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 L₂-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 L₂-gain analysis methods. Inspired by ideas from lifting we show that the stability and the contractivity in L₂-sense (meaning that the L₂-gain is smaller than 1) of the continuous-time hybrid system is equivalent to the stability and the contractivity in l₂-sense (meaning that the l₂-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 L₂-sense is the same as stability and contractivity in l₂-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 l₂-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 l₂-gain (and thus L₂-gain) conditions, despite the fact that the linearity assumption, which is usually needed in the lifting literature, is not satisfied.