Unsolved Problems in Mathematical Systems and Control Theory - Hardcover

Hardcover

ISBN 10: 0691117489 ISBN 13: 9780691117485

Publisher: Princeton University Press, 2004

Synopsis

This book provides clear presentations of more than sixty important unsolved problems in mathematical systems and control theory. Each of the problems included here is proposed by a leading expert and set forth in an accessible manner. Covering a wide range of areas, the book will be an ideal reference for anyone interested in the latest developments in the field, including specialists in applied mathematics, engineering, and computer science.


The book consists of ten parts representing various problem areas, and each chapter sets forth a different problem presented by a researcher in the particular area and in the same way: description of the problem, motivation and history, available results, and bibliography. It aims not only to encourage work on the included problems but also to suggest new ones and generate fresh research. The reader will be able to submit solutions for possible inclusion on an online version of the book to be updated quarterly on the Princeton University Press website, and thus also be able to access solutions, updated information, and partial solutions as they are developed.

About this title

About the Author

Vincent D. Blondel is Professor of Applied Mathematics and Head of the Department of Mathematical Engineering at the University of Louvain, Louvain-la-Neuve, Belgium. Alexandre Megretski is Associate Professor of Electrical Engineering at Massachusetts Institute of Technology.

From the Back Cover

"This is an extremely important book that presents, in a clear way, many important and stimulating mathematical problems in systems and control. It will be an important reference for both researchers and people outside the field."--William W. Hager, University of Florida

"This book covers a wide range of systems from linear to nonlinear, deterministic to stochastic, finite dimensional to infinite dimensional, and so on. It includes at least some set of problems that will interest any researcher in the field."--Kemin Zhou, Louisiana State University

From the Inside Flap

"This is an extremely important book that presents, in a clear way, many important and stimulating mathematical problems in systems and control. It will be an important reference for both researchers and people outside the field."--William W. Hager, University of Florida

"This book covers a wide range of systems from linear to nonlinear, deterministic to stochastic, finite dimensional to infinite dimensional, and so on. It includes at least some set of problems that will interest any researcher in the field."--Kemin Zhou, Louisiana State University

Excerpt. © Reprinted by permission. All rights reserved.

Unsolved Problems in Mathematical Systems and Control Theory

PRINCETON UNIVERSITY PRESS

Copyright © 2004 Princeton University Press
All right reserved.
ISBN: 978-0-691-11748-5

Contents

Preface.........................................................xiiiAssociate Editors...............................................xvWebsite.........................................................xviiPART 1. LINEAR SYSTEMS........................................1PART 2. STOCHASTIC SYSTEMS....................................65PART 3. NONLINEAR SYSTEMS.....................................87PART 4. DISCRETE EVENT, HYBRID SYSTEMS........................129PART 5. DISTRIBUTED PARAMETER SYSTEMS.........................151PART 6. STABILITY, STABILIZATION..............................187PART 7. CONTROLLABILITY, OBSERVABILITY........................245PART 8. ROBUSTNESS, ROBUST CONTROL............................265PART 9. IDENTIFICATION, SIGNAL PROCESSING.....................285PART 10. ALGORITHMS, COMPUTATION...............................297

Chapter One

Problem 1.1

Stability and composition of transfer functions

G. Fernndez-Anaya Departamento de Ciencias Bsicas Universidad Iberoamricana Lomas de Santa Fe 01210 Mxico D.F. Mxico guillermo.fernandez@uia.mx

J. C. Martnez-Garca Departamento de Control Automtico CINVESTAV-IPN A.P. 14-740 07300 Mxico D.F. Mxico martinez@ctrl.cinvestav.mx

1 INTRODUCTION

As far as the frequency-described continuous linear time-invariant systems are concerned, the study of control-oriented properties (like stability) resulting from the substitution of the complex Laplace variable s by rational transfer functions have been little studied by the Automatic Control community. However, some interesting results have recently been published:

Concerning the study of the so-called uniform systems, i.e., LTI systems consisting of identical components and amplifiers, it was established in a general criterion for robust stability for rational functions of the form D(f(s)), where D(s) is a polynomial and f(s) is a rational transfer function. By applying such a criterium, it gave a generalization of the celebrated Kharitonov's theorem, as well as some robust stability criteria under [H.sub.[infinity]]-uncertainty. The results given in are based on the so-called H-domains. As far as robust stability of polynomial families is concerned, some Kharitonov's like results are given in (for a particular class of polynomials), when interpreting substitutions as nonlinearly correlated perturbations on the coefficients.

