On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials

TL;DR

This paper characterizes Hurwitz-type matrix polynomials, derives explicit forms of associated Bezoutians, proves their Hurwitz stability, and proposes an extension method to generate new Hurwitz-type polynomials.

Contribution

It provides explicit formulas and proofs for the Hurwitz stability of Hurwitz-type matrix polynomials and introduces a method to extend non-Hurwitz polynomials to Hurwitz type.

Findings

Explicit form of the Bezoutian for Hurwitz-type polynomials

Proof that Hurwitz-type polynomials are Hurwitz matrix polynomials

Method to extend non-Hurwitz polynomials to Hurwitz type

Abstract

Every matrix polynomial $\mathbf{f}_n$ can be written in the form \[ \mathbf{f}_n(z)=\mathbf{h}(z^2)+z\,\mathbf{g}_n(z^2). \] 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 form of the Bezoutian associated with Hurwitz-type matrix polynomials. Using this explicit form, we provide an explicit proof that Hurwitz-type matrix polynomials are Hurwitz matrix polynomials. In [52], the Hurwitzness of Hurwitz-type matrix polynomials was also studied. Finally, we propose an extension of the class of Hurwitz-type matrix polynomials by adding to a non--Hurwitz-type matrix polynomial another matrix polynomial so that the resulting matrix polynomial is of Hurwitz type.