MTNS

International Symposium on the

Mathematical Theory of

Networks and Systems

Regensburg, August 2 - 6, 1993

Book of Abstracts

93

Regensburg anna 1493

Symposium Chairmen

U. Helmke, University of Regensburg, Regensburg H. Mcnnicken, {Tniversity of Regensburg, Regensburg

Organizing Committee

.I. Ackermann, Obcrpfaffcnhofen I). Bierlein, Hegcnsburg f. ('<)Ionius, Augsbnrg P. Dewilde, Delft L. Elsner, Bielefeld ,\. Fl'\tweis, 1:30clllll1l I. Gohberg, Tel Aviv D. Hinrichsen, Bremen I\I.A. Kaashoek. Amsterdam

H. Kimura, Osaka H.W. Knobloch, Wiirzburg S. Kodama, Osaka G. Kreissclmeier. Kassel .l.A. Kossek, Miinchen D. Pratzel- \Volters. Kais('J'slaut ern

A.C.:-1. Ran. Arnsterdalll J.H. van Schuppen, Amsterdalll

Program Committee

T. Ando, Sapporo A. ,\ntoulas, Houston .l.A. BaiL BlackslJllrg .1. Baras. College Park I{'\\'. Brockett, Cambridge (I\\A) C.1. Byrnes, St. Louis C;. (\)lltt'. Ancona 1'. E. Crouch, T(,lllIJl' H.F. Curtilin, (;roningen I\\. DeistieL Vienna P. Van Dooren, (l rbana II. Dym, Hehovot :\\. Fliess. Cif-sur-1'\'('11" B.A. francis, Toronto P.A. Fuhrmann, Beer SI"'\'a T. Fujii, Fukuoka :\\. (;,,\,ers. Louvain Ia. ;'lieu\'('

K. Glover, Cambridge 1\1.L..l. Hautus, Eindhoven 1\1. Hazl'winkel, Amsterdam .l.\V. Hellon. La .lolla S. 11(",)(" Nagoya ;-'1. Ikeda. 1';0[,,,

/\. Isidori. HOlllC T. Kailalh. Stanford T. Katayama. Kyoto

P.P. Khargonekar, Ann Arbor A.J. Kr('ner. Davis V. Kuc'era, Prague A. Kurzhansky, Laxenhurg

H. K wakernaak, Ensciwde A . .l. Lauh, Santa Barbara L. Lerer, Haifa 1\. Lindquist. St.ockholm L. Ljung, Linkiiping C.f. Martin, Lubbock S.K. Mitter. Cambridge (1\1:\) ,LB. :\\oore, Canberra A.S. Morse, J\ew Hawn

H. Kijmeijer. Ensched" It Ober. Dallas (;. Picci, Padova L. Rodlllan, \\,illiams[,lll'g

I\\. ShaYlllan. College Park E.D. Sontag, New Brunswick H. Sussmallll, :-Jew Brunswick T.J. Tarn, St. LOllis :\\. Vidvasagar, Ballga.!or(' J.C. Willelm, Croning(,11 Y. YaTllillTlotO. I';yo\.o

C. /':al1lcs, l\'Iuntl'Cill A.H. Z('lllanian, Stony Brook

....,

L

EDITORIAL

The Symposium Chairmen have the pleasure to present the Book of Abstracts of the 1993 International Symposium on the Mathematical Theory of Networks and Systems, MTNS 93, August 2-6, 1993, ill Regensburg.

In the first part of this book (pages 5-27) the reader finds the abstracts of the plenary lectures, the special topics lectures and of the presentations of the mini courses in alpha­bet.ical order. The second part (pages 31-.;344) contains the abstracts of the lectures from the invited and contributed sessions, also alphabetically ordered.

The Book of Abstracts shows that the International Symposium MTNS 93 offers an attractive program on a wide range of latest developments in systems and control the­ory, circuit theory, and signal processing. The program includes recent advances in linear/nonlinear systems and control theory, distributed parameter systems, stochastic control/filtering, robust control, H= control, optimal control, systems identification and modelling, adaptive control, circuits and signal processing, operator theory, numerical methods. algebraic-geometric met.hods, computer algebra, neural networks, and control theoretical aspects of artificial intelligence.

We wish all participants of MTNS 93 a successful conference.

U. Helmke R. Mennicken Regcnsburg, July 1993

2

Sponsors of MTNS 93

University of Regensburg

Bayerisches Staatsministerium fUr Unterricht, Kultus und Kunst

Deutsche Forschungsgemeinschaft (DFG)

Regensburger Universitatsstiftung Hans Vielberth

National Science Foundation (NSF)

International Science Foundation (ISF)

American Mathematical Society (AMS)

Canadian Mathematical Society (CMS)

Gesellschaft fur Angewandte Mathematik und Mechanik (GAMM)

Institute of Mathematics and its Applications (IMA)

Societa Italiana di Matematica Applicata e Industriale (SIMAI)

Wiskundig Genootschap (WG)

3

MTNS 93

Abstracts of

Plenary Speakers

Special Topics Speakers

and the Speakers in the Mini Courses

5

Robust control for active four-wheel steering of cars

Prof. J. Ackermann, DLR-FF-DR, 82230 Wessling

Conventional car steering involves feedback only via the driver commands at the steering

wheel. Due to the reaction delay time of the driver and uncertain operating conditions

(velocity, mass, tire-road contact) control is inaccurate or even dangerous if unexpected

yaw motions are excited by sidewind or braking on ice. A significant improvement in safety

can be achieved by robust automatic feedback control with the steering wheel command

as a reference input. Available actuators of such control systems are active front-wheel

steering (as used for automatic steering), active rear-wheel steering (as used in four-wheel

steering cars). Also sensors for states of the steering dynamics and for the uncertain

velocity are getting cheap enough for the use in cars.

Three robust feedback loops are designed in this paper for a simplified model of ca~ steering

dynamics:

1. A generic control law for yaw rate feedback to front wheel steering decouples robustly

the lateral motion of the front axle from the yaw dynamics. Interestingly this robust

decoupling property holds for arbitrary nonlinear tire characteristics.

2. A control law for yaw rate feedback to rear wheel steering yields velocity-independent

eigenvalues of the yaw subsystem. A controller parameter admits tuning of the yaw

damping.

3. A control law for lateral acceleration feedback to front wheel steering yields a velocity­

independent steering transfer function. A controller parameter admits tuning of the

time constant of the steering transfer function.

The control laws 2 and 3 are gain-scheduled by the velocity.

6

An overview of recent results on deterministic modeling

A.C. Antoulas, Rice University

The starting point is a set of input-output measurements (Ui,Yi), i E 11. We are seeking models for this data set, which are linear and time invariant. The basic idea emanates from the behavioral framework to system theory put forward by Jan Willems. It consists in considering the measurements without distinguishing between inputs and outputs. The data set D thus contains the time series Wi := (ul y!), i E 1l. As a consequence of this re-casting, the measurements are considered as generated by an autonomous system, and at first, we search for automomous models. The search for input-output (I/O) models, i.e. models which explain how the output Yi depends on the input Ui, is carried out at a later stage.

The above approach has been worked out for exact modeling in [1] and [2]. The main result is the existence of a unique (up to equivalence) autonomous system of minimal complexity denoted by 8*, which generates all other models 8, including the input-output ones. In system-theoretic terms, a parametrization of alI models is provided in terms of the following feedback configuration:

._--_ .. __ ... _-----_._---_._---:

: =W"------------..O. ~ ,

8

In [3] the above result is extended to the case of approximate modeling for arbitrary discrete-time measurements. The additional ingredient needed is a measure of the misfit between the data and the model, i.e. a measure by which we quantify the extent to which the models fail to match the data. Once such a measure is defined, and given a largest tolerated level of misfit, say f, the problem is to construct a fixed generating system 8*,,, such that all systems 8, obtained by means of the feedback configuration given above (with 8* replaced by 8*" and 8 by 8,) for arbitrary r, have misfit which is less than f.

The structure of the lecture is as follows. First the data set D will be introduced, which can be equivalently described by means of an arbirtary strictly proper rational matrix W with poles inside the unit disc. The set of exact models M turns out to be the kernel of the Hankel operator 1iw. It follows that every model 8 can be obtained as a linear combination of the rows of the fixed autonomous model 8*, which generates the kernel of 1iw. The misfit function 0, measuring the lack of fit between the given model and the data set, is an equation-type misfit involving 1iw. The main result states that the kernel of the Hankel-norm approximants of 1iw have the desired approximation property described above; the misfit level is given by the singular values of 1iw,

References

[1] A.C, Antoulas and J ,C. Willems, A behavioral approach to linear exact modeling, IEEE Trans­actions on Automatic Control, AC-38 (1993).

[2] A,C. Antoulas, Recursive modeling of discrete-time time series, IMA volume on Linear Algebra in Systems and Control (1993).

[3] A.C. Antoulas, A new approach to linear approximate modeling, Linear Algebra and Its Appli­cations, Third Special Issue on Linear Systems and Control (1994).

INTERPOLATION THEORY, FEEDBACK STABILIZATION AND HOC> CONTROL

J .A. Ball and M.A. Kaashoek

The mini-course will review recent developments in the theory of interpolation for rational matrix functions and the connections of the theory with feedback stabilization and HOC>_

control for linear, timee-invariant, finite dimensional systems. Highlights include: direct

translation of internal stability constraints to interpolation conditions, formulation and

solution of interpolation problems in the state space language, parametrization of solutions, recursive procedures, connections with J-inner-outer factorization, recent extensions to

time-variant and nonlinear systems.

7

8

Combinatorial Aspects of Differentiable Dynamical Systems

R.W. Brockett, Cambridge

At the heart of the most basic theories of formal computation is a certain semigroup

consisting of the set of finite words constructed from a given alpahebet. In this talk we will

identify an analgous object appropriate when one is attempting to do robust computations

with computers described by ordinary differential equations. Roughly speaking, we will

define an oriented version of the fundemental group. We will show that certain classes of

smooth dynamical systems of the input-output type define maps from the equivalence class

of all curves corresponding to a particular element of this semi group to the set of possible

steady state values. This allows us to establish a new class of relationships between analog

and digital computing.

r

STABILITY, OBSERVABILITY AND THE CONVERSE THEOREMS

OF LYAPUNOV FOR NONLINEAR SYSTEMS

Christopher 1. Byrnes

Dept. of Systems Science and Mathematics

Washington University

St. Louis, MO 63 130

USA

Abstract. In this talk we review some recent and sone new stability results for

nonlinear systems in the context of a unifying "LP invariance principle". This

invariance principle is similar in spirit to LaSalle's Invariance Principle, but is

instead based on an integral criterion which is particularly natural and easy to

use for certain classes of nonlinear control systems. The "LP invariance princi­

ple" is derived for stability about at tractors but can be considerably sharpened

in the case of stability of equilibria when suitable observability assumptions are

satisfied. In this case, one can also use the "LP invariance principle" to construct

Lyapunov functions for the controlled dynamical systems. Numerous applications and illustrations of the principle will be given.

9

10

The periodic Schur decomposition in control theory

P. Van Dooren, J. Sreedhar University of Illinois at Urbana-Champaign,

Coordinated Science Laboratory 1308 W. Main Str., Urbana, IL 61801

email:vdooren@maggie.csl.uiuc.edu

Recently a new unitary eigendecomposition for a sequence of matrices was derived, which

is called the periodic Schur decomposition. We show that it essentially plays the role

of the classical Schur decomposition but now applied to periodic systems of difference

equations. In this talk we discuss its existence and its application to the solution of

periodic difference equations arising in control and various control design problems. We

apply it to Lyapunov and Riccati equations, to pole placement and controllability and to

constructing the Floquet transform for periodic discrete time systems.

BACKWARDS SHIFTS, REALIZATIONS, REPRODUCING

KERNELS, AND INTERPOLATION; A TUTORIAL

Harry Dym

Department of Theoretical Mathematics

The Weizmann Institute of Science

Rehovot 76100, Israel

In this talk, we shall use the generalized back- wards shift operator f(z) to (f(z)­

f(w))/(z-w) as a tool to develop a number of important concepts in the theory of realization

and interpolation in a unified and simple way, both for the halfplane and the circle. The

talk is meant to be expository, and requires no prerequisites beyond some elementary

knowledge of Hardy spaces.

11

12

Numerical integration of POEs by discrete passive modelling of the underlying physical systems

Alfred Fettweis Lehrstuhl f. Nachrichtentechnik, Ruhr-Universitat Bochum,

Universitatsstr. 150, 0-4630 Bochum 1

Numerical integration of partial differential equations (POEs) can be achieved by directly modelling the original continuous-domain physic­al system by means of a discrete multidimensionally passive (MO-pas­sive) dynamical system /1-5/. The following features hold:

1. The POEs are first subjected to a space-time-domain coordinate transformation. This is done in such a way that the passivity and incremental passivity originally existing with respect to time become available in the multidimensional (MO) sense, i.e. with respect to all new coordinates. As a result, one can achieve not only full sta­bility with respect to the discretization in space and time but also full stability, and, more generally, full robustness with respect to the computational errors caused by the rounding/truncation and over­flow corrections and by extraneous sources.

2. Massive parallelism and full locality, which are inherent to all physical systems with finite propagation speed, are preserved in the resulting algorithm.

3. Arbitrarily changing parameters as well as arbitrary boundary shapes and conditions can be taken into account straightforwardly.

4. In order to achieve recursibility (computability), the simulation must be based not on the field variables appearing in the original partial differential equations, but on corresponding so-called wave variables. The resulting incident-to-scattering (cause-to-effect) re­lationships give rise to computational rules that exhibit the sequen­tial nature needed for obtaining an algorithm.

5. It appears easiest to apply the method by first representing the POEs resulting from the aforementioned coordinate transformation, by means of an MD Kirchhoff circuit. From this, the desired algorithm can be derived by applying the standard procedures known from the theory of MO wave digital filters /6/.

Some references: 1. A. Fettweis, "New results in wave digital filtering", Proc. URSI

Int.Symp.Sig.Syst.Electron. 17-23, Erlangen, Germany, Sept. 1989. 2. A. Fettweis and G. Nitsche, "Transformation approach to numeric­

ally integrating PEDs by means of WOF principles", Multidimension­al Systems and Signal Processing, vol.2,pp.127-159, May 1991.

3. A. Fettweis, "Discrete modelling of loss less fluid dynamic systems", Arch. Elek. Ubertr., vo1.46, pp.209-218, 1992.

4. A. Fettweis, "Discrete passive modelling of physical systems described by POEs", Proceedings, 6th Europ. Signal Processing Conference (EUSIPCO-92), pp. 55-62, Brussels, 1992.

5. A. Fettweis, "Multidimensional wave digital filters for dicrete­time modelling of Maxwell's equations", Int. J. of Numerical Mo­delling, vol. 5, pp. 183-201, 1992.

6. A. Fettweis, "Wave digital filters: theory and practice", Proc. IEEE, vol. 74, pp. 270-327, Feb. 1986.

-------------

On the Loo-L2 Interpolants for the Four Block Problem

Ciprian Foias Department of Mathematics

Indiana University Swain Hall East

Bloomington, Indiana 47405 USA

This is a report on a joint work with A.E. Frazho, W.S. Li and A. Tannenbaum on the general Kaftal-Larson-Weiss Loo-L2 type interpolants. Precisely, we will construct a specific controller in the four block problem for which we determined sharp estimates for both its Loo norm and the L2 Hilbert-Schmidt norm.

13

14

ON SEVERAL CLASSES OF CHARACTERISTIC FUNCTIONS AND

SOME APPLICATIONS

P. A. Fuhrmann*Department of Mathematics Ben-Gurion University of the Negev

Beer Sheva, Israel

June 14, 1993

Recently, Ober and the author did a systematic study of normalized coprime factorizations over Hoo, the normalization being with respect to different metrics. Such factorizations have been derived for several classes of functions. Combined with the Youla-Kucera parametrization this leads to the construction of several characteristic functions, all of them stable.

In this talk we will describe several diverse applications of the various characteris­tics. For the class of antistable functions the S-characteristic relates to the inversion of intertwining maps and hence to the inversion of Hankel operators, considered as restricted to maps from the cokernels to their image.

For the class of minimal systems the norm of the L-characteristic relates to robust stabilization under normalized coprime factor uncertainty. This is related tp previous work of Glover and McFarlane. We put the study of the L-characteristic in a wider context and develop around it a theory of duality between problems of robust stabilization and those of model reduction.

It is well known that some robust stabilization problems are solved by use of Nehari complements. We will describe how the B-characteristic, associated with the class of bounded real functions, can be used to derive a special suboptimal Nehari complement and furthermore to parametrize the whole set.

Finally, for the class of positive real systems, we relate the P-characteristic to the phase function associated with a stationary, Gaussian, stochastic process.

Much of .this work was done either in collaboration with Raimund Ober or owes much to it.

"Earl Katz Family Chair in Algebraic System Theory

F

Hoo control of nonlinear plants

J. William Helton

15

Department of Mathematics, University of California at San Diego, La Jolla, CA 92093 USA helton@osiris.ucsd.edu

This talk will deal with recent results on nonlinear H 00 control.

16

ABSTRACT

On stability of uncertain systems

Diederich Hinrichsen Institut fUr Dynamische Systeme

Universitiit Bremen 0-2800 Bremen 33 / F.R.G.

email: dh@matematik.uni-bremen.de

The aim of the paper is to give a selectiw overview of available tools and results for the robustness anal"sis of exponentiallY stable control systems. The focus is on linear state space models (both time-inyariant and time-varying) and structured perturbations of different types. Some important results concerning robust stability of input output systems will also be discussed. A number of open problems will be stated that mark the present state of knowledge in robust stability analysis.

Chain-Scattering Approach to Nonstandard Roo Control Problems

Hidenori Kimura and Xin Xin Osaka University

Abstract: The plant to be controlled is described by

The standard HOO control problem assumes that D21 is of full row rank and D12 is of full column rank. In many cases of practical importance, these assumptions do not hold. In this paper, we obtain a necessary and sufficient condition for the solvability of HOO control problems where these rank conditions on D21 and D12 fail.

We extend the chain-scattering formalism and J-lossless factorization for the standard problem [1] to non-standrd cases. Our method is based on the descriptor form of the chain-scattering representation of the plant

[

-sl + A Bl B2 0] G(s) = CHAIN(P) = (;2 D21 0 -I

C1 0 D12 0 o I 0 0

where (;2, D21 and D12 are augmented representations of C2 , D21 and D 12 ,

respectively. It is shown that G(s) has a (J, J/)-lossless factorization, Le., G(s) is

represented as a product

G(s) = 0(s)II(s)

where 0(s) is a (J, J/)-lossless matrix and II(s) is bi-stable, if and only if two generalized Riccati equations have solutions that satisfy certain conditions.

Based on the above factorization result, a complete parameterization of solutions of non-standard H OO control problem is given.

17

18

V. KOROLYUK, Kiev, Ukraine

E-mail: mathem@sovamsu.sovusa.com Fax: (044) 2252010

DIFFUSION APPROXIMATION OF MARKOVIAN QUEUEING SYSTEMS AND NETWORKS

Abstract. The diffusion approximation of Markovian queueing systems and net­works is obtained as special consequence of the pattern limit theorem about weak convergence of Markov process with locally independent increments on Euclidean finite-dimentional phase space RN. Theorem. Let ('(i) = ((k(i); 1 :; k :; N), E > 0, be family of Markov processes with generators

L;'P(x) = E-2Q;'P(x) - e'(i, x)'P'(x)

where Q~'P(x):= JRN Q;(x,dy)'P(x + EY) - q;(x)'P(x),

N

q;(x) = Q;(x,RN),e'(i,x)'P'(x):= 2>~(i,x)8'P(x)/8xk' k=l

Let there be following asymptotic representations as E -+ 0

e'(i, x) = E-1eO(i, x) + e(i, x) + 0"

E-1b'(i,x) = E-

1 r yQ;(x,dy) = t- 1bO(i,x) + b(i,x) + 0" JRN B'(i,x) = r y'yQ;(x,dy) = B(i,x)+ 0" JRN

where 10, I -+ 0 as t -+ O. Then under additional balance condition

and weak convergence of initial values ('(0) =} (Oast -+ 0, there is weak convergence

to the diffusion process (O(i) with the generator

L~'P(x) = [b(i,x) - c(i,x)l'P'(x) + ~B(i,x)'P"(x),

where B(i,.T)'P"(X):= 2::~r-1 Bkr(t,x)82'P(x)/8xk8xr. The application of this' tlleorem to the real Markov queueing system or network

consists in the calculation of asymptotic representation of the local characteristics.

.... F------~----

Necessary Conditions for Stabilizability and Detectability of Nonlinear Systems

Arthur J. Krener

Department of Mathematics

University of California, Davis, CA 95616-8633', USA

19

Abstract: A nonlinear system is said to be stabilizable around a critical point if there

exists a state feedback control law such that the closed loop system is asymptotically

stable to the crtical point. There are several variations on this definition, smooth or piece­

wise smooth feedback, local, regional or global stability, static or dynamic state feedback,

exponential or inverse polynomial convergence, etc.

In 1983, Brockett initiated the study of necessary conditions for nonlinear stabilizability

by giving his well-known three conditions. We shall review this as well as more recent

work in this area before turning to the dual problem of nonlinear detect ability.

A nonlinear system is said to be detectable around a positively invariant set if there exists

an observer with asymptotically stable error dynamics. Again there are several variations

on this definition, smooth or piecewise smooth observer, local, regional or global stability,

the type of error convergence, exponential, inverse polynomial or Lyapunov function, etc.

I will close with a presentation of joint work with Michael Zeitz on the dual for nonlinear

detect ability of Brockett's necessary conditions for nonlinear stabilizability.

20

The Nonlinear System Toolbox

Arthur J. Krener

Department of Mathematics

University of California, Davis, CA 95616-8633, USA

Abstract: During the 1960's and early 1970's there was tremendous progress towards a

comprehensive theory of design for linear controllers and observers. In the latter part of

the 1970's, approaches such as LQR and LQG were numerically vetted and incorporated

in easy to use software packages such as MATLAB's Control System Toolbox. As newer

theories for linear design have been developed, e.g. H-Infinity Control, they have quickly

been incorporated in software packages. These packages are easy to use, numerically stable

and have proven to be of great utility in linear design. More recently, researchers in the US

and in Europe have made impressive progress towards a comprehensive theory of nonlinear

design. The time has come to begin the next stage of the development process. These

theoretical approaches need to be implemented in user friendly code, numerically vetted

and applied to wide variety of models to determine their level and range of effectiveness.

For the past several years with support from AFOSR, I and some colleagues at Davis have

been engaged in the development of the Nonlinear Systems Toolbox, a MATLAB based

package of control and estimation algorithms for nonlinear systems using Taylor series

expansions of some nonlinear theories. Many recent theories of nonlinear design readily

lend themselves to series expansions because they are generalizations of linear theories and

lead to first order partial differential equations that can be solved term by term. Generally

a linear algorithm is used to find the linear coefficients, the higher coefficients are found

by successively solving a sequence of linear equations of Sylvester type.

We have made the Nonlinear Systems Toolbox available by anonymous FTP to researchers

throughout the world. The current version 0.5, can handle systems with up to cubic

nonlinearities, the next version 1.0 will handle systems of arbitrary degree subject to

memory limitations. I expect that these memory limitations will be greatly ameliorated

by the sparse matrix tools in MATLAB 4.0.

This workshop will be an introduction to the Nonlinear System Toolbox, its basic data

types and algorithms. Afterwards participants will be able to apply the existing algorithms

to the design of nonlinear compensators and use the toolbox to encode new algorithms for

these problems.

F

Abstract:

CONSTRAINTS, REDUCTION and CONTROL

P.S. Krishnaprasad

Department of Electrical Engineering & Institute for Systems Research

University of Maryland

College Park, MD 20742

email: krishna@src.umd.edu

21

In the realm of classical mechanics with exogenous variables, the Lagrange-d'Alembert

principle occupies a central place. Recent investigations of this principle in the setting of

nonholonomic constraints have lead to a deep understanding of the geometric underpin­

nings of the subject. Specifically, it has become possible to treat in an intrinsic manner,

questions of reduction in the presence of nonholonomic constraints with symmetries. The

reduced systems so obtained, are subject to both external forces and fictitious forces (e.g.

forces associated to the curvature of a connection that encodes the constraint). In this talk,

we present the general theory under two key hypotheses and discuss a variety of examples.

The theory of principal bundles with connections is used in an essential way in this theory.

While prior work in this area due to Koiller, and due to Marsden and Scheurle have been a

source of inspiration, the work reported here is joint with Rui Yang and W.P. Dayawansa,

and a basic reference is the 1992 Ph.D. thesis of R. Yang.

22

CAN WE PRESERVE THE STRUCTURE OF RECURSIVE BAYESIAN ESTIMATION IN

A LIMITED-DIMENSIONAL IMPLEMENTATION?

Rudolf Kulhavy

Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, P. O. Box 18, 182 08 Prague, Czech Republic

E-mail: kulhavy@utia.cas.cs

The basic operational tool of recursive Bayesian parameter estimation is propaga­tion of the conditional probability of uncertain quantities given observed data. Tbe propagation is specified by elementary rules of probability calculus which are simple conceptually but often unfeasible in practice. What we need is a limited-dimensional approximation of the propagation that would meet computational constraints and, at the same time, preserve maximum of the "mathematical structure" of the true Bayesian solution.

We demonstrate that differential geometry provides a natural framework for dealing with this challenging problem. The starting point is to specify a Lie group L of those transformations of a differentiable manifold P of probability distributions of unknown parameters which have the structure of Bayes-rule mappings.

Then we describe a way of reducing dimensionality which preserves the structure of the underlying manifold P under the L-action. Specifically, we find a partitioning of the manifold P (i.e. an equivalence relation on P) that is closed under the action of the Lie group L. The concept of a principal fibre bundle provides an elegant general tool for this step. An intuitively appealing interpretation of the partitioning through orthogonal projections of probability distributions onto a given submanifold of P is discussed.

Next we show how our limited information about the true posterior distribution is substantially improved by recognizing the repetitive structure of estimation. This structure accounts for a special role of the empirical distribution of data entering the model. A specific geometry of the manifold R of empirical distributions produced by the partitioning of P is described and a number of appealing dual features between the geometry on P and R are illustrated.

We deliberately consider a rather simple case of independent and identically distributed data with finite data and parameter spaces. Possible extensions are briefly discllssed the!l. Some ways of building a particular approximation of the true posterior distribution ba5ed on limited knowledge of the above type arc outlined in conclusion.

., ... -----------~-----~~--~---- ------------

Abstract:

ObservabiJity: theory and practice

Clyde F. Martin

Department of Mathematics

Texas Tech University

Lubbock, TX 79409

USA

23

The problem of recovering the orbit of a dyanmical system from some restricted set of

measurements is ubiquitous in applied mathematics and applied systems theory. The

applications in which it arise range from cryptology, to heat flow, to recovering water

content in soil from temperature measurements, to monocular and binocular vision to

mention but a few. The purpose of this talk is to consider an application in vision and

to show that the mathematics that arises from the model is highly nontrivial. The main

application will be the problem of perspective vision and we will examine the natural

problems of algebraic geometry that arise from this problem. Namely the classification of

a set of rank 2 bundles on two dimensional projective space. Other open problems related

to the mathematics of observability will be discussed.

24

MATRIX VERSIONS OF INTERPOLATION PROBLEMS

OF NEVANLINNA-PICK AND LOEWNER TYPE

A.A. Nudelman

Odessa Civil Engineering Institute

Didrikhson Str., 4, 270029, Odessa, Ukraine

e-mail: odessaeco@glas.apc.org

Abstract

Interpolation problems for matrix functions of certain classes, when interpolation data are

prescribed as inside as on the boundary of the given analyticity domain, are considered.

Let R be the class of m x m matrix functions w(z) which are analytic in the open upper

half-plane C+ and (w(z) - w(z)*)/i 2: 0 for z E C+. We start with the problem of finding

functions w(z) E R, having prescribed polynomial asymptotics when z --t Xi, i = 1,2, ... , I,

z E C+. For 1 =; 1 this problem reduces to the matrix version of the classical truncated

moment problem. This suggest the "correct" statement of the above mentioned problem:

the leading coefficients of the asymptotic polynomials must be given by inequalities, not

by equalities.

An analogous problem is considered in the class of those functions w( z) E R which are

regular and nonnegative on the negative semiaxis of the real axis. This problem is a gene­

ralization of the Stieltjes moment problem. It arises in the broadband matching problem

and in certain questions of molecular interactions.

The following generalization of the Akhiezer problem is considered for both classes Rand

S: find among solutions of the above mentioned problems those solutions that are also

solutions of an interpolation problem:

k=O k=O

where Zj E C+, j = 1,2, ... ,1 are given points, b~), c~j) are given m-dimentional rows.

The simplest rational solutions are investigated.

r

l

TIGHT FRAMES, ZEROS, AND THE ALGEBRAIC RICCATI INEQUALITY

Giorgio Picci Dipartimento di Elettronica e Informatica

Universita di Padova, Italy

Abstract

In this talk we shall illustrate some results obtained recently in collabo­ration with A. Lindquist and G. Michaletzky reported in detail in the paper Zeros of Spectral Factors, the Geometry of Splitting Subspaces, and the Al­gebraic Riccati Inequality (submitted for publication).

We show that the study and classification of solutions of the Algebraic Riccati Inequality can be done very naturally and economically in terms of the geometry of zeros and of the zero dynamics of the family of minimal (not necessarily square) spectral factors of the related spectral factorization problem. Through this concept the analysis can be made coordinate-free, and straightforward geometric methods can be applied.

It is very well known that spectral factorization plays an essential role in the modeling of stationary stochastic processes. The geometry of ze­ros and the zero dynamics of the family of minimal spectral factors of the spectral density matrix of the process is in turn very closely related to the geometry of splitting subspaces of the process. We introduce the notion of output-induced subspace of a minimal Markovian splitting subspace, which is the stochastic analogue of the supremal output-nulling subspace in geo­metric control theory. We show how the zero structure of the family of spectral factors relates to the geometric structure of the family of minimal Markovian splitting subspaces in the sense that the relationship between the zero dynamics of different spectral factors is reflected in the partial order­ing of minimal splitting subspaces. Finally, we generalize the well-known characterization of the solutions of the algebraic Riccati equation in terms of Lagrangian subspaces invariant under the corresponding Hamiltonian to the larger solution set of the algebraic Riccati inequality.

25

26

Adaptive Control without Parameter Identification

D. Pratzel-Wolters Fachbereich Mathematik Universitat Kaiserslautern Erwin-SchrOdinger-StraBe

W-6750 Kaiserslautern

e-mail: prwolt©mathematik.uni-kl.de

The problem of adaptively stabilizing classes of linear, finite dimensional plants with

unknown state dimension n, initial states Xo and system matrices A,B,G based on

controllers without identification mechanisms has been extensively studied since the

mid 1980's. The lecture surveys basic results in this research area related to the

concepts of high gain feedback and switching function controllers based on the

adaptation of a single gain parameter k(t). Aspects of stabilization, tracking, model

following and synchronization of interconnected linear systems are considered as

well as robustness properties of the adaptive controllers. Alternative control

strategies based on adaptation of multiple gains k(t) ERP are presented. For scalar,

minimum phase systems of relative degree one polynomial adaptive stabilizers are

constructed. The results are extended to multivariable minimum phase systems with spectrum either in ([+ or ([" The stabilizers can also be used in series connection

with an internal model to guarantee tracking of certain reference signals for the

same class of systems. Under the weaker assumption det GB>O a similar polynomial

multiparameter controller is given which stabilizes 2-input, 2-output minimum phase

systems. This simple controller is smooth as opposed to the known piecewise

continuous approach using a spectrum unmixing set.

THE BEHAVIORAL APPROACH TO SYSTEMS AND CONTROL

Jan C. Willems Mathematics Institute

University of Groningen

P.O. Box 800

9700 AV GRONINGEN

The Netherlands

Email: J.C.WilJems@math.rug.nl

Fax: + 31.50.633976

27

In system and control theory it is customary to depart from an input/output forma­

lism, often an input/state/output structure. Thus input/output maps and transfer functions

playa central role in the field, controllability is defined in terms of state transitions, ob­

servability refers to the possibility of reconstructing the state trajectory from input/output

measurements. The input/output paradigm has proven to be a very powerful and useful one

indeed.

However, a closer look at some physical examples suggests that the input/output structure

is often artificial. Physical laws usually state that certain events happen simultaneously, not

that certain events cause others, as suggested by the input/output framework.

In the behavioral approach to the modelling of dynamical systems the model simply consists

of a set of time trajectories (the behavior). This set is often specified by means of equations

(behavioral equations) or, more generally, by specifications which involve auxiliary (latent)

variables is addition to the (manifest) variables which the model aims at describing.

The purpose of this mini-course is to outline the highlights of this theory. We will concentrate

on:

(i) modelling issue .•. In particular, we will outline the behavioral approach as a basis of

distributed modelling. Further, we will discuss the elimination of latent variables, the

issue of state specification, the identification of free and bound variables, and various

parametrization questions.

(ii) controllability and oburvability and the representation of systems with these properties.

We shall see that these are closely connected to left and right co-prime factorizations of

transfer functions. In the behavioral framework, the controllability becomes an intrinsic

property of a dynamical system (and not a property of a particular representation, as

it is in the state space setting).

(iii) The behavioral approach as a basis to control problems. Particular attention will be paid

to questions of stabilization and LQ-control, and the resulting quadratic polynomial

matrix equations.

Reference: J.C. Willems, Paradigms and puzzles in the theory of dynamical systems, IEEE

Transactions on Automatic Control, Vol. 36,259-294, 1991.

-29

MTNS 93

A bstracts of

Speakers in Invited/Contributed Sessions

Control of Nonlinear Systems Near Fold Bifurcations

E. H. Abed

Dept. of Electrical Engineering and the Institute for Systems Research

University of Maryland College Park, Maryland 20742 USA

ABSTRACT

31

Consider a nonlinear system :i; = fl'(x, u) where /1 is a parameter and u is the control. Suppose that for a range of values of the parameter /1, this system possesses an equilibrium point xO(/1) at which the system normally functions. In this work, we study the design of control laws which maintain operation at or near the desired equilibrium even when this equilibrium is compromised by its proximity to a fold, or saddle node, bifurcation. We survey existing results on this problem and introduce a new system of ideas which allow one to make positive use of nonlinear dynamic phenomena that might be present in the open-loop system or introduced through feedback. The traditional method of controlling a nonlinear system susceptible to jump phenomena is to employ a limiting control law. In the control of aircraft at high angle-of-attack and of axial compressors, a limiting controller is referred to as a stall avoidance controller. This is a way of ensuring that system states remain a safe distance from the regime (jump boundary) in which sudden transitions can occur. There is one major difficulty with this approach. Namely, it is often the case in engineering that system performance improves significantly as one approaches the jump boundary. Thus, it is of importance to determine control strategies which facilitate system operation close to, or even beyond, the jump boundary. In this work, we study the use of nonlinear bifurcation control laws to extend the regime of operation. The bifurcation control laws can either take advantage of existing dynamic bifurcations near the fold, or introduce such bifurcations. Vie give analytical results on the introduction of Andronov­Hopf bifurcation to periodic solutions, bifurcation to invariant tori, and the possibility of introducing strange attractors.

References

[IJ J.M.T. Thompson and H.B. Stewart, Nonlinear Dynamics and Chaos, Wiley, Chich­ester, 1986.

[2J E.H. Abed and J.-H. Fu, "Local feedback stabilization and bifurcation control, I. Hopf bifurcation," Systems and Control Letters, Vol. 7, pp. 11-17,1986.

[3J E.H. Abed and J.-H. Fu, "Local feedback stabilization and bifurcation control, II. Stationary bifurcation," Systems and Control Letters, Vol. 8, pp. 467-4'13, 1987.

[4J V. Janovsky and P. Plechac, "Asymptotic analysis of perturbed Takens-Bogdanov . points," Journal of Computational and Applied Mathematics, Vol. 36, pp. 349-359, 1991.

32

SPARK IGNITION ENGINE MODELLING APPLICATION TO EXHAUST POLLUTANTS EMISSIONS

Jamil ABIDA*t Daniel CLAUDEt

A wide range of models has already been proposed to simulate the spark ignited engine (SIE) behavior under steady states and transients. These models are modular based conception. They describe the principle phenomena occuring in the SIE: air admission, fuel injection, air-fuel mixture formation, combustion, torque production and exhaust pollutants emissions. Nevertheless, one difficulty is always hidden: the SIE holds discrete and continuous phenomena. The engine geometry involves cyclic events. The piston of each cylinder meets the top dead center (TDC) twice a revolu­tion. We consider the TDC events as the natural time sampling. The time f'l. between two TDC is proportional to the inverse of engine speed. Now, the throttle angle and the intake pressure are continuous quantities. We consider the mean values of the continuous quantities on f'l.. Moreover, the time varying lags are f'l. multiple.

We propose a nonlinear discrete model with three state variables P, N, R : the mean intake pressure, the engine speed and the mean fuel-air ratio; three inputs B, 0', T : the mean throttle angle, the spark advance and the injection time; one disturbance Tr : the mean resistant torque and three outputs HC, NOx, CO: the mean values of the unburnt hydrocarbons, the azote monoxyde and dioxyde and the carbon monoxyde in terms of volumic concentrations. The fuel-air ratio model takes into account the fuel evaporation and the fuel film dynamics. In the exhaust gas emissions studying, we note the high influence of the fuel-air ratio. The classical emissions characteristics extrema and inflexion points positions depend on the engine speed, the intake mani­fod pressure and the spark advance values.

The data base is collected on a multiport full group injection 1.7 L. engine. The model is identified on a large running field (N, P, R, a) in steady states and tran­sients. Water cooling temperature, atmospheric pressure and temperature are taken constant. This model is used to predict the engine speed, the intake manifold pressure, the fuel-air ratio and the three exhaust pollutants concentrations. Easy implementa­tion and very good accuracy will allow facilities to study some control problems.

Note: This work is supported by RENAULT - Direction de la recherche.

'Laboratoire de Mecanique Physique. Universite de Paris 6. 78210 Saint-Cyr-L'Ecole Fran.ce. Tel:(33) (1) 39 25 4261

t Laboratoire des Signaux et Systemes. CNRS-ESE, Plateau de Moulon. 91192 Gif-sur-Yvette France. E-mail: abida@frese51.bitnet Fax (33) (1) 69413060. Tel:(33) (1) 69 418040.

ON STURM-LIOUVILLE BOUNDARY PROBLEM

DEPENDING NONLINEARLY ON THE

EIGENVALUE PARAMETER

The differential equation

V.Adamjan

y" +(2+-q-)y= 0 u-2

on the interval [0,11 with boundary conditions

yeO) = y(l) = 0

(1)

(2)

is considered in e(O,I). Here q and u are real continuous functions, q>O, and A,

is the eigenvalue parameter. Put a = inf u(x), /3= supu(x).

The eigenvalues of the boundary problem (1), (2) out of [a.,~1 form

semi bounded from below set of real numbers 0"0 with limit points at +00 and

maybe at a..

Suppose that the value a. of u is attained only at the finite number of points

Xl'····'x.,· The subset 0"0 n (-00, a) is finite if for all Xj

q(x) 1 lim(x-x)2 <-. x-u, J u(x)-a 4

The subset 0"0 n (-00, a) is infinite if for some Xj

q(x) 1 lim(x-xY >-. x-tx, u(x)-a 4

If for ye(a,/3) an interval (xpx2)c[0,1] exists such that

1) u( x) - y < 0, X e (-Xi, x2); 2) u( x, ) - y = 0 and q (x;) > 0,

if X, ;t 0,1, 1 = 1,2; 3) y< 1i(x2 - .x;)2,

then'Y is not an eigenvalue of the problem (1),(2).

Let fP/x),} = 1,2 ... be the normalized eigenfunctions of the boundary problem

(1), (2) for ail eigenvalues 2J > /3. If /3< 1i, then the set (fPJ (x)t is a basis in

LZ(O,l), which is quadratically close to orthonormal. This work was stimulated

by some recent works of H.Langer, R.Meunicken and M.Moller.

33

34

On the stabilization of planar homogeneous systems

D. Aeyels, R. Sepulchre and I. Mareels

The feedback stabilization problem of nonlinear control systems using (local) ho­mogeneous approximations has recently received a lot of attention in the literature (see for instance [2],[3],[4]). It generalizes the stabilization problem via the linear approximation in the sense that if a homogeneous approximation admits a homo­geneous stabilizing feedback, then the same feedback is (locally) stabilizing for the system itself. Moreover, in the particular case of planar affine control systems, it has been shown that the local controllability of the system always implies tl1P existence of a homogeneous approximation which is stabilizable by homogeneous feedback ([4]). As a consequence, every planar affine system which is locally controllable is also stabilizable.

In this paper we consider two particular examples of planar but not affine ho­mogeneous systems:

{X=.r+u

(E2)' . . 2 2 Y = .3y+xu,(x,y)EiR,UEiR.

Both systems are locally controllable and satisfy the degree condition which is known to be necessary for continuous stabilization (see (44)-(45) of [1]). We show that the two systems have different stabilizability properties. The system (E 2 ) is shown to be not stabilizable by homogeneous feedback but (semi-globally) stabilizable by (non homogeneous) continuous feedback while (Ed is not stabilizable at all using continuous feedback. It follows that only considering homogeneous feedbacks laws for stabilizing homogeneous systems is a limitation in some situations. This answers a recent question raised in [2]. The result also raises new questions regarding the necessary conditions for continuous stabilization (compare with [1], rpmark 7).

References

[1] Coron, J-M., "A necessary condition for feedback stabilization", Syst. & Con­trolletters 14(1989),227-2:32.

[2] Dayawansa, W.P., "Recent advances in the stabilization problem for low di­mensional systems", Proceedings of the NOLCOS'92, Bordeaux, 1-8.

[:l] Hermes, H., "Nilpotent and high order approximations of vector fields systems", SIAM Review :l:l(June 1991), 2:l8-264.

[4] Kawski, M., "Homogeneous stabilizing feedback laws", CTAT 6, n4 (1990), 497-516.

Abstract

Stability and Roundoff Noise Analysis

in 2-D Digital Filters

by

P. Agathoklis and A. Kanellakis

Dept. of Elec. & Compo Eng.

University of Victoria

P.O. Box 3055

Victoria, B.C.

CANADA V8W 3P6

35

The synthesis of I-D recursive digital filter structures with minimum roundoff noise un­

der an L2 scaling constrains has been considered by many authors. Hwang and Mullis

and Roberts have developed techniques for obtaining low roundoff noise structures and

Williamson showed later that the use of residue feedback can lead to lower roundoff noise

under certain conditions. Recently, the problem of extending this work and the concept

of balanced realization to the 2-D case has received considerable attention and 2-D low

roundoff noise structures were proposed.

In this paper the stability and roundoff noise of a finite wordlength implementation of 2-D

recursive digital filters is considered. In particular the case of using residue feedback is

discussed. In finite wordlength implementations the result of an arithmetic operation is

quantized to a certain number of bits. The presence of residue feedback has the effect

that the least significant bits, the quantization residue, are not discarded but kept for

subsequent operations. The residue is either added directly, or first multiplied by some

residue coefficient and then added to the state in order to reduce the error in the next

state iteration. The stability properties of such 2-D filter realizations are studied and

the extension to the 2-D case of the residue modes and their connection to the second

order modes is discussed. The effect of a 2-D block diagonal state- space transformation

to the variance of the output error signal is considered and the minimum roundoff error

realization for a 2-D filter is obtained. The connection of this realization to other existing

2-D minimum roundoff error realizations is also considered.

36

On Optimality of Abnormal Extremals for Variational Problems

A.A. Agrachev* A.V. Sarychevt

Abstract

We develop an approach to investigation of abnormal extremals for prob­lems of constrained extremum in a Banach space, and for Lagrange problem of Calculus of Variations:

<Ii = loT cp(q(t),u(t))dt -t min, q = J(q,u), q(O) = l, q(T) = ql.

Standard 1st-order necessary optimality condition for the problems re­stricts search of optimum to a set of extremals. Among them one can single out class of abnormal extremals, for which the sets of Lagrange multipliers are 'degenerate': the multiplier corresponding to the functional <Ii vanishes, and hence the functional does not enter the 1st-order optimality condition, in virtue of which the abnormal extremal is defined.

Defining 2nd variation along abnormal extremals we establish for them 2nd-order necessary/sufficient optimality conditions, which, as in normal case, amount to nonnegativeness/positive definiteness of the second variation. We demonstrate that the sufficient optimality condition implies actually 'rigidity' of abnormal extremal, that means that the last is an isolated "point" of the admissible set of the problem (defined by imposed constraints).

We go further with Jacobi-Morse-type theory of 2nd variation for abnor­mal extremals of Lagrange problem of Calculus of Variations and set Index and Nullity Theorems for abnormal extremals. These theorems provide pos­sibility of verification for the 2nd-order optimality conditions and also enable us to establish optimality and weak rigidity of having corank 1 small sub­arcs of abnormal extremals, whenever they satisfy Strong Legendre condition. This proves, that the abnormal extremals are quite worthy, although a bit pathologic, candidates to supply optimum to variational problems.

'" Steklov Mathematical Inst., Russian Acad. of Sciences, ul.Vavilova 42, Moscow,l17966,Russia. t Invited Professor, Dep. de Matematica, Universidade de Aveiro, 3800, Aveiro, Portugal. On

leave from: Inst. for Control Sciences, Russian Acad. of Sciences, Profsojuznaja ul. 65, Moscow, 117342,Russia.

~-- ---- ~~------------~

On the Worst Case Performance of the Least

Squares Algorithm *

Hiiseyin Ak<;ay § and Pramod P. Khargonekart

MTNS 1993

Abstract

The least squares algorithm, due to Gauss, is one of the most widely used al­

gorithms in science. It has been extensively studied and used for parametric system

identification. It is very well known that the least squares algorithm enjoys certain op­

timality properties under suitable stochastic assumptions about the exogenous noise.

In contrast, recently some papers have taken a worst-case deterministic approach to

identification. In particular, our work is most closely related to the work on time­

domain worst-case identification problems.

Our work has grown out of a need to make connections between the classical

identification theory and the more recent work in the area of robust identification.

Towards this goal, in this paper we have investigated the performance of the least

squares algorithm in the presence of worst-case bounded noise. In the result of this

paper 1 we derive a bound on the worst-case parameter estimation error using the

least squares algorithm in the presence of arbitrary bounded noise. This error bound

shows that if the input is chosen to be a pseudo random binary sequence, the worst­

case parameter estimation error decreases to zero as the noise bound decreases to

zero. While the problem formulation is motivated from the deterministic worst-case

identification theory, the techniques employed in this paper draw upon the results in

classical identification theory .

• § Dept. of Mechanical Engineering a.nd Applied Mechanics, The University of Michigan, Ann Arbor,

MI48109-4315.

t Dept. of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, 1.11

48109-2122. Tel. No. (313) 764-4328, Fax No. (313) 763-8041

37

38

CONTROLLABILITY OF DISCRETE-TIME NONLINEAR SYSTEMS Francesca Albertini(*)

Eduardo D. Sontag SYCON - Rutgers Center for Systems and Control

Department of Mathematics, Rutgers University, New Brunswick, NJ 08903 (908) 932-3072, E-mail: albertini@pdmatl.unipd.it.sontag@hilbert.rutgers.edu

(*) Also: Universita' di Padova, Dipartimento di Matematica, via Belzoni 7, 35100 Padova, Italy

ABSTRACT

We will study some controllability aspects of discrete-time nonlinear systems. More precisely we will consider systems of the type:

x(t + 1) = f(x(t), u(t)), t = 0,1,2, ... , (1)

where x(t) E X and u(t) E U. The state space X is a connected, second countable, Hausdorff, differentiable manifold of dimension n, and the control-value space U is a subset of lR. m such that U <:;; c10s int U. Given the system (1), one may introduce the forward-accessible set from a state x, which we will denote by R(x). This is the set of states to which one may steer x using arbitrary controls. We also introduce the orbit from x, which we will denote by O(x). This is defined as the set consisting of all states to which x can be steered using both motions of the system and negative time motions: a state z is in the orbit of x if there exists a sequence of states

such that, for each i = 1, ... ,k, either x. is reachable from X'_1 or X'_1 is reachable from x.. We will focus our attention on the relation between the notion of forward accessibility (i.e. intR(x) 1 0) and the weaker notion of transitivity (i.e. intO(x) 10). In analogy with the classical continuous-time "positive form of Chow's Lemma," we would like to see when transitivity implies forward accessibility.

Under the assumption that the system is analytic, U is connected, and that for all u E U the map f,. : X ..... X, with f,.(x) = f(x,u), is a diffeomorphism ("invertibility" assumption), we will prove the desired implication in several cases. More precisely we will establish this implication pointwise if the state x has a weak type of Poisson stability property, and globally if there exists a transitive state i E X which is a global "attract or" for E. These results strengthen considerably those obtained by the authors in a previous work (Siam J. Control, to appear).

We will also present some criteria for forward accessibility and transitivity when the invert­ibility assumption is relaxed. Finally various examples and counterexamples are provided.

r UNIQUENESS OF WEIGHTS FOR RECURRENT NETS

Francesca Albertini Universita' di Padova, Dipartimento di Matematica,

Via Belzoni 7,35100 Padova, Italy, Eduardo D. Sontag,

Department of Mathematics, Rutgers University, New Brunswick, NJ 08903

We will study the question of identifiability from input/output measurements of the weights of recurrent neural networks. A recurrent net with m inputs, p outputs, dimension n, and acti­vation function (1 is specified by a triple of matrices A, B, C, where A, B, and C are respectively real matrices of sizes n x n, n X 171 and p x n, and by a function (1 : JR -> JR. We use the notation

E = E(A,B,C,(1).

We will interpret the above data (A, B, C) as defining a controlled and observed dynamical system evolving in JR n by means of a differential equation

x = a(Ax+Bu), y = Cx (1)

in continuous-time (dot indicates time derivative), or a difference equation

x+ = a(Ax+Bu), y = Cx (2)

in discrete-time ("+" indicates a unit time shift). The notation a( x) denote the application of the activation function (1 to each coordinate of

the vector x; this function is often taken to be a sigmoidal-type map. The entries of the previous matrices are usually referred to collectively as the "weights" of the network.

Depending on the interpretation (1) or (2), one defines an appropriate behavior behE, map­ping suitable spaces of input functions into spaces of output functions. Given recurrent nets E and E (necessarily with the same numbers of input and output channels, i.e. with p = p and m = rh), we again say that E and E are equivalent (in discrete or continuous time, depending on the context) if it holds that behE = beh E.

It is clear that there are some transformations which bring a net E into an equivalent net E; for example, an interchanging of all incoming and outgoing weights between two neurons, a relabeling of neurons, or, for odd activation functions, flipping the signs of all incoming and outgoing weights at any given node. We will prove that these transformations are precisely the only possible ones.

More precisely, we will show that, under very weak genericity assumptions, the following is true. Assume given that two nets are given, if they are equivalent then necessarily they must have the same number of neurons and -except at most for sign reversals at each node- the same weights. Moreover, even if the activations are not a priori known to coincide, they are shown to be also essentially determined from the external measurements.

39

40

Symmetric Decomposition of a Matrix

Markus Ali, Jiirgen Gotze and Rainer Pauli Institute for Network Theory and Circuit Design

Technical University Munich Arcisstr. 21,8000 Munich 2, Germany

e-mail: maal@nws.e-technik.tu-muenchen.de

Matrix factorizations like the LR-, QR- or polar decomposition are employed to solve various linear algebra and linear system problems. No attention, however, has been drawn to the decomposition of a square matrix A into the product of two symmetric matrices Y and X-I. We analyze the properties, that is the existence and

. ambiguity, of such a symmetric decomposition and the related matrix equation AX -X AT = 0, the homogenous and "totally singular" case of AX -X B = C. An O(n3

)

algorithm for the symmetric decomposition, which initially computes a Hessenberg decomposition of A, is proposed. Applications of the symmetric decomposition with respect to the unsymmetric eigenvalue problem as well as with respect to network modelling are given.

Key words: symmetric matrix decomposition, eigenvalue problem, Jacobi algo­rithm, network modelling.

Existence and regularity of solutions of linear time-dependent and quasilinear wave equations on one-dimensional networks

F. Ali Mehnwti, Tt'chnisdlC' Hochschule Darmstadt

We consider scalar linear wave equations with time dependent coefficients (I) and quasi linear wave equations (2) on one-dimensional networks with linear transmission conditions at the nodes. In case (1) em-functions of time and space variables as coefficients and initia.l conditions satisfying a (recursively given, time dependent) compatibility condition imply existence and uniqueness of a lfm+l-solution. If the coefficients in case (2) are C m +1-fundions of the solution and its first time and space derivatives and the initial conditions satisfy an implicitely given (nonlinear) compatibility condition, we show the existencf' local in time and uniqueness of a solution in flm+l. In both casf'S the solution satisfies the compatibility condition for all times. These results (given in [1]) arc based on the application of the theory of T. Kato on quasilinear evolution equation of hyperbolic type [4J.

In the special case of the wave equation with constant coefficients on a cross we give an explicit d'Alembert-type solution.

A concept for nonlinear interactions of nonlinear f'volution phenomena on fami­lies of media is proposed in [2J. Loo-time d{~cay estimates for a transmission initial value problem with a tunnel type effect are given in [3J as essf'ntial steps to the existence of global small solutions.

[IJ Ali Mehmeti, F.: Technische Hochschule Darmstadt, Preprint-Nr. 1193, 1992.

[2J Ali Mehmeti, F., Nicaisf', S.: Nonlinear Interaction Problems; Nonlinear Analysis, Tlwory, Methods and Applications, Vol. 20, No.1, pp. 27-61, 1993.

[3J Ali Mehmeti, F.: Spectral theory and Loo-time decay estimates for Klein­Gordon equations on two half axes with transmission: the tunnf'1 effect; Tech­nische Hochschulc Darmstadt. Preprint-Nr. IS67, 199:3.

[4J Kato, T.: Linear and Quasilinear EquatioIls of Evolution of Hyperbolic Type; C. I.M.E., II cicio, Cortolla 1976, 125-191.

41

42

Speaker: D. Alpay

Title: State--space theory of automorphisms of rational matrix functions

Abstract: In this talk we report on joint work with J. Ball, l. Gohberg and L. Rodman. The state

space theory from linear control theory is used as a tool to describe the action of certain auto­

morphisms acting on rational matrix functions. The study consists of four pans: fust we focus

on degree preserving automorphisms. Then, we study the automorphisms representable as the

composition of a linear fractional change of variable together with the operations of inverse­

transpose, conjugation and :In inner automorphism. The next step is to study automorphisms

which preserve l-unitarity. Finally, we look at an application, namely interpolation in the

Stieltjes class.

p

Title: Reproducing kernel spaces associated to non stationary H2 inter­polation

(Daniel Alpay)

Let h be an analytic function in the unit disk 10. Then, as is well known, it is in the Hardy space H2 with norm less than one if and only if the kernel K(z,w) = 1/(1- zw·) - h(z)h(w) is positive in 10. Therefore, there exists an associated reproducing kernel Hilbert space of functions analytic in ID with reproducing kernel K. This space is the analogue of the de Branges­Rovnyak spaces associated to a Schur function. In this talk, we will discuss the analogues of these spaces in the nonstationary framework where H2 is replaced by the space 112 of upper triangular Hilbert-Schmidt operators. We will study also the relevance of these spaces to the Nevanlinna-Pick interpolation problem in 11 2•

43

44

Traces of Hardy functions and reproducing kernel Hilbert spaces

Daniel Alpayl and Juliette Leblond2

Abstract

We study Loewner-type intNpolation problems on the unit disk, using reproducing kernel Hilbert spaces methods. The special case of traces of the Hardy space 1I2 on" subset. of the unit circle is considered as an application.

Keywords: Loewner i,lterpolatioll probkms. reproducing kernel Hilbert spaces,

Hardy spaces.

pi

On the Giobal Stability of Stable Linear Systems with

Single Saturated Feedback

Jose Alvarez-Ramirez, Rodolfo Suarez and Jesus Alvarez

Division de Ciencias Basicas e Ingenieria

Universidad Autonoma Metropolitana-Iztapalapa

Apdo. Postal 55-534, 09340 Mexico, D.F., MEXICO

e-mail: rsua@snake4.uam.mx

In general, a stable linear system with a stabilizing feedback does not necessarily remain globally asymptotically stable (GAS) if control saturation occurs. Recently [1,2]' it has been proved that all trajectories of the saturated system converge to some compact set of zero volume, this compact set being the origin if the system is two dimensional. In this work, we provide sufficient conditions for an n-dimensional stable system to remain GAS despite saturation. The proof of sufficiency is based on an n-dimensional generalization [3J of the Bendixon Theorem.

[1J R. Suarez, J. Alvarez, and J. Alvarez, Stability Regions of Closed loop Linear Systems with Saturated Linear Feedback. IEEE Trans. Aut. Contr. (sub­mited).

[2J J. Alvarez, R. Suarez and J. Alvarez, Planar Linear Systems with Single Satu­rated Feedback, Syst. &; Control Lett. 1993, (to appear).

[3J R. A. Smith, An Index Theorem and Bendixon's Negative Criterion for Certain Differential Equations of Higer Dimension, Proc. Roy. Soc. Edinburgh, 104A, pp. 235-259, 1986.

45

46

n

Index and Stability of Differential-Algebraic Systems

Martin Arnold Rostock University, Department of Mathematics Postfach 999, D - 0 - 2500 Rostock, Germany

Tel. +49 381 369 372, e-mail: arnold@mathematik.uni-rostock.dbp.de

In contrast to the situation in the field of ordinary differential equations the (analytical) solution of higher index differential-algebraic systems (DA-systems) F(t, y(t), y'(t)) = 0 does not depend continuously on small perturbations 8(t) of the right hand side. This fact has a discrete analogue in the numerical solution of initial-value problems in DA­systems: small perturbations in the numerical solution (e. g. round-off errors, errors in the iterative solution of systems of nonlinear equations) can drastically be amplified during integration, an effect that leads to large errors in the numerical solution or the integration even totally breaks down. We give bounds for the influence of small pertur­bations on the analytical solution of DA-systems of index 2 and 3 in Hessenberg form that include 118(t)1I and 118'(t) II (for systems of index 2) and 118(t)1I , 118'(t)1I and 1W'(t)11 (for systems of index 3). In both cases the differential components of the solution vector are less influenced by small perturbations if the DA-system is linear in the algebraic variables. For implicit and half-explicit Runge-Kutta methods similar bounds are ob­tained for the influence of perturbations on the numerical solution. These bounds are used to analyze and to compare various approaches to the numerical solution of higher index DA-systems (e. g. stabilization and projection techniques, index reduction tech­niques). Numerical tests illustrate practical implications of the theoretically obtained results for the numerical solution of higher index systems by general purpose codes (e. g. DASSL, ODASSL).

p

On the Regular j-inner Mat~ix-Functions and the Regular ,-generating Matrices.

D.Z. Arov

Odtll. Ptd.,9,iu[ Inlfitult, Odella, tAe Ui,«int.

e-m«iL' ('IPOectlft".ia,.m,bu

The cla.u of the regular j-inner matrix-fundion. (m.fl W = [Wi1n with i = liag[I., -I.] (correspondelldy the c11S6 oC 'he regular 'l'-genera'ing ma'ricea U = [lliiU) had been introduced [1] a8 the claas of the matrice8 of the coefficient8 in the fractional transCorm which give the descrip'ion oC the set of the solutions Cor geners­JiBed bitangential interpolation problem in the matrix Schur clau (for matrix Nehary problem) in the completely indeterminate cue. We obtain a criterion that j-inner mJ. W (-r-generating matrix UI is regular which is different from given in [1]. Consider

X = W221W2J (X = U;,}U2J I, C = (I - xl(I + Xtl.

Then C is the m.f. of the Caratheodory clUII and W (or U) can be normalized 8uch that 0(0) = I •. H 17(,,) i. the apectral m.f. for C in the integral repreaent&tion then 41'(p) has left and right factorizations with the outer m.c. 'P and ,. in the Hardy class H!x.:

4.f.;

consider unitary valued m.f. f = [~·l-lcp. Theorem 1 j·inner m.l. Jf (l-,ener4ting metriz (f) i, "e,u/ar iff

(i) O'(p) " ab,tI/utei, continutllJ',

(ii) inc I = o.

For the scalar Nehary problem tbis criterion on the regular U is equivalent to a Sar&nlOll's reauU [2].

References

[1] D.Z. Arov. 'Y-generating matrices, j-inner matrix-functions and related extell8ion problema.

Teoriya funkt., funkt. anal. i ih pril. (Rul8i&n), Kharkov, 51, 62 (1989), 63(1990).

[2] D. Saraaon. Exposed points in HI, I. Operator theory: Adv. and Appl., ·n (1989).

47

48

On the Passive Scatterin, Linear Stationary Systems with ContlDuous Time.

D.Z. AroT, M.A. Nadelman

Odma Pti.,o,icalln.tiMt, Ode"., tile Uir.irar.

e-mail: [.[Ptla.oni •. iIJl.md .• fI

~: =.az + LIp- (z(O) = &I), Ip+ = N(z, Ip-),

where a(E [X,X_(Be)]) is the conjuga.te operdor for B"(E [X+(Be), X]), B is the geDerator of a CQ - semigroup T(t)(E [X]);

L E IN-, X_(B")]; N E ('DS,L,N'+],

'Dj,L = {(a,Ip-) : Btl + LIp- EX},

lI(tl,.p-)lr = lIalr + lIop-1r + lIaa + Lop-II'·

For anch a we consider ba.nsfer fnndions aa and the nolion8: conjngate &y8iem a', dilaLion, simple, minimal, CaJley transform. The syaiem III ia plLBBive BeAliering (a E 'PS) iff

2Re(aa + Llp-, a) S 1""-11' -IIN(a,Ip-HI' V(a,Ip-) E'Ds,£

and a is consen&ti1'e Buttering (a E C8) iff in this property we haTe C,=, but not ;'(" and such propeny has 01".

If a E 'PS than a· E 'PS and Sa belong to the cl&S6 S(}/-, }/+) holomorphic for Re~ ? 0 functions e(~) with values from [N'-,}/+] and a·(~la(AI S I. The .,.tem aO(E 'PS) i. called optimal if for any a(E 1'8) with a. = a ... we have for a = 0 a.nd for Any lAme input data op-( -) (E W~.:<}/-), op-(O) = 0) the inner abies Zae(t) of aO has the minimal norm:

liz ... U) II S Ilza(t)1I Cit ~ 0).

We UJe lhe Calley tra.nlform and corresponding reanl~s on lhe 8Y8teml wdh the di8-crete time [1] a.Dd prove that for a.ny 9 of the clasa S(N'- ,N'+) a simple a from Cs and a minim&! opt.imal a e from 'PS mit lIuch thAt

a .. lA) = 9 ... (AI = a(AI for Re~ > 0,

1hese Q and QO define by t.he 9 up 10 uui1a.ry equivalent.

References

[1] D.Z. Arov. Stable dissipative linear stationary dynamicaiscattering systems. Opera*or Theory 2 (1919), pp.95-12ti.

D

POSITIONING OF STICK-SLIP SYSTEMS AN ADAPTIVE BASED CONTROL

A. Astolfi and J. Chapuis Automatic Control Laboratory, Federal Institute of Technology

ETH-Ziirich, CH-8092 Ziirich, Switzerland FAX: CH-I-262'43'62, E-Mail: {astolfi,chapuis}@aut.ee.ethz.ch

May 19, 1993

The present paper is concerned with the design of a control law to improve the performance of a single mechanical axis. Its purpose is to point out and over­come all the difficulties connected with the practical implementation of a given control law.

In particular we take into account two different classes of limitations: the former, due to the struc­ture itself, leads to the presence of actuator satura­tioD, torque disturbances and friction torque; this sit­uation often arises when the given system and the control a.lgorithm are not. designed at the same time. The latter is connected to design choices and deals with the communication between system and world outside: the control system.

The desired performances of the controlled sys­tem for a prototype step response of ±3.5 mm are: no overshoot, Tgs 5 50 ms, em4% :5 0.04 mm.

The system can be described, using Euler­Lagrange formalism by the equation (x E JR)

M(x)x + C(x,x)x + e(x) =

sat_ •• (u) - h + [S'<Ck(X) + [Slid.(X) = u + d

where the r.h.s. consists of a known part (the con­trol signal u) and an unknown one, forming the total torque disturbance.

Even jf the closed loop specifications are given in term of the step response, we design a control law suitable for trajectory tracking. Therefore one of the task performed by the controller is to transform a given reference signal into an admissible trajectory. For such trajectory ( assuming all the states accessi­ble) it is possible to build ([4J and [1]) an adaptive control law ensuring Lyapunov stability of tracking error. The static error does not reach zero but it is possible to yield a bound. depending on the initial error and on the design parameters.

In the case under consideration we cannot access the velocity of the axis, therefore we have to estimate it. We propose in the sequel an adaptive observer for state estimation following the approach of [2J.

We turn now to the description of the critical points of the design procedure.

• the transfonnation of the reference step into an admissible trajectory requires the specification of the trajectory end time.

• To improve the perfonnance of the control sys­tem the initial state of the controller is not set to zero, but it is a function of the reference step. This results in a faster response to the plant, as the initial torque is not zero but tends to balance the effect of the load.

• To avoid limit cycles the integral action of the controller is switched. off when the position error is less than a given bound.

The previous discussion points out why and how a certain number of heuristic extensions of a pure math­ematical control law are necessary to obtain satisfac­tory performances for a real plant. However, we must stress that these extensions have to be introduced carefully and only when necessary, as they depend strongly on the parameters of the system and the working conditions. Therefore the designer should minimize every heuristic contribution to the control law, as they tends to derobusti/y the whole design.

As conclusion, even if our approach is not the only available (see [3J for different control law), it seems to be quite simple and general (application of the proposed a.lgorithm to a robotic arm is shown in [1]). In both cases, the obtained results are completely within the specifications and show low sensitivity to parameter variations.

References

[IJ A. Astolfi, Extended adaptive control of a rigid robot with friction, submitted to 199. CDG.

[2J G. Bastin and M.R. Gevers, Stable Adaptive Observers for Nonlinear Time-Varying Systems, IEEE 'lhlns. Automat. Contr., voL 33, pp. 650-657, July 1988.

[3J A.H. Glattfelder and W. Schaufelberger, Regelungssysteme mit Anschlagen und Begrenzungen: Eine Ubersicht und eine Fallstudie, Workshop GMA 1992, In­terlaken.

[4J J.-J.E. Siotine and W. Li, On the Adaptive Con­trol of Robot Manipulators, 1986 ASME Winter Annual Meeting, Anaheim, CA, 1986.

49

..

50

Invariance of Sector Domain Stability for Polynomials

with Perturbed Coefficients

T. Auba and Y. Funahashi Department of Mechanical Engineering

Nagoya Institute of Technology Showa, Nagoya 466, JAPAN

auba@eine.mech.nitech.ac.jp

Abstract

The purpose of this note is to show an algebraic aspect of Kharitonov's famous result and a generalization of the viewpoint to the sector domain stability problem.

Let D := {s E C ; 171" - arg sl < 71" - e}, where 71"/2 <:: 0 < 71", and A := {I:i=o QiSi E R[s] ; ai <:: Qi <:: ai, i = 0,···, n}, where 0i and ai are given numbers which determine the extent of A.

The following facts are well known: (i) all zeros of r(s) E R[s] belong to D if and only if all zeros of r(eies) are on the upper half plane, and (ii) all zeros of c(s) E C[s] belong to the upper half plane if and only if Bez(g, h) is positive definite, where C(5) := h(s) + ig(s), h(s),g(s) E R[s]' and Bez(g, h) is the Bezout matrix associated with g(s) and h(s).

Notice that the coefficients of f( eie s), f(s) E A, vary on spokes in complex number plane. The number of spokes, I, is n + 1 in general; however, reduc­tion of the number occurs in the case of special e, e.g., 1 = 2 when e = 71"/2, 1 = 3 when e = 271" /3, and so forth. As D-stability of a polynomial f( s) is de­stroyed, the corresponding polynomial f( e'"8) has some zeros on positive real line. Assuming that a polynomial for e'O s) := ho( s )+igo(s) has a positive zero (, we consider f(eiOs) = fo(eiOs)+ei¢p(s), p(s) E R[s]' i.e., the perturbation of coefficients on one spoke around fo(s). Then, we see that ('Bez(g, h)( = K· p((), where (= (1,(, ... ,("-1)', and K = g~(()cos¢ - h~(()sin¢. It is shown, because of the linearity of (' Bez(g, h)( with respect to p((), two extreme polynomials are not D-slable simultaneously. By using this prop­erty, it is proved that all polynomials of A are D-stable whenever specially constructed 21 polynomials are D-stable.

,..

Set-Valued Analysis, Viability Theory and Partial Differential Inclusions

Jean-Pierre Aubin & Helene Frankowska

Systems of first-order partial differential inclusions - solutions of which are feedbacks governing \'iable trajectories of control systems ~ are derived. A vari­ational principle and an existence theorem of a (single-valued contingent) solution to such partial differential inclusions are stated. To prove such theorems, tools of set-valued analysis and tricks taken from viability theory are surveyed.

Jean-Pierre Aubin & Helene ?~ankowska: CEREMADE

Universit~- of PariS-Dauphine Place Marecnal de Lattre de Tassigny F-75775 Paris Cedex 16 France

51

52

MTNS 93 Modelling and Estimation of Nonlinear Boundary

Value Processes

Arunabha Bagchi Department of Applied Mathematics

University of Twente P.O. Box 217, 7500 AE Enschede, The Netherlands

Phone 31-53-893452 Fax 31-53-340733

e-mail bagchi@math.utwente.nl

Abstract

Boundary value processes are stochastic processes satisfying ordinary or partial differential equations

with, possibly random, boundary conditions. For linear equations, existence of such processes and the asso­

ciated estimation problems have been extensively studied in the literature. The mathematical formulation

of such boundary-value problems for nonlinear equations is already a difficult one. This is due to the nature

of Ito integrals, which are defined forward in time. We study this problem when the input noise is modelled

directly as a finitely-additive white noise, instead of the usual approach where one works with the Brownian

motion. We consider two types of equations here. One involves a semilinear two-point boundary value problem

xCi) = A(t)x(t) + I(x(t» + net)

Fox(O) + F1X(1) = F

where A, Fa, Fl are m x m matrices, F is an m-dimensional random vector, n(-) is a finitely additive

Gaussian white noise, and I is bounded and Lipschitz continuous. The other one, in the simplest form, may

be described by

-ax(i) + I(x(t» = net)

x(O) = x(1) = 0

where a > 0, n(-) is again a finitely additive Gaussian white noise and I is continuously differentiable

satisfying

We show that in both cases, the solution process {x(-)} exists and is unique. The latter model can be

generalized to nonlinear elliptic partial differential equations. Finally, we consider the associated estimation

problem. Let the observation process be given by

yet) = x(t) + no(t)

where no(-) is another Gaussian white noise. We solve the smoothing problem of estimating xCi), for fixed

t, based on observing yes); 0 :S s :S I.