More recently, in, some results for proper and stable real rational SISO functions and coprime factorizations were proved, by making substitutions with [alpha](s) = (as + b) / (cs + d), where a, b, c, and d are strictly positive real numbers, and with ad - bc [not equal to] 0. But these results are limited to the bilinear transforms, which are very restricted.

In is studied the preservation of properties linked to control problems (like weighted nominal performance and robust stability) for Single-Input Single-Output systems, when performing the substitution of the Laplace variable (in transfer functions associated to the control problems) by strictly positive real functions of zero relative degree. Some results concerning the preservation of control-oriented properties in Multi-Input Multi-Output systems are given in, while deals with the preservation of solvability conditions in algebraic Riccati equations linked to robust control problems.

Following our interest in substitutions we propose in section 22.2 three interesting problems. The motivations concerning the proposed problems are presented in section 22.3.

2 DESCRIPTION OF THE PROBLEMS

In this section we propose three closely related problems. The first one concerns the characterization of a transfer function as a composition of transfer functions. The second problem is a modified version of the first problem: the characterization of a transfer function as the result of substituting the Laplace variable in a transfer function by a strictly positive real transfer function of zero relative degree. The third problem is in fact a conjecture concerning the preservation of stability property in a given polynomial resulting from the substitution of the coefficients in the given polynomial by a polynomial with non-negative coefficients evaluated in the substituted coefficients.

Problem 1: Let a Single Input Single Output (SISO) transfer function G(s) be given. Find transfer functions [G.sub.0](s) and H(s) such that:

1. G (s) = [G.sub.0] (H (s));

2. H (s) preserves proper stable transfer functions under substitution of the variable s by H (s), and:

3. The degree of the denominator of H(s) is the maximum with the properties 1 and 2.

Problem 2: Let a SISO transfer function G(s) be given. Find a transfer function [G.sub.0] (s) and a Strictly Positive Real transfer function of zero relative degree (SPR0), say H(s), such that:

1. G(s) = [G.sub.0] (H (s)) and:

2. The degree of the denominator of H(s) is the maximum with the property 1.

Problem 3: (Conjecture) Given any stable polynomial:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and given any polynomial q(s) with non-negative coefficients, then the polynomial:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is stable (see [3]).

3 MOTIVATIONS

Consider the closed-loop control scheme:

y(s) = G (s) u(s) + d(s), u(s) = K(s) (r (s) - y (s)),

where: P(s) denotes the SISO plant; K (s) denotes a stabilizing controller; u (s) denotes the control input; y(s) denotes the control input; d(s) denotes the disturbance and r(s) denotes the reference input. We shall denote the closed-loop transfer function from r(s) to y(s) as [F.sub.r] (G(s), K(s)) and the closed-loop transfer function from d(s) to y(s) as [F.sub.d](G (s), K(s)).

Consider the closed-loop system [F.sub.r](G(s), K(s)), and suppose that the plant G(s) results from a particular substitution of the s Laplace variable in a transfer function [G.sub.0](s) by a transfer function H(s), i.e., G(s) = [G.sub.0](H(s)). It has been proved that a controller [K.sub.0](s) which stabilizes the closed-loop system [F.sub.r][(G.sub.0](s), [K.sub.0](s)) is such that [K.sub.0] (H(s)) stabilizes [F.sub.r](G(s), [K.sub.0](H (s))) (see [2] and [8]). Thus, the simplification of procedures for the synthesis of stabilizing controllers (profiting from transfer function compositions) justifies problem 1.

As far as problem 2 is concerned, consider the synthesis of a controller K(s) stabilizing the closed-loop transfer function [F.sub.d](G(s), K(s)), and such that [parallel][F.sub.d](G(s), K(s))]parallel].sub.[infinity] < [gamma], for a fixed given [gamma] > 0. If we known that G(s) = [G.sub.0] (H (s)), being H (s) a SPR0 transfer function, the solution of problem 2 would arise to the following procedure:

1. Find a controller [K.sub.0](s) which stabilizes the closed-loop transfer function [F.sub.d] ([G.sub.0](s), [K.sub.0] (s)) and such that:

[parallel][F.sub.d]([G.sub.0](s), [K.sub.0](s))]parallel].sub.[infinity] < [gamma].

