Table Of ContentGeometrical Methods for the Theory of Linear Systems
NATO ADVANCED STUDY INSTITUTES SERIES
Proceedings of the Advanced Study Institute Programme, which aims
at the dissemination of advanced knowledge and
the formation of contacts among scientists from different countries
The series is published by an international board of publishers in conjunction
with NATO Scientific Affairs Division
A Life Sciences Plenum Publishing Corporation
B Physics London and New York
C Mathematical and D. Reidel Publishing Company
Physical Sciences Dordrecht, Boston and London
D Behavioural and Sijthoff & Noordhoff International
Social Sciences Publishers
E Applied Sciences Alphen aan den Rijn and Germantown
U.S.A.
Series C - Mathematical and Physical Sciences
Volume 62 - Geometrical Methods for the Theory of Linear Systems
(}eonaeUical~ethods
for the Theory
of Linear Systenas
Proceedings ofa NATO Advanced Study Institute
and AMS Summer Seminar in Applied Mathematics
held at Harvard University, Cambridge, Mass., June 18-29, 1979
edited by
CHRISTOPHER I. BYRNES
Harvard University, Cambridge, Mass., U.S.A.
and
CLYDE F. MARTIN
Case Western Reserve University, Dept. of Mathematics,
Cleveland, Ohio, U.S.A.
D. Reidel Publishing Company
Dordrecht : Holland / Boston: U.S.A. / London: England
Published in cooperation with NATO Scientific Affairs Division
Library of Congress cataloging in Publication Data
Nato Advanced Study Institute, Harvard University, 1979.
Geometrical methods for the theory of linear systems.
(NATO advanced study institutes series: Series C, Mathematical and
physical sciences; v. 62)
1. System analysis-Congresses. 2. Geometry-Congresses.
I. Byrnes, Christopher 1.,1949- II. Martin, Clyde. III. AMS
Summer Seminar in Applied Mathematics, Harvard University, 1979.
IV. North Atlantic Treaty Organization. V. American Mathematical Society.
VI. Title. VII. Series.
QA402.N33 1979 003'.01'512 80-20465
ISBN-13: 978-94-009-9084-5 e-ISBN-13: 978-94-009-9082-1
001: 10.1007/978-94-009-9082-1
Published by D. Reidel Publishing Company
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All Rights Reserved
Copyright © 1980 by D. Reidel Publishing Company, Dordrecht, Holland
Softcoverreprintofthe hardawer 15tedition 1980
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without written permission from the copyright owner
TABLE OF CONTENTS
PREFACE ix
C. BYRNES, M. HAZEWINKEL, C. MARTIN and Y. ROUCHALEAU /
Introduction to Geometrical Methods for the
Theory of Linear Systems 1
C. I. BYRNES / Algebraic and Geometric Aspects of the
Analysis of Feedback Systems 85
M. HAZEWINKEL / (Fine) Moduli (Spaces) for Linear Systems:
What Are They and What Are They Good for? 125
C. MARTIN / Grassmannian Manifolds, Riccati Equations
and Feedback Invariants of Linear Systems 195
Y. ROUCHALEAU / Commutative Algebra in System Theory 213
H. H. ROSENBROCK / Systems and Polynomial Matrices 233
P. A. FUHRMANN / Functional Models, Factorizations
and Linear Systems 257
J. C. WILLEMS and J. H. VAN SCHUPP EN / Stochastic Systems
and the Problem of State Space Realization 283
Index 315
DEDICATION
This volume is dedicated to Roger Brockett, Robert Hermann,
and Rudolph Kalman in acknowledgement of their pioneering contri
butions in the use of geometric and topological methods in control
theory.
PREFACE
The lectures contained in this book were presented at Harvard
University in June 1979. The workshop at which they were presented
was the third such on algebro-geometric methods. The first was
held in 1973 in London and the emphasis was largely on geometric
methods. The second was held at Ames Research Center-NASA in
1976. There again the emphasis was on geometric methods, but
algebraic geometry was becoming a dominant theme. In the two
years after the Ames meeting there was tremendous growth in the
applications of algebraic geometry to systems theory and it was
becoming clear that much of the algebraic systems theory was very
closely related to the geometric systems theory. On this basis
we felt that this was the right time to devote a workshop to the
applications of algebra and algebraic geometry to linear systems
theory. The lectures contained in this volume represent all but
one of the tutorial lectures presented at the workshop. The lec
ture of Professor Murray Wonham is not contained in this volume
and we refer the interested to the archival literature.
This workshop was jointly sponsored by a grant from Ames
Research Center-NASA and a grant from the Advanced Study Institute
Program of NATO. We greatly appreciate the financial support
rendered by these two organizations. The American Mathematical
Society hosted this meeting as part of their Summer Seminars in
Applied Mathematics and will publish the companion volume of con
tributed papers.
Many people were involved in the preparations for the meeting
and in the preparation of this volume. We would like to specifi
cally thank the conference secretaries, Joyce Martin and Maureen
Ryan, and a special thanks to Joyce who typed this volume, often
from almost illegible copy. As the editors of this volume, we
would like in particular to thank Roger Brockett who served with
us as codirector and provided the leadership that is necessary
for such a conference to succeed.
Christopher Byrnes
Harvard University
Clyde Martin
Case Western Reserve University
IX
c.l. Byrnes and C. F. Martin, Geometrical Methods for the Theory of Linear Systems, ix.
Copyright © 1980 by D. Reidel Publishing Company.
