Methods of Algebraic Geometry in Control Theory: Part II: Multivariable Linear Systems and Projective Algebraic Geometry

382 Pages·1999·8.28 MB·English

by  Peter Falb (auth.)

Peter Falb
Methods of Algebraic Geometry
in Control Theory: Part II
Multivariable Linear Systems and
Projective Algebraic Geometry
Springer Science+Business Media, LLC
Peter Falb
Division of Applied Mathematics
Brown University
Providence, RI 02912
Library of Congress Cataloging-in-Publication Data
Falb, Peter L.
Methods of algebraic geometry in control theory / Peter Falb.
p. cm.—
Includes bibliographical references
Contents: v. 1. Scalar linear systems and affine algebraic
geometry.
ISBN 978-1-4612-7194-9 ISBN 978-1-4612-1564-6 (eBook)
DOI 10.1007/978-1-4612-1564-6
1. Control theory. 2. Geometry, Algebraic. I. Title.
II. Series.
QA402.3.F34 1990
629.8'—dc20 90-223
CIP
AMS Subject Classifications: 14-01,14L17,14M15,14N05,93A25,93B27,93C35
Printed on acid-free paper.
© 1999 Springer Science+Business Media New York
Originally published by Birkhäuser Boston in 1999
Softcover reprint of the hardcover 1st edition 1999
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection
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tronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter devel
oped is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are
not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and
Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4612-7194-9
Typeset in iAlfeX by T^Xniques, Inc., Cambridge, MA.
9 8 7 6 5 4 3 21
Contents
Preface vii
Introduction 1
1 Scalar Input or Scalar Output Systems 7
2 Two or Three Input, Two Output Systems: Some Examples 35
3 The Transfer and Hankel Matrices 59
4 Polynomial Matrices 79
5 Projective Space 105
6 Projective Algebraic Geometry I: Basic Concepts 113
7 Projective Algebraic Geometry II: Regular Functions,
Local Rings, Morphisms 129
8 Exterior Algebra and Grassmannians 143
9 The Laurent Isomorphism Theorem: I 161
10 Projective Algebraic Geometry III: Products, Graphs, Projections 173
vi Contents
11 The Laurent Isomorphism Theorem: II 181
12 Projective Algebraic Geometry IV: Families, Projections, Degree 191
13 The State Space: Realizations, Controllability, Observability,
Equivalence 201
14 Projective Algebraic Geometry V: Fibers of Morphisms 223
15 Projective Algebraic Geometry VI: Tangents, Differentials,
Simple Subvarieties 231
16 The Geometric Quotient Theorem 243
17 Projective Algebraic Geometry VII: Divisors 259
18 Projective Algebraic Geometry VIII: Intersections 271
19 State Feedback 283
20 Output Feedback 313
APPENDICES
A Formal Power Series, Completions,
Regular Local Rings, and Hilbert Polynomials 329
B Specialization, Generic Points and Spectra 349
C Differentials 357
D The Space C~ 361
E Review of Affine Algebraic Geometry 367
References 375
Glossary of Notations 381
Index 383
Preface
"Control theory represents an attempt to codify, in mathematical terms, the
principles and techniques used in the analysis and design of control systems.
Algebraic geometry may, in an elementary way, be viewed as the study of the
structure and properties of the solutions of systems of algebraic equations. The
aim of this book is to provide access to the methods of algebraic geometry
for engineers and applied scientists through the motivated context of control
theory" .*
The development which culminated with this volume began over twenty-five
years ago with a series of lectures at the control group of the Lund Institute
of Technology in Sweden. I have sought throughout to strive for clarity, often
using constructive methods and giving several proofs of a particular result as
well as many examples.
The first volume dealt with the simplest control systems (i.e., single input,
single output linear time-invariant systems) and with the simplest algebraic
geometry (i.e., affine algebraic geometry). While this is quite satisfactory and
natural for scalar systems, the study of multi-input, multi-output linear time
invariant control systems requires projective algebraic geometry.
Thus, this second volume deals with multi-variable linear systems and pro
jective algebraic geometry. The results are deeper and less transparent, but are
also quite essential to an understanding of linear control theory. A review of
* From the Preface to Part 1.
viii Preface
the scalar theory is included along with a brief summary of affine algebraic
geometry (Appendix E).
Acknowledgements. Although he did not playa direct role in this work, I should
like to express my deep appreciation for the inspiration provided by my very
dear friend, the late George Zames.
There are a great many friends, colleagues, teachers, and students to whom
considerable thanks are due, but I should like especially to express my appre
ciation to Karl Astrom of the Lund Institute of Technology for my original
involvement in the precursor lectures and to the Laboratory for Information
and Decision Systems at M.LT. and its Director, Sanjoy K. Mitter, for the
