A Graphical Model Approach for Switched System with Switch Digraphs

This paper presents a graphical model approach for the stability check of switched systems with switch digraphs. To specify the restricted switching, we introduce the concepts of path-dependent average dwell time (PDADT) and path-dependent average switching frequency (PDASF), which identify three classes of graphical models. First, a complete switch digraph depicting the restricted switching is presented based on the PDADT. Together with the PDASF, a fundamental switch digraph is then developed originally, which is a subgraph of the complete switch digraph and rules out the essential switching paths. Finally, a clustering-based switch digraph can be further constructed by considering the periodic property of the fundamental switch digraph. Then, we derive stability criteria for a switched system under a set of time-dependent switching signals that obey the given switch digraph. By contrast to the existing results encapsulating all the possible inequality conditions, we provide a particular set of inequality conditions to guarantee the stability of the constrained switched system. A numerical example is given to illustrate the analysis.