Positive trigonometric polynomials for strong stability of difference equations

TL;DR

This paper presents a polynomial-based method using Hermite stability criterion and trigonometric polynomial matrices to analyze strong stability of linear difference equations with multiple delays, employing LMIs for certification.

Contribution

It introduces a converging hierarchy of LMIs to efficiently certify strong stability for systems with up to 4 or 5 delays, advancing computational methods in delay system analysis.

Findings

Certificates of strong stability are obtainable with reasonable computational effort.

The method is effective for systems with up to 4 or 5 delays.

Numerical experiments validate the approach's practicality.

Abstract

We follow a polynomial approach to analyse strong stability of linear difference equations with rationally independent delays. Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5.