L/sub 2/ gain analysis for switched systems with continuous-time and discrete-time subsystems

In this paper, we study L/sub 2/ 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 L/sub 2/ gain less than /spl gamma/, we discuss the L/sub 2/ gain that the switched system could achieve. First, we consider the case where a common Lyapunov function exists for all subsystems in L/sub 2/ sense, and show that the switched system has the L/sub 2/ gain less than the same level /spl gamma/ under arbitrary switching. As an example in this case, we analyze switched symmetric systems and derive the common Lyapunov function clearly. Next, we use a piecewise Lyapunov function approach to study the case where no common Lyapunov function exists in L/sub 2/ sense, and show that the switched system achieves an ultimate (or weighted) L/sub 2/ gain under an average dwell time scheme.