On generalization of classical Hurwitz stability criteria for matrix polynomials

TL;DR

This paper extends classical Hurwitz stability criteria to matrix polynomials by linking them with Stieltjes positive definite sequences, introducing new tests based on block-Hankel matrices and continued fractions.

Contribution

It introduces novel stability tests for matrix polynomials using positive definiteness and continued fractions, generalizing classical criteria.

Findings

Hurwitz stability linked to positive definiteness of block-Hankel matrices

Extension of classical criteria to matrix polynomials

New conditions involving block-Hankel minors and quasiminors

Abstract

In this paper, we associate a class of Hurwitz matrix polynomials with Stieltjes positive definite matrix sequences. This connection leads to an extension of two classical criteria of Hurwitz stability for real polynomials to matrix polynomials: tests for Hurwitz stability via positive definiteness of block-Hankel matrices built from matricial Markov parameters and via matricial Stieltjes continued fractions. We obtain further conditions for Hurwitz stability in terms of block-Hankel minors and quasiminors, which may be viewed as a weak version of the total positivity criterion.