Stability and ℒ₂ Gain Analysis for Switched Symmetric Systems with Time Delay

In this paper, we study stability and ℒ₂ gain properties for a class of switched systems which are composed of a finite number of linear time-invariant symmetric systems with time delay. We show that when all subsystems are asymptotically stable in the sense of satisfying an LMI, the switched system is asymptotically stable under arbitrary switching. Furthermore, we show that when all subsystems are asymptotically stable and have the ℒ₂ gains γ in the sense of satisfying an LMI, the switched system is asymptotically stable and has the same ℒ₂ gain γ under arbitrary switching. The key idea for both stability and ℒ₂ gain analysis in this paper is to establish a common Lyapunov function for all subsystems in the switched system.