Methods of Algebraic Geometry in Control Theory: Part I: Scalar Linear Systems and Affine Algebraic Geometry

211 Pages·2018·14.51 MB·English

by  Peter Falb

Modern Birkhäuser Classics
Peter Falb
Methods of
Algebraic Geometry
in Control Theory:
Part I
Scalar Linear Systems and Affine
Algebraic Geometry
Modern Birkhäuser Classics

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Methods of A lgebraic
G eometry in Control
Theory:Part I
Scalar Linear Systems
and A ffine A lgebraic
G eometry
Peter Falb
Reprint of the 1990 Edition
Peter Falb
Division of Applied Mathematics
Brown University
Providence, Rhode Island, USA

ISSN 2197-1803 ISSN2197- 1811 (electr onic)
Modern Birkhäuser Classics
ISBN 978-3-319-98025-6 ISBN978-3-319-98026-3 (eBo ok)
https://doi.org/10.1007/978-3-319-98026-3

Library of Congress Control Number: 2018952484

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Peter Falb
Methods of Algebraic
Geometry in
Control Theory: Part I
Scalar Linear Systems and
Affine Algebraic Geometry
1990 Birkhauser
Boston · Basel · Berlin
Peter Falb
Division of Applied Mathematics
Brown University
Providence, Rhode Island 02912
USA
Library of Congress Cataloging-in-Publication Data
Falb, Peter L.
Methods of algebraic geometry in control theory I Peter Falb.
p. cm.-(Systems & control; v. 4)
Includes bibliographical references (p.
Contents: v. 1. Scalar linear systems and affine algebraic
geometry.
ISBN 0-8176-3454-1 (v. 1. : alk. paper)
1. Control theory. 2. Geometry, Algebraic. I. Title.
II. Series.
QA402.3.F34 1990
629.8'312--dc20 90-223
Printed on acid-free paper.
© Birkhauser Boston, 1990
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, photo•
copying, recording or otherwise, without prior permission of the copyright owner.
ISBN 0-8176-3454-1
ISBN 3-7643-3454-1
Camera-ready copy prepared by the author using TeX.
Printed and bound by Edwards Brothers, Inc., Ann Arbor, Michigan.
Printed in the U.S.A.
9 8 7 6 5 4 3 2 1
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 these notes is to provide access to
the methods of algebraic geometry for engineers and applied scientists through the
motivated context of control theory.
I began the development of these notes over fifteen years ago with a series of
lectures given to the Control Group at the Lund Institute of Technology in Sweden. Over
the following years, I presented the material in courses at Brown several times and must
express my appreciation for the feedback (sic!) received from the students. I have
attempted throughout to strive for clarity, often making use of constructive methods and
giving several proofs of a particular result. Since algebraic geometry draws on so many
branches of mathematics and can be dauntingly abstract, it is not easy to convey its beauty
and utility to those interested in applications. I hope at least to have stirred the reader to
seek a deeper understanding of this beauty and utility in control theory.
The first volume deals 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). This represents the beginning of both system theory and algebraic
geometry. The classical frequency domain methods for control system design were
primarily for scalar linear systems and did not always extend readily to multivariable
systems. So, in a way, this first volume provides the algebraic geometry for understanding
the mathematical structure (in part) of classical scalar control systems. It also provides the
foundation for going further mathematically by developing the basic results of affine
algebraic geometry.
While affine algebraic geometry is quite satisfactory and natural for scalar systems,
the study of multi-input, multi-output linear time-invariant control systems requires the
introduction of projective algebraic geometry. Since an important role for algebraic
geometry to play in control theory is the description of the structure of multi variable linear
systems, it is quite necessary to go beyond the basic results of the first volume.
Consequently, the second volume of these notes is devoted to the study of multi variable
linear systems and projective algebraic geometry. A section, ("Interlude"), indicating in a
brief and heuristic fashion the flavor of Part II, is included in Part I.