STABLE MOTIONS OF MECHANICAL SYSTEMS

WITH OSCILLATORY INPUTS

J. Baillieul

Aerospace/Mechanical Engineering

Boston University

Boston, MA 02215 USA

(617) 353-9848

johnb@buenga.bu.edu

FAX (617) 353-5866

There are many classical and modern examples of mechanical systems forced by

oscillatory inputs whose motions are governed by nonholonomic effects. A general

theory of these forced motions is of particular interest in the design of micromecha­

nical systems since nonholonomic effects are independent of spatial scale. In this

paper, we study the dynamics of "super-articulated" mechanical systems-systems

having fewer inputs than configuration-space dimensions. Using classical avera­

ging theory, it is shown that the stability of motion for such forced systems may be

analyzed using energy methods together with the adroit introduction of dissipation

into the models. For systems where the control inputs are applied directly only to

cyclic coordinates, we define a simple but important quantity called the averaged

potential. It is shown that under certain reasonable assumptions, forcing by pe­

riodic inputs produces dynamic responses which are confined to neighborhoods in

the phase space associated with local minima of the averaged potential. While the

critical points of the averaged potential may not correspond to equilibrium points

of the original forced system, averaging theory implies that trajectories of the

forced system will "hover" around such points. Hence we may study the tangen­

tiallinearization of the original forced system about local minima of the averaged

potential. It will be shown by purely geometric means that Lyapunov stability

may be deduced from a critical point analysis of the averag"d potential. Hence,

although the role of the averaged potential in our stability analysis is initially

established by a classical averaging argument applied to an appropriate dissipa­

tive system, it is shown to be a geometric quantity which describes the effects of

anholonomy and is independent of the effects of dissipation.

S3

54

tn

PARAMETER IDENTIFICATION OF NON-STATIONARY SYSTEMS

VIA FUZZY ELLIPSOIDAL SETS METHOD

Bakan G.M., Kussul N.N. Glushkov Institute of cybernetics of Academy

of sciences of Ukraine 252207, Glushkova 40, Kiev-207, Ukraine

The set-theoretic approach is widely used for solving the problems of identifying the parameters of mathematical models of controlled plants with restricted but unknown noise and disturbances.

A robust identification algorithm based upon the theory of fuzzy sets is proposed in [1] for parameter estimation of linear objects if there isn't any apriory about unknown vector. The concept of the fuzzy set estimate is introdused by means of which the set of localization of the identified vector is approximated.

In this paper the idea of identification in the class of fuzzy sets is extended to the case of non-stationary object with vector input and scalar output where there is not any a priori information about identified parameter vector and vector of perturbations.

The developing algorithms is robust with respect to a possible violation of a priori assumptions about the estimating vector.

1. Bakan G. M., Kussul N. N. System identification of linear objects in the class of fuzzy ellipsoidal sets// International Symposium on the Mathematical Theory of Networks and Systems. - Kobe, Japan, June 17-21, 1991. - p. 31-32.

Hoo-control and interpolation for time-varying systems

Joseph A. Ball, Israel Gohberg and Marinus A. Kaashoek

55

Department of Mathematics, Virginia Tech, Blacksburg, VA 24061 USA ball@math.vt.edu

School of Mathematical Science, Raymond and Beverly Sackler Faculty of Exact Sciences,

Tel-Aviv Uriiversity, Ramat Aviv 69978 Tel-Aviv Israel gohberg@taurus.bitnet

Faculteit Wiskunde en Informatica, Vrije Universiteit, De Boelelaan 1081, (1081HV) Am­

sterdam, The Netherlands, kaash@cs.vu.nl

If F(s) is a bounded analytic function on the right half plane (RHP) of the form

F(s) == d+ fooo e-·t J(t) dt then the evaluation F(so) of F at the point So in the RHP can be

viewed in the time domain as the result (after a change of sign) of evaluation of the kernel

k(t,s) for the integral operator (ft - M'o)-ITF along the diagonal (F(so) == -k(t,t)), where M. o : ¢(t) -> so¢(t) is the operator of multiplication by the number So and where

TF is the integral operator TF : ¢(t) -> d¢(t) + f~= J(t - a)¢(a) da. This suggests the

notion of the evaluation 1'(so) of a more general integral operator T : ¢(t) -> d(t)¢(t) + f~oo J(t, a)¢( a) da on L2(R) along a " time-varying point" So E L=(R) with " time-varying

spectrum" in the RHP (i.e. solutions of x(t) == so(t)x(t) decay exponentially as t -> -00) by [1'( So )l( t) == - k( t, t) where k( t, s) is the kernel of the composite integral operator (ft -M.o )-l.T. Recently the authors have completed a thorough mathematical treatment of the

time-varying analogue of the matricial bitangential Nevanlinna-Pick interpolation problem

based on this more general notion of point evaluation; the development parallels the state

space method in the recent monograph Interpolation oj Rational Matrix Functions by Ball­

Gohberg-Rodman for the time-invariant case. A parallel theory also exists for the discrete­

time case (see the recent Birkhauser volume Time- Variant Systems and Interpolation edited

by 1. Gohberg). The purpose of this note is to apply this time-varying interpolaton theory

to give a simple direct solution of the standard H=-control problem for a finite dimensional,

linear, time-varying plant in the I-block case. Results along the same line for the sensitivity

minimization problem in the discrete time case were presented by the authors at the 1991

MTNS.

56

A factorization principle for stabilization of nonlinear, time-varying plants

Joseph A. Ball and Madanpal Verma

Department of Mathematics, Vir?;inia Tech, Blacksburg, VA 24061 USA ball@math.vt.ed

Department of Electrical Engineering, 3480 University St., McGill University, Montrea

Quebec, Canada H312A 7 verma(Cj)thunder.McRCIM.McGill.edu

We establish the equivalence of internal input-output stability for two feedback cor

figurations of a nonlinear, time-varying plant P for which a related plant G is assumed t

have a factorization G = 8R with both Rand R- 1 incrementally stable; this extends

factorization principle for stabilizability previously ?;iven only for the linear, time-invariar

case. As an application of a special case we recover a version of the Youla parametrizatio

of stabilizin?; compensators for the nonlinear case previously presentee! in the literature.

The equivalence of the robust performance measure and a certain 1{oo problem

Bassam Bamieh*, Mohammed Dahleht , and Petros Voulgarist

Abstract

We consider the robust performance measure in HOC, where it is defined as the worst case closed loop norm over all perturbations in the system. It is well known that robust performance is equivalent to a certain structured singular value (11-) problem.

In this paper we consider the robust performance measure of a system with 8180 perturbations directly. We show that it is equivalent to an Hoo norm of a function of several complex variables. The correspondence between functions of several complex variables and multi-dimensional systems is then used to show that the robust performance measure is actualy an Hoo norm of a certain infinite dimensional system. Finally we present a method for computing the robust performance measure by computing the Hoo norm of a standard finite-dimensional system. This system is constructed from the original nominal system with the perturbations replaced by certain all­pass functions. We show that as the order of the all-pass functions goes to infinity (in a certain way), this Hoo norm converges to the robust performance measure. The implications of this characterization for robust synthesis will be discussed.

"Dept. of Electrical and Computer Engineering, University of Illinois at Urbana­Champaign,Urbana, Illinois 61801, USA

tMechanical and Environmental Engineering, University of California, Santa Barbara, CA 93106, USA

I Aeronautical and Astronautical Engineering, University of Illinois at Urbana­Champaign,Urbana, Illinois 61801, USA

57

58

>

Control of Smart Material Structures

H. T. Banks Center for Research in Scientific Computation

North Carolina State University Raleigh, NC 27695·8205

e·mail: htbanks@crsc1.math.ncsu.edu Phone: 919-515-3968

Fax: 919-515-3798

ABSTRACT

In this presentation we will give an outline of recent advances in the control of smart material structures. We focus on the use of embedded or bonded piezoceramic patches as actuators and sensors. Models for identification and control of cylindrical thin shells, plates and beams with piezoceramic patch actuators have been derived in [1]. These models lead to new theoretical and computational challenges for control theorists.

More generally, such problems are a special case of general control problems for distributed parameter systems which can be abstractly written as

z(t) = Az(t) + Bu(t) in V*

where A generates a semigroup in abstract spaces V '-+ 1i '-+ V* forming a Gelfand triple and B E £(V, V*). An abstract LQR theory along with an approximation framework leading to computational methods has been developed recently [2]. This theoretical framework as applied to control of smart structures will be explained.

The above ideas have been tested computationally in two areas of applications. Time permitting, we will present some of our experiences in computing feedback gains for (1) nonlinear fluid/structure interactions arising in an active control noise suppression system for aircraft and (2) vibration suppression in linear and nonlinear beams.

Since control via piezoceramic actuators permits one to apply structure borne shear forces to a flow field (as well as the usual transverse wall forces through bending mo­ments), we shall discuss potential applications of smart material walls in controlling flow in chambers as well as flow over airfoils.

REFERENCES

[1] H.T. Banks, R.C. Smith, and Y. Wang, The modeling of piezoceramic patch inter­actions with shells, plates and beams, Quart. Appl. Math., to appear.

[2] H.T. Banks and K. Ito, Approximation in LQR problems for infinite dimensional systems with unbounded input operators, to appear.

n

UNITARY EQUIVALENCE: A NEW TWIST ON SIGNAL PROCESSING

Richard G. Baraniuk<> and Douglas L. Jones *

o Department of Electrical and Computer Engineering Rice University

P.O. Box 1892, Houston, Texas 77251-1892, USA E-mail: richb@rice.edu

Coordinated Science Laboratory University of illinois 1308 W. Main Street

Urbana, IL 61801, USA E-mail: jones@uicsl.csl.uiuc.edu

ABSTRACT

Multidimensional signal representations and decompositions play an essential role

in many aspects of signal processing, for their extra degrees of freedom allow them to

closely match the characteristics of many important classes of signals. However, there

still exist large classes of signals for which all current techniques - including the

wavelet transform, short-time Fourier transform, Gabor transform, and the Wigner

distribution - are ill-suited. Therefore, in this paper, we generalize the multidimen­

sional signal representations by utilizing unitary operators to warp the controlling

coordinates of their decompositions. The result is an infinite number of new signal

analysis and processing tools that are implemented simply by prewarping the signal

by a unitary transformation, performing standard processing techniques on the warped

signal, and then (in some cases) unwarping the resulting output. These unitarily equiv­

alent, warped signal representations are useful for representing many of the types of

signals for which current techniques are deficient. As an example of the application of

this technique, we generalize the time-scale and time-frequency analyses of the wavelet

and short-time Fourier transforms, in the process generalizing the concepts of time,

frequency, and scale.

·This work was supported by an NSERC-NATO postdoctoral fellowship, Ecole Normale Superieure de Lyon (France), the National Science Foundation, Grant No. MIP 90-12747, and the Joint Services Electronics Program, Grant No. NOOOI4-90-J-1270.

59

60

The "Look-ahead" Philosophy Applied to Matrix Rational Interpolation Problems

M.Van Barel and A.Bultheel K.U. Leuven, Dept. of Computing Science

Celestijenlaan 200a, 8-3001 Leuven (Heverlee) vanbarel@kuleuven.ac.be ade@cs.kuleuven.ac.be

The Lanczos algorithm is an iterative method to solve large systems of linear equations or large eigenvalue problems. Recently, a lot of work has been done to make the Lanczos algorithm also work in the case of breakdown or near breakdown. To do this, a look· ahead strategy is used. The same idea can also be applied to compute a well-conditioned diagonal or row subsequence of Pade approximants. Translating these results back to linear algebra, stable methods are derived to solve Hankel or Toeplitz matrices.

In this talk we want to generalize this look-ahead idea to matrix rational interpolation problems. Using a module-theoretic framework, all solutions satisfying a number of interpolation conditions and having a specified degree structure can be computed by a fast and elegant algorithm. An elementary step of the algorithm adds one new interpolation condition or changes the degree structure of the solutions, increasing or decreasing the importance of the degree of one component of the solution. We generalize these elementary steps into steps that can "jump" immediately to the solution set adding more than one new interpolation condition or changing the degree structure in a more arbitrary way than just increasing or decreasing the degree of one component. We also show how these results can be translated into linear algebra results.

Quotients of systems and differentiable spaces

Z. Bartosiewicz Politechnika Bialostocka

Wiejska 45, Bialystok Poland

E-mail: bartos@plbia111

K.Spallek Institut fiir Mathematik

Universitiit Bochum D-4630 Bochum, Germany

Abstract

The last twenty years have proved that differentiable manifolds form a good class of state spaces for nonlinear systems. But there is one transformation which causes problems: passing to a quotient system. This is especially important when the equivalence relation is the indistinguishability relation. Even in simple cases the quotient space does not admit a strudure of a differentiable manifold. We propose differentiable spaces, introduced by K.Spallek, as state spaces for nonlinear systems. Differpntiable spaces are locally isomorphic to subsets of Euclidean spaces.

Not every quotient space derived from a manifold is a differentiable space. The general criteria are still to be worked out. In this paper we study equivalence relations on manifolds given by a group action. Since we are mostly interested in equivalencf' relations ddined by families of functions (e.g. observation algebras), we distinguish families of functions for which we can find groups producing the same equivalence relations.

We say that a group G of diffeomorphisms on an CWmanifold M acts properly if for every x EM the ;;pt G x = {f E G : f(x) = x} is compact and the map G x M ->M x M : (g, x) f--t (g(x), x), is a closed map. If G acts properly on M, then tIlE' quotient space M / R( G) defined by G may be given a structure of a differentiable ;;pace.

Similarly, if a family <I> of real analytic functions on M is normal, (that is it defines the ;;am" equivallence relation like the group of diffeomorphism which preserve thi;; relation) then the quotient space M / R<J> with respect to <I> is a differentiable space. This lead;; to the following.

Theorem If the observation algebra of the system is normal and the cor­re;;ponding group acts properly on M then the quotient system's state space is a differential span>. Moreover all vector fields on tl1<> quotient space are locally integrable.

61

62

REALIZATION AND FACTORIZATION PROBLEMS

FOR J-CONTRACTIVE OPERATOR-VALUED

FUNCTIONS IN HALF-PLANE AND

SYSTEMS WITH UNBOUNDED OPERATORS

S.V.Belyi

Department of Mathematics

University of South Florida

4202 E.Fowler avo

Tampa, Fl 33620, USA

E-mail address:

E.R.Tsekanovskii

Department of Mathematics

State University of New York

91 Ramsdell ave.

Buffalo, NY 14216, USA

E-mail address:

bebyj@chuma.cas.usf.edu v 192ragh@ubvms.cc.buffalo.edu

In this paper realization problems for operator-valued R-functions acting in finite­dimentional Hilbert space E as linear-fractioiml transformations of the transfer operator­functions of linear stationary conservative dynamic systems (l.s.c.d.s.) 8 of the form

{(A - zI) = K ~~-

<P+ = <P- - 2111. X

(1m A = I\.J K*)

are investigated. In a system 8 an operator A is a bounded linear operator, acting from 5)+ into 5)- ,5)+ C 5) C 5)- is rigged Hilbert space, A ::::l T ::::l A, A* ::::l T* ::::l A, where A is Hermitian operator in 5), T is nonhermitian operator in 5), 1< is a linear bounded operator from E into 5), J = J* = J- 1 and this operator is acting in E, <P± E E, <P­is an input vector, <p+ is an output vector, x E 5)+ is a vector of an inner state of the system 8, an operator-valued function

Wo(z) = 1- 2iJ{*(A - zI)-1 K J

is a transfer operator-function of the system 8. It turns out, that not all operator-valued R-functions can be realized in the above mentioned sense and we give a criteria of such a realizability in this paper.In terms of realizable operator-valued R-functions we spe­cialize the subclasses of three types. Given classes of operator-valued R-functions allow us to define classes of J-contractive operator-valued functions in half-plane, which can be realized as a transfer mapping of the system 8 with, generally speaking, unbounded main operator.A problem when the product of J -contractive operator-valued functions from defined classes belongs to the same classes , is investigated. We consider also a problem of factorization of a realizable J-contractive operator-function in half-plane which is connected with invariant subspaces of the main operator T of a system 8. The theorem on constant J -unitary factor takes place in one of the mentioned above class. We investigate also a problem of ('Onnection between realizations of two transfer mappings differing in the constant J -unitary factor.

Typeset by AM5-'JEX

=

DICHOTOMY OF PERTURBED SYSTEMS AND STABILITY OF BOHL INDICES

Asher Ben-Artzi and Israel Gohberg

School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel email: benartzi.gohberg@math.tau.ac.il

Consider the system

(1) (n=O,l, ... ),

63

where (An):;O=o is a bounded sequenee of invertible r x r matriees with Supn IIA;;-lll < 00.

Associate with the system (1) the block weighted shift G = (8i,j+IAj)~o acting in f!;. The essential spectrum of G and its Fredholm properties can be described in terms of the system (1) as follows. First define the dic<crete Bohl exponeni8 K±(V) of a nonzero subspace V of Cr by the formulas

h:+(V) = lim [sup {(IIU"hrII/IIU"xll)l/j : ° ¥ x E V}] , )-00 H

and h:_(V) = lim [inf {(llUn+rrII/IlU".TII)I/j : ° ¥:r E V}] ,

]-00 11

where Un = An-I· .. Ao. This notion is a discrete analogue of the Bohl exponents of a differential equation, see [DK]. A projection P of C r is called a splitting projection of the system (1) if K+(Im P) < K_(I\ er Pl. It turns out that the essential spectrum of G is the union of at most r annuli or circles in the complex plane, centered at the origin. The radii of these annuli and the index of ),,1 - G are described using the Bohl indices of the kernels and the images of the splitting projections of (1), see [BG].

Perturbations of the system (1) to an analogue system :f,,+1 = A:,xn (n = 0,1, ... ), where limn~oo IIAn - A:,II = ° are studied. Such a perturbation induces perturbations of the splitting projections P of the system (1), and of the evolution projections UnPU;;1 (n = 0,1, ... ). Stability properties of the latter are proved. A generalization of previous results of M. Dubiner [D] in the ease wh"n the original system (1) is stationary, namely An = A (n = 0,1, ... ) is obtained. The case where the matriees An are singular is also considered.

References

[BG] A. Ben-Artzi and I. Gohberg, "Dichotomy, discI'et" Bohl exponents and spectrum of bloek weighted shifts", Integral Equations and Operator Theory, Vo!' 14, 613-677 (1991).

[D] M. Dubiner, "The product of a convergent sequence of matrices", Integral Equations and Operator Theory, Vol. 14, 359-372 (1991).

[DK] Ju. L. Daleekii and M. G. Krein, "Stability of solutions of differential equations in Banaeh space", Trans!. Math. Monographs No. 43, Amer. Math. Soc., Providence R.I., (1974).

64

A BIFURCATION APPROACH TO THE GAIN

SENSITIVITY ANALYSIS OF THE ROBOTIC

COMPUTED TORQUE TECHNIQUE

Dr Yasmina Bestaoui Laboratoire d'Automatique de Nantes, UA CNRS 823 Ecole Centrale de Nantes, 1 rue de la Noe, 44072 Nantes, France

Ojamila Benmerzouk Oepartement de Mathematiques, Institut des Sciences Exactes Universite de Tlemcen, BP 119, 13000 Tlemcen, Algeria

Abstract: Feedback design research has classically ties with Lyapunov stability theory

and with the classical theory of dynamical systems. Techniques from bifurcation theory

and differential dynamics have been also used successfully. This work presents a

sensitivity analysis of the computed torque technique applied to robotic control, using the

bifurcation approach.

Integrated Symbolic and Numerical Design of Nonlinear Control Systems

G. L. Blankenship Electrical Engineering Department and Institute for Systems Research

University of Maryland College Park, Maryland 20742

April 1.5, 1993

Abstract

This paper describes integrated computer algebra and numerical analysis algorithms for the analysis and design of nonlinear control systems. Feedback equivalence among nonlinear systems is used to linearize and thereby control certain classes of nonlinear control sys­tems. Left and right invertibility of nonlinear systems (partial feedback linearization) is used to solve the output tracking problem. The sym­bolic programming algorithms include functions that perform basic differential geometric computations, modules for study and analysis of nonlinear control systems, and packages for the design of nonlin­ear controllers for the output tracking problem. Tools for design of adaptive nonlinear controllers are described. Applications in several different areas, including design of military vehicle suspensions, con­trol of multibody dynamics and process control are given to illustrate the tools.

65

66

Hamiltonian Flows in Networks and Control

Anthony M. Bloch

Department of Mathematic." The Ohio State University

Columbu." OH 43210

email: bloch@function.mp.l. ohio-.ltate. edu

In this talk, which includes joint work with R. Brockett, T. Ratiu and P. Crouch,

we discuss the nature of Hamiltonian flows which arise in certain optimization and

control problems. In optimization and identification problems it is natural to find

the optimum by considering gradient flows, and the analysis of such flows and

their relationship to discrete algorithms has received attention from a number of

researchers recently. As we have shown, in certain cases these gradient flows have

a dual Hamiltonian and gradient character. In particular the gradient flow may

occur on the level set of an integrable Hamiltonian system, the integrable flow

being characterized in this case by the fact that the level sets are noncompact.

Here we contrast this flow with flows that arise in other problems in networks and

control - the determination of constrained minima, flows in electrical networks,

and the Hamiltonian flows that arise in optimal control problems - and we show

that these flows have a number of geometric and qualitative properties in common.

In problems with constrained mininm one often needs to determine saddle point

equilibria. Flows with such equilibria may be Hamiltonian and gradient on the

entire phase space. By changing the metric, one can determine a flow in which

the saddle point equilibria become asymptotically stable. ( This is related to the

Arrow-Hurwicz-Uzawa algorithm for constrained variational problems.) Moreover

there is an interesting relationship of these flows to flows in nondissipative electrical

networks. In this case one obtains Hamiltonian flows with periodic orbits which

are gradient with respect to an indefinite metric.

In the optimal control problem, Hamiltonian flows arise from the maximum prin­

cipal. The flows here thus are derived from a variational problem. If there are

sufficiently many integrals in the problem, the flows may be integrable, but again

often of a periodic nature rather than having asymptotically stable behavior. On

the other hand, we show that when there is sufficient symmetry in the problem

these flows have some of the structural features of flows that arise in the optimiza­

tion problems discussed above.

The silllultaneous stabilizability question of three linear systems is rationally

undecidable

V. Blondel (+) and M. Gevers (++)

October 28, 1992

Abstract

We show that the simultaneous stabilizability of three linear sys­tems. that is the question of knowing whether three linear systems are simultaneously stabilizable, is rationally undecidable. By this we mean that it is not possible to find necessary and sufficient conditions for simultaneous stabilization of the three systems that involve only a combination of arithmetical operations (additions, substractions, mul­tiplications and divisions), logical operations (,and' and 'or') and sign test operations (equal to, greater than, greater than or equal to, ... ) on the coefficients of the three systems.

Key words: simultaneous stabilization, decidability, decidable question.

(+) University of Oxford, Mathematical Institute, 24-29 St Giles', OX] 3LB Oxford, UK. Tel: +44-865-270509. Fax: +44-86;'j-2i05].S. E-mail: blondel@maths.oxford.ac.uk (++) lfniversite Catholique de Louvain, Cesame, Place du Levant 2, B-l:3'!8 Louvain- La';'-1euve, Belgium. Tel: +:32-10-.1i2590. Fax: -'-:32-10-478667. E-mail: gevers@auto.ucl.ac.be

67

68

Summary

A Closed Form Design Method for Recursive 3-d Cone Filters

Michael Bolle Robert Bosch GmbH, Hildesheim I

FAX: 049-5121-492520 Tel: 049-5121-493902

In 2-d data processing, fan or velocity filters have received wide attention because of the applications in fields such as processing of geological and seismological data in oil exploratic and beamforming applications. In 3-d data processing the analogy to 2-d fan filters are 3· cone filters. This type of filters may be used to separate waves according to their propagatic velocity. In contrast to the design of 2-d fan filters, no analytical design approach to tl design of 3-d cone filters is known. From there the design and realization of these filte represents an unsolved and interesting problem.

In our paper we will present an analytical approach leading to a closed form design methc for recursive 3-d cone filters. The resulting filters may be realized using the well know 3-d wave digital filter (3-d WDF) structures with their associated good properties, as e.; low sensitivity to coefficient perturbations, good dynamic range and stability under finil arithmetic conditions. In addition to this the proposed filters offer full parallelism in a space coordinates because recursion is only necessary in time. The resulting filter structul is highly modular, which makes very efficient implementations possible.

The paper is organised as follows. The first part gives a brief summary on two closed fon design methods for so called ±45° fan filters which are the starting point in the design of 3-cone filters. We show that the transfer function of these kind of filters can be parametrised i the form H = H(ZI, COS(W2)), where H is a 2-d rational function, if the filter owns symmetri! which can easily guaranteed in the design procedure. In the second part we make use of th 2-d McClellan transformation to derive the transfer function of a 3-d cone filter. We examin the accuracy of this transform and give extensions to higher dimensional cone filters. Th third part focusses on concrete passive and therefore unconditional stable realizations fc the derived cone filter transfer functions. In addition to the well known WDF structun we introduce an interesting new class of multidimensional filter structure to realize the con filter transfer functions. In the fourth and last part a complete design and realization cycl is demonstrated. A computer simulation of the resulting filter shows the use of a 3-d con filter to separate circular waves with different propagation velocities.

I formerly with the Lehrstuhl fur Nachrichtentechnik der Ruhr-Universitiit Bochum

Discrete-time dynamic models of piezoelectric materials

Basilio Bona, Marina Indri and Antonio Tornambe Dipartimento di Automatica e Informatica, Politecnico di Torino

Corso Duca degli Abruzzi, 24, 10129 Torino, Italy Fax +39 11 5647099, E-mail: bona@polito.it, indri@polcg.polito.it

Abstract

The purpose of this paper is to study the possibility of designing a discrete-time dynamic model of a thin piezoelectric element, which is assumed to have a sandwich structure, constituted by two piezoelectric films mounted on a thin support with known elastic characteristics. The piezoelectric structure can be used both for sensing and for distributed actuating [1, 2J. The dynamic model of the element is determined considering the applied electric potential as exogenous variable (the input of the system) and minimizing the action functional of the system, according to the integral Hamilton principle. In order to obtain an approximate model particularly suitable for control purposes, the Ritz- Kantorovich method is used to represent the kinematics of the element in a polynomial form. The approximate discrete-time motion equations of the structure are obtained via the direct discretization of the action functional [3J, once the piezoelectric and flexure phenomena of the element have been characterized by the introduction of the piezoelectric and flexure energy densities. Under some assumptions about the structure of the mechanical system, the approximate discrete­time model thus obtained is linear and time-invariant.

References

[IJ C. -K. Lee and F. C. Moon, "Modal Sensors/Actuators", ASME J. Applied Me­chanics, 57, 434-441,1990.

[2J H. S. Tzou and C. I. Tseng, "Distributed Structural Identification and Control of Shells using Distributed Piezoelectrics: Theory and Finite Element Analysis", Dynamics and Control, 1,279-320, 1991.

[3J S. Nicosia, P. Tomei and A. Tornambe, "Discrete-Time Modeling and Control of Robotic Manipulators", Journal of Intelligent and Robotic Systems, 2, 411-423, 1989.

69

h

70

Robustness of feedback systems under simultaneous plant and controller perturbation

Peter M.M. Bongers Mechanical Engineering Systems and Control Group,

Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands e-mail:bongers<Otudw03.tudelft.nl

A perfect model of the real plant, if available, will in general be non-linear and of extremely high order. In engineering practice the plant will be approximated by a low order linear model. The discrepancy between the nominal model and the plant is then described by a set of plant uncertainty models. In the next step a controller will be synthesized in such a way that it robustly stabilizes the nominal model and the set of plant uncertainty models, for a pre-specified performance. When dealing with industrial processes the control computers are calculating in finite word length arithmetics or even integer arithmetics, while for the controller limited time and space on the computer is available. Therefore the implemented controller is only an approx­imation of the designed controller. The discrepancy between the designed controller and the implemented controller can be described by a set of controller uncertainty models. In [2J gap-metric robustness under simultaneous plant and controller perturbations has been studied. In the gap-metric robustness the nominal plant (controller) is factorized in normalized coprime factors. The difference between a perturbed plant (controller) and the nominal plant (controller) is described by perturbations on the normalized coprime factors of the nominal plant (controller). Robustness of the closed loop for a class of perturbed plants and controllers is guaranteed if the norm of the perturbations on the normalized coprime factors is small enough. The maximum allowable norm of the perturbations is determined by the infinity norm of the feedback system. This means that in the gap-metric robustness only crude information about the nominal feedback system is taken into consideration. The main idea behind the new and less conservative robustness margin [1], to be considered in this paper, is to take into account more information about the nominal feedback system. One can think of this information -as refinement of the infinity norm of the feedback loop to frequency dependent singular values of the feedback loop, and the directionality of the feed back loops in multi variable systems. In order to account for the closed loop characteristics a normalized coprime factorization of the nominal controller (plant) is used to define a specific coprime factorization of the nominal plant (controller). The difference between a simultaneously perturbed plant and controller (P~, C~) and the nominal plant and controller (P, C) is now described by perturbations on the specific co­prime factors of the nominal plant and nominal controller which includes detailed informa­tion about the nominal controller and plant respectively.

[IJ Bongers P.M.M., "On a new robust stability margin", in Recent Advances in Math­ematical Theory of Systems, Contml, Networks and Signal Processing, Pmc. of the Int. Symp. MTNS-gl, H. Kimura, S. Kodama (Eds.), pp.377-382, 1992.

[2J Georgiou T.T., M.C. Smith, "Robust control of feedback systems with combined plant and controller uncertainty", in Pmc. Amer. Contr. Conf., pp.2009-2013,1990.

keywords: robustness, simultaneous perturbations, coprime factorizations, gap-metric

Graph-Theoretic and Several Variable Result. in Neural Networks t

N. K. Bos ... The Spatial and Temporal Signal Processlng Center Department of Electrical and Computer En~ineerinli

The Pennyy[vania State University UnIversity Park, PA 16802

U.S.A.

Neural network design, jike any network design, involves topology and

components. The topology involves structure and interconnectIons while

components involve neurons havln~ transfer characteristics which realize

nonlinear input/output maps. The structures. interconnections, and

71

interconnection weights may be conveniently cl.assified and studied by applying

graph-theoretic fundamentals. ,"ven a new approach to training has been

lIIotivated from results in graph theory. On the other hand, the Hilbert-

Kollllogorov results on vector mapPlngs of 9. vector varjable over any compact

multidimensional region in mUltidimensional space have only indIcated about

possibilities instead or stimulating the development of constructIve

procedures for neural network desi!!n. This presentation critically analyzes

the impact of graph theory as well as results involving contInuous maps of

several variables in the search towards a formal realIzability theory for

trhi~ research was supported by the National Science Foundatlon Grant MIP-~lH99i .

72

Robust Computation with Pulses

R. W. Brockett and Jeff Kosowsky

At certain energy levels and over certain epochs analog computation can be done "directly" exploiting physical principles such as Ohm's law etc. Usually this kind of technique can only give accurate results at low values of time-energy product. When one wants to generate a signal that will last for a longer time andlor have the energy necessary to interact with a number of other units, a different representation of data is needed. Pulse frequency and pulse phase are often cited as possibilities in the study of biological systems. From this point of view it is too naive to ask for an analog method for realizing a certain computation. Instead, one needs to recognize that function evaluation at one

level of I'1Tt1E may require completely different techniques than function evaluation at a

different level of I'1T&E. In a biological setting it may happen that an ion pump in a cell might displace only a few hundred ions, but in so doing effect an important part of a "calculation". The time-energy product might be as small as 100 mv times 10-14

coulombs times 50 ms. or 5x 10 -14 erg-seconds. On the other hand, the energy in a fully developed action potential traveling down a large pyramidal nerve may exceed this by a factor of greater than 1000. In a serious study of analog computing one needs to distinguish between the problem of computing a function at one time energy product from that of computing it at another. In a typical digital processor the representation of an 8 bit number on a 50 megahertz bus would require lO-llerg seconds.

In this paper we build on our earlier work on pulse computation presenting models that will regenerate standard pulses from distorted ones and showing how one can use low erg-second level physical effects to produce larger erg-second level signals. We will also illustrate how to carry OUI level based computations using pulse-like signals for communication. These results are based on a generalization of earLIer results on dH/dt = [H,[H,N(t)J) that require fewer communication channels ..

r ,

Computer algebra and its uses

B. Buchberger Research Institute for Symbolic Computation (RISC)

Johannes Kepler University A-4040 Linz

Austria

Rough sketch of the talk:

- What is computer algebra?

- Some examples of CA methods

(underlying math and resulting algorithms)

- Some examples of applications in engineering.

73

74

Title: A condensed form for the solution of the symplectic eigenvalue problem

by:Angelika Bunse-Gentner

For the Hamiltonian eigenvalue problem, aa it arises for instance in the aolution ofthe continous time optimal conlrol problem, II fast, structure preserving QR-like method bAIled on similarity transformations with eympleclic matrices is known. The basia for this method ie a symplectic similarity transformation of a 20 x 2n Hamiltonian matrix 10 II condensed form having only 4n - 1 po •• ibly different nonzero entries. A recently developed look.ahse.d technique for its computation indicates that the proce .. can be numerically stabilized which may make this method more interesting for practical purpose. In thle t,e.lk we .how that in complete analogy a corresponding condensed form can al80 be achieved for the 20 x 20 symplectic eigenvalue problem, which ariles in the discrete time optimal conlrol problem. Here it is e. symplectic matrix with only 60 - 1 nontrivial entries, depending on 4n - 1 parameters. This reduction can serve liS a basis to developed a fast aymplectic QR-like method analogoll.l to the method for Hamiltonian matrieea, which preservee this symplectic condensed form. The numericaletability of the proc8s8 can be monitored as in the Hamiltonian case.

Fftchbereleh Mathem"tik und Inrannatik Univerailit Bremen Paott&e!t 33 04 40 D 2800 Bremen 33 FnO tllectrotUc maUl anleUkaGm&~hamatik.wu.Bremen,d.e

A New Computational Algorithm for Optimization Based Design and Control of Fluid/Structure Interactions

JOHN A. BURNS

Virginia Polytechnic Institute & State University

Blacksburg, VA 24061-0531

Interdisciplinary Center for Applied Mathematics

Department of Mathematics

75

In this paper we first describe a optimal design problem that involves the construction

of a geometric shape to minimize the difference between a given fluid flow field and the fluid

flow over the shape. This problem falls into a class of infinite dimensional optimal control

problems known as shape optimization. We review several approaches to the solution of

these problems and discuss some mathematical issues associated with the combination of

optimization and simulation algorithms. A standard approach to such problems is to use

a CFD code as a "black-box" function evaluator for some optimization code. In problems

with flow discontinuities (shocks, etc.), this approach can produce some unexpected results.

In particular, we show that it is possible for a high order scheme to introduce extraneous

local minima that can cause the optimization algorithm to fail. Examples and the results

from numerical experiments will be given to illustrate the ideas. Finally, we present a new

hybrid algorithm designed to overcome some of the difficulties described above.

76

THE TIME OPTIMAL CONTROL OF PREDATOR-PREY SYSTEM WITH INTRASPECIFIC STRUGGLE

BUYVOLOVA A.G. Department of Cybernetics. "IE".

1091n, Novospassldi, 3-1-20, Moscow I Russia. Phone: 7-(095)2723156

KOLMANOVSKII V.B. Department of Cybernetics, MIEM,

1091n, Novospassldi, 3-'-20, Moscow I Russie. Phone: 7-(095)2723156

KOROLEVA N.I. Department of Cybernetics, MIEM,

123458, Tvardovski j St., 18-2-126, Moscow I Russ;a Phone: 7-(095)9448070

ABSTRACT

The time optimal control problem of predator-Prey. system with intraspecific competition ~s considered. The optimal control problem consists of attainment of the system to its equilibrium state from arbitrary initial state at the minimum time. Existence of an admissible control is established and existence of the optimal one is proved by the aid of the maximum principle. switching curves of the optimal control are constructed for all cases. Dependence of the optimal time on the system parameters is investigated numerically.

Controllability Radii in Descriptor Systems

Ralph Byers Department of Mathematics 405 Snow Hall University of Kansas Lawrence, Kansas USA

byers@math.ukans.edu

We investigate numerical methods for measuring the distance from a con­trollable descriptor Ex = Ax + Bu system to the nearest uncontrollable system. The distance is related to the conditioning of a variety of computa­tional control problems. Arbitrary perturbations of the system may not be physically meaningful and may make give unrealistically small distance to uncontrollability. It is natural to limit consideration to a class of allowable perturbations. We identify four different sets of allowable perturbations that might be reasonable in different circumstances. In some cases, the compu­tational problem of measuring the distance reduces to a single real variable optimization problem.

77

78

Stable Rational Interpolation

S. Cabay, R. Meleshko Department of Computer Science, University of Alberta

Edmonton, Alberta T6G 2Gl, Canada cabay@cs.telecom.canada.ca ron@cs.telecom.canada.ca

and

M.H.Gutknecht IPS, ETH Zurich, CH-S092 Zurich

mhg@ips.ethz.ch

Recently, a fast (weakly) stable look-ahead algorithm has appeared for the computation of Pade approximates and the inverses of Hankel matrices (Cabay and Meleshko, to appear). The ideas contained in this algorithm can be extended to the rational interpolation problem. Using Newton series for the interpolants, we describe this extension and discuss results obtained about the stability of this technique. The stability results provide the basis for the look-ahead portion of the algorithm. Included are results of numerical experiments with the algorithm.

3

On the Convergence of the Control Riccati Differential Equation for Stabilizable Time-Invariant Systems

Frank M. CALLIER and Joseph WINKIN Facultes Universitaires Notre~Dame de la Paix

Department of Mathematics 8 Rempart de la Vierge

B~5000 NAMUR, BELGIUM E-mail: FCL@math.fundp.ac.be;JWK@math.fundp.ac.be

Abstract: We report a necessary and sufficient condition for the solution of the time~invariant Riccati differential equation to converge towards the strong solution of the corresponding algebraic Riccati equation, when the system is stabilizable and the Hamiltonian matrix may have eigenvalues on the imaginary axis (or equivalently there may be critical unobservable modes), [1]. The condition is a generalization of an earlier one established in [2]. Our proof revises an earlier one in [3] (where the result is derived under a more restrictive condition, namely controllability), leading to additional information of what can be assumed without loss of generality in our context. It is also shown that the convergence is not always exponential and that the presence of critical unobservable modes may slow down but does not prevent the convergence of the solution of the Riccati differential equation. The impact of the condition on Linear~Quadratic optimal control is briefly discussed. The theory is illustrated by a simple example

Keywords: Time-invariant Riccati differential equation, strong solution, conver­gence, stabilizability, LQ-optimal control.

References:

[1] F.M. Callier and J. Winkin, Convergence of the time-invariant Riccati differential equation towards its strong solution for stabilizable systems, Facultes Universi­taires de Namur (Belgium), Department of Mathematics, Report No. 92-21, 1992; submitted.

[2] F.M. Callier and J.1. Willems, Criterion for the convergence of the solution of the Riccati differential equation, IEEE Trans. Auto. Control, Vol. AC-26, pp. 1232-1242, 1981.

[3] P. Faurre, M. Clerget and F. Germain, Operateurs rationnels positifs, Dunod, Paris, 1979.

79

80

b

ADAPTIVE RLS IDENTIFICATION WITH GENERALIZED FORGETTING TECHNIQUES

M. CAMPI and F. LoRITO

Diparlimenlo di Elellronica e InJormazione, Polilecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano - Ila~y

Ab,tract. For the identification of time·varying systems, a widely used tecbnique is the Recursive Least Squares algorithm with Forgetting Factor (RLS·FF). The basic idea behind RLS-FF is as follows: as a new datum becomes available, discOlUlt all data collected in the past of the same (possibly time-varying) rate ).1<1.

The discount mechanism of RLS·FF is critically analyzed in the present paper. It is shown that such a mechanism is not flexible enough in certain situations in that it does not alluw to achieve a trade· off between tracking capabilities and disturbance rejection features. On the grounds of these considerations, a new recursive forgetting algorithm is proposed which incorporates a generalized discounting device. The analysis, which is mainly based on an approximate frequency characterization of the estimation error, shows the increased flexibility of the new algorithm and puts into light how it can lead to better performances with respect to classical Forgetting Factors tecbniques.

Transform-based image compression optimized for some characteristics

of the human visual system

S. ('an,llo D.E.E.!., {"niN/'"ity of Trils/(

"I a A. \"a!uio. IIJ . .341 17 Tii"If. HA.L}'

81

Image rOJl1))fPf:.:"iOll is prf'sf'ntiy OIlP of !he lllu~t (-Ictiyf' fidds in Illlil1idimE'nsionai :.-ignal prores:;ing, thanks to the large number of applications ill hoth re~{'ar(h alld industry f'llvironmenb. ~Iany difTnf'llt techniquf's havp been present.ed in literature [1]; however. the murt" commOll Ollt'S ('Ire almost always baseci on linear transforms. According to this approach. all imagf' i:-:; suhdivided into hlock::.; E'cH"h hlo('k is treatf'd as a \'(-'('tor, which is linearly transformed into anotiU't" vector of reduced sizp in order to COllllH't'SS thp original informatioll. Anot.ilf'l' transform is then needf'd to expand tllf' H'ctor 10 the original sizt': tlw llwan of tlIP squarf'd difff'rt'ncf' hetwE'f'n the original and the rpconstl'uC"tf'd vector is tllf' cOlJlprf's~ion ('ITor. Dup t.o the subdivision of thf' image illto blocks, thf' so-caliNI blockillg dfect is in gellPl'ni pre~f'nt in the Pl'oct':-;~ed illlagf'S, which i~ VNy disturhing for tllf' viewer; moreOVN, no aspf'rt of the Humall Yisual Systelll (IIVS) i:-. takpll int.o accoullt, so that these transforms are not. very efff'ctive if a subjective test.· -in:-:.tead of comprpssioll errol' -is w .. ;ed for their f'valuation.

In this paper we consider the prohlelll of filldillg a llllP:ll" trall;.;fol'm with optill1(1ll)f'rformances with J'e-SPf'Ct to some characteristics of the HVS [1]. In porticular:

• the HVS if, particularly ;';('lI~it.i\"t"> to artifact:-" J.('. to ('nOff, which hen!' ~OIlH' :-,tfucture; with no doubt, tlw most import ant dist urhanrp ill t ransforlll- ha.':i{-,d IPrhn iqUt'~ is t Iw alrpcHly lllPllt iOIl('d hlocking effect,

• the l\lodulcltion Tl'ansfE'l' FUllction C'ITF) of t hf' HVS ha:-; a 100\"PI" gain at. low spat.ial frpquen("ies thaH at intermediate ones; this lllf'311:-; t.hat. t IIC eye is l1Ior<' SE'II:-..it.i\f' to \'ariatiolls of the I4ra,v 1(>\,(,1 in an image t lIall to tlw if'v('llht'it;

• t.he I"f'SIWII: .... (' of thE' (,yf' is lIot litwar: in partirular, ,",Illal] djfft'rpllcP~ in lUlllill<lIlCP (Iff' perceptuaily Jliore

important if t.lw nlt'an gray Ipvf'l is low thall if it is high. due to the \\'el)f'r's Law.

These issues are taken into account by t.he traw .. form operator Wf' df'scribp in t.he paper: tilt' resulting com­pressed/expanded images can have a higlwr conlJ)JT~sion elTOl', if cOltlpan.>d \vit h images oht ained using st andard techniqups, but are of higher visual quality. Our approach ('xtelld:-; tl1f' onp proposf'd in [2), wherE" a linear neural network is trained in order to optimizp the pcrformancp ill tf'l"lllf:> of comlHPssioll PITor only. OUf system, in turn, If'arns how to presen'p t.lw hest. image qUnlity with r(>~pf'('t to tiIe (lhOH' l1lPntioiled ("iIal'act.f'ristics of tht' HVS

In order to find such operator. (I fUllftioIl i:'i iJlt]"oduc~'d which t'valuatps tllP quality of tlIP recollstruded images consioe-ring tlIP citf'd aSpt'cts of tlw HVS. A gradit'llt dcsr(,lIt techlliqlle is thell llsed ill ordt'f 1.0 find tlw coefficients of tllP linear transform which minitllizt' tlw error according to that [ullctioll. The computation of the gradient implie:-. tilt' f'\alu(ltioll of tilt' dt'riv<lti\"("s of tilt' fU!lctioll with I'PSlwct to all tlw copfficiPllts of tlw transforlll; ill ordt'l' to simplify tillS ("valuation. Wt' ll."t' tlip (;akaux del"l\·a.tiw." [:1], which i . .., a simpk hut powprfui tool gt'lwrally llsed ill calculw; of variation.

Tlw propost'd transform ha:;; IH'Pll cOlllpaJ"t'd with a f(,\\· ~talldaJ"(1 0111':-.; :-,OllW ll11llwrinti r('Hllts alIt! som(' proct"ssed images ;'Uf' shoWJl in the paper. which cOllfirlll tilt' \alidity of til(' approach.

References

[1] A. K .. Jain. FUlldall/(lItal.'i oj dl.Qifail11ul(j( /J/·O(( . .,'J/lI/J Eng,!c"'ood ('tiff;.;, :\.J: Pl'enticp-H<lll Intf'l"llntion(ll, Inc .. HI~9.

[2] I. [{usso and E. C. Ht'al. "Illlag" cOlllprc"ion u,illg all <Jllter prodllct 1It'liral 11<'1 work."' ill Pm(. [("ASSP-.9.!. (San Francisco). PI'. II :l77" II -:lKO, ~Iar("1i 19~12.

[3] JY(Jllll1l((I/'jllllrtto/loi a 11 a I.IJ." , . .., (llId upp[/((/fuJ/I . ..,. (L. B. HnlL ('d.). "cad~'I1Iic Prt·,s:-" lQil.

82

L

Continuous-Time Stochastic Approximation*

Han-Fu Chen

Institute of Systems Science, Academia Sinica,

Beijing, 100080, People's Republic of China

E-Mail: BMADIS@CA.BEIJING.CANET.CN

FAX: 2568364 Tel: 2551965

ABSTRACT

A continuous-time stochastic approximation algorithm is proposed to

search the zero xO of a regression function which is observed with noise. It

is proved that the estimate Xt given by the algorithm converges to XO and its

average xt = t 10t x,ds is an asymptotically efficient estimate for Xo. The

paper is characterized by that i) no growth rate restriction is made on the

regression function; iiJ the slowly varying gains and the randomly varying

truncation technique are used; iii) the results are proved by a direct method

which is different from the classical methods such as martingale(1] , 0 D £[2,3]

and weak convergence[4]. This method was first developed for discrete-time

algorithms in [5,6].

Keywords: Continuous-time, stochastic approximation, efficiency, slow gain, av-

eraging, random truncations .

• Work supported by the National Natural Science Foundation of China

Chaos in digital filters: a global picture

Leon O. Chua ., Zbigniew J. Galias $, Ljupco Kocarev., Madej J. Ogorzalek $, Chai Wah Wu •

*Department of Electrical Engineering and Computer Sciences, University of California,

Berkeley, CA 94720, USA e-mail: ljupco@fred.berkeley.edu

$ Department of Electrical Engineering, University of Mining and Metallurgy

al.Mickiewicza 30, 30-059 Krakow, Poland e-mail: maciej@uci.agh.edu.pl

83

In this paper, we consider a second order digital filter realized in the direct form using either a modulo-2 [1], [2J or saturation [3], [4J arithmetic for the overflow rule. The digital filter is modelled by a two-dimensional discrete-time dynamical system with the state equations:

where f(x) is the chosen overflow nonlinearity.

In the first part we consider the modulo-2 characteristic. In this case we have the following main result: Let (X,I1,jl) be a normalized measure space and S: X -> X a measure-preserving map such that S(A) E 11 for each A E 11. If limn _ oo jl(sn(A)) = 1 for every A E 11, jl(A) > 0, then S is called exact.

Theorem 1:

If a and b are integers such that b 0/ a + I, b 0/ -a + I and b 0/ 0, ± 1, then the map F is exact.

In the second part of the paper we consider digital filter with saturation arithmetic. In this case we have the following main result: Let us denote w(x) the limit set of point x under F, Qo = {(a; b) : b::; -1, b < I -Ial}. Theorem 2: If (a, b) if. Qo then "Ix, w( x) is a periodic orbit of period 1 or 2. If (a,b) E Qo then w(x) is either periodic or dense on a set homeomorphic to a circle. It is possible to find parameters (a, b) corresponding to an orbit of any arbitrarily long period.

[IJL.Kocarev, L.O.Chua "On chaos in digital filters: Case b = -1" Memo UCB/ERL M92/74, University of California Berkeley, 1992.

[2JC.W.Wu, L.O.Chua "Properties of admissible symbolic sequences in a second order digital filter with overflow nonlinearity". Memo UCB/ERL M92/18, University of California Berkeley 1992.

[3JM.J.Ogorzalek "Complex behavior in digital filters" International Journal of Bifurcation and Chaos, Vol.2, No.1, pp.11-29, 1992.

[4JM.J.Ogorzalek, Z.Galias "Limit sets of trajectories in a nonlinear digital filter". Proc. MTNS'91, Mita Press 1992, pp.389-394.

84

Title:

Abstract:

Rational Spectral and J-Spectral Factorization using Descriptor Forms

A number of optimal control design methods essentially reduce to a spectral or J-spectral factorization of a rational power spectral density. The most common case is that for which the spectral density is proper, so that the spectral factors are also proper. The usual method of solution is state-space based in that, via the solution of a Riccati equation (or something similar), the factors are expressed in state-space form.

In recent years, polynomial design methods have become more popular and the corresponding polynomial spectral factorization problems then arise. The factors are also polynomial and can therefore be represented in generalized state-space (or descriptor form.

We discuss a general state-space based algorithm which can be used to solve the spectral factorization problem for any spectral density, whether it be proper, improper, or polynomial. The algorithm starts with a descriptor form representation of the give spectral density and produces factors also expressed in descriptor form. The method is basically a generalization of the well-known Schur method for proper spectral densities, and can handle any rational spectral density. There is no restriction on the locatio or mUltiplicity of poles and zeros, other than that implied by inertia restrictions on the imaginary axis.

We also discuss some of the numerical problems that occur, for example, when poles and/or zeros occur on the imaginary axis.

Address:

David Clements School of Electrical Engineering University of New South Wales P.o Box 1 Kensington, NSW, 2031 AUSTRALIA

Phone: + 61 2 697 4015 Fax: + 61 2 662 2087

E-mail: D.Clements@unsw.edu.au

FACTORIZATION THEORY OF LINEAR PERIODIC SYSTEMS

P.Colaneri

Politecnico di Milano

Dipartimento di Elettronica e Informazione

Piazza Leonardo da Vinci 32, 20133 MILANO <11

FAX ++39.2.23993587

Tel. ++39.2.23993615

email COLANERI@IPMEL1.POLIMI.IT

Abstract

In this paper, a couple of factorization problems for a given periodic

system are addressed, both for the continuous and discrete-time cases.

The spectral and inner-outer factorizations are considered in special

but important case. Among other things, the problems call for the

properties of periodic Riccati equations.

Key words: Factorization, periodic systems, periodic Riccati equations

85

86

Gaussian Reciprocal Processes and Their Associated Conservation Laws

Jerry Coleman Graduate Group in Applied Mathematics

551 Kerr Hall, University of California, Davis, CA 95616 email: coleman@ike.ucdavis.edu phone: 916-752-0827 fax: 916-752-6635

Abstract

A stochastic process x(t) is called reciprocal if for any two times to, tl, the process interior to [to, tJ] is independent of the process exterior to [to, td, given x(to) and x(td.

Krener, Frezza and Levy have considered Gaussian Reciprocal Processes (GRPs) whose covariances satisfy certain smoothness assumptions. In [I] and [2] it is shown that such G RPs satisfy for 0 :; t leqT a 2nd order stochastic differential equation of the form -d2x + G(t)d1xdt + F(t)dOxdt 2 = Q(t)~(t)dt2, where ~(t) is a generalized noise process. We refer to T as the process lifetime and to F, G, and Q as the reciprocal invariants of the process. In [I] it is shown that GRPs have conditional moments which satisfy a system of conservation laws similar to those of continuum mechanics. These conservation laws form a system of quasilinear PDEs. They are the analogy for GRPs of the Fokker­Planck equation satisfied by the density of a Markov process.

This presentation will summarize certain results from [3] concerning the con­ditional moments and conservation laws. It turns out that a GRP is determined by its lifetime, its reciprocal invariants and the initial values of its conditional moments. A succinct characterization can be given of those functions that are allowable as the initial values for the conditional moments of a GRP having a given lifetime and given reciprocal invariants. Furthermore, given any reason­able F, G, and Q and any proposed initial values (having a prescribed functional form)' there is a T and a GRP x(t) su ch that x(t) has lifetime T, has invari­ants F, G, and Q and has conditional moments whose initial values are those proposed. Finally, it can be shown that the conservation laws are hyperboli c for either all or none of the process lifetime.

(1] A. J. Krener, Reciprocal Diffusions and Stochastic Differential Equations of Second Order, Stochastics 24 (1988), pp 393-422.

{2] A. J. Krener, R. Frezza, B. C. Levy, Gaussian Reciprocal Processes and Self-Adjoint Stochastic Differential Equations of Second Order, Stochastics and Stochastic Reports, Vol. 34 (1991), pp 29-56.

[3] J. M. Coleman, Gaussian Reciprocal Processes and Their Associated Conserva­tion Laws, Masters Thesis, Graduate Group in Applied Mathematics, University of

California, Davis, Calif ornia.

Exponential Asymptotic Null-Controllability of Bilinear Systems with Control Constraints

Fritz Colon ius Institut fiir Mathematik Universitiit Augsburg 8900 Augsburg Germany fax: +821-598-2200 e-mail:colonius@uni-augsburg.de

Wolfgang KIiemann Department of Mathematics Iowa State University Ames, Iowa 500ll U.S.A. fax: +515-294-5454 e-mail:s2.whk@ISUMVS.bitnet

We consider bilinear control systems of the form

x(t) = (Ao + LUi(t)Ai)x(t), u(t) = (ui(t» E U,

where U c Rm is convex and compact and Ao ,.,Am are constant matrices of appropriate dimension. The exponential growth rate (or Lyapunov exponent) ofa trajectory x(t,x,u(,», t E R, of this system with initial state x E Rd at time t=O and admissible control function u(·) is given by

A(Uo.X) = lim SUPt~oo Inlx(t,x,u(·»I.

Obviously, an initial state x is exponentially asymptotically controllable to the origin iff there exist an admissible control function u(·) such that A(U(')'x) < O. We characterize precisely those initial points for which there exists an admissible control function u such that A(U('),x) < O. The proofs use that the bilinear control system above corresponds to a linear flow (the

associated control flow) on the (trivial) vector bundle RdxU, where U is the set of admissible control functions. Then spectral theory of these flows together with an analysis of the controllability properties of the control systems induced on projective space allows us to give the announced characterization.

87

., i

88

OOMlPOSITTITON Of IT/O lRlElHIA VITORAL SYSTlEMS

G. Conte Dipartimento ill Elettronica

eAutomatica Universita di Ancona via Brecce Bianche

60131 Ancona - ITALY

A.M. Perdon Dipartimento ill Matematica

"Vito Volterra" Universita ill Ancona via Brecce Bianche

60131 Ancona - ITALY

ABSTRACT In the behavioral approach to system theory developed by J. Willems (see [IJ and

the references therein), an input/output dynamical system is described in general by a quadruple

(T,U,Y, ffi), where T is the time axis, U is the input signal space, Y is the output signal space

and ffi is a subset of (U X y)T satisfying specific axioms. Denoting respectively by fl: ffi ~ UT and by f2: ffi ~ yT the restrictions of the canonical projections from (U X y)T maps

(fl,f2) with common domain. In an abstract algebraic framework (like the one described in

[2]), two pairs of maps (fl,f2) and (gl,g2) as above can be "composed", provided f2 and gl

have the same codomain. The composition turns out to be a pair of maps (fJ,f2), with fl:

ffi'~ UT and f2: ffi'~ ZT, where ZT denotes the codomain of g2 and ffi' is a subset of (U

X Z)T. In this paper it will be shown that such notion of composition can be used to define

series composition of I/O behavioral systems. The invariance of dynamical properties under

series composition will then be investigated. In the case in which the I/O system L is described

by behavioral equations, that is ffi is defined as the set {(u,y) E (U X Y) T such that!i (u) =!2

(y)}, it may be natural to associate to L the pair of maps ifi J2) having the same codomain,

say E. From an abstract algebraic point of view, this gives us a situation dual to the previous

one. Pairs of maps with common codomain can also, under suitable hypothesis, be composed,

obtaining as a result a pair of map of the same kind. It will be shown that this sort of

composition can be used to define series composition of I/O behavioral systems described by

behavioral equations. This is the case, in particular, of linear I/O dynamical system. The

invariance of dynamical properties under series composition of systems described by behavioral

equations will then be investigated.

REFERENCES

[IJ J. C. Willems - Paradigms and pu;:zles in the theory of dynamical systems - IEEE Trans.

Autom. Control 36 (1991)

[2] S. MacLane - Category for the working mathematician - Springer Verlag (1971)

Isospectral matrix flows, implement able on a network structure, with application to neural network learning.!

Jeroen Dehaene2, Joos Vandewalle

Katholieke Universiteit Leuven Department of Electrical Engineering, ESAT-SISTA Kardinaal Mercierlaan 94, B-3001 Heverlee, Belgium. e-mail:jeroen.dehaene@esat.kuleuven.ac.be

We study isospectral matrix flows (preserving the eigenvalues) and flows de­rived from isodirectional flows (preserving the eigenvectors), which can be im­plemented on a network structure in the following sense. The matrix H is stored in the connections of a network by associating each entry hi.; to the connection between node i and node j. If a vector signal p(t) is supplied to the nodes of the network (component Pi to node i), the vectors Hp and H2p are easily obtained, by propagating the components of p over the network and multiplying them with the entries of H. We consider matrix flows for which hi,; obeys a uniform

local law hi,; = Lk xlk)yY), where x(k) and y(k) are vectors available in the

nodes. In matrix notation we obtain if = Lk X(k)y(k)T

The flow if = (H2p)pT - 2Hp(Hp)T + p(H2pf where H is symmetric and p is a time varying, externally applied vector signal is a special case of the dou­ble bracket flow if = [H, [H,NJ] where N = ppT and is therefore isospectral. We can prove that if p evolves (and keeps evolving) through a k-dimensional subspace P of R" and an initial matrix H 0 is supplied with a principal eigen­value A with multiplicity k, the flow converges to a matrix whose eigenspace corresponding to A is the subspace P, described by the signal p.

Using a discrete approximation of the flow, the isospectrality is lost. Extra locally implementable terms can be derived from isodirectional flows to ensure attraction to a prespecified spectrum.

The global behavior of the system can be useful for subspace tracking applica­tions and in neural network learning. A design rule for recurrent neural models, stabilizing binary patterns in a linear subspace, can be implemented as an adap­tive learning rule, governed by p as a teaching signal.

1 This research work was carried out at the ESAT laboratory of the Katholieke Uni­versiteit Leuven, in the framework of a. Concerted Action Project of the Flemish Com­munity, entitled Applicable Neural Networks. The scientific responsibility is assumed by its authors. 2 Research assistant of the Belgian National Fund for Scientific Research.

89

90

IDENTIFICATION OF LINEAR SYSTEMS FROM NOISY DATA *

M. DEISTLER AND W. SCHERRERt

Abstract. In this paper linear dynamic errors-in-variables models with mutually uncorrelated noise components are considered. A main complication in identification here is that the systems are not uniquely detennined from the (ensemble) second moments of the observations. In this paper we analyze certain properties of the set of all observationally equivalent systems. In addition we derive continuity results for the relation between the spectral densities of the observations and the sets of observationally equivalent systems. Finaly we describe the sets of spectral densities corresponding to a given Frisch-coran!<. The results obtained are of importance for developing and analyzing identification algorithms.

Key words. errors-in-variables models, linear dynamic systems, identification, differentiable manifolds

, Nonlinear Disturbance Rejection by

Quasi-Static State Feedback

e-mail:

E. DELALEAU M. FLIESS

Laboratoire des Signaux et Systemes C.N.R.S. - E.S.E

Plateau de Moulon F - 91192 Gif-sur-Yvette Cedex

FRANCE Phone: (33) (1) 69 418040

Fax: (33) (1) 69 413060 delaleau@frese51.bltnet e-mail: fliess@frese51.bitnet

The nonlinear disturbance rejection problem via dynamic feedback has been re­cently investigated by several authors [1,2,3].

We propose here a solution which utilizes quasi-static feedback. This new type of feedback has been introduced by Delaleau and Fliess [4] and does not necessitate the integration of any differential equation. Quasi-static feedbacks are rich enough for decoupling any right-invertible input-output system [4].

The main contribution to this communication consists of finding a necessary and sufficient rank condition for disturbance rejection and to show that this rejection can always be achieved via this new class of feedback loops.

The mathematics we are using permit the construction of the feedback.

References

[1] L. Cao and Y.-F. Zheng, "Disturbance decoupling via dynamic feedback", Int. J. Systems Sci., 23, 1992, pp. 683-694.

[2] H.J.C. Huijberts, H. Nijmeijer and L.L.M. van der Wegen, "Dynamic disturbance decoupling fm' nonlinear systems", SIAM J. Control and Optimization, 30, 1992, pp. 336-349.

[3] A.M. Perdon, Y.-F. Zheng, C.H. Moog and G. Conte, "Disturbance decoupling for nonlinear system: a unified approach", in Proc. 2nd IFAC Workshop"System structure and Control", Prague, Sept. 1992.

[4] E. Delaleau and M. Fliess, "Algorithme de structure, filtrations et decouplage", C. R. Acad. Sci. Paris. Serie I, t. 315, 1992, pp. 101-106.

91

92

Structured total least squares and L2 approximation problems

Bart De Moor 3

ESAT - Department of Electrical Engineering Katholieke Universiteit Leuven Kardinaal Mercierlaan 94 B-3001 Leuven Belgium

tel: 32/16/220931 fax: 32/16/221855 email: demoor@esat.kuleuven.ac.be

Let B( T) = Bo + T} B} + ... + TnBn E BRP x q be an affine matrix function of the parameter vector T where Bi, i = 0,1, ... , n are fixed given matrices. Let a E BRm be a data vector and w be a given vector of weights. A problem that often occurs in systems and control applications is to find a rank deficient matrix in the affine set B( T) such that a given quadratic function [T, a, wg of the parameters Ti is minimized. This will be called the Structured Total Least Squares problem, which can be formalized as:

min r E BRn [r, a, w H subject to B(T)y = 0 y'y = 1

(1 )

The solution of the general problem is generated by the triplet (u, (J, v) corresponding to the minimal (J that satisfies

Av = Dvu(J

A'u = Duv(J

u'Dvu = 1

v'Duv = 1 (2)

where Du (Dv) is a positive or nonnegative definite matrix, the elements of which are quadratic func­tions of the elements of u (v) and linear functions of the weights w. The structure of these matrices depends on the matrices Bi of the affine matrix function B( T), which will also determine the structure of the matrix A which is affine in the elements of the data vector a. It is shown how structured (Hankel, Toeplitz, etc .. ) and weighted total least squares and 12 ap­proximation problems lead to this 'nonlinear' generalized singular value decomposition. An inverse iteration scheme to find a local minimum is proposed. Applications in systems and control include total least squares with relative errors and/or fixed elements, inverse singular value problems, an errors-in-variables variant of the Kalman filter, infinite impulse response realization from noisy data, H2 model reduction, H2 system identification and calculating the largest stability radius of uncertain linear systems. Numerical examples will be given.

References

[1] De Moor B. Structured total least squares and 12 approximation problems. Internal Report ESAT­SISTA 1992-33, Department of Electrical Engineering, Katholieke Universiteit Leuven, Belgium, August 1992 (also IMA Preprint Series nr 1036, Institute for Mathematics and its Applications, University of Minnesota, September 1992), Accepted for publication in the special issue of Linear Algebra and its Applications, on Numerical Linear Algebra Methods in Control, Signals and Systems (eds: Van Dooren, Ammar, Nichols, Mehrmann), Volume 188, July 1993.

[2] De Moor B., Van Overschee P., Schelfhout G. H2 -model reduction for SISO systems. Internal Re­port ESAT-SISTA 1992-30, Department of Electrical Engineering, Katholieke Universiteit Leu­ven, Belgium, July 1992. Accepted for the 12th IFAC World Congress, Sydney Australia 1993. (also IMA Preprint Series nr 103.5, Institute for Mathematics and its Applications, University of . Minnesota, September 1992).

[3] EI Ghaoui L. Fast computation of the largest stability radius for a two-parameter linear system. IEEE Transactions on Automatic Control, Vo1.37, no.7, July 1992, pp.l033-1037.

3Research Associate of the Belgian National Fund for Scientific Research (NFWO).

DISCRETE OBSERVABILITY OF THE HEAT EQUATION ON BOUNDED DOMAINS IN EUCLIDEAN SPACE

Alisa DeStefano

Department of Mathematics College of the Holy Cross

Worcester, MA 01570 USA

Fax: (508)793-3030 e-mail: destefano@hcacad.holycross.edu

telephone: (508)793-23.50

Key Words: discrete observability, control theory, heat equation

The study of observability is the study of deducing information about the stat.e of a dynamical system from incomplete measureInents. Discrete ohserv­ability asks if we can recover the initial data with only a discrete set of measure­ments. Discrete observability of the heat equation has been studied for bounded domains in Euclidean space by Gilliam, Li and Martin. They use a sampling scheme which consists of measuring the temperature for a discrete but infinite set of times at a finite number of spatial points. Wallace and Wolf studied dis­crete observability of the heat equation on the sphere, and more generally, the evolution equation on compact homogeneous spaces. Their sampling scheme consists of measuring the temperature at. an infinite but discrete set of spatial points for one time. This type of sampling may be more practical for real-time control problems.

This paper examines discrete observability of the heat equation on bounded domains in Euclidean space using the sampling scheme employed by Wallace and Wolf. The techniques used in this case are different than those used by Gilliam, Li and Martin. The main results are similar to those of Wallace and Wolf. However, they exploited the representation theory of the spaces involved, and we do not have that structure, so a new approach is needed. We show that the general problem of discrete observability (determining under what conditions these samples uniquely determine the solution of the heat equation) depends only on the spectrum of the elliptic operator in the heat equation. The main results involve describing the conditions under which discrete observability holds for a given set of samples and proving the existence of such samples.

93

94

Systematic Reduction of Linear Models Using Interpolation

Theory

P. Dewilde and Aile-Jan van der Veen

Delft University of Technology

Department of Electrical Engineering

Delft, the Netherlands

Tel: +31-15-786234, Fax: +31-15-623271

email: dewilde@dutentb.et.tudelft.nl

The last three to five years a flurry of new results on reduced modeling for linear systems of com­

putations have appeared. These results generalize and subsume the classical Schur, Nevanlinna-Pick

and AAK theories for transfer functions in the complex plane. However, the new theory is completely

algebraic and is therefore not dependent anymore on the 'time invariance' of the classical models. Key

to model reduction and approximation is the conversion of the problem to an interpolation problem

and the generation of the solution via symplectic matrices or operators. There is a strong relation

with lossless inverse scattering as well. The paper will be of a tutorial nature and give a survey of the

state of the art in the field. We shall also devote some attention to computational aspects and present

practical examples.

The Factorization of Noncommutative Substitutionary Matrices with not Prime Order on the Sequence of

Algebraic Surfaces and its Application

Irene Dmitriyeva

Mathematical Analysis Department, Odessa State Pedinsitute, Yakira Street 17, Flat

126, 270076 Odessa, Ukraine

Let {T,L} be the set of commutative (but not mutually) monodromy groups which n-l

are defined by substitutions T = U Tl with order mi, (1' = O,n -1), and open i=O

n-l

curves L = U Li. Substitutions {Td7:~ are fixed on the edges of corresponding (=0

open curves {Ld7:~, and L(, (1' = 0, n - 1), are situated on the compact Riemann

surface R( with the genus p( > 0 if l' > 0 (if l' = 0 it's quite possible that po = 0). The

main purpose of this research is the construction of all noncommutative monodromy

groups with order mi· m(-l ..... mj, (j = 0,£ - 1; l' = 0, n - 1), and which are raised

by initial set {T, L}.

In language of vector-matrix boundary Riemann problem the above mentioned result

means the constructive factorization of noncommutative substitutionary matrix with

order m( . ml-l ..... mj on compact Riemann surface Rj with the genus Pj < Pi,

(j = 0, l' - 1; l' = 0, n - 1). Such very difficult factorization becomes attainable be­

cause of well known factorization of the corresponding commutative substitutionary

matrix with order m( on compact Riemann surface R(, (1' = 0, n - 1).

These results are considerable and rather important generalization of paper [1 J.

[1J 1. Yu. Dmitriyeva, St. Petersburg Math. Journal, V. 4, No.2 (1992) 129-140.

95

96

Some results on optimization of general input-output systems

Vaclav Dolezal Department of Applied Mathematics and Statistics

State University of New York at Stony Brook Stony Brook, New York 11794-3600, U.S.A.

