Eva Zerz

Primeness of multivariate polynomial matrices

Abstract Different characterizations of primeness of polynomial matrices that are mutually equivalent in the one-dimensional case of univiriate polynomials lead to various primeness concepts in the multivariate situation. Applied to systems theory, this fact gives rise to a more refined view to observability and controllability of multidimensional systems. Algorithms for testing primeness properties are given based on computer algebraic techniques.

A Constructive Solution to Interconnection and Decomposition Problems with Multidimensional Behaviors

By a multidimensional behavior, we mean the solution space of a linear constant-coefficient system of partial difference or differential equations. Within the behavioral framework, a natural concept of interconnection has been introduced by J. C. Willems. The regular interconnection problem can be formulated as follows: Given a behavior (the plant), find---if possible---another behavior (a controller) such that their interconnection is not redundant and is equal to a certain desired behavior. We give a constructive solution to this problem, which is in terms of polynomial matrix equations that can be solved using the theory of Grobner bases. We also study a dual problem concerning the existence of direct sum decompositions of multidimensional behaviors. Finally, we present a unified framework for studying both problems as special cases of constructing relative complements in the lattice of behaviors.

Notes on the definition of behavioural controllability

We take another look at the behavioural definition of controllability for discrete 1D systems, and its extension to multidimensional (nD) systems defined on

Multidimensional behaviours: an algebraic approach to control theory for PDE

\Z^n

Extension modules in behavioral linear systems theory

or

LFT representations of parametrized polynomial systems

\N^n

Discrete multidimensional systems over Zn

. We suggest that the current definition for nD systems is inappropriate for systems defined on the domain

Computing Quasi-Steady State Reductions

\N^n

On strict system equivalence for multidimensional systems

, and that care has to be taken even with the 1D definition. We propose a new definition which applies to all standard classes of discrete nD systems.

Behavioral Systems Theory: A Survey

This paper gives an overview of the algebraic theory of multidimensional behaviours, and it contains some new aspects as well. Some known results are restated with a new proof. After a short introduction to behavioural systems theory, we briefly describe the mathematical background of the algebraic approach. Then we study structural properties such as autonomy and over-determined systems on the one hand, and controllability and flat systems on the other.