Kronecker weights for instability analysis of Markov jump linear systems

TL;DR

This paper introduces a new criterion for analyzing the exponential mean instability of continuous-time Markov jump linear systems using Kronecker products and matrix weights, enabling more effective instability verification.

Contribution

It presents a novel instability criterion based on auxiliary systems and spectral optimization, filling a gap in existing stability analysis methods.

Findings

Effective instability verification method demonstrated through numerical examples.

Spectral optimization approach enables tighter instability bounds.

Auxiliary Markov jump linear systems facilitate analysis of system instability.

Abstract

In this paper, we analyze the instability of continuous-time Markov jump linear systems. Although there exist several effective criteria for the stability of Markov jump linear systems, there is a lack of methodologies for verifying their instability. In this paper, we present a novel criterion for the exponential mean instability of Markov jump linear systems. The main tool of our analysis is an auxiliary Markov jump linear system, which results from taking the Kronecker products of the given system matrices and a set of appropriate matrix weights. We furthermore show that the problem of finding matrix weights for tighter instability analysis can be transformed to the spectral optimization of an affine matrix family, which can be efficiently performed by gradient-based non-smooth optimization algorithms. We confirm the effectiveness of the proposed methods by numerical examples.