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.

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 [

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 [

Let

One of standard tasks carried out in systems theory is to transform a given system representation into a simpler form before applying any analytical or numerical method. The transformation involved must of course preserve relevant system properties if conclusions about the reduced system are to remain valid about the original one. An equivalence transformation used in the context of multidimensional systems is unimodular equivalence. This transformation can be regarded as an extension of Rosenbrock’s equivalence [

Let

The Smith form

Let

Now we state the necessary and sufficient conditions for the reduction of a class of polynomial matrices to the Smith form.

Let

Now let

Introduce the canonical form given in [

Equating the coefficients of the polynomials

The matrix in the canonical form

Consider the matrix

The following result based on the Smith form establishes the connection between a polynomial matrix

Let

By Theorem

Let

Using the method given by Boudellioua and Quadrat [

In this paper, the Smith form of a bivariate polynomial matrix together with symbolic computation techniques is used effectively to compute the equivalence transformations that reduce a class of 2D polynomial matrices to a canonical form. The classes of matrices considered are those amenable to be reduced by unimodular equivalence to a single equation in one unknown function. These matrices arise from 2D Roesser systems which are strongly controllable.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author wishes to express his thanks to Sultan Qaboos University (Oman) for their support in carrying out this research and Dr. Alban Quadrat for his help with the OreModules Maple package.