Polyhedral regions of local stability for linear discrete-time systems with saturating controls

The determination of polyhedral regions of local stability for linear systems subject to control saturation is addressed. The analysis of the nonlinear behavior of the closed-loop saturated system is made by dividing the state space in regions of saturation. Inside each of these regions, the system evolution can be represented by a perturbed linear system. From this representation, a necessary and sufficient algebraic condition relative to the positive invariance of a polyhedral set is given. In a second stage, a necessary and sufficient condition to the contractivity of such a positively invariant set is stated. Consequently, the polyhedral set can be associated to a Lyapunov function and the local asymptotic stability of the saturated closed-loop system inside the set is guaranteed. An algorithm based on linear programming is proposed to generate homothetic expansions of a positively invariant and contractive polyhedral set w.r.t. closed-loop saturated system