Table Of ContentISNM
International Series of
Numerical Mathematics
Vol. 121
Managing Editors:
K.-H. Hoffmann, Munchen
D. Mittelmann, Tempe
Associate Editors:
R.E. Bank, La Jolla
H. Kawarada, Chiba
R.J. LeVeque, Seattle
C. Verdi, Milano
Honorary Editor:
J. Todd, Pasadena
Stability Theory
Hurwitz Centenary Conference
Centro Stefano Franscini, Ascona, 1995
Edited by
R. Jeltsch
M. Mansour
Birkhauser Verlag
Basel· Boston· Berlin
Editors:
Rolf leltsch Mohamed Mansour
Seminar for Applied Mathematics Automatic Control Laboratory
Ern Zentrurn ETII Zentrum
8092 ZOrich 8092 ZUrich
Switzerland Switzerland
A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA
Deutsche Bibliothek CataJoging-in-Publication Data
Stability theory I Hurwitz Centenary Conference, Centro
Stefano Franscini, Ascona, 1995. Ed. by Rolf Jeltsch ;
Mohamed Mansour. -Basel; Boston; Berlin: Birkhiiuser, 1996
(International series of numerical mathematics; Vol. 121)
ISBN-13:978-3-0348-9945-1 e-ISBN- 13:978-3-0348-9208-7
001: 10.1007/978-3-0348-9208-7
NE: Jeltsch, Rolf [Hrsg.); Hurwitz Centenary Conference <1995,
Ascona>; Centro Stefano Franscini; GT
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the
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© 1996 Birkhiiuser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland
Printed on acid-free paper produced from chlorine-free pulp. TCF 00
Cover design: Heinz Hiltbrunner, Basel
ISBN 3-7643-5474-7
ISBN 0-8176-5474-7
987654321
Contents
Preface .................................................................. VII
Stability theory
A. Beghi, A. Lepschy and U. Viaro
The Hurwitz Matrix and the Computation of Second-Order
Information Indices ...................................................... 1
B. Bemhardsson, A. Rantzer and L. Qiu
A Summary on the Real Stability Radius and Real
Perturbation Values ...................................................... 11
J. Garloff and B. Srinivasan
The Hadamard Factorization of Hurwitz and Schur Stable Polynomials .... 19
Y. V. Genin
On the Cauchy Index of a Real Rational Function and the Index Theory
of Pseudo-Lossless Rational Functions .................................... 23
H. Gorecki, M. Szymkat and M. Zaczyk
A Generalization of the Orlando Formula-Symbolic
Manipulation Approach .................................................. 33
L. Atanassova, D. Hinrichsen and V.L. Kharitonov
On Convex Stability Directions for Real Quasipolynomials 43
E.I. Jury
From J.J. Sylvester to Adolf Hurwitz: A Historical Review 53
F.J. Kraus, M. Mansour and M. Sebek
Hurwitz Matrix for Polynomial Matrices 67
H.C. Reddy, P.K. Rojan and G.S. Moschytz
Two-Dimensional Hurwitz Polynomials................................... 75
E.D. Sontag and H.J. Sussmann
General Classes of Control-Lyapunov Functions 87
R. Strietzel
Towards the Stability of Fuzzy Control Systems .......... . . . . . . . . . . . . . . . . . 97
M. Vidyasagar
Discrete Optimization Using Analog Neural Networks with
Discontinuous Dynamics ................................................. 107
v
VI Contents
Robust Stability
B.D. O. Anderson and S. Dasgupta
Multiplier Theory and Operator Square Roots: Application to
Robust and Time-Varying Stability ....................................... 113
R. Lozano and D.A. SudTes
Adaptive Control of Non-Minimum Phase Systems Subject to
Unknown Bounded Disturbances ......................................... 125
M. Mansour and B.D.O. Ander-son
On the Robust Stability of Time-Varying Linear Systems 135
W. Sienel
On the Computation of Stability Profiles ................................. 151
Q.H. WU and M. Mansour
Robust Stability of Family of Polynomials with 1-Norm-Bounded
Parameter Uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 163
E. Zeheb
On the Characteri;mtion and Formation of Local Convex Directions
for Hurwitz Stability ..................................................... 173
Numerics
G.E. Collins
Application of Quantifier Elimination to Solotareff's
Approximation Problem .................................................. 181
R. leltsch
Stability of Time Discretization, Hurwitz Determinants
and Ordp,r Stars .......................................................... 191
R. Liska and S. Steinberg
Solving Stability Problems Using Quantifier Elimination .................. 205
M.R. Trummer
Stability of Numerical Methods for Solving Differential Equations ......... 211
1. Sreedhar, P. Van Dooren and A.L. Tits
A Fast Algorithm to Compute the Real Structured Stability Radius 219
Some Open Problems .................................................... 231
Apendix: Original Article by A. Hurwitz
A. Hur-witz
tiber die Bedingungen, unter welchen eine Gleichung nm
Wurzeln mit negativen reellen Theilen besitzt ............................. 239
Preface
This book contains the historical development of the seminal paper of Adolf
Hurwitz, professor in mathematics at ETH (1892~1919), and its impact on other
fields. The major emphasis, however, is on modern results in stability theory and
its application in the theory of control and numerics. In particular, stability of
the following problems is treated: linear, nonlinear and time-dependent systems,
discretizations of ordinary and partial differential equations, systems with time
delay on multidimensional systems. In addition robust stability, pole placement
and problems related to the stability radius are treated.
