Almost sure stability of discrete-time switched linear systems

In this paper, we study the stability of discrete-time switched linear systems via symbolic topology formulation and the multiplicative ergodic theorem. A sufficient and necessary condition for μA-almost sure stability is derived, where μA is the Parry measure of the topological Markov chain with a prescribed transition (0,1)-matrix A. The obtained μA-almost sure stability is invariant under small perturbations of the system. The topological description of stable processes of switched linear systems in terms of Hausdorff dimension is given, and it is shown that our approach captures the maximal set of stable processes for linear switched systems. The obtained results cover the stochastic Markov jump linear systems, where the measure is the natural Markov measure defined by the transition probability matrix. We further show that if the switched system is periodically switching stable, then (i) it is almost sure exponentially stable for any Markov probability measures; (ii) the set of stable switching sequences has the same Hausdorff dimension as the one for the entire set of switching sequences.