Operators, Systems and Linear Algebra: Three Decades of Algebraic Systems Theory

223 Pages·1997·8.56 MB·German

by  Uwe Helmke

European Consortium for
Mathematics in lndustry
U. Helmke, D. Prätzel-Wolters
and E. Zerz (Eds.)
Operators, Systems,
and Linear Algebra
European Consortium for
Mathematics in Industry
Edited by
Leif Arkeryd, Göteborg
Heinz Engl, Linz
Antonio Fasano, Firenze
Robert M. M. Mattheij, Eindhoven
Pekka Neittaanmäki, Jyväskylä
Helmut Neunzert, Kaiserslautern
Within Europe a number of academic groups have accepted their responsibility
towards European industry and have proposed to found a European Consor
tium for Mathematics in Industry (ECMI) as an expression of this responsi
bility.
One of the activities of ECMI is the publication of books, which reflect its
general philosophy; the texts of the series will help in promoting the use of
mathematics in industry and in educating mathematicians for industry. They
will consider different fields of applications, present casestudies, introduce
new mathematica\ concepts in their relation to practical applications. They
shall also represent the variety of the European mathematical traditions, for
example practical asymptotics and differential equations in Britian, sophisti
cated numerical analysis from France, powerful computation in Germany,
novel discrete mathematics in Holland, elegant real analysis from ltaly. They
will demoostrate that all these branches of mathematics are applicable to real
problems, and industry and universities in any country can clearly benefit from
the skills of the complete range of European applied mathematics.
Operators, Systems,
and Linear Algebra
Three Decades of
Algebraic Systems Theory
Edited by
Uwe Helmke, Universität Würzburg
Dieter Prätzel-Wolters and Eva Zerz,
Universität Kaiserslautern
83
Springer Fachmedien Wiesbaden GmbH 1997
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Operators, systems and linear algebra : three decades
of algebraic systems theory 1 ed. by Uwe Helmke ...
(European Consortium for Mathematics in Industry).
ISBN 978-3-663-09824-9 ISBN 978-3-663-09823-2 (eBook)
DOI 10.1007/978-3-663-09823-2
© Copyright 1997 by Springer Fachmedien Wiesbaden
Originally published by B. G. Teubner Stuttgart in 1997
Ali rights reserved
No part of this book may be reproduced by any means, or transmitted, or translated
into a machine language without the written permission of the publisher.
Cover design by Peter Pfitz, Stuttgart.
Dedicated to Paul A. Fuhrmann
on the occasion of his 60th birthday
Preface
This book contains the plenary lectures presented at the Workshop 'Operators,
Systems and Linear Algebra: Three Decades of Algebraic Systems Theory,' held
in Kaiserslautern, Germany, September 24-26, 1997. It is a Festschrift honoring
the impact of the work of Paul Fuhrmann in operator and control theory. The book
includes essays written by prominent scientists to present their views on some of
the most recent developments in the area of mathematical systems theory and its
applications. The papers cover a wide range of theoretical and applied topics,
with emphasis on computational aspects and relations to engineering problems.
The impact of Paul Fuhrmann's work can be traced through many parts of the
volume. Polynomial models and shift realizations, the algebraic structure of linear
state feedback, partial realization theory and the recursive inversion of Hanke!
and Toeplitz operators, spectral factorization, and parametrizations of classes of
rational functions are some of the topics in the book, where explicit references are
made to his work.
It is impossible to give in this short preface any substantial review of Paul Fuhr
mann's numerous and major contributions to finite and infinite dimensionallinear
systems theory. Instead we like to emphasize that his work demonstrates in a
clear way the underlying attempt to unreveal the basic unity of mathematics.
His work is guided by the paradigm 'forest before trees.' As a striking example
for this we mention the unifying concept of polynomial models, which led to a
synthesis of Kalman's module-theoretic approach to the realization problem, state
space models and the polynomial matrix method of Rosenbrack for linear systems
in differential operator representation. Polynomial models have their roots in
operator theory. Paul Fuhrmann's elegant book Linear Systems and Operators
in a Hilbert Space established the basic relations between Operator and systems
theory in a novel and general way. The recent monograph A Polynomial Approach
to Linear Algebra presents the areas of linear algebra and linear systems theory
