Transformation of a PMD into an implicit system using minimal realizations of its transfer function matrix in terms of finite and infinite spectral data

A simple method is given which uses the notions of finite and infinite Jordan pairs from operator theory in such a way to find the minimal realization of the inverse of a polynomial matrix. Specifically, given the finite and infinite Jordan pairs of a polynomial matrix A(s), we shall find a realization for its inverse A−1(s) which in general is a rational matrix. Then a method which finds the realization of the least possible degree, i.e. the so-called minimal realization of the matrix A−1(s) is presented. In the sequel we propose a method which transforms a given Polynomial Matrix Description (PMD) into a generalized state space (implicit) system using the above analysis of minimal realization for the transfer function matrix of the PMD in terms of finite and infinite Jordan pairs.