On Stability of Switched Discrete-Time Singular Systems

In this article, we prove multiple criteria for the stability of switched discrete-time linear singular (SDLS) systems. First, we show that if the Lie algebra generated by the flow matrices associated with an SDLS system, consisting of stable subsystems, is solvable, then the SDLS system is globally uniformly exponentially stable. Most results in the literature are based on commutativity and the Lie algebraic results of this note generalize the existing results. Furthermore, using the first result, we prove a Lie algebraic criterion involving the system matrices. We also prove a Lyapunov function-based sufficient condition for the exponential stability of SDLS systems and show that this result is equivalent to the existing Lyapunov function-based sufficient condition in the literature. Using this result, we show that an SDLS system with a common descriptor matrix satisfying the Lie algebraic criterion admits a common quadratic Lyapunov function. Finally, we extend the commutativity-based result for SDLS systems involving two subsystems and a common descriptor matrix in the literature to SDLS systems involving finitely many, but arbitrary number of subsystems.