On the reduction of 2-D polynomial systems into first order equivalent models

In this paper we propose a novel approach for the reduction of a 2-D rectangular polynomial matrix of arbitrary degree, to first-order matrix pencils of the form \(sE_{1}+zE_{2}+A\), utilizing the framework of zero coprime equivalence (ZC-E). The proposed approach is in turn employed to derive a series of ZC-E matrix pencils, which can be obtained “by inspection” of the coefficients of the original bivariate polynomial matrix. Improving similar constructions of first order pencils available in the literature, our approach results in matrix pencils whose size increases linearly with the degrees of the indeterminates of the original polynomial matrix. From a system-theoretic point of view, the proposed method, provides the algebraic tools to transform a high order bivariate linear system, into a zero coprime system equivalent first order representation. Notably, one of the proposed transformation techniques gives rise to generalized 2\(-D\) Roesser models.