An NNth-step set invariance approach to the analysis and design of discrete-time linear systems subject to actuator saturation

For a discrete-time linear system subject to actuator saturation, an analysis and design approach based on NNth-step set invariance is proposed. An NNth-step invariant set is a set from which all trajectories will enter a contractively invariant set in NN steps and thus can be used as an estimate of the domain of attraction. Such an estimate is less conservative than an estimate obtained on the basis of contractive invariance of ellipsoids, which excludes initial conditions from which the trajectories “overshoot” outside the ellipsoid before converging to the origin. In comparison with the existing approach based on conventional set invariance, the proposed approach results in a larger domain of attraction and better performance, both in the analysis of the closed-loop system under a given feedback gain and in the design of feedback gain. In comparison with the saturated and nonsaturated (SNS) invariant set approach, which is an analysis approach, the proposed approach can be used for control design even in the presence of plant uncertainties.