1992 that the ideas behind this re-
search were first conceived. We are also indebted to Abbie Feintuch who
suggested a means for comparing the entropy operator to the quadratic
control problem.
We are most grateful for the financial support of the National Science
Foundation, under grant number ECS-9309387.
We would also like to thank the staff at Birkhauser for their excellent
cooperation.
Finally, we would like to thank our friends and families for their sup-
port and interest over the years. Above all, we are grateful to Elsbeth and
Elizabeth, to whom we dedicate this book, for their patience and encour-
agement.
Marc A. Peters
Pablo A. Iglesias
Baltimore, Maryland
August 1996
Introduction 1
Minimum entropy control provides a means of trading off some of the fea-
tures of other control problems, namely 1-£2 optimal control and 1-£00 control.
For continuous-time, linear time-invariant (LTI) systems, a complete solu-
tion to the minimum entropy control problem is found in the monograph
by Mustafa and Glover [64].
This book is devoted to the extension of this theory to time-varying
systems. Using concepts from operator theory, we provide a time-domain
interpretation for the entropy. This notion, equivalent to the frequency
domain version of [64], allows us to consider more general classes of systems.
In this chapter we provide an introduction to the minimum entropy
control problem. We do so by first considering two related optimal control
problems: the 1-£2 and 1-£00 control problems. These two optimal control
theories were proposed as means of achieving conflicting goals in the system:
performance and robustness, respectively. We will show how minimum
entropy control allows the designer to trade off these goals.
Finally, since the word entropy is usually associated with the fields of
thermodynamics and information theory, we also present a brief introduc-
tion to the connection that exists between these notions of entropy, and the
system entropy that is considered in this book.
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M. A. Peters et al., Minimum Entropy Control for Time-Varying Systems
© Birkhäuser Boston 1997
