%0 Journal Article
%T Computation of a Canonical Form for Linear 2D Systems
%A Mohamed Salah Boudellioua
%J International Journal of Computational Mathematics
%D 2014
%R 10.1155/2014/487465
%X Symbolic computation techniques are used to obtain a canonical form for polynomial matrices arising from discrete 2D linear state-space systems. The canonical form can be regarded as an extension of the companion form often encountered in the theory of 1D linear systems. Using previous results obtained by Boudellioua and Quadrat (2010) on the reduction by equivalence to Smith form, the exact connection between the original polynomial matrix and the reduced canonical form is set out. An example is given to illustrate the computational aspects involved. 1. Introduction Canonical forms play an important role in the modern theory of linear systems. In particular, the so-called companion matrix has been used by many authors in the analysis and synthesis of 1D linear control systems. For instance, Barnett [1] showed that many of the concepts encountered in 1D linear systems theory such as controllability, observability, stability, and pole assignment can be nicely linked via the companion matrix. Boudellioua [2] suggested a matrix form which can be regarded as a 2D companion form for a class of bivariate polynomials. These polynomials arise in the study of 2D linear discrete state-space systems describing, for example, 2D image processing systems, as suggested by Roesser [3]. However in that paper, the author did not establish the exact connection between the original matrix and the reduced canonical form. In this paper, using symbolic computation based on the OreModules [4] Maple package the connection between the original polynomial matrix and the canonical form is established. 2. Polynomial Matrices Arising from Linear 2D Systems A 2D system is a system in which information propagates in two independent directions. These systems arise from applications such as image processing and iterative circuits. Several authors (Attasi [5], Fornasini and Marchesini [6], and Roesser [3]) have proposed different state-space models for 2D discrete linear systems. However, it has been shown that RoesserĄ¯s model is the most satisfactory and the most general model since the other models can be embedded in it. The model of Roesser is one in which the local state is divided into horizontal and vertical states which are propagated, respectively, horizontally and vertically by first order difference equations. The model has the form: where is the horizontal state vector, is the vertical state vector, is the input vector, and , , , , , and are real constant matrices of appropriate dimensions. System (1) can be written in the polynomial form: where represents an advance operator
%U http://www.hindawi.com/journals/ijcm/2014/487465/