Qualitative analysis of switched discrete-time descriptor systems

In this paper, we consider stability analysis and design for switched systems consisting of linear discrete-time descriptor subsystems. When all descriptor subsystems are stable, we show that if the descriptor matrix and all the subsystem matrices are commutative pairwise, then the switched system is stable under arbitrary switching. We also extend the result to the case where all subsystems have different descriptor matrices. Under the same commutation condition, we show that in the case where none of the descriptor subsystems is stable, if there is a stable combination of the unstable descriptor subsystems, then we establish a class of switching laws which stabilize the switched descriptor system. All the results are natural but important extensions to the existing results for switched systems composed of state space subsystems.