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Safety and invariance for constrained switching systems

We study discrete time linear switching systems subject to additive disturbances. We consider two types of constraints, namely on the states and on the switching signal. A switching sequence is admissible if it is accepted by an automaton. Contrary to the arbitrary switching case, stability does not imply the existence of an invariant1 set. In this article, we propose a generalization of a bounded invariant set, namely, the notion of an invariant multi-set and show its significance in terms of dynamical systems. Under standard assumptions, we provide an iterative algorithm to approximate the minimal invariant multi-set with a guarantee of accuracy and an algorithm to compute the maximal invariant multi-set. Application of the established framework to switching systems with minimum dwell time reveals potential computational benefits and allows formulations of more refined notions.