The book is an outgrowth of the international conference "Centennial
Hurwitz on Stability Theory" which was held to honor Adolf Hurwitz, whose arti
cle on the location of roots of a polynomial was published one hundred years ago.
The conference took place at the Centro Stefano Franscini, Monte Verita, Ascona,
Switzerland, on May 21~26, 1995. This book contains a collection of the papers
and open problem:; discussed all that occasion. Leading researchers from allover
the world working on stability theory and its application were invited to present
their recent results. In one paper the historic development initiated by Hurwitz's
article was discussed.
The interaction between the two major groups of participants, researchers
in control theory and mathematics, as well as the excellent setting of the Monte
Verita strongly contributed to the success of the meeting. We thank the Centro
Stefano Franscini of the Swiss Federal Institute of Technology (ETH) in Zurich
and the Swiss National Science Foundation for financial support. Our thanks go
also to the publisher Springer which allowed us to reproduce the original article
by Adolf Hurwitz in this volume. We also thank the members of the Seminar for
Applied Mathematics (ETH) for their help in the successful organization of the
meeting, in particular Ms M. Kramer and M. Pfister. In addition we thank the
secretaries of the Centro Stefano Franscini, Ms K. Bastianelli and F. Tewelde, for
their support.
We express our deep gratitude to the staff of Birkhauser Verlag for their
excellent cooperation in producing this volume. Our biggest thanks go to our
secretaries, Ms. M. Kramer and M. Pfister, as well as to our systems people, Ms.
E. Copeland and Dr. P. Scherbel, who all went to great effort to produce the
excellent electronic version of this book.
We hope that this volume will be helpful to engineers and mathematicians
working in the area of control theory and numerics.
Rolf J eltsch
Mohamed Mansour
Zurich, Switzerland
October 1995
International Series of Numerical Mathematics, Vol. 121, © 1996 Birkhauser Verlag Basel
The Hurwitz Matrix and the Computation
of second-order Information Indices
Alessandro Beghi, Antonio Lepschy and Umberto Viaro
Department of Electronics and Informatics
University of Padova, via Gradenigo 6/ A, 35131 Padova, Italy.
Abstract. An all-pole transfer function Q(8) = l/P(s), where pes) is a monic
Hurwitz polynomial of degree n, is uniquely characterized by the energies (second
order information indices) of q(t.) = LT~1{Q(8)} and of its first n - 1 derivatives.
These can be obtained by solving a set of linear equations whose coefficients matrix is
the standard Hurwitz matrix for pes) or by using the entries of its Routh table. Any
strictly proper transfer function W(s) = N(s)/P(s) is characterized by n first-order
information indices, e.g., Markov parameters, and by n second-order information
indices, e.g .. the energies of the related impulse response and its n - 1 successive
derivatives; these are simply obtainable from the energies of q(t). This fact can be
exploited to construct reduced-order models that retain both first- and second-order
information indices of a given original system. The extension of this approach to
multi-input multi-output systems clmcrihccl hy a matrix fraction is analysed.
1. Introduction. As is known, the Hurwitz matrix as well as the Routh array
were conceived for stability analysis and zeros separation problems but have found
interesting applications in diffcrent fields too. In particular, they have been used to
compute the integrals along the imaginary axis of the square magnitude of rational
functions of a complex variable, which are related to the energies of certain signals.
