Asymptotic Stability of Piecewise Affine Systems With Filippov Solutions via Discontinuous Piecewise Lyapunov Functions

Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. In this article, the feasible Filippov solution concept is introduced by characterizing single-mode Caratheodory, sliding mode, and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable, and Caratheodory boundaries. It is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. Several examples illustrate the theoretical arguments and the effectiveness of the stability result.

I. Introduction

Lyapunov theory has been widely used for the asymptotic stability analysis of continuous-time piecewise affine (PWA) systems defined over a polyhedral partition of the state space [1], [2]. When the vector fields are not continuous on the boundaries, which is the case considered in this article, the stability problem becomes more challenging due to the possible occurrence of sliding mode and Zeno behaviors [3], [4]. For this class of discontinuous systems, to find a global Lyapunov function is a nontrivial issue [5]–[7] and its existence is not ensured either [8], [9].