Fax # (516) 632-8490

We consider the optimization of general input-output systems

over extended spaces which need not be of the feedback type.

Assuming that the system given by an input-state-output

description depends on a parameter A that belongs to a specified

subset a of a normed linear space J:, we try to minimi ze on a a given performance functional f:c?~Rl which is determined by the

optimization problem considered. Since a minimizer need not

exist in general, our theorems give conditions under which an

€-minimizer can be found as a minimizer of f on an appropriate

subset c?n of a. The performance functionals discussed are

determined by optimal tracking without and/or with presence of

outer disturbances, and by minimization of sensitivity with

respect to disturbances. Moreover, since c!t' is frequently the

linear space of stable convolution operators, we describe

fundamental sequences in J:::' --an assumption needed in the main

theorems.

a Hllbert

vaclav Dolezal Department of Applied Mathematics and Stati&tics

State University of New York at Stony Brook Stony Brook, New York 11794-3600, U.S.A.

Fax # (516) 632-8490

In this paper we will study the behavior of a Hilbert network

li.e., a finite or countably infinite network whose variables are

In a Hilbert space) under perturbations of its elements.

~ore specifically, we will give estimates for the change of the

:urrent distribution daused by (a) perturbations of the nominal

letwork elements when the voltage sources are fiied, (b) a change

)f voltage sources in a network whose elements are perturbed.

:ases (a) and (b) entail insensitivity and robust stability of

:he nominal network, respectively. The results will be

,llustrated by an example of an infinite network.

97

98

Commutant Lifting When the Intertwining Operator

is Not Necessarily a Contraction

MICHAEL A. DRITSCHEL *

November 1992

Necessary conditions are formulated in the commutant lifting problem so that a lifting may be found that preserves the number of squares of the intertwining operator and its adjoint. These conditions are also shown to be sufficient in the case that the intertwining operator is a contraction and in some more general circumstances. The approach taken is an adaptation of the so-called "coupling method" due to Arocena from the Hilbert space to the Krein space setting.

INTERPOLATION FOR UPPER

TRIANGULAR OPERATORS

Harry Dym

Department of Theoretical Mathematics

The Weizmann Institute of Science

Rehovot 76100, Israel

99

Analogues of the classical interpolation problems of Nevanlinna-Pick and Caratheodory

Fejer will be formulated and solved in a general setting of upper triangular operators. A

large part of this talk is based on assorted collaborations with D. Alpay and P. Dewilde.

100

Explicit formulas for optimally robust controllers for delay systems!

Harry Dym2 , Tryphon T. Georgiou3 , Malcolm C. Smith4

We consider the class of delay systems of the form P( s) = e-sr Po( s) where Po( s) is a strictly proper rational function and T > 0 is a time-delay. We present a closed form formula for controllers which are optimally robust with respect to perturbations measured in the gap metric. This formula is a generalization of one given in [4] for the case of a first order Po(s). The approach builds on earlier work [5] which developed state-space formulas for computing optimal Hoo-performance for such systems.

The form of the controllers represents a synthesis of state-space techniques, as­sociated with the rational part Po( s), together with the exploitation of a certain set of infinite dimensional "distributed delays". These latter elements are expressed in terms of a certain commutative algebra of operators, acting on the delay element e-sr

. These operators are defined as follows: 8" f := (J( s) - f (Ci)) / (s - Ci) (for Ci E C and f(s) any suitable meromorphic function). Some early uses of these operators ([1] and [2]) were in the context of reproducing kernel Hilbert spaces and scattering theory-see [3] for a more recent account, numerous references, and applications to interpolation theory. A number of interesting properties of these operators is exploited in the derivation of the optimal controllers.

[1] L. de Branges, Some Hilbert spaces of analytic functions I, Trans. Amer. Math. Soc., 106, pp. 445-468, 1963.

[2] L. de Branges and J. Rovnyak, Canonical Models in Quantum Scattering Theory, in Perturbation Theory and its Applications in Quantum Mechanics, edited by C. H. Wilcox, John Wiley, 1966.

[3] H. Dym, J-Contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, Regional conference series in mathematics/Conference Board of the Mathematical Sciences; no. 71, 1989.

[4] T. T. Georgiou and M. C. Smith, Topological Approaches to Robustness, Lecture Notes in Control and Information Sciences, 185, pp. 222-241, Springer-Verlag, 1993.

[5] J.R. Partington and K. Glover, Robust stabilization of delay systems by approx­imation of coprime factors, Systems fj Control Letters, 14, pp. 325-331, 1990.

lThis work was supported in part by the NSF, AFOSR, SERC and the Nuffield Foundation.

2Dept. of Theoretical Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel. e-mail: MTDYM@weizmann.weizmann.ac.il, Tel: +972-8-342902.

3Dept. of Electrical Engineering, Universitr of Minnesota, Minneapolis, MN 55455, U.S.A. e-mail: georgiou@ee.umn.edu, Tel: +1-612-625-3303.

4Dept. of Engineering, University of Cambridge, Cambridge, CB2 IPZ, U.K. e-mail: mcs@eng.cam.ac.uk, Tel: +44-223-332745.

Filtrations and Hilbert polynomials in control theory

S. EI Asmi

Laboratoire des Signaux et Systemes

C.N.R.S. - E.S.E

Plateau de Moulon

F-91192 Gif-sur-Yvette Cedex

FRANCE

J. Rudolph

Institut fUr Systemdynamik und RegelWlgstechnik

Universitat Stuttgart

Postfam 801140

D(W)-70511 Stuttgart

GERMANY

E-mail: rudolph@isr.Wl.i-stuttgart.d400.de

The differential algebraic approach to linear and nonlinear systems has allowed to gain new insight in the system structure. In the present contribution, we would like to underline the usefulness of two concepts stemming from this branch of mathematics, namely filtrations of differential field extensions (resp. differential modules) and the associated Hilbert polynomials. These polynomials, which have been introduced in differential algebra by Kolchin and Johnson, carry a lot of dimensional information of the filtrations and the field extensions.

Using different filtrations for the same system, important structural information on the system can be gained. Examples are the number of independent inputs and the state dimension. Further examples are the calculation of the differential output rank, the essential orders, and the observability indices. The concepts have also been successfully used for the interpretation of the structure algorithm and of system properness. Finally, they permit to characterize new system properties, as for example flatness or well formedness, and to define feedback.

References

[1] E. Delaleau and M. Fliess, Algorithme de structure, filtrations et decouplage, C. R. Acad. Sci. Paris, t. 315, Serie J, pp. 101-106, 1992.

[2] S. El Asmi, An algebraic approach to essentiallity, in Proc. ECC-91, Grenoble, France, July 2-5 1991, Hermes, Paris, 1991, pp. 468-473.

[3] S. El Asmi and M. Fliess, Formules d'inversion, in "Analysis of Controlled Dynamical Systems", B. Bonnard, B. Bride, J.P. Gauthier, 1. Kupka eds., Birkhiiuser, Boston, 1991, pp.201-210.

[4] J. Johnson, Differential dimension polynomials and a fundamental theorem on differen­tial modules, Amer. J. Math., 91, 1969, pp. 239-248.

[5] E.R. Kolchin, Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.

[6] J. Rudolph and E. Delaleau, On Properness of Linear Time- Varying Systems and its Generalization to the Nonlinear Case, European Control Conference ECC'93, Gronin­gen, The Netherlands, June 28 - July 1, 1993.

11 i

101

102

An algorithm for computing the distance to uncontrollability

by L. Elsner* and C. He**

*FakulUit fur Mathematik, Universitiit Bielefeld, W-4BOO Bielefeld 1, Germany email: elsner@mathematik.uni-bielefeld.de

**Fakultiit fur Mathematik, TU-Chemnitz, 0-9010 Chemnitz, Germany email: he@mathematik.tu-chemnitz.de

Abstract: In this paper, we present an algorithm to compute the dis­tance to uncontrollability. The problem of computing the distance is an optimization problem of minimizing O'(x,y) over the complete plane. This new approach is based on finding zero points of grad O'(x, y), for which 0'( x, y) is shown to be differentiable at its minima. We obtain the explicit expression of the derivatives of grad O'(x,y). The Newton's method and the bisection method are applied to approach these zero points. Numerical results show that these methods work well.

r

CONSTRUCTION OF WAVELETS IN THE FREQUENCY DOMAIN by

Nurgun Erdol and Feng Bao Department of Electrical Engineering

Florida Atlantic University Boca Raton, Florida 33431

USA e-mail: erdol@acc.fau.edu

103

Construction of continuous-time orthogonal wavelets from discrete-time filter banks for multiresolution analysis have been explored by many researchers. In this paper we propose to construct a family of discrete-time filters from a known sequence hO(k) with a I-periodic fourier transform HO(t), via the following operation:

H(t) is said to generate a scaling function ¢(t)if it satisfies the dilation equation

(11 2)¢(t / 2) = L h(k)¢(t - k) k

Further, if some regularity conditions are met, then

ip(f) = n H(f / 2j)

j=!

(1)

(2)

(3)

where ip(f) is the Fourier transform of the scaling function. It is shown that the family of filters constructed using the equation (1) imply the following propositions:

Proposition 1. If Hi-) (f) satisfies the regularity conditions, then the relationship of equation

(1) yields a family of functions Hi (f) that also satisfies these conditions.

Proposition 2. If two functions H oo (f) and HOI (f) both satisfy the regularity conditions, then a flexible family of discrete -time ftIters can be generated from each one using different forms of

(4)

The family of filters Hi(f) possess the following properties as a function of the iteration number i:

> increasing smoothness at f=O and f= 1/2. > improved frequency response characteristics both in the pass bands and the stop bands. > the transition bandwidth decreases with the iteration number i.

Design examples and illustrations of the algorithm are also provided.

104

ON THE SPECTRAL THEORY OF AN ELLIPTIC BOUNDARY

VALUE PROBLEM INVOLVING AN INDEFINITE WEIGHT

M. FAIERMAN

We are concerned here with the boundary value problem

Lu = '\w(x)u in 0,

Bu=O on r,

where L is a linear elliptic operator of the second order defined in a bounded region 0 C

1R", n ~ 2, with boundary r, B is a linear differential operator of the first order defined

on r, and w is a real-valued function in Loo(O) which assumes both positive and negative

values. If we let A denote the closed operator in )/ = L2 (0) induced by Land B and let T

denote the operator of multiplication in)( induced by w, then we arrive at the pencil A-AT.

The spectral theory of this pencil has been the subject of investigation in various works for

the case where A is selfadjoint (cf. IF], IFLJ). In this talk we discuss the spectral theory fo<

an operator A which is not selfadjoint and under certain assumptions concerning T, and

in particular we present some results concerning the completeness of the principal vectors

in certain function spaces as well as concerning the angular and asymptotic distribution of

the eigenvalues.

REFERENCES

IF] M. Faierman, Elliptic problems involving an indefinite weight, Trans. Amer. Math.

Soc. 320 (1990), 253-279.

IFL] J. Fleckinger and M.L. Lapidus, Eigenvalues of elliptic boundary value problems with

an indefinite weight fucntion, Trans. Amer. Math. Soc. 295 (1986), 305-324.

105

"" Simple Approach to Simultanec;>us Assignment of Finite and Infinite Eigenvalues for Descriptor Systems

L

Chun-Hsiung Fangt and Fan-Ren Changt

Abstract

A very simple method for simultaneously assigning finite and infinite eigenvalues of

descriptor systems, using constant-ratio proportional and derivative (CRPD) feedback of

the state, is proposed in this paper. To assign a mixture of finite and infinite eigenvalues,

only two constant matrices which could be chosen at random are required in the proposed

approach.

Keywords: descriptor systems, feedback, eigenvalue assignment.

tDepartment of Electrical Engineering, National Kaohsiung Institute of Technology,

415 Chien-Kung Road, Kaohsiung 807, TAIWAN, R.O.C.

tDepartment of Electrical Engineering, National Taiwan University, Taipei 107,

TAIWAN, R.O.C.

106

Regional Eigenvalue-Assignment Robustness of Linear Uncertain Systems

Chun-Hsiung Fangt and Jyh-Horng Chout

Abstract

A simple approach is proposed to ensure that all eigenvalues of the uncertain system

are placed in prescribed regions. The explicit bounds on linear time-invariant perturbations

with highly structured information are obtained. Under these allowable highly structured

perturbations, both stability robustness and certain performance robustness will thus be

ensured. The merit of the simple approach proposed is demonstrated by given examples

where the results achieved are much better than the ones reported recently. An important

contribution of this paper is that we extend the approach to study robust control problem

of singular systems with structrued perturbations. In the past literature, little effort has

been devoted to dealing with singular system robust control. As far as we are aware, this

paper is the first one to discuss such problems .

Keywords: Robust stability, regional pole assignment, singular systems,

structured perturbations.

tDepartment of Electrical Engineering, National Kaohsiung Institute of Technology,

415 Chien Kung Road, Kaohsiung 807, TAIWAN, R.O.C.

tDepartment of Mechanical Engineering, National Yulin Institute of Technology,

Touliu, Yunlin 640, Taiwan, R.O.C.

Title: On the Rutishaueer' approach to eigenvalue problema

BY:L.Faybusovich ,Department of Mathematics,University of Notre Dame, Notre Dame,P.O. box 3ga,In ,46556,USA

The work of H.Rutishauser had a profound influence on the development of the nu­merical aMlyeie. Yet the part of this work fits perfectly to the echeme of polynomial realization theory of P.Fuhrmann. In this paper we discuss in detail the quotient­difference algorithm from this point of view. We eetablish relatiolllhips with both the classical work of Stiltjes and very recent results of Flaschka. The applications t.o coding theory are given.

107

108

CONTROL OF ONE-LINK FLEXIBLE MANIPULATORS *

DE-XING FENG

Institute of Systems Science, Academia Sinica Beijing 100080, Fax: 261-256 8364, Phone: 256 1965

HUI-ZHEN WANG

Computing Center, Academia Sinica, Beijing 100080, China

Abstract. A controlled model of one-link flexible manipulator with load at the end

of link is considered. A boundary feedback control law is proposed. It is showed that

the closed loop control with this control law is asymptotically stable. Finally, some

.imulation results are given.

Keywords. One-link flexible manipulator, boundary feedback control, asymptotic

stability.

....,

Complete parametrization of solutions of the rational spectral factorisation problem

Augusto Ferrante, Dipartimento di elettronica e informatica Universita di Padova, via Gradenigo 6/ A, 35131 Padova, ITALY

e-mail: augusto@paola.dei.unipd.it

Abstract The spectral factorization problem plays an important role in many fields of system

theory. In this work we deal with matrix-valued real rational coercive spectral densities, i.e. square matrix (say m x m) of real rational functions s.t.:

i) 4>(s) = 4>(-sf

ii) 4>( s) is analytic and non-negative definite on the imaginary axis including infinity.

iii) 3 c E R s. t. 4>(jw) > cI > 0 Vw E R.

In the literature there are parametrizations of special classes of spectral factors starting from the classical results of J. C. Willems [4] and Finesso and Picci [2] where different parametrizations of stable spectral factors are given. More recently, Picci and Pinzoni in [3] gave a parametrization of all minimal spectral factors with a given structure of poles or zeroes via the solutions of two A.R.E ..

The aim of this talk is to present a unique parametric description of all minimal square spectral factors of such spectra, i.e. of m x m rational matrix W(s) S.t.:

1. W(s)W(-sf = 4>(s)

2. W(s) has minimal McMillan degree among the functions that satisfy 1.

We give two different solutions to this problem showing that the set W of mimimal square spectral factors of 4>(s) is in one to one corrispondence with the lattice of the left inner divisors of a certain inner function which is computable from 4>( s) and also with the lattice of the solutions of an A.R.E .. In this talk besides reviewing the above results obtained in [1], we present explicit formulas for the driving noises and the state processes of the corresponding minimal stocastic realizations.

References

109

[IJ A.Ferrante, G.Michaletzky and M.Pavon, Parametrization of all minimal spectra.! fac­tors, Submitted for pubblication.

[2] L.Finesso and G.Picci, A characterization of minimal spectral factors, IEEE Trans. Aut. Control AC-27 (1982), 122-127.

[3J G.Picci and S.Pinzoni, Acausal models of stationary processes Proceedings of the First European Control Conference, July 1991.

[4J J.e.Willems, Least square stationary optimal control and the algebraic Riccati equation, IEEE Trans. Aut. Control AC-16 (1971), 621-634.

110

Splitting subspaces and generalized spectral factors

Augusto Ferrante, Dipartimento di elettronica e informatica, Universita Padova,

Gyiirgy Michaletzky, Department of Probability Theory and Statistics, Eiitviis Lorand University,

Michele Pavon, Dipartimento di elettronica e informatica, Universita Padova.

In this paper we are going to consider a generalization of the theory of splitting subspaces used for the

construction of state space realizations of stationary time series. The usual construction provides forward or

backward realizations corresponding to stable or unstable spectral factors. We would like to study spectral

factors without specification on the pole (or zero) structure and to show that the usual" geometric" approach

(using splitting subspaces) can be extended to this more general setting.

Assume that H+ and H- are subspaces of a Hilbert space H and X is a splitting subspace for H+ and

H-. The next step in the usual construction (cf. Lindquist - Picci (1985)) is to associate two perpendicularly

intersecting subspaces 5 and S to X such that X = 5 n Sand H- C 5, H+ C S We can observe that the second conditions imply that the theory based on this construction investigates

only stable or unstable realizations.

In our study we do not want to extend the notion of splitting subspaces i.e. we are going to analyse

Markovian realizations, but we do not use these assumptions.

So let us assume that H = A Ell B, H+, H- C H. Define

where EA H+ denotes the closed subspace generated by the projection of H+ onto A. Let X = A+ Ell B-.

In this case X is a splitting subspace for H+, H-.

If U is a shift operator acting on H, such that U- 1 H- C H-, U- 1 A C A, U H+ C H+, U B C B then

X is a Markovian splitting subspace.

Minimality. It Can be shown that if X is a nonminimal splitting subspace then minimality reduction

can be carried out in such a way that from the triplet (A, B, X) we get (AD, Bo, X o), where Xo eX, Bo C

B, AD ::l A or vice versa (AD C A, Bo ::l B).

Time reversing. Observe that B- C X is invariant under the operator EX U. We are going to prove

that there exists a realization with the same zeros but with poles "flipped" with respect to the unit circle if

and only if there exists a complementary (with respect to B-) invariant subspace of X.

Also we are going to analyse - at least in the internal case - the connection between the triplets (A, B, X)

and the subspaces X ~ (5,S).

References:

A. Ferrante - Gy. Michaletzky - M. Pavon: Parametrization of all minimal square spectral factors,

(1993) Systems and Control Letters, to appear.

A. Lindquist - G. Picci: On the stochastic realization problem, SIAM J. Control and Optim. 23 (1985)

pp. 365-389.

G. Picci - S. Pinzoni: Acausal models of stationary processes, Proceedings of the First European Control

Conference, July 1991.

,

f III

Shape Matching in Smart Material Structures

Ben G. Fitzpatrick

In this talk we consider the problem of matching desired shapes in smart material ~tructures

via piezoceramic actuators. This problem involves the determination of attainable shapes

given a patch configuration as well as the control problem of bending a given structure

into a desired shape. The shape matching problem can be posed as a minimum norm

problem in a Hilbert space. Existence of solutions and convergence of approximations are examined.

112

Integral Manifolds

in Nonlinear Control Theory

D.Flockerzi

Mathematisches Institut der Universitiit Wiirzburg, Am Hubland, 8700 Wiirzburg, Germany, Tel.: 0221-470-2602, e-mail: flockerzi@vax.rz.uni­wuerzburg.dbp.de

We attempt to give answers to some non-local problems for affine control.systems. The main tool hereby is the theory of non-local integral manifolds that are attractive and allow an asymptotic phase. The control will be used twofold: first to generate the necessary dichotomy, secondly to render a given manifold invariant. We present some applications, e.g. given a non-local solution of the state-feedback regulator problem we construct a nonlinear observer solving the non-local error feedback regulator problem.

p

On the asymptotic dynamics of 2D positive systems

E.Fornasini Dept. of Electronics and Compo Sci.,

via Gradenigo 6/a, University of Padua, 35131 Padova, Italy e-mail meme@paola.dei.unipd.it

20 positive systems are multidimensional state models described by the following equa­tion

for which the state and the input variables are always positive, or at least nonnegative, in value, i.e. x{h, k) E R~ and u{h, k) E R';' for every (h, k) E Z x Z.

Such systems naturally arise in modelling physical, biological and ecological processes whose state variables may not have meaning unless they are nonnegative. A noteworthy example of homogeneous positive 2D systems is provided by 2D Markov chains, obtained from (I) by assuming BI = B2 = 0; Al = ",P,A2 = (I - ",)Q with 0 ~ '" ~ 1, P,Q stochastic matrices and x{h, k) a probability vector for all x{h, k) in Z x Z. For this kind of systems a satisfactory theory of their asysmptotic behaviour is already available in literature,

The aim of this paper is to present the theory of homogeneous positive 20 systems in a fairly broader context, by assuming that the initial local states of system (1), x{ i, -i), i E Z meet only the following regularity condition

1 h+n

lim --IILx{i,-i)-mll=O, 'VhEZ n-(X) n + 1 i=h

where m belongs to R~ and no particular hypothesis is made on Al and A 2 , except positivity. Under these assumptions, we will investigate some aspects of the asymptotic dynamics. In particular, we will determine equivalent conditions on the variety of the characteristic polynomial det{I - Aizi - A 2z 2) which guarantee the existence of a vector p E R~ such that the local states x{h, k) become parallel to p, as h + k diverges to +00.

Under the hypothesis of a periodic pattern for the initial local states, i.e. x{i, -i) = x{ i + N, -i - N), 'Vi E Z the structural constraints on the 20 characteristic polynomial correspond to a set of spectral properties of the 10 system (of dimension nN) which represents the periodic dynamics.

113

114

SOME ISSUES ON GENERALIZED RICCATI EQUATIONS

ARISING IN STOCHASTIC CONTROL*

Marcelo D. Fragosot and O.L.V. Costa:j:

Abstract. In this paper a generalized version of the well-known matrix Riccati equations which

arise in certain stochastic optimal control problems is studied. It is assumed that the systems are

not necessarily detectable including those having non-observable modes on the imaginary axis.

Via the concepts of mean square stabilizability and mean square detectability we improve previous

results on both the convergence properties of the generalized Riccati differential equation and the

solutions of the generalized algebraic Riccati equation. The results are derived under relatively

weaker assumptions and include, inter alia, the following: a) An extension of Theorem 4.1 of

[24] to handle systems not necessarily observale, b) Existence of strong solution, subject only to

mean square stabilizability assumption, c) Conditions for existence and uniqueness of stabilizing

solution for systems not necessarily detectable; d) Relixing the assumptions for convergence of

the solution of the generalized Riccati differential equation, including replacing observability

by mean square detect ability, and deriving new convergence results for systems not necessarily

detectable.

Keywords. Generalized Riccati equation; strong solution; mean square stabilizability; nonob­

servable systems; stochastic control.

* This work was supported in part by CNPq (Brazilian National Research Council) and

FAPESP (Research Council of the State of Sao Paulo). t National Libora tory for Scientific Computing - LNCC/CNPq, Department of Research

and Development, Rua Lauro Muller 455, 22290-160, Rio de Janeiro, RJ - Brazil. E-mail:

USERFRAG@LNCC.BITNET :j: Escola Politecnica da Universidade de Sao Paulo, Departamento de Engenharia Eletronica,

05508-900, Siio Paulo, SP - Brazil. E-mail: OLDVCOST@BRUSPVM.BITNET

NONSYMMETRIC MATRIX RICCATI EQUATIONS WITH APPLICATIONS

G. FREILING

lIS

Lehrstuhl II fur Mathematik - RWTH Aachen, Templergmben 55, D-5100 Aachen, Germany, email: freil@math2.rwth-aachen.de

We consider matrix Riccati differential equations

(RDE)

and the corresponding algebraic Riccati equations

(ARE)

where W, Ell, E 12 , E 21 , En are matrices of dimensions m x n, n X n, n xm, m x nand m xm respectively. It is known that matrix Riccati equations are playing an important role in many branches of applied mathematics and in particular in systems theory; nonsquare matrix Riccati equations appear for example in Nash and Stackelberg control problems, where the solutions of Riccati equations are used to determine the optimal open loop strategies.

In the first part of the lecture we represent a general representation formula for all solu­tions of (RDE) with constant coefficients E;] and we show that a similar formula can be obtained if these coefficients are T -periodic functions or are polynomially dependent on t or on a complex parameter A. Further we explain how our representation formula can be used for the description of the phase portrait of (RDE) and for a parametrization of all solutions of (ARE).

In the second part of our talk we apply our results to the investigation of coupled matrix Riccati equations of the form

1\1 = _AT/(I - /(IA - QI + /(1 5 1/(1 + /(152/(2,

1\2.= _AT/(2 - J\2A - Q2 + /(251/(2 + /(251/(1, (1)

appearing in open-loop Nash differcntial games. Further we givc necessary conditions for the existence of constant stabilizing solutions of (1).

116

Nearness Problems in Control Theory and the Algebraic Riccati Equation

Pascal Gahinet

INRIA Rocquencourt, BP 105

78153 Le Chesnay Cedex, France

email: gahinet@colorado.inria.fr

MTNS '93

Abstract

System properties such as stability, controllability, stabilizability, absence of in­variant zeros, etc. can be characterized in terms of the solutions of certain algebraic Riccati equations (ARE) or inequalities (ARI). Such ARE or ARI reformulations have the advantage of providing tests which are more tractable and more reliable from a numerical standpoint. In addition, the near loss of controllability, stabilizability, etc .. is reflected as the near failure of the corresponding algebraic conditions. The latter is much easier to detect in general since this only involves simple tests on entities such as eigenvalues, singular values, norms, etc. Applications to "nearness to ... " problems in system theory and to state-space H 00 synthesis are discussed. The limitations of this approach are also addressed as well as questions open for future research.

References

[I] Cahinet, P., and A .. J. Laub, "Algebraic Riccati Equations and the Distance to the Nearest Uncontrollable Pair," SIAM J. Contr. Opt., 30 (1992), Pl'. 765-786.

[2] Cahinet, P., "On the Came Riccati Equations Arising in Hoo Control Problems," to appear in SIAM J. Contr. Opt., 1992.

[3] Hewer, C. and C. Kenney, "The Sensitivity of the Stable Lyapunov Equation," SIAM J. Contr. Opt., 26 (1988), pp. 321-344.

Tht' m-D Cayley-Hamiltoll theorem

Krzysztof Galkowski Insti tute of TelecOmlll11I1ic:ation and Acoustics

Tpchnical Cniversity of \Vrodaw uI. WyiJl'z('zf' \iVys]>imiskiq~() 27

;')0-370 Wrodaw, POLAND

In n'cpnt YPitrS tlIP Ill-I) (:ayipy-Hiillliiton theorPlll is considf'red exteusively in noD systelll theory. The I\f'lwralizatioll of tllis wpll kllowll frolll the classical linear

algf'bra tlworpm for the lllultidilll('IISiollal cas(' has IWPll prf'S("lltpd hy sf'vpral authors,

for example Thpodorou. Also tlIP ,!',f'llf'ralizatioll for siu,!',ulitr systpms receives an increasing attpntion, spe for exalll"lp tlIP works of I\iiczorek alld Chang Clwn. Tlw

basic version of tlIP Cavi<-.y-Hallliitoll tlworPlll ill lll-D casp rpquirf's using the m-D powers of squarp matrix whicll vi"lds rpCIIIT('nt dppPlldellcc'. In this lIot" tlw compact, non-rpcursivp statpllwllt of this tlworpm is prpsPllted. For this purposp, tlw special

matrix rppnespntiltiou of all lll-D jlolYllomial is introducPd, and tllP multi way matrices or the tpnsor products of lllatrices are usc,d.

117

liS

ON SOME OPERATOR-VALUED FUNCTIONS ON BIDISC

AND PREDICTION THEORY

by

D. Galjpar, N. Suciu and 1. Valll§escu

(University of Timi§oara and Institute of Mathematics

of Romanian Academy Bucure§ti)

Address for correspondence: Prof.Dr.Dumitru Galjpar,

University of Timi§oara, Department of Mathematics, Bul. V.parvan nrA, RO-1900 Timi§oara, ROMANIA,

Fax: +40 6 115633 or +40 6 116722

Abstract:

Factorization theorems was crucial points in solving prediction problems for

stationary processes. Such a way the Szego factorization for the scalar case

solved the univariate prediction problem, while factorizations for matrix­

valued functions was necessary for the multivariate prediction. For infinite­

variate prediction, factorization- theorems for operator- valued functions

are necessary. We mention here only those of H. Helson- D. Lowdenslager

and of I. Suciu- I. Valllliescu. Also the Wold decomposition theorem for an

isometry and variants of it for couple of isometries (1. Suciu, M. Slocinski,

P. Muhly, G. Kaliampur- V. Mandrekar) played an important role.

An approach for the study of infinite- variate case in the context of the com­

plete correlated action was made by I. Suciu and the third named author.

Here, using a four- fold decomposition for the space of a process on Z2 and

the factorization of an appropiate operator- valued function on the bidisc,

the problem of stationary processes on Z2, in complete correlated actions is

aborded.

Key words: Opemtor- valued functions, factorizations, Wold- type function,

complete correlated actions, stationary processes on Z2.

r THE ALGEBRAIC RICCATI EQUATION AND SINGULAR OPTIMAL CONTROL:

THE DISCRETE-TIME CASE

Ton Geerts

Tilburq University, Department of Economics, P.O. Box 90153, NL-5000 LE Tilburq, e-mail: geerts@kub.nl

Consider the standard discrete-time system E:

x(i+1) Ax(i) + Bu(i),

y (i) Cx (i) + Du (i) ,

(lola)

(1.1b)

with u(i) e ~m and x(i) e ~n for all i ~ 0 and x(O) xo' together with

the auxiliary output z(i)

where z(i) e ~s for all i

Sx(i) + Tu(i), (1.1c)

O. All matrices are real and constant. Also,

let J(x o' u) := r y' (i)y(i), i=O

where u = iu(i) 'i denotes any control sequence.

(1. 2)

We are interested in I1eCf:''S~arr dnri .;:;uff icient condi t 111"9 for

solvability of the Linear·Quadratic Control Problem ~'ith z·stabilitF

(LQCP)z: For all Xo e ~n, determine

Jz(x o) := infiJ(x o' u) ju is such that lim z(i) i_

and if "'xo

e ~n : Jz (x o) ( 00,

0' ,

then find (if possible) for every Xo e ~n a control sequence u

(1. 3)

(1.4)

iu (i) , . 1

for which J(x o' u) equals Jz(x o)' This problem is called the LQCP ~'ith

stability if T = 0 and ker(S) = 0, and it is called the LQCP ~·ithout

stability if s = O. Suppose that there exists a real symmetric matrix Kz

2 0 such that, for all xo' Jz(x o) = xo'KzX o. Then it is well known that

[C'C + A'KA - K C'D + A'KB]

F(K z) 20, with F(K) = D'C + B'KA D'D + B'KB ' (1. 5)

and, in fact, even ~(Kz) = 0, where ~(K)

C'C + A'KA - K - (C'D + A'KB) (D'D + B'KB) '(D'C + B'KA), (1. 6)

H' denoting the Moore-Penrose inverse of matrix H.

Now it will proven that (1.4) is equivalent to a certain subspace

condition, and that Jz (x o) = Xo 'KzX o for some Kz 2 0 if this condition

is satisfied. We will specify this Kz ln terms of (1.5) rather than

(1.6) and thus, in particular, provide a new characterization of the

optimal cost for the LQCP without stability. Finally, we will present a

condition for existence of optimal control~ and show that theSe controls

can be giVen in terms of f~edback laws, if r is left invertible.

119

120

OUTPUT CONSISTENCY AND WEAK OUTPUT CONSISTENCY

FOR CONTINUOUS-TIME IMPLICIT SYSTEMS

Ton Geerts

Tilburg University, Department of Economics, P.O. Box 90153, NL-5000 LE Tilburg, e-mail: geerts@kub.nl

Recently (1), issues such as solvability, consistency and "eak

consistency were investigated for linear systems of the form

Ex (t) = Ax (t) + Bu (t) , (1)

with t E [0, 00), x(O-) = Xo E IRn , E, A E 1R1xn , B E 1R1xm This was done

by redefining (1) in terms of distributions: (l)

Eo *X=Ax+Bu+Exoo, (2)

where 0, 0(1) denote the Dirac delta distribution and its derivative, *

denoteS convolution of distributions, and u E cm x E cn ,the m- and imp' lmp

n-vector versions of cimp ' the commutative algebra of implllsil'e-smooth

distributions (2). Now, Suppose that a linear system E is characterized by (2) and the output equation y = Cx, (3)

with C E IRrxn. Then we will call a point Xo € IRn output ",,'nsistent if

there exists an input u E em and a trajectory x € S(x o' u) such that y sm

€ er

and y(O+) = CX o. Here, e c c, stands for the subalgebra of sm sm lmp

smooth distributions (functions that are smooth on [0, 00)) and

S(x o' u) = Ix E e~ I[Eo(l) - M) * x = Bu + Exool. (4) lmp

Next, a point Xo will be called "eakl\' output conslstent if there exists

a control u E em and a trajectory x E S(x o' u) such that y € er . Then sm sm

output consistency and weak output consistency are generalizations of

(state) consistency and weak (state) consistency (1).

In this talk we will present nt>cessan' and sufficlt>nt conditions

for global (weak) output consistency (Le., for all points in the state

space to be (weakly) output consistent), and we will establish in

particular that these conditions reduce to those for global (weak) state consistency [1) if C = I.

[1) T. Geerts, "Solvability conditions, consistency and weak consistency for linear differential-algebraic equations and

time-invariant singular systems: The general case", Lin. _Ilg . .'lppl.

181, pp. 111-130, 1993.

(2) M.L.J. Hautus, "The formal Laplace trasnform for smooth linear

systems", Lecture Notes In Econ. & Math. Syst., vol. 131, pp. 29-46, 1976.

EQUILIBRIUM POINTS, LIMIT CYCLES AND COMPLEX DYNAMICS:

SOME STRUCTURAL LINKS IN A BASIC CLASS OF NONLINEAR SYSTEMS

Roberto Genesio, Alberto Tesi and Francesca Villoresi

Dipartimento di Sistemi e Informatica, Universitd di Firenze Via di Santa Marta 3 - 50139 Firenze - Italy

Tel.: (39) 55-4796263; Fax: (39) 55-4796363; email: atesi@ingfi1.cineca.it

Abstract. The class of dynamical systems defined by the differential equation

q(D)y(t) + p(D)n[y(t)] = 0 (1)

121

is considered, where D == dldt, q(.) and p(.) are two polynomial operators and n(·) is a nonlin­ear single valued function. These systems describe important processes, especially in the area of Systems and Control Theory where they are also called "Lur'e systems" [1]. It has been recently recognized that equation (1) can exhibit complex and chaotic dynamics, even for reduced degree of q (2 3) and p (2 0) [2][3]. In particular, the well-known Chua's circuit belongs to this class [4].

The purpose of this paper is to investigate for the above systems two general phenomena of complex dynamics, that is homo clinic orbits and period-doubling cascades, in order to explain them and their relations in terms of the main elements characterizing the system behaviour. The idea is that the structural simplicity of equation (1) can allow one to enlight the more significant aspects of such dynamics and the ingredients they need.

The essential tool employed in this analysis is the harmonic balance principle which leads to a qualitative description of the periodic solutions of systems (describing function method) with an accuracy which can be suitably validated and controlled [5]. By such a technique, recently used also for predicting chaos [3][6], the structural links connecting the equilibrium points and the conditions for periodicity to the studied complex phenomena are expressed in form of approximate relations depending on the parameters of the system.

Examples concerning a simple third order differential equation with polynomial and piece-wise linear characteristics are developed to give specific insight to the general obtained results.

References

[1] Mohler, R. R. [1991]: Nonlinear Systems (Vol. 1, Dynamics and Control), Prentice-Hall, Englewood Cliffs.

[2] Mees, A. I. [1986]. Chaos in feedback systems. in Chaos, ed. Holden, Manchester Un. Press, Manchester.

[3] Genesio, R. and A. Tesi [1992]. Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica, 28,531-548.

[4] Chua, L. 0., M. Komuro, and T. Matsumoto [1986]. The double scroll family. IEEE Trans. on Circ. Syst., CAS-33, 1072-1118.

[5] Mees, A. I. [1981]. Dynamics of Feedback Systems. John Wiley, New York.

[6] Genesio, R. and A. Tesi [1992]. A harmonic balance approach for chaos prediction: the Chua's circuit. Int. J. of Bifurcation and Chaos, 2, 61-79.

122

Abstract

Balanced realizations and system identification

Laszlo Gerencser and Raimund Ober

Control and Systems Laboratory Computer and Automation Research Institute

Hungarian Academy of Sciences H-1518 Budapest

Center for Engineering Mathematics University of Texas at Dallas

Richardson, TX 75083 U.S.A.

A survey will be given of statistical issues in system identification using balanced realizations. Special attention will be given to pathwise behaviours of prediction performance and stochastic complexity issues in general. The research is partly motivated by the need for high accuracy control in micro robotics.

l

Approximation by tranlsates

Federico Girosi

Artificial Intelligence Laboratory, M.LT. 545 Technology Square #788

Cambridge, Massachusetts, 02139 e-mail: girosi@ai.mit.edu

Abstract

We consider the problem of approximating a function belonging to some function space 4> by a linear combination of n translates of a given function G. This approximation technique seems to be very natural if the functions belonging to 4> have an integral representation of the convolution type. In fact, using a lemma by Jones (1990) and Barron (1991) we show that it is possible to define function spaces and functions G for which the rate of convergence to zero of the error is O(,in) in any number of dimensions. The apparent avoidance of the "curse of dimensionality" is due to the fact that these function spaces are more and more constrained as the dimension increases. Exam. pIes include spaces of the Sobolev type, in which the number of weak derivatives is required to be larger than the number of dimensions. We give results both for approximation in the L2 norm and in the Loo norm. The interesting feature of these results is that, thanks to the constructive nature of Jones' and Barron's lemma, an iterative proce. dure is defined that can achieve this rate. Similar results are obtained for approximation by translates and dilates of a given function.

123

124

ON THE ROLE OF FILTRATIONS IN THE

ALGEBRAIC THEORY OF LINEAR SYSTEMS

Steven 1. Giust , Bostwick F. Wyman

Mathematics Department The Ohio State University

Columbus, Ohio 43210, USA wyman@mps.ohio-state.edu

(614)292-4901

ABSTRACT

. A general transfer function, described by a possibly improper rectangular matrix of rational functions, gives rise to a global pole space. The global pole space is the direct sum of a polynomial module which describes the classical poles in the finite plane, and a module over the ring of proper rational functions which describes the pole structure at infinity. The global pole space itself carries no canonical module structure. On the other hand, it is isomorphic to the Wedderburn-Forney space of its "graph" in the product of the input and output spaces. This Wedderburn-Forney space carries two natural dual filtrations which correspond to global controllability and observability filtrations on the global pole space, and the systematic study of these filtrations provides a useful substitute for a global module structure. Additional filtrations for spaces of zeros and unreachable or unobservable improper systems will also be studied.

Constructive Algebra and Control Theory

S. T. Glad K. Forsman

Dept. of Electrical Engineering Linkoping University

S-581 83 Linkoping, Sweden

In many physical situations, a dynamic system is described by implicit differen­tial equations and/or static relations. If such a system is described by polynomial relations between the physical variables and their derivatives, then there is a de­scription in terms of so called autoreduced sets, which can be arrived at in a finite number of algebraic calculations. This description generalizes traditional state space descriptions.

In some situations the autoreduced sets can be shown to be characteristic sets that define prime differential ideals. From the characteristic sets it is possible to de­duce many properties like the differential transcendence degree (number of "degrees of freedom", number of "inputs"), the system order (number of "initial conditions"), observability etc. It is also possible to calculate explicitly nonlinear regulators that give a desired performance.

For systems in state space form, many of the problems addressed above in a differential algebraic setting can be recast in a commutative algebraic framework. This is particularly interesting for constructive issues, i.e. basically, elimination.

For many of the important problems in state space theory we can get a bound on the number of differentiations that have to be performed on the original equations before the problem can be considered as one of commutative algebra. This is the case for questions relating to observability and determining the external behavior of a system given in state space form.

From a practical point of view the consequence is that we can use Grabner bases instead of characteristic sets to do elimination.

125

126

Various topologies for spaces of linear systems

Heide Gliising-LiierBen Institut fiir Dynamische Systeme

U niversitat Bremen Postfach 330 440, D-2800 Bremen 33, Germany,

E-mail: heide@mathematik.uni-Bremen.de

Abstract

Via polynomial coprime factorizations and the Hermann/Martin map (cf. [4,2]) transfer matri­ces in IR( s )pxm can be viewed as rational maps from the Riemann sphere S2 into the Grassmannian 9m(Ck

) of m-dimensional subspaces in Ck, where k = p+ m. In this paper we will study the space

Ik,m of all such rational maps. This space can be viewed, in some sense, as the space of all ARMA-systems of fixed dimensions. It contains the proper and non-proper rational matrices. The Ik,m can be endowed with various topologies: uniform convergence on the full sphere or on some suitable subset, e.g. on the right half-plane C+. These topologies can be formulated via the gap-metric for finite-dimensional spaces. For the case of uniform convergence on the right half-plane this topology was recently studied by Qiu/Davison [5], they call it the pointwise gap-metric.

On the other hand, the space Ik,m can be equipped with the graph-or gap-topology as it was done for proper systems by Vidyasagar [6] resp. by Zames/EI-Sakkary [7]. For the space of rational matrices Qiu/Davision [5] recently proved the coincidence of graph-topology and topology of uniform convergence on C+ by using the fact, that both topologies are the weakest in which closed-loop stability is a robust property. The equality of these topologies (on Ik,m) can also be proven directly, see De Does et al. [1]. On the subset Ik,m of all systems having fixed McMillan degree r these topologies also coincide with the topology of uniform convergence on S2. Finally, there is no difference to the well-known proper case (Georgiou [3]) to show the uniform equivalence of the gap-metric (in the H2 (C+)-sense) and the graph-metric on Ik,m'

References

[1] J.De Does, H. Gliising-LiierBen, and J. M. Schumacher. some connectedness properties of spaces of linear systems. In Proceedings of the Symposium on Implicit and Nonlinear Control, Ft. Worth, 1992. to appear.

[2] F.De Mari and H. Gliising-LiierBen. The space of improper rational matrices and ARMA­systems of fixed McMillan degree. Report 267, Institut fiir Dynamische Systeme, Universiitat Bremen, 1992. submitted.

[3] T.T. Georgiou. On the computation of the gap metric. Syst. Contr. Let., 11:253-257, 1988.

[4] C. Martin and R. Hermann. Applications of algebraic geometry to systems theory: The McMillan degree and Kronecker indices of transfer functions as topological and holomorphic system invariants. SIAM J. Contr.& Opt.,16:743-755, 1978.

[5] L. Qiu and I.J. Davison. Pointwise gap metrics on transfer matrices. IEEE Trans. Auto. Cont., AC-37:741-758, 1992.

[6] M. Vidyasagar. Control System Synthesis: a Factorization Approach. MIT Press, Cambridge, MA,1985.

[7] G. Zames and A.K. EI-Sakkary. unstable systems and feedback: the gap metric. In Proc. 16th Allerton conf., pages 380-385, 1980.

1

pi

MINIMAL FACTORIZATION OF INFINITE MATRICES AND TIME-VARYING SYSTEMS

I. Gohberg Tel Aviv University

Ramat Aviv Tel Aviv 69989

Israel

M.A. Kaashoek Vrije Universiteit De Boelelaan 1081a 1081 HV Amsterdam

The Netherlands

~-"-_L~LIT Technion-Israel Institute

of Technology Haifa 32000

Israel

The present paper is concerned with the problem of minimal

factorization of a double-infinite lower triangular block matrix T

[tkJ~,i=-co (tki = 0, if k < i) with mkxPi matrix entries. We associate

with T its degree sequence {OJ.(T)}COj·=_co' where o.(T):= rank[t. . I . k]~ k I j j+l- ,j- I, =

and we say that T is of local finite type if ° .(T) < co, j E 71.. If T j

factorizes as T = T?z' where TI and Tz

are also double infinite lower

triangular matrices, we say that the factorization T TT 1 z is minimal if

° H) = ° H )+0 H), j E 71.. J J 1 J 2

The problem of minimal factorization of infinite lower triangular block

matrices is closely related to the problem of representing a time-varying

linear system as a minimal cascade connection of "simpler" systems. This

connection stems from the fact that any infinite lower triangular block

matrix T of local finite type can be realized as the input-output map of a

time-varying finite dimensional linear system.

In this paper we prove a geometrical criterion for minimal

factorization of infinite matrices of finite local type in terms of a

sequence of certain supporting projections. Particular attention is given

to the case of m-banded matrices T = [\J~,i=-co' i.e., such matrices that

tki 0 for < k-m, > k, where m is some positive integer. In

particular, we find some conditions and algorithms which allow factorizing

an m-banded infinite matrix T into a product of m I-banded matrices. In

the case when the matrix T is totally positive, factorizations of this type

appear to be important in approximation theory. We also note that in the

case when the given m-banded matrix is periodic, the above factorization is

closely related to factorization of certain matrix polynomials into a

product of linear factors of a special form.

127

128

Complexity of Structured Matrix Multiplication

I.Gohberg and V.OIshevsky School of Mathematical Sciences, Tel-Aviv University

Ramat-Aviv gohberg@math.tau.ac.il vadim@math.tau.ac.il

While the standard rule for multiplication of two arbitrary n x n matrices requires O(n3)

multiplications and additions, Strassen [S] showed that this problem can be solved in a fewer number O(n1og,7) of arithmetic operations. Subsequently this result was improved by different authors. In [G03] the authors showed that in the case when one of the matrices considered is arbitrary and the second is structured, the above estimates can be reduced to O(n2 log2 n) operations and in some cases to O(n2 log n) operations.

In the present report it will be shown that if both matrices involved are structured, no more than O(n 2

) operations are needed for the computation of all n2 entries of the product. Here the multiplication of Toeplitz, Vandermonde and Cauchy matrices is considered, as well as the multiplication of Toeplitz-related, Vandermonde-related and Cauchy-related ma­trices. The latter objects are defined using the concept of displacement of a matrix [KKM] [HR]. In the estimates of the complexity of matrix multiplication obtained here, the hid­den coefficient is proportional to the corresponding displacement ranks of both factors. The proposed methods for the multiplication of two structured matrices are based on the results from [GOI] concerning the circulant displacement, on the fast algorithms from [G02] and [G03] for multiplication of structured matrices with vectors, and on an auxiliary proposition about the displacement rank of a product, similarly to one proposed in [CKL].

References

[CKL] Chun J., KaiIath T. and Lev-Ari H., Fast parallel algorithms for QR and triangular factorization, SIAM J. Stat. Comput.,8(No. 6) (1987),899 -913.

[GO 1] Gohberg I. and Olshevsky V., Circulants, displacements and decompositions of ma­trices, Integral Equations and Operator Theory, 15 (No.5) (1992),730 -743.

[G02] Gohberg I. and Olshevsky V., Fast algorithms with preprocessing for multiplication of transpose Vandermonde matrix and Cauchy matrix with vectors, submitted.

[G03] Gohberg I. and Olshevsky V., Complexity of multiplication with vectors for structured matrices, submitted.

[HR] Heinig G., Rost K., Algebraic methods for Toeplitz-like matrices and operators, Oper­ator Theory, vol. 13, Birkauser, Basel, 1984.

[KKM] T.Kailath, S.Kung and M.Morf, Displacement ranks of matrices and linear equa­tions, J. Math. Anal. and Appl., 68 (1979), 395-407.

[S] Strassen V., Gaussian elimination is not optimal, Numer. Math., 13 (1969), 354 -356.

On reduced order controllers

Andrea Gombani LADSEB-CNR

Corso Stati Uniti 4 35020 Padova, ITALY

gombani@ladseb.pd.cnr.it

Abstract

We investigate here how some results appeared in Fuhrmann [1] and Fuhrmann and Ober [2] can be used to obtain reduced order con· trollers. The idea is to apply the principle of Nehari extension, which generates optimally robust stabilizing controllers, to lower degree con· trollers obtained using extensions associated to different Schmidt pairs. Since the singular values and Schmidt vectors of the Hankel operator with symbol a normalized coprime factorization of a plant can be given an explicit representation in terms of the plant, of its optimally robust controller and of the Schmidt pairs of another scalar Hankel operator, the whole problem can be reduced to the study of the Schmidt pairs of a scalar Hankel operator.

References

[1] Fuhrmann, P.A. (1991) A Polynomial Approach to Hankel Norm Bal­anced Approximations. Linear Algebra and its Applications 146, pp.

133-220.

[2] P.A.Fuhrmann, R.Ober, A functional approach to LQG balancing, to

appear.

129

130

GENERALIZED BEZOUTIANS FOR ANALYTIC OPERATOR

FUNCTIONS AND INVERSION OF STRUCTURED OPERATORS

G. Gomez and L. Lerer

Technion-Israel Institute of Technology

Haifa 32000, Israel

In this paper we introduce a notion of a generalized Bezoutian for a

family of operator functions and we develop a general inversion scheme for

certain classes of infinite dimensional operators. This scheme provides

formulas for inversion of operators in terms of solutions of a finite

number of vector equations. A basic result of this type is the well known

Gohberg-Semencul (and Gohberg-Heining) formula for inversion of an integral

operator with a difference (matrix valued) kernel on a finite interval.

Other results of this nature were obtained by L. Sakhnovich; T. Kailath,

L. Ljung and M. Morf; G. Heining and K. Rost; B. Kon for inversion of

operators whose structure is close in some sense to the displacement

structure of the integral operator with a difference kernel. We develop a

new approach to the inversion problem which is based on the concept of a

generalized Bezoutian for a family of analytic operator functions given in

realized form and on identification of solutions of operator Sylvester

equations as such Bezoutians. Our approach provides an effective tool for

developing new inversion formulas and adequate interpretation of the known

ones. The finite dimensional predecessor of this approach has been

developed by L. Lerer and M. Tismenetsky.

r

SOlliE SL,C~;S OF TES'l' ImD G.r.;If.sR.ALI~.r.;D V',,:C'fORS 01<' A CLOSED

OP.c:fuI.TOR AND THEIR AI'PLICA1' IONS TO INWSTIGA2.'ION OF SYS­

'l'ElIIS OF DIFFERENTIAL E,tUATIONS

Gorbachuk M. and Gorbachuk V.

Institute of Mathematics Ukrainian Academy of Sci.

131

Repina str. 3 ,Kiev 252601, Ukraine; e-mail: mathem @ sovamsu.

savusa.com.

Our aim is to give a brief exposition of the principal lllomrnts

of the test and generalized vectors theory of a closed opera _

tor on a Banach space and to show S01;1e of its applications.

In particulur we investigate a general form of solutions of

differential equutions with operator coe.fficients which are

smooth inside the interval,boundary values of such solutions

at the ends of the interval and their behavior near il~inity

in the Case of asymptotic, not eXllOnential, stabili ty.

132

CONTROLLABILITY PROPERTIES OF SINGULARLY PERTURBED CONTROL

SYSTEMS

G6tz Gramme! Institut fur Mathematik cler Universitiit Augsburg

Universitiitsstr. 8, D-8900 Augsburg, Germany Telefon (821) 598-2190, Telefax (821) 598-2200

e-mail: grammel@Uni-Augsburg.DE

Abstract

We are interested in singularly perturbed control systems of the form

;,(t) f(x(t),y(t)) + L f;(x(t), y(t))u;(t) i=1

'li(t) g(x(t), y(t)) + Lg;(x(t),y(t))u;(t) i=l

where e is a small parameter and u a control function. So we have a slow motion x(t) and a fast motion y(t). We want to describe the behaviour of the slow motion when the small parameter tends to zero. It turns out that it is possible to approximate the slow motions corresponding to given initial conditions uniformly on compact time intervals by solutions of an autonomous limit differential inclusion under controllability conditions on the fast motion. The right hand side of this differential inclusion is con­structed by an averaging method.

Furthermore we define regions of controllability for the slow motion as projection of so called control sets of the whole singularly perturbed control system and in a similar way regions of (chain-) controllability for the differential inclusion. We show that there are connections between these regions of controllability when the small parameter tends to zero. In particular limit points of projected control sets lie in regions of chain­controllability for the limit differential inclusion.

133

Robust ripple-free tracking and regulation for sampled-data systems under structured parameter variations*

Abstract

Osvaldo Maria Grasselli, Antonio Tornambe Dipartimento di Ingegneria Elettronica, Universita di Rama "Tor Vergata"

via della Ricerca Scientifica, 00133 Roma, Italy e-mail tornambe@tovvxl.ccd.utovrm.it

Sauro Longhi Dipartimento di Elettronica e Automatica, Universita di Ancona

via Breece Bianche, 60131 Ancona, Italy e-mail longhi@anvax2.cineca.it

The problem of the robust asymptotic tracking and disturbance rejection of a linear multi variable system subject to unmeasurable disturbances was studied by many authors. In most of these contributions the kind of parameter uncertainties that were taken into account consists of small - or, possibly, large - independent perturbations of all the elements of matrices describing the system: it is required the compensator to maintain stability, asymptotic tracking and output regulation in spite of them. Recently, the problem of the asymptotic tracking and output regulation under uncertainties or perturbations of "physical" parameters affecting the description of the system was solved [1,2J; it was shown that robust solutions may exist even when no solution exists for wholly independent variations of the entries of matrices describing the system, i.e. when the Davison condition [3J is not satisfied.

If the problem of the asymptotic tracking is faced for a co"'.'nuous-time plant making use of a digital control system, the undesiderable ripple which may arise between sampling instants can be unacceptable if the sampling time is large, and/or if a deadbeat error response is required at the nominal parameters, and should be avoided. It is known that this can be obtained if a continuous-time internal model of reference signals is included in the forward path of the feedback control system [4,5J; however, these papers give solutions of such a problem under the implicit assumption of independent perturbat ions of the elements of the matrices describing the plant to be controlled, hence under the Davison condition.

Here a method for obtaining such a continuous-time internal model for both reference signals and disturbance functions is derived from [1,2J, for the case when the only uncertainties about the description of the plant concern the values of some "physical" parameters, each of which possibly affects all the elements of the matrices appearing in the state-space description of the plant. Such a method allows the control requirements to be robustly satisfied, at least in a neighbourhood of the nominal "physical" parameters of the plant to be controlled, and, in particular, a continuous-time null steady-state error response to be g"~rallteed for all the values of the physical parameters in such a neighbourhood of the nominal ones. This can be ou,ained under conditions which are weaker than the Davison one [3), and are expressed in terms of the description of t.he original continuous-time plant.

References

[IJ O. M. Grasselli, S. Longhi, and A. Tornambe, "Robust tracking and performance for multivariable systems under physical parameter uncertainties," Autolllalica, vol. 29, no. I, pp. 169-179, 1993.

[2J O. M. Grasselli and S. Longhi, "Robust output regulation under uncertainties of physical parameters," Systems and Control Letters, vol. 16, pp. 33-40, 1991.

[3] E. J. Davison, "The robust control of a servomechallism problem for linear time- invariant multivariable sys­tems," IEEE Trans. Aut. Call 1"01, vol. AC-21, PI'. 25-34, 1976.

[4] R. Doraiswami, "Robust cont.rol st.rategy for a linear time-invariant. multi variable sampled-data servomechanism problem," lEE Proceedillgs, PI. D, vol. 129, pp. 283-292,1982.

[5J G. F. Franklin and A. Emami-Naeini, "Design of ripple-free multivariable robust servomechanism," IEEE Trans. Automatic Control, voL AC-31, no. 7, pp. 661-664,1986.

·This work was supported by Ministel'o Pubblica Isll'llzionc, Ministem Universita. Ricerca Scientifica Tecnologica and Consiglio N azionale Ricerche.

134

Parametric Sensitivity of Nonlinear State Space Realizations

W. Steven Gray

Department of Electrical and Computer Engineering

Drexel University

Philadelphia, PA 19104 email: steven...gray41ece. drexel. edu

Erik 1. Verriest School of Electrical Engineering

Georgia Institute of Technology

Atlanta, GA 30332

email: erik. verriest41ee. gatech. edu

Abstract

Given an input-output mapping whose formal power series yields a system Hankel

matrix of finite Lie rank, PL, it is well established that an affine nonlinear state-space

realization of dimension n = PL must exist for the system. Furthermore, any two such

realizations are known to be locally diffeomorphic [1). The objective of this paper is to

explore the problem of synthesizing minimal realizations of a given input-output system

whose parametric sensitivity is minimized with respect to a naturally defined sensitivity

measure. This is commensurate to finding a nonlinear coordinate transformation that in

some sense optimally scales a known parametric representation of the internal dynamics.

The problem is solved in a geometric context where the set of state space realizations can be

given the structure of a Riemannian differential manifold and the sensitivity measure and

its gradient are defined in terms of the metric and the Riemannian connection compatible

with the metric.

This paper presents the first extension of the sensitivity theory recently developed for

linear time-invariant systems in [2).

[1) A. Isidori, Nonlinear Control Systems, Berlin: Springer-Verlag, 1989.

[2) E. 1. Verriest and W. S. Gray, 'A Geometric Approach to the Minimum Sensitivity

Design Problem,' SIAM Journal on Control and Optimization, under review.

Differential Elimination and Taylor Series

Thomas Grill

NWF-I Mathematik

Universitiit Regensburg

Universitiitsstr. 31

D-93053 Regensburg, Germany

135

In this lecture a comparison is made between solution sets of systems of algebraic (partial)

differential equations and associated algebraic sets. Elements of those algebraic sets are

refered to as quasi-solutions.

For a coherent system of algebraic differential equations it is shown that every quasi­

solution together with a fixed initial condit on determines a unique solution of it. Further­

more the coherence property of the system allows one to calculate explicitly the formal

Taylor series expansion of a solution associated to a pair of quasi-solution and initial con­

dition.

The cohere~nce property of a system can be tested by a slight generalisation of Ritt's method

reducing differential polynomials (see below).

The problem now is to obtain in a effective way from a given system of differential equa­

tions a coherent one with the same set of solutions.

In general by Seidenberg's elimination method every system can be decomposed in a equi­

valent set of coherent systems of differential equations and differential inequalities.

However for some special kinds of systems it is not necessary to do all the computations

needed in the Seidenberg elimination. For this an analysis of the reduction method used

by Ritt and Kolchin leads to a method to calculate from a given arbitrary system a co­

herent system without changing the differential ideal given by the differential polynomials

corresponding to the system started with. For example in this calculations reduction steps

which have to be made for computing characteristic sets are not necessary.

The method is obtained by transfering the notions of reduction relation and subtraction po­

lynomial, known from the theory of Grabner bases, in a appropriate way to the differential

context and then applying Buchbergers algorithm.

Parameter Identification of Nonlinear Mechanical Multibody Systems in Descriptor Form

Abstract

Friedemann Grupp * DLR German Aerospace Research Establishment

Institute for Robotics and System Dynamics

D-8031 Oberpfaffenhofen Germany

Phone: (08153) 28 429 Fax: (08153) 28 1441

E-Mail: df21@vm.op.dlr.de

August 14, 1992

An off-line identification method is presented for estimating unknown system parame­ters in mechanical multibody systems. The method is based on the minimization of a cost-function with respect to the unknown parameters. Under several assumptions this approach represents a maximum likelihood estimation.

Mechanical systems can be given in descriptor form, i.e. in the form of differential­algebraic-equations (DAE). For a given multibody model structure, the task is, to es­timate the unknown system parameters from noisy, time-discret measurements. Let the measurement noise be white, gaussian and with zero mean and unknown, time-in­dependent covariance. Often the initial conditions are unknown too. Therefore, they have to be estimated together with the parameters and the unknown covariance. The differ­ence from the model output and the measurements defines the output-error. Depending Dn the output-error and the covariance a cost-function is formulated. Then the parameter identification problem is as follows:

Determine the parameters (initial values and covariance, respectively) in such a way, that the cost-function is minimized with respect to the parameters and under the condition that the solution fulfills the differential-algebraic system equations.

Por solving this problem a numerical method will be presented. It makes use of the struc­ture of the multibody system equations in descriptor form. The approach will be compared with other methods and it will be demonstrated for problems in vehicle dynamics.

"This work is supported by the Volkswagen-Stiftung within the general research program "Fachiiber­greifende Gemeinschaftsprojekte in den Ingenieurwissenschaften".

Abstract

FRISCH SCHEME IDENTIFICATION:

RESULTS AND OPEN PROBLEMS

R. Guidorzi University of Bologna, Italy

fax: +3951 6443073, e-mail: guidorzi@ingbo1.cineca.it

137

While the introduction of the Frisch scheme for the estimation of the parameters of a linear model dates back to 1934, its application to real problems and its presence in the area of scientific research have been modest for many years despite the advanced conceptual framework under which it has been formulated and its potential advantages over some of the more diffused schemes.

One of the reasons for the modest interest in the Frisch scheme is probably due the generality of its assumptions (lack of "prejudices") which has a counterpart in the fact that it leads to a whole family o( models compatible with the available (noisy) data. Another reason could be found in the marly theoretical questions opened by the Frisch scheme, some of them still waiting for a solution.

More recently, the analysis of R.E. Kalman on the implicit assumptions behind many commonly used estimation schemes and the interest in the sector of Errors-in-Variables schemes have conveyed new attention and new researches on the Frisch scheme. This has led to an understanding of some complex algebraic problems connected to its use and to its extension to the identification of dynamic systems.

The first part of this presentation describes some of these recent results and, particularly, those regarding the identification of dynamic systems where the Frisch scheme leads, differ­ently from the algebraic case, to a single model which describes exactly the additive noises on the inputs and outputs and the system behind the data. The second part deals with the problems connected with the application of the Frisch scheme to real processes. In this case the (few) assumptions behind the scheme are violated and the (nice) theoretical results re­garding model uniqueness are no longer valid so that consistent criteria must be introduced to select, in a whole family of possible models, the single model leading to the best description of the noisy data in terms of deterministic sequences and of additive white noises.

Some final considerations will regard a short description of the results given by the Frisch scheme in the identification of a dynamic process, a comparison of the Frisch scheme with Total Least Squares and with the Output Error method on a simulated system with levels of additive noise up to 100% and the congruence between identified models and Kalman filtering.

This research has been supported by the Ministry for University and Scientific and Technologic Research, Rome, under project Model Identification, Control Systems and Signal Processing.

138

Abstract

ON A MODEL REDUCTION PROCEDURE

BASED ON THE FRISCH SCHEME

R. Guidorzi University of Bologna, Italy

fax: +39516443073, e-mail: guidorzi@ingbol.cineca.it

A. Stoian Polytechnic Institute of Bucharest, Roumania

Model order reduction can be seen as the substitution of a known model of a dynamical system with a simpler model preserving some relevant characteristics of the original one. This operation can be performed with reference to many different aspects of the model behaviour (that can not always be easily quantified) and this is one of the reasons for the presence, in the literature, of several model reduction techniques whose philosophy and applicability may be oriented to specific applications.

The Frisch scheme, in dynamic system identification, assumes that the available in­put/output sequences are corrupted by unknown amounts of additive noise and allows to compute the admissible points in the noise space for every assumed model order. Such points lie on curves (for SISO systems) which have, once the correct order is reached, a common point corresponding to the exact variances of the noises and to the exact model of the process.

The procedure proposed in this paper for model order reduction uses the same tools as Frisch identification procedures but in a different context. Using the (exactly known) model of the system to be reduced and an (exactly known) input sequence (e.g. white noise) the corresponding output sequence is generated. The curves in the noise space associated to

models with increasing order under the Frisch assumptions are then computed. Since no additive noise is present on the input/output sequences the curve corresponding to the exact model order will collapse onto the origin of the noise space while the previous curves will describe in terms of additive input and output white noises the data/model mismatch due to the reduction in the model orders. The distance of such curves from the origin can be considered as a measure of the approximation associated to the corresponding model while the distance between two different curves can be considered as a measure of the difference between the performances of the associated models.

This procedure has been compared with the well known Model Balancing method and with a Prediction Error Method on four order 5 systems with different pole and zero patterns. The results of these simulations show how different criteria can lead to scattered performances in model order reduction problems and how such performances can be influenced by the position of the poles of the system to be reduced. The proposed procedure has shown, in some cases, a very good performance and can thus be, thanks also to its flexibility, a further choice in a field where, apparently, no procedure can be considered as clearly superior to all others.

This research has been supported by the Ministry for University and Scientific and Technologic Research, Rome, under project Model Identification, Control Systems alld Signal Processing.

Decomposition of Skew Symmetric Matrices, Spectral Inequalities and Nonholonomic

Systems Leonid Gun'its

Siemens Corporate Research, 755 College Rd. East, Princeton, :-;J 08540

Let us consider the follmving problem. PI. ~Iotion planning for nonholonomic systems with drift. For a given system x = A(X)U, construct a control U(t), (0 :S t :S 1) such that x(O) =;", x(l) = f. Also \\'e consider asymptotic time period feedback. In our consideration) surprisingly there is no essential difference between open-loop and feed back solutions. P2. Optimal decomposition of ske\v symmetric matrix. For a given skew symmetric matrix 5 and positive numbers (Co, ... , Cn, ... J, to solve the following optimization problem.

00

5= LXil\Yi i=l

L C,(II Xi II' + II y, II') - inf(min) 1=1

P3. For a given skew symmetric matrix 5 with elements belonging to a com­mutative algebra to decompose 5 = L;=1 Xi 1\ Yi such that k is minimal. P4. Let us define the following function for a finite square matrix A.

I"c(A) = (L Ci ),;) 1/2 .=1

where C, > 0, )', 2: )'" ... ,2: ),n are singular values of A. When inequality I"c(A + B) :S I"c(A) + I"c(B) holds. In this paper we will explain how these problems are interconnected. Below we Just give some examples to illustrate the central idea. Ex I. Using optimal control arguments we prove subadditivity in (P4) for C = (1,2, ... , n, -dots) and C = (1,3,5, .. '). Ex 2. Using (P2) we give simple analytical solution for the following problem of optimal control. x = u, S = x 1\ U, X E ~N, s = -s·, s(27r) - s(O) = fl, x(O) = 0 and (fa" < u, U > dt) - min. Here we do not specify the value of x(27r).

139

How Finite Automata can check Absolute Stability

Leonid Gurvits Siemens Corporate Research,

;.).) College Rd. East. Princeton, :'IJ 08540

Let us consider a finite set of linear operators (square matrices) X = {At, ... , Am}, At : ~N ----0 :}tN, X- 1 = {All, ,..1;;/} (if corresponding inverse exists).

For a word", = (it,. "Ie) we define Aw = A" .. {i,_, ... "Ai" 1",1 = [{. The following quantities play crucial role in this paper:

"'k(X) = maxlwl=K IIAw 11,3dX) = minlwl=K II Aw II. ,dX) = maxlwl=tp(Aw ), I'dX) = minlwl=tp(.4.w)

Here II II is some operator norm induced by a norm II II in !J(N, p(.) is a spectral radius.

The following algorithmic problems arise in many areas such as control the­ory (absolute and stochasttc stability), wavelets, nonstationary Markov chains, numerical methods:

To find out an effective algOrithm to check whether or not for some integer k:

(PI) "'k < I; (P!) 13k < I Theorem 1. limk_= (/odX) = infk<= \/ilk(X-t)

This theorem does not hold in infinite dimensional case. Below we assume without loss of generality that if A E X then also -A E X. Theorem 2. "'k(X) < I for some k iff the following two conditions hold. I) there exists I' > 0 such that eO(Aw , 1",1 $ 1') = eO(Aw , 1",1 $ I' + I). (Here eO(·) is a convex hull.) 1) p(Aw) < I for all w such that Iwl $ F(I', m, S), where F is well defined and very rapidly grO\ving functions.

:\,ow let us consider the case of a bounded semigroup: there exists norm such that II.4i 11$ I for aliA, E X. Theorem 3. There is a polynomial (P(mN')) algorithm to check whether or not infk ilk < 1.

Let us now introduce a language L(X.II . II) = {w :11 Aw 11= I}. The crucial observation which leads eventually to theorem 2 is that for some good norms (polytope, Hilbert, .. ) this language is regular. In view of this fact to solve (PI) one has to check whether or not the language L(X.II II) is finite. It is obvious that both problems (PI) and (P2) are nontrivial because of noncommutativity. Since it is quite natural to consider intermediate nilpotent case: Ai = exp(B;) and Liealgebra generated by B, is nilpotent. In this case there is an effective algorithm to solve (PI). Theorem 4. In a nilpotend case "'k(X) < 1 for some k iff there exists positive definite P > 0 such that A: PA, < P for all Ai E .\'.

A Reliable Stability Test for Exponential Polynomials *

Luc Habets Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513

NL-5600 MB Eindhoven The Netherlands

Email: luch@bs.win.tue.nl

Abstract

Exponential polynomials (or quasi-polynomials) are analytic functions of the form n-1

I(z) = zn + L:: Pi(e- T", ••• , e- T

"). zi i;:;Q

(z E C),

where for all i E {O, 1, ... , n - I} the function Pi(e- rlz , ... ) e-"'/cZ) is a polynomial in the variables e- T

", ••• , e- T", with real coefficients, and all Tj > 0 (j = 1, ... , k). An

exponential polynomial is called stable if it has no zeros in C+, the closed right half plane. In this way exponential polynomials are useful to test stabilizability of linear systems with time-delays, and for the design of stabilizing compensators for such systems.

Although exact analytic conditions to test the stability of exponential polynomials are known, they are often too difficult to check, certainly in concrete problems. Therefore mostly a graphical test is used, based on the well-known circle-criterion, With help of the image under the exponential function I of the imaginary axis, the number of RHP-zeros of f can be determined. However, in the actual application of this method, one encounters several practical problems.

In this talk we present a concrete algorithmization of the stability test using the circle­criterion. A stop-criterion is introduced that indicates that one only has to search along a finite part of the positive imaginary axis. This yields enough information to decide on the stability question. Moreover, for the search along this part of the imaginary axis, a method is given to compute a (variable) step-length. This criterion is based on the curvature of the image of the imaginary axis under the exponential polynomial I considered in the complex plane, and assures that small steps are taken when necessary, and larger steps when this is possible. In comparison with ordinary linear search, our stability-test is much more reliable. However, the computational costs of these two methods are almost the same.

