Controller Design For Hybrid Systems With Nonlinear Sub-Systems Using Common Lyapunov Functions

Abstract

Hybrid systems are common in real life and have been studied in many different fields. However, due to the interaction of different sub-systems, a hybrid system is much more complex than a mono-dynamic one, where great challenges are confronted when seeking stable controllers either for linear or nonlinear systems. The traditional design scheme for such a system, namely, to design a controller for each sub-system separately, cannot yield satisfactory performance, since the switches between sub-systems are not specifically considered. In fact, the controllers constructed in this way are usually of poor performance, or even unstable, as a disastrous effect caused by the alternation of sub-systems. In this paper, considering the aforementioned problem, a control scheme is proposed to design suitable controllers for hybrid systems to achieve overall stabilization by taking account of the switching behaviors of the system, as well as its sub-systems carefully. The design scheme consists of two steps, wherein the first step aims to design sub-controllers for different sub-systems, usually with different Lyapunov functions, while a common Lyapunov function candidate is composed in the second step to modify the previously designed sub-controllers correspondingly. Following this design scheme, not only the stability, but also the performance of the closed-loop hybrid systems, is successfully guaranteed. Some simulation results are provided to show the satisfactory performance of the proposed design scheme.