Robust stability of polytopic time-inhomogeneous Markov jump linear systems

The transition probabilities of jumps between operational modes of discrete-time Markov(ian) jump linear systems (dtMJLSs) are generally considered to be time-invariant, certain, and often completely known in the majority of dedicated studies. Still, in most real cases the transition probability matrix (TPM) cannot be computed exactly and is time-varying. In this article, we take into account the uncertainty and time-variance of the jump parameters by considering the underlying Markov chain as polytopic and time-inhomogeneous, i.e., its TPM is varying over time with variations that are arbitrary within a polytopic set of stochastic matrices. We show that the conditions used for time-homogeneous dtMJLSs are not enough to ensure the stability of the time-inhomogeneous system, and that perturbations on values of the TPM can make a stable system unstable. We present necessary and sufficient conditions for mean square stability (MSS) of polytopic time-inhomogeneous dtMJLSs, prove that deciding MSS on such systems is NP-hard and that MSS is equivalent to exponential MSS and to stochastic stability. We also derive necessary and sufficient conditions for robust MSS of dtMJLSs affected by polytopic uncertainties on transition probabilities and bounded disturbances.