L2-Gain Analysis for a Class of Hybrid Systems With Applications to Reset and Event-Triggered Control: A Lifting Approach

In this paper we study the stability and L-2-gain properties of a class of hybrid systems that exhibit linear flow dynamics, periodic time-triggered jumps and arbitrary nonlinear jump maps. This class of hybrid systems is relevant for a broad range of applications including periodic event-triggered control, sampled-data reset control, sampled-data saturated control, and certain networked control systems with scheduling protocols. For this class of continuous-time hybrid systems we provide new stability and L-2-gain analysis methods. Inspired by ideas from lifting we show that the stability and the contractivity in L-2-sense (meaning that the L-2-gain is smaller than 1) of the continuous-time hybrid system is equivalent to the stability and the contractivity in l(2)-sense (meaning that the l(2)-gain is smaller than 1) of an appropriate discrete-time nonlinear system. These new characterizations generalize earlier (more conservative) conditions provided in the literature. We show via a reset control example and an event- triggered control application, for which stability and contractivity in L-2-sense is the same as stability and contractivity in l(2)-sense of a discrete-time piecewise linear system, that the new conditions are significantly less conservative than the existing ones in the literature. Moreover, we show that the existing conditions can be reinterpreted as a conservative l(2)-gain analysis of a discretetime piecewise linear system based on common quadratic storage/Lyapunov functions. These new insights are obtained by the adopted lifting-based perspective on this problem, which leads to computable l(2)-gain (and thus L-2-gain) conditions, despite the fact that the linearity assumption, which is usually needed in the lifting literature, is not satisfied.