Table Of ContentAlgebraic and
Differential Topology
of Robust Stability
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Algebraic and
Differential Topology
of Robust Stability
Edmond A. Jonckheere
University of Southern California
Department of Electrical Engineering—Systems
and Center for Applied Mathematical Sciences
Los Angeles, California
with 79 pictures,
computer generated by
Chih-Yung Cheng, Chung-Kuang Chu, and Murilo G. Coutinho
New York Oxford • Oxford University Press 1997
Oxford University Press
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Copyright © 1997 by Oxford University Press, Inc.
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All rights reserved. No part of this publication may be reproduced,
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without the prior permission of Oxford University Press.
Library of Congress Cataloging-in-Publication Data
Jonckheere, Edmond A., 1954-
Algebraic and differential topology of robust stability / Edmond A. Jonckheere :
with over 50 illustrations by Chih-Yung Cheng, Chung-Kuang Chu, Murilo G. Coutinho.
p. cm.
Includes bibliographical references and index.
ISBN 0-19-509301-1
1. Control theory. 2. Algebraic topology. 3. Differential topology. I. Title.
QA402.3.J66 1996
629.8'312'015142—dc20 95-49073
135798642
Printed in the United States of America
on acid-free paper
In conclusion I would like to suggest that
algebraic topology has now reached a sufficient
degree of maturity so that it should be regarded
as a tool available for use in appropriate
branches of analysis. At least I hope it will
be the natural thing for a mathematician to ask:
if I vary the coefficients or parameters of my problem,
what sort of a topological space do I get -
is it for example contractible and if not,
what is the significance of its topological invariants?
M.F. ATIYAH
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PREFACE
In this book, two seemingly unrelated fields of intellectual endeavor—
algebraic/differential topology and robust control—are brought together.
The terminology "seemingly unrelated" might be somewhat inappropriate,
because this link already appears at the most fundamental level of multi-
variable theory. It is indeed well-known, at least to control engineers, that
the stable multivariable loop transmission L(s) remains stable after closing
the loop if and only if det(I + L(j )) 0, together with the condition
that the plot of det(I + L(j )) does not circle around 0 + j . What is less
well-known, at least to control engineers, is that the latter is (generically)
nothing other than a necessary and sufficient condition for invertibility of
the Toeplitz operator I + T . Invertibility of a Toeplitz operator relies
L
crucially on the concept of index of a Fredholm operator. The problem of
finding an index for a family of Fredholm operators—in other words, an
"uncertain" Fredholm operator—historically led to the so-called (topolog-
ical) K-theory. It is fair to say that K-theory has been one of the most
recent and important developments in algebraic topology. The fact that
index of a family of Fredholm operators was a milestone in the historical
development of K-theory was very convincingly argued by Atiyah in a 1969
monograph and reiterated in a 1977 survey paper. There, in an effort to
provide a most natural motivation for K-theory, Atiyah introduced what in
the control jargon would be called the "uncertainty" by defining a family
I + T of operators, indexed by A running in some topological space
L
D, the "uncertainty space." To make the problem of mathematical inter-
est, the family is restricted to be Fredholm, which in essence means that
whenever I + T is invertible for one single , then it is invertible for all
L
's in the same connected component. In other words, Atiyah restricted
the system to be robustly stable. He then showed that the fundamental
K-theory concept of Grothendieck group of vector bundles over the uncer-
tainty space, K°(D), comes out most naturally as the isomorph of the set
of homotopy classes of maps from the uncertainty space into the space of
Fredholm operators. It is therefore fair to say that the robust stability
problem, disguised as a "Fredholm family of Toeplitz operators" problem,
has been one of the driving motivations of the whole K-theory (besides
algebraic geometry). We hasten to say that, in this book, we will enter
the algebraic topology arena from a much more applications-oriented path
approach, although we will return to the Fredholm family problem.
More recently, about three decades after the emergence of K-theory, the
robust stability problem—that is, the problem of checking stability for a
viii PREFACE
great many uncertain parameter values—has received several ad hoc solu-
tions, gravitating around the idea of breaking the set of uncertainties D into
pieces and then localizing the stability boundary within some pieces. What
does not seem to have been perceived is the fact that breaking a domain of
uncertainty into pieces is a basic technique of algebraic topology—usually
referred to as triangulation—.and has roots tracing back to Poincare and
Lefschetz. In such a "structured" problem as the multivariable phase mar-
gin, the uncertainty space becomes a nontrivial manifold, the n-torus or
the unitary group, depending on the formulation. Triangulating a manifold
and then looking at the combinatorial properties of the resulting assembly
of simplexes is the basic technique for computing such global properties of
manifolds as number of connected components, loops, holes, twists, and so
on. The same triangulation technique is also standard when one has to
ascertain how loops, holes, twists, and so on, of a manifold are mapped
by a continuous function into loops, holes, twists, and so on, in the image
manifold.