in an excitingly new coherent way, stressing the fundamental role played by shift
operators and polynomial models in these contexts.
It is a pleasure to thank all the authors for the enthusiasm with which they ac
cepted the invitation to contribute to this volume. Their work covers diverse top
ics in systems and control theory, such as Toda Lattices, Noncommutative Power
Series in Nonlinear Control, Computer Vision, the Partial Realization Problem,
Lyapunov Theory, Discontinuous Dynamics, 2D Systems, Spectral Factorizations,
Supervisory Control, Modeling and Interpolation. The papers in this book present
an excellent perspective on prominent trends in systems and control theory, and
their relations to a variety of mathematical areas. It also underlines the point of
view Paul Fuhrmann has expressed so clearly in 1981:
8
'Mathematics has always benefited from the transplanting of ideas and motivations
from other fields. It seems to me that system theory besides being intellectually
exciting is today one of the richest sources of ideas for the mathematician as well
as a major area of application of mathematical knowledge.'
It has been a great honor and privilege to have the opportunity of celebrating the
60th birthday of Paul Fuhrmann by holding this workshop in Kaiserslautern. With
admiration and affection we honor a colleague and friend of enormous creativity,
energy, courage and mathematical strength. Paul Fuhrmann has been most influ
ential on the development of the German systems and control community. We are
convinced that he will continue to work vigorously for many years to come and we
extend to him on this occasion our best wishes for future health and happiness.
The editors thank the 'Graduiertenkolleg Technomathematik' and the 'Institut
für Techno- und Wirtschaftsmathematik' at the University of Kaiserslautern for
generaus support of the workshop. We are grateful to the 'European Consortium
for Mathematics in Industry' and Dr. Peter Spuhler at Teubner Verlag for their
fast and encouraging response to our planning of this volume. Cordial thanks
are especially due to Prof. Joachim Rosenthai who helped us in writing parts of
the introduction. Finally, we wish to express once more our gratitude towards all
authors, for their willingness to participate in this meeting and for providing their
manuscripts in a timely way.
Kaiserslautern, September 1997
Uwe Helmke, Dieter Prätzel-Wolters, Eva Zerz
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
On the Approximation of Hankel Matrices
A. C. Antoulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
The Reduction of Multiplicity in Systems Modeling
T. Björk, A. Gombani . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A Rational Flow for the Toda Lattice Equations
R. W. Brockett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
On Output-Stabilizability and Detectability of Discrete-Time
LQ-Optimal Control Systems
F. M. Callier .............................................................. 45
Matrix Pairs and 2D Systems Analysis
E. Fornasini, G. Marchesini, M. E. Valeher ................................. 56
Spectral Minimality of J-Positive Linear Systems of Finite Order
A. Gheondea, R. J. Ober .................................................. 70
State Feedback in General Discrete-Time Systems
M. Hautus ................................................................ 81
Nonblocking Supervisory Control of Nondeterministic Systems
M. Heymann, F. Lin ...................................................... 96
Noncommutative PowerSeriesand Formal Lie-Algebraic Techniques
in Nonlinear Control Theory
M. Kawski, H. J. Sussmann ............................................... 111
Recent Progress on the Partial Stochastic Realization Problem
A. Lindquist .............................................................. 129
Generalized Partial Realizations
W. Manthey, U. Helmke, D. Hinrichsen .................................... 138
Oblique Splitting Subspaces and Stochastic Realization with Inputs
G. Picci ................................................................... 157
Lyapunov Revisited: Variations on a Matrix Theme
H. Schneider .............................................................. 175
10
Linear Systems and Discontinuous Dynamics
J. M. Schumacher ......................................................... 182
Differential Invariants and Curvature Flows in Active Vision
A. Tannenbaum, A. Yezzi, Jr .............................................. 196
On the Convergence of the Newton Iteration for Spectral Factorization
J. C. Willems ............................................................. 214