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

384 Pages·2018·24.56 MB·English

by  Peter Falb

Modern Birkhäuser Classics
Peter Falb
Methods of
Algebraic Geometry
in Control Theory:
Part II
Multivariable Linear Systems and
Projective Algebraic Geometry
Modern Birkhäuser Classics

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Methods of Algebraic
Geometry in Control
Theory: Part II
Multivariable Linear Systems
and Projective Algebraic
Geometry
Peter Falb
Reprint of the 1999 Edition
Peter Falb
Division of Applied Mathematics
Brown University
Providence, Rhode Island, USA

ISSN 2197-1803 ISSN 2197-1811 (electronic)

Modern Birkhäuser Classics
ISBN 978-3-319-96573-4 ISBN978-3-319-96574-1 (eB ook)
https://doi.org/10.1007/978-3-319-96574-1

Library of Congress Control Number: 2018951899

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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
with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, elec
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 freelyb y anyone.
ISBN 978-1-4612-7194-9
Typeset in iAlfeX by T^Xniques, Inc., Cambridge, MA.
9 8 7 6 5 4 3 2 1
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
vii
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.
ix
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