Robust Exponential Stability and Disturbance Attenuation for Discrete-Time Switched Systems Under Arbitrary Switching

In this note, the global exponential stability of discrete-time switched systems under arbitrary switching is investigated. First, for discrete-time switched nonlinear systems, the global exponential stability is found to be equivalent to the existence of an <inline-formula><tex-math notation="LaTeX">$M$</tex-math></inline-formula>-step sequence with sufficient length and a family of Lyapunov functions, and then a stability criterion is proposed for the nominal linear case in the framework of quadratic Lyapunov function. In order to extend the stability criterion to handle uncertainties, an equivalent condition which has a promising feature that is convex in system matrices is derived, leading to a robust stability criterion for uncertain discrete-time switched linear systems. Moreover, also taking the advantage of the convex feature, the disturbance attenuation performance in the sense of <inline-formula> <tex-math notation="LaTeX">$\ell _2$</tex-math></inline-formula>-gain is studied. Several numerical examples are provided to illustrate our approach.