In this regard, wc may recall Krasovskii's formula (cf. [1, p. 262]) for computing
the 8O-called ISE (integral of the square error in the step response) which uses a
matrix similar to the Hurwitz one, the procedure and related table given by Phillips
[2], the method independently derived by Kac [3] and Mersman [4] which requires
the computation of a number of determinants formed from the coefficients of the
considered rational function, the expression suggested by N ekolny and Benes [5],
and those obtained by Effertz [6] along lines similar to those followed by Astrom
et al. [7] and by Astrom [8]; the problem has also been considered by Lehoczky
[9], by Csaki and Lehoczky [10]' by Lepschy et at. [Il] and by the present authors
[12].
The above-mentioned techniques can usefully be applied to compute the so
called second-order information indices of a dynamic system, such as the entries
of the associated impUlse-response Gramian. In fact, its diagonal elements are the
energies of the impulse response and of some of its derivatives, and the off-diagonal
elements can be obtained by adding to such energies terms that simply depend on
suitable first-order information indices, e.g., Markov parameters; [or this reason
the energies can be regarded as the essential second-order information.
The idea of characterizing a linear time-invariant dynamic sysLem of order n
by means of n first-order and n second-order information indices goes back to the
seminal paper by Mullis and Roberts [13] in which they also suggest a procedure
for constructing simplified models that retain an equal number of first- and second-
2 A. Beghi, A. Lepschy and U. Viaro
order information indices of an original system. This approach has been adopted
by various authors [14]-;.-[25] for both single-input single-output (8180) and multi
input multi-output (MIMO) systems in discrete and continuous time.
This paper is concerned with the above topics; more precisely, it deals with the
following problems: (i) the computation of the energies associated with a transfer
function by using Hurwitz-type matrices or the Routh table; (ii) the use of the
considered information indices for characterizing and simplifying 8180 continuous
time systems; (iii) the extension to the case of MIMO systems and the comparison
of the related features with those of the 8180 case.
2. Computation of the energies. Let us consider a monic Hurwitz polynomial of
degree n with real coefficients
(1) P(s) = sn + an_ISn-1 + ... + ao
and the associated all-pole (stable) transfer function
1
(2)
Q(s) = P(s) .
The corresponding impulse response q(t) = LT-I{l/P(s)} satisfies the homoge
neous differential equation
nL-l °
(3) q(n)(t) + aiq(i)(t) =
i=O
for the initial conditions q(i)(O+) = 0, i = O,l, ... ,n - 2, and q(n-l)(o+) = 1.
Multiplying successively eqn. (3) by (-l)j q(j) (t), j = 0,1, ... ,n - 1, integrating
from 0+ to 00, and denoting by ei the energy of q(i)(t):
(4) i = 0,1, ... , n - 1
we arrive for n even at the following set of equations
aoeo - a2el + a4e2 - ... - (_1)(n/2)-lan_2e(n/2)°_1 = (_1)(n/2)-len/2
{ -aIel + a3e2 - a5e3 + ... + (_1)n/2an_len/2 =
(5) .
al (_1)(n/2)-len/2 - (_1)n/2a3e(n/2)+1 + ... - an-l en-l = -~
and for n odd at a similar set. These sets are linear in the coefficients ai and in
the energies ei. If the latter are given and coefficients ai are unknown, these can
uniquely be determined by solving the matrix equation
(6) Ga=g
The Hurwitz Matrix and the Computation of second-order Information Indices 3
where G (impulse-response Gramian) is the nonsingular matrix
,J
eo 0 -e1
0 el 0
(7) G= -e1 0 e2
and
]T
(8) a [ ao, a1, ... an-1
{
[ (-I)(n-2)/2en/2, 0, ... en-I, ~f n even
(9) g =
[ 0, ( _1)(n+1)/2e(n+l)/2, 0, en-I, ~f n odd
Note that, by exploiting the structure of (7), eqn. (6) can be decoupled into two
matrix equations whose unknown are the even-order coefficients a2i and odd-order
coefficients a2i+l, respectively.
If, instead, coefficients ai are given and energies ei are unknown, these can
uniquely be determined by solving the matrix equation
(10) He= h
where H is the nonsingular Hurwitz matrix associated with the stable system (2)
an-1 an-3 an-5
1 an-2 an-4
0 an-l an-3
(11) H=
0 1 an-2
ao
and
(12) e [ en-1 -en-2 en-3 -en-4 eo ]T
of·
(13) h [ 1 0 0 0
'2
Therefore, a and e are alternative parameterizations of the all-pole transfer func
tion (2).
Eqn. (10) can be rewritten as
(14) Ke=k
where K is the matrix considered by Krasovskii for the computation of the ISE
ao -a2 a4
o
a1 -a3
o
-ao a2
(15) K=
