gain analysis for switched systems with continuous-time and discrete-time subsystems

In this paper, we study gain property for a class of switched systems which are composed of both continuous-time LTI subsystems and discrete-time LTI subsystems. Under the assumption that all subsystems are Hurwitz/Schur stable and have the gain less than γ, we discuss the gain that the switched system could achieve. First, we consider the case where a common Lyapunov function exists for all subsystems in sense, and show that the switched system has the gain less than the same level γ under arbitrary switching. As an example in this case, we analyse switched symmetric systems and establish the common Lyapunov function explicitly. Next, we use a piecewise Lyapunov function approach to study the case where no common Lyapunov function exists in sense, and show that the switched system achieves an ultimate (or weighted) gain under an average dwell time scheme.