2-D POLYNOMIAL AND RATIONAL MATRICES, AND THEIR APPLICATIONS FOR THE MODELING OF 2-D DYNAMICAL SYSTEMS.

摘要

The systems considered in this work are those that can be described by a rational transfer function depending on two variables. Such systems appear for example in image theory, partial difference equations in two variables, delay-differential systems as well as networks containing lumped and distributed elements. The objective of this study is to generalize to these systems some of the concepts (state-space, controllability, observability, feedback) that have proven to be important for the analysis of one-dimensional systems.;These results are subsequently exploited to study the internal behaviour of 2-D systems from a PMD as well as a state-space point of view. To do so, we associate to a 2-D system an abstract state-space, in general infinite dimensional, and we use this state-space to characterize the poles and zeros and the controllable and observable modes of this system. Unlike in the 1-D case, 2-D systems can be either exactly or approximately controllable (observable) and it is shown here that this distinction leads to some important differences in the solution of the 2-D pole placement problem.;The problem of obtaining some state-space (i.e., first order) models for 2-D systems is also considered, and we introduce here a class of models obtained by linearization of polynomial matrix operators. These state-space models include those considered earlier by Roesser and Fornasini-Marchesini, and it is shown here a large number of the properties of 1-D state-space realizations (including the existence of controller and observer forms) can be extended to these systems.;The previous results are then specialized to the study of delay-differential systems, and in this context a state-space realization algorithm is presented, as well as a reinterpretation of some results of Morse and a solution of the dynamic compensation problem. Finally, we discuss the generalization of the results of this thesis to the N-D case. However, several results valid for 2-D systems do not hold in this case.;The point of view developed is mainly algebraic and relies on the theory of 2-D polynomial and rational matrices, which is exposed here in detail. Several new concepts, such as the notions of primitive factorization and of factor and zero coprimeness of 2-D polynomial matrices, are introduced and are used to generalize to the 2-D case the properties of 1-D polynomial and rational matrices. The results that we obtain in this context include the existence of general factorizations for 2-D polynomial matrices, the study of polynomial matrix descriptions (PMD) of 2-D transfer functions, a procedure for the extraction of the greatest common right divisor of 2-D polynomial matrices as well as a theory of equivalence for 2-D polynomial and rational matrices.