2. The composed controller K (s) = [K.sub.0] (H (s)) stabilizes the closed-loop system [F.sub.d](G (s), K (s)) and: [parallel][F.sub.d](G(s), K(s))]parallel].sub.[infinity] < [gamma]

(see [2], [4], and [5]).

It is clear that condition 3 in the first problem, or condition 2 in the second problem, can be relaxed to the following condition: the degree of the denominator of H (s) is as high as be possible with the appropriate conditions. With this new condition, the open problems are a bit less difficult.

Finally, problem 3 can be interpreted in terms of robustness under positive polynomial perturbations in the coefficients of a stable transfer function.

Problem 1.2

The realization problem for Herglotz-Nevanlinna functions

Seppo Hassi Department of Mathematics and Statistics University of Vaasa P.O. Box 700, 65101 Vaasa Finland sha@uwasa.fi

Henk de Snoo Department of Mathematics University of Groningen P.O. Box 800, 9700 AV Groningen Nederland desnoo@math.rug.nl

Eduard Tsekanovskii Department of Mathematics Niagara University, NY 14109 USA tsekanov@niagara.edu

1 MOTIVATION AND HISTORY OF THE PROBLEM

Roughly speaking, realization theory concerns itself with identifying a given holomorphic function as the transfer function of a system or as its linear fractional transformation. Linear, conservative, time-invariant systems whose main operator is bounded have been investigated thoroughly. However, many realizations in different areas of mathematics including system theory, electrical engineering, and scattering theory involve unbounded main operators, and a complete theory is still lacking. The aim of the present proposal is to outline the necessary steps needed to obtain a general realization theory along the lines of M. S. Brodskii and M. S. Livsic, who have considered systems with a bounded main operator.

An operator-valued function V(z) acting on a Hilbert space [??] belongs to the Herglotz-Nevanlinna class N, if outside R it is holomorphic, symmetric, i.e., V(z)* = V([??]), and satisfies (Im z)(Im V (z)) [greater than or equal to] 0. Here and in the following it is assumed that the Hilbert space [??] is finite-dimensional. Each Herglotz-Nevanlinna function V (z) has an integral representation of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where Q = Q*, L [greater than or equal to] 0, and [summation](t) is a nondecreasing matrix-function on R with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Conversely, each function of the form (1) belongs to the class N. Of special importance (cf. [15]) are the class S of Stieltjes functions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where [gamma] [greater than or equal to] 0 and [[integral].sup.[infinity].sub.0]d]summation](t)/(t+1) < [infinity], and the class [S.sup.-1] of inverse Stieltjes functions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [alpha] [less than or equal to] 0, [greater than or equal to] 0, and [[integral].sup.[infinity].sub.0]d]summation](t)/[(t.sup.2]+1) < [infinity].

2 SPECIAL REALIZATION PROBLEMS

One way to characterize Herglotz-Nevanlinna functions is to identify them as (linear fractional transformations of) transfer functions:

V (z) = i[[W(z) + I].sup.-1][W(z) - I]J, (4)

where J = J* = [J.sup.-1] and W(z) is the transfer function of some generalized linear, stationary, conservative dynamical system (cf. [1], [3]). The approach based on the use of Brodskii-Livsic operator colligations [THETA] yields to a simultaneous representation of the functions W(z) and V (z) in the form

[W.sub.[THETA]](z) = I - 2iK*[(T - zI).sup.-1]K J, (5)

V.sub.[THETA](z) = K*[([T.sub.R] - zI).sup.-1]K, (6)

where [T.sub.R] stands for the real part of T. The definitions and main results associated with Brodskii-Livsic type operator colligations in realization of Herglotz-Nevanlinna functions are as follows, cf. [8], [9], [16].

