Computation of the inverse of a polynomial matrix and evaluation of its Laurent expansion

Abstract

An algorithm is given which constitutes a generalization of the algorithm for the inversion of a pencil sE-A due to Mertzios (1984) in the case of general systems described in differential operator form. Recursive formulae are obtained for the calculation of both the coefficient matrices of the adjoint of the polynomial matrix, as well as for the coefficients of the characteristic polynomial of the polynomial matrix. A simple method is also presented that allows the evaluation of the Laurent expansion for the inverse of a polynomial matrix of any order. Specifically, an ARMA model is given for the computation of the coefficient matrices of the Laurent expansion in terms of the coefficient matrices of the polynomial matrix, without inverting the latter. The Laurent expansion of a polynomial matrix is used for the analysis and synthesis of the polynomial matrix descriptions that constitute a further generalization of the singular (or generalized) systems.