Stability analysis of switched linear systems defined by graphs

We present necessary and sufficient conditions for global exponential stability for switched discrete-time linear systems, under arbitrary switching, which is constrained within a set of admissible transitions. The class of systems studied includes the family of systems under arbitrary switching, periodic systems, and systems with minimum and maximum dwell time specifications. To reach the result, we describe the set of rules that define the admissible transitions with a weighted directed graph. This allows to express the system dynamics as a time invariant difference inclusion. In turn, a modified version of the forward reachability set mapping is utilized to analyze global exponential stability. The developed framework leads to the establishment of an iterative stability verification algorithm.