Although our method can be applied to low order exponential polynomials, it is espe­cially suitable to test the stability of high order exponential polynomials. These are often needed for the stabilization of time-delay systems. The method proposed in this talk is designed to carry out this test in a reliable and efficient way.

-Research supported by the Netherlands Organization for Scientific Research (N\VO)

141

142

Testing Reachability and Stabilizability of Systems over Polynomial Rings using Grabner Bases*

Luc Habets Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513

NL-5600 MB Eindhoven The Netherlands

Abstract

Linear systems over polynomial rings can be seen as a rather straightforward general­ization of ordinary systems over fields. They are useful for the modeling of several classes of systems, for example systems with varying parameters and time-delay systems.

From the control point of view, the properties of reachability and stabilizability are very important for systems over polynomial rings. Several interesting control problems, e.g. pole-placement and input-output decoupling, have (partly) been solved under the assumption that the system under consideration is reachable. The importance of stabiliz­ability is self-evident. In the algebraic setting, stability is defined with help of a so-called Hurwitz-set, which describes the set of all stable systems. In a particular application, the Hurwitz-set is defined in such a way that it matches our intuitive notion on stability.

Conditions for reachability and stabilizability are well-known in the literature. For several interesting applications, especially for time-delay systems, these properties can be described by a rank condition on the system defining matrices. In this way, these conditions have very much in common with the well-known PBII-test, but unfortunately they are still very difficult to check, in general.

In this talk we explain how the rank conditions for reachability and stabilizability can be reformulated into terms of polynomial ideals. This can be done in several different ways. But in either case, the Grabner Basis technique from constructive commutative algebra makes it possible to determine these polynomial ideals explicitly. Once this is done, conclusions on the reachability and stabilizability of the system under consideration are easily derived.

Each method proposed in this talk has its own merits: the computational efficiency depends on the application in mind. Moreover, one of the methods has a very important by-product: it can be used to obtain a right- or left-inverse of an arbitrary non-square polynomial matrix. This is very interesting from the control point of view because such in­verses are often needed in the design of feedback compensators. Based on these properties, the performances of the proposed methods will be compared.

-Research supported by the Netherlands Organization for Scientific Research (NWO)

CONTROL OF DYNAMICAL SYSTEMS NEAR BIFURCATION POINTS

Gerhard Hackl' Institut fur Mathematik

Universitat Augsburg Universitatsstr. 8

8900 Augsburg Germany

Klaus Sc:hneidert

Institut fur Angewandte Analysis und Stoc:hastik

Hausvogteiplatz 5-7 0-1086 Berlin

Germany

Abstract

We study the controllability properties of parameter dependent control affine systems with constrained control values.

If we assume an accessibility condition, then control sets (i.e. re­gions of complete controllability) will appear around the limit sets of the uncontrolled system.

The bifurcation behaviour of the uncontrolled system causes a bi­furcation of the control sets, where the control range appears as an additional bifurcation parameter.

We investigate theoretically and numerically the situation of a con­trolled Takens-Bogdanov-Singularity. In this case the homoclinic bi­furcation of the uncontrolled system causes a global bifurcation of control sets .

• e-mail: Haeckl@Uni-Augsburg.de te-mail: Schneider@IAAS-Berlill.dbp.de

143

Nonlinear Controller Design of a Spatial Servohydraulic Multi-Axis Test Facility based on Symbolic Computer Languages

H. Hahn, K.-D. Leimbach

[n.titut fUr Mep- und Automati.ierun9.technik - Univer6itcit Ka .. el - Ge6amthoch.chule, Moncheberg.trape 7, D-3500 Ka .. el, Germany, Tel: (561) 804 !768, Faz: (561) 804 !330

Industrial vibration tests of e.g. space structures or structures and components of power plants and nuclear power plants are performed by spatial multi-axis test facilities. In order to accomplish the test requirements concerning different kinds of test specimen and high frequency ranges up to 200 Hz sophisticated control concepts have to be implemented [1]. However, such a complex system is of high order and strongly nonlinear in both, the mechanical equations of motion and the equations describing the physics of the servohydraulic actuators. These system properties lead to well known unwanted physical effects like overturning moments and off-center loads. An overall contr0l strategy has to take into account all these phenomena.

The nonlinear control theories developped during the past decades provide an access to deal with such systems [2]. In case of the system considered the exact state linearization technique is applied to derive suitable controllers. This approach is based on sophisticated mathematical component models of the test facility including \

- detailed models of the servohydraulic actuators, - detailed models of the servovalve dynamics and - detailed spatial models of the rigid test table and rigid payload.

The strategy for deriving the nonlinear controller includes the following steps: - test of exact linearization, - derivation of a nonlinear diffeomorphism and its inverse, - transformation of the system into a nonlinear normal form and - nonlinear controller design.

Both, the detailed nonlinear system modeling and the nonlinear control system design make extended use of symbolic computer languages (as e.g. MACSYMA, ALTRAN, REDUCE) [3]. The calculated symbolic control algorithms are extrem lengthy and unwieldy. They have been suitiably represented in order to provide insight into the system and have been condensed by substitutions to be suitable for online implementations of industrial practice. Simulation results of a controlled nonlinear multi-axis servohydraulic test facility controlled by transient signals from industrial practice show excellent system performance concerning both, the nonlinear decoupling behaviour and the tracking behaviour.

References

[1] Hahn, H. and Raasch, W. (1986). Multi-Axis Vibration Test On Spacecraft Using Hydraulic Exciters. AGARD Conference Proceedings No.397, pp. 23-1 to 23-23.

[2] Isidori, A. (1989). Nonlinear Control Systems: An Introduction. Springer Verlag, pp. 234-288.

[3] Birk, J., Zeitz, M. (1989). Computer-Aided Design of Nonlinear Observers. IFAC Symposium Nonlinear Control Systems, Capri/Italy.

ROBUST CONTROLLERS FOR THE TIME-VARYING SYSTEMS

Professor Aristide HALANAY University of Bucharest, Mathematics Department

14 Academiei str., 70109 Bucharest, ROMANIA

Let A, B, C be a time varying plant, with (A, B) uniformly stabilizable and (C,A) uniformly detectable. The construction of Glover and MacFarlane of a stabilizing controller, suboptimally robust with respect to the perturbations in the normalized coprime factorization are carried out for this case. The subop­timal solution of the time-varying Nehari problem is used.

The final result is expressed in terms of the bounded on the real axis stabi­lizing solutions to the Riccati equations

With

R' + A'(t)R + RA(t) - RB(t)B'(t)R + C'(t)C(t) = 0

S' = A(t)S + A'S - SC'(t)C(t)S + B(t)B·(t).

l' = sup p(S(t)R(t», , (p(M) is the spectral radius of M),

l' > 1/v'i+V,a2 = 1- 1;..,,2,

a suboptimal robust controller is

x' {A(t) + B(t)B'(t)R(t) - S(t)[0-2 1- 1;..,,2 R(t)S(tW1C'(t)C(t)}x -S(t)[0-2 1- 1;..,,2 R(t)S(t)t1C'(t)u

Y = B'(t)R(t)x

For antistable plants a suboptimal robust controller is proposed for additive perturbations, by using solutions to Lyapunov equations.

145

f6

SPECTRAL TRANSFORMS & PROBABILITY PROPAGATION FOR DIGITAL SYSTEMS.

A. G. HALL Department of Electronic Engineering, University of Hull, HULL, HU6 7RX, U.K.

Tel. (0)482465891, Fax. (0)482466666; email AG.Hall@uk.ac.huILee

Acknowledgement: this work was supported by SERC grant GRiG 19336

Keywords: digital systems, probability, spectral transforms, VLSI design, testability, (DSP, information theory).

Continuous aspects of probability are integrated with discrete aspects of digital logic in a standard matrix algebra to produce an explicit mathematical representation of the digital transfer action, vis:

q =T.p

q and p are vectors of the probabilities of complete digital patterns on the input and output respectively of the digital system. T is a digital transfer matrix, whose sparse representation is given by the truth table of the system. For definite logic the probability of an input pattern is set to I.

Complete Hadamard-Walsh spectral transformation of this transfer equation gives

IjI = S.Ql where IjI = H.q <p = H.p S = H.T.H-1

H is the hadamard matrix. Components <p[a], \jf[13] of the spectral vectors are given by the difference between the probabilities of exor combinations on groups of individual variables (or nets in a circuit) indicated by the indices a, 13 E.g.

<p[a] =p(xif' = 0) -p(xif' = I)

a= ijk

The spectral line-set index a is equivalent to a binary string, pattern or number, which has individual variables Xi, X;· . ·Xk set to value 1 and all others O. These indices involve groups

of all numbers of variables (or nets) from zero upwards. Thus the spectral transfer equation gives a hierarchical expansion in terms of probabilities on correlated sets of variables (nets) This will allow expansion up to the required degree of correlation, and controlled approximation. Full dimensionality is only implicit in the spectral form, and this opens the possibility of extension to sequential systems.

Block analysis via subsystems is possible. Serial connection is by multiplication of the subsystem transfer matrices. Parallel combinations are given by the logical convolution

S[l3I132Ia] = LSI[13llaEBy]S2[132Iy] y

Thus the T and S transfer matrices perform a similar role in digital theory to the response function - transfer function pair in analogue systems.

These techniques are applicable to a wide range of digital system theory, especially VLSI design, and testability. They are also of interest to DSP of stochastic signals, and information theory and coding.

T !

On the Lower Bounds of the Uniformly Optimal Sensitivity with Robust Stability

for Nonlinear Time-Varying Controllers

Kazuyasu Hamada*, and Masayuki Suzuki**

* Faculty of Engineering, Gifu University Yanagido 1-1 Gifu Gifu 501-11, Japan

** Dep. of Aeronautical Engineering, Nagoya University Furou-Chou, Chikusa-ku Nagoya 464, Japan

Abstract: It is shown under some weak assumptions in the problem of uniformly (or Hoo-) optimal control of a linear time-invariant

continuous-time plant maintaining the robust stability for its unstructured families (i. e., the 2-disk problem when we use linear, time-invariant controllers only), there exists a non-zero lower bound for the optimal sensitivity and arbitrary nonlinear, time­-varying controllers can not reduce the sensitivity beyond this bound without destroying the robust stability. This means that the adaptive control technique can not achieve the tracking error zero without violating its robust stability for the unstructured un­certainty.

Key words: Adaptive control, Robust control, Hoo-control theory.

147

Overlapping block-balanced canonical forms and parametrizations in the multivariable case.

Bernard Hanzon Dept. Econometrics

Free University Amsterdam De Boelelaan 1105

1081 HV Amsterdam Fax +31-20-6461449 E-mail: bhnz@sara.nl

Abstract

Ie have investigated the problem of how to construct an atlas of overlapping canonical 'rms that is an extension of a balanced canonical form (which is not overlapping). The ,able SISO case was treated in [2]. Here we consider the multivariable case. In the SISO lSe we were able to find an atlas which was a g [1] In the multivariable case we first ~consider the problem of d~fining a balanced canonical form fro stable systems and we )m all-pass systems of fixed degree. Remarks will be made about how one can construct a atlas of overlapping canonical forms for several other classes of systems.

teferences

:1] R. Ober, Balanced realizations: canonical form, parametrization, model reduction, Int. Journal of Control 46. No.2 (1987), pp.643-670.

:2] Bernard Hanzon and Raimund J. Ober, OverlaPEin!l block-balanced canonical forms and parametrizations: the stable SISO case, CDC, Tucson, Arizona, Dec. 1992 and submitted to IEEE Trans. Aut.Control.

:3] Bernard Hanzon, Overlapping Balanced Canonical Forms for Stable Multivariable AI/­Pass Systems in: P. Dewilde, M. Kaashoek, M. Verhaegen, "Cl,allenges of a Gener­alized System Theory", Essays of the Royal Dutch Academy of Arts and Sciences, Amsterdam, 1993 (forthcoming).

,

l

Chaotic Synchronization Techniques appplied to Modulation and Demodulation of Chaotic Carriers

Martin Hasler', Michael Peter Kennedy $, Herve Dedieu •

Abstract

*Department of Electrical Engineering, Swiss Federal Institute of Technology, CH 1015 Lausanne, SWITZERLAND

$ Department of Electronic and Electrical Engineering, University College Dublin,

Dublin 4, IRELAND

149

The design of auto-synchronizing systems driven by chaotic signals has been addressed in very recent studies. As a consequence of these fundamental studies, new strategies for the trans­mission of digital signals by means of the modulation/demodulation of analog chaotic carriers have been proposed. However, although circuits have been built to demonstrate the efficiency and robustness of such strategies, these circuits operate in quasi optimal conditions and there is a need to know the influence of disturbing conditions. This paper reviews some of the existing techniques and circuits for the modulation/ demodulation of a chaotic carrier and deals with the problem of the robustsness of the synchronization. We analyze the sensitivity of the different techniques to imperfections of the components values and distorsion effects (filtering, loss) due to the transmission channel. Simulation results as well as experimental results are presented.

RECURSIVE LINEAR ESTIMATION IN KREIN SPACES WITH APPLICATIONS TO Hoc PROBLEMS

BABAK HASSIBI , ALI H. SAYED, AND THOMAS KAILATH

Information Systems Laboratory, Stanford University, Stanford, CA 94305

e-mail: hassibi@rascals.stanford.edu, phone [415] 723-1538, FAX [415] 723-8473

Classical results in linear least-squares estimation and Kalman filtering are based on the nimization of the H2-norm of the estimation error, and require apriori knowledge of the .tistical properties of the noise signals. In many applications, however, one is faced with )del uncertainties and lack of statistical information on the exogenous signals, which has I to an increasing interest in minimax estimation, with the belief that the resulting so­led HOC algorithms will be more robust and less sensitive to parameter variations. It is o known that the Kalman filter minimizes a certain quadratic form, and recently there has ~n an increasing interest in an alternative so-called exponential cost function. This is a k sensitive criterion in the sense that it allows to attribute more or less weights to higher smaller errors. The corresponding filters have been termed risk-sensitive and have natural mections with the HOC filters. With the purpose of simplifying, unifying, and extending several of the existing results

H2, Hoc, and risk-sensitive filtering and control theory, we have recently developed a self­ltained theory for linear estimation in Krein spaces, viz., vector spaces with indefinite inner )ducts. We have shown, from first principles, that the results constitute, apart from certain eresting and revealing facts, the natural generalization of conventional Hilbert space-based imation. The arguments are based on a simple extension of the notion of projections and d to a recursive algorithm that computes the stationary points of general second-order quadratic) forms. When specialized to signals with an underlying state-space model, the

ursive procedure collapses to a Krein space generalization of the conventional Kalman er and yields recursive estimates ~f the state vectors. More interesting, it readily follows that the existing Hoc and risk-sensitive filters are

:cial cases of the Krein-space Kalman filter. Consequently, by further pursuing these mections we can show how several numerically and computationally effective square-root i Chandrasekhar array algorithms, which have been developed in the context of Kalman er theory, can be extended rather directly to the Krein-space setting. We also consider a ther application and introduce HOC adaptive filters. It turns out that the well-known LMS :ast-Mean Squares) adaptive algorithm is nothing but the standard Hoc adaptive filter. is fact provides a theoretical justification for the widely observed excellent robustness 'perties of the LMS algorithm. In other words, the LMS algorithm, which has long been arded as an approximate least-mean squares solution, is in fact a minimizer of the Hoo "ill and not the H2 norm.

l

A Stochastic Multivariable Adaptive Pole Placement Controller

with Parameter Reduction by Restricted Use of Inputs

Xiang-Hong He

Department of Electrical Engineering, Shenyang Polytechnic University, Shenyang 110023,

Loaning, P.R. China.

Tian-You Chai

Research Center of Automation, Northeastern University, Shenyang 110006, Liaoning, P.R.China.

Tel. 393000-4064

Abstract

Elliott and Wiolvich pointed out: We feel that the critical issue for

application of multivariable adaptive control is reduction of the size

of the parameter estimation problem.

In the paper, a new parameter reduction scheme adopting constant

linear compensator by restricted use of inputs for stochastic

multivariable adaptive control systems be presented. A stochastic

multivariable adaptive pole placement controller with parameter

reduction is proposed based on the scheme. Al though the number of

inputs are changed in new model, we use the pole placement controller

and that algorithm to assure that n outputs of the system track n

reference inputs. The results of simulations show that the algorithm

proposed is effective.

The theoretical analysis and simulation results show that the pole

placement adaptive control algorithm for parameter reduction proposed

has the following advantages: I t can reduce the parameters to be

estimated greatly; it restricted use of inputs, need not equal

transformed C( z -1); I t can achieve arbitrary adaptive pole placement

for general stochastic multivariable systems; it can track the

time-varying reference inputs; it can handle systems having arbitrary,

unknown and/or time-varying interactor matrices; it is able to control

open-loop unstable and nonminimum phase system; It can avoid solving

pseudo commutative matrix equations on line.

151

;2

A behavioural approach to 12-approximation

Christiaan Heij and Berend Roorda Econometric Institute and Tinbergen Institute, Erasmus University Rotterdam,

P.O.Box 1738, 3000 DR Rotterdam, The Netherlands fax: +31-104527746 tel.: +31-104081269 Email: HEIJ@ect.few.eur.nl

~t BB denote the class of all linear, time invariant, finite dimensional systems. If w is a multivariate ne series observed on the finite time interval T, then the Euclidian distance from this observation a system BE BB is measured by d( w,B) := inf{llw - 'Ibli; wEB IT}. Similarly, if w is an infinite,

uare summable series then the i2-distance is given by d( w, B) := inf {liw - w 12; wEB}. The ,timal i2-identification problem consists of determining the system with minimal distance from the ne series under a constraint on the allowable complexity of the system. For example, if B( m, n) 'notes the class of all systems in BB with m inputs and n states, then the optimali2-model is given 'B:= argmin{d(w,B);B E BB(m,n)}. In our paper we briefly describe an iterative algorihm for identification. This method uses an isometric state-driving variable representation of the form

lere (~ ~) is an isometry from BRn+m to BRn+m to BRn+q. This identification method is

lstrated by some examples. In particular we will investigate the following aspects The treatment of finite data by means of estimating the initial and final state.

) The use of the algorithm in i2-approximation of a systems impulse response. i) The performance for observations from stochastic linear systems and for some empirical time ·ies.

eferences

B. Roorda and C. Heij, On i2-optimal approximate modelling of vector time series, Report 9257, Econometric Institute, Erasmus University Rotterdam, The Netherlands, 1992.

J .C. Willems, From time series to linear system, part I : Finite dimensional linear time invariant systems; part II : Exact modelling; part III : Approximate modelling, Atuomatica 22/23,561-580, 675-694,87-115,1986,1987.

...

Inversion of Toeplitz-like Matrices Via Generalized Cauchy Matrices and Rational Interpolation

G.Heinig Universitiit Leipzig, FH Mathematik/Informatik

D-O-7010 Leipzig heide@mathematik.uni-Ieipzig.dbp.de

The standard algorithmns for the inversion of Toeplitz and related struc­tured matrices behave well in the positive definite case but they may suffer from instabilities in the indefinite or nonsymmetric case. The usual way to come out of such a situation is to apply a look-ahead strategy. This was first proposed by Cabay /Meleshko and T.Chan/Hansen and further developed by Freund/Zha, Gutknecht, Hojanczyk/Heinig and others.

In this talk another proposal is presented, which is based on the transfor­mation of Toeplitz and related matrices via discrete Fourier transforms into generalized Cauchy matrices, which are matrices of the form

where Ci and dj are numbers and Zi and Yj are vectors. For generalized Cauchy matrices fast inversion algorithms can be constructed with the help of intepretations as rational interpolation problems. The advantage of the Cauchy structure is that it permits the application of pivoting techniques. In that way one obtains also stable inversion algorithms for Toeplitz and related matrices.

153

On the Numerical Solution of the Discrete-Time Periodic Riccati Equation '*

J.J. Hench Institute of Information and Control Theory Academy of Science of the Czech Republic

Pod vodarenskou vezi 4 128 08 Praha 8

phone: (42)(2) 815-2342 e-mail: hench®opte. utia. cas. cs

A.J. Laub Department of Electrical and Computer Engineering

University of California Santa Barbara, CA 93106-9560

phone: (805) 893-3616 fax: (805) 893-3262

e-mail: laub®ece. ucsb. edu

Abstract

In this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic Riccati equation. This method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the Euler-Lagrange difference equations. In so doing, it is possible to extract a basis for the stable deflating subspace of the extended pencil, from which the Riccati solution is obtained. This algorithm is an extension of the standard QZ algorithm and retains its attractive features, such as quadratic convergence and small relative backward error.

*This research was supported by the Air Force Office of Scientific Research under Grant No. AFOSR-91-0240, and by the National Science Foundation under Grant No. ECS-9120643.

-

OPERATOR INDEPENDENT SYSTEM REPRESENTATION M.A. HERSH

DEPARTMENT OF ELECTRONICS AND ELECTRICAL ENGINEERING, UNIVERSITY OF GLASGOW, GLASGOW G12 8QQ, SCOTLAND

Tel: 041 3398855 extn. 4906· Fax: 041 3304907 Email: mhersh@uk.:lc.gla.elec

155

Keywords: system representation, continuous and discrete time systems, generalised operator, generalised transform

Traditionally analysis and algorithm development for discrete time and continuous time systems have been considered totally separately from each other. However the functional form of many algorithms is identical for both continuous and discrete time systems and this artificial separation is an unnecessary complication.

The use of, for instance delta operator discrete time system representations has gone some way to lcsolve ihe problem vr this dftificial barrier between uiscrete dOd .;ontinuous time :,ystem theory in giving behaviour which approximates that of the associated continuous time system as the sampling time tends to zero. However the delta operator and analogous operators, such as the w operator, are discrete time operators which give representations which converge to the associated continuous time representation as the sampling time tends to zero.

The approach in this paper goes several steps further. A unified representational theory for continuous and discrete time systems is developed using an llperator-based approach lO system modelling, allowing continuous time, discrete time and discretised systems to be represented as

A(V')y(T) = B(V')U(T) + C(V')e(T)

where V' is a general operalOr. which could be the differential. backwards shift or delta operalOr.

and t is the associated 'time' i.e. t. n or (n + E)L'> respectively for t and n real and integer variables, 0::::; E::::; I, and the remainder of the notation is standard.

A class of associated transforms, which includes transforms in common use such as the Laplace, Fourier and z-transforms, is also derived. It is shown [hat the class of operators so-defined with [he associated transforms possess a range of desirable properties and is therefore contained in an appropriate class of admissible operators for use in system representations.

Further work will extend this approach to [he derivation of a unifying framework fO" control algorithms.

s

On Orthogonal Basis Functions that contain System Dynamics

Peter s.c. Heuberger!, Paul M.J Van den HoP and Okko H. Bosgra2

1 National Institute for Public Health and Environmental Protection (RIVM), P.O. Box 1,3720 BA Bilthoven, The Netherlands. e-mail: cwmheub@rivm.nl

2 Mechanical Engineering Systems and Control Group, Delft University of Technology, Mekelweg 2,2628 CD Delft, The Netherlands. e-mail: vdhof@tudw03.tudelft.nl

In many areas of signal, system and control theory, orthogonal functions play an important rol in issues of analysis and design. Especially in the field of system identification these functions have gained renewed attention in the past decade, See e.g [1,2]. The orthogonal functions which are considered in the literature are generally of a classical type, such as Hermite and Laguerre polynomials, Walsh and Block pulse functions etc. In this paper it will be shown that orthogonal functions are intrinsic to linear systems, in the Sense that every linear system gives rise to several sets of orthogonal functions. Each of these spts, referred to as system based orthonormal functions, constitutes a basis for the space f2 of square summable functions. In [I] a theory on these system based orthogonal functions was developed, based on balanced state space realizations. In the present paper these results are generalized to descriptions in terms of input/ output transfer functions, and hence representation independent. It is shown that every square all-pass (inner) transfer function induces two orthonormal bases of f2 in a natural way. Moreover it is shown that the ordinary pulse functions and the classical Laguerre polynomials are special cases in this theory. With this concept it will be shown that any connection between a linear system and an alI­pass function, e.g. through inner/outer, normalized coprime or inner/unstable factorization, leads to bases of specific system based orthonormal functions. An important property of these orthonormal functions is that they - to some extent - incorporate the dynamic behavior of the underlying system. A generalized orthonormal basis gives rise to an alternative series expansion of rational transfer functions and induces transformations of signals and systems to an orthogonal domain. Properties of these transformations are analysed, leading to interesting relations between orthogonal system representations and the corresponding basis functions. The application of these functions to the problem of approximate system identification leads to promising results. The underlying idea is that the orthogonal basis which is used should be shaped according to the behavior of the data generating system. One of the resulting algorithms is based on the alternative series expansion of rational transfer functions and it will be shown that this approach is a generalization if the estimation of for instance finite impulse response (FIR) and Laguerre models.

[1 ] P.S.C. Heuberger (1991). On Appmximate System Identification with System Based O,·thonormal Functions. Dr. Dissertation, Delft University of Technology, The Nether­lands, 1991.

[2 ] B. Wahlberg (1991). System identification using Laguerre models. IEEE Trans. Automat. Contr., 36,551-562.

r

Abstract

The real stability radius and spectral value sets under structured real perturbations

D. Hinrichsen

The real stability radius r(A, B, C) of a stable real matrix A under per­turbations of the structure (D, E) is by definition the norm of the smallest real perturbation ll. such that A + Dll.E is unstable (where D and E are given real matrices). To date there is no computable general formula available for the real stability radius but some progress has been made. In this paper we give a survey of the available results and apply them to the recent concept of spectral value set or pseudospectrum.

157

ORBIT CLOSURE UNDER THE ACTION OF THE GENERALIZED FEEDBACK GROUP

liederich Hinrichsen lstitut fUr Dynamische Systeme rniversitat Bremen 1-2800 Bremen 33 Germany h@mathematik.uni-Bremen.de

:iven a time-invariant linear system

L) Ex = Ax + Bu

Joyce O'Halloran Department of Mathematical Sciences Portland State University Portland, OR 97207--0751 USA hmjo@psuorvm

msider the following transformations of the system:

Left multiplication:

State change of coordinates:

Input change of coordinates:

State feedback:

LEx = LAx + LBu

ERic: = ARx + Bn

Ex = Ax + BWu

Ex = (A+BF)x + Bu

he generalized feedback group G consists of all transformations which are combinations

'these. We say that two systems (1) are G--equivalent (or belong to the same G-orbit)

hen one can be transformed to the other via a transformation in G. A complete set of

.variants and a canonical form for G--equivalence were established in [LOMK91J. Here we

idress the problem of identifying the topologoical closures of the G-orbits, a problem

hich arises in the context of high gain feedback. This problem has been solved for the

~e of controllable systems (see [G-L91]); i.e. there are criteria on the invariants of the

:tion of G which describe the controllable systems in the topological closure of the

-orbit of a given controllable system. In [H091aJ we established necessary conditions for

system to lie in the closure of the G-orbit of a given system; in [H091bJ we showed that

lese necessary conditions are sufficient in the case of state space systems. Here we

tablish a (possibly complete) range of cases for singular systems in which the necessary

,nditions given in [H091aJ are sufficient for G-orbit closure.

References

:-L91J H. Gliising-Liierssen, Gruppenaktion in der Theorie singularer Systeme, 1.D. Thesis, Institut fUr Dynarnische Systeme, Universitat Bremen, 1991.

[091aJ D. Hinrichsen, J. O'Halloran, The orbit closure problem for matrix pencils: lcessary conditions and an application to high gain feedback, in: New Trends in Systems ~eory, Ed. G. Conte, A.M. Perdon, B. Wyman, Birkhauser, Boston, Basel, Berlin, 1991, 6-392.

:091bJ , A note on the degeneration of singular systems under ncil equivalence, Proceedings of the 30th IEEE Conference on Decision and Control, 9l.

OMK91J J.J. Loiseau, K. Ozcaldiran, M. Malabre, N. Karcanias, Feedback canonical rms of singular systems, Kybernetika 27 (1991),289-305.

POSITIVE REAL BALANCING AND ORTHOGONAL POLYNOMIALS

J. Hoffmann Fachbereich Mathematik

Universitat Kaisers1autern Kaiserslautern, Germany

hoffmann@mathematik.uni-kl.de

P. A. Fuhrmann' t

Department of Mathematics Ben-Gurion University

of the Negev Beer Sheva, Israel

paf@indigo.bgu.ac.il

Positive real functions are of importance in many areas of system and control theory, e.g. in stochastic system theory or in adaptive control. By associating a certain type of Riccati equation with positive real systems, the so-called positive real Riccati equations, Ober [1991] defines balanced realizations for positive real systems by ba­lancing the minimal solutions of these equations. The corresponding canonical form for positive real balanced functiqns gives a parametrization for this system class. Observe that balancing is usually introduced on the state space level; however, it is clear that balanced realizations exhibit certain system invariants. Thus it is of interest to explore the links between these invariants and the external, i.e. the input/output properties of the system. In Hoffmann and Fuhrmann [1993a], a balanced realization in canonical form of a proper, antistable rational transfer function is constructed as the matrix representation of the abstract shift realization. The required basis is constructed as a union of sets of polynomials orthogonal with respect to weights given by the square of the absolute values of minimal degree Schmidt vectors of the corresponding Hankel operator. Here we take a similar approach for positive real functions. In Fuhrmann [1993] in the bounded real case there was established a tight connection between the Hankel operator with the B-characteristic as its symbol and the Hankel operator associated with a JB-normalized coprime factorization. Now the Caley transform is used to

e obtain the corresponding result for a positive real function 9 = d' By further analizing

this relationship in a similar way as in Hoffmann and Fuhrmann [1993b] we can construct an orthogonal basis of Xd such that the matrix representation of the shift realization with respect to this basis is in positive real balanced canonical form as derived in Ober [1991}.

References [1993J P.A. Fuhrmann, 'The bounded real characterist.ic function and Nehari extensions', submitted [1993aJ J. Hoffmann and P.A. Fuhrmann, 'Remarks on orthogonal polynomials and balanced reali­zations', submitted [1993bJ J. Hoffmann and P.A. Fuhrmann, 'A polynomial approach to bounded real balancing', in preparation [1991J R. Ober, 'Balanced parametrization of classes of linear systems', SIAM J. Gontr. and Opti­mization , 29, 1251-1287

• Earl Katz Family Chair in Algebraic System Theory Ipartially supported by the Israeli Academy of Sciences under Grant No. I 184

159

Recent Results in Neural Network Approximation

Kurt Hornik Institute fur Statistik und Wahrscheinlichkeitstheorie

Technische Universitat, Wien Wiedner Hauptstr 8-10/1071

A-I040 Wien Austria

The rigorous analysis of approximation and learning by multilayer per­ceptrons has been receiving very broad research interest within the last few years. In particular, many contributions have been made towards obtaining very general results on uniform functional approximation capabilities. We give refined results on the well-known fact that "basically", single hidden layer feedforward networks are universal approximators provided that suffi­ciently many hidden units are available and that the activation function is nonpolynomial. The results show in particular that for the case of uniform approximation, continuity-type assumptions on the activation function can be relaxed to Riemann integrability, and that the input-to-hidden weights can be constrained to very small sets. In fact, for certain activation functions, including the logistic and arctangent squashers, a universal bias will suffice. We also discuss rates of approximation results (i.e., how the approximation error scales with the number of hidden units) with particular emphasis on smooth (Sobolev-type) approximation.

Feedback Control of Linear Descriptor Systems

M. HOll and P. C. Muller Sicherheitstechnische Regelungs- und Mefitechnik

Bergische Universitiit-GH Wuppertal D-42119 Wuppertal, Germany

Telephone: 0202-439-2336; Telefax: 0202-439-2901 e-mail: tahboub@cyber.urz.uni-wuppertal.dbp.de

Abstract: We consider descriptor systems of the form

Ex = Ax + Bu, y = Gx

161

(1)

where x u and yare the descriptor, control and measurement vectors, respectively; E, A E Rqxn, B E Rqx, and G E Rmxn are known matrices. E and A are not necessarily square and rankE < n; no assumption is made on the rank of the matrix pencil >'E - A. A set of this kind of equations arises very naturally and :onveniently in describing a plant.

This paper deals with the synthesis of the output feedback control u = Ky regularizing system (1) In the sense that the matrix pencil >'E - (A + BKG) has full column rank. The design of the feedback :ontrol u = Kx stabilizing system (1) is also considered. The discussion on the admissibility of feedback :ontrol shows some important points concerning feedback control in descriptor systems with singular matrix pencils.

Definition 1: System Ex = Ax is called asymptotically stable if x(t) --+ 0 as t --+ 00.

Definition 2: If f3 = rank(>.E - A) for almost any finite>. E C, f3 is called the rank of the matrix oencil >'E - A. . Definition 3: A finite>: E C is called the finite eigenvalue of the pencil >'E - A if rank(>:E - A) < rank(>.E - A) .

. Denote by a( >'E - A) the set containing all finite eigenvalues of >'E - A. Main results of this paper He as follows.

Lemma (stability). System Ex = Ax is asymptotically stable iffrank(>.E-A) = nand a(>.E-A) E C-.

Theorem 1 (rank assignment). For a given integer nk with rank(>.E - A) ::; nk ::; min{ q, n}, there ~xists a gain matrix K such that rank[>.E - (A + BKG)] = nk, iff

min{rank [ >'E - A B l, rank [ >'E C A ]} 2: nk (2)

Theorem 2 (stabilization). There exists a feedback control u = Kx such that the system (1) is tsymptotically stable iff the system is stabilizable, i.e.

rank [ >'E - A B 1 2: n, V>. E C+ (3)

Definition 4: A feedback control is called admissible if the closed-loop system adopts any initial con­litions Ex(O_).

Theorem 3 (admissibility). The feedback control u = Ky is admissible if and only if

rank [E A + BKG 1 = rank [ >'E - (A + BKG) 1 fhe feedback control is restrictively admissible for the given initial condition Ex(O_) if and only if

rank [ >.E-(A + BKG) Ex(O_) 1 = rank [ >'E - (A + BKG) 1

(4)

(5)

The results in Theorem I and Theorem 2 can be considered as the generalizations of the existing ones of cletcher (Int. J. Syst. Sci., vol 27, pp. 843-847, 1986) and Dai (Singular Control Systems, Berlin: Springer, .989) for the corresponding problems. The explicit design procedures of the feedback gain matrices are lfovided in the proofs of the sufficient parts of the theorems.

)2

On Maximizing the Stability Radius and iIO

Minimizing the H -norm

Ting-Shu HU Dept. of Automatic Control Shanghai Jiao Tong University Shanghai 200030 , P.R. China

BSTRACT: This paper will present a method to maXlmlze the real tability radius for system with only one uncertain parameter. The ethod can be extended to minimize the H~-norm of some transfer atrices with state feedback.

Denote the stability radius as p(F), where F is the state feedback atrix. The continuity and differentiability of p(F) with respect o F will be studied, and the partial differential ap(F)/aF will e derived. Thus a gradient method can be generated to find the o cal maxima' or the stationary points of p (F) •

By parametrizing all the F that assign the eigenvalues of A+BF s a function of U mxn () . . .

~R ,p F can be maxlmlzed under the constralnt f pole assignment ,since ap(F)/aU can be obtained.

The above result will be utilized to minimize the H~-norm of some ransfer matrices under the constraint of pole assignment. :ey words: Stability radius, state-feedback, differentiability

gradient method, H~-norm

-

On Optimizing A Class of Performance

Indexes Related to Robustness

Ting-Shu HU Dept. of Automatic Control Shanghai Jiao Tong University Shanghai 200030, P.R. China

163

)TRACT: This paper will provide a general approach to robust design

solving a class of optimization problems,

inf TT a-2[ Si (F, V)]

s.t.: V-l(A+BF)V=Al

lre Al is a diagonal matrix with the desired eigenvalues, and S1(~'V) a matrix function of F and V.

Typical performance indexes are: J=t a-2[V].a-2[V-l ], t ~2[HV].~2[V-1G1 r2[DHVR].~2[RV-1GD-1J , and t~2[PJ , where P is the solution to:

.BF):'P+P(A+BF)=-2In ,etc. Through robust analysis, this paper derives

reral other performances that reflect robustness. Systems with stru­lred,unstructured and parameter-dependent uncertainties are concerned.

The optimization problems are solved by gradient method. At first,

i constraint (**) is relaxed by parametrizing those F and V that

;isfy ( ... 4:) as a function of a free parameter UERmxn • Thus J is a

Lctional of U. The gradient oJ/aU is computed by using some proper-

is of Sylvester equation and kroneker product •

. words: robust design, gradiend method, pole assignment, structured­uncertainty, parameter dependent uncertainty

64

Nonlinear Output Regulation and Feedback Stabilization for a Class of Globally Nonminimum Phase Systems

Abstract:

Xiaoming Hu Optimization and Systems Theory

Royal Institute of Technology 100 44 Stockholm, Sweden

Email: hucgmath.kth.se

In this paper, we consider the following system:

i = jo(z,17d

liJ = 172

Y = 171

the prohlems of feedback stabilization and output regulation on compact" are studied. High gain control laws are given to solve output regulation on compacta for the case where exact tracking with internal stability may he impossible. With some modification, that result is applied to feedback stabilization on compacta for nonlinear systems whose zero dynamics may be glohally unstahle.

>

BILINEARIZA TION OF A NONLINEAR DISTRIBUTED PARAMETER SYSTEM

Xiang-Ming Hua

Institute of Automatic Control,

East China Uni. of Chemical Tech.,

Meilong Road 130,

Shang~ai 200237, P. R. China

Abstract: In the paper, bilinear modelling of a nonlinear distributed

parameter system, with a typical laboratory-scale fixed-bed reactor as

an example purpose, via a parameter estimation techniques is presented.

A bilinear continuous-time state model suitable for control studies is

considered as a reduced model of the reactor, and its coefficient

matrices are determined by means of the recursive estimation algorithm

based on block-pulse functions. Two bilinearization approaches via

parameter estimation are suggested, one based on the lumped model of

the reactor and the other directly based oil the detailed distributed

model of the reactor. In both cases, the temperature variables at the

axial orthogonal collocation points only are chosen as states. The

approaches are applicable for bilinearization of nonlinear distributed

parameter systems wi~h or without multiplicative input. Through

comparisons between a detailed model and the reduced bilinear model of

the reactor, the validity of bilinear modelling over a relatively wide

region of operation is demonstrated.

Keywords: modelling; nonlinear distributed parameter systems; bilinear

sys.tems; parameter estimation; chemical reactors.

165

6

Isospectral Matrix Flows Discretized by Lie-Poisson Integrators, with Application to Nonlinear Signal Processing

Knut Huper, Steffen Paul, and Rainer Pauli Technical University Munich Institute for Network Theory and Circuit Design D-8000 Munich 2, Arcisstr. 21, Germany Tel.: 0049 89 21 05 85 29 Fax.: 00498921 05 85 04 e-Mail: knhu@nws.e-teclmik.tu-muenchen.de

In this paper, we reformulate isospectral matrix flows as nonlinear Hamiltonian dynamical systems on Poisson manifolds:

A = [A, B] {:} z = K'V H(z)

Isospectral matrix flows can be interpreted as continuous generaliza­tions of iterative algorithms known from numerical linear algebra. We are interested in analog and digital circuits for modelling this kind of differential equations, i.e. performing eigenvalue calculations and sorting operations. For digital realizations the dynamic al system has to be discretized. The Hamiltonian formulation offers a systemati­cal method to develop structure preserving iteration schemes. Using' the formalism of generating functions implicit algor ithms can be con­structed. A Lie algebraic treatment of the dynamics leads to explicit integration schemes. Similar to symplectic integrators, which preserve symplectic structures, such discrete time maps can be constructed to exactly preserve the degenerate Poisson brackets for the general cate­gory of Lie-Poisson dynamical systems. We present numerical results using different algorithms.

167

Relationships between Structured TLS and Constrained TLS, with applications to Signal Enhancement

Sabine Van Huffel 1 Bart De Moor 2 Hua Chen ESAT, Department of Electrical Engineering, Katholieke Universiteit Leuven

Kardinaal Mercierlaan 94,3001 Leuven, Belgium tel: 32/16/220931 fax.: 32/16/22 1855 e-mail: vanhuffel@esat.kuleuven.ac.be

The Total Least Squares (TLS) method has been devised as a more global fitting method than Least Squares (LS) for solving overdetermined sets of linear equations Ax "" b in which A, as well as b, are noisy [4]. From a statistical point of view, TLS operates under the assumption that the errors in A and b are independently and identically distributed with zero mean and equal variance. If there is correlation among the errors, a noise whitening transformation can be applied or a Generalized TLS problem [4] can be solved and the error norm is appropriately modified. However if there is a linear dependence among the error entries in [t>.A; t>.b] -which is the case when the data matrix is linearly structured: Hankel, Toeplitz, ... - then the TLS solution may no longer yield optimal statistical estimators.

To get more accurate estimates of x, Abatzoglou and Mendel [1] extended the classical TLS method to incorporate the algebraic dependence of the errors in [A; b] and called their extension "constrained TLS". In this paper, we show how CTLS is a special case of the Structured TLS (STLS) problem recently presented by B. De Moor [3]. Although both methods use a different formulation and solve the problem in a quite different way, there are some nice similarities between both approaches. Based on these we could simplify the straightforward inverse iteration based algorithm outlined in [3] for solving the CTLS problem in cases where the data matrix C = [A; b] is a linearly structured unweighted matrix. Numerical examples are given in which one wishes to approximate a Hankel matrix by one of lower rank. This problem is a key issue in the (partial) realization problem and in the enhancement of sinusoidal and exponentially modeled signals. Applications occur in system identification, modal analysis, biomedical signal processing such as Nuclear Magnetic Resonance spectroscopy [5], etc. The STLS and CTLS approaches are compared to currently used suboptimal approaches described in [2,5].

References

[1] T.J. Abatzoglou, J.M. Mendel and G.A. Harada, The constrai1wl totallcast squares technique and its application to harmonic superresolution. IEEE Trans. Signal Processing, Vol. SP-39, No.5, May 1991, pp.l070-lOR6.

[2] J. Cadzow, Signal enhancement: a composite prope1·ty mapping algorithm, IEEE Trans. on Acous­tics, Speech, and Signal Processing, Vol. ASSP-3G, No.1, January 1988, pp. 49-62.

[3] De Moor B. Structured total least squares and L2 approximation problems. Internal Report ESAT­SISTA 1992-33, Department of Electrical Engineering, Katholieke Universiteit Leuven, Belgium, August 1992 (also IMA Preprint Series nr 1036, Institute for Mathematics and its Applications, University of Minnesota, September 1992), Accepted for publication in the special issue of Lin­ear Algebra and its Applications, on Numerical Linear Algebra Methods in Control, Signals and Systems (eds: Van Dooren, Ammar, Nichols, Mehrmann), Volume 188, July 1993.

[4] S. Van Huffel and J. Vandewalle, The total least squares pmblem : computational aspects and analysis, Frontiers in Applied Mathematics series, Vo1.9, SIAM, Philadelphia, 1991.

[.5] S. Van Huffel, Enhanced Resolution Based on Minimum Variance Estimation and Exponential Data Modeling. Signal Processing), Vo1.33, no.3, to appear.

1 Research Associate of the Belgian N.F. W.O. (National Fund for Scientific Research) 2Research Associate of the Belgian N.F.W.O. (National Fund for Scientific Research)

)8

Controlled invariance of nonlinear systems: nonexact forms speak louder than exact forms

H.J.C. Huijberts Dept. of Mathematics and Compo Sci.

Eindhoven Univ. of Techn. P.O. Box 513

5600 MB Eindhoven, The Netherlands Email: hjch@bs.win.tuc.nl

C.H. Moog Lab. d'Automat. de Nantes Ecole Centrale de Nantes/

Universite de Nantes U.A. C.N.R.S. 823 1, Rue de la Noe

44072 NANTES Cedex 03, Fritncc Email: moog@lanOl.ensm-nantes.fr

The extension of controlled invariance to nonlinear systems was initiated at the end of the 70's by Brockett. It rapidly appeared that, while the notion was well established for linear time-invariant systems, the situation is less clear for nonlinear systems. At the beginning of the 80's, numerous authors worked on the notion of a controlled invariant distribution, which became a major and standard notion in nonlinear systems theory. It gave rise to a main stream of research during the last decade and helped to solve numerous control problems via regular static state feedback.

Since 1985, an increasing interest in dynamic compensator solutions to control problems attracted an increasing number of researchers. From that time on, severe pathologies were highlighted and the controlled invariant distributions seemed to be hardly suitable for the analysis of control problems when dynamic compensator solutions are sought.

This paper is devoted to a generalization of the notion of controlled invariance, which encompasses the notion of a controlled invariant distribution and which allows a geomet­ric interpretation of control schemes involving dynamic compensation. The involved state feedback in general belongs to the class of quasi-static state feedbacks which was recently introduced by Delaleau and Fliess.

Nonlinear dynamic disturbance decoupling and linearization

H.J.C. Huijberts Dept. of Mathematics and Compo Sci.

Eindhoven Univ. of Techn. P.O. Box 513

5600 MB Eindhoven, The Netherlands Email: hjch@bs.win.tue.nl

H. Nijmeijer* Dept. of App!. Mathematics

University of Twente P.O. Box 217

7500 AE Enschede, The Netherlands Email: h.nijmeijer@math.utwente.nl

/

In recent years a lot of work has been done towards establishing a rather complete picture about the disturbance decoupling problem for nonlinear control systems of the form :i; = f( x) + g( x)u + p(x)q, y = h(x) with x, u, q, y respectively the state vector, input vector, disturbance vector and output vector. By now it is known under which conditions one can locally construct an invertible static state feedback u = a(x) + j3(x)v or a dynamic static state feedback i = a(x, z) + j3(x, z)v, u = ,(x, z) + 8(x, z)v that will achieve decoupling of q to the output y. In the latter situation, the so called dynamic disturbance decoupling problem, a new phenomenon not known from linear control theory, has been treated in [IJ. The remarkable observation is that for nonlinear systems, in contrast to linear systems, we can enlarge the class of systems for which the disturbance isolation can be achieved by allowing for dynamic state feedback. In order that the above (nonlinear) work on disturbance decoupling becomes nlOrc practically applicable, one would impose in addition that the nominal closed loop system is locally asymptotically stable. For linear systems this disturbance decoupling problem with stability is well understood.

The purpose of the present paper is twofold. First, we will review (see [2]) the connection between the nonlinear dynamic disturbance decoupling problem about an equilibrium and the corresponding linear static disturbance decoupling problem for its linearization. Under generic conditions it can be shown that each of the above problems is solvable if and only if the other is. In the second part of the paper we will extend the idea of relating the linearization with the original nonlinear system by also studying the (dynamic) disturbance decoupling problem with stability. In particular it will be shown, again under generic conditions, that the nonlinear dynamic disturbance decoupling problem with local asymptotic stability is solvable if and only if the linear disturbance decoupling with stability for the linearization is.

References

[IJ Huijberts, H.J.C., H. Nijmeijer and L.L.M. van der Wegen, Dynamic disturbance decoupling for nonlmear systems, SIAM J. Control Optimiz., 30 (1992), pp. 336-349.

[2J Huijberts, H.J .C., II. Nijmeijer and A.C. Ruiz, Nonlinear disturbance decoupling and lineariza­tIOn: a partial interpretatIOn of integral feedback, COSOR Memorandum 92-41, Department of Mathematics and Computing Science, Eindhoven University of Technology, submitted for publication .

• Author to whom all corespondence should be sent.

169

o

A Servo Design of Decentralized Adaptive Control

Kenji IKEDAt, Seiichi SHIN+,and Toshiyuki KITAMORI+

Depart.ment of Information Science and Intelligent SYStf'IllS,

University of Tokushillla, MillamijoSftlljimft 2-1, Tokllshitna 770. Japan phone: +81,886,23,2311 ext,,4726 e-mail: ikeda!Q)is.toknshima-u.ac.jp

DepartllH>nt of Mat.hematical Engineering and Information Physics, University of Tokyo, Hongo, Bunkyo,ku, Tokyo 113, Japan

Abstract

We present a decentralized adaptive {'ontrol system. which stabilizes interconnect.pel sllhsystCIlls by means of local feedback controls. Decf'ntralized cont.rollers an' lls('d for mORt real plant.s. not only becanse it is impossible to make a preci:;;c model of the whole system, whkh is indispensable to the centralized con to} based 011 the modern control theory, hilt becanse the decciltariizcd control system is considered to prevent local failures frolll ext.ending to the overall syst.em.

In order to analyze the fault tolerance of thr decelltralizrd control systeIll, we proposf' a plant model composed of first order subsystems and stable dynamical interconnections. A class of failures which can be modeled as a zero cont.rol input of a stable snh..;;ysteIll is considered. Aft('r snch failures. the failed sllbsystem is no more lllodeit'd a...:: a subsystem but is dist.rihuted into a part of rach interconnection. It is shown that a decentralized syst.em bas('d 011 t.his plant model has an invariant form under this kind of failures. When the int.erconnections sat.isfy t.he range conditioll and each subsystem sat.isfy t.he robust. high gain stabilizability condit.ion. the overall syst.elll ran lH' represented as the proposing plant model. Furt.hermore, t.he error syst.em in a servo df'sign with PI controllers can also be reduced t.o t.he proposing model.

Another requirement. for t.he designing of cont.rol syst.f'IIlS may he thf' small ff'pdha('k gains from thf' viewpoint. of energy t'ffidency. If t.he sensit.ivit.y funct.ioll of ra('h suhsyst.em was srt small enough for fault. t.olerance. f'adl local loop gain might. he high gain. In order to rf'ali?'f' a fauH tolerant system without high gain feedbacks. we construct. a decelltraliz.f'd adaptive control ~ystel1l has('(l on the proposing plant. model. Each local hef'flhack gain is tnlled by thc (T-Illodified adaptive law to attain small output.s with small ff'edba('k gains.

Since tht' plant model is iuvariant Hnder the failures and the d('c('ntraliz.ed adaptivf' control SySt,f'1ll hased on thiR plant model is proved to he stahle. we can conrimlf' that tlH~ proposing df'cf'1It.raliz('d adapt.ive S(,fVO Rystem is fanlt tolf'fallt.. Whrll t.he controller of a stahlf' snhsyst.f'Ill bn'aks down. the feedback gains of the ot.hf'r snhf.lyst.f'll1S will he ohsC:fved t.o become larger t.han hrforf' fail1irf' automat.ically to maint.ain t.he overall stability.

Key words: fault. t.olerance'. servo sYSt.CIll. decentralized cont.rol. adaptvip cont.roL st.abilit.y.

Factorizations of Transfer Matrices for Linear Systems over Unique Factorization Domains

Hiroshi Inaba and Wei ben Wang

Department of Information Sciences, Tokyo Denki University

Hatoyama-Machi, Hiki-Gun, Saitama 350-03, Japan

Abstract

Various factorizations of transfer matrices for linear systems defined over the real

number field have been studied and applied for several important system problems.

In particular, the factorization approach using stable transfer matrices has been thor­

oughly studied and has played an important role to develop a new control theory,

called the H", control theory.

On the other hand, linear systems defined over rings have been extensively studied

in the last two decades. Linear systems over rings are a natural generalization of

those over the real number field. For instance, linear systems over rings can be used

for modeling systems characterized by parameters, systems described by time-delay

defferential equations, systems involving integration operators, etc.

Tlus paper prescnts a preliminary result on generalizing the idea of previous fac­

torization approaches for linear systcms over the real number field to those for linear

systems over rings. More precisely, restricting our rings to the class of U1uque fac­

torization domains (UFD's), and introducing the notion of denonunator sets, it will

be shown that any transfer matrix H(z) over a UFD can be factorized as H(z) =

W(z)V(Z)-1 = W(zf1V(zfl where W(z), V(z), W(z) and V(z) are some transfer

matrices such that all of their denominators belong to a given denominator set 'D.

Further, various properties of such factorizations will be presented.

171

GENERALIZED TIME-VARYING POPOV-YAKUBOVICH THEORY: THE

DISCRETE-TIME CASE

Professor Vlad IONESCU 3, Emile Zola str., 71272 Bucharest Romania

Tel.: 40-1-633.63.78 Fax: 40-1-312.24.00

General conditions for the existence of a stabilizing solution to the so called Kalman-Szego-Popov-Yakubovich (KSPY) system which, in turn, is equivalent to the discrete-time Riccati equation are presented. The well-known Popov­Yakubovich "positivity condition" is replaced with a more general one expressed in terms of the invertibility of an adequate toeplitz-like operator family. this allows to relax the usual condition imposed to the DTRE's coefficients as well as to consider the game-theoretic situations. concerning this last subject a relevant application of the above mentioned theory is presented. It concerns necessary and sufficient solvability conditions for the disturbance attenuation problem, with infinite time-horizon, in the time varying discrete case. In this context, the connections between the contracting property of the resultant closed-loop input­output operator and the existence of the stabilizing solutions to certain KSPY systems in "J" form, playa central role. Other topics such as the time-varying analogue of the "extended" symplectic pencil associated to the time-invariant DTRE is also to be considered.

/L-Analysis with Two Performance Objectives and Its Application to Controller Reduction

Hiroshi Ito , Hideto Watanabe, Hiromitsu Ohmori and Akira Sano

Department of Electrical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223, Japan

hiroshi@sano.elec.keio.ac.jp

Abstract

The focus of this paper is analysis of linear feedback control systems with multiple objectives. To deal with any two objective functions WI and W2 associated with single feedback system, a criterion in the form of II[WI W2111 :<::: 1 is often used for control design. The paper however, directly considers individual conditions IlwI11 :<::: 1 and IIw211 :<::: l.

The paper begins by proposing a /1 test to assess robustness of stability and two perfor­mance objectives associated with single feedback systems consisting of constant matrices, where the performance is measured by the largest singular values of two objectives which are subject to structured bounded perturbations. It is shown that robust performance of multiple objectives can be assessed by single test if repeated nonscalar blocks are adequately introduced in structure of /1. The /1 test is applied to analysis of linear time-invariant dy­namic systems. A necessary and sufficient condition for ?too robust performance of two objectives is obtained in the form of single /1 test. The /1-analysis is believed to be one step to the further research on /1-synthesis. Through the proposition of the /1 test, the importance of the repeated nonscalar block is pointed out. An upper bound of /1 under the structure including repeated nonscalar blocks is clarified using the Kronecker product. Then, the convexity property of the upper bound is proved.

The second part of the paper deals with controller reduction for robust control systems. In order to ensure the robust performance of simultaneous two objectives associated a reduced order controller, our /1 test is utilized to choose an effective weighting function in the controller reduction procedure as well as to assess robustness of resulting systems. A sufficient condition for the robust performance of reduced order systems is obtained. Utilizing the condition, an effective reduction procedure is proposed.

Finally, an example is provided that illustrates the analysis and controller reduction techniques presented.

173

74

Block Triangular Decoupling for Linear Systems over a Principal Ideal Domain

Naoharu Ito and Hiroshi Inaba

Department of Information Sciences, Tokyo Denki University

Hatoyama-Machi, Hiki-Gun, Saitama 350-03, Japan

Abstract

The block triangular decoupling problem for linear systems over the real number field

was first studied by Wonham and Morse within the so-called geometric approach, and necessary and sufficient conditions for the block triangular decoupling problem

to be solvable were obtained in terms of the largest elements in families of reachability

subspaces satisfying certain conditions. This paper studies the block triangular decoupling problem for linear systems over

a commutative principal ideal domain with identity, also in the geometric approach. Systems over a principal ideal domain are a natural generalization of systems over

the real number field and are used as abstract descriptions of, for instance, systems

with a parameter or with a time-delay operator. Such systems have been extensively

studied in the last decade. The purpose of the present investigation is to extend the results on the triangular

decoupling problem for systems over the real number field to systems over a princi­

pal ideal domain. First, various properties of reachability submodules are discussed.

Next, for a given system (A, B, {Gi}:=l)' assuming that the largest reachability sub-i-1

module 4i; contained in nKerGj (i = 1,2,···,k) exists, it will be shown that i=l

sufficient conditions for the block triangular decoupling problem to be solvable are

given as

4i; + KerGi = X

Clx ( 4i;) c 4i;_1 (i=1,2, ... ,k),

(i=2,3, ... ,k)

where Clx ( 4i;) denotes a closure of 4i; in X. Further, it will be claimed that the

decoupled system is pole assignable if and only if all 4i; are closed in X.

A}T OPERATOR APPROACH TO THE INTERPOLATION PROBLEms SOLVING

Ivanchenko T., Sakhnovich L.

The operator identity

(1)

suggests the following interpolation problem:

Describe the set of ca(t)€ [ and }~o,

vlhich gives the representations <X>

S = J ( [ - j/ t f j n/ d 6'( t ) J n2.* ( E - JI *0-1

+ f F /1-; -00 00 _1 .1_

n1

= -i JUI(£-tflJ + j~tzEJ!7~db(t)TL·[!7~ci+FJ.zJ.

This general interpolation problem contains the clas­

sical interpolation problems (IJevan1inna-Pick, Caratheodory,

Schur, Hamburger and etc.) and the new interpolation prob­

lems (the problem about the representation of operators

wi th W - Difference Kernels, the problem of De Branges L.)

as special cases.

We give the conai tions vlhich are guarantee that the

interpolation problem is equivalent to the abstract analo­

gue of the Potapov Fundamental Matrix Inequalities and

write down the description of the set of all solutions in

this general setting.

The receiving results are used by the investigation

of the spectral problems of the canonical system and by

the investigation of the nonlinear integrib1e equations.

175

'6

Robust Stability of Delay-Difference Equations

Anatoli F. Ivanov Institute of Mathematics

Ukrainian Academy of Sciences Kiev, Ukraine

Erik I. Verriest· School of Electrical Engineering Georgia Institute of Technology

Atlanta, GA 30332-0250

This paper deals with the stability of delay-difference systems. In the first part, the linear difference-delay equation of the form XHI = AXk + BXk-n, where A and B are :onstant m x m matrices, is discussed. Such a system is said to be robustly stable if it is ~ymptotically stable for all values of n E N. The following are established: The above system is robustly stable, if there exist a triple of positive definite matrices [P, R, W) satisfying either of the following Riccati-equations:

i) A'PBW-1B'PA + A'PA + W + B'PB + R = P

ii) B'PAW-1A'PB+ B'PB+ W + A'PA+ R = P

Several other conditions are derived. One generalizes the Lyapunov condition for the no­:lelay case (but is only sufficient): The above system is robustly stable, if either A or Bare nonsingular, and there exists positive scalars p and q such that ~ + ; = 1, and positive definite symmetric matrices X and Q satisfying a generalized Lyapunov equation pA' X A + qB' X B + Q = X.

In the second part we consider the singular perturbation of a parameter dependent :lifference equation with delay (dd): p. AXk = -Xk+1 + F(Xk-n) where F : Rm -+ Rm is a continuous nonlinear map, p. is a positive parameter, and AXk is the forward difference at time k. We suggest an approach which allows to study the global stability phenomenon as a singularly perturbed problem. The limit case p. = 0 gives an equationxHl = F(Xk_n) whith dynamics characterized by the discrete map F. The global stability in the limiting discrete system implies the global stability of the original equation for all positive values of the parameter. In particular, we established the following:

i) If 0 is a closed convex invariant domain (under the map F) then for arbitrary Xi E 0, with i = -n, ... , 0 the corresponding solution Xk of equation (dd) satisfies Xk E 0, k ~ 1.

ii) Assume that the image of any convex set by the map F is convex in the domain 0 and x, E 0 is globally attracting fixed point of the map F. Then limk_oo Xk = x, for every solution of equation (dd) with the initial conditions Xi E 0, i = -n, ... , 0, and every p. > o.

'Corresponding author: FAX: (404) 897-2997, E-mail: erik.verriestlDee.gatech.edu

177

Speaker: M.A. Kaashoek

Title: Column reduced rational matrix functions with given null-pole data in the complex plane

Abstract: Explicit formulas are given for rational matrix functions which have a prescribed

null-pole structure in the complex plane and are column reduced at infinity. A full parametriza­

tion of such functions is obtained. The results are specified and developed further for matrix

polynomials. Connections with Wiener-Hopf factorization will be discussed. The work reported

on is joint work with l.A. Ball. G. Groenewald. and l. Kim.

8 REDUCTION OF 2-D PERIODIC SYSTEMS TO CONSTANT

2-D SYSTEMS BY STATE-FEEDBACK AND FLOQUET-~APUNOV TRANSFORMATION

Tadeusz Kaczorek

W'arsaw University of Technology

Institute of Control and Industrial Electronics

00-662 Warszawa, Koszykowa 75, Poland

Consider a periodic 2-D 1 inear system descrlhed by the equation

..... here

x .. E Rn

is the local state vector :-it the point (l,j) lJ

U .. E nm is the input vector F~ F2 G~lj' G~. real IJ I J . i j' IJ

matrices of appropriate dimensions with entries depending on i

and j, Z+ is the set of nonnegative integers

It is assumed that

Fk . k

l+t 1P l·J+t 2 P2 F ij

ror k 1,2 and t I' t2 E Z+ C;k .

Ck

. l+tIP1,y·t;::p;:: 1 J

and witl10ut loss or gener'al it y that

r'ank C;k ("or' k ~ I,;> and all i ,j E N + 1 J

The mi nimal pair (PI' P2 1 or nonnegat i ve integers ( for

Pl+P2 is minimall is called the period of the system (1),

Let the state feedback be described by

ll .. 1,1

K .. x .. ~v I J IJ !.I

! , J E N j.

where K .. E nmxn and v .. is a new input vector I J I J

(2)

which

(3 )

ronsider thp. Op.w local st"te vpctor' i j defIned by t.he equation

i i -: Iii i,i

where L .. E Rnxri

is nonsingular for all lJ

the period (Pl,P2l, i,e.

L i +P1

,j+P2

::: Lij

(1 I

li,j), and periodic with

(5)

The problem under

the matrices F~., 1 J

periodic a matrix

consider'alian can be stated as follows. Given k

Gij for k ::: 1,2 of tl}, find a feedback gain

that the

equat ion

K .. lJ

transformed

and a nonsingular periodic matrix Lij such

closed-loop system Is descrIbed by the

zi +1, jQ:::A\7.i+l, j+A2 z l , J+l +8 1 Vi +1, j+82 v i . j+1 (61

wIth constant (Independent of 1 and j) real matrices A1

,A2

,B1

,B2

,

Necessary and suffIcient conditions for the exstence of a

solution to the problem will be established. An algorithm for

the reduct ion of 2-D periodic systems to their canonical forms

wIll be presented and Illustrrtted by an numerical example.

------------............ Time-Variant Displacement Structure:

Theory and Applications

THOMAS KAILATH

Information Systems Laboratory, Stanford University, Stanford, CA 94305

e-mail: tk@rascals.stanford.edu, phone [41.5J 723-3688, FAX [415J 723-8473

179

The displacement structure concppt provides a powerful and unifying tool for exploiting

the inherent structure in diverse problems in signal processing and mathematics. In this talk,

we show how to extend the notion to the time-variant setting and describe several applica­

tions. In particular, we describp a time-variant recursivp (square-root or array) algorithm

that allows for an efficient time-updating of the Cholpsky factor of a time-variant structured

matrix. The algorithm leads naturally to a time-variant discrete transmission-line cascade,

i. e., a cascade of first-order sections, each composed of a rotation matrix followed by a storage

element and a tapped-delay filter. As expected on physical grounds, the cascade has certain

blocking properties that turn out to be equivalent to time-variant interpolation conditions.

This fact allows us to give a nice computational solution to a general time-variant interpola­

tion problem of the Hermite-Fejer type, as well as to several matrix completion problems such

as the band problem and the strong Parrott problem, and to certain model validation prob­

lems in system identification and control. A feature of the approach is the strong parallel to

the simple (Gaussian elimination plus displacement structure) approach in the time-invariant

case. In fact, in sOlIle sense, the time-invariant results arp morp naturally seen as specializa­

tion of the time-variant results.

We also describe applications in adaptive filtering. More specifically, we show that the

time-variant array algorithm, when applied to the structured tin1P-variant coefficient matrix of

the normal equations, collapses to the now widely studied QR algorithm, with the additional

bonus of providing a parallel (systolic) procedure for extracting the desired weight vector.

Connections with time-variant structured state-space modpls and the efficient Chandrasekhar

recursions are the basis for our study of adaptive filtering. Somewhat ironically, we now

recognize that it was the time-variant displacement structurp that first arose in our work in

1972 on the Chandrasekhar equations as an alternative to the Riccati pquations of the Kalman

filter. This talk is based on work done with A. H. Sayed, H. Lev-Ari, and T. Constantinescu.

A

CYCLE GENERATING EQUATIONS AND A NEW PROOF OFALEVY CONJECTURE

CONCERNING THE POSITIVENESS OF TRANSITION PROBABILITIES

Sophia Kalpazidou

Aristotle University, Faculty of Sciences,

Department of Mathematics, 54 006 Thessa1oniki, Greece rtL~'''.~''-: 003011- 6>216,!-

'Al<. 00 3~31- l..fr4/7

Abstract

~yc1e generating equations are defined to be those relations

iefining directed cycles cE'(f; and positive numbers {wc,CE"G'}

such that Kirchhoff.'s current law holds. We show that the

cransition probabilities Pij(t) of denumerable continuous

Jarame ter Marko v processes accep t i ng in vari an t measures

Jrovide cycle generating equations, and then apply this

lpproach to prove a conjecture of Paul Levy concerning the

)ositiveness of the Pij(t)'s .

r I

Feedback control problems in Banach spaces M.1. Kamenskii

Dep. of Mathematics, Voronezh State University, 394693 Voronezh (Russia).

P. Nistri Dip. di Sistemi e Informatica, Universita di Firenze, via S. Marta n. 3,50139 Firenze (Italy).

Fax(+55) 4796363. Phone(+55) 4796356. E-mail: pnistri at ifiidg.bitnet.

V.V. Obukhovskii Dep. of Physics and Mathematics, Voronezh State Pedagogical Institute, 394611 Voronezh (Russia).

P. Zecca Dip. di Sistemi e Informatica, Universita di Firenze, via S. Marta n. 3,50139 Firenze (Italy).

Fax( +55) 4796363. Phone( +55) 4796256. E-mail: pzecca at ifiidg.bitnet.

ABSTRACT. We will consider the following semilinear feedback control system

f yi(t) = Ai(t)Yi(t) + fi(t, Yl(t), .. , Yn(t), Ul(t), .. , um(t)), Yi E Xi, t E [0,1]'

1 Uk(t) E Vk(t, YI(t), .. , Yn(t)) C Gk t E [0,1], k = 1, .. , m,

i = 1, .. , nj

181

(1 )

where Xi, i = 1, ... , n, and Gk, k = 1, ... , m are Banach spaces. (The spaces Xi are assumed to be separable).

Given an acyclic compact set K eX, our goal is to provide conditions on Ai, /;, Vk and on Fi , given by Pitt, Xl,"" Xn) = f;(t, Xl,· .. , Xn, VI(t, Xl,.··, Xn)"'" Vm(t, Xl, ... , xn)), in order to guarantee the existence of a mild trajectory y(t) = (YI(t), ... , Yn(t)), t E [0,1]' of (1) (with corresponding controls Uk(t) E Vk(t,YI(t), ... ,Yn(t)), t E [0,1], k = l, ... ,m), which satisfies the boundary condition

y(l)-y(O) E K. (2)

The approach is based on an abstract existence result (Theorem 2.3 of [1]) concerning the solvability of inclusions in Banach spaces. In fact, we first convert the nonlinear boundary value control problem (l )-(2) into an equivalent system of multivalued equations

{ Y E Dx,y) X E G(x,y)

where F and G are suitable multivalued maps associated to (1) and (2) respectively, Y E Y is the trajectory y(t), t E [0,1]' and X E X is its initial condition y(O).

Then we solve the first inclusion with respect to Y considering X as a .fixed parameter in a suitable ball 13(O,r) C X. Replacing the obtained solution set S(x) to Y in the second equation we get a multivalued map T : 13(0, r)-<>X, given by T(x) = G(x, S(x)), and we look for the fixed points of T which represent the solutions of (1 )-(2).

REFERENCES [1] Nistri P., Obukhovskii V.V. and Zecca P., On the solvability of Systems of Inclusions

Involving Noncompact Operators, submitted for publication.

2

Nonnegative-Definite Solutions of Algebraic Matrix Riccati Equations

with Nonnegative-Definite Quadratic and Constant Terms

Hiroyuki Kano l and Toshimitsu Nishimura2

1) Department of Information Sciences, Tokyo Denki University, Ishizaka, Ha­toyama, Hiki-gun, Saitama 350-03, Japan. (e-mail) kano@j.dendai.ac.jp (fax) +81-492-96-6403.

2) Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229, Japan.

Abstract

In this paper, algebraic matrix Riccati equations of the following form,

FP + PFT + PHT HP+GGT = 0

are analyzed with reference to the stability of closed-loop system F + P HT H. First,

a necessary and sufficient condition is established for the existence of stabilizing

solution P, (not necessarily P, :2: 0). It is then shown that P, :2: 0 exists if and

only if F is asymptotically stable and the H 00 norm of transfer matrix is less than

one. For the existence of P, > 0, we need an additional condition that (F, G) is

controllable.

Then, starting with the stabilizing solution P, :2: 0, the existence of other

nonnegative-definite solutions P are established including the anti-stabilizing so-

lution Pm. Here the observability of (H, F) plays the key role. The solutions P,

and Pm are shown to be the minimal and maximal solution respectively, and all

the solutions are proved to constitute lattice structure. The solution structure of

this equation is thus clarified which, in a sense, is in contrast with that ari~ing in

the standard LQG problems. For the analysis, we employ the so-called algebraic

method which is based on the eigenvalue problem of Hamiltonian matrix.

