Control of discrete-time periodic linear systems with input saturation via multi-step periodic invariant set

This paper studies local control of discrete-time periodic linear systems subject to input saturation by using the multi-step periodic invariant set approach. Multi-step periodic invariant set refers to a set from which all trajectories will enter a periodic invariant set after finite steps, remain there forever, and eventually converge to the origin as time approaches infinity. A couple of problems including the (robust) estimation of domain of attraction, (robust) local stabilization, and disturbance rejection are considered. Compared with the conventional periodic invariant set approach which has been used in the literature for local stability and stabilization of discrete-time periodic linear systems subject to input saturation, this new invariant set approach is capable of significantly reducing the conservatism by introducing more auxiliary variables in the set invariance conditions. Moreover, the new approach allows to design (robust) stabilizing periodic controller whose period is the same as the open-loop system, which is different from the existing periodic enhancement approach by which the period of the controller is multiple times of the period of the open-loop system. Several numerical examples are worked out to show the effectiveness of the proposed approach.