Exponential stability and hybrid [formula omitted]-gain analysis of hybrid systems via max functions

Abstract

This paper investigates exponential stability and hybrid L2h-gain performance for a class of hybrid systems. A novel Lyapunov function composed of two max functions, one describing the flow-storage function and the other describing the jump-storage function, is presented to analyze these problems. Due to the effect of flow set, jump set and a timer, the novel Lyapunov function is allowed to increase along flows with a bounded increasing rate and is required to decrease to compensate the possible increasing energy, which relax the constraint on non-increasing requirement of existing common quadratic functions in literature. Then, a stability criterion of hybrid systems is obtained, which can also be used to judge the stability of hybrid systems with unstable flow behaviors. Furthermore, this paper studies the effect of two external disturbances in flows and jumps of hybrid systems, while most literature only study external disturbances existing in flows. The concept of hybrid L2h-gain is first presented to hybrid systems. By analyzing hybrid L2h-gain performance of hybrid systems, one can obtain that L2-gain performance for flows or l2-gain performance for jumps can be improved by degrading the other one. Finally, two numerical examples are provided to illustrate the effectiveness and reliability of the results.