In the early robust stability techniques, it never came out quite clearly
what is being accomplished by breaking the domain of uncertainty into
pieces. These robust stability techniques followed in the footsteps of the
"Horowitz template" approach. The domain of uncertainty D is mapped
into the complex plane by means of the Nyquist map to yield the template
N = f(D). In algebraic topology, decomposing the domain of definition of
such a continuous map as the Nyquist map / is a process aimed at deriving
an approximate map / that has the remarkable property that, restricted
to each simplex of the triangulation, it "commutes" with the boundary,
in the sense that f( ) f( ). The latter has the computationally
interesting property that, if we need to check the boundary of the image,
f( ), which is typical of a homotopy procedure, it suffices to check the
boundary of the domain, f( ). This decomposition technique culminates
in the celebrated simplicial approximation theorem that asserts that / can
be chosen arbitrarily close and homotopic to /.
Horowitz should be credited as being probably the first who perceived
this boundary issue. He indeed pointed out that if the perimeter of a
rectangular domain of uncertain parameters is mapped into the perimeter
of a polygonal Horowitz template— f = f in our notation—then the
robust stability check can be substantially simplified.
To proceed more formally, assume that, after an application of the sim-
plicial approximation theorem, the approximate Nyquist map / commutes
with the boundary over each simplex of the domain of uncertainty. Since
the map commutes with the boundary, it suffices to check stability for all
parameter values on the boundary to ensure stability for all parameters.
Unfortunately, if we follow the path of a direct application of the simpli-
cial approximation theorem, because of the triangulation it entails, we are
likely to end up with what has usually been referred to as combinatorial
PREFACE ix
explosion. Indeed, the drawback of the simplicial approximation theorem
is that the higher the required accuracy of the approximation, the more
finely the manifold of uncertainty should be triangulated. Clearly, requir-
ing a high accuracy simplicial approximation could lead to prohibitively
complicated gridding, if a uniform gridding procedure is adopted.
Along parallel lines of investigation, a substantial amount of effort was
devoted to the avoidance of the combinatorial explosion, until it was shown
in [Coxson and DeMarco 1991] that such a representative problem as the
computation of the real p-function is NP-hard.
A procedure to alleviate the odds of combinatorial explosion stems from
the observation that it is not necessary to further triangulate a big chunk
of the uncertainty manifold, if we can make reasonably sure that no insta-
bility will develop in that region. This leads us to the idea of implementing
a variable grid refinement restart algorithm: Start with a coarse triangu-
lation, check for stability in each piece using a boundary test, and refine,
in a nested fashion, only those simplexes where an instability could de-
velop. Operations researchers have been very successful at implementing
these kinds of schemes, following in the footsteps of the celebrated Sperner
lemma, and leading to the so-called simplicial algorithms. Many other
computationally intensive mathematical problems, such as the Brouwer
fixed-point theorem, have received solutions along this line of ideas. Our
approach to the structured stability problem makes no exception to this
rule.
Of course, since we are dealing with NP-hard problems, some sort of
combinatorial nastiness must appear somewhere. It turns out that the
bottleneck of the variable grid refinement restart procedure is to guarantee
Sperner properness. Nevertheless, with a bit of heuristics, the procedure
can be made to work very well.
In the course of writing this book, it became evident that we could not
possibly develop the algorithms all the way to the details of their imple-
mentation and prove anything about their complexity status. This would
require two other books and a few more years of research. In this book,
we are limiting ourselves to show the topological roots of the simplicial
algorithms and their combinatorial nastiness, and we are leaving the de-
tails of the implementation and the complexity analyses to our students.
The [Cheng 1994] dissertation contains the details of the simplicial algo-
rithms on V-triangulation and the so-called vector labeling algorithm. The
stochastic complexity of simplicial algorithms along with the root lattice
properties of regular pattern triangulations have been developed by [Chu
1996].
The above computational implementations were developed as Matlab
m-files, relying on matrix data bases and primitive instructions. How-
ever, since the fundamental problem is geometric, it appeared more nat-
ural to develop an implementation based on geometric data bases and
