Contractive polyhedra for linear continuous-time systems with saturating controls

The study of the contractivity properties of polyhedral sets with respect to (w.r.t) 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. In each one of these regions, the system evolution can be represented by one of a linear system with an additive disturbance. From this representation, a necessary and sufficient algebraic condition for the contractivity of a polyhedral set w.r.t the control saturating system is stated. Consequently, a Lyapunov function can be associated with the polyhedral set and the local asymptotic stability of the closed-loop system inside the set is guaranteed. The conditions for polyhedral contractivity are used to state an algorithm, based on linear programming, to compute polyhedral regions of local asymptotic stability for the closed-loop saturated system.