Let T [member of] [??], i.e., T is a bounded linear mapping in a Hilbert space [??], and assume that Im T = (T - T*)2i of T is represented as Im T = KJK*, where K [member of] [??], and J [member of] [??] is self-adjoint and unitary. Then the array

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

defines a Brodskii-Livsic operator colligation, and the function [W.sub.[THETA]](z) given by (5) is the transfer function of [THETA]. In the case of the directing operator J = I the system (7) is called a scattering system, in which case the main operator T of the system [THETA] is dissipative: Im T [greater than or equal to] 0. In system theory [W.sub.[THETA]](z) is interpreted as the transfer function of the conservative system (i.e., Im T = KJK*) of the form (T - zI)x = KJ][psi].sub.-] and [[psi].sub.+] = [[psi].sub.-] -2iK*x, where [[psi].sub.-] [member of] [??] is an input vector, [[psi].sub.+] [member of] [??] is an output vector, and x is a state space vector in [??], so that [[psi].sub.+] = [W.sub.[THETA]](z)][psi].sub.-]. The system is said to be minimal if the main operator T of [THETA] is completely non self-adjoint (i.e., there are no nontrivial invariant subspaces on which T induces self-adjoint operators), cf. [8], [16]. A classical result due to Brodskii and Livsic states that the compactly supported Herglotz-Nevanlinna functions of the form [[integral].sup.b.sub.a]d]summation](t)/(t - z) correspond to minimal systems [THETA] of the form (7) via (4) with W(z) = [W.sub.[THETA]](z) given by (5) and V(z) = [V.sub.[THETA]](z) given by (6).

Next consider a linear, stationary, conservative dynamical system [THETA] of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Here [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a rigged Hilbert space, A [subset] T [subset] A, [A.sup.*] [subset] [T.sub.*] [subset] A, A is a Hermitian operator in [??], T is a non-Hermitian operator in [??], K [member of] [??], J = [J.sup.*] = [J.sup.-1], and Im A = KJ[K.sup.*]. In this case [THETA] is said to be a Brodskii-Livsc rigged operator colligation. The transfer function of [THETA] in (8) and its linear fractional transform are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The functions V (z) in (1) which can be realized in the form (4), (9) with a transfer function of a system [theta] as in (8) have been characterized in [2], [5], [6], [7], [18]. For the significance of rigged Hilbert spaces in system theory, see [14], [16]. Systems (7) and (8) naturally appear in electrical engineering and scattering theory.

3 GENERAL REALIZATION PROBLEMS

In the particular case of Stieltjes functions or of inverse Stieltjes functions general realization results along the lines of remain to be worked out in detail, cf. [4], [10].

The systems (7) and (8) are not general enough for the realization of general Herglotz-Nevanlinna functions in (1) without any conditions on Q = [Q.sup.*] and L [greater than or equal to] 0. However, a generalization of the Brodskii-Livsic operator colligation (7) leads to analogous realization results for Herglotz-Nevanlinna functions V (z) of the form (1) whose spectral function is compactly supported: such functions V (z) admit a realization via (4) with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where M = [M.sub.R] + iKJ[K.sup.*], [M.sup.R] [member of] [[??]] is the real part of M, F is a finite-dimensional orthogonal projector, and [THETA] is a generalized Brodskii-Livsic operator colligation of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

see [11], [12], [13]. The basic open problems are:

Determine the class of linear, conservative, time-invariant dynamical systems (new type of operator colligations) such that an arbitrary matrix-valued Herglotz-Nevanlinna function V (z) acting on [??] can be realized as a linear fractional transformation (4) of the matrix-valued transfer function [W.sub.[THETA]](z) of some minimal system [theta] from this class.

Find criteria for a given matrix-valued Stieltjes or inverse Stieltjes function acting on [??] to be realized as a linear fractional transformation of the matrix-valued transfer function of a minimal Brodskii-Livsic type system [THETA] in (8) with: (i) an accretive operator A, (ii) an [alpha]-sectorial operator A, or (iii) an extremal operator A (accretive but not -sectorial).

The same problem for the (compactly supported) matrix-valued Stieltjes or inverse Stieltjes functions and the generalized Brodskii-Livsic systems of the form (11) with the main operator M and the finite-dimensional orthogonal projector F.

There is a close connection to the so-called regular impedance conservative systems (where the coefficient of the derivative is invertible) that were recently considered in [17] (see also [19]). It is shown that any function D(s) with non-negative real part in the open right half-plane and for which D(s)/s -> 0 as s -> [infinity] has a realization with such an impedance conservative system.

(Continues...)

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Edition details

• Publisher: Princeton University Press
• Publication date: 2004
• Language: English
• ISBN 10: 0691117489
• ISBN 13: 9780691117485
• Binding: Hardcover
• Number of pages: 352
• Editor: Blondel Vincent D., Megretski Alexandre