Lecture Notes in
Control and
Information Sciences
Edited by M.Thoma and A.Wyner
1118
L. Dai
Singular Control Systems
Springer-Verlag
Berlin Heidelberg New York
London
Paris Tokyo
Series Editors
M. Thoma • A. Wyner
Advisory Board
U D. Davisson • A. G. J. MacFarlane ° H. Kwakernaak
J. L. Massey • Ya Z. Tsypkin • A. J. Viterbi
Author
Liyi Dai
Division of Applied Sciences
Harvard University
Cambridge, MA 02138
USA
ISBN 3-540-50724-8 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-50724-8 Springer-Verlag NewYork Berlin Heidelberg
Library of Congress Cataloging in Publication Data
Dai, L. (Liyi)
Singular control systems / L. Dai.
(Lecture notes in control and information sciences ; 118)
Bibliography: p.
Includes index.
ISBN 0-387-50724-8 :
1. Automatic control. 2. Control theory. 3. Linear systems.
I. Title. I1. Series.
TJ213.D25 1989
629.8-dc19 89-4091
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© Springer-Verlag Bedin, Heidelberg 1989
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Binding: B.Helm, Berlin
2161/3020-543210
To Yun Wang
PREFACE
Singular systems are found in engineering systems (such as electrical circuit
network, power systems, aerospace engineering, and chemical processing), social sys-
tems, economic systems, biological systems, network analysis, time-series analysis,
singular singularly perturbed systems with which the singular system has a great deal
to do, etc. Thei F form also makes them useful in system modeling. In many articles
singular systems are called the descriptor variable systems, the differential/algeb-
raic systems, the generalized state space systems, etc. Singular systems are governed
by the so-called singular differential equations, which endow the systems with many
special features that are not found in classical systems. Among these are impulse
terms and
input
derivatives in
the
state response, nonproperness of transfer matrix,
noncausality between input and state (or output), consistent initial conditions,
etc., making the study of singular systems more sophisticated than of classical li-
near systems. In my opinion, this is the reason why singular systems have attracted
interest in recent years. Currently, although much effort has been made in the explo-
ration of special properties for singular systems, the studies are mostly confined to
the
generalization
of classical system theory; of course, this is also an important
task in control theory.
There are mainly two approaches in the studies: geometric and algebraic. The
latter approach is adopted in this book. The primary purpose of writing this book is
summing up the development of singular control system theory and providing the
control circle with a systematic theory of the systems. Some acquaintance with linear
algebra and linear system theory is assumed. Those not familiar with these prerequi-
sites can read the book admitting the results in appendices. The systems are all li-
near and time invariant. Technically, the book focuses on the mathematical treatment
of singular systems from the point of view of the system, or, in other words, I make
an exclusive effort to discuss the analysis and design of singular control systems.
This makes possible the
systematic
design of singular control systems in practical
applications. Some examples are used only to illustrate the results and design proce-
dures in this book.
The book is organized as follows. Chapter 1 discusses the state response structure
for singular systems and the state space equivalent forms needed for later dlscu-
sion. Chapter 2 provides some fundamental concepts in the system analysis, such as
Vl
reachability, controllability, observability, realization theory, transfer matrix,
and so on. These conc:epts are essential for system analysis and design. Chapter 3
deals with the control problem. The closed-loop behavior is given under state feed-
back controls and detailed information on finite or infinite pole placement may be
obtained. Chapters 4 and 5 investigate some aspects of control system design for
singular systems; the state observation problem is discussed in Chapter 4 and the dy-
namic compensation problem is treated in Chapter 5. Results in these two chapters
show that under fairly general conditions, we may design normal observers and dyna-
mic compensators which may be realized as in linear ~ystem theory. Chapter 6 is devo-
ted to a special problem --- the structural stability, or, more precisely, the robus-
tness of stability, which is not as crucial in classical linear system theory as in
the singular system acse. It is shown that to guarantee structural stability, great
care must be excercised in singular systems. Chapter 7 presents results on the system
analysis and synthesis through the transfer matrix method. Three problems are solved:
the transfer matrix structure under state feedback control~ decoupling control, and
input function observers. Since the digital technique holds an important position in
practical system design. Chapter 8 explores discrete-time singular systems. Only
basic design concepts are studied. Chapter 9 contains basic results on optimal con-
trol theory and Chapter IO contains a preliminary discussion on topics for farther
research.
Liyi Dai
Beijing, PRC
August, 1988