use of a quiet office (without which this book would never have been written).
Thanks are also due to Elizabeth Loew for the excellent computer preparation
of the manuscript.
Finally, I dedicate this work to my (long suffering) dear wife, Karen.
Peter Falb
Cambridge, 1999
Introduction
We recall from Part I that "the overall goal of these notes is to provide an
introduction to the ideas of algebraic geometry in the motivated context of
system theory." We shall suppose familiarity with the development in Part I and
we adopt the same general approach and conventions. (See also Appendix E.)
Part II deals with multivariable (Le., several input, several output) linear sys
tems and projective algebraic geometry. This represents, in essence, the second
stage of both system theory and algebraic geometry. The results extend the
material in Part I, but, as we shall see, the extension is not entirely straightfor
ward.
We begin with a brief review of the scalar theory and extend it directly to sys
tems with either a single input or a single output. We recall and introduce seven
representations, namely: (1) a strictly proper rational meromorphic vector; (2)
a vector of coprime polynomials; (3) a block (with vector blocks) Hankel matrix
of finite rank; (4) a triple of matrices; (5) a curve in a projective space; (6) a pair
of matrices modulo equivalence (the observability, controllability pair); and, (7)
a causal k[z]-module homomorphism of power series. The representations (1),
(2), (3), (4) extend those developed in Part I. The representations (5) and (6)
were sketched in Chapter I.23 and the representation (7) is developed here. We
treat the case of a single input p x 1 with p ~ 1, system here. We develop the
entire theory (with the exception of the pole placement theorem) including the
appropriate transition theorems and the geometric quotient theorem. We also
2 Methods of Algebraic Geometry in Control Theory: Part II
introduce some notation for indices and the idea of a p-partition of n. The single
input theory essentially mimics the scalar theory of Part 1.
In order to motivate the general theory, we treat a number of examples of
two or three input, two output systems in detail. We observe that the degree
of such a system need not be the length of the shortest recurrence, that a co
prime matrix representation involves equivalence under the unimodular group;
and, that the curve in a projective space actually lies in a Grassmannian. We
give an explicit algebraic structure to Rat(n, 2, 2) and Rat(n, 3, 2) involving a
set of degree conditions and some Grassmann equations. Critical is a natural
embedding into a Grassmannian. Next we examine the Laurent map and de
velop an explicit algebraic structure for Hank(2, 2, 2) and Hank(2, 3, 2). We also
"prove" a geometric quotient theorem for these low order cases. It should be
clear from this critical Chapter 2 that the general extension of the theory is not
straightforward.
Chapters 3-8 are devoted to some basic concepts of system theory and pro
jective algebraic geometry. The transfer and Hankel matrices are introduced
using the module formulation ([F -3]) and the notion of a strictly proper ratio
nal meromorphic transfer matrix is defined. Block Hankel matrices are defined
and the key structure lemma of Risannen (Lemma 3.42) is established. The sys
tem spaces Rat(n,m,p) and Hank(n,m,p) are defined and shown to depend
on a finite number of parameters via an embedding in A~(mp+l). Polynomial
matrices are studied in Chapter 4. The unimodular group, the Hermite normal
form and the concepts of divisibility (right or left) and coprimeness are de
veloped. The representation (P(z), Q(z)) by coprime matrices is analyzed and
minimal realizations defined. The degree and Kronecker set of a system are also
introduced. The Transfer Lemma (Lemma 4.50) which is the transition theorem
to the (P(z), Q(z)) representation is also established.
We begin the work on projective algebraic geometry with a brief account of
projective space. Linear subspaces, dimension, homogeneous and affine coordi
nates, the notion of general position and the projective group, PGL(N), are
developed. The join of subspaces is analyzed. The critical concept of a projec
tion with center a linear subspace is also treated. In Chapter 6, we deal with
some basic concepts revolving around the notion of a projective algebraic set.
Graded rings, homogeneous ideals and the Hilbert polynomial are commutative
algebra ideas we need. The (homogeneous) ideal of V, the Zariski topology and
the homogeneous coordinate ring are defined and the projective Nullstellensatz
is proved. Some ::;ignificC1nL differences with the affine case occur; notably, (i)
the homogeneous coordinate ring is not an invariant for a projective variety
(it depends on the embedding in projective space); and (ii) the "irrelevant"
ideal in the Nullstellensatz. The concept of dimension for projective varieties is
treated and several results established including a theorem (Theorem 6.57) on