Kronecker weights for instability analysis of Markov jump linear systems
For control theorists and engineers, this provides a first systematic method to certify instability in Markov jump linear systems, filling a methodological gap.
This paper introduces a novel criterion for verifying exponential mean instability in continuous-time Markov jump linear systems, addressing a gap where stability criteria exist but instability verification lacks methods. The approach uses Kronecker products with matrix weights and transforms weight optimization into spectral optimization, validated through numerical examples.
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.