RELATIONSHIP BETWEEN GRASSMANN AND KRONECKER INVARIANTS OF MATRIX PENCILS

Nicos Karcanias and John Leventides Control Engineering Research Centre City University, Northampton Sq, London EC1V OHB E.MAIL: N.Karcanias@uk.ac.city

ABSTRACT

183

The theory of invariants and structural properties of linear state space type systems is intimately related to the theory of Kronecker invariants [1} of appropriate system pencils [2}-[6]. Another important family of invariants of linear systems, are the different types of rational Grassmann Representatives (R(s)-GRs) and associated PWcker matrices [7} of different rational vector spaces associated with the system; these invariants are introduced by the classical PWcker embedding [8}, and provide a complete characterisation of rational vector spaces, which is equivalent to the echelon basis characterisation of rational vector spaces [9]. The Grassmann invariants are essential ingredients of the algebraic geometry framework, developed for the study of a number of control problems [10}, [11]. The aim of this paper is to establish the links between the Kronecker and PWcker invariants of matrix pencils: this work will provide the means for an alternative characterisation of various properties of singular systems [12}, in terms of the properties of the corresponding PWcker invariants, as well as enable the extension of the algebraic geometry framework for determinantal type control problems [8} defined on singular systems.

For a general pencil SF-GE Rmxn[s} with p=rank.r,j (sF-G) and X/F, G) and X ,(F,G) column, row rational vector spaces, the notion of column-Grassmann Representative [13}, [7} ((F,G)-cGR) g, (F,G), row-Grassmann Representative (F,G)-rGR) g, (F,G) are introduced, as well as, the pfijcker invariants P/F,G), P,(F,G) defined as the coefficient matrices of the g, (F,G) g, (F,G) polynomial multivectors respectively. The family of all (F,G)-cGR ((F,G)-rGR) are module R(s) equivalent and a coprime g/F, G) (g,(F,G)) is called canonical, as well as the corresponding PWcker matrix. It is shown that all PWcker matrices P/F,G), (P,(F,G)) (canonical or not) have the same column (row) space and its dimension is uniquely defined by the set of column (row) space and its dimension is uniquely defined by the set of column (row) minimal indices of the penCil. A canonical factorisation of the p-th compound [13} of the pencil Cp(F, G) ACp(sF-G) in terms of the finite zero polynomial and the canonical g/F, G) g,(F,G) vectors; this factorisation allows a complete parameterisation of all GR's and leads to a simple new characterisation of the various families of matrix pencils (each family is characterised by certain combination of Kronecker invariants). Central to this classification are the degree properties g,(F,G) g/F,G) as well as the ranks of the associated PWcker matrices. This new classification of matrix pencils provides the means for an alternative characterisation of system properties based on the properties of the Grassmann invariants of the corresponding pencils.

REFERENCES

1. Gantmacher, F.R. 1959, "The Theory of Matrices", Vol. 2, Chelsea, New York. 2. Kalman, R.E. 1972, "Kronecker invariants and feedback" In Weiss L. (ed), Ordinary Differential Equations, Academic Press, New York, pp 459-71. 3. Rosenbrock, H.H. 1970, "State Space and Multivariable Theory", Nelson, London 4. Thorp, J.P. 1973, "The singular pencil of linear dynamical systems" Int J Control, Vol. 18, pp 577-96. 5. Karcanias, N. and MacBean, P. 1980, "Structural invariants and canonical forms of linear multivariable systems", Proc. 3rd, IMA Conf. Control Theory, Academic Press, London pp 257-82. 6. Jaffe, S. and Karcanias, N. 1989, "Matrix Pencil Characterisation of almost (A, B)-invariant subspaces: A classification of geometric concepts", Int. J. Control, Vol 33, pp 51-93. 7. Karcanias, N. and Giannakopoulos, C., 1984., "Grassmann invariants, Almost zeros and the determinantal zero pole aSSignment problems of linear systems", Int. J. Control, Vol. 40, pp 673-698. 8. Griffiths, P. & Harris J, 1978 "Principles of Algebraic Geometry", John Wiley & Sons, New York. 9. Fornbey, G.D. 1975, "Minimal bases of rational vector spaces with applications to multivariable linear systems", SIAM J. Control and Optimization, Vol. 13, pp 493-520. 10. Brochett, R.W. & Byrnes, C.I., 1981, "Multivariable Nyquist criteria, Root Loci and Pole Placement: A geometric viewpoint". IEEE Trans. Aut. Control, Vol. AC-26, pp271-285. 11. Giannakopoulos, C. & Karcanias N. 1984, "Pole assignment of strictly proper and proper linear systems by constant output feedback", Int. J. Control, Vol. 42, pp 543-565. 12. Lewis, F.L. 1986, ':.4 survey of linear singular systems", Circuits Systems & Signal Process, Vol. 5, pp 3-36. 13. Marcus, M. 1973, "Finite Dimensional Multilinear Algebra" in two parts, Parcel & Deker, New York.

GrsKro 11

184

A Calculation of Minimal Basis in Polynomial Vector Space

vVataru KASE

Dept. Mech. Eng. Shizuoka lnst. Sci. & Tech. 2200-2, Toyosawa, Fukuroi, Shizuoka, 437 Japan E-mail:kase@me.sist.ac.jp

Katsutoshi TAMURA Peter N. NIKIFORUK

Dept. Mech. Eng. Sophia Univ. 7-1, Kioi-cho, Chiyoda-ku, Tokyo, 102 Japan

Abstract

Dean Mech. Eng. Univ. Saskatchewan Sa.skatoon, Saskatchewan, S7W OWO Canada

The problem discussed in this presentation is of finding a left coprime polynomial matrix pair D(s) and N(s) satisfying D(s)N(s) = N(s)D(s) for given matrices D(s) and NIiJ. This ~n be easily formulated as a problem of finding a minimal orthogona1 basis [N(s) - D(s)] for [DT(s) NT(s)f. It is well known that the row degrees of D( s) are given by observability indices of an associated state space realization of D-l(S)N(s), so the indices are also obtained from a realization of N(s)D-l(S). Let (A,B,C) be a controllable realization of N(s)D-l(S). A main result of this paper is to show that a real annihilating matrix of the observability matrix of (A, C) will provide the minimal orthogonal basis [N(s) -D(s )]. A systematic way to obtain the real annihilating matrix is given. According to the choice of the observability indices, D( s) is determined as a both row and column proper matrix or as a column proper lower triangular marix.

This method is also applicable to polynomia1 matrix operations such as obtaining a greatest common right (ldt) divisor of two matrices and a solution of a Bezout identity.

•

185

.tudy on a new criterion of the stability of structured uncertainty

Takeshi KAWAMURAt & Masasuke SHIMA!

t Department of Electrical Engineering, Kitami Institute of Technology, 165 Koen-cho,

Kitami, Hokkaido, 090, Japan.

E-mail address:kawamura@yoshida.elec.kitami-it.ac.jp

t Department of Precision Engineering, Faculty of Engineering, Hokkaido University,

North 13 West 8, Sapporo, Hokkaido, 060, Japan.

ABSTRACT The structural uncertainty is always expressed by Interval

matrices or Interval polynomials. Several papers focused on the stability of

Interval polynomials, but their conditions were no more than the multilin­

ear parameter condition or the linear parameter condition. If parameters,

describing structured uncertainty, appear multilinearly in the coefficients of

characteristic polynomial, we can use some tools as the Mapping theorem

for the stability test. But parameters usually appear nonlinearly in the

characteristic polynomials of Interval matrices. So we can't use convenient

tools for the stability test of the characteristic polynomials.

In this paper, we present a new criterion of the stability of Interval poly­

nomials. Defining the monotonicity of multi variable function, the Mapping

theorem is extended to the general case where parameters appear nonlin­

early. Using this extended mapping theorem, we can check the stability of

interval polynomials where parameters appear nonlinearly. For one param­

eter functions and multi-parameter functions, we will show how to check

the monotonicity. And if the function is monotone of parameters, we can

check the stability of structured uncertainty using all endpoints of inter­

val parameters. Using this result, it is possible to check the stability of

characteristic polynomials of Interval matrices.

~6

Dynamical aspects of nonlinear observation

H.W.Knobloch

The purpose of the lecture is to present a new angle on the design of nonlinear

observers. The formal observer shape is the usual one, i.e. the differential equation

governing the observer dynamics is made up of the plant equation plus a prediction­

correction term proportional to the estimation error e. The choice of the proportional

factor which we propose however does not follow traditional lines in observer design.

Our motivation is the desire to create a global attractive invariant manifold for the

closed loop system which exhibits a geometric property ('existence of an asymptotic

phase'). If advanced invariant manifold theory can be applied one then can draw

conclusions concerning the behaviour of e(t) which come close - and in one respect

even go beyond - familiar results of the linear theory.

There is a wide range of applications including regulator design and on-line param­

eter identification.

ON ASYMPTOTIC RELIABILITY FUNCTIONS OF SOME SERIES-PARALLEL SYSTEMS

KRZYSZTOF KOLOWROCKI DeplU'tment of Mathematics, Maritime Univel'Bity

Morska 83,81-962 Gdynia, PolBDd

ABSTRACT

In the reliability investigation of the large scale systems the problem of the complexity of their reliability functions appe­ars. This problem may be approximetly solved by the 8B8umption that the number of the system components tends to infinity and finding the limit reliability function of this system. In this paper a three-element closed cl8B8 of limit reliability func­tions for regular homogeneous series-parallel system is fixed. The system is such that the number of its series components is of less order than the logarithm of the number of its parallel compo­nents. Moreover , one example of the considered system and its limit reliability function is given. The result may be useful in the reliability evaluation of large ho­mogeneous systems and may be the origin in the reliability inve­stigation of large systems with unalike components.

KEY WORDS - Limit Reliability Function, Series-Parallel System

187

:8

OPTIMAL FEEDBACK SWITCHING METHOD FOR LINEAR CONTROL SYSTEM

KondraLenko Y.P., Timchenko V.L. Nikolaev Shipbuilding InsLiLuLe, Ukraine.

The reporL deals wiLh Lhe meLhod of ploLLing a conLrollable LrajecLory of a dynamic sysLem movemenL deseribed by a seL of linear differenLial equaLions wiLh changeable parameLers

X=A(L)X+B(L)U, where X-vecLor of coordinaLes;

A(L) ,B(L)-array of coefficienLs; U-vecLor of conLrolling effecLs.

(1)

The deri vaLion of Lhe full-conLrollabi liLy condi Lions of Lhe non-sLaLionary sysLem (1) based on Lhe expansion of, Lhe vecLor' X aL every poinL of Lhe conLrol inLerval [O,T] inLo a Taylor series and allowing Lo exLend Lhe noLion of conLrolla­b iIi Ly noL onl y on Lhe componenLs of Lhe vecLor X bUL aiso on i LS der i vaL i ves

(i,)

X (i=1,2 ... ,m) .. C, where C - consL. The moL ion in Lhe dynamic sysLem (3) is represenLed by

Taylor series as a LransiLion from Lhe veCLor X reached a sLeady-sLaLe value relaLively Lo Lhe derivaLive of Lhe i-Lh order LO Lhe one reached a sLedy-sLaLe value relaLively Lo Lhe deri vaLi ve of Lhe i+1 order.

The conLrol effecL in Lhe feedback sysLem will Lake Lhe

form: U. (p)=W (p)X(p), , ,

• -1 2 W2(p)=-(AB+pB+B) (A +A),

G=1,2,3), (2)

(X=O);

(X=O);

Wg(p)=-(A 2B+2AB+B+pAB+pB+p2B) -1 (A3 +3AA+A ), (X=O); B,B,A,ii.- Lhe firsL and second derivaLives of maLrices B

and A, reprecLively. The opLimizaLion of following kind is examing

I(U)= I 12[Xl (L)VX(L)+I'U2(L)]dL, (3) 1

1

where V-posiLive-definiLe maLrix; 1'>0; L l' L 2 - Lhe momenLs of swi Lching conLrol funcLion.

The funcLion of coordinaLes change for known insLanL L1 (sLeady-st.at.e mot.ion relat.ively t.o t.he derivat.ive of t.he i-t.h order) and conLrol funct.ion are defined at. t.he i·irst. st.ep of OpL i mizat.i on.

The equat.ion of following kind I (U) .. O

for insLanL L is det.ermined aL Lhe second st.ep. Thus, we i-ecei ve Lhe pi ece- opt. i mal moLion LrajecLory of

cont.rollable object. according t.o crit.erion (3).

Key words: linear syst.em, swi Lching met.hod, opLimal conLrol

189

Gnostical Modelling of Uncertainty

Pavel Kovanic

UTIA CSAV P.O.Box 18 18208 Prague, Czechoslovakia

fax: +42 2 847452 tel.: +4228152230 Email: kovanic@cspgasl1.bitnet

The gnostical theory of uncertain data is a nonstatistical mathematical theory of uncertainty developed

to cope especially with small samples of strongly dispersed data having no a priori known statistical

model. This theory is based on elementary axioms:

• data structure - Cartesian product of two Abel groups,

• uncertainties - analytical operators over the data structure,

• data composition law - additive with respect to data entropy.

Theoretical results include the following:

• Theory of an individual uncertain datum: entropy increase caused by uncertainty, information

loss caused by uncertainty, entropy <-+ information conversion, distribution function of the uncer­

tainty ("probability" distribution), Riemannian metrics for uncertainty, geodesics and variation

theorems =} ideal cycle of data transformation.

• Theory of small data samples.

Practical consequences of the theory are

• maximization of information,

• robustness:

- with respect to a priori assumptions of statistical nature (none applied),

- with respect to outliers - inherent, natural Riemannian metrics,

• it really works.

The last statement will be documented by results from difficult applications such as robust modelling of

economial processes, reliability and survival modelling, random quality control and intelligent sensors.

I

j

190

SECOND-ORDER INFORMATION IN RECURSIVE STABILITY-TEST ALGORITHHS

WIESLAW KRAJEWSKI

Systems Research Institute, Polish Academy of Sciences

ul. Newelska 6, 01-447 Warsaw, Poland

and

ANTONIO LEPSCHY UMBERTO VIARO

Department of Electronics and Informatics, University of Padova

via Gradenigo 6/A, 35131 Padova, Italy

Tel. +39(0)49 8287603 - Fax +39(0)49 8287699

Abstract - The paper is concerned with the recursive algorithms for the

generation of sequences of polynomials of descending or ascending

degree. In fact, many signal processing, circuit theory and control

procedures may be formulated in these terms (cL, e.g., the Levinson

algorithm and the Routh algorithm). With reference to the general form

of these algorithms, we analyse the relationship among the second-order

indices (energies or autocorrelation coefficients) that may be

associated with the polynomials in the sequence. More precisely, a

recursive formula is derived which expresses the indices for the

polynomial of degree i-1 as linear combinations of the indices for the

polynomial of degree i. This allows us to prove some interesting

properties of the mentioned algorithms in an efficient way.

Key words: Linear systems, Zero-location procedures, Model reduction,

Digital signal processing.

Address for correspondence:

ANTON IO LEPSCHY Department of Electronics and Informatics, University of Padova Via Gradenigo 6/A, 35131 Padova (Italy) Tel. +39(0)49 8287612 - Fax +39(0)49 8287699

191

Control under a shortage of information

N.Krasovskii, A.N.Krasovskii

This report is devoted to problems of controling a dynamical system in conditions of in­

complete information on the phase states of the plant and on disturbances. The considered

systems are described by ordinary differential equations. The quality index of the process

is a functional of the realization of motion of the object, of the realization of the control

actions and of the disturbances. The considerations are concerned mostly with the cases

in which the quality index is the so-called positional functional whose strict definition is

given. In these cases the problem is reduced to control actions which guarantee the mini­

max of the given quality index. The proposed approach formalizes the process of control

as a differential game. Two main classes of positional strategies are defined: pure and

mixed strategies. A pure strategy forms deterministic control actions basing on current

information about the realized phase states of the system. A mixed strategy generates

control actions using a suitable stochastic mechanism. We formulate theorems of the exi­

stence of value and saddle point for the considered differential games in the chosen classes

of strategies. The main topic of the report is one effective method for constructing optimal

feedback control actions. This method is given for a somewhat restricted but rather typical

class of problems. The idea of this method is prompted by the so-called program stochastic

synthesis. In order to solve the original problem of feedback control we construct some au­

xiliary problems of open-loop control for appropriate stochastic systems. We construct in

this way convex hulls for a sequence of some functions that are obtained from the solutions

of the auxiliary problems. This sequence provides the basis for constructing the required

optimal control actions. Two cases are considered in detail. The first of them is the case

of an integral functional. In the second case the quality index estimates the deviation of

the motion from a given trajectory in the whole time interval of the process. Computer

simulation results for illustrative model systems are presented.

192

(short version without proofs)

Parameter Adaptive Control: A Solution to

the Overmodeling Problem

Gerhard Kreisselmeier')

Abstract

This paper presents a parameter adaptive controller that will stabilize and asymptot­ically regulate any unknown linear time-invariant, controllable and observable plant, given only an upper bound on its order.

o)Dept. of Electrical Engineering, University of Kassel, Kassel, Germany. Phone: 0561/804-6367; Fax: 804-6327

9 November 1992

Symbolic Computation and Nilpotent Systems *

Arthur J. Krener

Department of Mathematics

University of California

Davis, CA 95616·8633

USA

Temporary address to 6/93: Institute for Systems Research

A. V. Williams Bldg University of Maryland

College Parle, MD 207.2-3311, USA

Tel: 301-.05-6553

Fax: 301-31.-9920

email: lerener@src.umd.edu

Nilpotent control systems are particularly well-suited to the use of sym­

bolic computational packages such as MAPLE and MATHEMATICA. This is because the integral curves of a family of nilpotent vector fields can be

reduced to simple quadratures. Moreover since arbitrary control systems can be approximated by nilpotent control systems, the insights gained by study­

ing the latter can be generalized to the former. We illustrate this by a study

of the set of accessible of a sequence of low-dimensional systems leading to

chattering phenomenon of A. T. Fuller .

• Abstract submitted to MTNS Regensburg 1993 for invited session "Computation and Nonlinear Control"

193

194

Lag indices of a dynamical system

Margreet Kuijper Mathematics Institute

University of Groningen Postbus 800

9700 AV Groningen The Netherlands

email address:margreet@math.rug.nl

In recent publications [2,3J the concepts of controllability and observability have been defined in terms of a system's behavior (= the set of time-trajectories of relevant system variables). In this setting controlla­bility expresses an intrinsic property of the behavior that reflects the ability to influence its trajectories. Unlike the classical notion, the above type of controllability is not defined in terms of a specific kind of representation; in particular, it does not relate to absence of redundancy in the representation, as is the case in the classical set-up.

Similar to the notions of controllability and observability, the concepts of controllability indices and ob­servability indices are classically defined in terms of a standard state space form. The purpose of this talk is to give these indices a much larger system-theoretic scope by introducing them as integer invariants relating to the behavior of a (linear time-invariant) system. In doing this, we consider the (nonrestrictive) case in which the behavior is represented by a set of high-order differential/difference equations acting on the relevant variables. The minimal lag structure that is implicit in such a representation gives rise to a set of lag indices that are invariants of the behavior. When the behavior can be represented by a standard state space representation the lag indices turn out to be equal to the classical observability indices. When the behavior can not be represented by a standard state space representation, as e.g. in the case of a nonproper transfer function, the lag indices, being invariants of the behavior 1 can of course still be defined. This enables us to generalize the notion of observability indices: the lag indices are considered generalized versions of the observability indices. The question arises how these indices can be computed from representations other than the standard state space representation. We will answer this question for various types of " generalized" first-order representations, including the well-known descrip­tor representation. With minor nonredundancy conditions on the representations, the expressions take a particularly nice form in terms of Kronecker indices of matrix pencils.

In an analogous but not completely dual way we define the notion of controllability indices in a behav­ioral setting and derive similar results. In particular, the expressions in [IJ that generalize controllability indices for a (limited) descriptor form are obtained as a special case.

REFERENCES

1. Kucera, V. and P. Zagalak (1988) Fundamental theorem of state feedback for singular systems, AutomatIca, Vol. 24, No.5, pp. 653-658

2. Willems, J .C. (1986) iFrom time series to linear system. Part I: Finite-dimensional linear time invariant systems, A utomatica, Vol. 22, pp. 561-580

3. Willems, J.C. (1991) Paradigms and puzzles in the theory of dynamical systems, IEEE Trans. Aut. Control, Vol. AC-36, pp. 259-294

-

Abstract

Lattice Structure for the Realization of the Instrumental Variable Method

Anton Kummert

Atlas Elektronik GmbH POJtjach 44 85 45. D-2800 Bremen 44, Germany

195

In the literature, the least squares (LS) method has been introduced for static and for dynamic models in the context of adaptive filtering and system identification. It is easy to apply but has a substantial drawback: the parameter estimates are consistent only under restrictive conditions. To overcome this drawback, the LS method has been modified in different ways which led to the class of instrumental variable (IV) methods. The general idea is to modify the so-called normal equations in such a way that consistent parameter estimates can be obtained under less restrictive conditions.

One of the main problems involved with IV methods is the efficient inversion of a nonsymmetric correlation matrix. In the present paper. a lattice-type algorithm is introduced which solves this problem for certain applications of the IV method in an order- and time-recursive manner. It can be considered as a generalization of the so-called least squares lattice (LSL) algorithm.

The IV method was introduced by Reiersol and has been applied by different authors to various problems in control engineering. The name IV method is not only applied to a certain algorithm but it comprises a whole class of parameter estimation techniques.

In the context of this paper, we consider linear systems of equations of the type

R(k)9(k) = -r(k), (1 )

with

[ 1 [ 1

T k w( 1 - 1) y(! - 1)

R(k) = L.\k-.: : .=0 w(t - S) Y(l - S)

.VE{1.2 .... }. (2)

k [ w(i -1) 1 r(k) = L .\k-i . : y(i).

.=0 w(! - .V)

(3)

and 9(k) = [6dk) ..... 6.v(k)]T. (4)

where .\ is an exponential weighting factor. with 0 < .\ < 1, w( k) and y( k) are known causal discrete-time signals, and 9(k) is the unknown parameter vector. A direct inversion of R( k) in order to solve the linear system of equations (1) would require O( _V3) operations at every time-instant k. This computational complexity usually cannot be accepted for real time applications. Therefore, a lattice-type algorithm is proposed which solves (1) order- and time-recursively. Its computational complexity is proportional to N.

196

OBSERVING DISTRIBUTED SYSTEMS THROUGH FINITE-DIMENSIONAL

SENSORS

A.B. Kurzhanskii Moscow State University, Moscow Russia

e-mail kurzhan@lvk.cs.msu.su kurzhan@kur.ac.msk.su

30 March 1993

One of the basic problems of observability theory for distributed systems is to estimate the ,tate·space variable (a distribution) through on-line measurements corrupted by unknown but bounded noise. A typical situation motivated particularly by mathematical modelling for ecol­Jgy and technology is when the measurements are generated by sensors with finite-dimensional Jutputs.

This presentation deals with the solvability of the problem in a stronger or a weaker sense for parabolic and hyperbolic systems (the observability issue), when the Qbservations arrive both in a finite and in an infinite horizon.

In general, the observability property holds for the considered type of systems only when the sensors are nonstationary, particularly, of a pointwise "scanning" nature.

A numerical computation of the "guaranteed estimates" faces difficulties, as the original prob­lem, especially in the absence of measurement "noise", is ill-posed and requires regularization.

A scale of regularization schemes is naturally introduced here through perturbation techniques, by constructing an array of auxiliary estimation problems for systems with perturbed parameters and constraints on the unknown but bounded "noise".

A particular regularizing scheme arises through the problem of optimalizing the observation by selecting, in an open-loop or feedback mode, of the measurement curves that minimalize the measurement error. Under observability assumptions and some additional requirements, with the measurement error tending to zero, these optimal curves may stably converge and yield a stable convergence of the respective estimates to the solution of the original ill-posed problem.

References

[1] A.B.Kurzhanskii, A. Yu Khapalov The State Estimation Problem for Parabolic Systems. Birkhauser Series in Numerical Mathematics v.100. Birkhauser, Boston, 1991.

[2] A.B.Kurzhanski, I.F.Sivergina Variational Regularization Schemes; Quasiinvertibility and the Guaranteed Estimation Problem for Noninvertible Systems. Journal of Computat. Mathematics and Mathematical Physics, N ll, 1992.

[3] A.B.Kurzhanskii Observability Theory for Distributed Systems. Proc. Conf. on Control Theory for PDE's, Trento, Italy, 1993.

Abstract

Quasiinversion, Regularization and the Observability Problem

A.B. Kurzhanski 1. Sivergina

197

This paper deals with the problem of estimating the initial state of a distributed field on the

basis of measurements generated by sensors. The original ill-posed problem is regularized here

through an auxiliary "guaranteed estimation" problem. This yields a stable numerical procedure

and also allows to establish a unified "systems· theoretic" framework for treating regularizers

in general. Particularly the important point is that for finite dimensional sensor outputs a

necessary condition for the existence of a stable numerical solution is the observability property

which ensures existence of solution in the absence of measurement noise.

A.B. Kurzhanski: MoscoW, Russia, V-234

I. Sivergina:

Moscow State University (MGU) Faculty of Computational Mathematics & Cybernetics (VMK)

Institute for Mathematics and Mechanics Sofia Kovalevski st. 16 GSP - 384, 620066 Ekatherinburg (Sverdlovsk) Russia

98

A Fast, Reliable and Stable Algorithm for the Inversion of Mosaic Hankel and Toeplitz Matrices

George Labahn University of Waterloo, Dept.of Computer Science

Waterloo, Ontario, N2L 3Gl, Canada glabahn@daisy.uwaterloo.ca

Stan Cabay University of Alberta, Dept.of Computer Science

Edmonton, Alberta T6G 2Gl, Canada cabay@cs.telecom.canada.ca

There are a number of fast algorithms for the inversion of matrices having a mosaic Hankel or Toeplitz structure. In most cases, these algorithms as­sume exact arithmetic. However, these algorithms have efficiency problems when being implemented in exact computation domains (such as MAPLE for example) because they do not consider coefficient growth of numerics. On the other hand when these algorithms are implemented in fixed precision numeric domains stability problems occur.

In this talk we describe an inversion algorithm that is both fast and (weakly) stable. The techniques used rely Oil stable computation of multi­dimensional Pade-like systems. These are certain matrix rational approxi­mants to a matrix of power series which completely determine the inverse of the mosaic Hankel or Toeplitz matrix.

Eigenvalue problems for operator Nevanlinna functions

HEINZ LANGER

Technische Universitat Wien Institut fiir Analysis, Technische Mathematik

und Versicherungsmathematik

We consider eigenvalue problems L (A) x = 0 for an operator function L (A) of the form

here B is a bounded, C a bounded and selfadjoint and A is a possibly un­bounded selfadjoint operator in some Hilbert space 'H. Further, we suppose that there exists some AO E IR, such that A » Ao, C «: Ao. Then in the integral representation of the operator N evanlinna function - L (A) -I:

'" -L(A)-I = J dF(t)

t - A -'"

'" the operator J dF (t) is strictly positive. It follows that if e.g. 0' (A) is >'0

discrete (that is the resolvent of A is compact) then the eigenvectors of L corresponding to the eigenvalues in [AO, 00) form a complete system or a basis (in some sense) in 'H. Corresponding statements about the spectrum of Lin (-00, AO) are proved.

As a "linearization" of L the operator

plays some role. The results can be applied to boundary eigenvalue problems

of the form ~y" (x) -AY(X) - c:(~)Y~'j = 0 on [0,1]' y(O) = y(l) = 0, where

e.g. c(x) 2'O,u(x) <;Oandu,cEL"'(O,l).

(Joint work with V. M. Adamjan)

199

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200

Smart Material Based Control Schemes for Vibration Suppression of Dynamical Systems with Uncertain Excitation

George Leitmann College of Engineering

University of California at Berkeley Berkeley, CA 94720, USA

Phone: + 1-510-642-3984, Fax: + 1-510-642-6216

Eduard Reithmeier Automation & Medical Systems Technology

Bodenseewerk Geriitetechnik GmbH Postfach 10 11 55, 7770 Uberlingen, GERMANY

Phone: -t- 49-7551-89-2344, Fax: + 49-7551-89-2822

April 20, 1993

Abstract

Some smart materials are known to change their mechanical properties if sub­jected to certain external stimuli. An example of such materials are so-called ER Fluids; their stiffness and damping can be varied very rapidly (microseconds) within prescribed bounds by applying an electrical field. An appropriate change in these properties may be used for attenuation of undesired vibrations of dynamical systems over wide frequency ranges. Based on a semi active control scheme acting on such smart materials, a vibration suppression method is proposed for dynamical systems subjected to unknown but bounded disturbances. On the one hand, a semiactive control scheme is derived by employing Lyapunov stability theory. For one dimen­sional cases Lyapunov stability theory can be employed readily to investigate the stabilizing properties of the controller; and, given the bounds of the unknown ex­citation forces of the system, a worst case analysis yields a guaranteed region of ultimate boundedness for any realization of the excitation force. On the other hand, there is a possibility of designing a control scheme by optimizing the ratio between the energy flowing into and out from the system. Examples of mechanical systems are presented for one and two-dimensional cases, and simulations are carried out for discontinous and stochastic excitation forces. The different approaches to designing a control scheme are also discussed by means of these examples.

Abstract

BEZOUTIANS AND FACTORIZATIONS OF

RATIONAL MATRIX FUNCTIONS, AND

MA1RIX EQUATIONS

Leonid Lerer Department of Mathematics Technion -Israel Institute of Technology 32000 Haifa, Israel

Leiba Rodman Department of Mathematics The College of William and Mary Williamsburg, VA 23187-8795, USA e-mail: LXRODM@WMVMl.BITNET

201

The concept of Bezoutian of two rational matrix functions is introduced, thereby extending this concept (previously studied in the framework of matrix and operator polynomials and analytic functions) beyond the class of analytic functions. In the previous paper by the authors, basic properties of the Bezoutian have been studied, and a key result has been proved providing a description of common zeros (suitably understood) of two regular rational matrix functions in terms of the kernel of the Bezoutian.

Here, we focus on the connections between the Bezoutian on the one hand, and matrix equations and factorizations of rational matrix functions on the other hand. It turns out that the Bezoutian can be characterized as a solution of certain system of linear matrix equations, as well as a solution of a quadratic matrix equation. This latter connection allows us to express factorizations of rational matrix functions (given in realized fohns) in terms of the Bezoutian.

202

An invariant manifold approach for robust control design and applications

J. LEVINE' P. ROUCHONt Y.CREFFt

Abstract

In this paper, we consider a class of nonlinear systems having two invariant manifolds in open loop, corresponding to a gap in the spectrum of the tangent linear approximations of the controlled vector field at each point of an open subset of the equilibrium manifold. One of these submanifolds may be interpreted as the fast one, assumed to be hyperbolically stable, and the other one is slower, not necessarily stable and possibly containing stationary bifurcations of low codimension, when parametrized by constant controls. This last subman­ifold is assumed first order controllable. The whole state is not measured but measurements of the slow state are available. More precisely, we assume that the measurements contain a complete information on the slow state, that is when the fast part is at its steady-state, the observation is diffeomorphic to the slow state.

We aim at using this invariant manifold structure in open-loop for the design of a robust

partial state feedback in order to:

• make an open and bounded subset of the equilibrium manifold quasi-statically reach­

able;

• follow trajectories in the slowest manifold without affecting the existence and stability

of the fast one.

We prove that there exists such a robust control law which is realized by a dynamic output feedback, the dynamics of which may be interpreted as a filter of the fast dynamics

in the observations. Two classes of examples are shown to resort from this approach: pseudo-binary distillation

columns and chemical reactors subject to thermal runaways.

'Centre Automatique et Systemes, Ecole des Mines de Paris, 35 rue Saint-Honore, 77305 Fontainebleau Cedex, RANCE. Tel. (33) 1 64 69 48 58. Fax: (33) 1 64 69 47 01. E-mail: levine@ensmp.fr

tCentre Automatique et Systemes, Ecole des Mines de Paris, 60, Bd. Saint-Michel, 75272 Paris Cedex 06, RANCE. Tel. (33) 1 40 51 91 15. Fax: (33) 1 43 54 18 93. E-mail: rouchon@cas.ensmp.fr

ICentre Automatisation et Informatique, Centre de Recherche, ELF Antar France, B.P .. 22, 69360 Solaize, RANCE. Tel. (33) 78 02 60 95. Fax: (33) 78 02 60 95.

DISCRETE-TIME GAUSS-MARKOV PROCESSES WITH FIXED RECIPROCAL DYNAMICS

Bernard C. Levy Department of Electrical and Computer Engineering

University of California, Davis, CA 95616. e-mail: bclevy@ucdavis.edu

and Alessandro Beghi

Department of Electronics and Computer Science University of Padova

via Gradenigo 6/ A, 35131 Padova, Italy. e-mail: beghi@paola.dei.unipd.it

Abstract

203

Motivated by a problem considered earlier by Schrodinger [1], Jamison [2]-[3] and others, we examine the construction of Gauss-Markov processes with fixed reciprocal dynamics. Given the class of reciprocal processes specified by a second-order model, a procedure is described for constructing a Markov process in the class with preassigned marginal probability densities at the end points. The procedure relies on a characteriza­tion of the class of boundary conditions of second-order models corresponding to Markov processes, and requires finding the solution of an algebraic Riccati equation. The problem of changing the final density of a Gauss-Markov process while remaining in the same recip­rocal class is also considered. Unlike the continous-time case [4], [5], it is shown that this problem does not admit a stochastic optimal control interpretation, but an alternative, more general, interpretation is given in terms of an estimation problem.

References

[!] E. Schrodinger, "Uber die umkehrung der naturgesetze," Silzungsber. Preuss. Akad.

Wissen. Phys. Math. /(1., pp. 144-153, 1931.

[2] B. Jamison, "Reciprocal processes," Z. Wahrscheinlichkeitstheorie verw. Gebiete,

vol. 30, pp. 65-86,1974.

[3] B. Jamison, "The Markov processes of Schrodinger," Z. Wahrscheinlichkeitstheorie verw. Gebiete, vol. 32, pp. 323-331, 1975.

[4] P. Dai Pra, "A stochastic control approach to reciprocal diffusion processes," Applied

Math. Optim., vol. 23, pp. 313-329, 1991.

[5] A. Wakolbinger, "A simplified variational characterization of Schrodinger processes," J. Math. Phys., vol. 30, pp. 2943-2946, Dec. 1989.

204

A MINIMUM VARIANCE PREDICTOR FOR TIME-VARYING SYSTEMS

ZHENGLI

Department of Electrical and Electronic Engineering, University of Melbourne,

Parkville, Victoria 3052, Australia

e-mail: zhengli@mullian.ee.mu.oz.au

ABSTRACT

This paper studies problems in minimum variance prediction for a class of SISO linear

time-varying discrete-time systems. Consider plants which can be described by

y(k)=z(k) + C 1 (k)z(k-l )+c2( k)z(k-2 ) ... +cm(k)z(k-m)

z(k)+a l(k)z(k-l )+a2(k)z(k-2) ... +an(k)z(k-n)=w(k)

where y(k) is the plant output produced by a zero mean, independent and possibly

nonstationary stochastic noise w(k), z(k) is an unmeasurable internal state.

The objective is to design a predictor which generates a sequence of estimates (9(k+d!k)}

that satisfies

J(k+d)=E{[y(k+d)-9(k+dlk)TID(k)}=min

where d>O is the prediction range, D(k)={y(k) y(k-l) ... } is set of output measurement

data up to and including time k and9(k+dlk) is the d-step-ahead minimum variance

prediction of the plant output at time k.

It is shown that when the time-varying plant parameters are known for all time and the

mapping form (z(k)} to (y(k)} is exponentially stably invertible a minimum variance

predictor can be designed which meets the prediction objective.

ON THE ABSOLUTE STABILITY

OF MONO MEASURABLE CONTROL SYSTEMS

M.R.Liberzon, V.V.Alexandrov, N.Kh.Rozov

27 Petrovka Str., MATI, Moscow, 103767 Russia

Tel.(095)3076341. Fax. (095)9732136. E-mail alirm@asdi.msk.su

205

New aproach for absolute stability analysis of nonstationary liner control systems is shown.

This approach is based on results from the Inner's Theory, the Oscillation Theory, the

Optimal Control Theory, the Stability Theory, variational methods.

Control systems wich are described by ordinary monomeasurable differential equation

is investigated. All coefficients are numbers, exept the last one, which depends on function

ot time v(t). There is no complit information about function V. It is known only that

function v belongs to the given functional set V. The problem is to find algebraic condition

of absolute stability: asymptotic stability in the global of the trivial solution of given equation for any choice of permissible function v from set V.

It is suggested to divide the set of equations which is determined by functional set

V into n subsets. Criteria of absolute atability for control systems from some subsets are

obtained. It was done with the help of Inner's Method, using the Pontriagin's principle of

maximum and solving the corresponding problem of Cauchy. The question about absolute

stability of systems from some other subsets is open. It is formulated as a new problem of absolute stability.

Key words: control systems, absolute atability, inner, oscillation.

206

Adaptive Control System for a Plant with Input Amplitude Constraints and Bounded External Disturbances

Sheng-Fuu Lin Nai-Wen Lu

Institute of Control Engineering, National Chiao Tung University Hsinchu, Taiwan, R.O.C.

Abstract

We study the tracking problem for stable minimun phase systems with input amplitude constraints and bounded external disturbances. We also explore how the disturbances and constraint on the control input signal influence the control system. We find the difference between the control input signal whose system parameters are unknown and control input signal whose system parameters are known is relative to the magnitude of disturbance. We get a sufficient condition so that the tracking error falls a given interval. Key Words: Adaptive control, input amplitude constraint, minimum phase system

0:'1 THE :'IO:'lLI:-lEAR DYNAMICS OF KALMA:'I FILTERING

Abstract: In this te.lk we report on 80me work (joint with C. I. Byrnes and Y. Zhou) which addresses a fundam~ntal open problem in linee.r filtering and estimation, vi •. what is the .teadY-8tate or asymptotic behavior of the

Kalman filter, or the Kalman gain, when the observed .tationary stochastic procelle i. not generated by a finite-dimensionalstochutic system, or when it is generated by II stochaetic sy.tem baving linee.r dimensional unmodelled dynamics. For example, some time ago Kalman pointed out that the usual positivity conditions assumed in the classical situation are not in fact necessary for the Kalman filter to converge. Using a "fast filtering" e.lgorithm, which incorporate the statistics of the

observation proce.s as initial conditions, (rllther than coefficient parameters) for a dynamical system, this question is analyzed in terms of the phase portrait of a "uni· veual" nonlinear dynamical system. This point of view hM additional advantages as wen. since it enables one to use the theory of dynamice.l syetem. to study the sen­sitivity of the Kalman filter to (smaU) changes in initial conditions; e.g. to change in the statist.ics of the underlying proces •. This is eapecially important since these sta.tistics are often either approximated or estimated. In our work, for II scalar ob· servation proces. we derive neces ... ry and sufficient condition for the Kalman filter to converge, using methode from stochastic systems and from nonlinear dynamice -especially the use of stable, unstable and center manifolds. We Illso show that, in non con vergent c ... es, there exist periodic points of every period p, p ;:: 3 which Ilre arbitrarily clooe to initial conditions having unbounded orbits· rigorously demon· strating that the Kalman filter r.an also be "sensiti"e to initial conditions" .

Professor Anders Lindquist Optimi~ation and Systems Theory Royal Institute of Technology 100 44 Stockholm, Sweden

Telefone: +46-8-7907311 (office)

+46-8-7.)64864 (home)

Telef",,: +16-8-202398

E-mail: alq@math.kth.se

alq@sekth.bitnet

207

208

ON "SUBSPACE METHODS" IDENTIFICATION AND STOCHASTIC REALIZATION THEORY

Anders Lindquist Optimization and Systems Theory

Department of Mathematics Royal Institute of Technology S-10044 Stockholm, Sweden

email ALQ@math.kth.se

Giorgio Picci Dipartimento di Elettronica e Informatica

Universita' di Padova via Gradenigo 6/ A

35131, Padova, Italy email PICCI@paola.dei.unipd.it

Recently there has been a renewed interest in Identification algorithms based on a two step pro­cedure which roughly can be described as covariance estimation followed by stochastic realization. These algorithms are very general, in the sense that they can naturally accomodate multivariate processes, possibly with purely deterministic componenets, and offer the major advantage of con­verting the nonlinear parametre estimaton phase which is necessary in traditional ARMA models identification, into the solution of a Riccati equation, a much better understood problem for which efficient numerical solution techniques are available.

This approach was pioneered by Faurre [1], and successively popularized by Aoki, [2], especially in the econometric community. Recent papers by B. De Moor and co-workers, e.g. [3], offer very sophisticated and accurate numerical procedures to carryon the computations, making the method even more attractive. It should be noted however that some potential difficulties of the proced ure seem to have passed unnoticed, especially in the more recent literature. There is in fact a fundamental requirement of positivity on the covariance estimates, which if not satisfied can inficiate the whole identification procedure. Positivity is in particular, the natural condition ensuring solvability of the underlying Riccati equation. Also some aspects of the so-called partial stochastic realization problem are simultaneously involved, since to match an increasing number of covariance lags, typically the algorithm produces a sequence of state-space models of increasing order.

In this paper we shall analyze the above class of stochastic state-space identification methods in the light of recent results on stochastic systems theory and on the analysis of the phase portrait of the discrete time Riccati equation. Based on standard results of stochastic realization theory, a very streamlined derivation of the algorithms is presented and a special regard is paid to the positivity issue mentioned above. ContinUity arguments basically guarantee solvability of the Riccati equation provided sample variability in the covariance estimates is sufficiently small.

References

[I] Faurre P.: Stochastic realization algorithms. in System Identification: advances and case studies, R.K. :Vlehra and D.L.Lainiotis eds. Ac. Press 1976.

[2] Aoki l\L: Slale-Space Modellinq of Time Series, Springer 1987.

[31 P. van Overschee. B. de Moor: Subspace Algorithms for the Stochastic Identification Problem. Preprint.

Alternative Expressions of some HP Norms and

Their Implications in Optimal Control

Kang-Zhi Liu * Zheng-Hua Luo t Tsutomu Mita*

Keywords: 7-{oo norm, 1{2 norm, induced norm, modeling of disturbance and noise, optimal control, weighting function

Abstract The purpose of this paper is to give new physical explanations for }iP (p = 2, (0)

norms used in optimal control and to provide a quantitative guideline for the de­termination of cost function and weighting functions. It is first argued that we can convert the input space to the space of impulse vectors or white-noise process since all lumped deterministic disturbances are generated from their generators by impulse vectors and all lumped stationary stochastic noises are generated from their gener­ators by the white-noise vector. It is then proved that the }ioo norm of a transfer function is both the worst-case frequency output response to all unit impulse vector inputs and the supremum of the maximal singular value of the square root of the output power spectrum to unit white-noise vector input, and the}i2 norm is both the square root of the supremum of the sum of output energy to all sets of orthonomal impulse vector inputs and the square root of the average output power to unit white­noise vector input. It is argued based on these facts that the transfer functions from all disturbances and noises to the controlled plant output must be penalized and the worst-possible disturbance (having the largest magnitude of frequency response over the whole frequency domain in the class of disturbances) and noise, as well as the plant uncertainty, must be modeled and used as weighting functions in optimal control design so as to yield good tradeoff between various system performances and robust stability. Further, a modified two-degree-of-freedom control scheme is proposed in this spirit.

"Department of Electrical & Electronics Engineering, Chiba University, 1-33 Yayoi-cho, Inage-ku,

Chiba 263, Japan. Tel 043-251-11 11 Ext 2842 , Fax 043-251-7337, E-mail liu@cute.te.chiba-u.ac.jp. tDepartment of Control Engineering, Faculty of Engineering Science, Osaka University, Toyonaka,

Osaka 560, Japan.

209

Adaptive Stabilization of Infinite-Dimensional Discrete-Time Systems

Hartmut Logemann Bengt Martensson

Institute for Dynamical Systems University of Bremen

P.O. Box 330 440 D-2800 Bremen

F.R.G. logemann@mathematik.uni-Bremen.de

bengt@mathematik.uni-Bremen.de

In this contribution, we propose a universal adaptive stabilizer for a class of linear, time-invariant, discrete-time systems on Banach-spaces. The only a priori knowledge required is the order of any stabilizing, linear, proper, time-invariant controller. The proposed controller is a so-called switching function controller. It turns out that this is by no means a trivial translation of previous results by the authors and their col­leagues, in particular dill' to to the non-connectedness of the discrete-time trajectories. Recent results on the stability and stabilizability of infinite-dimensional discrete-time systems plays an important role in the analysis.

Destabilization of M ultivariable Infinite-Dimensional Feedback Systems by Small Time-Delays in the Loop

Hartmut Logemann * Institut fur Dynamische Systeme

Universitiit Bremen Postfach 330440 2800 Bremen 33

Germany

Richard Rebarber Dept. of Mathematics University of Nebraska

836 Oldfather Hall Lincoln, NE68588-0323

U.S.A.

Summary

Set lea := {s E Ie : Re( s) > a}, a E IP!., and let Ma denote the field of meromorphic functions defined on lea: The algebra of all bounded holomorphic functions on lea will be denoted by H;:' and we set HOC := Hg". The spectral radius of a matrix M E I[;"xm will be denoted by Q(M).

The paper contains the following frequency-domain results on destabilization by small delays in the feedback loop.

THEOREM 1. Suppose that G is in (H;:,)mxm for' some a > ° and

lim G(s) = D ,,-00,3Eli+

exists. Then the following statements hold: (i) If

Q(D):S 1, limsup Q(G(s)) > 1 and G(I + G)-' E (Hoc)mxm, l"I-CX),lECo

then there exist numbers ° < On < cn, cn ~ 0, such that the closed-loop transfer matrix G(s)(I + exp(-cs)G(s))-l has a pole p, E!Co for all c E UnEN(cn - On,Cn + on).

(ii) If {!(D) > 1, then there exist numbers ° < On < cn, cn ~ 0, such that the conclusion of statement (i) holds true. Moreover, there exist poles Pn of the closed-loop transfer matrices G(s)(I + exp( -cns)G(s))-l satisfying limn_oc Re(Pn) = 00.

THEOREM 2. Suppose that GEM;;' xm. If G(I + G)-l E (HOC )mxm and if there exists a > Osuch that

lim sup Q(G(s» > 1 and liminf {!(G(s» < 1, 1"I-co, O<Re(" )<a 1"1-00, O<Re(" )<a

then the conclusions of Theorem 1 (i) hold true.

Theorem 1 will be used in order to show that the standard stabilization scheme for neutral systems with infinitely many unstable poles pruduces a closed-loop system which can be destabilized by arbitrarily small delays in the loop. Moreover, a sufficient condition for the existence of a dy­namic compensator ensuring that closed-loop stability is robust with respect to small delays in the feedback-loop will be given. The interrelation between Theorems 1 and 2 and previous results in the literature will be discussed.

·Tel: +49-421-218-2763, Fax: +49-421-218-4235, E-mail:logemann@mathematik.uni-bremen.de

211

2

Conditions for robustness of the stability of S1S0 feedback systems with respect to

small delays III the feedback loop

Hartmut LOGEMANN Institut fur Dynamische Systeme Univ. Bremen, Postfach 330.140 2800 Bremen 33, Germany logemann@mathematik.uni-bremen.de

Richard REBARBER Dept. of Mathematics and Statistics, Univ. of Nebraska at Lincoln, Lincoln, NE 68588, U.S.A.

re bar ber@hoss.unl.edu

George WEISS Department of Electrical Engineering Ben- Gurion University 84105 Beer-Sheva, Israel weiss@bguvm.bgu.ac.il

Consider the linear feedback system shown in the figure, where u is the input function and y is the output function, both complex-valued. H is the open loop transfer function, which we assume to be well posed and regular. Well posedness means that H is bo,.nded and analytic on some right half-plane Re s > Q, and regularity means that the limit limA~oo H(>.) exists, where>. is positive. These are quite natural assumptions on the transfer function of a (possibly infinite dimen­sional) linear system. The block with transfer function e-<! represents a delay by E:, where E: :::: O. The transfer function for the closed loop system is given by

G«s) H(s)(I + e-<!H(s))-l .

y We say that G< is L2-stable if it is a bounded analytic function on the right open half-plane Re s > O. (Indeed, then u E L2 implies y E L2.) We say that G O is robustly stable with respect to delays if there is a a > 0 such that for any E: E [0,0], G' is L2-stable.

Suppose GO is L2-stable. Then it is easy to see that H is a meromorphic function on the half-plane Re s > 0 (even if, initially, it was defined only on a smaller half­plane). Therefore, the number

-y lim sup IH(s)1 l'l~oo

Re .>0

is well defined. Our main result is the following: If -y < 1 then GO is robustly stable with respect to delays, and if I > 1 then it is not. We have no results concerning -y = 1. We give several examples and explain how our results generalize earlier work on delay destabilization, and how they relate to w-stability.

Eigenstructure Assignment in Linear Uncontrollable Systems

J. J. LOISEAU! P. ZAGALAK2

! Laboratoire d'Automatique de Nantes, Ecole Centrale de Nantes-Universite de Nantes, 1, Rue de la Noe, 44072 Nantes Cedex 03 - France. E-mail: loiseau@lanOl.ensm-nantes.fr

2 Institute of infor~ation Theory and Automation, Czech Academy of Sciences, P.O. Box 18, 18208 Prague, Czech Republic, E-mail: zagalak@opte.utia.cas.cs

The problem of eigenstructure assignment by state feedback in linear systems described by

Ex(t) = Ax(t) + Bu(t) (1)

Where E, A E R n.n and B E R n.m, is of long history. The first result in this direction was established by Rosenbrock [1] in early seventies. It concerns the case where the system (1) is controllable and rankE = n.

Generalizations of the Rosenbrock's result go in a few directions. For implicit and con­trollable systems of the form (1), i.e., rankE :::; n, there are now plenty of results available. The most achievable one is due to Zagalak and Loiseau [2]. Another direction in which the Rosenbrock's theorem is generalized consists in considering the uncontrollable case of (1). For the system with rank E = n, the result has been established by Zaballa [3].

We give Necessary conditions and sufficient conditions for the existence of a static state feedback assigning to prescribed values both the finite invariant factors and the infinite pole orders of the system (1). These conditions generalize both the results of [2] and [3]. More generally, these conditions coincide and get necessary and sufficient conditions if the system (1) is regularizable and has no non-proper almost controllability indices.

References

[1] H.H. Rosenbrock, State Space and Multivariable Theory, Wiley, New- York, 1970.

[2] P. Zagalak and J.J. Loiseau, Invariant Factors Assignment in Linear Systems, Proc. SINS'92, Dallas, Texas, 1992.

[3] 1. Zaballa, Interlacing Inequalities and Control Theory, Linear Algebra and its Appli­cations, 87, 1987, 113-146.

213

214

A perspective estimation problem in machine vision: Is it enough to have just two eyes?

E. P. Loucks and Bijoy K. Ghosh" Department of Systems Science and Mathematics

Washington University St. Louis, MO 63105, U.S.A.

ghosh@cecl.wustl.edu

Abstract

We consider the problem of motion and shape estimation of a moving body with the aid of a monocular camera. We show that the estimation problem reduces to a specific parameter estimation of a perspective dynamical system. Surprisingly, the above reduction is independent of whether or not the data measured is the brightness pattern which the object produces on the image plane or whether the data observed are points, lines or curves on the image plane produced as a result of discontinuities in the brightness pattern. Many cases of the perspective parameter estimation problem has been analyzed. These cases include a fairly complete analysis of a planar textured surface undergoing a rigid flow and an affine flow. These two cases have been analysed for orthographic pseudo-orthographic and a more general image-centered projection. The basic procedure introduced for parameter estimation is to subdivide the problem into two mod ules, one for spatial averaging and the other for time averaging. The estimation procedure is carried out with the aid of a new "realization theory for perspective systems" introduced for systems described in discrete time and in continuous time. Finally many of our analysis has been substantiated by computer simulation of specific algorithms developed in order to explicitly compute the parameters.

The main results of this paper is now summarized. For a visual system with its focal length permanently fixed at infinity, parameters of a planar motion with an affine flow are recovered via orthographic projection upto a surface of dimension 4. On the other hand if the focal length can be varied, then parameters can be recovered via perspective projection and in this case parameters can be identifiable upto a one parameter ambiguity. For a biological system, this justifies the need for a pair of eyes with a capability to vary its focal length at will in order to decipher a non rigid environment. For a rigid motion, whether the projection is orthographic or not, parameters are always recovered (with a possible sign ambiguity) up to one parameter even when f approaches infinity. Thus in order to decipher an environment with only rigid objects, it is enough to have a pair of eyes or visual systems without the capability to vary its focal length. Likewise for a nonrigid phenomena happening closeby, the projection model is essentially perspective, it is enough to have two eyes to disambiguate one parameter uncertainty. However for a nonrigid phenomena happening at a distance, for which the projection model is orthographic and the uncertainties in the parameters are of dimension 4, do we need five eyes?

'Partially supported by DOE under grant No. DE-FG02-90ER14140

Continuous Relation between Models

and System Performances

-A Case Study for Optimal Servosystems-

Hajime MAEDA t , and Shinzo KODAMA tt

tDepartment of Communication Engineering,

ttDepartment of Electronic Engineering,

Osaka University

2-1, Yamada-oka,Suita-shi,Osaka 565,J APAN

e-mail h-maeda@oucom.osaka-u.ac.jp

215

Abstract This paper is concerned with the continuous relation between

models of the plant and the predicted performances of the designed sys­

tem. Let P( s) be transfer matrix of a plant model, and let A( s) be the

transfer matrix of interest of the resultant system designed through a

specified control scheme, e.g. LQR. Such an A( s) is regarded as a per­

formance measure for evaluating the designed responses. A( s) depends

upon P( s) and is written as A = A( P). In this paper we consider the lin­

ear quadratic optimal servosystem with integrators scheme as the design

methodology, and prove that A(P,,) converges to A(P) if and only if the

modified plant transfer matrix (s + 1) / sP" (s) converges to (s + 1) / sP( s)

in graph topology. A numerical example is given for illustrating the re­

sult.

216

ON SYSTEM IDENTIFICATION AND MODEL VALIDATION VIA LINEAR PROGRAMMING

P.M. Miikilii and T.K. Gustafsson, Abo Akademi University, Department of Engineering, SF-20500 Abo, FINLAND, fax: +35821 654479,

tel. +35821 654311, e-mail: pmakila@abovax.abo.fi

Abstract

Recently there has been a lot of interest in problems of modelling of uncertain systems for the purpose of robust control design, see e.g. the special issue [2] and [1,5,8,4,3,6]. This field is concerned with the development of systematic methodologies for obtaining the modelling information needed for successful robust control design.

In the present work linear programming methods for discrete [1 approximation are used to provide solutions to problems of approximate modelling and identification with state space models and to problems of model validation for stable uncertain systems. Several types of model and error priors are easily included. Choice of model structure is studied via the Kolmogorov n-width concept and a related n-width concept for state space mod­els. The task of modelling of uncertain systems from noisy data is decomposed into two stages. The first stage consists of identification of nominal models as best approximations within a chosen model class. The second stage consists of feasibility analysis of quantitative uncertainty models [7,6]. Simulation examples illustrate the methods.

References

[1] Helmicki, A.J., C.A. Jacobson and C.N. Nett (1991). Control-oriented system identifi­cation: A worst-case/deterministic approach in Hoo. IEEE Trans. Automat. Control, 36, 1163-1176.

[2] Kosut, R.L., G.C. Goodwin and M.P. Polis, Guest Editors. (1992). Special Issue on System Identification for Robust Control Design. IEEE Trans. Automat. Control,37, 899-1008.

[3] Lin, L., L.Y. Wang and G. Zames (1992). Uncertainty principles and identification n-widths for LTI and slowly varying systems. Proc. 1992 American Control Conf., Chicago.

[4] Miikilii, P.M. and J.R. Partington (1992). Robust identification of strongly stabilizable systems. IEEE Trans. Automat. Control, 37, 1709-1716.

[5] Partington, J .R. (1991). Robust identification and interpolation in Hoo. Int. J. Control, 54,1281-1290.

[6] Poolla, K., P. Khargonekar, A. Tikku, J. Krause and K. Nagpal (1992). A time-domain approach to model model validation. Proc. 1992 American Control Conf., Chicago.

[7] Smith, R.S. and J.C. Doyle (1992). Model validation: A connection between robust control and identification. IEEE Trans. Automat. Control, 37, 942-952.

[8] Tse, D.N.C., M.A. Dahleh and J.N. Tsitsiklis (1991). Optimal asymptotic identification under bounded disturbances. Proc. 30th IEEE Conf. Decision Control, Brighton.

Output Regulation for Systems Linear in the Input

Bob Mahony Iven Mareels Guy Campion George Bastin

Centre for Systems Engineering and Applied Mechanics Universite Catholique de Louvain

Batiment Maxwell Place du Levant 3

B1348 Louvain-Ia-Neuve Belgium

First two authors are on leave from

Department of Systems Engineering

Australian National University

GPO Box 4 ACT 2601 Australia

All correspondence to:

lven Mareels, Department of Systems Engineering, Australian National University,

GPO Box 4, ACT 2601, fax int+61+6+2490506, e-mail iven@faceng.anu.edu.au

November 11, 1992

Acknowledgement: The first two authors would like to acknowledge the financial and human sup­port of the Centre for Systems Engineering and Applied Mechanics of the Universite Catholique de Louvain.

Abstract

We consider nonlinear systems which are linear in the input. The central question we are concerned with is output regulation (driving the output function to zero) via smooth static state feedback. Several output regulation properties are formulated. We demonstrate that when the system may be input-output decoupled output regulation can be ensured.

Keywords: stability, partial state stability, nonlinear systems, output regulation 1991 Mathematics Subject Classification: 93099 Subject area: Theory

217

218

The Partial Model Matching or Partial Disturbance Rejection Problem:

Geometric and Structural Solutions

Michel MALABRE Juan Carlos MARTINEZ GARCIA * Laboratoire d'Automatique de Nantes c!'ms URA 823

Ecole Centrale de Nantes-Universite de Nantes

1 rue de la Noe, F-44072 Nantes Cedex 03 (FRANCE)

Phone: (+) 40371651: Fax (+) 40 37 25 22

Email: malabre@lanOI.ensm-nantes.fr martinez@lanOl.ensm-nantes.fr

Abstract

The kiJ2. order Partial Model Matching has been introduced in [1]. It amounts to finding a compensator such that the first k Markov parameters of the compensated plant exactly match those of a prespecified model. Partial Disturbance Rejection is defined in a similar way: reject the effect of the disturbance on the first k Markov parameters of the compensated system. We give here both geometric and structural solutions to these partial problems: the conditions are expressed in terms of the first k steps of the controlled invariant subspace algorithm (as in [2]) or, equivalently, in terms of the first k smallest orders of the zeros at infinity.

The solution to the Exact Model Matching and Exact Disturbance Rejection Prob­lems are directly deduced from these partial versions: perfect matching (or perfect rejection) is possible if and only if partial matching (or partial rejection) is solvable for the order k = sup{n,}, where {n,} is the list of the orders of the zeros at infinity of the system.

References

[1] Emre, E. and Silverman, L. H.: Partially!odel Matching of Linear Systems. !EEE Trans. Automat. Contr., Vol. AC-25, pp. 280-281,1980.

[2] Wonham, Murray W.: Linear Multivariable Cont1'01: A Geometric App1'Oach. Third Edition. New York, Springer-Verlag, 1985.

·Supported by the National Council of Science and Technology of Mexico and by the Advanced Studies and Research Center of the IPN of Mexico.

Theory of Power in Electrical Systems and Networks

and Decomposition of the Hilbert Transform

Dr.rer.nat. Wolfgang Marten* and Prof.Dr.-Ing.habil. Wolfgang Mathis"

University of Wuppertal

'DMV ** VDE/ITG, IEEE(CAS), DPG

Adress: Department of Electrical Engineering, Fuhlrottstr. 10 D-5600 Wuppertal 1

Telephone: +49-202-43930011, FAX: +49-202-439-3040 E-Mail: Mathisatcyber.URZ.UNI-WUPPERTAL.DBP.DE

Abstract:

In this paper we present a decomposition theorem of Hilbert transforms in Hilbert spaces of real functions which can be used in the theory of power in electrical systems and networks. To this end we give an overview about some previous results of the authors and formulate an abstract concept of power.

Keywords: power concepts, average power, apperent power, complex power, Hilbert transform, Hilbert spaces, Hilbert sum, Hilbert integral,

219

220

Title: Continu&tion Method. for Exponential Interpolation

By: Clyde F. Martin and John Miller, Department of Mathematics, Texa. Tech LJniversity, Lubbock, Texa.

Abotract: There hal been an exten.ive body of literature devoted to the otudy of exponentl"l interpolation. That is trying to fit data to a function of the form

" 1(1) = L Qi exp(>.,t) i_l

It has been observed that for small data aets continuation technique. are effective provided t.hat the weight. are pooitive. In thia paper we will examine that phenomena and show that there are underlying topological and geom.tric r.aeono why positive weights are necessary to guarantee convergence.

StochBBtic Regularization of the Observability Problem for Linear and Nonlinear Diffusion Equations

C. Martin, Fellow and

V. Shubov Department of Mathematics

Texas Tech University Lubbock, TX 79409-1042

Telephone: (806) 742-1511 Fax: (806) 742-1389

E-mail: gqcfm@ttacsl.ttu.edu

Abstract

We consider a diffusion type equation of the form

Ut:: div[a(u) 'V II + btu)]'

where'tl :: u(x, f), t ;:: 0, x E JRd, a(u) is a positive matrix, btu) E JRd, boLh a and b have bounded derivatives. Assuming that u(x, f) is the solution of an initial-boundary problem for the above equat.ion (with boundary conditions of Borne type) in a bounded domain n c JRd, we are interested in the problem of reconstructing the initial function IIO(X) = u(x,O) from measurements of 'tl at a given point 1'0 E n for all moments of time t ;:: D. This problem is ill­posed and even in the simplest case of the 2-dimensional heat equation (a == ], b == 0, d =: 2) the ill-posedness lea.ds to results which are not natural from the point of view of applications. E.g., if n is a rectangle with the sides £1 and £2, then the observability of the initial function 1.10(1') depends on whether £del is a rational or an irrational number. To avoid such difficulties we give a stochastic reformulation of the above observability problem using the known result on hydrodynamicallimits. In the case when n is a cube in JR" and the boundary conditions are periodic we replace the original equation by an infinit.e sequence of' diffusion processes which approximates the equation in a certain sense. For this sequence of processes we formulate and prove lin

observa.biJity result which is free of the above mentioned difficulties.

221

222

Endogeneous feedbacks and equivalence

Philippe Martin Centre Automatique et Systemes, Ecole des Mines de Paris 35 rue Saint-Honore, 77305 Fontainebleau Cedex, FRANCE

Email: martin@cas.ensmpJr

September 1992

Abstract

In [2] was intoduced in the differential algebraic framework a state-independent concept of equivalence between systems, called equivalence by endogeneous dynamic feedback. The important case of flat systems, i.e. systems equivalent to a linear one, was singled out, which led to a new formulation of the dynamic feedback linearization problem. Nevertheless, the links between equivalence, flatness and dynamic feedback were merely outlined, and no definition of endogeneous feedbacks was given.

The aim of this note is to fill this gap. Endogeneous feedbacks are defined and compared to static and quasistatic feedbacks introduced in [1]. The main result is then that equivalence between systems ~l and ~2 simply means that ~2, extended by pure integrators, can be obtained from ~l by the action of an endogeneous feedback and a coordinate change. Naturally, since equivalence is a symmetric relation, the roles of ~l and ~2 can be exchanged; for flat systems, one can even make do with a quasistatic feedback. As a corollary, one gets the fundamental property that endogeneous feedbacks are "revertible" up to pure integrators, thus generalizing a well-known property of (invertible) static feedbacks.

Of course, these results, which clarify the notion of equivalence, make sense only for systems described by state and input (nothing so precise can be said when a system is given simply as a differential field extension).

References

[1] E. Delaleau and M. Fliess. An algebraic interpretation of the structure algorithm with an application to feedback decoupling. In Proceedings of [FAG NOLGOS Gonf.,

Bordeaux, France, 1992.

[2] M. Fliess, J. Levine, Ph. Martin, and P. Rouchon. On differentially flat non linear systems. In Proceedings of IFAG NOLGOS Gonf., Bordeaux, France, 1992.

Systems without drift and flatness

Ph. MARTIN" P. ROUCHONt

October 1992, submitted to MTNS 93

Abstract

A necessary and sufficient flamess condition for systems without drift having two control variables is given. The proof is based on a result. due to E.Cartan [4l. on Pfaffian systems of n - 2 equations in n variables. This result is illustrated by a nooholonomic system describing a kinematic car with one trailer.

223

Key words: dynamic feedback linearization. systems without drift. flamess. Pfaffian systems. nonholonomic control systems.

9

224

System-theoretic properties of port-controlled Hamiltonian systems

B.M. Maschke' A.J. van cler Schaft t

In our previous paper [1] it has been shown how by using a (generalized) bond graph formalism the dynamics of non-resistive physical systems (pertaining to different domains, i.e., electrical, me­chanical, hydraulical, etc.) can be given an intrinsic Hamiltonian formulation of dimension equal to the order of the physical system. Here "Hamiltonian" has to be understood in the generalized sense of defining Hamiltonian equations of motion with respect to a general Poisson structure. The Poisson structure is fully determined by the network structure of the physical system (called "junction structure" in bond graph terminology), while the Hamiltonian equals the internally stored energy. A striking example is the direct Hamiltonian formulation of (nonlinear) electrical LC-circuits [2], where (in the case of independent capacitors and inductors) the Poisson structure is simply given by a constant skew-symmetric matrix determined by the network topology. Sub­sequently in [3] the interaction of non-resistive physical systems with their environment has been formalized by including external ports in the network model, naturally leading to two conjugated sets of external variables: the inputs u represented as generalized flow sources and the outputs Y which are the conjugated efforts. This defines the port-controlled Hamiltonian systems

i:=XHo(X)+Lgj(X)Uj, Yj=<dHo,gj>(x), j=I,"',m, j=l

(1)

where x are local coordinates for a Poisson manifold M (the space of energy variables), XHo is the Hamiltonian vectorfield with respect to the Poisson structure on Af and t.he internal pnergy H 0, and gj, j = 1, .. " m, are a priori arbitrary input vectorfields modelling the external ports. Note that if we assume the dynamics (1) to be Hamiltonian for all inputs u then the input vector­fields gj are necessarily Hamiltonian, and we are back to the Hamiltonian control systems studied e.g. in [4], [5]. However, this assumption is quite restrictive and excludes several interesting ex­amples (LC-circuits with external ports, several types of controlled mechanisms). On the other hand, one may naturally assume that the input vectorfields are Poisson structure preserving. In the present paper we will study systemtheoretic properties of systems (1). In particular we will show that the Poisson structure is determined by the external behavior of (1) in case the input vectorfields are Poisson structure preserving and the system is observable. Furthermore, the ob­servability and controllability properties of (1) will be analyzed continuing [3].

[1] B. Maschke, A.J. van der Schaft, P.C. Breedveld, an intrinsic Hamiltonian formulation of network dynamics: non-standard Poisson structures and gyrators, J. Franklin Institute, 329, pp. 923-966, 1992. [2] B. Maschke, A.J. van der Schaft, P.C. Breedveld, in preparation. [3] B. Maschke, A.J. van der Schaft, Port-controlled Hamiltonian systems: modelling origins and system-theoretic properties, pp. 282-288 in Proceedings NOLCOS '92 (ed. M. Fliess)' 24-26 June 1992, Bordeaux, France. [4] R.W. Brockett, Control theory and analytical mechanics, in Geometric Control Theory (eds. C. Martin, R. Hermann), Math Sci Press, Brookline (1977), pp. 1-46.

[5] A.J. van der Schaft, System theory and mechanics, in Three Decades of Mathematical System Theory (cds. II. Nijmeijer, J.M. Schumacher), LNCIS 135, Springer, Berlin (1989), pp. 426-452.

·Control Laboratory, Conservatoire Nationale des Arts et Metiers, 21 Rue Pinel, 75013 Paris, France 'Dept. of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands, Fax

31-53-340733, TeJephone 31-53-893449/3370, e-mail twarjan@math.utwente.nJ

....

Viscothermal loss.es in wind J' ~truments a non-Integer mo el

D. Matignon and P. Depalle IRCAM, groupe Analyse/Synthese,

31 rue Saint Merri, 75004 Paris, France. matignon@ircamJr, phd@ircamJr

B. d'Andrea-Novel Centre Automatique et Systemes, Ecole des Mines de Paris,

35 rue Saint Honore, 77305 Fontainebleau, France. andrea@cas.ensmpJr

A. Oustaloup Laboratoire d'Automatique et de Productique, ENSERB,

351 cours de la Liberation, 33405 Talence, France.

225

The purpose of this paper is to present a new discrete-time model of a wind instrument that has viscothermal losses, using the realization of non-integer derivators performed by A. Oustaloup.

In musical sound synthesis, recent works developed at IRCAM have led to a new point of view, positioned halfway between physical-modelling (explicit simulation of the physical behaviour of the instrument) and signal-modelling (signal processing techniques, such as Fourier transform and linear prediction coding, being employed to represent the sound). These works aim at describing connected acoustic tube networks as linear state space models, in order to simulate and control wind instruments.

As the sounds computed with a model derived from the classical \-D wave equation sound very unrealistic, we need a better approximation. Taking the viscothermal losses into account leads to a modified wave equation of non-integer order, the structure of which takes the following form:

(act + (a:' - ax)(acd (a:' + ax) <II = 0

where ( is a small geometric parameter. In continuous time, these effects introduce p! terms in the transfer function between the ingoing and outgoing waves in a cylindrical tube. The non-integer derivative appears in numerous situations in the field of physics, and is particularly frequent when geometry conditions a phenomenon governed by a differential equation.

The paper is organized as follows: we briefly recall the lossless model of acoustical tubes, and then present the tube description with energy loss. We focus on the definition and synthesis of a frequency-limited non-integer derivator, and finally illustrate this with a worked-out example which gives realistic-sounding results.

