Table Of ContentSystems & Control: Foundations & Applications
Series Editor
Christopher I. Byrnes, Washington University
Associate Editors
S.-1. Amari, University of Tokyo
B.D.O. Anderson, Australian National University, Canberra
Karl lohan Astrom, Lund Institute of Technology, Sweden
Jean-Pierre Aubin, CEREMADE, Paris
H.T. Banks, University of Southern California, Los Angeles
John S. Baras, University of Maryland, College Park
A. Bensoussan, INRIA, Paris
John Bums, Virginia Polytechnic Institute, Blacksburg
Han-Fu Chen, Beijing University
M.H.A. Davis, Imperial College of Science and Technology, London
Wendell Fleming, Brown University, Providence, Rhode Island
Michel Fliess, CNRS-ESE, Gif-sur-Yvette, France
Keith Glover, University of Cambridge, England
Diederich Hinrichsen, University of Bremen, Federal Republic of Germany
Alberto Isidori, University of Rome
B. Jakubczyk, Polish Academy of Sciences, Warsaw
Hidenori Kimura, University of Osaka, Japan
Arthur J. Krener, University of California, Davis
H. Kunita, Kyushu University, Japan
Alexandre Kurzhansky, IIASA, Laxenburg, Austria and Academy of Sciences,
U.S.S.R.
Harold J. Kushner, Brown University, Providence, Rhode Island
Anders Lindquist, Royal Institute of Technology, Stockholm
Andrzej Manitius, George Mason University, Fairfax, Virginia
Clyde F. Martin, Texas Tech University, Lubbock, Texas
Sanjoy Mitter, Massachusetts Institute of Technology, Cambridge
Giorgio Picci, University of Padova, Italy
Boris Pshenichnyj, Glushkov Institute of Cybernetics, Kiev
H.J. Sussman, Rutgers University, New Brunswick, New Jersey
T.J. Tam, Washington University, St. Louis, Missouri
V.M. Tikhomirov, Institute for Problems in Mechanics, Moscow
Pravin P. Varaiya, University of California, Berkeley
Jan C. Willems, University of Groningen, Sweden
W.M. Wonham, University of Toronto
TamerBa~ar
Pierre Bernhard
HOO -Optimal Control
and Related
Minimax Design Problems
A Dynamic Game Approach
1991 Springer Science+Business Media, LLC
TamerBa~ar Pierre Bernhard
Coordinated Science Laboratory Inria
University of Illinois Unite de Recherche Sophia-Antipolis
Urbana, Illinois 61801 Valbone Cedex
USA France
Printed on acid-free paper.
© Springer Science+Business Media New York 1991
Originally published by Birkhiiuser Boston in 1991
Softcover reprint of the hardcover 1st edition 1991
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photocopying,
recording or otherwise, without prior permission of the copyright owner.
ISBN 978-1-4899-3563-2 ISBN 978-1-4899-3561-8 (eBook)
DOI 10.1007/978-1-4899-3561-8
Camera-ready copy prepaired by the author using TeX.
987654321
Preface
One of the major concentrated activities of the past decade in control
theory has been the development of the so-called "HOO-optimal control
theory," which addresses the issue of worst-case controller design for linear
plants subject to unknown additive disturbances, including problems of
disturbance attenuation, model matching, and tracking. The mathematical
symbol "HOO" stands for the Hardy space of all complex-valued functions
of a complex variable, which are analytic and bounded in the open right
half complex plane. For a linear (continuous-time, time-invariant) plant,
the Hoo norm of the transfer matrix is the maximum of its largest singular
value over all frequencies.
Controller design problems where the HOO norm plays an important
role were initially formulated by George Zames in the early 1980's, in the
context of sensitivity reduction in linear plants, with the design problem
posed as a mathematical optimization problem using an (HOO) operator
norm. Thus formulated originally in the frequency domain, the main tools
used during the early phases of research on this class of problems have been
operator and approximation theory, spectral factorization, and (Youla)
parametrization, leading initially to rather complicated (high-dimensional)
optimal or near-optimal (under the HOO norm) controllers. Follow-up work
in the mid 1980's have shown, however, that the maximum McMillan de
gree of these controllers is in fact in the order of the M eMil/an degree of the
overall system transfer matrix, and further work has shown that in a time
domain characterization of these controllers (generalized) Riccati equations
of the type that arises in linear-quadratic differential games playa key role.
These findings have prompted further accelerated research on the topic, for
more direct time-domain (state-space) derivation of these results - a direc
tion which has also led to more general formulations, including time-varying
plants and finite design horizons.
