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

The study and the determination of polyhedral regions of local stability fo r linear systems subject to control saturation is addressed. The analysis o f the nonlinear behavior of the closed-loop saturated system is made by div iding the state space in regions of saturation. Inside each of these region s, the system evolution can be represented by a linear system with an addit ive disturbance. From this representation, a necessary and sufficient condi tion relative to the contractivity of a given convex compact polyhedral set is stated. Consequently, the polyhedral set can be associated with a Lyapu nov function and the local asymptotic stability of the saturated closed-loo p system inside the set is guaranteed. Furthermore, it is shown how, in som e particular cases, the compactness condition can be relaxed in order to en sure the asymptotic stability in unbounded polyhedra. Finally, an applicati on of the contractivity conditions is presented in order to determine local asymptotic stability regions for the closed-loop saturated system.