226

Spectral properties of Hankel operators

A.Megretskii. V.V. Peller and S.R.Treil

Abstract

The main result is the following: Let A be a self-adjoint operator on Hilbert space. Then A is unitarily equivalent to a Hankel operator if and only if the following conditions are satisfied

(CI) A is non-invertible;

(C2) the kernel of A is either trivial or infinite-dimensional;

(C3) if A is represented as multiplication by the independent variable on the direct integral

J H(t)dJ.l(t),

where J.I is a scalar spectral measure of A and I/(t) = dimH(t), then

II/(t) - I/(-t) I~ 2

almost everywhere, and II/(t) -I/(-t) I~ I

on the singular spectrum of A.

Riccati inequalities and completion problems in control

Volker Mehrmann Fachbereich Mathematik

Technische U niversitiit Chemnitz-Zwickau D-O-9022 Chemnitz Fed. Rep. Germany

For the solution of linear constant coefficient descriptor control problems of the form Ex = Ax + Bu, y = ex it is often required that the corresponding matrix pencil aE - f3A is regular index 1. This can be achieved using output proportional and/or output derivative feedback control u = FiJ + Gy + v. In order to make the regular­ized system robust against perturbation it is useful to choose the feedback to improve the conditioning. We show that this leads to a matrix completion problem. The solution of this problem can be ob­tained via a matrix Riccati inequality, that we can solve explicitely. We will discuss this inequality and its numerical solution.

This is joint work with L. Elsner and C. He

227

8

Control of linear descriptor systems which are not controllable at infinity

Volker Mehrmann Fachbereich Mathematik

Technische Universitiit Chemnitz-Zwickau D-O-9022 Chemnitz Fed. Rep. Germany

For the solution of linear constant coefficient descriptor control prob­lems of the form Ex = Ax + Btl. it is usually required that the system is controllable at infinity. This condition guarantees that the corre­sponding matrix pencil aE - f3A can be made regular index l.

Now there exist practical examples where this is not the case. We will present such examples.

We will then give a theoretical analysis that shows that the con­dition of controllability at infinity may be dropped from the assump­tions and we give necessary and sufficient conditions such that the system can be regularized via proportional and derivative state or output feedback.

We will show that from the theoretical point of view a higher index part that is uncontrollable does not lead to an unstable part of the solution. Under the influence of noise or roundoff errors, these higher index parts may lead to severe instabilities, however. We will discuss this topic and give numerical examples.

This is joint work with R. Byers, Univ. of Kansas and T. Geerts, Kath. Univ. Brabant

On systems of differential operators

R. Mennicken Regensburg, Germany

Various spectral problems which arise in magnetohydrodynamics, in fluiddynamics and in astrophysics can be reduced to the investigation of a linear operator repre­sented by a matrix

ILa = (~ ~) defined on the product XI X X2 of Banach or Hilbert spaces XI and X2. We consider the case when A is an operator with compact resolvent in XI, D is bounded in X 2, B : X2 -> XI and C : XI -> X2 are closable, moreover, D(C) :::J D(A). In applications A, B, C are differential operators and D is a multiplication. The operator ILa is well defined on D(ILa) = D(A) x D(B) but not necessarily closed nor even closable. In abstract terms we find natural conditions (which can be verified in applications) when the operator S = C A -I B admits a bounded closure. (We assume for simplicity that 0 E e(A)).

Our main result is the following

Theorem. If both operators S = C A-I Band F = A-I B which are defined on D(B) admit a bounded closure, then ILa is closable. Moreover, if S = CA-2B admits a compact closure in X2, then the essential spectrum of ILa and D - S coincide.

Some other results and applications will be discussed.

(Joint work with F.V. Atkinson/Toronto, H. Langer/Vienna and A.A. Shkali­kov /Moscow)

229

230

EXTERNAL AND INTERNAL MATRIX LOSSLESSNESS IN PERFECT RECONSTRUCTION M-BAND FILTER BANKS

by B.G. MERTZIOS1 and A. FETTWEIS2

October 28, 1992

ABSTRACT

In this paper the concept of losslessness and its implications to perfect reconstruction M-band filter banks is considered. The external losslessness, the internal losslessness conditions and the losslessness of E(z), are discussed. The external losslessness requirement may ensure only the input-output amplitude, phase and aliasing cancellation and does not ensure the perfect signal reconstruction in the case where an analysis or synthesis filter of the filter bank is not lossless. The losslessness of E(z) does not have any conceptual relation to the system's losslessness requirement and the inherent properties of a perfect reconstruction filter bank. Internal losslessness implies external losslessness, while the opposite does not hold (in correspondence to the internal and output stability). In the literature the losslessness of the analysis polyphase matrix E(z) of the filter bank has been extensively used to derive easy solutions for FIR filter

banks.

1Automatic Control Systems Laboratory, Department Engineering, Democritus University of Thrace, 67 100 Fax: (30) 541-20275, 26947e-mail: mertzios@xanthi.uucp 2Lehrstuhl fur Nachri ctentechni k, Ruhr Uni vers i tat Bochum, GERMANY, Fax: (234) 709-4100.

of Electrical Xanthi, GREECE,

Bochum, 0-4630

A HYBRID ANALYTICAL - INTELLIGENT APPROACH TO FAULT TOLERANT CONTROL SYSTEM DESIGN

B.G. MERTZ lOS Department of Electrical Engineering, Democritus University of Thrace, Xanthi, Hellas Tel.: +30-541-26473, Fax: +30-541-26947 e-mail: mertzios@xanthi.uucp

G. VACHTSEVANOS School of Electrical Georgia Institute of Tel: +404-894 6252,

Engineering, Technology, 30332 Atlanta, Georgia, USA Fax: +404-853-9171

231

Modern dynamical systems, such as aircraft and space vehicles, nuclear power plants and industrial and manufacturing processes, are complex systems that have a heavy burden on control strategies, performance monitoring and status evaluation. The complexity and sophistication of these systems require the use of robustand reliable automated techniques in order to detect failures, identify the faulty component and finally restructure the system and reconfigure the control functions, so that the operational integrity of the system is maintened. The structural information is expressed in terms of simple components or subsystems and can be expressed analytically. The system is considered to exhibit an hierarchical structure. The fault diagnosis methodology is based upon the combination of signal redundancy techniques and fuzzy logic. Causal or qualitative reasoning is used to address the issues of the behavior of the givern system. The concept of vector Lyapunov functions is used in order to describe the energy conservation.

The elimination problem in differential algebraic systems

Juraj Michalik & Jan C. Willems Mathematics Institute

University of Groningen P.O. Box 800

9700 AV Groningen The Netherlands

Fax +3150 633976 Email J.Michalik@math.rug.nl

Email J.CWillemS@math.rug.nl

In the polynomial differential algebra there is an algorithm (due to Ritt and

Seidenberg) for deriving the equations of the projection of a differential­

algebraic set along some of the coordinates. It is known that the projection of a

differential-algebraic set along these coordinates is in general not closed. We

will present an algorithm for the computation of the equations of the closure of

the above projection in a finite number of steps and derive a necessary and

sufficient condition for the projection to be closed (i.e. to be itself a differential­

algebraic set). We will illustrate the relevance of these ideas in system theory.

ORTHOGONAL SYSTEMS OF ROOT FUNCTIONS FOR SYMMETRIC OPERATORS

Manfred Moller

Let E be a Banach space which is contained continuously and densely in a Hilbert space F. We can identify the anti-dual E* of E with a space containing F. Let V be an open subset of the complex plane which is symmetric with respect to the real axis. For each ,X E V let T('x) : E -> F be a continuous Fredholm operator. We assume that T depends holomorphically on 'x. T is called symmetric if

T*(:\)IE = T('x) (A E V).

A number p, E V is called an eigenvalue of T if N(T(p,)) = T(p,)-I(O) of {O}. A root function Y of T at p, E V is a holomorphic vector function Y : V -> E such that y(p,) of 0 and (Ty)(p,) = O. The multiplicity of the zero of Ty at p, is called the order v(y) of y. There are root functions YI, ... ,Yr such that YI (p,), ... , Yr(P,) is a basis of N(T(p,)), and such that, for mi := V(Yi), the numbers ml ~ m2 ~ ... ~ mr > 0 are maximal. The numbers mi are called the partial multiplicities of T at p,.

It is well-known that the principal part of T- I at a pole p, can be represented in terms of biorthogonal root functions of T and T*. For real p, we can take orthogonal root functions:

THEOREM. Let p, E V n R be an eigenvalue of T. Then there are polynomials Yj : C -> E of degree less than mj, numbers cj E {-I,l}, j = I, ... ,r, and a holomorphic operator function D : VI -> L(F, E) in a suitable neighbourhood VI of p, such that

T

T-I(,X) = L Cj(A - p,)-m'Yj(A) ® Yj(:\) + D(A) j=1

for all ,x E VI \ {p,}. The root functions are orthogonal, i. e.,

I dl

If d,XI(1]ih,llJ(P,) = Ci~ij~m.-h,l

(I :S i :S r, I :S h :S mi, I :S j :S r, O:S I:S mj - I)

REFERENCES

M. Moller, Orthogonal systems of eigenvectors and associated vectors for symmetric holomorphic operator functions, Math. Nachr., to appear.

, 233

234

Inversion of rational maps and identifiability of dynamical systems

T. Mora Department of Mathematics

University of Genova Genova, Italy

Identifiability of a dynamical system is related with the question whether the map

associating the input-output data of a dynamical system to its internal

parameters is one-to-one (or locally such). If one is interested in structural identifiability (Le. whether being one-to-one or

locally one-to-one holds except for a set of measure zero), one can reduce (in

several ways) the question to the same problem for a rational map between two

affine spaces over the complex numbers. Such an algebraic version of the problem can then benefit by the large body of algorithms in Effective Algebraic Geometry and has therefore several algorithmic

solutions. The talk will discuss the prototype of these algorithmic solutions (the

killer tag variables algorithm by Sweedler) and some recent variants and

improvements by Ollivier and Audoly et al.

Orientation and Stabilization of a Rotating Flexible Structure

Orner Morgiil

Dep:trl.ll1cnt of Electrical ami EI"clronics Engineering

Bilkcnl Ullivl'rsity

OG.'i:n. Bilk",,!, Allbr<t, lTHJ\lcY

C'-lllf1ij morgul(iitrlJilllll.hitJIr!t

fax: 'JO-4-266 41 27

Abstract

\\'E' consider a ftexiblt' ",true-ture \\'hirh consist..:; of a rigid hody \ ..... ith a flpxih!e b .... ~1!Jj danllwd

to it at. one end, the other (,Ild of the heall) is fret' \Vc a:--.~U!lJ(- tbat the celltl~r I)f !nas:-. of t.he

rigid body IS fixed III all ilwrtial fr;\!l'l'_' ':~IH! that tht" ":!Iok structure pnfonns plauar :notioIJ. To

contr0J tlJe structure. Wf' ~L<';SIIJl1e tha[ a (onlrollorqllf' is applied t.o the rigid bod! ar:'~ boulldary

cOlltrol force and mOnl('Ilt an~ applied to !h~' fr('E" PHd of t he bt:alll. The c.ontrol prn!d· III we st.ndy

IS t.o design a cOlltrol Jaw to a('hif'\4~ all arbit.rary orielitatlOn for the rigId hody ;l::tj sllppr('~s

tIl(' 1JE:>am vibratIOns. \Ve I'ropo:,f' a (,()lltrol'law \\·hidl .... ohr-. ..; this problem. By tll',,!ifyillg t.his

control law we ohtain a ;-;1.ahilizlng r\lntro! Jaw for tliP lll>:\ihl,-· .;;truclure considf'l'pd ll"'rl..'.

hPywurus: Flexiblp Span:, Strunllre. Boundary (>_ll1lr(1\. Distributed ParalllPlr-r SystE'lliSI

Sell1igroups. StabilIty L) ;lPUIIO\ FUllctlons

235

°This resea.rcl! ha...5 been '>llpported by the SCientific aud T ..... ch11 iC1.1 fh::,c;Lrch Council of Turk(:y Hilder the gr;1nt TBA(;-1116.

6

A.. Gain Matrix Decomposition and Some of Its Applications*

A. S. Morse Department of Electrical Engineering

Yale University New Haven, Ct., 06520-1968

October 7, 1992

Abstract

Any real square matrix M can be written as

M = U(I + L)S

where U is a matrix of D's, 1's and -1's having exactly one nonzero element in each row and column, L is a strictly lower triangular matrix, and S is a {symmetric}, pos­itive semi-definite matrix. The aim of this paper is to demonstrate the utility of this easily derived fact. This is done in two ways. First, the decomposition is used to develop an identifier-based solution to a simplified multivariable adaptive stabilization problem solved previously using nonindentifier-based methods. Second, it is briefly ex­plained how to use the decomposition together with hysteresis switching and a certain "lifted" discrete-time system representation, to obtain an excitation-free, identifier­based, adaptive stabilizer for the entire class of n-dimensional, siso, controllable, ob­servable, discrete-time linear process models. This is accomplished by exploiting a new method of discrete-time parameter adjustment called "pseudo-continuous" tuning.

Keywords: Adaptive control; identification; multivariable control; matrix factorization; switching logic.

"This research was supported by the National Science Foundation under Grant No. ECS-9012551 and by U. S. Air Force Office of Scientific Research under Grant No. F49620-92-J-0077

Linear-Quadratic Optimal Control of Mechanical Descriptor Systems

Peter C. Miiller

Bergische lJuiversitiit - Gesamthochschule Wuppertal GauJ3str. 20, D-42119 Wuppertal, FRG

237

lear mechanical descriptor systems may be governed by a set of f linear time-invariant regular 'erential equations of second order

Mi(t) + (D + G)z(t) + (K + N)z(t) = Tu(t) + FTA(t) (1 )

I of p holonomic constraints

Fz(t) = o. (2)

~ f-dimensional vector z denotes the displacements and u is the r-dimensional vector of control uts. The p-dimensional vector A characterizps the constraint forces corresponding to the constraints . The matrix of inertia M is assumed to be symmetric and positive definite. The matrices D, G, K, 're alternately symmetric or skew-symmetric and they are related to damping, gyroscopic, stiffness,

circulatory forces of the mechanical system. The differential-algebraic equations (1,2) define a criptor system of index 3. ~ feedback control of the system (1,2) will be designed minimizing the quadratic cost fu~ctional

It>'> [Q Z][x] . J = 2' Jo [xT

uT 1 ZT R u dt ~ nun (3)

nning an at least positive semidefinite weighting matrix with submatrices Q = QT 2 0 and

= RT > O. The descriptor vector is defined as xT = [zT zT AT]. Without looking for any uls in a first step, formally the solution may be expressed as

(4)

I a suitable solution P of the algebraic Riccati equation

ETp(A-BR-IZT)+(A-BR-IZT)TpE-ETPBR-IBTPE+Q_ZR-IZT =0. (.5)

en the formal solution (4,.5) does make sense for mechanical descriptor systems? In the contribu­the following theorem will be proved. The linear-quadratic optimal regulator problem (3-5) for

hanical descriptor systems (1,2) is properly stated iff

sIz = o. n the Riccati equation (5) has a solution P with

xTETpEx 2 0

Ie system (1,2) is stabilizable and the detectability condition

. [8E-(A-BR-IZT)] _ rank S[(Q _ ZR-IZT) - n

(6)

(7)

(8)

s. In this case the optimal control (4) feeds back only z, z. The closed-loop control system is Iptotically stable. tis result the matrices SI, S2 define a matrix S = [SI S2 1 of a transformation to the Kronecker 'nical form of (1,2). The matrix SI characterizes the slow subsystem while S2 is related to the subsystem. The matrix E, A, B result from the descriptor form Ex = A x + B u of (1,2). In the Ir the transformation of the system (1,2) to its Kronecker canonical form is explicitly given such the conditions (6,8) can be checked. By the matrices SI, S2 the Riccati equation (5) is reduced to :ular Riccati equation of dimension 2(1 - p) (instead of 2f + pl. Thprefore, for linear mechanical ,iptor systems the complete solution of the linear-quadratic optimal control problem is at hand

238

Modelling and Cont.rol of one diffusion process

Leonid A. Muravei

TSiolkovsky State Tochnical University (Moscow Aviation TQchnolB~Y Irl~llLltL\:i), 1OS707. Pell'ovka 2'. Mos(:()w. ~ussia;

PAX: (095) 973-21-36; Tel.: (095) 227-87-'72; E-mail: muravey @ msk.su

Abst.ract.. This (so named get.tering) process applied (VLSI Techniligy. Ed. by S.Sze. New-York, 1983) in cleaning of silicon circui.t. board from nondesirable admixtures. Mathematicaly I t is described by parabolic equation ut = kCt.)uxx ' 0 < t < T. 0 < x < 1, with a special non-local boundary value condilion on right side x = 1 of silicon board

t

f uCT)uxCT,l)dT = pouo(1) - pCt)u(t.l), t. E (O,T). o

where uo(x) is initial distribulion of admixture and kel) and pel) are diffusion and segregation coefficients - control functions. ~. ~ords. Diffusion equalion. non-local boundary value condItion. coefficient control problem. existence class of functions having bounded variation. Main result. A characteristic feature of considered problem Is the presence of discontinuity of the control functions. The correspondi ng theorem establ i shes the sol uti on of optimlzati on problem in a class of discontinious functions having bounded variation. I dea of proof. The ground ana I yti ca 1 d iff i cu It Y is to extend the old comparison theorem of H.Westphal (1949) on the non-smooth case and to give the replacement of t.he parabolic maximum prinCiple on following estimates

exp{-lnf'(O~~)~(TJV[O,t)(P)} :!: u(l.x) :!: MeXP{tnf'[:~~)~tT)V[O,tJCP)}, where m = min uoCx), M = max uoC;:) and V[O,t]Cp) is a variation

of pC T) on [0, t.l.

Stabilization using low-gain PI control: 5150 case

Denis Mustafa Department of Engineering Science, University of Oxford

Parks Road, Oxford, OX1 3PJ, England email: Denis.MustafalOeng.ox.ac . uk

Abstract

It is shown how to construct all stabilizing low-gain PI controllers for SISO sys­

tems. The existence work was done by Lunze (1985) and Morari (1985): For a SISO

system, there exists a stabilizing neighbourhood around the origin in PI gain-space

if and only if the dc gain is positive. We show how to construct the largest such

neighbourhood - that is, the largest region around the origin in PI gain-space such

that all gains in the region are stabilizing. Closed eigenvalue formulae are derived for the boundary of the maximal region.

The technique exploits results of Fu and Barmish (1988) on stability of perturbed

matrices, and Saydy, Tits and Abed (1990) on stability analysis using guardian maps.

The key to obtaining closed formulae is to use the recently-defined block Kronecker

sum of Hyland and Collins (1989). As pointed out by Mustafa (1993) the method is

also applicable to finding the maximal stability region of singularly perturbed systems.

References

M. Fu and B. R. Barmish (1988). Maximal unidirectional perturbation bounds for stability of polynomials and matrices. Systems and Control Letters, 11:173-179.

D. C. Hyland and E. G. Collins (1989). Block Kronecker pruducts and block norm matrices in large-scale systems analysis. SIAM Journal on Matrix Analysis and Applications, 10(1):18-29.

J. Lunze (1985). Determination of robust multivariable I-controllers by means of experiments and simulation. Systems Analysis, Modelling, and Simulation, 2(3):227-249.

M. Morari (1985). Robust stability of systems with integral control. IEEE Transactions on Auto­matic Control, 30(6):574-577.

D. Mustafa (1993). How much integral action can a control system tolerate? To appear in Linear Algebra and its Applications (Special Issue on Systems and Control).

L. Saydy, A. L. Tits, and E. H. Abed (1990). Guardian maps and the generalized stability of parametrized families of matrices and polynomials. Mathematics of Control, Signals, and Systems, 3:345-371.

239

240

Stabilization of Feedback Linearizable Systems Using B.P. Neural Network

KWANGHEE NAMt

Abstract

The feedback linearizability of the system, i: = f(x) + ug(x) can be easily checked by the linearly independence of span{g,ad/g, ... ,ad'j-l g} and the involutiveness of span{g, ad/g, ... , ad,-2g}. But, even though we know the system is feedback linearizable, It is, in general, very difficult to obtain a linearizing feedback and a coordinate transformation which has the vital importance in the construction of stabilizing or tracking controller. The main obstacle lies in the diffi­culty in obtaining an integrating factor for an annihilating one form of the involutive distribution, span{g, ad/g, ... , adj-2 g}. We present a constructive method for obtaining the required integrating factor by utilizing a three-layered back propagation (b.p.) neural network and also propose a stablizing control scheme.

p

ORBITS HOMOCLINIC TO RESONANCES IN

EXTERNALLY EXCITED NONLINEAR OSCILLATORS

N. SRI NAMACHCHIVAYA AND NARESH MALHOTRA

Department of Aeronautical and Astronautical Engineering University of Illinois at Urbana-Champaign

Urbana, IL 61801

ABSTRACT

In this paper, we examine the global dynamics associated with a generic two­degree-of-freedom, coupled nonlinear oscillator of the following form:

iii + /3,<iI + h(q"q':I") = pcosvt

ii, + /3,q, + h(q"q':I") = 0

where (q" q,) represent the generalized coordinates, and the first mode is harmo­nically excited while the second mode is unforced. In these equations, we assume that the potential terms satisf y the following condition:

The method of averaging is used to obtain the second order approximation of the response of the system in the presence of external and one-one internal resonance. This system can describe a variety of physical phenomenon such as the motion of an initially deflected shallow arch, nonlinear oscillations of an auto-parametric vibration absorber, nonlinear response of suspended cables etc .. Using a perturba­tion technique developed by Kovacic and Wiggins (1992), we show the existence of Silnikov type homoclinic orbits which may lead to chaotic behaviour in this syst em.

241

242

Authors : Mahesh Nerurkar Department of Mathematics Rutgers University. NJ 08102. E-Matl: NERURKAR@CANCER.RUTGERS.EDU Tel (609)-757-6132. FAX (609)-757-6495 Hector Sussmann Department of Mathematic', Rutgers univerSity. NJ. 08903.

Abstract

Dynamics of the generic Rabl OscWator

The motion of a spin t particle in the external time dependent

magnetic field feo(t) is described by the equation

2 where x E C Jw:R -+ C is continuous. A E R, ro E n are parameters. Typically fw(t) = F(ro1), ro E Q, where (ro,t) -t rot Is a flow on nand F:O -+ C ls continuous. Physicist and Englneers have numerically studied this model for specific chOices of f. Very recently existence of almost periodic trajectories for this system was proved uslng the K.A.M. technique (for suitable A,f and for the suitable Irrational rotation flow (n,R)). The fundamental matrix solution of above equation defines a SU(2.C) valued co cycle and it generates a skew product flow on SU(2,C) X n. USing methods of Dynamical Systems and Control Theory. we show that every orbit of this flow Is dense and there are no almost periodic trajectories for a generic cholce of F. The methods developed prove a result about accessibtltty properties of a control system with time dependent. left invariant vector fields on a compact Lie group in the presence of a drift term.

Robust eigenstructure assignment in systems subject to structured perturbations

Abstract:

Dr. N.K. Nichols University of Reading, UK

243

The problem of robust eigenstructure assignment by feedback in a linear, multi variable, time-invariant system which is subject to jtructu.red perturbations is investigated. A mea­

sure of robustness, or sensitivity, of the poles to a given class of 'perturbations is derived,

and a reliable and efficient algorithm is presented for constructing a feedback which assigns

the prescribed poles and optimizes the robustness measure.

244

Observer Design for Descriptor Systems with Application to Gas Dynamical Networks

N. K. Nichols and S. M. Stringer

Abstract

A technique is often required in practice for estimating the inputs to a dynami­cal system from its measured outputs. In this paper one approach to this problem is investigated. An augmented system matrix is used, together with a dynamic observer, or state-estimator. Two techniques for constructing robust observers are employed: robust eigenstructure assignment and singular value assignment. This approach is successfully applied to the estimation of flow demands in a gas network with sparse pressure telemetry.

We consider discrete generalized state-space, or descriptor, systems of the form

Ex(k + 1) = Ax(k) + BJu(k + 1) + B2U(k)

y(k) = Cx(k)

(1)

(2)

We partition the inputs u(k) = [uf(k) uf(k)JT such that u2(k) contains the inputs to be estimated. An augmented system is then formed by incorporating u2(k) into the vector of state variables and introducing extra dynamical equations of the form

(3)

Conditions are derived that guarantee the complete observability of the aug­mented system. In practice the difference equations (3) give reasonable approxi­mations, provided the inputs u2(k) vary slowly relative to the sampling time.

Observers based on the augmented model are designed using two techniques. The robust eigenstructure assignment technique constructs an observer with given eigenvalues and an appropriate set of eigenvectors such that the eigenvalues are insensitive to small perturbations in the system. The eigenvalues are chosen to ensure convergence of the observer. The singular value assignment technique is used for generalized state-space systems together with feedback at the present time level in order to ensure robust regularity of the system.

Application of these techniques to the estimation of gas network flow demands is described. The results show that the use of these design tt;dmiques can signif­icantly reduce the effects of modelling error on the eSLimated inputs.

Numerical solution of Timoshenko's equations using wave digital filters

Gunnar Nitsche

Robert Bosch GmbH C/FOH Robert-Bosch-Str. 200

0-3200 Hildesheim, Germany

Phone: +49 5121 493962 FAX: +49 5121 492538

E-Mail: nitsche@hievn.bosch.de

245

The numerical solution of partial differential equations (POEs) implies the discretization of both the independent variables (coordinates) and the dependent ones (field quantities). Methods known from multidimensional digital signal processing are a powerful tool to

deal with the resulting discrete systems. Physical systems described by POEs are usually passive due to conservation of energy and positiveness of stored energy. Furthermore, the propagation speed in proper models of physical systems can be assumed to be finite.

Both properties are preserved in the discrete simulation if the wave digital filter (WOF)

approach to approximate continuous systems is applied. The method yields discrete passive dynamical systems that are inherently stable and allow massive parallelism.

The WOF approach is quite general and has been applied to a variety of PDEs from different areas of applications. These include electrodynamics, acoustics and even nonlinear problems in fluid dynamics. The focus of this talk will be the applicability to the so-called

Timoshenko's equations describing the dynamics of a bendable bar. Bending waves are subject to strong dispersion that may cause stability problems in the numerical solution. Contrary to other methods these problems can fully be solved by applying the WOF

approach.

246

A Design of Strongly Stabilizing Controller

and Loop Transfer Recovery

Shohei Niwa and Masayuki Suzuki Department of Aeronautical Engineering, Nagoya University

Furocho Chikusaku, Nagoya, 464 Japan Tel: +81-52-781-5111 ext.4418, Fax: +81-52-781-4094

E-mail: a40418a@nucc.cc.nagoya-u.ac.jp

Abstract

The control system design utilizing modern state space theory sometimes produces unstable controllers. Unstable controllers have poor robustness for the plant parame­ter variations and the physical realization of them are very difficult. Stability problem of controller or compensator appears when modern state space control theories such as linear quadratic optimal control theory, observer theory, Kalman filter theory, Hoo robust control theory, etc. are applied, because the controllers designed by such mod­ern theories are often open loop unstable although they stabilize the original plants with loop closed. In the design of controllers for complicated systems such as robots, aerospace vehicles and space structures etc., classical design methods of compensator are not applicable and only the modern techniques can be utilized, where the number of states, inputs and outputs are very large. In such cases, controllers should include high dimensional complex dynamics. Therefore the problem of strong stabilization i.e. the design of stable controller, is especially important.

For a scalar transfer function so called parity interlacing property (p.i.p.) is well known as a necessary and sufficient condition of strongly stabilizability. A stable stabilizing compensator for the plant can be constructed based upon this property. However the method relying on p.i.p. can not be practically used because this is not consistent with the commonly used optimal control design such as linear quadratic Gaussian method.

The purpose of this paper is to present a method to obtain a stable observer based controller. A method is presented for the design of strongly stabilizing controllers. This method is consistent with the standard state space design methods such as LQG, Hoo robust control, etc .. The proposed method is derived by applying the quadratic stabilization method which is developed for the robust stabilization of uncertain sys­tems. This method utilizes the quadratic stabilization techniques which is developed for the robust control design for the systems with structured uncertain parameters.

We will show a strong relation between the design method of observer based stable controller and that of loop transfer recovery (LTR). The LTR method can also produce stable observer based controllers. However the controllers obtained by the LTR method include rather high observer gain. High gain controller is not favorable because of its too broad band width and the problem of saturation. The value of gain obtained by the proposed method is less than what is required in LTR method.

On ripple-free deadbeat control of sampled-data systems

~

by Eitaku Nobuyama

Department of Control Engineering and Science

Kyushu Institute of Technology

Iizuka, Fukuoka 820, JAPAN

e-mail: nobuyama@ces.kyutech.ac.jp

This paper is concerned with deadbeat control in sampled-data systems. Deadbeat

control achieves finite-time settling (deadbeat settling) at sampling instants, but there

may exist error called ripple between sampling instants even after the response is settled

at sampling instants. The author have given a parametrization of all ripple-free deadbeat

controllers (controller which achieve deadbeat settling without ripple) for the unit step

reference. The objective of this paper is to extend the result to the general reference case.

The procedure for deriving a parametrization of ripple-free deadbeat controllers

consists of the next three steps.

(I) Parametrization of stabilizing controllers (the Youla parametrization).

(2) Parametrization of deadbeat controllers.

(3) Parametrization of ripple-free deadbeat controllers.

Each step requires us to solve a polynomial Bezout equation; hence, we obtain a

parametrization of ripple-free deadbeat controllers by solving three polynomial Bezout

equations.

247

248

A new look at realization theory

by R. Ober Center for Engineering Math

The university of Texas at Dallas P.O. Box 830688

Richardson, Texas 75083-0688 E-mail: ober@utdallas.edu

P. Fuhrmann Department of Mathematics Ben Gl'rion University of the Negev

Beer Sheva, Israel E-mail: paf@bengus.bitnet

Abstract

A realization procedure will be given for continuous-time finite dimensional systems. A

generalized Hankel map is defined on spacewise continuous functions. No restrictions on

the stability of the system are necessary. The realization is in terms of the left shift on a

space of continuously differentiable functions. A transfer function version of this approach will also be given.

Abstract

Boundedness properties of infinite dimensional balanced realizations

Raimund Ober and Yuanyin Wu

Center for Engineering Mathematics University of Texas at Dallas

Richardson, TX 75083

In general infinite dimensional continuous-time balanced realizations have unbounded input and output operators. Also the generator of the Co-semigroup of the realiza­tion is in general unbounded. In this paper results will be presented that give conditions for these operators to be bounded. The conditions are given in terms of properties of the transfer function. The proofs involve methods based on the theory of shift invariant subspaces in the Hardy space H2.

249

250

VARIA nONS ON THE FUNDAME~T AL PRINCIPLE FOR LINEAR SYSTEMS

OF PARTI,\L DIFFERE!'iTIAL OR DIFFERENCE EQUATIONS WITH

CONSTA:-JT COEFFICIENTS (for ~1TNS 1993 • September 1992)

Ulrich Oberst . Institut fiir Mathematik . Universitiit [nnsbrock

Technikerstrasse 25/7 . A - 60::0 [nnsbrock . Osterreich( Austria)

Around 1960 Ehrenpreis . ~laJgrange and Palamodov (see [E ] and [p]) proved

in very important work that many iunction spaces satisfy the fundamental

principl" . and thereby solved the existence problem for the solution of linear

systems of partial differential equations with constant coefficients. In [0] I

proved the liuge cogenerator property, for these function spaces and applied

them to multidimensional system theory. Here I construct new .. smaller"

spaces of "locally finite" or .. rational" functions and sequences which satisfy

the fundamental principle and the cogenerator property too and which are also

interesting for analysiS . numerical mathematics and system theory . this work

was initiated by a hint of A.Dress that my paper [0] is useful for the theory

of multivariate ,plines wher!' the sclutions of certain sy:;tcm:; of part!:d , differential and difference equations and their properties are also discussed .

-

A BRIEF SURVEY OF CANONICAL FORMS OF SINGULAR SYSTEMS

Kadri Oz<;aldlran

Department of Electrical and Electronics Engineering Bojjaziri University

80815 Bebek-Istanbul, Turkey e-mail:ozcal@trboun.bitnet

251

Over the last decade considerable attention has been paid to systems whose dynamics can be modelled in the form:

Ex' = Ax + Bu ; y = Cx

where x E nn, Ex E n r, u E nm and y E np

• Systems which admit representations of the sort above have been referred to as descriptor systems, singular systems, implicit systems and generalized state-space systems.

Group analysis of singular systems has been considered over the last four-five years and a number of canonical forms have been obtained. To wit, define E by

and define two subsets of E by

6. Eco = {(C,E,A,B) EEl C = O}

Eco ~ {(C,E,A,B) EEl C = 0 and B = O}

Define a number of transformations as follows:

TI (C,E,A,B) -> (CV, W-IEV, W-IAV, W-IB) (IWI i' 0, IVI i' 0) T2 (C,E,A,B) -> (C,E,A,BQ) (IQI i' 0) T3 (C,E,A,B) -> (C,E,A + BF,B) T4 (C,E,A,B) -> (C,E+BK,A,B) T5 (C,E,A,B) -> (PC,E,A,B) (IPI i' 0) T6 (C,E,A,B) -> (C,E,A+JC,B) T7 (C,E,A,B) -> (C, E + LC, A, B)

Let 9j denote the transformation group generated by transformations TI , .•• , Tj • Some of the actions defined by these transformation groups are meaningful from both mathe­matical and system-theoretic points of view. More specifically, the main actions of interest are the following:

1. Actions of 91 and 92 on the sets E, Eco, Ec;;.

2. Actions of 93 and 94 on the sets E, Eco.

3. Actions of 96 and 97 on the set E.

In this paper, we shall survey the results available in the literature on the classification via canonical forms of some of the actions defined above. We shall also underline the importance of some of the remaining actions which have not been studied in detail and remark on the virtual difficulties involved in studying them.

252

Strategies for controlling chaos in Chua's circuit

M.J.Ogorzalek, A.D~browski, Z.Galias

Department of Electrical Engineering, University of Mining and Metallurgy,

al.Mickiewicza 30 30-059 Krakow, Poland

e-mail: maciej@ucLagh.edu.pl

Abstract. Recent results concerning control of systems operating in a chaotic regime [2J-[3J encouraged us to study possibilities of achieving control of chaos in Chua's circuit [1 J. We describe circuit applications of the OGY technique for stabilisation of unstable pe­riodic orbits embedded in the chaotic attractor and extensions needed in the case of a real midium frequency circuit. We describe both simulation and experimental results in the case of full description of the system (ie. all state variables are accessible and measurable) and results based on measurement of a single variable using time-delay method. Influence of measurement errors on the performance of the control algorithms is studied. We de­veloped a software package consisting of programs for handling automatic measurements in the electronic system, characterisation of the system on the basis of measured experi­mental time series using measurements of single variable and delay coordinates technique or using measurements of all state variables form the process. A suplementary package is included for generating data via numerical integration of circuit equations thus making it possible to carry out simulation tests. Chaotic signals are further characterised (iden­tified) in terms of unstable periodic orbits. Advanced algorithms are used to compute Poincare sections of trajectories, unstable periodic points of the Poincare map, eigenval­ues and eigendirections of periodic points. These characterisations are further used for generation of the control signals for stabilisation of chosen periodic state.

The software is fully interactive - the user takes the decisions at each step of the procedure (eg. length of the measured interval, choice of orbits to compute, accuracy, choice of the controlled orbit etc. It is an extremely important feature of the system that the user has full control over the procedure - controlled periodic state can be chosen in a systematic way among the identified orbits - this is in contrast to many attempts of controlling chaos in which the accessible "goals" are found by trial and error experiments in the breadboarded system.

Results of experiments of controlling a real Chua's circuit are described. A comparison of the results obtained with known results obtained in purely electronic on-line control are included. The developed methods and algorithms are suitable for studies of any chaotic systems measurable electronically provided the operating frequencies are in the kHz range and the dimensionality is not high.

References [1J Chua, L. O. [l992J. The Genesis of Chua's Circuit. Archiv fur Elektronik und

Ubertragungstechnik, Special Issue on Nonlinear Networks and Systems, vol. 46,pp. 250-257.

[2J Ott, E., C. Grebogi and J.A. Yorke (1990a). Controlling chaotic dynamical systems. Published in: CHAOS - Soviet-American Perspectives on Nonlinear Science, D.K. Campbell ed., American Institute of Physics, New York, pp.153-172.

[3J Ott, E., C. Grebogi and J.A. Yorke (1990b). Controlling chaos. Phys. Rev. Letters, vo1.64, No.Il, pp.Il96-1199.

Dualistic Differential Geometry of Positive Definite matrices and Its Applications to Optimizations and Matrix Completion Problems

A. Ohara*, N. Suda* & S. Amari**

*Osaka University Department of Systems Engineering

1-1, Machikane-yama, Toyonaka, Osaka 560, JAPAN Tel. +81-6-844-1151 ext.4633 e-mail ohara@ce.osaka-u.ac.jp

** University of Tokyo Department of Mathematical Engineering and Information Physics

7-3-1, Hongo, Bunkyo-ku, Tokyo 113, JAPAN Tel. +81-33-812-2111 ext.6910

253

Abstract: Positive definite matrices often appear to play important roles in various aspects of systems theory, e.g., Lyapunov theory, Riccati equation and covariance analysis. Hence utilizing the geometrical knowledge of the set of positive definite matrices brings a possibility to give new insights on the theory and advances on optimization techniques on the set.

It is well known that manifold of positive definite matrices is qualified as a Rieman­nian symmetric space [1] by the usual Riemannian (Levi-Civita) connection. Although this space has nice properties, it is not sufficient to use for system theoretical analysis. However, by introducing dual connections, which were extensively studied by Amari [2] in statistics, we can give abundant differential geometic structures on the manifold.

In this paper, we first discuss dualistic differential geometric structures of positive defi­nite matrices. As a result, dual coordinate systems and pseudo distances called divergences are determined. They are fundamental tools of geometrical analysis of positive definite matrices. Next, we apply the theory to matrix (positive definite) completion problems [3]. Geodesic lines are shown to be essential to construct the solutions. Finally, we discuss how the matrix completion problem can be utilized for controller design problems via parametrization of stabilizing state feedback gains [4], [5].

References [1] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York

( 1962) [2] S. Amari, Differential-Geometrical Methods in Statistics Springer-Verlag, (1985). [3] C. R. Johnson, "Positive Definite Completions: A Guide to Selected Literature", L.

Auslander et. al eds. Signal Processing Part I: Signal Processing Theory, Springer Verlag (1990)

[4] A. Ohara & T. Kitamori, Geometric structures of stable state feedback systems, IEEE Trans. AG, Vol. 38, No.7 (1993).

[5] A. Ohara & S. Amari, "Differential Geometric Structures of Stable State Feedback Systems with Dual Connections," Proc. of IFAG Workshop on System Structure and Control, 176-179 (1992).

254

Stabilization of a Flexible Beam Subject

to Random Disturbance

Akira Ohsumi, * Yoshiaki f{usunoki,tand Zhi Yang:j:

Control and System Science Group Department of Mechanical and System Engineering

Kyoto Institute of Technology Matsugasaki, Sakyo, Kyoto 606, Japan

Abstract

In this paper the state feedback stabilization problem is studied for a class of second-order linear systems of flexible structures subject to distributed white noise. As a measure of stabilization, the total energy function of the system is considered, and its upper and lower bounds are evaluated by introducing a new Lyapunov function to show that the stationary value of the energy depends on the noise intensity as well as the feedback gain.

Furthermore, it is shown that by the appropriate choice offeedback gain it is possible to minimize the stationary upper bound and to maximize the decay rate of total energy of the system. To illustrate the theory, numerical aspects are provided.

* E-mail: ohsumi@ipc.kit.ac.jp t Present Address: Mitsubishi Electric Corporation, Tsukaguchi, Amagasaki City 661, Japan. + Present Address: Tsubakimoto Chain Co., Shinkou, Hannou City, Saitama 357, Japan.

.,

....

Use of computer algebra for the motion planning problem

N.E. Oussous & M. Petitot

We are using the combinatorical properties of nilpotent Lie algebras to study the Motion Planning problem in nonlinear control.

The main part of these techniques is implemented in the Computer Algebra

System Axiom. An example will be presented to illustrate the link with recent appeared methods: flat systems, and systems in chained form.

255

256

2D SYSTEMS THEORY AND ITERATIVE LEARNING CONTROL WITH EXTENDED MEMORY - File mtns932d.tex

David H.Owens, Centre for Systems and Control Engineering, University of Exeter, North Park Road, Exeter EX4 4QF, England

September 18, 1992

Summary: In iterative learning control (ILC) [1] the basic problem for the control algorithm is to learn inputs that generate required outputs from a dynamical system by repeated trials and updating of control inputs from trial to trial. More formally, given a dynamical system SeA, B, C} in Rn and a desired output signal ret}, 0 :s; t :s; T, construct a causal sequence of experiments that generates a sequence of input signals (Uk(t}}t2:0 and outputs (Yk(t})no with the property that limk_oo(r(.) - Yk(')) = 0 with respect to a chosen topology in a suitable linear space -;'f output signals.

The new learning law used in this paper has the form

(1)

where I< is a causal "learning" operator feeding back the current trial error. The inclusion of relaxation pa­rameters "'; allows for the possibility of controlling the convergence rate of the scheme to ensure convergence in ,orne acceptable sense whilst guaranteeing small limit errors. The purpose of this paper is to demonstrate the applicability of the 2D systems theoretical techniques of [2], [3] to the problem and to completely characterize the convergence of the algorithms and the limiting behaviour in terms of "'I, "2 and systems structure. The following illustrates the form of result in the paper.

Theorem 1: (Lp - Convergence of lLC) The learning algorithm converges to a limit if and only if p(z} = ,2 _ "IZ - "2 has roots 1-'1,1-'2 in the unit circle only.

:Note: V.t.c. the error will be shown to converge in the norm topology of Lp(O, T} to a limit eoo and that the ,rror sequence is also bounded by a geometric expression}

rhe counter-intuitive aspect of the result is that the stability oflearning is independent of (A, B, C}. Convergence .. ates will however depend on such systems data. More precisely, convergence is predicted to be rapid if", is 'mall, converging roughly as ",k. Also, the limit error is described by a feedback system with effective control [('11= I-al-a I<. There is a conftict in this observation. This conftict has implications on systems and control ,tructure - that' small learning errors will require high effective gain and that G I< is stable under such high ~ains. This can be described briefty as a requirement that G I< is minimum-phase and relative degree one or two. rhe paper will explore the implications of this observation and illustrate the application of the ideas to simple J.umerical examples.

lteferences

1] Padieu and Su:"An Hoo approach to learning control", Int. J. Adapt. Control and Sig. Proc., 1990(4), pp 165-474.

2] J .B.Edwards and D.H.Owens:" Analysis and control of multipass processes", Research Studies Press, Taunton, 'ngland, 1982.

3] E.Rogers and D.H.Owens:"Stability analysis for linear repetitive processes", Springer-verlag, Berlin, April 992.

The singular values of a class of boundary control systems

Pandolfi, L.· Politecnico di Torino

Dipartimento di Matematica Corso Duca degli Abruzzi, 24

10129 Torino - Italy Tel. +39-11-5647516

E-Mail LUCIPAN@ITOPOLI.BITNET

October 3, 1992

Abstract

We present a new result on the singular values of systems with boundary control and distributed observation. This result shows a relation between singular values and Fredholm equations.

257

258

A NOTE ON SUFFICIENT CONDIDIONS OF OPTIMALITY FOR

DISCRETE-TIME OPTIMAL CONTROL PROBLEMS WITH

CONTROL INEQUALITY CONSTRAINTS (1)

J .F.A. DE O. PANTOJA (2)

ABSTRACT

A stagewise version of wei I-know second order sufficient

conditions for characterizing isolated local minimizers to

conventional mathematical programming problems is presented

and shown to be sufficient for characterizing isolated 10

cal minimizers to discrete-time optimal control

with control inequality constraints.

problems

Keywords. Discrete-Time Optimal Control; Control Constra

ints; Sufficient Conditions of Optimal ity; Isolated Local

Minimum.

0) This research was partially supported by the National Research Counci I/CNPq and the IBM, Brasi I, whi Ie the au thor was visiting the National Laboratory for ScientT fic Computing/LNCC.

(2) Departamento de Matemat ica Universidade Federdl do Maranh~o Sao Luis, MA - Brasil FAX: 55-21-2958499 e-mail: USER FRAG LNCC.BITNET

ON THE "BANG-BANG" PRINCIPLE FOR

NONLINEAR EVOLUTION INCLUSIONS

N.S. PAPAGEORGIOU

National Technical University Department of Mathematics Zografou Campus and Athens 15779, GREECE

Florida Institute of Technology Department of Applied MathematiC8 150 West University Blvd. Melbourne, Florida 92901-6988, USA

ABSTRACT: In this paper we establish the existence of extremal solutions for a class of nonlinear evolution inclusions defined on an evolution triple of Hilbert spaces. Then we show that these extremal solutions are in fact dense in the solutions of the original system. Subsequently we use this density result to derive nonlinear and infinite dimensional versions of the "bang-bang" principle for control systems. An example of a nonlinear parabolic distributed parameter system is also worked out in detail.

259

o Worst-case identification in f2

Jonathan R. Partington

School of Mathematics, University of Leeds,

Leeds LS2 9JT, U.K.

Let h = (h(O), h(I), ... ) E f1 be the impulse response of an unknown linear discrete­

time system and let u = (u(O),u(I), ... ) E foo be a given bounded input. Given

corrupted data

k

y(k) = 'E h(j)u(k - j) + T/(k), for k = 0, ... , n - 1, j=O

where T/, unknown, is a bounded noise sequence, the worst-case identification problem

in f2 is to construct an approximation hn(Y(O), ... ,y( n - 1)) to h such that

Using the ideas behind a recent very general result [5] we give a solution to this

problem and suggest how to choose inputs efficiently. In contrast to the f1 problem,

where to identify n coefficients accurately may require much more than n measurements

[3,4,6], we note that in the f2 case one only requires O(n) measurements. This makes

use of various results on the size of polynomials [1,2].

References

[1] G. Benke, On the minimum modulus of trigonometric polynomials, Proc. Amer.

Math. Soc. 114 (1992), 757-761.

[2] F.W. Carroll, D. Eustice and T. Figiel, The minimum modulus of polynomials with

coefficients of modulus one, J. London Math. Soc. (2) 16 (1977), 76-82.

[3] M.A. Dahleh, T. Theodosopoulos and J.N. Tsitsiklis, The sample complexity of

worst-case identification of F.I.R. linear systems, System and Control Letters 20

(1993), 157-166.

[4] B. Kacewicz and M. Milanese, On the optimal experiment design in the worst­

case f1 system identification, Proc. 31st IEEE Conference on Decision and Control

(1992), 56-61.

[5] J.R. Partington, Interpolation in normed spaces from the values of linear function­

als, Bull. London Math. Soc., to appear.

[6] K. Poolla and A. Tikku, On the time complexity of worst-case system identification,

IEEE Trans. Auto. Control, to appear.

On Markovian and non-Markovian Schrodinger processes

MICHELE PAVON

Dipartimento di Elettronica e Informatica,

Universita di Padova, via Gradenigo 6/A and LADSEB-CNR35100 Padova, Italy

Tel. (39 )-49-828-7604 Fax: (39)-49-828-7699. E-mail: pavon@ladseb.pd.cnr.it

261

~et Pt be the one-time density of the Markov diffusion process Xt, and an let q be a positive nvariant density. In [1], [5] the process Yj := In(pt/q)(Xt ) was shown to be a reverse­

ime submartingale with respect to the filtration induced by the "future" of the process

r, thereby strenghtening the classical H-theorem. The submartingale property was also

:onnected to a backward-in-time stochastic variational problem, [2]. The latter was then

elated to the recently developed theory of Schrodinger bridges, and to large deviations of

he empirical distribution, [3J-[5]. Similar monotonicity results were also obtained replacing

he relative entropy distance with the Hellinger pseudo-distance. The purpose of this talk

; to review the above results and to show that the theory of Schr?dinger processes may be xtended, under mild assumptions, to non-Markovian diffusion processes. Moreover, when

n everywhere positive invariant probability measure exists, and und~r a certain crucial

everse-time purely nondeterminiticity property, an H-theorem may be establish also for

ilis class of processes.

References

1. M.Pavon, Stochastic control and nonequilibrium thermodynamical systems, Appl.

Math. & Optimiz. 19 (1989), 187-202. 2. F.Guerra and M.Pavon, Stochastic variational principles and free energy for dissipa­

tive processes, Analysis and Control of Nonlinear Systems, Proc.8th Internat. Sympo­

sium on MTNS-87, Phoenix,AZ, Usa, C.Byrnes, C.Martin and R.Saeks Eds., North­

Holland, Amsterdam,1988, 571-578. 3. P.Dai Pra and M.Pavon, On the Markov processes of Schr?dinger, the Feynman-Kac

formula and stochastic control, in Realization and Modeling in System Theory - Proc.

1989 MTNS Conf., M.A.Kaashoek, J.H. van Schuppen, A.C.M. Ran Eds., Birk?user,

Boston, 1990,497-504.

4. M.Pavon and A.Wakolbinger, On free energy, stochastic control, and Schr?dinger

processes, Modeling, Estimation and Control of Systems with Uncertainty, G.B. Di

Masi, A.Gombani, A.Kurzhanski Eds., Birk?user, Boston, 1991, 334-348.

5. M.Pavon, On Lyapunov functionals, and stochastic variational principles in nonequili­

brium thermodynamics, in Probabilistic Methods in Mathematical Physics, F.Guerra,

M.Loffredo and C.Marchioro Eds., World Scientific, 1992, pp. 334 -348.

c

262

Non square spectral factors and non internal stochastic realizations: a dilation problem and a conjecture*

Michele Pavon Dipartimento di Elettronica e Informatica

Universita di Padova and LADSEB-CNR 35100 Padova, Italy

Abstract

Let <1>(s) be an mxm, full-rank, rational spectral density matrix function. Let W be a minimal, mxp, stable spectral factor of <1>, and let W _ be the minimum phase stable spectral factor. Then Q:= W_-lW is a stable mxp function isometric on the imaginary axis (Q is inner if and only if p=m). In [I, Section 5.3] it is observed that constructing the driving noise of a minimal stochastic realization with transfer function W is equivalent to finding an inner dilation of Q.

The purpose of this talk is to present one particularly natural solution to this problem. We shall also show that this particular dilation is connected with a larger-dimensional spectral factorization problem. The latter leads naturally to a conjecture regarding the possibility of parametrizing all minimal, stable, square and non square spectral factors of <1> by means of the left inner divisor of the inner function that solves the dilation problem. This result would extend those of [2]-[4].

References

I. A.Lindquist and G.Picci, A geometric approach to modelling and estimation of linear stochastic systems, I.Math.Systems, Estimation, and Control I (1991), 241-333.

2.G.Ruckebush, Factorisations minimales de densites spectrales et representations markoviennes, Proc.1re Colloque A FCET-SMF, Palaiseau, France, 1978.

3.A.Lindquist and G.Picci, A Hardy space approach to the stochastic realization problem, Proc. 1978 CDC Conj, San Diego, 933-939.

4. L.Finesso and G.Picci, A characterization of minimal spectral factors, IEEE Trans.Aut.Control AC·27 (1982), 122-127.

* Research partially performed at tbe Mathematisches Seminar der Universitat Kiel with support provided by an Alexander von Humboldt Foundation fellowship. Tel.(39)-49-828-7604.Fax: (39)-49-828-7699.E­mail: Pavon@pdadrI.pd.cnr.it

A balanced canonical form for discrete time stable all-pass systems

Ralf L.M. Peeters and Bernard Hanzon Dept. Econometrics

Free University Amsterdam De Boelelaan 1105

1081 HV Amsterdam Fax +31-20-6461449

E-mail: bhnz@sara.nl

Abstract

263

In the work of Ober (d. e.g.[ID a balanced canonical form for continuous time all-pass systems plays an important role. The discrete time case is treated by using the well-known bilinear transformation that transforms a stable continuous time system to a stable discrete time system. The transformation transforms continous time all-pass systems to discrete time all- pass systems. However it turns out to be possible to apply ideas similar to those used in the continuous time case to obtain a different canonical form in the discrete time case. The dynamical matrix in this form is closely related to the one studied in [2, 3]. In this canonical from the Schur parameters play a prominent role. The canonical form is such that truncation leads to a stable system. Using the canonical form it can be shown that the set of all stable SISO all-pass systems of order smaller than or equal to n can be parametrized by the unit sphere in (n+l)-dimensional Euclidean space. This sphere is in fact obtained as the quotient space O(n + l)jO(n), where O(n) stands for the orthogonal group of n x n matrices. We treat also the complex case and if time permits, the connections between the Euclidean algorithm and the Schur algorithm are commented upon and the connection with Schur parametrization of AR models is explored.

References

[1] R. Ober, Balanced realizations: canonical form, parametrization, model reduction, Int. Journal of Control 46, No.2 (1987), pp.643-670.

[2] B.D.O. Anderson, E.!. Jury, M. Mansour, Schwarz matrix properties for continuous and discrete time systems, Int. Journal of Control 23, No.1 (1976), pp.I-16.

[3] B.D.O. Anderson, E.!. Jury, M. Mansour, On model reduction of discrete time systems, Automatica, 22, No.6 (1986), pp.717-721.

264

Superoptimal Hoo approximations of matrix functions

V.V.Peller

The talk is devoted to our joint work with N.J.Young. It is well known that for a scalar Hoo + C function on the unit circle T there is a

unique best approximation by Hoo functions. However for HOO + C matrix functions a best approximation is almost never unique. To make it unique one has to impose additional conditions.

Consider the following minimization problem. Let <fl be an m x n matrix function in Hoo + C. Cosider the sets

no <!t {F E H OO ; 111> - Flloo = distLoo(1), H OO )},

nj = {F E nj _ 1 ; F minimizes SUpSj(1)(() - F(())}, 1 5:j 5: min(m,n)-1 (ET

(here sj(A) is the jth singular value of a matrix A, so(A) = IIAII). A function Fin Hoo is called a superoptimal Hoo approximant of 1> if F E nmin(m,n)-l'

Theorem. If 1> E HOO + C, then a superoptimal Hoo approximant is unique. Moreover all singular values of 1>( 0 - F( 0 are constant on T.

The proof allows us to elaborate an algorithm to find the superoptimal approximant. We also study properties of the so-called very badly approximable functions, i.e. those for

which the superoptimal approximant is 0 and obtain certain special factorizations for them. We obtain sharp inequalities between the singular values of the Hankel operator H~ and

the superoptimal singular values. We also study hereditary properties of the (non-linear) operator of superoptimal approx­

imation and obtain analogues of Peller-Khrushchev results for scalar functions.

1

265

The Inverse Spectral Problem for Self-adjoint Hankel Operators

V.V.Peller' and S.R.Treilt

The talks are based on a joint work with A.V.Megretskii. The main result of the talks characterizes the self-adjoint operators on Hilbert space that

are unitarily equivalent to a Hankel operators. This problem is a refinement of the problem to characterize the positive operators that are unitarily equivalent to the modulus of a Hankel operator which was posed by Khrushchev and Peller in connection with a problem in prediction theory and was solved by Treil with the help of the method (suggested by Ober) based on linear systems with continuous time.

Any self-adjoint operator is characterized (modulo unitary equivalence) by its scalar spectral measure and its spectral multiplicity function. The following theorem is the main result.

THEOREM 1. Let r be a selfadjoint operator on Hilbert space, /l a scalar spectral measure of it, and v its spectral multiplicity function. Then r is unitarily equivalent to a Hankel operator if and only if

(1) r is non-invertible; (2) Ker r = {O} or dim Ker r = 00; (3) Iv(t) - v(-t)l:S: 2, /la-a.e. and Iv(t) - v(-t)l:S: 1, /ls-a.e.,

where /la and /l. are the absolutely continuous and the singular component of /l. The proof of the sufficiency is based on balanced realizations with discrete time. THEOREM 2. Let r be a self-adjoint operator satisfying (1), (2), and (3). Then there ex­

ists a balanced SISO linear system with discrete time such that the Hankel operator associated with it is unitarily equivalent to r.

The construction of a desired system is complicated and is based on the Kato-Rosenblum theorem on the stability of the absolutely continuous spectrum under trace class perturba­tions.

However the following theorem shows that linear systems {A, B, C} with continuous time which involve bounded operators A, B, and C cannot produce enough Hankel operators to solve the problem completely.

THEOREM 3. Let r be a positive self-adjoint operator with multiple spectrum. Then there exists no balanced SISO linear system with continuous time which involves bounded A, B, C such that the Hankel operator associated with the system is unitarily equivalent to r.

However linear systems with continuous time can be used to construct a Hankel operator with prescribed spectral properties in the case when Iv(t) - v( -t)1 :s: 1 almost everywhere. In that case the formulae are simpler that in the case of discrete time.

THEOREM 4. Let r be a selfadjoint operator satisfying (1), (2), and the inequality Iv(t)­v( -t)1 :s: 1, /l-a.e. Then there exists a balanced SISO linear system with continuous time such that the Hankel operator associated with it is unitarily equivalent to r.

'Department of Mathematics, University of Hawaii, Honolulu, HI, USA tDepartment of Mathematics, Michigan State University, East Lansing, MI, USA

266

PROBLEMS ON MULTILAYERED NEURAL NETWORK

LEARNING

V.Petridis

Dep. of electronic and computer engineering.

School of electrical engineering.

Faculty of engineering. Aristotle University of Thessaloniki.

BOX 438 54006 Thessaloniki - Greece.

e-mail: petridis@vergina.eng.auth.gr

ABSTRACT

The back propagation (BP) training method of artificial neural

networks (ANN) has received much attention and many variations of

the basic method have been proposed.

The main drawbacks of BP are: a) it gets trapped at local minima b)

it converges relatively slowly c) it exhibits scaling problems and d)

it cannot be implemented in hardware easily.

In this paper we present very simple and efficient techniques for

training ANNs, what we call the feedforward (FF) techniques . They

involve only forward passes. Extensive simulation has established that

they work very well as ANN training laws for multilayered and

recurrent ANNs; they are fast and they can be easily implemented in

parallel architectures. They do not require calculation of the

derivative of the activation function, a feature that makes them very

useful in the case of unmodelled activation functions, and their signals

propagate only in the forward direction. Moreover, their hardware

implementation is very easy.

FEW REFERENCES

1. Jacobs, R.A. (1988). Increased rates of Convergence through

Learning Rate adaptation. Neural Networks ,vol. 1 pp 295-307.

2. Tollenaere T., (1990). Super SAB: Fast Adaptive

Backpropagation with Good Scaling Properties. Neural Networks,

vol.3, pp.561-573.

3. Samad T. (1991). Back Propagation With Expected Source

Values. Neural Networks, vol.4, pp.615-618.

THE FULL MARTIN BOUNDARY OF THE BI-TREE

MASSIMO A. PICARDELLO AND WOLFGANG WOESS

Dipartimento di Matematica, Universita di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy

Dipartimento di Matematica, Universita di Milano, Via Saldini 50, 20133 Milano, Italy

ABSTRACT

Let T] and T2 be two homogeneous trees with degrees £] and £2 :::: 3, respectively. On the bi-tree, i.e., their Cartesian product T] X T2 , consider the aperiodic simple random walk or, more generally, a random walk obtained by taking a convex com­bination of aperiodic simple random walks on each of the Tj . Let p be the critical eigenvalue of the transition operator.

The minimal Martin boundary (i.e., set of minimal t-harmonic functions) is well understood, and so is the Poisson boundary.

On the other hand, the description of the full Martin compactification has remai­ned an open problem. This problem was proposed by Guivarc'h and Taylor, who gave the answer only for the critical eigenvalue (where the behaviour is different).

We solve the problem by making use of recent work of Lalley, who has given a local limit theorem for aperiodic, nearest-neighbour random walks on homogeneous trees, i.e., an asymptotic estimate of the n-step transition probabilities, which is uniform both in time and space We use this uniform estimate to derive a kind of "renewal theorem" for our random walk on T] x T2 , i.e., to describe the asymptotic behaviour in space of the Green function (resolvent) of the random walk for t > p. Our method is a bi-dimensional discrete variant of Laplace's method for determining the asymptotics of parametrized integrals.

As an immediate corollary, we find the directions of convergence of the Martin kernels. The full Martin boundary M(t) is then what was to be expected from previous evidence: if n j denotes the space of ends of T j , then

where St is a closed line segment in ]R2 which collapses to a single point when t = p. In particular, the Martin boundary of the product is considerably larger than the product of the boundaries.

Typeset by AW-1J;;X

267

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268

IDENTIFICATION OF FACTOR ANALYSIS MODELS

Giorgio Pieci DEI, Universita di Padova,

via Gradenigo 6/A, 35131 Padova, Italy Email PICCI@paola.dei.unipd.it.

Stefano Pinzoni LADSEB-CNR,

Corso Stati Uniti 4, 35020 Padova, Italy.

In this paper we will discuss some issues about the identification problem of Factor Analysis models of the form

y=Ax+e, y,eERm, xERn.

The aim of these models is to provide an explanation of m observed variables y = [YI . Y= I', describing them as linear combinations of n (with n < m and possibly minimal) factors x = [XI . Xn I', plus inde-pendent "noise". As a part of the specification of the model, it is assumed that the vector X and the m components of the error e = [el . e= I' are zero-mean and mutually independent random variables. It is well known that to each covariance matrix A := E yy' there correspond infinitely many minimal Factor Anal­ysis models, which provide equivalent (second order) descriptions of the data. We will consider a particular identtfiable class of Factor Analysis models where it is postulated that all error variables e, have the same variance af = a2 , i = I, ... 1 m.

For these models we shall compare two possible identification paradigms. The first is based on direct estimation from the data of an external description (such as the joint covariance matrix or the joint probability distribution of the observations) and then doing stochastic realization to obtain the (unique) corresponding Factor Analysis model with the chosen identifiable structure. On the other hand, one could instead identify directly the "internal" model parameters A E R=xn and (72, starting from the observed data. In this estimation problem, the latent variable x of the model appears in general as a "nuisance" parameter and must be treated in an appropriate way. However, the values XI,"" xn yielded by the identification scheme may be regarded as (Bayesian) estimates of the factors, as if they were computed on the basis of the identified model.

We shall compare the two methods referring to Maximum Likelihood estimates. An intelligent imple­mentation of the associated Total Least Squares algorithm permits to elucidate the relation between the two procedures.

Finally, we will consider some possible extensions to the dynam,e case, with special emphasis on the input-output approach.

-

Positive Linear Algebra

for Stochastic Realization of Finite-valued Processes

Abstract

G. Pieci Dipartimento di Elettronica e Informatica, Universita di Padova

Via Grademgo 6/a, !lS1!!1 Padova, Italy

J.R. van Sehuppen CWI, P.O. Box 4079, 1009 AB Amsterdam, The Netherlands

The stochastic realization problem of finite-valued processes asks for the classification of all minimal finite stochastic systems such that the output process of such a system equals a given process in distribution. This problem is motivated by the use of stochastic models in signal processing, communication, and control. The problem of characterizing the minimal number of states in this problem leads to a factorization problem for positive matrices and hence to the study of positive linear algebra.

In an earlier paper [4J the classification of primes in the positive matrices has been formulated and motivated. This classification reduces to the investigation of fully indecomposable primes in the set of doubly stochastic matrices. This problem is partially solved. In the lecture the most recent results on this problem will be presented.

The negation of a matrix being a fully indecomposable prime in the doubly stochastic matrices is that it admits a specific factorization. This factorization can, by Birkhoff's characterization of the set of doubly stochastic matrices as a convex polytope, be shown to be equivalent to the solvability of a linear relation over a Latin square. Conditions for the solvability of the latter equation may be derived by use of the specialization order. The fully indecomposable primes in the doubly stochastic matrices are a subclass of the positive circulant matrices.

For an entry to the literature on the stochastic realization problem for finite-valued processes see [3, 4J and on positive linear algebra see [1, 2J.

References

[1] A. Berman, M. Neumann, and R.J. Stern. Nonnegative matrices in dynamic systems. John Wiley & Sons, New York, 1989.

[2J A. Berman and R.J. Plemmons. Nonnegative matrices in the mathematical sciences. Academic Press, New York, 1979.

[3J G. Picci. On the internal structure of finite-state stochastic processes. In Proc. of a U.S.-Italy Seminar, volume 162 of Lecture Notes in Economics and Mathematical Systems, pages 288-304. Springer-Verlag, Berlin, 1978.

[4J G. Picci and J.H. van Schuppen. Stochastic realization of finite-valued processes and primes in the positive matrices. In H. Kimura and S. Kodama, editors, Recent advances in mathematical theory of systems, control, networks, and signal processing II . Proceedings of the International Symposium MTNS-91 , pages 227-232, Tokyo, 1992. Mita Press.

269

270

ALMOST SURE STABILITY OF DISCRETE TIME JUMP LINEAR PROCESSES

D. L. Pinto Jr., J. B. Ribeiro do Val

Universidade Estadual de Campinas - UNICAMP, Fac. de Eng. EItitrica, Depto. de Telematica,

C.P. 6101, 13081 - Campinas, SP, Brazil. fax: 055-192-391395

e-mail address:jbosco@dt.fee.unicamp.br

M. D. Fragoso Laboratorio Nacional de Computaciio Cientifica,

LNCC/CNPq, R. Lauro Muller 455, 22296 - Rio de Janeiro, RJ, Brazil.

Correspondence should be sent to the second author

October 30, 1992

Abstract

The stability of discrete time linear systems taking a number of distinct structures is studied in this paper. The changes among structures or forms occur randomly, and are described by an underlying finite state Markov chain that enumerates each form. The main result refers to the stochastic stability of these systems in the almost sure sense. When stability is simultaneously required in the terms of definitions 1.1 and 1.2, the equivalence to mean square stability follows. Also, a well known test of stability in use in the literature of these processes in fact assures stability in the almost sure sense. These results are based in the regenerative property of the state process, with respect to a se­quence of stopping times.

Key words: Stochastic stability, Discrete time jump linear systems, Markov processes.

DIVISIBILITY IN A CLASS OF RATIONAL MATRIX FUNCTIONS

Marek Rakowski

Department of Mathematics

Ohio State University

231 West 18th Avenue

Columbus, OH 43210

Leiba Rodman

Department of Mathematics

College of William and Mary

Williamsburg, VA 23187-8795

Abstract

271

A minimal factorization of an mxn rational matrix function W has been defIned to be the factorization

(1)

where the number of columns of WI is equal to the normal rank of W and ~(W) = ~(W 1) + ~(W 2)· Here ~(*) denotes the McMillan degree of a rational matrix function *. In this paper we

study divisibility in the class G = G (m,k) of rational matrix functions with m rows and normal

rank k which admit a minimal factorization.

We will consider the following problem. Given two functions W, WI e G, when there

exists a rational matrix function W2 so that (1) holds and ~(W) :: ~(W I) + ~(W 2)? If such W 2

exists, we will say that WI is a de&ree minimal left divisor of W. Our approach is based on state­

space methods. We fInd a minimal factorization WI = V LV R. Main result: W I is a degree

minimal left divisor ofW if and only ifVL and W have the same left kernel polynomial and a left

spectral triple of V L is a restriction of a left spectral triple of W. This extends results known in

the literature for certain special cases.

Complexity of Zero-Defect Generalized Inverses of Transfer Functions

Marek Rakowski and Bostwick F . Wyman

Dppartnwnt of ivIathematics

ThE' Ohio Statp l'niwrsity

Columbus, Ohio 43210

Abstract

DenotE' by R the field of scalar rational functions. and let G E Rmx" be a transfer

nction of a linear. time im'ariant systpm. A generalized in'verse of G is any function

x E R"xm which satisfies

and (1)

particular. a generalized in\"erse of G is a one-sided inverse of G whenever G is invertible

L one side.

vVe are interested in the complpxity of a generalized inverse G x of G measured by

; McMillan degree S( G X), that is. the (limension of its global pole space. Denote by Z( *)

Ld P( *) the glohal pole and zero spaces of a function *. There exist epimorphisms

and (2)

once. dimZ(G) is a lower bound for S(G X). If G X is not invertible. no upper bound for

G X) exists. Suppose. however. that the defect of G X pquals O. Then the dimensions of

(G X) and P( G X

) coincide and. since the second map in (2) is epic. 8( G X) :S 8( G).

vVp show that there exists a zero defect generalized inverse G X of G such that

G X ) = b'( G). vVp also show how to construct a zero defect generalized inverse of G of

inimum McMillan degree in tIlt" case when G is inn'rtible on one side. The construction

n be applied to transfer functions ill the class of rational matrix functions which admit

minimal factorization.

273

Speaker: A.C.M. Ran

Title: Contractiv~ rational m:mix functions. characteristic functions and Darlington synthesis

Abstract: Factorization results for functions of the form identity plus a contractive function will

be discussed as the first topic. A major role is played here by matrices which are dissipative in

an indefinite inner product space. As a second. related topic characteristic functions for such

matrices will be discussed. 'An application to Darlington synthesis will be given. Most of the

results were obtained in joint work with l. Gohberg.

274

High Gain Robustness of Distributed Parameter Systems

R. Rebarber S. Townley

We consider general abstract boundary control systems of the form discussed in Salamon [2]: itt) = ~z(t), rz(t) = u(t), y(t) = Kz(t). We assume here that the state z(t) is in a Hilbert space Z, the control u(t) is in a Hilbert space U, and the observation y(t) is in a Hilbert space Y for time t ~ O. For suitable ~ the nature of the boundary operators r and A will determine whether this system is regular in the given in Weiss [3]. If this is the case the system is equivalent to a system of the form itt) = Az(t) + Bu(t), y(t) = Cz(t) + Du(t), for z(t) in a Hilbert space X ;2 Z, and Az = ~z on D(A) = {z E D(~) I rz = O}.

We are interested in the high gain behavior of the closed loop system, with semigroup 5.(t), obtained by the feedback u(t) = -ky(t), where k > O. Since this feedback is equivalent to y(t) = (-I/k)u(t), it is reasonable to expect that under certain circumstances the closed loop system with high gain will behave as k -+ 00 like the system itt) = ~z(t), /{ z(t) = O. This system is known as the zero dynamics of the original system, and - -is formally equivalent to a system of the form itt) = A z(t), where D(A) = {z E D(~) I /{z = O}. In Byrnes and Gilliam [I] it is shown that for a large class of parabolic systems the eigenvalues of the high gain system approach the eigenvalues of the zero dynamics. In this paper we discuss the high gain behavior of some feedback systems which are not covered in [I].

