L2-gain Analysis for a Class of Hybrid Systems with Applications to Reset and Event-triggered Control A Lifting Approach

Abstract: 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  £2-sense  (meaning  that  the  £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 L2-sense is the same as stability and contractivity in 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 £2-gain analysis of a discrete-time 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 £2-gain (and thus L2-gain) conditions, despite the fact that the linearity assumption, which is usually needed in the lifting literature, is not satisfied.