While there are many friends, colleagues, teachers, and students, to whom
considerable thanks are owed, I should like especially to express my appreciation to the
vii
PREFACE
Vlll
Laboratory for Information and Decision Systems at M.I.T. and its Director, Sanjoy K.
Mitter, for the use of a quiet office (without which this book would still not be finished.)
Thanks are also due to Ann Kostant for the excellent computer preparation of the
manuscript.
Finally, I dedicate this work to my daughters, Hilary and Alison.
'Iable of Contents
Preface vii
O. Introduction.................. . . 1
1. Scalar Linear Systems over the Complex Numbers . . 5
2. Scalar Linear Systems over a Field k . . 9
3. Factoring Polynomials 15
4. Affine Algebraic Geometry: Algebraic Sets 20
5. Affine Algebraic Geometry: The Hilbert Theorems 24
6. Affine Algebraic Geometry: Irreducibility ..... 29
7. Affine Algebraic Geometry: Regular Functions and Morphisms I 32
8. The Laurent Isomorphism Theorem . . . . . . . . . . . . . . 39
9. Affine Algebraic Geometry: Regular Functions and Morphisms II . 45
10. The State Space: Realizations . . . . . . . . . . . . . . 50
11. The State Space: Controllability, Observability, Equivalence 58
12. Affine Algebraic Geometry: Products, Graphs and Projections 65
13. Group Actions, Equivalence and Invariants 71
14. The Geometric Quotient Theorem: Introduction 76
15. The Geometric Quotient Theorem: Closed Orbits 83
16. Affine Algebraic Geometry: Dimension 87
17. The Geometric Quotient Theorem: Open on Invariant Sets . 103
18. Affine Algebraic Geometry: Fibers of Morphisms . . . . .. . 105
19. The Geometric Quotient Theorem: The Ring of Invariants . 113
20. Affine Algebraic Geometry: Simple Points . . 128
21. Feedback and the Pole Placement Theorem . 137
22. Affine Algebraic Geometry: Varieties . 145
23. Interlude . . 151
Appendix A: Tensor Products . . 168
Appendix B: Actions of Reductive Groups 175
Appendix C: Symmetric Functions and Symmetric Group Actions 179
Appendix D: Derivations and Separability . 185
Problems . . . . . . 190
References . . . . . . . . . . . . . . . 200
ix
0. Introduction
The overall goal of these notes is to provide an introduction to
the ideas of algebraic geometry in the motivated context of system
theory. While there are a number of excellent mathematical works
on algebraic geometry (e.g. [D-1], [H-2], [K-4], [M-3], [S-2] etc.), these
books do not deal with applications to system theory and are not,
by virtue of their abstraction, always as accessible to engineers and
applied scientists as the potential utility of the concepts warrants. We
seek to provide a bridge for those interested in applications.
While we do not assume considerable mathematical knowledge
beyond the basics oflinear algebra, some simple notions from topology,
and the elementary properties of groups, rings and fields, (see e.g. [B-
1], [J-1], (Z-3]), we do suppose some knowledge of system theory and
applications such as might be provided by a course on linear systems.
Thus, the motivation for such ideas as, say, controllability will not
be considered in detail here nor will we discuss questions of design.
In addition, we deal only with the algebraic side of the subject and
not with the analytic (or differential geometric) side. The comparable
analytic development is examined, for example, in [H-4], [H-6], etc.
Basic material on system theory can be found, for example, in [A-1],
[B-4], [K-2], [R-1], and (W-1].
Part I deals with scalar (i.e. single input, single output) linear
systems and affine algebraic geometry. This represents, in essence, the
beginning of both system theory and algebraic geometry and is the
subject of this volume.
We begin by introducing the four representations of a scalar linear
system, namely: (i) a strictly proper rational meromorphic function
(the transfer function); (ii) a pair of relatively prime polynomials (the
differential operator); (iii) the Hankel matrix (the impulse response);
and, (iv) a triple of matrices (the state space). A key complex of ideas
revolves around the transition theorems from one representation to
another. The first transition theorem (Hankel's theorem 1.4) relates
rationality of a meromorphic function to the finiteness of the rank of
a Hankel matrix.
1