Invariance and Contractivity of Polyhedra for Linear Continuous-time Systems with Saturating Controls

The study of the positive invariance and 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 the that of a linear system with an additive constant disturbance. From this representation, a sufficient algebraic condition relative to the positive invariance of a polyhedral set is given. In a second stage, using the same system representation, a necessary and sufficient condition to the contractivity of a compact polyhedral set with respect to the trajectories of the system stated. In this case, it is shown that a Lyapunov function can be associated with the polyhedral set and the local asymptotic stability of the saturated closed-loop system inside the set is guaranteed. From these results, an algorithm based on linear programming is proposed to determine such positively invariant and contractive polyhedral sets.