If the roles of y(t) and u(t) are reversed, the resulting system is itt) = ~z(t), /{z(t) = u(t), y(t) = rz(t). - - -If this system is regular it is equivalent to a system of the form itt) = A z(t) + Bu, y = Cz(t) + Du(t). We refer to this system as the reverse system. In the paper we discuss examples which are regular in their foward realization only, regular in their reverse realization only, and regular in both realizations.

Let A. be the generator of 5.(t), 5 00 (t) be the semigroup generated by A, and H be the transfer function for the reverse system. The following result follows immediately if the reverse system is regular:

Theorem 1 For every t > 0, 5.(t) -+ 5 00 (t) uniformly as k -+ 00.

The following simple result discusses high gain continuity of individual branches of the spectrum:

Theorem 2 Suppose A is a zero of order n of H, and for some p > 0 H is analytic on {s lis - AI ~ p} and nonzero on {s I 0 < Is - AI ~ pl. Then there exists m such that for alii'll < m there are n zeros, counting

mulitplicity, of H(s) - '1 in {s lis - A < pl·

If we wish to get continuity which is uniform for all of the branches, we need to verify that H satisfies certain technical, but checkable, conditions. In this case the following theorem is also true.

Theorem 3 Suppose {iJj }jEf are the eigenvalues of A. Then for every r > 0, there exists M > 0 such that A. has exactly one eigenvalue aj such that laj - Ilj I < r for k > M.

If the faward system is not regular and the reverse realization is regular) we show that it is often easier to analyze semigroup generation and stability of the closed loop system using y(t) = (-I/k)u(t) in the reverse system rather than y(t) = -kurt) in the original system. Furthermore, we will investigate system properties that are inherited under the operation of 'reversing the system'.

References

[I] C.1. Byrnes, D.S. Gilliam, Boundary feedback design for nonlinear distributed parameter systems, Proc. of IEEE Can/. on Dec. and Control Brighton, England 1991.

[2] D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: a functional analytic approach, Trans. Amer. Math. Soc., 300, pp. 383-431, 1987.

[3] G. Weiss, The Representation of Regular Linear Systems on Hilbert Spaces, Proceedings of the Conference on Distributed Parameter Systems, Vorau, Austria, July 1988, Birkhiiuser, Basel, 1989.

Duality Aspects between Rational H2 and Hankel Norm Approximations

Phillip A. Regalia Departement Electronique et Communications, Institut National des Telecommunications,

9, rue Charles Fourier, 91011 Evry cedex France Tel. : +33-1 60 764631 Fax: +33-1 60764431 e-mail: regalia@galaxie.int-evry.fr

275

Rational 112 approximation intervenes in many aspects of control and signal processing, and despite the conceptual simplicity of the approximation criterion, the problem presents many theoretically unresolved aspects upon approximating with rational functions. Recently, attention has refocused on characterizing the properties of the error function which result from first-order necessary conditions [IJ, [2J. This talk recasts the problem into a Hankel operator approach. Consider the Hankel operator associated to the function to be approximated, constrained to act upon an adjustable Mth order causal all-pass (or inner) function. If the result lies in the shift-invariant subspace generated by the all-pass function, then a critical point of the 112 approximation problem has been found, and vice-versa. The poles of the approximating function match those of the all-pass function in question, while the zeros can be determined from linear interpolation constraints determined from the all-pass function. From this, it is shown that to each critical point of the 112 approximation problem corresponds the optimal Mth order Hankel norm approximation to a nearby function, and vice-versa. As a result, the 12 error of the approximation is precisely the M + 1 st singular value of the nearby Hankel operator.

References

[IJ P. A. Regalia and M. Mboup, "On the approximation character of some recursive identification schemes," 3rd IMA Conf. Mathematics in Signal Processing, Warwick, UK, December 1992.

[2J De Moor B., Van Overschee P., Schelfhout G. H2-model reduction Jor SISO systems. Internal Re­port ESAT-SISTA 1992-30, Department of Electrical Engineering, Katholieke Universiteit Leu­ven, Belgium, July 1992. Accepted for the 12th IFAC World Congress, Sydney Australia 1993.

Acausal State Representation of a 2D System

Paula Rocha and Sandro Zampieri

September 23, 1992

Abstract

In this paper we study the problem of the construction of acausal state representations of an arbitrary 2D system. In the behavioural approach a system E is defined as the triplet (T, W, B), where T is the time set, W is the attributes space and B is a subset of the signal set WT that is called behaviour and specifies the system fixing the admissible trajectories. In this context we define 2D system a system with Z2 as the time set.

Following Willem's philosophy, the Markovian properties are characterized by means of con­catenability of trajectories. In [IJ the concept of adirectional Markov property is introduced. A system E = (Z2, W, B) is said to be adirectional Markovian if, taken any finite or infinite interval in Z2 and any partition T_, To, T+ of it, such that To separates T_ and T+, we have that for all WI, W2 E B such that wqTo = w21T, there exists wEB such that

WIT, = wqT, = w2lTo' WIT- = wIIT_ and WIT+ = w2IT+'

However other definitions of acausal Markov property have been proposed in the literature on random fields [2,3J. The so called global and local Markov property ,are two important examples. Their deterministic traslations are both weaker versions w.r. to the adirectional Markov property. In the global Markov property the same conditions as the adirectional Markov property are valid but only when the interval I is Z2 Finally the local Markov property assumes moreover that either T_ or T+ must be a finite set.

In this paper we investigate the connections between these various kinds of Markov properties under the hypothesis that the system is linear, shift-invariant and complete. In [IJ it is shown that any 2D system always admits an adirectional state representation and that any adirectional Markov behaviour can be canonically described by a first order difference equation. Any local I\larkovian system can be described by a nearest neighbour model. This was shown by Woods in the stocastic case under certain hypothesis [3J. We show that the same is valid in the deterministic case by choosing a suitable extension of the nearest neighbour model. We also investigate on the possibility of characterizing the global Markov property by another special form of difference equations.

As a second step we study the problem of the state representation of a generic linear shift­invariant system.

In [IJ it is shown how to construct the adirectional Markovian state representation for a generic system. We investigate the possibility of doing the same by the other two Markov characterizations. Finally we show the connections between the three Markov properties when the behaviour is externally controllable, as specified in [4J.

1. Rocha, P. and J .C. Willems (1989) : "State for 2-D Systems", Linear Algebra and its Appli­catIOns, Vols. 122/123/124, pp. 1003-1038.

2. Goldstein, S. (1980) : "Remarks on the Global Markov Property", Comm. in Mathematical Physics, Vol. 74, pp. 223-234.

3. Woods, J.W. (1972) : "Two-Dimensional Discrete Markovian Fields", IEEE Trans. Inr. Th., Vol. IT-18, no. 2, pp. 232-240.

4. Rocha, P. and J.C. Willems (1991) : "Controllability of 2-D Systems". IEEE Trans. on Aut. Cont., Vol. AC-36, pp. 413-423.

277

STABILITY THEORY FOR A CLASS OF 2D LINEAR SYSTEMS WITH INTERPASS

SMOOTHING EFFECTS

ABSTRACT

E. Rogerst and D.H. Owens*

t University of Southampton, U.K. * University of Exeter, U.K.

The essential unique characteristic of a repetitive, or multipass, process is a series of

sweeps, or passes, through a set of dynamics defined over a finite duration - the pass

length. On each pass an output - the pass profile - is produced which acts as a forcing

function on, and hence contributes to, the dynamics of the next pass profile. This is

the source of the unique general control problem for these processes in that the

output sequence can contain oscillations which increase in amplitude from pass to

pass.

In the simplest case, therefore, the output at any point on a given pass is a function

of the inputs/disturbances at that point and the output at the same point on the

previous pass - so-called 'point' processes. Recent work has developed a rigorous

stability theory for 'point' processes using an abstract model in a Banach space

setting.

Examples also exist where the 'point' assumption is no longer valid due to dynamic

action between passes, termed interpass smoothing, which severely distorts the

previous pass profile. This paper will extend the currently available stability theory

to processes with the state-space description.

X k+ /t) = AX k+ /tl + BU k+ ,(t) + B 0 J: K(t, L) Y kIt) de

Here a < + 00 is the pass length, Xk+,(t) ER" is the current pass state vector, Yk+,(t)

ERn. is the current pass profile vector, UH,(t) ERe is the current pass input vector and

the interpass interaction term

represents a smoothing out of the previous pass profile in a manner governed by

properties of the kernel KU, 1).

278

Computer-Algebra Program for Analysis and Design of Nonlinear Control Systems *

R. Rothfufi, J. Schaffner, M. Zeitz

Institut fUr Systemdynamik und Regelungstechnik - Universitiit Stuttgart, PostJach 801140, D-70511 Stuttgart, Germany

Tel: (711) 685 6313, Fax: (711) 685 6371, email: {ralJ.hanna.zeitz}@isr.uni-stuttgart.dbp.de

March 31, 1993

At the University of Stuttgart, the symbolic computing language MACSYMA has been used for a five year developement of a program for the computer-aided analysis and design of nonlinear control systems x = J(x,u), y = h(x,u) [1, 2J. This tool makes available a number of custom-made functions required for the stability, controllability and observ­ability analysis or for the recognition of normal forms of nonlinear system. Moreover, several controller and observer design methods which are known from recent literature are implemented. For numerical calculations and simulation, there exist interfaces to the external software packages MATLAB and ACSL, respectively. The program is menu driven and enables user interactions to various degrees. The dialogue between the user and the program is supported by a large explanation component. Furthermore, the pro­gram contains a rule base for the selection of 11 observer design methods to advise the user by the implemented system-theoretic and heuristic knowledge [3J. The abilities of the program are realized on one hand side with plain MACSYMA func­tions, on the other hand side with specific mathematical basic functions that enable facilities exceeding the MACSYMA standard, e.g. the calculation of a jacobian matrix or of Lie derivatives. These mathematical basic functions, which are written in plain MACSYMA, are an important tool in the program concerning a modular implementation and the extension of complex analysis and design procedures; however, the functions are also available for the user.

References

[1 J J. Birk, M. Zeitz: Program for symbolic and rule-based analysis and design of non­linear systems. In: G. Jacob et al. (ed.), Proceedings First European Conference Al­gebraic Computing in Control, Paris 1991. Lecture Notes in Control and Information Sciences 165, Springer-Verlag 1991, 115-128.

[2J J. Birk: Rechnergestiitzte Analyse und Losung nichtlinearer Beobachtungsaufgaben. VDI-Fortschritt-Berichte 8/294, VDI-Verlag, Diisseldorf 1992 (in German).

[3J J. Birk, J. Schaffner, M. Zeitz: Rule-based selection of nonlinear observer design methods. In: M. Fliess (ed.), Proceedings IE'AC-Symposium Nonlinear Control Sys­tems Design, Bordeaux 1992, 251-257 .

• Abstract for Invited Session 'Computation and Nonlinear Control' of MTNS Regensburg 1993

Abstract:

ON THE USE OF C k SPLINE FUNCTIONS

MARC ROUFF

# Laboratoire de Genie Electrique de Paris Ecole Superieure d'Electricite Plateau du Moulon F-91192 Gil sur Yvette Cedex, FRANCE 33-1-69-41-80-40 (Office) 33-1-69-41-83-18 (FAX)

Ck spline functions i.e. functions wich give a k times derivable and continu­ous approximated solution are presented. After a brief review of their computations and their algebraics proprieties, we give several exemple of use, in identification and discretization problems ..

279

280

A Canonical Form under Quasi-Static State Feedback

J. Rudolph Institut fUr Systemdynamik und Regelungstechnik, Universitiit Stuttgart

Postfnch 801 LID, D - 70511 Stuttgart, Germany e-mail: rudolph@isr.uni-stuttgart.dbp.de

The question of equivalence under feedback and the existence of corre­sponding canonical forms is one of the corner-stones of system analysis. In the present contribution we are interested in the equivalence of nonlinear dynamics under quasi-static state feedback. This kind of feedback is characterized by the fact that the new input can be calculated from the original one, its time derivatives and the state, and conversely (see Delaleau and Fliess)l.

In the differential algebraic approach state representations may depend on time derivatives of the input. In this context we define the well formed dynam­ics which are the generalization of the usal dynamics of the type :i; = f(x, u) with full rank jacobian matrix ~ to which they are equivalent by quasi-static state feedback.

The flat systems recently introduced by Fliess et aU are equivalent to a linear controllable one via endogeneous feedback. They admit so-called flat outputs.

Our principal result concerns dynamics which are well formed and flat: The Brunovsky form, well known in the linear theory, is a canonical form for the state representations of these flat well formed dynamics. This canonical form can be constructed from any flat output. Moreover, we give a simple condition for the well-formedness of flat dynamics which shows that most flat systems are well formed. Our result is new also in the linear case, because the flat outputs are just the bases of the system module associated to a linear controllable system.

A chemical reactor example shows the usefulness of the result for applica­tions.

1 E. Delaleau and M. Fliess, An Algebraic Interpretation of the Structure Algorithm with an Application to Feedback Decoupling, Proc. IFAC-Symposium NOL COS '92, Bordeaux, M. Fliess ed., 1992, pp. 489-494.

2 M. Fliess, J. Levine, P. Martin and P. Rouchon, On Differentially Flat Nonlinear Systems, ibidem, 1992, pp. 408-412.

Some Lyapunov methods for robust contol

David L. RUSSELL

Department of MathematicJ Virginia Tech

BiackJburg, VA 24061 U.S.A.

russell@mthunx.math.vt.edu tel. 703-2316171

Abstract

The general acceptance of HOO methods as the paradigm of choice for .obust control design has obscured some earlier formulations within the Lya­punov framework. We will review some of these approaches and relate them to current research.

The talk will stress practical, computational uses of Lyapunov methods to enhance system robustness.

281

282

High Gain Regulator Problem Nonlinear Control Systems

. In

Noboru Sakamoto, Yoshinori Ando and Masayuki Suzuki Department of Aeronautical Engineering, Nagoya University

Furo-Cho, Chikusa-ku, NAGOYA 464-01, JAPAN Tel.+81 52 781 5111, Fax.+81 52 782 9318

Abstract

In this research, a high gain feedback control is concerned in general affine multivari­able nonlinear control systems. While a high gain feedback has many advantages, the systems with the high gain feedback sometimes are unstable or have sharp peaks even if the systems are stable. Therefor it is important to clarify the class.. of the systems to which a high gain feedback can be applied. In linear systems, the solutions for this problem were obtained by various approaches. The main purpose of this research is to classify nonlinear systems in which the high gain feedback is allowed and to design the high gain regulator for such a class of nonlinear control systems, in which the cost function is the U norm of the output.

The methods to be used in this research are based on the nonlinear control theory according to the differential geometric approach and the multiple time scale singular perturbation method. First, we derive the normal form of general nonlinear control systems. Secondly we apply the noninteracting control method to the normal form systems. At that time the output is a state of linear subsystems. It is known that a fixed internal dynamics exists in the noninteracted systems, which has plural in­put channels and can not be removed even by the dynamic compensator. In order to guarantee the stability of the nonlinear subsystems including fixed modes, it is assumed that the original system has the minimal phase property. At last we con­struct the high gain feedback regulator by using the singular perturbation method. In this process the multiple time scale design technique is applied for each subsystem respectively. But, because the terms with the high gain parameters which has outputs approached to zero rapidly may make the nonlinear part unstable, the parameters of the gain have to be chosen properly.

The high gain regulator for the linear system needs the minimal phase property and the right invertibility condition. In case of nonlinear systems the minimal phase property is replaced the stability of zero dynamics and the invertibility condition is translated into the condition of the controllability distributions. Thus the result obtained here is an enhancement of the linear high gain regulator problem to nonlinear control systems.

;3PBCTRAL PROBLE!.IS FOR THE SYSTEi.:S

OF EQUATIOnS ON THE AXIS

L.A.Sakhnovich

1. We investigate a spectral problem on the axis

(-00,00

equations

for tIle canonical systems of differential

cL?J =- i l.J:;e (a:..) ?J( a: l.) cl.x. ' )

?Jro z.):. E, I c:hl! )

(1)

where

[ 0 Errz.]

J= . Em. 0 ' :Jt(x).~ 0

)

Ve reduce the problem, which is considered to the investi­

gated proble!:l on the half-axis (0, <>0 ) ( 1]:

where 0 {; :x: L 0<)

283

if (X z): T J T' ,,[W(X Z) 0 J o ) 0 ?J"(-,x l) )

J •

(X): T T. If [:Ie/X) 0 J-o .1{f-X}

THZOrtI~i,; 1. The f8!:.ilies of the 8,)(?ctral matrix: - functions

<[(u) of system (1) and of corres;:>oncinG sy8ten: (2)

coincide.

284

Geometric Concepts in Acausal Stochastic Realization Theory

Jan-Ake Sand Division of Optimization and Systems Theory

Royal Institute of Technology 10044 STOCKHOLM, Sweden

Fax: +46-8-22 53 20, e-mail: sand@@math.kth.se, Phone: +46-8-790 72 20.

Abstract: In this paper we discuss some problems in acausal stochastic realization theory. Geometric concepts analogous to those of Lindquist and Picci [3J are intro­duced. As an example of an acausal realization, consider a random field {y(t); t E 22} defined as y(t) = Cx(t) where {x(t) : t E 22} is a Gaussian Markov random field having a state-space representation [1,6J. It is believed that a realization theory for random fields could be useful to the area of image processing.

We generalize the results in [4,5J concerning reciprocal realizations on a circle to the setting of Markovian realizations on 22. It turns out that in this setting the splitting subspaces are indexed by closed curves and that for a fixed closed curve '"Y the splitting subspace X b) is minimal if and only if the realization is interiorly and exteriorly observable on the interior and exterior of '"Y. Moreover, if a splitting subspace is minimal for some '"Y then the realization is minimal in the sense that the vector x(t) has the least possible dimension.

Finally, we connect the geometric theory to the spectral theory of Markovian realizations on 22 and study some special cases.

References

1. B.C. Levy, Noncausal Estimation for Discrete-time Gauss-Markov Random Fields, Proc. MTNS-89, Birkhiiuser Boston Inc., 1989.

2. B.C. Levy, R. Frezza and A.J. Krener, Modeling and Estimation of Discrete­time Gaussian Reciprocal proceses, IEEE Trans. on Automatic Control 35 No.9 (1990), 1013-1023.

3. A. Lindquist and G. Picci, A Geometric Approach to Modeling and Estimation of Linear Stochastic Systems, Journal of Mathematical Systems, Estimation and Control 1 (1991), no. 3,241-333.

4. J-A Sand, A Geometric Approach to the Reciprocal Realization Problem, Pro­ceedings of the 31st IEEE-CDC 1992.

5. J-A Sand, A Geometric Approach to the Reciprocal Realization Problem on the Circle (revised August 1992},Technical report, Division of Optimization and Systems Theory, KTH, STOCKHOLM, SWEDEN, 1992.

6. J.W. Woods, Two-Dimensional Discrete Markovian Random Fields, IEEE Trans. on Information Th!'ory, vol. IT-18, No.2, March 1972,232-240.

APPROXIMATELY-FINITE MEMORY AND THE

CIRCLE CRITERION

Irwin W. Sandberg Department of Electrical and Computer Engineering

The University of Texas at Austin, Austin, TX 78712, USA

e-mail: sandberg@emx.utexas.edu

Abstract

285

In recent work a complete characterization is given of those input-output maps G that

can be uniformly approximated by the maps of certain simple structures. The criterion

is that G must satisfy certain continuity and approximately-finite-memory conditions.

It is proved here that the conditions are satisfied by the input-output maps of feedback

systems of a familiar type containing a (possibly distributed) linear part and a sector

nonlinearity for which the circle condition for stability is met. In particular, this shows

that such feedback systems, with inputs drawn from a certain large set of bounded

functions, can be input-output approximated arbitrarily well by a structure that takes

the form of a feedforward dynamical neural network.

286

Displacement Structure and Rational Interpolation Theory

Ali H.Sayed Information Systems Laboratory, Stanford University

Stanford CA 94305 sayed@rascals.stanford.edu

The displacement structure concept provides a powerful and unifying tool for exploiting the inherent structure in diverse problems in signal processing and mathematics. In this talk, we describe an efficient recursive (square-root or array) algorithm for factoring the structured matrices that arise in these applications, and emphasize how naturally transmission-line structures, tri­angular arrays, and embedding relations arise in this context. We stress the fact that the transmission-line cascades have useful physical characteristics such as causality, energy conservation, and blocking properties. The first two have been exploited previously. The blocking property, viz., that signals propagating through the cascade at certain frequencies get annihilated, is here exploited to derive efficient recursive solutions to general rational in­terpolation problems, which arise in many applications in circuit and system theory. The main feature of the derivation is that it requires very little math­ematical background, being based largely on combining a simple Gaussian elimination step with displacement structure. Moreover, the derivation is exclusively carried out in the matrix domain, working with matrix quanti­ties only, and not in the function domain. This allows us to nicely extend the matrix-based derivation very smoothly to the time-variant setting, and to solve general time-variant interpolation problems. This talk is based on work done with T./(aiiath, H.Lev-Ari, and T. Constantinescu.

Inner-outer factorization of nonlinear state space systems

A.J. van cler Schaft * J.A. Ball t

Abstract

287

In a number of problems in analytic function theory the technique of inner-outer factorization has become a standard tool. In linear control theory the inner-outer factorization (or, more generally, J-inner-outer factorization) of rational matrices has been crucially applied in the theory of Hoo optimal control. In a series of papers, see e.g. [1], Ball and Helton have investigated inner-outer factorization of nonlinear (input-output) operators, partly motivated also by the problem of Hoo optimal control for nonlinear control systems. In the present paper we will study inner-outer factorization of nonlinear state space systems, largely inspired by the recently developed approach to Hoo optimal control of such systems and, more generally, the theory of dissipative and lossless nonlinear systems. Concretely, given a nonlinear system

x f(x)+g(x)u, uERm

y h( x) + Du, y E RP

on a state space manifold M, we want to represent its external (input-output) behavior by the external behavior of the series interconnection of two nonlinear state-space systems of the same type, the first being a stable and minimum-phase nonlinear system (i.e. the system and its inverse are internally asymptotically stable), while the second system is a lossless nonlinear system. A crucial tool we will employ is the notion of the Hamiltonian extension of a nonlinear state space system as given in [2]. Indeed, for stable systems we will be able to give a nonlinear analogue of the canonical factorization [3] of the product of a transfer matrix G( s) and its adjoint GT ( -s). For unstable systems there are some intrinsic problems, which, in general, do not seem to admit easy solutions. Finally for stable nonlinear systems, we will investigate the minimal representation of the stable and minimum-phase (outer) and lossless (inner) factor, in the spirit of the treatment of the corresponding linear problem in [4J.

References

[IJ .LA. Ball, J.W. Helton, Inner-outer factorization of nonlinear operators, J. Funct. Anal., 104 (1992),363-413.

[2J P.E. Crouch, A.J. van der Schaft, Variational and Hamiltonian Control Systems, LNCIS 101, Springer, Berlin, (1987).

[3J H. Bart, I. Gohberg, M.A. Kaashoek, Minimal Factorization of Matrix and Operator Functions, Birkhiiuser, Basel (1979).

[4J M. Green, On inner-outer fac/orization, Systems & Control Letters, Vol. 11 (1988),93-98.

'Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands, hone 31-53-893449, Fax 31-53-340733, e-mail twarjan@math.utwente.nl

tDepartment of Mathematics, Virginia Tech., Blacksburg, VA 24061, U.S.A.

288

A Paradigm Shift in the N eurodynamics of Spatio-Temporal Neural Networks:

The Coherent Wavelet Revolution in Linking Neural Network Architectures,

Quantum Holography, and Chaos

Walter Schempp Lehrstuhl fiir Mathematik I

University of Siegen D-5900 Siegen, Germany

schempp@hrz.uni-siegen.dbp.de

Dedicated to Walter J. Freeman

who applied nonlinear dynamical systems theory in neuroscience

Abstract

The purpose of this paper is to demonstrate that phase coherence forms a self-organizing principle of neurodynamics which leads to quantum neurodynamics. This is a converse of the current tendency to recast quantum physics in terms of information theory with the goal to in­clude semantics and finally to include phenomenal consciousness into quantum physics. In particular, the neurodynamical functional con­nectivity model of stimulus-response association by coherent neural wavelets leads to quantum holography which includes deterministic chaos as well as a basic quantum dynamical limitation of knowledge that goes far beyond the standard Heisenberg uncertainty principle of quantum mechanics. The quantum holographic model of spatio­temporal linking neural networks admits an implementation for real time application that operates from a transputer multiprocessing plat­form.

The Nonstrict Algebraic Riccati Inequality

Abstract

Carsten W. Scherer'

Mathematisches Institut

Am Rubland

D-8700 Wiirzburg

Germany

scherer@vax.rz.uni-wuerzburg.dbp.de

289

If (A, S) is stabilizable, it is well-known how to verify the solvability or the existence of

positive definite solutions of the the nonstrict algebraic Riccati inequality (ARI)

A'X+XA-XSS'X+Q::::O.

This paper serves to demonstrate in how far it is possible to generalize these results if

(A, S) may have uncontrollable modes on the imaginary axis. We prove a new reduction

principle which may be formulated as follows: The ARI has a solution iff a certain reduced

order Riccati equation (for a stabilizable system), a linear equation and a reduced order

nonstrict Lyapunov inequality (corresponding to the uncontrollable modes on the imagi­

nary axis) have solutions. Basically, the quadratic Riccati inequality is hence reduced to

a linear Lyapunov inequality. We discuss how this principle may be applied to actually

check the solvability of the ARI. As the main motivation of our work we briefly explain

the consequences of these results for the general state-feedback R",-optimization problem.

Keywords

Algebraic Riccati inequality, Lyapunov inequality, positive definite solutions, state­

feedback Hoo-optimal control.

'supported by Deutsche Forschungsgemeinschaft Sche 402/1-1.

290

Balancing of the normalized right coprime factorization of a nonlinear system

J.M.A. Scherpen* and A.J. van der Schaft Department of Applied Mathematics, University of Twente

P.O. Box 217,7500 AE Enschede, The Netherlands e-mail: scherpen@math.utwenle.nl

For model reduction of linear systems we often use the balanced realization of the system. Some of the ideas to balance linear systems turned out to be useful for a set up of balancing nonlinear systems. In [2] we have set up such a method to balance a stable nonlinear system. Like for linear systems we considered the minimum amount of control energy required to reach a state and the amount of output energy generated by this state. If the nonlinear stable system is in balanced form, we can decide when a state component is important or not by means of this control and output energy and therefore use balancing of the nonlinear system as a tool for model reduction, see [2].

This talk concerns balancing of unstable nonlinear systems and, like above, we make use of the ideas from the theory for linear systems. Balancing for unstable linear systems can be found in e.g. [1], where the normalized right coprime factorization of the unstable linear system is used to obtain the (stable) state space realization of the corresponding graph operator. Then the order of the original unstable system is reduced based on the balanced realization of this graph operator.

In a similar way, balancing of an unstable nonlinear system can be done by considering the normalized right coprime factorization of the system. In order to do so, we first introduce the concept of an inner nonlinear system by considering the Hamiltonian extension of a system. The right coprime factorization of an unstable nonlinear system is defined by the state space representation of two stable nonlinear systems where a serial connection of the first system and the inverse of the second system gives exactly the original unstable nonlinear system. Moreover, the factorization is called normalized if the combination of both systems is an inner system. Furthermore, we give some properties of the normalized right coprime factorization, we deal with the control energy and output energy of the combined system and we deal with balancing of the linearization.

References

[1] D.G. Meyer, A fractional approach to model reduction, Proceedings Amerzcan Control Conference, Allanta (1988) 1041-1047.

[2] J .M.A. Scherpen, Balancing for nonlinear systems, (Memorandum No. 1071, Faculty of Applied Math­ematics, University of Twente, 1992). to appear in Syst. [j Contr. Leiters (1993).

[3] J .M.A. Scherpen and A.J. van der Schaft, Balancing of the normalized right coprime factorization of a nonlinear system, (Memorandum No.1096, Faculty of Applied Mathematics, University of Twente, 1992).

·Supportcd by the Dutch Systems and Control Theory Network

Dr. W. Schmidt Technische Universitat MOnchen Mathematisches Institut Barer Strafie 23 D-8000 MOnchen 2

Limit cycle computation of oscillating electric circuits

A well-established method for mathematical simulation of electric circuits

is the modified nodal voltage analysis, which is based on Kirchhoff's laws. In

general, this method leads to a system of nonlinear, implicit differential equa­

tions. In the special case of oscillating electric circuits the describing system is

often of a simpler form, namely a system of stiff differential equations.

Computation of periodic solutions of electric circuits with constant input

requires the determination of the unknown period and a set of adequate initial

values. Moreover, the limit cycle depends also on some specific electric para­

meters. That means limit cycle computation is a special case of bifurcation.

For numerical analysis it is useful to formulate bifurcation problems as two­

point boundary value problems. Their solution can be done with the well-known

mUltiple shooting technique. By coupling the stiff integration method and the

multiple shooting algorithm, computation time cun be reduced drastically.

291

At last simulation results of some realistic circuits are presented. For example

the limit cycles and bifurcation diagrams of a ring oscillator and a colpitts

oscillator is given.

2

Efficient approximation with neural networks, a comparison of gate functions

Georg Schnitger Fachbereich Mathematik-lnformatik,

Universitaet Paderborn

and

Department of Computer Science Penn State University

We compare different gate functions in terms of the approximation power of their circuits. Evaluation criteria are circuit size s, circuit depth d and the approximation error e(s,d). We consider two different error models, namely "tight" approximations (i.e. e(s, d) = 2-') and "more relaxed" approxima­tions (i.e. e(s, d) = s-d). Our goal is to determine those gate functions that are equivalent to the standard sigmoid a(x) = l+ex~(-x) under these two error models.

We first consider the approximation of real-valued functions. For error e(s, d) = 2-', the class of equivalent gate functions contains, among others, (non-polynomial) rational functions, (non-polynomial) roots and most radial basis functions. Newman's approximation of I x I by rational functions is obtained as a corollary of this equivalence result. Provably not equivalent are polynomials, the sine-function and linear splines.

For error e(s, d) = s-d, the class of equivalent activation functions grows considerably, containing for instance linear splines, polynomials and the sine­function. This result only holds if the number of input units is counted when determining circuit size. The standard sigmoid is distinguished in that relatively small weights and thresholds suffice.

Finally, we consider the computation of boolean functions. Now the bi­nary threshold gains considerable power. Nevertheless, we show that the standard sigmoid is capable of computing a certain family of n-bit functions in constant size, whereas circuits composed of binary thresholds require size at least proportional to log n.

ON THE RELATIONSHIP BETWEEN EXTENDED ZEROS AND WEDDERBURN-FORNEY SPACES

Cheryl B. Schrader Division of Engineering

The University of Texas at San Antonio San Antonio, Texas 78249, USA

(210) 691~'ij96

Michael K. Sain Electrical Engineering

University of Notre Dame :"Jotre Dame, Indiana 46556, USA

(219) 631~6538

ABSTRACT

Ordinary zeros are related to places where rank drops occur, either on the input~ output level in the transfer function matrix, or on the internal level in terms of the system matrix or other system description, However, extended zeros, which have been defined in a module~theoretic context, include also drops in rank which occur everywhere. These extended zeros have been shown to be intrinsically involved with the question of fixed zeros in solutions to transfer function equations, with the block~ ing of inputs to a system by certain initial conditions, and with the key subspaces in geometric control theory. Recently, they have also been employed to give broad gen~ eralizations of the zero effects upon feedback loops brought about by pole dynamics in the feedback path-a classic adage in the single~input, single~output case-and to explain in general terms just how feedback affects zeros in a loop. Moreover, in the case of general dynamical feedback compensations, they have also been employed to examine precisely how feedback affects system zeros and to indicate the exact cir~ cumstances under which certain types of zeros can be affected by closing the loop. These ideas extend classical notions on the subject. On an input~output level, it is known that Wedderburn~Forney spaces measure, in a certain way, the zeros of the transfer function matrix which are not captured in the ordinary or lumped zeros, both finite and at infinity. The Wedderburn~Forney construction has also been employed to demonstrate the role of kernels and cokernels in completing the study of poles and zeros of a rational transfer function matrix. This investigation examines the relationship between extended zeros and Wedderburn~Forney spaces, and determines the relative roles of the two ideas-modules and spaces-in representing key issues associated to questions in linear dynamical systems.

293

'94

RffiMANNIAN GEOMETRY AND NONLINEAR

OPTIMAL FEEDBACK CONTROL

Lorenz Schumann, H.P. Geltring, ETH Z1lrich

A global setting of an optimal control problem requires severallocalIy trivial

iberhundles. ThetotaJ space of the state, time and control fiberbundle is itself the base

ipace of a cotangent vector subbundle on which lives canonically a presymplectic

rtructure. The flI'st order necessary cnndition of all optimal control problem, in either

LMayer, Lagrange or Boiza [1] shape, appear IlB a Hamiltonian V'ectorfie1d restricted

o a closed submanifold ofthetotaI space of this ootangential vector Bubbundle.A BOlu­

ion of the optimal control problem is maximally a cross-eection of the state, time and

!:Introl tibel'bundle, which is in abstract tel'Illl! called an optimal control strategy.

The necelisary condition of the optimal control problem under consideration

an be naturally embedded in a Hamiltonian vectorfield on a cot&ngentialbundle [21.

'he imagiruu-y part of the Hermitian structure on this cotangentialbundle restricted

I} the submanifold of the optimal control problem generates canonically the men­

ioned presymplectic structure. The real part of the Hermitian structure, correspond-

19iy, generates a tensorfield on this pr\:!l:Iymplectic submWlifold, Which is under the

tahility assumptions on the nonlinear optimal feedback controller Wld its action on

Ie nonlinear plant equivalent to a Rif'.mllnnian metric.

Firuilly, the connection between this Riemannian metric and the integrability

)oditions as well as the transversality conditions r2] is going to be exhibited. This en­

bles one to give the clear tmderstanding of the characteristics in the context of geodf!S­

s associated to the Riemannian metric.

[1] Lamberto Cesari, Optimization - Theory and Applicationa Jroblems with Ordinary DiITerential EquationH), Springer 17, 1988, New York, SA

[2J Lorenz Schuma.nn, Symplectic Geometry and Nonlinear Opti· lal Feedback Control, PhD - Thesis, ETH, 1992, ZUrich, Switzerland

Some new methods for the algebraic anlaysis of nonlinear systems

H. Schwarz, F. Svaricek and T. Wey UniversiUit Duisburg

FB 7, FG 8 Postfach 101503

W-41 00 Duisburg 1 Germany

New methods for an efficient computer aided analysis of nonlinear control

systems which are based on a digraph representation of the system are

presented. The application and performance of these new methods is discussed and demonstrated on a technical plant.

295

296

Hain-Liist problem: the eigenvalue accumulation

at the end points of the essential spectrum

A.A. Shkalikov

Moscow State University

Department of Mathematics

119899 Moscow, Russia

K. Hain and R. Lust showed in the 50-th that the investigation of spectral properties of

the force operator in a linear magnetohydrodynamic model can be reduced to the study of

the spectral problem for the second order differential equation

(1) -(r(x, >.)y')' + p(x, >.)y = 0, x E [0,1].

For Hain-Lust equation functions r(x, >.) and p(x, >.) are rational on >.. Spectral problems

for (1) with Dirichlet boundary conditions has been investigated in numerious papers,

although many results has been proved in a nonrigorous fashion.

We consider spectral problem (1) with boundary conditions

(2) {Cl:o(>')Y(O, >.) + /3o(>')Y'(O, >.) = 0

Cl:J(>.)y(l, >.) + /3J(>.)y'(l, >.) = 0

and assume that the spectral parameter of (1), (2) runs through an interval A = (/1, v) c IR.

Let all the coefficients of (1), (2) are real valued continuous functions, moreover,

rex, >.) > 0 for all {x, >'} E [0,1] x A,

for j = 1,2, either /3j(>') =I 0 or /3j(>') == 0 and Cl:J(>,) =I 0 for all >. E A.

Theorem. Suppose there exists a point a E [0,1] and f > 0 such that the functions r(x, >.) and p( x, >.) are continuous on (a - f, a) X A (or on (a, a + f) X A) and

lim Ix - al-"Yr(x, v) = ra < 00 x_a

lim Ix - aI 2-"Yp(x, v) = Pa < 00 x_a

for some, E IR. If the condition

4pa < -(1 -,)2ra

holds, then v is the accumulation point of the spectrum of problem (1), (2). If all the

coefficients in (1), (2) are analytic on A, then v is the accumulation point of isolated eigenvalues.

Some applications of this result will be discussed.

(Joint work with R. Mennicken and H. Schmidt).

~ • ~ • • • • M • W ~ ~ ACADEMIA SINICA

INSTlTUTE OF SYSTEMS SCIENCE BEIJING H.ou8J, TIlE PEOPLE'S REPUBLIC OF CHINA

CABLE ADDRESS: 8El JING 47Gl

IE-mAil BI.1AI)lS @ leA ScI :rINEr CANE T eN

Transition to Market Economy in China

Deng Shuhui Jin Xin Chen Bin Wang Shilin

(Institute of Systems Science, Academia. Sinica)

Abstract: The present operational mechanism of China's na­

tional economy is a combination of planned economy with market

economy, that is, complex economy mechanism. In this paper we

study the existence of the equilibrium points in the complex econ­

omy and the stability of the transaction to market economy. We also

developed a decision support system for the economy analysis.

Key words: Complex economic mechanism, Gene~al equilibrimn. De­

cision support system.

In the past decade,the quantitative study of China's macro-econorr.y has

made big progress:1- 4i , and some monographs on its applicationis ,6J have come

out. In addition, the Blue Book analyzing and predicting China's economic

development has been published two years in succession.

In spite of all the progress in the study, there are still problems. This pa-

per will address these problems some new results, and the future development

of the study.

*J Tois work partially support by National Natural Science Fundation in China

297

~98

The Drazin inverse and its nonlinear extension for constrained mechanical systems *

B. Simeon

Department of Mathematics Munich University of Technology Arcisstr. 21, D-8000 Miinchen 2

The numerical integration of constrained mechanical systems is based on the differential­algebraic formulation of the equations of motion. First, a thorough analysis of the equa­tions of motion in linearized form is presented in order to get new insight in the stability propertit's of the different formulations as index 1, 2, 3, or Baumgarte system. It turns out that the structural information is sufficient to employ the general solution theory of systems of singular linear differential equations without explicit knowledge of Jordan canonical forms or other decompositions. The main tool is the Drazin inverse, a gener­alized matrix inverse, which preserves the eigenvalues. The Drazin inverse enables the formulation of an equivalent system of ordinary differential equations, the Drazin ODE, which offers various applications in the numerical analysis.

The second part introduces the fundamental idea for an extension to nonlinear systems. The Lagrange multiplier technique is applied to both the original and the differenti­ated constraints of the system, which leads to a redundant second order formulation of the equations of motion. This second order system can be transformed into an index 1 system where certain projections guarantee that the solution satisfies all constraints. Fur­thermore. the Drazin ODE of the linear case can be derived in this way. The numerical integratioll clearly profits from the index 1 property of the extended formulation. How­ever, an additional curvature expression has to be supplied, which is cheap to compute for multibody systems with absolute coordinates. Numerical experiments with the nonlinear and the linearized version of a truck model demonstrate the stabilizing effects and the efficiency of this approach.

The technique discussed here is restricted to constrained mechanical systems or, more general, variational problems subject to constraints. The other classes of problems where DAEs arise, e.g. electrical network analysis, show different structures and properties.

'This work has been supported by the Stiftung Volkswagenwerk within the research project Identijizierungs-, A nalyse- und Entwurfsmethoden fur mechanische Mehrkorpersysteme in Deskrip­torform.

New algorithms for optimization using central paths with applications to feedback

design

G. Sonnevend Eotvos University, Dept. of Numerical Mathematics

H1088,Budapest, Muzeum krt 6-8. e-mail:sonnevend@ludens.elte.hu

May 15 1993

The need for efficient optimization algorithms in the design of feedback control for dynamic, uncertain systems is well known. We shall present re­sults concerning new algorithms (both using analytic centers and central paths) for several classes of problems, we demonstrate (both by theoretical and numerical test resul ts) that the new class of methods are essentially more powerful the ones used hitherto. First a new extrapolation method for fol­lowing the central path by increasingly high (up to 20·th) order exrapolation is constructed, which has connections (i.e. in a special case it reduces) to a vectorial generalization of rational Pade ·type approximation to Stieltjes (Nevanlinna-Pick, ... ) functions. Our numerical experiments on classes of random linear and convex (generalized) quadratic programs demonstrate a reduction of complexity (and cputime) - compared to recent, standard algo­rithms - between 3 . 40 times and more. A preliminary report ,were the semiinfinite case is also studied is given in [1 J.

As a second application of analytic centers we report on a new method for computing feedback controls in the analogons of Hoc> optimization problems for (not only linear) systems with hard, pointwise (saturation) bounds on controls, disturbances and states (the latters confined e.g. to a polyhedron). A detailed report with test resuls for several differential games is given in [2J.

1. G. Sonnevend, High-order extrapolation algorithms for following the central paths in linear and convex, semiinfinite programs, in "Control Applications of Optimization", eds. R. Bulirsch and D. Kraft, forth­coming in the series ISNM, Birkhauser, 1993.

2. G. Sonnevend, Constructing feedback control in differential games by the use of central trajectories, DFG Report Nr. 385, Schwertpunktpro­gramm Anwendungsbezogene Optimierung und Steuerung, lnst. fiir Angewandte Mathematik Univ. Wiirzburg, July 1992.

299

)

:-11 X ED Hl/HCX) CONTROL FOH AN i:NVERTED PENDULU~1

y /lWi/lNE ~iOI1Y

[) (' P (I r'l rn (~n l () r f"1 a' 1\ 1"11 ;J I 1(";

I ;,lcull y or ~ic len;:I", LJ n I v (~r~; I 1 Y 0 r (c 'l'! ~ ,'y

Conakr'y, Guinea (We';I, M,'j(::I!

,~~ S'LR AJ l'

T~ilJrOMU ISlliii," r~,\()t ,r I 1/~r'1 ,'ic~-ll r I\("r,,,,nr"nn

IJI'IVCI'illy o! I!)(' ilylJ~yU~i G" ,n;~\N"!j (~():) ('! , I;H"~n

Mixed 1I,/IiCX) control theory and il~ application to the Inverted

ldulum System in both the state-feedback and output-feedback cases

~ considered, A mixed II~/HCX) controi problem is the problem of fin-

Ig an internally stabilizing control;~r that minimizes the II,-norm

a closed-loop transfer function sutject to an inequality constraint

the HCl-norm of another closed-loop transfer function,

n order to apply this control theory to a gjven plant, the augmer-

p I ant m u s t b e k now n, II ere .' e h a v e p;' oro sed the mix e d II 2 / H CX) aug -

ted plant construction for both the ;tilte-feedback and output-feed-

k problems, furthermore the paramete:s and the weighting Functions

can v e n i en t I y c has en i n order to a btl in!. h e more rob u s t control I er s

our plant,

inally, according to the results of simulations, we understood that

mar e per for man t con t r 0 'I I e r for our i' Jan t J S the s tat e - r ceo b a c "

troller, the more robust being the JlJ:put-fecdback controller,

CONTROL OF EPIDEMIOLOGIC DISEASES WITH OPTIMIZA nON OF THE INVESTMENT IN EDUCA nONAL CAMPAIGNS

J. A. M. Felippe De Souza

Universidade da Beira Interior Dept. de Engenharia Electromecanica

6200 - Covilha - PORTUGAL

Tel.: (351-75) 314207. Fax: (351-75) 26198.

e-mail: F _SOUZA @ UBl.bitnet

Takashi Y oneyama

Instituto Tecnol6gico de Aeronautica Divisao de Engenharia Eletr6nica - CT A - ITA - IEE

12225 - Sao Jose dos Campos - S~ - BRAZIL

Fax: (55-123) 22.9195. e-mail: ITA@ BRFAPESP.bitnet

ABSTRACT

The main objective of this work is to present two quantitative methods for the optimization of the investment spent in educational campaigns in the control of Epidemiologic Diseases. An extension of the Nerlove Arrow equation, used originally in the field of marketing, is adopted as the mathematical model for the description of the degree of motivation of the general population towards elimination of still water used by the mosquitoes for reproduction.

The investment policy is computed in such a way as to minimize the overall cost functional, which includes the cost of educational campaigns, the cost of sanitary campaigns (i.e., direct spending with insecticides) and the incurred social costs when individuals acquire the disease. Two procedures to solve the optimization problem are shown. One using optimal control theory and another using a sob-optimal approach. For the latter we present some numerical results by using a genetic algorithm.

Results obtained by computer simulation show significant variations on the overall cost as the investment policy is changed. Moreover, because these variations are of non­trivial nature, the use of algorithms of numerical optimization may be valuable.

Also, the mathematical model presented in the present work can be used in specialist courses, as a tool integrating a Computer Aided Educational System. This could provide an easy visualization of the consequences of changes in the investment policy or in some parameters of the model.

1 301

Simulation of Internal Air Systems of Aero Engines

Thomas M. Speer MTU Motoren- und Turbinen-Union Munchen GmbH

Dachauer StraBe 665 D-8000 Munich 50

Tel. 089 / 1489-2116

The secondary air system of an aero engine serves for the preven­tion of leakage of mainstream air from the thermodynamic cycle and the cooling of components. Each turbine blade is cooled by air flowing within a complex cooling system in the interior of the blade.

For the purpose of steady state and transient simulation, both air systems are modelled as networks. Arcs represent flow paths con­taining the flow elements which describe certain types of ge­ometry. Nodes represent junctions or branches' of several flow paths. The elements are modelled by nonlinear algebraic or stiff differential equations which combine inlet and outlet pressures and temperatures with mass and energy flows. The task is to compu­te nodal pressure and temperature values such that conservation of mass and energy at the nodes is guaranteed by the resulting flows. For the steady state case, this gives rise to a nonlinear system of algebraic equations, for the transient simulation a system of stiff and differential-algebraic equations is to be solved.

This talk describes the numerical solution algorithm, which in­cludes nested iteration schemes, a special method for the rapid solution of the stiff flow element ODEs, a continuation method with rotation velocity as homotopy parameter, and special features for choking flows. The algorithm uses a modification of Powell's Hybrid Method for the steady state and DASSL for the transient computation.

303

A SKELETON-BASED HIERARCHICAL SYSTEM FOR LEARNING AND RECOGNITION

Iraklis M. spiliotis, Dimitris A. Mitzias and Basil G. Mertzios Department of Electrical Engineering

Democritus University of Thrace Tel +30-541-79108, 26473

Fax +30-541-26947 e-mail mertzios@xanthi.uucp

Abstract

In this paper we present an hierarchical system for learning :md recognition of image objects. The system consists of three independent levels. The first level is based on a novel advanced East thinning algorithm. The algorithm extracts the critical ?oints of the object and all the necessary information concerning :he links among these points. All the data in this level constitute :he image primitive features, which actually are then used for the ~xtraction of a set of subpatterns, suitable for structural inalysis of the object. In the second level the extracted ;ubpatterns are classified in classes, using simple an.alytical :ormulae to obtain geometric features. The selected classes are ;hosen to represent the shape and also the orientation of the ;ubpatterns. In the case where a subpattern is not simple, a ;econdary system of this level forms two or more new subpatterns, ,hich are suitable for classification. In a higher level another :lassifier is used to combine the available information about the ;hape, the orientation of the subpatterns and the connections lmong them, in order to classify the object. The learning of this 'inal classifier is a difficult task, due of the variety and the :omplexity of the possible subpattern combinations. The desired )ehavior of the system is satisfied and a high recognition and earning rate is aChieved, depending on the application. Also the :ime complexity of the system is low. The proposed system may be lsed for applications such as optical inspection in industry printed circuit board inspection), medical imaging (chromosome nalysis), finger print identification and handwritten recognition.

......

-

14

Ie Caratheodory-Toeplitz Interpolation Problem for Almost Periodic Functions

Ilya M. Spitkovsky and Hugo J. Woerdeman (speaker)

Department of Mathematics

The College of William and Mary

Williamsburg, VA 23187

ilya@cs.wm.edu hugo@cs.wm.edu

n this paper we consider the Caratheodory-Toeplitz positive extension problem in the

Viener algebra of almost periodic functions. We derive necessary and sufficient conditions

Jr the existence of a positive (band) extension, which is also characterized by a maximum

ntropy principle. In addition a linear fractional parametrization for the set of all positive

xtensions of a given function is obtained.

,n important tool in the solution of the problem is the general algebraic scheme for positive

xtension problems (the band method), which was initiated by H. Dym and I. Gohberg,

nd further developed by I. Gohberg, M.A. Kaashoek and H. J. Woerdeman.

EQUIVALENCE CONDITIONS FOR CONTROL PROBLEMS IN LINEAR DISCRETE SYSTEMS

(abstract)

T. A. Stoilov, A. E. Gegov

Bulgarian Academy of Sciences, Central Laboratory of Control Systems,

Bl.l07, P.O.Box 79, Sofia 1113, Bulgaria

Optimality is one of the most important requirements of modern control

systems because it usually guaranteers "their best bahaviour" in a

certain sense. For this reason optimal control has been intensively

investigated by many researchers. In spite of that there exist certain

problems in this field which are due mainly to the fact that accurate

optimal solutions can not be always obtained. As an alternative, some

suboptimal procedures are used.

One of the most frequently used suboptimal procedures is sequential

optimization. It is characterized by simplification of the system design

as a result of which the computing requirements are reduced. The paper

considers the conditions under which the solutions of linear discrete

optimal control problems and suboptimal sequential optimization problems

coincide. The procedure for providing this coincidence is called

"parametrization" because it leads to the defining of certain parameters

in the respective sequential optimization problems. To carry out the

parametrization effectively, two methods are proposed whose main

advantage is the reduced dimensions of the problems being solved. These

methods are further developed as algorithms which are applied in an

off-line mode.

305

)6

NONITERATIVE COORDINATION IN MULTILEVEL SYSTEMS

K. P .Stoilova, T. A. Stoilov,

Bulgarian Academy of Sciences, Central Laboratory of Control

Systems,Acad.G.Bontchev str., bl.107, 1113 Sofia, Bulgaria, tel. (02) 713

2774, Fax 0359 2 723787

ABSTRACT: Open loop control strategy in multilevel, hierarchical systems

is introduced. Naniterative coordination strategy is performed.

Preposition - correction/agreement sequence is worked out which decrease

the iteration of computation up to one. The global optimization problem,

resolved by the multilevel system is assumed in a block diagonal form

(1)

n

g 1 (x) = 0 = E

E F i(X i) ~, i = 1

(i) g1 (x i)'

i = 1

g (x) m

o

h (x ) = 0 n n

x = (X" •• ,Xn

), 9 (9" ... ,9m), F,Fi,gi,9

ij,h;-scalar functions.

The subproblems and the solutions, solved by the local subsytems are

(2 ) h. ( x . ) =0, } , i = 1, n. 1 1

where ~. is the dual optimal value of (2) and A are zero dual values of

the corresponding g(x) constraints of (1). The local subsystems compute

(2), skipping the interconnected constrai~ts. The coordination unit uses

the prepositions x.(O,/) and the global solution xopt

is found as a 1

opt * . function x = ~(Xi (O,~ ) ,1=1,n. Analytical relation of ~ is defind as

(3) X(A,~)

where 10,/ denotes evaluation ~f x~ and x~ at point 0,/. Explicit

relations for of x' x' and ' op are obtained. The coordination unit A' ~ "

agrees or corrects the local prepositions X(O,~·) and the global optimal

solutions xopt

are obtained from (3) without iterrative computations.

One step coordination policy is performed. This coordination is

applicable for real time control in distributed systems, as it decreases

the information exchange between the control levels. This coordination

policy founds on the linear - quadratic approximation of the nonexplicit

analytical relations concerning the right and inverse Lagrange problems.

The discrete-time Hoo control problem with measurement feedback

Anton A. Stoorvogel • Department of l\lathematics

and Computer Science

Eindhoven University of Technology

P.O. Box 513

5600 MS Eindhoven

the Netherlands

Ali Saberi 1

School of Electrical Engineering

and Computer Science

Washington State G'niversity

Pullman. \VA 99164·2752

U.S.A.

Ben M. Chen t

Department of Electrical Engineering

State t'niversity of New York

at Stony Brook

Stony Brook, .\:Y 11794-2350

U.S.A.

Abstract: This paper is concerned with the discrete-time If:0 control problem with measurement feedback. \'Ve extend previous results by havillg weaker assumptIOns on the system parameters. \Ve also show explicitly the structure of Hev controllers. Finally. we show that it is in certain cases possible, without loss of performance. to reduce the J,:-nall1ical oruer of the controllers.

Keywords: Discrete time systems. Hoo control prohlem. Reduced order controllers.

307

OPERATOR MODELS FOR PROCESSES WITH PARAMETRIC PERIODICITY IN CORRELATIONS

by ION SUCIU

Institute o~ Mathematics, Romanian Academy C.P.1-764, 70700 BUCURESTI, ROMANIA,

e-mail: isuciu@imar.ro

Abstract

Let V O'v1 , ••• ,V., be a string o~ succesively recived states

longing to a stationary process Suppose the

tocorrelation (Fk)~ is a pozitive de~inite ~unction on Z

king values operators on the Hilbert space E - the parameters

ace. Let

d suppose it satisfies the following range condition:

(.) Im RkcKeI R;n ... rlR;_l' 2~k~n

;?n the predictible part Q""l of'

near" filter

" Q ... 1=!:Ak V ... 1- k , n~l

.t-1

V".l can be obtained by the

;?re the filter coefficients can be determined by the ~ollowing ~orithm:

k-l

~"Fl' Ak"Fk-J"A~j_l' 2~k~n 1:f

~ prediction error operators verif'y the recurrent formula

Et-I-F;Fl' ~-Ei-l-Ei-l-A;,Ak' 2a~ . Examples of pozitive def'inite functions verifying (*, are

,en. They verify a parametric periodicity of the form

F ... k1'k-F.,Pk' nEZ, k~l

a sequence

'ameters space.

of' orthogonal projections on the

309

External Descriptions and Staircase Forms in Implicit Systems

Vassilis Syrmos Department of Electrical Engineering

University of Hawaii at Manoa 2540 Dole St., Holmes Hall

Honolulu, HI 96822 hellas@wiliki.eng.hawaii.edu

Petr Zagalak and Vladimir Kucera Institute of Information Theory and Automation

Czechoslovak Academy of Sciences P. O. Box 18, 182 08 Prague 8

Czechoslovakia zagalak@lan01.ensm-nantes.fr

Submitted to MTNS '93 September 23, 1992

Abstract

The use of input output descriptions in system theory has a long history. Kuceral has extensively study structural properties of linear systems using the polynomial approach as a tool. Furthermore, he showed how to formulate design problems in terms of Diophantine equations the solutions to which characterize feedback controllers. Although, this body of work focused to the detailed structure and solution of Diophantine equations, the computational aspects of the problem were not addressed.

The results proposed in this paper tackle the following problem. Given an implicit realization {E, A, B} how can we compute computationally efficient a minimal polynomial basis for its control­lability pencil. If this minimal basis also satisfies some further properties it will be called a normal external description of the system. To further motivate these results we draw some attention to the pole placement problem in implicit systems using state feedback. The most detailed and unified approach up-to-date is given by Zagalak and Loiseau2 • The results there are based strictly on the polynomial approach and the solution of a Diophantine equation. Therefore, our results can be viewed as a first step in solving the same problem from a computational point of view. Another, application of the normal external description is the computation of coprime matrix fraction de­scriptions in implicit systems. Finally, the proposed approach can be used for computing transfer functions of multi-input multi-output regular implicit systems. These applications are not discussed in this paper, since such results are still far from complete.

Two are the important tools in this problem; the staircase (reachability, controllabilty) form of the implicit system {E, A, B} and the use of embedding techniques. In particular, we embed the controllability pencil to a square unimodular matrix, which we invert. The proposed embedding technique ensures the minimality of the controllability and reachability chains of the system, and therefore the feedback invariants are preserved under the embedding. In addition the structure of the computed normal external description shows how to compute the proper and nonproper controllability indices of the system using the length of the stairs of the reach ability Hessenberg form.

Finally, we show the diferences and similarities of the controllability and reachability Hessenberg form in computing a normal ewxternal description. All these ideas are clarified in a simple example.

I V. Kucera, "Analysis and design of discrete linear control systems", Prentice-Hall, 1991. 2p. Zagalak and 1.1. Loiseau, "Invariant factors assignment in linear systems") SINS'92 Texas, 1992.

ROBUST IDENTIFICATION USING DISCRETE FOURIER TRANSFORM : SOME IMPROVEMENTS

M. TADJINE *, M. M'SAAD * and L. DUGARD *

Laboratoire d'Automatique de Grenoble B.P. 46 E.N.S.LE.G, 38402 - SAINT-MARTIN-D'HERES

e-mail: tadjine@lag.grenetJr, m 'saad@lag.grenetJr

ABSTRACT. There has been recently a growing interest in the robust control theory. The latter allows to design control systems that are able to maintain specified stability robustness /

performance properties in the presence of state disturbances and unmodelled dynamics. Unlike the usual control design where only a nominal plant model is required, the robust

control needs also precise knowledge of the uncertainty bounds arround this nominal plant

model. Both the nominal model and the uncertainty bounds are usually assumed to be

given in the frequency domain, in the form of Nyquist plot of the nominal model with uncertainty bounds arround it. Though the nominal model can be obtained as the result of

an identification experiment, the available identification theory is not able to come up with

the uncertainty bounds under realistic assumptions. Indeed the state of system identification

for robust control design is still in its infancy but many developements are likely to arise from better understanding of the interaction between identification and robust control. Of

particular importance for the current paper is the work of Lamaire et al (1991). In the latter, a nominal parametric model is fitted to the empirical transfer function estimate (ETFE), for

which hard error bounds are derived. The main assumptions that are a certain degree of stability (smoothness of the frequency function) the true frequency response and bounds

on the Gibbs effect due to the finite data windowing used in calculating the ETFE. The

resulting bounds have been slightly improved in de Vries et al (1992) using a partial

periodic input signal and an interpolation algorithm. The former makes it possible to reduce the noise to signal ratio, while the latter allows to get a continuous error bound rather than a

commonly obtained discrete error bound. The motivation of this paper is threefold. Firsly, it is shown that it is possible to have more tight transfer function error bounds than those

given in Lamaire et al (1991) and de Vries et al (1992). Secondly, it is shown that ellipsoidal error bound may be obtained instead of circular error bound when using the

Discrete Fourier Transform (OFT). And finally, it is shown that it is possible to

characterize the parameter set in which the nominal plant model parametrers lies from the

derived frequency model error bound. REFERENCES.

Lamaire, R. 0., Valvani, L., Athans, M. and G. Stein. (1991). A frequency domain estimator for use in adaptive control systems. Automatica, vol. 27, I, pp. 23-38.

de Vries, D. K and P. M. J., Van den hoff (1992). Quantification of model uncertainty from data. Selected Topics in Identification, Modelling and Control, 4, pp. 1-10.

311

The Beurling-Lax Theorem And Nevanlina-Pick Interpolation: A Continuous Time LTV System Theoretic Perspective

Gilead Tadmor

Department of Electrical and Computer Engineering, 409 Dana Building, Northeastern University, Boston, MA 02115, USA, e-mail: tadmor@northeastern.edu

In this talk we propose and establish a linear time varying (LTV), continuous-time, system theoretic interpretation of two central results: The Beurling-Lax theorem and the solution of the Nevanlina-Pick problem. It is noted that several other researchers (Alpay, Ben Arzi and Gohberg, Ball, Gohberg and Kaashoekh, Dewilde and Dym, Feintuch and Francis, and Sayed, Constantinescu and Kailath, to name just a few in a sample and partial list) have already addressed time varying interpolation pro blems, with very interesting results.

The concept of a unilateral shift is central in these developments. In the linear time invariant (LTI) context, the property of unilateral shift invariance means both time invariance and causality. In the LTV setting we focus only on the causality aspect. The unilateral shift is thereby defined not on a single space, but rather as a mapping between members of a family of subspaces, parametrized by time. Our "Beurling­Lax" theorem states that such a family is "shift-invariant", it satisfies a certain uniform exponential decay constraint and has some uniform Krein space properties, if and only if it is obtained as the set of images of the compressions to the respective positive rays of a J-unitary operator with a uniformly controllable and observable, exponentially stable (integral) system realization.

Ball and Helton established that the solutions to a family of classical (function theoretic) and modern (operator theoretic) interpolation problems stem from the generalized LTI Beurling-Lax result. Generalized wondering subspace techniques are fundamental in that analysis. In the current case the Beurling-Lax theorem is established by system theoretic, geometric arguments, and it serves more as a backdrop to the interpolation problem. In essence the Nevanlina-Pick problem is interpreted as follows: One looks for a stable LTV system of minimal induced I/O norm, given some (feasible) specifications on the compressions of the I/O mapping to a certain parametrized family of input and output subspace-pairs. This interpretation seems as a natural extension ofSarason's approach. The "LTV Beurling-Lax Theorem" focuses our attention on the appropriate compression subspaces. Yet the solution is based on a reduction to a one block model matching problem that have already been solved by Tadmor and Verma, using LQ variational state space, time domain methods.

It is interesting to note that in related studies by this author and by others tight connections have been established between unitary and J-unitary systems and LQ optimal control and game theoretic problems. Indeed, as it turns out unit ary and J-unitary systems invariably represent the mappings from the input deviation from feedback optimality to the controlled output variable in some associated optimization pro blem or differential game.

As in other LTV extensions of LTI resuits, the familiar LTI pattern is recovered. The main massage is therefore not in the mere statement of results, but rather in the precise concepts and the intuitions one brings to bear. Perhaps in a somewhat twisted way, but not altogether wrong, this study can be thus viewed as a small part of a "revisionist" effort: If so far we used function theory as an allegory, an underlying well established field, to explore and explain various properties of linear systems, now one atternpts to use linear systems and control tools to shed new light on time-known function theoretic facts.

12

Full-Order Estimators with a Weighted Hoo Error Bound

Kiyotsugu Takaba and Tohru Katayama

Department of Applied Mathematics and Physics Faculty of Engineering, Kyoto University

Kyoto, 606-01, JAPAN E-mail: takaba@kuamp.kyoto-u.ac.jp

Hooestimation has received a considerable attention as an effective method for the estimation for plants driven by the disturbances with uncertain spectral density. A standard Hooestimation problem has been considered from various points of view, e.g. bounded rpal condition, game theory, time domain LQ optimization, etc. Re­cently, Limebeer and Shaked gave a state-space solution to a weighted Hooestimation problem as an extension of the standard Hooestimation problem. In the weighted Hooestimation problem, the error signal is penalized by a weighting function for fre­quency shaping. A polynomial approach is also employed to give sufficient solutions of the weighted Hoo estimation and fixed-lag smoothing problems by Grimble. In general , the orders of the estimators given by the previous works are higher than that of the original plant. This can be a disadvantage from the computational point of view.

In this talk, we give a design method of a full-order estimator which satisfies the weighted Hooerror bound. By 'full-order', we mean that the state dimension of an

estimator is equal to that of the original plant. In the present design method, the weighted Hoo estimation problem reduces to a problem of determining a filter gain of an observer-type estimator. We derive a necessary and sufficient condition for the existence of a solution based on the bounded real condition and an algebraic Ric­cati equation. Furthermore, we consider a weighted Hd Hoo estimation nroblem. By employing Lagrange multiplier approach, we reduce the weighted H2/ Hoo estimation problem to a certain nonlinear optimization problem. We derive a necessary and sufficient condition for the existence of a solution to the optimization problem. The condition is given by a coupled system of a modified Riccati equation and Lyapunov equation, and we also develop an iterative algorithm for solving the optimization problem based Newton-Raphson method. Finally, we give a numerical example to show the effectiveness of the present design method.

Parametrization of All Stable Unbiased H 00 Estimators Based on Model Matching

Kiyotsugu Takaba and Tohru Katayama

Department of Applied Mathematics and Physics Faculty of Engineering, Kyoto University

Kyoto, 606-01, JAPAN E-mail: takaba@kuamp.kyoto-u.ac.jp

Recently considerable attention has been directed to Hooestimation problems. The estimation with Hoocriterion is appropriate when there is significant uncertainty in the spectral density of disturbance. This problem has been considered by various ap­proaches, e.g. bounded real condition, time domain LQ optimization, game theory, etc. However, less attention has been paid to the parametrization of all Hoo estimators than that of Hoo controllers. It may be noted that although a parametrization of all Hooestimators is contained as a solution to OE problem in the famous DGKF paper, it cannot be directly applied to an unstable system due to the restriction of internal stability. Recently, Limebeer and Shaked considered a minimax terminal estimation and Hoofiltering problem by employing a game theory and the duality between esti­mation and control. In particular, for the steady state case, they derived a necessary and sufficient condition for the existence of stable estimators for a possibly unstable system based on bounded real condition. Fernandes et al. have suggested design techniques for robust estimators based on a parametrization of all stable unbiased estimators.

In this talk, we derive a state-space representation of all solutions to steady state Hoo estimation problem based on a model matching approach. Since Hoo estimation problem is not a standard model matching problem when the plant is unstable, we first reduce Hooestimation problem to a standard unilateral model matching problem by using a parametrization of all stable unbiased estimators. From the model matching point of view, the duality between estimation and state-feedback (or full-information) control does not hold since state-feedback or full-information control problem is a singular bilateral model matching problem. Next, we derive a necessary and sufficient condition for the existence of a solution based on Nehari's theorem. Although the present result is the same as that by Limebeer and Shaked, the proof is straightforward and is based on a purely frequency domain approach. Moreover, the present proof clarifies the relations between H2 optimal estimator and Hoo estimators and between H2 type Riccati equation and Hoc type Riccati equation.

Finally we develop an LFT (linear fractional transformation) representation of all solutions to Hoc estimation problem.

313

-

4

A Study on Trade-off Problems for the Control of Uncertain

Systems

Masahiro Tanaka

Department of Information Technology

Faculty of Engineering

Okayama University

Tsushima-naka 3-1-1, Okayama 700, Japan

Fax +81 86 255 9136

E-mail tanakalOmathpro.it.okayama-u.ac.jp

This paper deals with some practical implementation problems arising in the control synthesis

for uncertain systems [IJ, [2J. The system under consideration is a linear continuous control

system with disturbances. The control is bounded, and the disturbances are also assumed to be

in known bounded sets. The goal of the problem is to guide the state variable into a prescribed

target set at a fixed time.

Based on ellipsoidal techniques on guaranteed control with unknown but bounded distur­

bances [2J, [3J, the following kinds of practical issues will be discussed.

• The sampling rate versus the size of solvability set trade-off.

• The worst and the best case solvability sets under uncertainty.

• On-off control problem for the system.

• Application to control of nonlinear system using fuzzy model.

• Decision support system for designing control strategy.

Especially, numerical examples will be given to show the usefulness of the method.

Keywords. control synthesis, guaranteed control, ellipsoidal approximation, computer

simulation, application

References

[IJ Krasovskii, N.N. and A.I. Subbotin: Game-Theoretical Control Problems, Springer-Verlag,

1988.

[2J Kurzhanski, A.B. and I. Vatyi: Evolution and control of uncertain systems, IIASA Tutorial-

92-1, Laxenburg, Austria, 1992.

[3J Kurzhanski, A.B. and I. Valyi: The problem of control synthesis for uncertain systems:

Ellipsoidal techniques, in G.Di Masi et al. (eds.) Modelling and Control of Uncertain

Systems, Birkhiiuser, Boston, 1991.

On the Identification of Slowly-varying Systems *

Ashok Tikku and Kameshwar Poolla t

June 17, 1993

Abstract

The focus of this paper is the investigation of the fundamental limits of system identification in a deterministic worst-case setting.

The modelling uncertainty attendant with all physical models has two distinct components. The first component results from the situation in which we deliberately choose to approximate the plant by a simplified model for reasons of expediency. The second, more fundamental component results from modelling errors inherent to system identification. The modelling un­certainty here represents a part of the plant model that resists any attempt at identification. We characterize this fundamental component of modelling uncertainty via upper and lower bounds.

These bounds suggest that it is the interplay between measurement noise and the rate of variation of the plant model that limit our ability to identify the model. It is this interplay that results fundamentally in modelling uncertainty.

At this point several clarifying remarks are in order in order to motivate the idealized iden­tification problem considered above. The essential problem we are concerned with is that of regulation of an uncertain time-varying system. This problem necessarily involves simultaneous identification and control. Indeed, we cannot first identify the plant model to within desirable tolerances and then proceed with controlling the plant based on this model. As a consequence, the hypothesis that we can conduct arbitrary input-output experimentation is suspect. The exciting input signals must be generated in feedback.

We are nevertheless interested in fundamental limits on identification accuracy for the time­varying plant. These limits can be regarded as a lower-bound on the identification accuracy for the regulation problem for an uncertain time-varying plant described above. A knowledge of these fundamental limits would enable us to determine a priori whether or not accurate identifi­cation is possible. In the event these fundamental limits are unacceptable, it becomes necessary to conduct better modelling from first principles or better experimentation with superior sensors.

315

·Supported in part by the National Science Foundation under Grant ECS 89-57461, and by gifts from Rockwell International. t Tel. (510) 642-4642, Email: tikku@jagger.berkeley.edu, poolla@jagger.berkeley.edu

16

The Kamke problem - properties of the eigenfunctions

CHRISTIANE TRETTER

NWF I - Mathematik, University of Regensburg, D-93040 Regensburg

We consider the following spectral problem on the finite interval [O,IJ,

N(y) = >.P(y),

Uj(Y)=O, j=I,2, ... ,n, (1)

(2)

where Nand P are ordinary differential operators of orders n > p 2: ° and Uj(Y) are two-point boundary conditions of orders :s n - I.

