ℒ₂ gain analysis for switched systems with continuous-time and discrete-time subsystems

In this paper, we study ℒ₂ 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 ℒ₂ gain less than 7, we discuss the ℒ₂ gain that the switched system could achieve. First, we consider the case where a common Lyapunov function exists for all subsystems in ℒ₂ sense, and show that the switched system has the ℒ₂ gain less than the same level 7 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 ℒ₂ sense, and show that the switched system achieves an ultimate (or weighted) ℒ₂ gain under an average dwell time scheme.