Stability and Invariance Analysis of Uncertain Discrete-Time Piecewise Affine Systems

This note proposes a method to analyze uniform asymptotic stability and uniform ultimate boundedness of uncertain piecewise affine systems whose dynamics are only defined in a bounded and possibly non-invariant set chi of states. The approach relies on introducing fake dynamics outside chi and on synthesizing a piecewise affine and possibly discontinuous Lyapunov function via linear programming. The existence of such a function proves stability properties of the original system and allows the determination of a region of attraction contained in chi. The procedure is particularly useful in practical applications for analyzing the stability of piecewise affine control systems that are only defined over a bounded subset chi of the state space, and to determine whether for a given set of initial conditions the trajectories of the state vector remain within the domain chi.