Stability analysis of piecewise affine systems with sliding modes

This paper proposes new sufficient conditions for stability analysis of Piecewise Affine (PWA) systems. The conditions are based on a convex combination of Piecewise Quadratic (PWQ) Lyapunov functions and are given in terms of Linear Matrix Inequalities (LMIs), which can be solved efficiently using available software packages. There are three contributions of the new conditions presented in this paper. First, the conditions guarantee exponential stability of the state dynamics even in the presence of non-destabilizing sliding modes of all possible dimensions smaller than the dimension of the state space. Second, the conditions can handle the important case where the equilibrium point is located at a boundary between affine subsystems. Third, sufficient conditions for stability of systems independently of the parametrization of the boundary surfaces are derived as a corollary. The new method presented in this paper leads to a unified methodology for stability analysis of switched affine systems and piecewise affine systems with sliding modes.