Among different time-domain approaches to this class of worst-case
design problems, the one that uses the framework of dynamic (differen-
v
tial) game theory seems to be the most natural. This is so because the
original HOO-optimal control problem (in its equivalent time-domain for
mulation) is in fact a minimax optimization problem, and hence a zero-sum
game, where the controller can be viewed as the minimizing player and
disturbance as the maximizing player. Using this framework, we develop
in this book a complete theory that encompasses continuous-time as well
as discrete-time systems, finite as well as infinite horizons, and several dif
ferent measurement schemes, including closed-loop perfect state, delayed
perfect state, sampled state, closed-loop imperfect state, delayed imperfect
state and sampled imperfect state information patterns.
It is our belief that the theory is now at a stage where it can easily be
incorporated into a second-level graduate course in a control curriculum,
that would follow a basic course in linear control theory covering LQ and
LQG designs. The framework adopted in this book makes such an ambi
tious plan possible, and indeed, both authors will be teaching such courses
during the Spring 1991 semester. The first author will incorporate this
material into an optimal control course at the University of Illinois, and
* **
the second author will be teaching such a course at ESSI and ISlA in
Sophia Antipolis, and at Ecole Poly technique in Paris.
For the most part, the only prerequisite for this book is a basic knowl
edge of linear control theory, at the level of, say, the texts by Kwakernaak
and Sivan (1972) or Anderson and Moore (1990). No background in dif
ferential games, or game theory in general, is required, as the requisite
concepts and results have been developed in the book at the appropriate
level. The book is written in such a way that makes it possible to follow the
theory for the continuous- and discrete-time systems independently, so that
a reader who is interested only in discrete-time systems, for example, may
skip Chapters 4 and 5 without any disruption in the development of the
theory. On the other hand, for the benefit of the reader who is interested
in both continuous- and discrete-time results, we have included statements
in each relevant chapter, that place the comparisons in proper perspective.
The general framework adopted in this book, and the methodologies de
veloped, should also make it possible for the interested reader to extend
*
Ecole Superieure des Sciences de l'Informatique, University of Nice.
**
Institut Superieur d'[nformatique et d'Automatique, Ecole des Mines
de Paris.
VI
these results to more general classes of systems, such as nonlinear plants
and/or distributed-parameter systems. We, in fact, present in the book
some preliminary results on the former class of problems.
This project was conceived about six months ago at an International
Symposium on Differential Games and Applications, held in Helsinki, Fin
land, and the collaboration between the two authors was facilitated by a
ten-day visit of the second author to Urbana in October, and by one of the
most incredible technological advances of the past decade - the electronic
mail. The first author would like to thank his colleagues at the Decision and
Control Laboratory of the University of Illinois (UI), Urbana-Champaign,
for their encouragement and support in the writing of this book. Garry
Didinsky, a Ph.D. student at UI, read parts of the manuscript, and besides
providing valuable input, also contributed to the development of some of
the results. Parts of the manuscript have also benefited from the comments
of the participants at a one-day short course on the topic, offered by the first
author at the Coordinated Science Laboratory of the University of Illinois,
on October 2, 1990. The second author wishes to acknowledge, beyond
the support of INRIA and the steadfast encouragement of its President,
his share of the enthusiasm generated at the Coordinated Science Labora
tory whose support and warm welcome during his stay in Urbana has been
deeply felt. Finally, our thanks go to Francie Bridges, at the Decision and
Control Laboratory, for her excellent assistance in word processing, which
has transformed the hand-written version into the final form seen here.
Tamer Ba§ar Pierre Bernhard
Urbana, Illinois, USA Sophia Antipolis, Valbonne, France
January 2, 1991 January 2, 1991
VII
To
Tangiil et Annette
Contents
Preface v
1 A General Introduction to Minimax (Hoo.Optimal) Designs 1
1.1 A Brief History . . . . . . . . . . . . . . . . . . . . . . . .. 1
1.2 A Relationship Between Hoo.Optimal Control and LQ Zero-
Sum Dynamic Games ........ . 4
1.3 Discrete- and Continuous-time Models 7
1.4 Organization of the Book ...... . 9
1.5 Conventions, Notation and Terminology 10
2 Basic Elements of Static and Dynamic Games 13
2.1 Static Zero-Sum Games .... 13
2.2 Discrete-Time Dynamic Games 16
2.3 Continuous-Time Dynamic Games 24
3 The Discrete·Time Minimax Design Problem With Perfect
State Measurements 28
3.1 Introduction......................... 28
3.2 The Soft-Constrained Linear-Quadratic Dynamic Game 29
3.2.1 Open-loop information structure for both players. 30
3.2.2 Closed-loop perfect state information for both players. 32
3.2.3 Closed-loop I-step delay information for both players. 38
3.2.4 An illustrative example ................ 42
