A generalized Hurwitz stability criterion via rectangular block Hankel matrices for nonmonic matrix polynomials

TL;DR

This paper introduces a generalized Hurwitz stability criterion for nonmonic matrix polynomials using rectangular block Hankel matrices, extending classical methods to broader polynomial classes.

Contribution

It develops a novel stability criterion that applies to nonmonic matrix polynomials by redefining Markov parameters and constructing structured matrices, expanding classical stability analysis.

Findings

Establishes a link between polynomial inertias and structured matrices.

Shows Hurwitz stability corresponds to positive definiteness of Hankel matrices.

Provides a new framework for stability analysis of nonmonic matrix polynomials.

Abstract

We develop a Hurwitz stability criterion for nonmonic matrix polynomials via column reduction, generalizing existing approaches constrained by the monic assumption and thus serving as a more natural extension of Gantmacher's classical stability criterion via Markov parameters. Starting from redefining the associated Markov parameters through a column-wise adaptive splitting method, our framework constructs two structured matrices whose rectangular Hankel blocks are obtained via the extraction of these parameters. We establish an explicit interrelation between the inertias of column reduced matrix polynomials and the derived structured matrices. Furthermore, we demonstrate that the Hurwitz stability of column reduced matrix polynomials can be determined by the Hermitian positive definiteness of these rectangular block Hankel matrices.