<inline-formula> <tex-math notation="LaTeX">$\mathcal{L}_{2}$</tex-math> </inline-formula>-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 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.