Problems of this form frequently appear in mathematical physics. Apparently first who paid attention to them was E. Kamke [KJ. Contributions to this field are due to W. Eberhard, G. Freiling, G. Heisecke, F.-J. Kaufmann, R. Mennicken and M. Moller.

Following the paper of A.A. Shkalikov [SJ, we introduce the spaces Wu(O, I) C W'(O,I) where the properties of the eigenfunctions {Yd of (I), (2) are investigated. These spaces are natural for this problem and the consideration of concrete examples confirms this point.

We also define the concepts of almost regularity and regularity for the problem (1), (2). These concepts are modifications of the former ones given in [TJ. For p = ° the concept of regularity coincides with Birkhoff-regularity.

Under the assumption of almost regularity, we prove a theorem on completeness and mini­mality of {yd in Wu(O, 1) for some s 2: r where ° :s r :s p and r depends on the coefficients of (I), (2) (in the general case {yd is not minimal in L2(0, I)). The main result is the following:

Theorem. If the problem (I), (2) is regular and P : wt(O, 1) ...... L 2(0, I) is injective, then {Yd form a Riesz basis in Wt(O,I).

The work reported is joint with R. Mennicken, Regensburg, and A.A. Shkalikov, Moscow.

References:

[KJ Kamke, E.: Differentialgleichungen I; Akademische Verlagsgesellschaft Geest & Portig, Leipzig 1969.

[SJ Shkalikov, A.A.: Boundary value problems for ordinary differential equations with a parameter in the boundary conditions; J. Sov. Math., n. 6, 1986.

[TJ Tretter, Chr.: On A-nonlinear boundary eigenvalue problems; Akademie Verlag, Berlin 1993.

Geometry on the Space of Linear Systems Based on Gramians

Kouji Tsumurat and Toshiyuki Kitamoritt

t Department of Information and Computer Sciences, Faculty of Engineering, Chiba Uni­versity, Yayoi-cho 1-33, Inage-ku, Chiba 263, Japan tt Department of Mathematical Engineering and Information Physics, Faculty of Engi­neering, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan

Abstract

In many conventional investigations about the geometry on the space of linear

systems, only the transfer functions of the systems have been discussed. In this

paper, we introduce two geometric structures, i.e. distance and divergence, on the

space of the realizations of linear systems and analyze them.

We take up controllability and observability Gramians because they represent the

power effects on the state variables from the inputs and on the outputs from the state

variables, and propose a Riemannian metric on the manifolds of Gramians of linear

systems, which holds mathematical and physical conditions, that is invariance for

basis transformation and so on. The metric tensor is related to Fisher information

matrix, and a distance and a divergence are derived from this metric. We take

the summation of the distance between controllability Gramians and the distance

between observability Gramians of two systems and regard this summation as a

distance between the systems. The same summation of the divergences is also defined

in the same way. The individual metric, distance and divergence are invariant for

basis transformation, although the summations of them between two systems are

not invariant.

The main object of this paper is to analyze the geometric structures of the real­

izations. Concretely, we present the realizations which attain the minimum values

of the distance and the divergence between two systems whose transfer functions

are given. The balanced realization is one of the realizations which attain the min­

imum values and the values can be derived from Hankel singular values. We give

some examples of approximation problem of the realizations and show that these

consequences are effective in designing robust systems.

317

18

Eigenvalue and Singular Value Distribution for Structured Matrices

E.E. Tyrtyshnikov Dept.of Numerical Mathematics, Russian Academy of Sciences

Ryleeva 29, Moscow 119034 tee@node.ias.msk.su

A unifying approach is proposed to studying the distributions of eigenval­ues and singular values of Toeplitz matrices associated with a Fourier series, and multilevel Toeplitz matrices associated with a multidimensional Fourier series. Extensions of the Szego and A vram-Parter theorems are obtained for generating functions not belonging to L2 or to Loo. Analogous extensions are given for multilevel Toeplitz matrices. In particular, it is proved that if f(x\, ... , x p ) belongs to L2 , then the p-Ievel (complex) Toeplitz matrices allied with f have their singular values to be distributed as ABS(f(x\, ... , xp)). The distribution results about the Cesaro (optimal) circulants are granted even if f belongs L\. Also new theorems on clustering will be presented that have to do with the preconditioning of multilevel Toeplitz matrices by multilevel circulants.

The approach is spread over to the case when a matrix family is con­structed from some Toeplitz matrix families via basic matrix algebra op­erations, such as addition, multiplication, and inversion. If function f is obtained from the generating func- tions (that correspond to the Toeplitz matrix families at hand) with the help of similar but functional operations, then the singular values of the new structured matrix family are distributed as ABS(f), provided that certain conditions on the initial generating functions are fulfilled. The eigenvalues of Hermitian components of the matrices from the new family are distributed as ~(f) while those of the scew-Hermitian parts do as ir;J(f), where i is the imaginary unit. Among other things, these results seem to give a simplest possible way to explain nice properties of the circulant preconditioning.

Signal processing and polynomials of discrete argument

L.1. Vainerman and N.B. Filimonova

(Kiev University, Zabolotny street No.38, apt. 61, Kiev,252I87, Ukraine; e.mail: wain%agrosys.kiev.ua@relay.USSR.EU.netj phone number 478-89-00.)

The approach to the invariant signal processing system design was proposed in rI]. It was based on the applicat.ion of the Hermit polynomials. This paper is intended to show how to use the ideas and the methods described in [1] for the creation of corresponding computer-oriented model. In many cases it is essential that the decision about attributing the processed signal to one or another class would not depend on possible transformations of the sig.nal in the input of recognition system. The approach which we propose is based on the usage of the expansion of a signal by the special polynomials of discrete argument. These polynomials, orthogonal on the set of points 0, 1, ... M - 1 with respect to the binomial distribution, were intr04uced by M.P.Krawtchouk [2]. The binomial distribution has the form p(i) = Mlp'(1 -p)M-i/il(M - i)l, where natural M and real p E (0,1) are given beforehand (i =. O, ... ,M-I). SO we have the relations of orthogonality for Krawtchouk polynomials

M

E K~J(i, M)K;;J(i, M)p(i) = o ... "Ml[P(I - p)]" /nl(M - 11.)!.

We used the Krawtchouk functions F,t)(i, M) which are connected with Krawtchouk polinomials K~) ( i, M) by the following relations

F¥J(i, M) = K:t')(i, MhIPW/jMI[P(I- p)]"/nl(M - n)!.

The properties of the Krawtchouk functions make it possible to select a complete system of the informative features of signals, which are invariant with respect to the linear and some nonlinear transforma.tions. The algorythm of the selection of the informative features of a signal is the basis of the designed software for the compression and the restoration of the signals as well as the computer program package for the recognition and the classification of the signals.

References [1] Vainerman L.1.(I992). Signal processing and harmonic analysis of generalized shift

operators. Recent Advances in Mathematical Theory of Systems, Control, Net­works and Signal Processing. Volume 2. Proceedings of the International Sympo­sium MTNS-9I. MITA PRESS, Tokyo, pp.557-56I.

[2] Krawtchouk M.(I929). Sur une generalisation des polynomes d'Hermite. C.R. Acad. Sci., Paris, v.I89,pp.620-622.

319

c

Numerical issues and parallelism in adaptive beamforming methods

Filiep Vanpoucke, Marc Moonen, Joos Vandewalle

ESAT - E.E. Dept. - K.U. Leuven K. Mercierlaan 94, 3001 Heverlee, Belgium

vpoucke@esat.kuleuven.ac.be moonen@esat.kuleuven.ac.be

vandewalle@esat.kuleuven.ac.be

The adaptive beamforming problem has been a fruitful research topic dur­ing the last decade. This is not surprising since it will have important applications, e.g., in future telecommunications systems. Typically in these applications the data rates are extremely high, and, other hand, the same computational steps have to be repeated possibly infinitely many times. Therefore one needs to develop algorithms that are highly parallel and at the same time exhibit good numerical properties. Linear algebra plays an important role here. The adaptive beamforming problem can be reformulated as a recursive least-squares problem with one or multiple linear constraints. These constraints impose a prescribed gain in certain look directions, put zeros in interference directions or influence the shape of the main lobe. The main point of focus here is the implementation of adjustable constraints. Under certain circumstances small errors in the constraint coefficients may severely degrade the SNR at the output of the beamformer. An adjustable constraint can then alleviate this problem. An overview will be given of different approaches to the adaptive beamforming problem. It is shown how existing methods can or can not be used for the adjustable constraint prob­lem. The McWhirter and Shepherd method, which converts the constrained least-squares problem to an unconstrained canonical least-squares problem by the use of a constraint preprocessor, the 'blocking matrix' method, and the Yang and Biihrne method with constraint postprocessing and hyperbolic rotations, are compared to a newly developed parallel algorithm, in terms of parallelizability and numerical properties.

321

COMPUTATIONAL METHODS FOR STABILIZATION OF DESCRIPTOR SYSTEMS

Andras Varga DLR - Institute for Robotics and Systems Dynamics

Oberpfaffenhofen, D-8031 Wessling, F.R.G. Tel.: (08153)28461, Fax: (08153)281441

February 15, 1993

• botract. The paper propol.,. numeriea.lly relia.ble lputational methods for the stabiliution of a linear :riptor sYltem with or without aimultaneoua elimina­L of it. impul.ive behavior, Two buic stabilization ap­~ches are disculled. The firat approach relies on meth­which represent generalizations of direet stabilization Lniques for standard state-Ipaee ,yatema. The aecond roaeh is based on a reeuraive generamed Schur algo­m for pole _Ignment. Both approach., are bued uaively on numerieally reliable proeedures and can e for robu.t software implementations. :eyword.: Descriptor syatcmlj atabililation meth­pole ulignmentj numerical algorithm..

Introduction :onsider the follOWIng linear tim&-invariant descriptor

E>.::(t) = A:(t) + BU(/) (!) 'e::(t) eRn, u(t) e Rm,E,A E R"',B e Ro,m and 'ither the differential operator dl dt for a continuoua­oystem or the .. dvMce operator "'(t) "" .. (t+1) for a ete-time system. Throughout the paper we denote a m of the form (1) hy the triple (E, A, B). Generally natrix E can be singular and let r = rank(E). The f polea of the system (1) (finite and infillite) i, de­I with ~(A, E) and represents the lOt of generalized value. of the pair (A, E). The syetern (1) is called ar if det(~E - A) 'f O. We say that the system (1) ia e if >'(A, E) E C- , where C- denotee the Itability n of tho complox plane C &lid it either the left open ,lex half-plane for a continuoul-time ayetem or the or of the unit circle for II diacrete-time ay,tem. The bility region of C i8 the complement of C- and is ~ed by C+, e following 8tabiliaation problems are coneidered.

i-Stabili:ation Problem (SSP): For the system (1) lotermine a .tate feedbaek matrix F e Rm ,,, in the ontrollaw

u(t) '" F,,(t) + vet) (2)

such that the cIoled-loop IYltem

E>.:I(t);:;; (A+ BF):r(t) + Bv(t)

i. regular and haa ex&etly r stable finite poles,

(3)

2, R-Stobilizotioll Problem (RSP): For the regular sys­tem (1) determine a atate feedback matrix Fe i1!",n in the control law (2) auch that the elosed-loop sys­tem (3) i. regular &lid all its polO!l (at most ~) are stable.

The e_ntial difference between the .. problems ie that while in the SSP we have to move all impulsive modes of the open-loop .y.tem to finite locationa, this .. peet in the RSP i. not important and thus the closed-loop sys­tem may have &II impul.ive behaviol provided the num­ber of dOled-loop finite poles is 1_ than r. The follow-ing theorem. give necessary and lufllciem condition. for the existenee of solution. of the formulated .tabilization problems.

Theorem 1 [8] A aolutioll F of the SSP .:;'1, if and olily if the .yltem (E, A, B) i. S-8tabi/i,abl, (llrongly Ita­bi/i:able), that i. the matriee. E, A alld B ,atil/lJ the [ollowing condition.:

1. rank([>.E - A Bll;:;; n for all finite>' E C+.

B. rank([E AS ... Bn = n, where the eolumlU of Soc 'pan the null .pace of E.

The flrat eondition expr_es the requirement for the controll .. bility of unIt able poles while the seeond eon­dition charaeterizes the eontroll&bility of the impulsive modes and the regularizability of the .y.tem by state feedbaek. In what follows when we concern with the SSP we ahall _ume that the above conditione are fulfilled. Note that the regularity of the open-loop IYltem is not nee-ary to be a.seumed for the exiJtence of the 8Olution of the SSP.

Theorem 2 [5) A solution F of file RSP frill, if and only if the ay.tem (E, A, B) is ,tabili.able, that i, the rnalncel E, A alld B sati./lJ th. condition:

r&llk([ >'E - A B)) = n for all flnit. >. E C+.

Cauchy Matrices and Multiple Interpolation Nodes

Matrices of the form

Zdenek Vavfin Math. Inst. Acad.

Zitmi 25 115 67 Prague

Czech Republic kurzweil@csearn.bitnet

Ci~zJ are usually called Cauchy matrices. They form an interesting class of struc­tured matrices and have deep connections to interpolation. This topic was studied by Finck, Heinig and Rost recently and a new class of so-called Cauchy - Vandermonde matrices was introduced which corresponds to an interpolation problem including infinity as one of the interpolation nodes in a certain sense. The author introduced a generalization of Cauchy and Cauchy - Vandermonde matrices corresponding to interpolation problems with multiple nodes. In the present lecture the basic properties of these matrices will be described, especially the form of the corresponding inter­polation problems, the formulas for inverse matrices and an application for factorization of Loewner matrices. Some numerical remarks will be joint.

Parametrization of Hankel-norm approximants of time-varying systems

Aile-ian van der Veen and Patrick Dewilde Delft University of Technology Department of Electrical Engineering

2628 CD Delft, The Netherlands

tel: (+31 IS) 781442, fax: (+31 IS) 623271

email: allejan@dutentb.et.tudelft.nl

323

We consider the Hankel-norm approximation problem for (bounded) upper triangular operators: generalized

f 2-operators T with matrix representations f T'j]:' such that T;j = 0 (i > j). Here, each Tu is a matrix with dimensions M; x Nj , where M;, Nj are finite integers (possibly zero). Associated to T is a sequence of Hankel­

operators H, = f T'_,+I.'+j]O' (k = ~ ... =), which are submatrices of T corresponding to its top-right parts. The rank of these operators plays an important role in realization theory: if d, = rank H, < =, then there exist

minimal time-varying realizations with system order d, at point k:

such that [. . Yo YI ... ] = [ Uo UI .. ·]T.

A,: d, x dk+1 , C,: d, xN ..

B,: M,xdhl D,: M,xN,

(I)

The Hankel norm of T is defined to be II TIIII = sup II H, II. This definition is a generalization of the time­invariant Hankel norm and reduces to it if all H, are the same. Let r = diag(y,) be an acceptable approximation

tolerance, with y, > O. If an operator Ta is such that

(2)

then Ta is called a Hankel norm approximant of T, parameterized by r. We are interested in Hankel norm approximants of minimal system order. In [I], we proved that if the number of Hankel singular values of r-l T

that are larger than I is equal to N, at point k, then there exists a Hankel norm approximant Ta whose system

order is equal to N, at point k, assuming none of the singular values are equal to I.

In the construction of such Hankel-norm approximants Ta, two additional operators playa role. The first is U: the inner (i.e., upper and unitary) factor of Tin a coprime factorization l' = ~'U (where ~ is upper). The second

is 8: a i-unitary operator (8'i18 = h, 8h8' = ii, where ii, h are signature matrices) such that

[U' -r1J8 = fA' -B'] (3)

consists of two upper operators A " B '. This equation describes an interpolation problem. 8 exists under certain conditions and can be constructed explicitly, and the resulting signature matrices are determined by the singular

values of the Hk- In fact, the system order of the strictly upper part of 8:;:1 is at each point k equal to Nk •

Let SL he an upper contractive operator. Then

(4)

is contractive, and the strictly upper part of 1" = 1'+ rs' U js a Hankel norm approximant. Conversely, for each

T' of which the strictly upper part Tn is a Hankel norm approximant, there is an upper contractive operator SL

such that r (1" - 1) = s' U, where S is given by the above expression.

[I] P.M. Dewilde and AJ. van der Veen, "On the Hankel-Norm Approximation of Upper-Triangular Operators

and Matrices," accepted for Integral Equations and Operator Theory, September 1992.

=leducing the computational sensitivity of the (ey step in solving the stochastic realization lroblem

ilichel Verhaegen

ctober 1992

ru Delft

:Ift University of Technology

Department of Electrical Engineering Network l1Jeory Section

P.O. Box 5031. NL-2600 GA Delft. The Netherlands Tel. (31-15) 78 1442 Fax. (31-15) 62 3271

e-mail: verhaege@dutentb.et.tudelft.nl

Abstract

In this short paper we propose an instrumental variable solution to the stochastic realization problem in a state space setting. In a first part. it is highlighted that the key step in most existing approaches. namely the determination of the extended observability matrix. generally leads to nu­merically very sensitive computations. In a second part. a particular type of instrumental variables based on a Kalman filter design is proposed. In general this design requires the full solution to the stochastic realization problem. A simplified design is proposed that only requires the knowledge of the column space of the extended observability matrix. The usefulness of this simplified design is demonstrated in a numerical simulation study.

words: Stochastic realization. LQ factorization. numerical robustness. Kalman filter.

325

Sensitivity Minimization for causal, linear, dis­crete time-varying systems through Nevanlinna­Pick interpolation and outer-inner factorization

Michel Verhaegen and Patrick Dewilde

October 1992

TU Delft

Delft University of Technology

Abstract

Department of Electrical Engineering

Network Theory Section

P.O.-Box 4, NL-2600 GA Delft, TIle Netherlands

Tel. (31-15) 78 1442 Fax. (31-15) 62 32 71 e-mail verhaege@dutentb.et.tudelft.nl

The sensitivity minimization problem for discrete time-varying systems is treated in a state space framework. Given an inner-outer factorization of the plant, the solution consists in solving a time-varying Nevanlinna-Pick interpolation problem. The latter solution is characterized by a

Lyapunov type of equation.

Left fractional representations of nonlinear systems

Madanpal S. Verma and Louis R. Hunt

)artment of Electrical Engineering, 3480 University St. McGill University Montreal,

~bec, Canada H312A 7, verma@thunder.McRCIM.McGill.edu

grams in Mathematical Sciences, P.O. Box 830688, University of Texas at Dallas, Ri­

rdson, TX 75083-0688 USA

Fractional representation theory has played a significant role in analysis and synthesis

inear feedback systems. During the preceding decade, attempts have been made to

~lop an analogous theory for nonlinear -systems. In doing so, differences with fractio­

representation theory for linear systems have been observed and attempts have been

Ie to find characterizations of coprimeness which are independent of the requirement

nearity. In particular, the asymmetry between right and left fractional representations

.1Onlinear systems has been noted. While the concept of right coprime fractional repre­

,ation and related construction techniques have been well developed, the treatment of

left fractional representation theory is less complete.

In this work , an approach believed to be sui table for defining left coprime fractional

'esentations for nonlinear systems is presented. For this prupose, input-output repre­

ations of nonlinear systems are considered. Let U and Y be linear spaces consisting of

:tions of time and let G : U --> Y be an input-output map representing some physical

em of interest. Subspaces of U and Y, representing bounded inputs and outputs, are

)ted by Us and Y" respectively. A fractional representation G = D-1 N is said to

, left fractional representation over the space of causal, stable maps if Nand Dare

;al and stable and the mapping [D N 1 : [~:] --> Ys is onto. In the linear case, this

lirement is equivalent to the rational matrix [D(8) N(s) 1 having no zeros in the right

plane.

More generally, we define a mapping L: [b] --> Y which is causal and stable, whose

.el is {[ ~u] : u E U} and which is onto from [~:] to Ys . This is believed to be

quivalent of coprime fractional representation. The relevance of this notion to issues

,edback design is noted. The problem of constructing such representations for input­

mt maps arising from a state variable description is considered. Recent results on

~ observers in the presence of inputs are utilized to provide left coprime fractional

esentations for some classes of nonlinear systems.

327

Balancing for Robust Modeling and Uncertainty Equivalence

Erik 1. Verriest

School of Electrical Engineering, Georgia Tech Atlanta, GA 30332-0250

Phone (+1) (404) 894-2949, FAX:(+l) (404) 894-2997, email: erik.verriest@ee.gatech.edu

Two applications of balanced realizations are introduced. Both contribute to a new approach to

the robust modeling of systems. A first robustification of a nominal model is obtained by embedding

the nominal model in a whole class of extended systems. The balancing framework lends itself nicely

as this embedding is based on an "inverse" to the balanced model reduction technique (projection

of dynamics). The problem of balanced system extension is in a sense a matrix completion problem.

The second application is in a sense complementary to the first: Here one tries to find a stochastic

reduced order model of a given deterministic system. The philosophy behind this is that by discarding

some of the dynamics, one throws away available information, hence introduces uncertainty. The

reduced order model should reflect this uncertainty by introducing stochastics (hence the uncertainty

equivalence). Ultimately, it is an approach similar to the one which gives a thermodynamic description

as a simpler model for the large dimensional microscopic dynamics.

Whereas in earlier work, one started from a given nominal state space model, in this paper we will

show how these concepts can be used directly from the available input/output data at the modeling

stage itself via LQG-balancing. As entry point the subspace algorithms for state space modeling will be

discussed. It is suggested that a model is retained, whose dimenson is the smallest possible dimension,

larger than the real dimension of the data-derived (i.e., the nominal) system. The "deleted part" is

modeled as a stochastic input, and is due to two contributions: One is the uncertainty equivalence,

from a reduced subspace algorithm. The other contribution takes the uncertainties (confidence) in

obtaining the nominal model from the data into account. This obviously depends on the size of

the available data, and is in a sense a smoothing operation on the noisy data. A tradeoff between

smoothness and available information is made, resulting in fewer a priori structure assumptions, by

intertwining confidence and robustness.

8

2D Systems and Realization of Bundle Mappings on Compact Riemann Surfaces

Victor Vinnikov Department of Theoretical Mathematics

Weizmann Institute of Science Rehovot 76100, Israel

We consider a commutative two-operator colligation, i.e. a two-dimensional (2D) .ynamical input-output system presented by state-space equations of the form

(1)

[ere f = f(tl, t2) is the state with values in the Banach space H, u = U(tl' t2) and = V(tl' t2) are respectively the input and the output with values in the Banach space

; (dimE < 00), and A I ,A2 : H ------+ H, 'ljJ: E ------+ H, q,: H ------+ E are bounded linear lappings. If we require both the system (1) and its inverse to be consistent for any choice f initial condition f(O, 0) = fo and the zero input we must have

(2)

here we set

(3)

two-operator colligation satisfying (2) is called commutative. It has been shown by Livsic and Kravitsky that a commutative two-operator colliga­

:m (more precisely, a slightly more complicated object called a commutative two-operator ,ssel) determines an algebraic curve X in the complex projective plane, the discriminant ~rve, and a pair of vector bundles on X, the input and the output vector bundle. Fur­ermore, the transfer function of the system (1) can be viewed as a mapping from the put to the output vector bundle holomorphic in a neighbourhood of the points of X at finity.

We consider the inverse problem of realizing a given mapping between two vector mdles on a compact Riemann surface (that we embed as an algebraic curve in the ojective plane) as a transfer function of a commutative two-operator colligation. The 1lization theorem that we establish allows to extend the basic interplay between systems d matrix functions on the complex plane to 2D systems and bundle mappings on a mpact Riemann surface.

Robust D-stability test of polynomial polytopes

WANG En-ping and AN Sen-jian

Institute of Systems Science, Academia Sinica

Beijing 100080, People's Republic of China

ABSTRACT: In this paper, we first consider the robust D-stability

of the convex combination of two real polynomials, where D is the

stability region whose boundary aD is formulated by a union of a

number of rational curies in the complex plane. An algebraic criterion

which can be realized by solving the real roots of polynomials, and

a geometric criterion which can be realized by graphical method are

given for testing the D-stability of the convex combination of two

polynomials. Then by using the famous 'Edge Theorem' we can test

the robust D-stability of polynomial polylopes.

329

Schubert Calculus and Dynamic Pole Placement

Xiaochang Wang 1

Texas Tech University Lubbock, TX 79409

mdxia@ttacsl.ttu.edu

M.S. Ravi University of Notre Dame

Notre Dame, IN 46556 msravi@cartan.math.nd.edu

Abstract

Joachim Rosenthal 2

University of Notre Dame Notre Dame, IN 46556

Joachim.Rosenthal@nd.edu

It was proved by Brockett and Byrnes that the pole assignment problem with static com­msators is an intersection problem in Grassmann manifold Grass( m, m + p). In making the ,nnection to the classical Schubert calculus they were able to show that there are

1!2!··· (p - I)!(mp)! d(m,p)=degGrass(m,m+p)= 1( )1 ( )1 m. m + 1 .... m + p - 1 .

(I)

mplex static output feedback laws which assign each set of poles for a nondegenerate m-input, output linear system of McMillan degree n = mp. In particular if the number d(m,p) is odd, lIe assignment by real static feedback is possible. It follows that when d(m,p) is odd, the generic stem has the arbitrary pole assignability if and only if mp 2: n.

People have been looking for similar results for the dynamic pole assignment problem for a long ne. Recently Rosenthal proved that the pole assignment problem with dynamic compensators :ain is an intersection problem in a projective variety K'/n,p and there are d( m, p, q) = deg K'/n,p 'mplex dynamic feedback compensators of order q which assign each set of poles for a q­mdegenerate m-input, p-output linear system of McMillan degree n = q(m + p - I) + mp. 'hen d( m,p, q) is odd, the generic system has the arbitrary pole assignability by dynamic feed­Lck of order q if and only if n :'0 q(m + p - I) + mp. But unlike the Grass(m, m + p), little e known about K'/n,p and its degree. The goal of our paper is to use the technique of Schubert Jculus to derive the formula for deg K'/n,p. Let

,d define a partial order in S:

(QI, ... ,Qp):'O (f3I, ... ,f3p ) ¢} Qi:'O f3,vi.

hen our main result states

Theorem: The deg KZ,rn is equal to the number maximal totally ordered subsets of S.

A formula of deg KZ,rn is given in terms of degrees of

deg Kt2 = 22q+l ,

[q/21 ( .) deg Kt3 = 5 L q ~ I 11 q-2i,

1==1

deg Ki,4 = ~(33(q+l) + (-1 )q),

degKj3 = ~(26(q+l) -I). , 3

lSupported in part by the Research Enhancement Fund from College of Art and Sciences, Texas Tech University. 'Supported in part by NSF grant DMS-9201263.

Robust Internal Model Control

Keiji Watanabe and Kou Yamada

Depatment of Electrical and Infonnation Engineering, Faculty of Engineering Yamagata University, Jonan 4-3-16, Yonezawa, 992, JAPAN

Tel. +81-238-22-5181, Fax. +81-238-24-2752 E-mail kw@eie.yz.yamagata-u.ac.jp

331

Abstact The purpose of this paper is to break through the dilemma between the sensitivity and the robust stability of internal model control systems. A design mehtod of robust intenal model control systems with low sensitivity is presented.

The internal model control system contains the model of the plant, which is connected to the plant in parallel. The error between the output of the plant ant that of the model is feed backed to the input of the plant via the controller. The sensitivity can be easily assigned to be very small by adjusting the controller such that the product of the transfer function of the controller and the model is near one. If the control system is stable, then influence of disturbances and perturbations can be reduced very small and the desired response can be obtained. The model error, however, frequently makes the control system unstable. As the sensitivity becomes lower, smaller model error destroys the stability of the control system. Furthermore, the internal model control systems, which are designed by conventional methods, suufer for the internal unstability, if the plant is unstable.

In this paper, we concentrate our attention on the stability under zero sensitivity. The stability criterion for zero sensitivity is derived. It is shown that if the model does not contain any unstable poles, and if the relative degree of the model and the number of the unstable zeroes of the model are equal to those of the plant respectively, then the control system is stabilizable with low sensitivity in the presence of model error, whether the plant is stable or not. A design method of robust internal model control system with desired sensitivity is presented.

Key Words Internal Model Control, Robust Stability Criterion, Low Sensitivity

llancing for Identification and Model Approximation of Dissipative Dynamical Systems

ract

Siep Weiland Department of Electrical Engineering Eindhoven University of Technology P.O. Box 513; 5600 MB Eindhoven

the Netherlands

A.C. Antoulas Department of Electrical and Computer Engineering

Rice University P.O. Box 1892; Honston, TX 77251

U.S.A.

the introduction in 1981, balanced representations oflinear systems have proved to be extremely I in a wide range of applications in system analysis, model reduction, signal processing, controller rr, system identification and problems related to data reduction. Apart from the traditional pt of Lyapunov balancing [1J which amounts to equating and diagonalizing the controllability )bservability gramians of a state space system, we also distinguish the concepts of LQG and )alanced coordinates as discussed in [2J and [3], respectively. In this paper we introduce a n of balancing that is particularly useful for dissipative dynamical systems. We will show that otrinsic property of dissipativeness can be used to associate a set of input-output characteristic , with the system that reflect the dissipated energy of internal states of the system. Two algebraic ti equations are needed to compute these characteristic values and to associate a balancing formation with it. We will apply the so called balanced truncation method to derive reduced systems by discarding those states that dissipate the least amount of energy. Reduced order

llS obtained in this way turn out to be dissipative again. We moreover discuss how this concept of cing can be used for the purpose of model identification and approximate modeling of dissipative ns. In this application, the state trajectory of the system is the primary object to be identified ,asuring the rate of dissipation in past and future data sets. The system parameters can be easily :nined from the time evolution of the state trajectory.

rences

B.C. Moore, "Principal Component Analysis in Linear Systems: Controllability, Observability and model Reduction," IEEE Trans. A.C., Vol. 26, No.1, pp 17-32, 1981.

E.A. Jonckheere and L.M. Silverman, "a New Set of Invariants for Linear Systems -Application to Reduced Order Compensator Design," IEEE Trans. A.C., Vol. 28, No. 10, pp. 953-964, 1983.

D. Mustafa and K. Glover, "Controller Reduction by Hoo Balanced Truncation," IEEE Trans. A.C., Vol. 36, No.6, pp. 668-682,1991.

HIGH ORDER VARIATIONAL PRINCIPLES

Jan C. Willems*

Abstract

Let L : )Rn X lRn ~ ... x lR~ -> lR be a functional and consider the problem of finding

(k+I)-times the extremal trajectories for the integral

1.tl dq dkq

L(q'-d ""'-d k)dt to t t

The purpose of the talk is to derive the behavioral differential equations which specify these extremals. Also, we will consider the question of which differential equations have as their solution set the set of extremals for a suitably chosen L. One of the difficulties encountered is to make precise what it means for a trajectory q : t E lR >-> q(t) E lRn to be an extremal.

Let us illustrate the type of results which will be derived for the case n = 1 and for quadratic functionals L. Denote

k

L(qO,qI,.··,qk) by L Lrs qrq3 r,s=O

and assume that Lrs = Lsr. Introduce the two-variable polynomial

k

L(~,71) = L Lrs('713

Then the extremal trajectories are those which satisfy the differential equation

Conversely, every differential equation

satisfying the time-reversibility condition

R(~) = R(-O

can be obtained this way. The results will finally be interpreted in the context of the classical Euler-Lagrange

equations, and of mechanical and Hamiltonian systems.

333

"Mathematics Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands, Telephone: +31.50.633984, FAX: +31.50.633976, Email: J .C. Willems@math.rug.nl

Semidefinite solutions of the continuous-time algebraic Riccati equation

Harald K. Wimmer Mathematisches Institut

U niversitiit Wiirzburg D-S700 Wiirzburg, Germany

Abstract. We consider the algebraic Riccati equation

R(X) = A'X +XA+XBB'X - C·C = 0 (1)

and give a complete description of the set T = {X I R(X) = 0, X:::: O} of negative­semidefinite solutions of (1). No assumptions on controllability or stabilizability are made. The main tool of our investigation is a decompositions theorem. With respect to a suitable basis the matrices A, B, C have the form

(2)

where all eigenvalues of Ao are purely imaginary and where AT> BT> Cr satisfy con­ditions which involve controllable and unobservable subspaces. To (2) corresponds a decomposition X = diag(Xo, X r ) of solutions X E T, and Xo satisfies a Lya­punov equation A~Xo + XoAo = 0, and Xr is the solution of an indecomposable Riccati equation A;Xr + XrAr + XrBB; - C;Cr = O. We focus on the solution set S = {X E TIX = diag(O,Xrn which is a closure of T. The map X ...... KerX is an order-preserving bijection from S onto a well-defined set of A-invariant subspaces.

Linearly Constrained Contractive Extension Problems

Hugo J. Woerdeman Department of Mathematics

The College of William and Mary Williamsburg, VA 23187

335

In this talk we present a linear fractional description for the set of all positive semi­definite extensions of operator band matrices. In addition, we derive a maximum entropy principle and a "zero-inverse" characterization in this setting. Subsequently, the results are applied to the linearly constrained contractive extension problem which is a general­ization of the Strong Parrott introduced in [1]. Consequently, we obtain a linear fractional parametrization for the set of all solutions, and maximum entropy and "zero-inverse" prin­ciples for the linearly constrained extension problem.

The linearly constrained extension problem is also considered in the general framework of nest algebras, and the results on generalizations of the Nehari and Corona are obtained.

This talk is based upon joint work with Mihaly Bakonyi.

[1] C. Foias and A. Tannenbaum, Proc. of the AMS, Vol. 106, 1989, pp. 777-784.

An upper bound on the number of linear relations identified from noisy data by the Frisch scheme

Keith G. Woodgate

DepClrtment of Aeronautics Imperii.! ('ollegE' of SciE'ncC'. Technolog,\' and i\/pdici!lC'

Prine£' Consort Road London SW7 28't'. England. email: kwoodgCQae.ic.ac . uk

April :29. 199:3

Abstract

The paper pre.-:ent.-: an upper hound on m((~) defined alj t.he maxi­

lIlal corank of ~ - i.J. when- 2..: is a 91t'en non-singular co\'ariancE' matrix and ~ i~ an IInblOll'11 diagonal co\'ariance matrix !'.uch that ~ - ~ rE'­

maill~ a co\'arianct' matrix. To date no general procedure for determining

mc(~) ilj il\'ailahle. Thi:-; optimization prohlem is, in e~f'ence, the Fn'lch

~chf'lllt· fOJ identif.\'ing linear relat.ion:" from noi:,,~' data reprE':"ented hy ~ by ext.racting ,6. a~ the covariancE' matrix of ~ome additive, ullcorrelat.ed

noi.-:e. If ~"'''.,. i~ a maximizer then ~ - .i1f}l<1J." is the covariance matrix of a lIl(uil1l(l1 nllmhel I1IC(~) of hlleal relations ident.ifted. The upper hound i:-:; derived using the theory of Schur complements in conjullction with a

re.'lIit for thE' mc(~) = 1 ca~e. It i:-. .-:hOWll how, for prohlems of low di­

IlH;'lI:-.ionali t.\', mc(~) and ~"l<" Citll be det.ermined u:-:;ing the.-:e ideits. The po:-.:-.ibilit.\' of dcve\opillg a gelleral pron:durt" to determine /Ilc(~) i .... i11~o

hriefl.\· di~Cl1:-.~('d.

Keywords idelltiftcatioll of linear relatioll:-'. Fri:-;ch ~cheme, positivE'

:-.emi-defiltitt· 11latrict":-., Schur cOlllplemE'nt:-..

Practical Internal Stability of nD Discrete Systems

Li XU, Osami SAITO and Kenichi ABE

Toyohashi University of Technology, Tempaku-cho, Toyohashi 441, Japan

e-mail: xuli@mizar.tutkie.tut.ac.jp

nD (multidimensional) signal processing has many applications in such areas as

seismic, sonar and image processing. In practical situations, the independent variables

ii, ... , in of an nD signal x( ii, ... , in) are usually spatial variables bounded in finite

domains, except that perhaps one variable is a unbounded temporal variable. Taking

this feature into account, Agathoklis and Bruton introduced the concept of practical­

BIBO stability for nD discrete systems, which proved to be less restrictive and more

relevant for practical applications than the conventional BIBO stability.

It is also well-known that a large class of iterative learning systems and linear

multipass processes can be described by 2D system model. As a common feature

of these systems, it is observed that there is no special boundary restriction on the

iterations, while the iterated passes are almost always restricted in a finite discrete-time

interval. Motivated by the control problems of these systems, the authors have recently solved the nD feedback practical-stabilization problem based on the practical-BIBO

.stability concept. It is noted, however, that one has to consider first the asymptotic

stability problem in the practical sense [Agathoklis and Bruton, 1983J if one tries to solve the tracking and regulation problems for the above systems.

In this paper, we will consider the internal stability problem for nD discrete sys­tems whose input signals are unbounded in, at most, one dimension. Based on nD

Roesser state-space model, we introduce the concept of practical internal stability and derive a necessary and sufficient condition. This condition is, in fact, equiva­

lent to the stabilities of n 1D systems, just like the case shown by Agathoklis and

Bruton for practical-BIBO stability. Namely, suppose n(zl, ... , zn)/d(ZI,"" zn) is

the transfer function of an nD system described by Roesser state-space model, then this system is asymptotically stable in the practical sense of [Agathoklis and Bruton,

1983J iff d(O, .. . , Zk, ... , 0) -=f 0, 'v'IZkl ::::: 1, k = 1, ... , n. In contrast with this, when

n = 2, the system n(zl, Z2) / d( ZI, Z2) is asymptotically stable in the conventional sense iff d(ZI, Z2) -=f 0, 'v'(ZI, Z2) E [;2 where [;2 is the unit bidisc [Fornasini and Marchesini,

1979J. By these results, we see that the conventional definition of internal stability is

also unnecessarily restrictive for many practical applications, and expect to be able to establish a simple and unified design method for iterative learning control systems and

linear multipass processes.

337

338

A Global Convergence Analysis for Self-Tuning Pole-Assignment Controller

Toru Yamamoto*, Yoshiyuki Sakawa* and Sigeru Omatu**

* Department of Systems Enginering, Faculty of Engineering Science, Osaka University, 1-1 Machikaneyama-Cho, Toyonaka-City, 560 Japan Tel. +81-6-844-1151 ext. 4612 Fax. +81-6-857 -7664 E-mail yama@sys.es.osaka.-u.ac.jp

** Department of Information Science and Intelligent Systems. Faculty of Engineering, University of Tokushima. 2-1 Minamijohsanjima-Cho, Tokushima-City, 770 Japan Tel. +81-886-23-2311 ext. 4731 Fax. +81-886-26-4739

Abstract Design methods of the self-tuning control(STC) for the system with unknown

parameters have been studied up to now. Among them. the STC methods based on the minimization of the generalized output have been reported, for example, gen­eralized minimum variance control(GMVC) and generalized predictive control(GPC). These methods are useful for the nonminimum-phase system. However, for such systems it is necessary to find the weighting factor or polynomials so as to make the closed loop stable. On the other hand, there is a pole-assignment technique as a method to overcome this problem. Many literature in regard to self-tuning pole-assignment control have been reported. However, almost of these method real­ize the pole-assignment from the view point of the output response for the desired value, that is, the robustness for the disturbance is not considered.

In this paper, we construct a pole-assignment self-tuning controller with two­degree-of-freedom structure which is easily to realize for the real plants. This control method enables us to design control system by considering the transient property for the reference signal and the robustness for the disturbance. respectively. We treat the stchast.ic system with st.ep distubance in this papaer. Therefore, in order to remove st.ep disturbance and realize the robusr tracking property we insert the integral action to the control law. And also, so as not disturb the control system by the variation of step disturbance. we present a modified algorithm for the usual least squares method. Furthermore. we present a global conYergence analysis for the proposed pole-assignment self-tuning control algorithm. Finally, we illustrat.e some numerical simulation results in order to show the effectiveness of the proposed control algorithm.

Frequency Response and Its Computation for Sampled-Data Systems

Yutaka Yamamoto' Division of Applied Systems Science

Faculty of Engineering Kyoto University

Kyoto 606, JAPAN

Abstract

While the notion of frequency response plays a crucial role in the study of linear

continuous-time time-invariant systems, it is not well understood for sample-data

systems, especially when intersample behavior is taken into account. This is due to

the fact that sampled-data systems with intersample behavior present time-varying

characteristics, so that the notion of steady-state response is difficult to justify. The

present work shows a natural way of introducing the notion of frequency response,

and that it is in fact viewed as the extension of the classical notion of the steady

state response against sinusoidal inputs.

To be more specific, let h be a fixed sampling period and G(z) a stable transfer

matrix operator induced by lifting. Then for every w, G(eiwh ) is an operator mapping

£2[0, h] into itself. Furthermore, if the input is the sinusoid eiwt , then the correspond­

ing output will asymptotically approach eiwkhG( eiwh )( eiw9 ), k = 0,1, ... This implies

that as k -> 00, the gain characteristic is invariant and given by IG(eiwh)(eiw9)1.

Therefore, the induced norm sup IIG(eiwh)vll/llvll is defined to be the gain of Gat

w. A computational procedure for the frequency response is also given in terms of

a finite-dimensional generalized eigenvalue problem.

"Research supported in part by the Tateishi Science and Technology Foundation.

33~

Synthesis of Nonlinear Hoo Output Feedback Controllers Maximum Dissipation Rate Approach

Ciann-Dong Yang Institute of Aeronautics and Astronautics

l\ ational Cheng E ung C niversity Tainan. Taiwan, Republic of China

Chee-Fai Yung and Fang-Bo Yeh Institute of Applied Mathematics

Tunghai Cniversity Taichung, Taiwan, Republic of China

Abstract

In this paper, the nonlinear analog of DGEFs results is fully de­loped for input affine nonlinear systems. The solution presented here es only elementary ideas beginning , .... ith dissipative systems. Conditions lder which separation principle can be applied to nonlinear case is ad­essed first and the problem of nonlinear Hoo output feedback (NHOF) ntrol problem is then reformulated as an optimization problem wherein timal controller, which has a standard structure of state feedback plus Lte estimator, is found to maximize a 'defined dissipation rate. Through lS optimization process, the necessary and sufficient conditions for the .vability of NHOF problem is derived in terms· of a Hamilton-Jacobi in­llality which, when applied to linear case, is automatically reduces to the o standard Riccati inequalities occurred in the DGEF's result.

Adaptation, Identification and Complexity: An overview by G. Zames

Department of Electrical Engineering McGill University

3480 University, Montreal, PQ. H3A 2A7 e-mail: jennifer@mcrcim.mcgill.edu

Abstract

341

The growth of the H- theory of feedback is forcing a reexamination of traditional areas of control. In particular, it provides a new perspective on the problem of system identification for feedback design, and the question 'What is adaptive control?'. We will focus on three recent developments which relate the problems of identification and adaptation to the theory of metric complexity:

1. The conventional stochastic approaches to identification emphasize the filtering out of additive noise. However, recent work on worst-case identification makes clear that this filtering inevitably comes at the cost of speed of identification. Although speed may not always be important in non-adaptive control, it becomes a key parameter for adaptive design. The problem of fast identification, i.e., of characterizing inputs which identify a system to a tolerance £ in minimum time, has been solved for a variety of data sets in the H- and L 1 metrics. Identification speed has been shown to be proportional to the complexity of the a priori data sets, as measured by their n-width. For slowly varying data, the inability to identify quickly leads to uncertainty principles which imply that perfect accuracy in control is not achievable even in the absence of additive noise.

2. Feedback can reduce the complexity of a priori data which is needed to control to a tolerance £. However, it has turned out that the relative reduction in, e.g., n­width, approaches 0 as £-"70.

3. The question of What is adaptation? has remained elusive, and adaptation has frequently been confused with the use of feedback or the presence of nonlinearity. However, from the point of view of information-based complexity, a precise answer to this question is possible, which makes a clear distinction between adaptation, feedback, and nonlinearity.

2

A Maximum Principle for Node Voltages in a Finitely Structured, Transfinite, Electrical Network

A.H.Zemanian State University of New York at Stony Brook

Stony Brook, N.Y. 11794-2350 U.S.A

Email: zeman@sbee.sunysb.edu

Abstract - In general, unique node voltages with respect to a given ground node do not exist in a transfinite electrical network. An example illustrating this fact is given. However, under certain circumstances, unique node voltages will exist. In this work, we establish a maximum principle for those voltages in the case where the transfinite network is "finitely structured" and "perceptible." The "finite structure" is a generalization to transfinite networks of local finiteness for ordinary infinite net­works. "Perceptibility" means that between every two nodes of whatever rank there is an (in general, transfinite) path whose resistances sum to a finite amount. The maximum principle asserts that in any sourceless (possibly transfinite) subnetwork of the given network, the maximum and minimum node voltages will occur at boundary nodes of the subnetwork. The proof of this result is considerably more complicated than that for ordinary networks because we must now take into account voltages at infinite extremities of transfinite subnetworks.

rt

Periodic control with singular perturbations. Zharikova E.N. Samara College

443099 Samara, Chapaevskaya, 186, 33-28-23 Fax 8462-332823

Abstract.

The method of integral manifolds for the matrix systems of the differential equations with fast and slow variables based on singular perturbation theory was developed to transform the singularly perturbed systems to the regularly perturbed systems of lower dimension.

In this paper the method of the replacement of the boundary value problem by the initial value problem and calculation of initial values is generalized on the systems w1th per1od1c matrix coeff1c1ents and a small pos1t1ve parameter mult1plying by some of the der1vat1ves.

Combining these methods, we obtain the equat10n for tne init1al value and reduce 1t's dimens1on.

We present here the method, containing the following three steps:

1) constructing the in1 t1al value problems for the R1ccat1 equations and for the linear equat1ons,

2) reducing the systems to a lower dimens10n system, 3) computing the 1nit1al values.

Keywords: singular perturbat1ons, linear control systems, period1c solut1on, Riccat1 equat1ons.

343

Maximal Jtoo-norm of Transfer Functions Consistent

with Prescribed Finite Input-Output Data

Tong Zhou and Hidenori Kimura

Dept. of Mechanical Engineering for Computer-Controlled Machinery, Osaka University, 2-1, Yamada-oka, Suita, Osaka 565, Japan.

E-mail: shu@watt.ccm.osaka-u.ac.jp/kimura@watt.ccm.osaka-u.ac.jp.

ABSTRACT: Recent success in robust control theory also results in a renewed interest in system identification. For the compatibility with robust control theory, both the plant [lOminal model and its error uncertainty bound are required as system identification results. On the other hand, it seems to be essential to obtain the smallest model set which is defined by a nominal model and uncertainty bound and includes all of the Jossible transfer functions that can be induced from the prior information about the Jlant and identification experiment data. To emphasis the intended application, this dentification problem is referred to as robust control oriented system identification.

[n this paper, the following. problem is investigated which is closely related to robust :ontrol oriented system identification. For given noise-free plant input-output data, uo, Yo), (Ul, yd, ... , (un' Yn), and non-negative numbers M 2: 0, 0:::::: p < 1, define two ,ransfer function sets,

9, := {g(z) 1 Yo + y,z + ... + Ynzn + ... = g(z)(uo + u,z + ... + unzn + ... )},

92 := { g(z) I g(z) = ~g,Z" Ig,j:::::: Mp' } .

[he objective is to find

J = sup Ilg(z)lloo, and g(,IE91 n9,

!d = {g(z) 1 g(z) E 9, n 92 , Ilg(z)lloo = J}.

lased on classical extrapolation theory, the desirable J and the transfer function set !d .re obtained.

The Nehari Problem for tllt' Pritchard-Salamon dass of

Infinite-Dimensional Lineal' S~'stellls: a direct approach

IL'lls Zwart Faculty of ;\pplied :'IIatlwlllatics. l"lIi,·,·rsity of TWellt".

P.O. Box 21,. ,:iOO AE Ens'-/J(-'d", TIl<' .\<>tllerlallds fax: :ll-:j:l-:140,:n. tt>!: :ll·:j:l-"~:14(i-l. ,·-Illail: lwllillls"lllillh.lJtwt'nte.nl

R1Jth (:1Jrtaill Dept. ~IatlIf'lllil.tics, l'nin>rsity of (;wlJilJ.!?;'·Il.

P.O. Box 800, 9,00 AV (;rollin~"Il. Tlw :'\erlwrlilllds. fax: :11<iO-6;J:l9'fi. «>1: :ll-~)O-():I:l')A:-l. "-lIlail: r.f.ClJrtilill f'lllath.m.'!;.nl

Key Words :,\£'hari ProhJt"1I1. lIi11"dy ;'''P;'I(''':--, Irdillilc'-dillll'II:-.I'lIlal lilH'dr :,y~IPIJIS. J-s/WC! ral fac­torizat.ions, Lyapullov f"qll<ttioll .... , f1i.1t!kl~1 ...;ill~llh,. \;dIH':--.

Abstrac.t :\ C'olllpif'tf> solution i.-; ohlailwd IC, tilt" :"t->hari prulllc'llI fur ::;Ylllhol::; which hasp a r("­alizal.ion as all exponentially 5tahlt" Prildl<lrd-SiliaIlH)JI .... y . ..;t('11l ~(_-I. B,('), TI,is allows for tilt' possibility that Band (' bt>- UlihOlllldt'd. Tlw approach j~ ro :.,ol\·t> all t>quiv<lll'llt .I-spt'clral factor­ization problem for this particuim n'alizal iOtl.

345

347

List of Contributions

Abed, E.H. . .......................... 31 Brockett, R.W. . .. , ..... , .. , .......... ,8 Abida, J., Claude, D. . ................ 32 Brockett, R,W" Kosowsky, J. .., ..... 72 Ackermann, J. . ....................... 5 Buchberger, R. . ... , .. , ............... 73 Adamjan, V. . ........................ 33 Bunse-Gerstner, A. . .................. 74 Aeyels, D., Sepulchre, R., Mareels, I. .. 34 Burns, J.A. . .................. , ...... 75 Agathoklis, P., Kanellakis, A. . ....... 35 Buyvolova, A.G., Kolmanovskii, V.B., Agrachev, A.A., Sarychev, A.V. . ..... 36' Koroleva, N.I. , ...... , ........... , 76 Ak~ay, H., Khargonekar, P.P. . ........ 37 Byers, R. . .. , . , , . , , ............. , , ... 77 Albertini, F., Sontag, E.D. . ....... 38-39 Byrnes, C.l. ........... , ............... 9 Ali, M., Gotze, J., Pauli, R. . ......... 40 Cabay, S., Meleshko, R., Ali Mehmeti, F. . ..................... 41 Gutknecht, M.H ...... "., ......... 78 Alpay, D. . ........................ 42-43 Callier, F,M., Winkin, J, ............. 79 Alpay, D., Leblond, J. . ............... 44 Campi, M., Lorito, F, .......... , .. , .. 80 Alvarez-Ramirez, J., Suarez R., Carrato, S .. " ............. " ......... 81

Alvarez, J ......................... 45 Chen, H.-F. , ............. , .. , ........ 82 Antoulas, A.C. . ....................... 6 Chua, L.O., Galias, Z.J., Kocarev, L., Arnold, M. . .......................... 46 Ogorzalek, M.J., Wu, Ch.W ... ,.,.83 Arov, D.Z. . .......................... 47 Clements D. . ....... , ............ , ... 84 Arov, D.Z., Nudelman, M.A. . ........ 48 Colaneri, P. , , . , ...................... 85 Astolfi, A., Chapuis, J. . .............. 49 Coleman, J. . ............ ,., ......... ,86 Auba, T., Funahashi, Y ............... 50 Colonius, F., Kliemann, W, ......... ,87 Aubin, J.-P., Frankowska, H. . ........ 51 Conte, G., Perdon, A.M. . .... , ....... 88 Bagchi, A. . .......................... 52 Dehaene, J" Vandewalle, J. . .......... 89 Baillieul, J. . ......................... 53 Deistler, M., Scherrer, W. .., ......... 90 Bakan, G.M., Kussul, N.N. . .......... 54 Delaleau, E., Fliess, M ....... " ....... 91 Ball, J.A., Gohberg, I., De Moor, B, . , , ...................... 92

Kaashoek, M.A ................... 55 DeStefano, A, ....... , .. , ............. 93 Ball, J.A., Kaashoek, M.A. . ........... 7 Dewilde, P., Veen van der A.-J. . ...... 94 Ball, J.A., Verma, M. . ............... 56 Dmitriyeva, 1. .... , ............ , , . , ... 95 Bamieh, B., Dahleh, M., Voulgaris, P. 57 Dolezal, V. . .... , .. , ............. , 96-97 Banks, H.T. . ......................... 58 Baraniuk, R.G.', Jones, D.L. , .. , ...... 59

Dooren, P. van, Sreedhar, J. , ......... 10 Dritschel, M.A, ...................... 98

Barel, M, van, Bultheel, A. . ........ ,.60 Dym, H ............. , .......... , .. 11,99 Bartosiewicz, Z., Spallek, K. . .. "., ... 61 Dym, H., Georgiou, T.T., Belyi, S.V., Tsekanovskii, E.R. , .. , ... 62 Smith, M.C, ............ " ....... 100 Ben-Artzi, A., Gohberg, I. ..... , .. " .. 63 EI Asmi, S., Rudolph, J. . ... , ....... 101 Bestaoui, Y" Benmerzouk, D, , ....... 64 Elsner, L., He, C. .., ... , ............ 102 Blankenship, G.L ..... , .. , .......... , .65 Erdol, N., Bao, F. .. ................. 103 Bloch, A.M. . ........... , ... , ......... 66 Faierman, M. . .. , .. , , .......... , .... 104 Blondel, V., Gevers, M. . .... ,." ..... 67 Fang, Ch.-H., Chang, F.-R. .. ........ 105 Bolle, M .......... ,., ............... ,.68 Fang, Ch.-H., Chou, J.-H. .." ....... 106 Bona, B., Indri, M., Tornambe, A. .,.,69 Faybusovich, 1. ........... , .... ' .... 107 Bongers, P,M,M, ........... , .. " ..... 70 Feng, D.-X., Wang, H.-Z. . ........... 108 Rose, N.K. . ............... , .. , ....... 71 Ferrante, A. . ....... , ............. ,.,109

8

Ferrante, A., Michaletzky, G., Heij, Ch., Roorda, B. . ...... '" ..... 152 Pavon, M ........................ 110 Heinig, G. . ................. '" ..... 153

Fettweis, A. . ......................... 12 Helton, J.W. . ........................ 15 Fitzpatrick, B.G. . ................... 111 Hench, J.J., Laub, A.J. . ............. 154 Flockerzi, D. . ....................... 112 Hersh, M.A. . ....................... 155 Foias, C. . ............................ 13 Heuberger, S.C., Hof., P.M.J. van den, Fornasini, E. ........................ 113 Bosgra, O.H ..................... 156 Fragoso, M.D., Costa, O.L.V. . ...... 114 Hinrichsen, D. . .................. 16, 157 Freiling, G. . ........................ 115 Hinrichsen, D., o 'Halloran , J. . ...... 158 Fuhrmann, P. A. . .................... 14 Hoffmann, J., Fuhrmann, P.A. . ...... 159 Gahinet, P. . ..... '" ................ 116 Hornik, K. . ......................... 160 Galkowski, K. . ...................... 117 Hou, M., Miiller, P.C. . .............. 161 Gaspar, D., Suciu, N., Valu§escu, 1. .. 118 Hu, T.-S. . ...................... 162-163 Geerts, T. . ..................... 119-120 Hu, X. . ............................. 164 Genesio, R., Tesi, A., Villoresi, F. . .. 121 Hua, X.-M. . ..... '" ......... '" .... 165 Gerencser, L., ObeL R. . ..... " ..... 122 j Hiiper, K., Paul, St., Pauli, R. . ...... 166 Girosi, F. . ........ '" ............... 123 Huffel, S. van, De Moor, B., Chen, H. 167 Giust, St.J., Wyman, B.F. . ......... 124 Huijberts, H.J.C., Moog, C.H. . ...... 168 Glad, S.T., Forsman, K. . ............ 125 Huijberts, H.J.C., Nijmeijer, H. . .... 169 Gliising-LiierBen, H. . ............... 126 Ikeda, K., Shin, S., Kitamori, T. . .... 170 Gohberg, 1., Kaashoek, M.A., Inaba, H., Wang, W. . ............... 171

Lerer, L. ......................... 127 Ionescu, V. . ........................ 172 Gohberg, 1., Olshevsky, V. . ... " .... 128 Ito, H., Watanabe, H., Ohmori, H., Gombani, A. . ....................... 129 Sano, A .......................... 173 Gomez, G., Lerer, L. . ............... 130 Ito, N., Inaba, H. . .................. 174 Gorbachuk, M., Gorbachuk, V. . ..... 131 Ivanchenko, T., Sakhnovich, L. . ..... 175 Grammel, G. . ....................... 132 Ivanov, A.F., Verriest, E.l. .......... 176 Grasselli, O.M., Tornambe, A., Kaashoek, M.A. . .................... 177

Longhi, S ........................ 133 Kaczorek, T. . ....................... 178 Gray, W.St., Verriest, E.l. ........... 134 Kailath, Th. . ............. ; ......... 179 Grill, Th. . .......................... 135 Kalpazidou, S. . ..................... 180 Grupp, F. . .......................... 136 Kamenskii, M.l., Nistri, P., Guidorzi, R. . ....................... 137 Obukhovskii, V.V., Zecca, P ...... 181 Guidorzi, R., Stoian, A. . ............ 138 Kano, H., Nishimura, T. . ........... 182 Gurvits, L. . .................... 139-140 Karcanias, N., Leventides, J. . ....... 183 Habets, L ....................... 141-142 Kase, W., Tamura, K., Hackl, G., Schneider, K. . ............ 143 Nikiforuk, P.N ................... 184 Hahn, H., Leimbach, K.-D. . ......... 144 Kawamura, T., Shima, M. . .......... 185 Halanay, A. . ........................ 145 Kimura, H., Xin, X. . ................. 17 Hall, A.G. . ................... " .... 146 Knobloch, H.W. . ................... 186 Hamada, K., Suzuki, M. . ............ 147 Kolowrocki, K. . ..................... 187 Hanzon, B. . ....................... 148 Kondratenko, Y.P., Timchenko, V.L. 188 Hasler, M., Kennedy, M.P., Korolyuk, V. . ........................ 18

Dedieu, H ........................ 149 Kovanic, P. . ........................ 189 Hassibi, B., Sayed, A.H., Krajewski, W., Lepschy, A.,

Kailath, Th ...................... 150 Viaro, U .......................... 190 I

He, X.-H., Chai, T.-Y. . ............. 151 Krasovskii, N., Krasovskii, A.N. . .... 191 I

1

Kreisselmeier, G. . ................... 192 Krener, A.J. . ................ 19~20, 193 Krishnaprasad, P.S. . ................. 21 Kuijper, M. . ........................ 194 Kulhavy, R. . ......................... 22 Kummert, A. " ................... " 195 Kurzhanskii, A.B. . .................. 196 Kurzhanskii, A.B., Sivergina, 1. ...... 197 Labahn, G., Cabay, St. ............. 198 Langer, H. . ......................... 199 Leitmann, G., Reithmeier, E. . ....... 200 Lerer, L., Rodman, L. . .............. 201 Levine, J., Rouchon, P., Creff, Y ..... 202 Levy, B.C., Beghi, A. . .............. 203 Li, Z. . .............................. 204 Liberzon, M.R., Alexandrov, V.V.,

Rozov, N. Kh .................... 205 Lin, S.-F., Lu, N.-W ................. 206 Lindquist, A. . ...................... 207 Lindquist, A., Picci, G. . ............. 208 Liu, K.-Z., Luo, Z.-H., Mita, T. . .... 209 Logemann, H., Martensson, B. . ..... 210 Logemann, H., Rebarber, R. . ........ 211 Logemann, H., Rebarber, R.,

Weiss, G ......................... 212 Loiseau, J.J., Zagalak, P ............. 213 Loucks, E.P., Ghosh, B.K. . .......... 214 Maeda, H., Kodama, S. . ............ 215 Miikilii, P.M., Gustafsson, T.K ....... 216 Mahony, B., Mareels, 1., Campion, G.,

Bastin, G ........................ 217 Malabre, M., Martinez Garcia, J.C. .218 Marten, W., Mathis, W. . ........... 219 Martin, C. F. . ....................... 23 Martin, C.F., Miller, J ............... 220 Martin, C.F., Shubov, V ............. 221 Martin, Ph. . ........................ 222 Martin, Ph., Rouchon, P. . ........... 223 Maschke, B.M., Schaft, A.J. van der .224 Matignon, D., Depalle, P.,

d'Andrea-Novel, B., Oustaloup, A. 225 Megretskii, A., Peller, V.V.,

Treil, S.R ........................ 226 Mehrmann, V. . ................. 227~228 Mennicken, R. . ..................... 229 Mertzios, B.G., Fettweis, A. . ........ 230 Mertzios, B.G., Vachtsevanos, G. . ... 231

349

Michalik, J., Willems, J.C. . ......... 232 Moller, M. . .................... ',' ... 233 Mora, T. . ........................... 234 Morgul, O. . ........................ 235 Morse, A.S. . ........................ 236 Miiller, P.C. . ......... '" " ...... '" 237 Muravei, L.A. . ...................... 238 Mustafa, D. . ........................ 239 Nam, K ............................. 240 Namachchivaya, N.S., Malhotra, N ... 241 Nerurkar, M., Sussmann, H. . ........ 242 Nichols, N.K ......................... 243 Nichols, N.K., Stringer, S.M. . ....... 244 Nitsche, G. . ..................... '" 245 Niwa, S., Suzuki, M. . ............... 246 Nobuyama, E. . ..................... 247 Nudelman, A.A ....................... 24 Ober, R., Fuhrman~, P. . ............ 248 Ober, R., Wu, Y. . .......... " . " ... 249 Oberst, U. . ......................... 250 Oz<;aldiran, K. . .............. '" .... 251 Ogorzalek, M.J., Di}browski, A.,

Galias, Z .. , ... ' .................... 252 Ohara, A., Suda, N., Amari, S. . ..... 253 Ohsumi, A., Kusunoki, Y., Yang, Z. .254 Oussous, N.E., Petitot, M. . ......... 255 Owens, D.H. . ....................... 256 Pandolfi, 1. ......................... 257 Pantoja, J.F.A. de 0.. .......... ~ .... 258 Papageorgiou, N.S. . ................. 259 Partington, J.R. . ................... 260 Pavon, M. . ..................... 261~262 Peeters, R.L.M., Hanzon, B. . ........ 263 Peller, V. V. . ........................ 264 Peller, V. V., Treil, S.R. . ..... " ..... 265 Petridis, V. . '" ..................... 266 Pi cardello, M.A., Woess, W. . ........ 267 Picci, G. . ............................ 25 Picci, G., Pinzoni, S. . ............... 268 Picci, G., Schuppen, van J.H. . ...... 269 Pinto Jr., D.L., Val, J.B.R. do,

Fragoso, M.D .................... 270 Priitzel-Wolters, D. .. ................. 26 Rakowski M., Rodman, 1. ........... 271 Rakowski M., Wyman, B.F. . ........ 272 Ran, A.C.M. .. ...................... 273 Rebarber, R., Townley, S. . .......... 274

~egalia, P.A. . ....................... 275 ~ocha, P., Zampiere, S. . ............ 276 ~ogers, E., Owens, D.H. . ........... 277 ~othfuB, R., Schaffner, J., Zeitz, M. .278 ~ouff, M. . .......................... 279 ~udolph, J. . ........................ 280 ~ussell, D.L. . ....................... 281 akamoto, N., Ando, Y., Suzuki, M. .282 akhnovich, L.A. . ................... 283 and, J.-A. . ........................ 284 andberg, 1. W. . ..................... 285 ayed A.H. . ........................ 286 chaft, A.J. van der, Ball, J.A. . ..... 287 chempp, W. . ...................... 288 cherer, C.W. . ...................... 289 cherpen, J.M.A.,

Schaft, A.J. van der ............. 290 chmidt, W. . ....................... 291 chnitger, G. . ....................... 292 chrader, C., Sain, M.K. . ........... 293 chumann, L., Geering, H.P. . ....... 294 chwarz, H., Svaricek, F., Wey, T. . .. 295 hkalikov, A.A. . .................... 296 huhui, D., Bin, Ch., Shilin, W. . .... 297 imeon, B. . ......................... 298 onnevend, G. . ..................... 299 ory, Y., Ishida, T. . ................. 300 ouza, J.A.M.F. de, Yoneyama, T. ..301 peer, Th. M. . ..................... 302 piliotis, LM., Mitzias, D.A.,

Mertzios, B.G .................... 303 pitkovsky, LM., Woerdeman, H.J. ..304 toilov, T.A., Gegov, A.E. .......... 305 toilova, K.P., Stoilov, T.A. . ........ 306 toorvogel, A.A., Saberi, A.,

Chen, 8.M ....................... 307 uciu, I. ............................ 308 yrmos, V., Zagalak, P., Kucera, V. .309

Tadjine, M., M'Saad, M., Dugard, L. 310 Tadmor, G. . ........................ 311 Takaba, K., Katayama, T. . ..... 312-313 Tanaka, M. . ........................ 314 Tikku, A., Poolla, K ................. 315 Tretter, C. . ......................... 316 Tsumura, K., Kitamori, T. . ......... 317 Tyrtyshnikov, E.E. .................. 318 Vainerman, L.L, Filimonova, N.B. . .. 319 Vanpoucke, F., Moonen, M.,

Vandewalle, J .................... 320 Varga, A. . .......................... 321 Vavrin, Z ............................ 322 Veen, A.-J. van der, Dewilde, P. . .... 323 Verhaegen, M. . ..................... 324 Verhaegen, M., Dewilde, P. . ......... 325 Verma, M.S., Hunt, L.R. . ........... 326 Verriest, E.L ........................ 327 Vinnikov, V. . ....................... 328 Wang, E.-P., An, S.-J. . .............. 329 Wang, X., Ravi, M.S., Rosenthal, J. .330 Watanabe, K., Yamada, K. .......... 331 Weiland, S., Antoulas, A.C. . ........ 332 Willems, J. C. . .................. 27,333 Wimmer, H.K. . ..................... 334 Woerdeman, H.J. . .................. 335 Woodgate, K.G. . ................... 336 Xu, L., Saito, 0., Abe, K. . .......... 337 Yamamoto, T., Sakawa, Y.,

Omatu, S ........................ 338 Yamamoto, Y. . ..................... 339 Yang, C.-D., Yung, C.-F., Yeh, F.-B. 340 Zames, G. . ......................... 341 Zemanian, A.H. . .................... 342 Zharikova, E.N. . .................... 343 Zhou, T., Kimura, H. . .............. 344 Zwart, H., Curtain, R. . ............. 345

351

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Maple V Library Reference Manual 1991. xxv, 698 pp. Hardcover DM 82,- ISBN 3-540-97592-6

Springer All prices for books and journals include 7% VAT. In EC countries the local VAT is effective. d&p.B71MNT/F/1

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Progress in Systems and Control Theory

M.A. Kaashoek. Vrlje Universlteit Amsterdam, The Netherlands / J.H. van Schuppen. Centre for Mathematics and Computer SCience, Amsterdam, The Netherlands / A.CM. Ran. Vrije Universiteit Amsterdam, The Netherlands (Eds)

PSCT 3 Realization and Modelling in System Theory Proceedings of the International Symposium MTNS-89 Volume I 1990. 572 pages. Hardcover sFr. 148.- / DM 168.-ISBN 3-7643-3469-X

PSCT4 Robust Control of Linear Systems and Nonlinear Control Proceedings of the International Symposium MTNS-89 Volume II 1990. 680 pages. Hardcover sFr. 148.- / DM 168.-ISBN 3-7643-3470-3

A. Bensoussan, IN.RIA, Le Chesnay, France / G. Da Prato, Scuola Normale Superlore, Pisa, Italy / M.C Delfour, Universite de Montreal, Quebec, Canada / S.K. Mitter, M.IT., Cambridge MA. USA

Representation and Control of Infinite Dimensional Systems Volume2

1993. 330 pages. Hardcover Approx. sFr. 128.- / DM ISBN 3-7643-3642-0

This reference rTl9I'Iograph 'deals with the quadratic cost optimal cont'~I\~r:oblem fcir.systems described by linear ordmary dlft'1ilreri~L equations It will be sueful for math~mi'ltlciarlii:'!IfI{n~retlcal engineers in the field of control. . ... pu~tillnpaii!,: septernbt,;1993

volu~1" . 1992.330 pages. Hardcover sFr. 128- / DM 148.-ISBN 3-7643-3641-2

PSCT 5 Signal Processing. Scattering and Operator Theory. and Numerical Methods Proceedings of the International Symposium MTNS-89 Volume III 1990. 592 pages. Hardcover sFr. 148.- / DM 168.-ISBN 3-7643-3471-1

Vols 3, 4 and 5 are also available as a set: sFr. 378.- / DM 438.-ISB~j 3-7643-3468-1

Progress in Systems and Control Theory PSCT 15 K. Bowers / J. Lund, Department of Mathematics, Montana State University, Bozeman, Montana, USA (Eds)

Computation and Control III, Proceedings of the Third Bozeman Conference. 1992 1993.416 pages. Hardcover sFr. 178.- / DM 198.-ISBN 3-7643-365frD

J. Zabczyk

Mathematical Control Theory 1992. 272 pages. Hardcover sl'r. 108.-1 DM 124.-ISElN 3-7643-3645-5

Mathematical control theory IS a separate branch of mathematics that has, over a span of 150 years, de­veloped an extensive literature covering its various ideas and applications. Direct access to the theory, In a self contained and precise way, IS proVided by this presentation of basic concepts and results In the field. It therefore represents an Introduction to mathema­tical control theory, requiring only a knowledge of basic facts from linear algebra, differential equations, and calculus, With a few more advanced concepts required for the final part of the book. It has the further advantage of covering all deterministic theory in one, concise volume

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Mathematische Nachrichten Mathematical News

6 volumes per year (approx. 1920 pp. yearly) 170 mm x 240 mm ISSN 0025-584X Annual subscription price 1994 DM 1600,-; abroad DM 1618,-

The journal Mathematische Nachrichten has been founded in 1948 by Erhard Schmidt. Scope and purpose, as established by the founder and editors, provide for the publication of original papers with new results and methods concerning the substantial progress of mathematics and its applications.

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