Stability analysis and design for switched descriptor systems

In this paper, we consider stability analysis and design for switched systems consisting of linear descriptor systems that have the same descriptor matrix. When all descriptor systems are stable, we show that if the descriptor matrix and all the system matrices are commutative pairwise, then the switched system is stable under arbitrary switching. This is an extension of the existing well known result in [1] for switched linear systems with state space models to switched descriptor systems. Under the same commutation condition, we also show that in the case where all descriptor systems are not stable, if there is a stable convex combination of the unstable descriptor systems, then we can establish a class of switching laws which stabilize the switched system. We finally make some discussion about obtaining the stable convex combination by solving a matrix inequality efficiently, and about relaxing the commutation condition for stabilizability of the switched system.