New results on the robustness of discrete-time Markov jump linear systems

This paper investigates necessary conditions for robustness of Markov jump linear systems. It is proven that the version of the small-gain theorem available in the current literature of this class of systems may sometimes yield an arbitrarily conservative robustness margin. Such conservatism, which does not exist in the classical linear time-invariant or linear stochastic scenarios, indicates a general lack of knowledge on how robust Markov jump linear systems can be, at least from the viewpoint of the scaling techniques currently available in the literature. A key step in the paper is the introduction of adjoint LMIs which, in the same situation, attain the maximal degree of robustness. In addition, a spectral approach is proposed and its effectiveness is investigated. An exact characterization for the stability radii is also given in the scalar case.