On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials

Abstract

Every matrix polynomial ${\mathbf f}_n$ admits a decomposition of the form \[ {\mathbf f}_n(z)={\mathbf h}_n(z^2)+z\,{\mathbf g}_n(z^2). \] The matrix polynomial ${\mathbf f}_{2m}$ is said to be of Hurwitz type if the expression ${\mathbf g}_{2m}(z){\mathbf h}_{2m}^{-1}(z)$ admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial ${\mathbf f}_{2m+1}$ is of Hurwitz type if $\frac{1}{z}{\mathbf h}_{2m+1}(z){\mathbf g}_{2m+1}^{-1}(z)$ has the same property. We derive an explicit representation of the Bezoutian associated with Hurwitz-type matrix polynomials. Using this representation, we obtain a direct proof that every Hurwitz-type matrix polynomial is Hurwitz. The Hurwitz property of this class was also investigated in [52]; our approach is based on an explicit Bezoutian representation. This provides a constructive connection between continued-fraction representations, matrix Bezoutians, and Hurwitz stability. We also develop a completion procedure that associates with a given matrix polynomial a Hurwitz-type matrix polynomial of higher degree. As a consequence, whenever such a completion exists, the original polynomial is Hurwitz. The proposed construction is illustrated by examples.