INTRODUCTION TO GEOMETRICAL FOR THE THEORY OF LINEAR
~1ETHODS
SYSTEMS
C. Byrnes, M. Hazewinkel, C. Martin, and Y. Rouchaleau
In this joint totally tutorial chapter we try to discuss
those definitions and results from the areas of mathematics which
have already proved to be important for a number of problems in
linear system theory.
Depending on his knowledge, mathematical expertise and inter
ests, the reader can skip all or certain parts of this chapter O.
Apart from the joint section, the basic function of this chapter
is to provide the reader of this volume with enough readily avail
able background material so that he can understand those parts of
the following chapters which build on this--for a mathematical
system theorist perhap,s not totally standard--basic material.
The joint section is different in nature; it attempts to explain
some of the ideas and problems which were (and are) prominent in
classical algebraic geometry and to make clear that many of the
problems now confronting us in linear system theory are similar
in nature if not in detail. Thus we hope to transmit some intui
tion why one can indeed expect that the tools and philosophy of
algebraic geometry will be fruitful in dealing with the formid
able array of problems of contemporary mathematical system theory.
This section can, of course, be skipped without endangering one's
chances of understanding the remainder of this chapter and the
following chapters.
The contents of this introductory chapter are:
1. Historical prelude. Some problems of classical algebraic
geometry.
1.1 Plane algebraic curves
C. L Byrne, and C. F. MfUtin, Geometrical Methoth for the Theory of Linf!llr SyBtemB, 1-84.
Copyright © 1980 by D. Reidel PubliBhing Company.
2 C. BYRNES ET AL.
1.2 Riemann surfaces and fields of meromorphic func
tions
1.3 Invariants
2. Modules over Noetherian rings and principal ideal
domains
2.1 Noetherian rings and modules: fundamental results
2.2 Examples of Noetherian rings
2.3 On duality and the structure of modules over
Noetherian rings
2.4 Modules over a principal ideal domain
3. Differentiable manifolds, vector bundles and Grassman
nians
3.1 Differentiable manifolds
3.2 Partitions of unity
3.3 Vector bundles
3.4 On homotopy
3.5 Grassmannians and classifying vector bundles
4. Varieties, vector bundles, Grassmannians and Intersec
tion theory
4.1 Affine space and affine algebraic varieties
4.2 Projective space, projective varieties, quasi pro
jective varieties
4.3 Grassmann manifolds, algebraic vector-bundles
4.4 Intersection theory
5. Linear algebra over rings
5.1 Surjectivity of linear transformations. Nakayama's
lemma
5.2 Injectivity of linear transformations. Solving
Tx = y. Localization.
5.3 Structure of linear transformations. The Quillen
Suslin theorem (formerly the Serre conjecture)
1. SOME OF CLASSICAL ALGEBRAIC GEor1ETRY
PROBLE~1S
The purpose of this section is to give insight into certain
of the problems and achievements of 19th century algebraic geome
try, in a historical perspective. It is our hope that this per
spective, which for several reasons is limited, will go some of
the distance towards explaining some natural interrelations
between algebraic geometry and analysis, as well as a natural con
nection between algebraic geometry and linear system theory.
GEOMETRICAL METHODS FOR THE THEORY OF LINEAR SYSTEMS 3
1.1 Plane algebraic curves
To begin, perhaps the most primitive of algebraic
object~
geometry are varieties, e.g., p1an2 curves in [ (say the vari
ety defined by the equation y = x ), and the most primitive rela
tions are those of incidence, e.g., the intersection of varieties.
To fix the ideas, ~et us consider the problem of describing all
plane curves in [ and the problem of describing their intersec
tions. Since any two distinct irreducible (i.e., the polynomial
f(x,y), whose locus is the curve, is irreducible) curves inter
sect in finitely many points, the first problem of describing such
an intersection is to compute the number of such points in terms
of the two curves.
Now, whenever one speaks of a scheme for the description or
classification of objects, such as plane curves, one has in mind
a certain notion of equivalence. And, quite often, this involves
the notions of transformation. For example, if SL(2,[) is the
group of 2x2 matrices with determinant 1, then g E SL(2,[)
acts on [2 by linear change of variables and it has been known
since the introduction of Cartesian coordinates that a linear
change of coordinates leaves the degree of a curve invariant.
That is, if f(x,y) is homogeneous, then
(1.1.1)
has the same degree as f. So, for homogeneous f, we may begin
the classification scheme by fixing the degree. Now any f which
is homogeneous of degree 1 is a linear functional, and these
are well understood. If f is homogeneous of degree 2, then
one can check that the discriminant
l',(f) = b2 - 4ac,
where
f(x,y) = ax2 + bxy + cy2
is invariant under SL(2,[); i.e.,
l',(f) = l',(fg) , for all g E SL(2,[) • (1.1.2)
This explains, in part, why the discriminant is so important in
analytic geometry, but there really is a lot more to the story.
First of all, (1.1.2) asserts that the discriminant of f, l',(f) ,
is the same regardless of the choice of coordinates used to
express f (provided we allow only volume preserving, orienta
tion P2ese2ving changes of coordinates). But this is also true
for l', ,l', + 3, etc. In 1801, Gauss [2,4J proved an important
Description:The lectures contained in this book were presented at Harvard University in June 1979. The workshop at which they were presented was the third such on algebro-geometric methods. The first was held in 1973 in London and the emphasis was largely on geometric methods. The second was held at Ames Resear
