Interpolating Control with Periodic Invariant Sets

This paper presents a novel low-complexity interpolating control scheme involving periodic invariance or vertex reachability of target sets for the constrained control of LTI systems. Periodic invariance relaxes the strict one-step positively invariant set notion, by allowing the state trajectory to leave the set temporarily but return into the set in a finite number of steps. To reduce the complexity of the representation of the required controllable invariant set, a periodic invariant set is employed. This set should be defined within the controllable stabilising region, which is considered unknown during the design process. Since periodic invariant sets are not traditional invariant sets, a reachability problem can be solved off-line for each vertex of the outer set to provide an admissible control sequence that steers the system state back into the original target set after a finite number of steps. This work develops a periodic interpolating control (pIC) scheme between such periodic invariant sets and a maximal admissible inner set by means of an inexpensive linear programming problem, solved on-line at the beginning of each periodic control sequence. Theorems on recursive feasibility and asymptotic stability of the pIC are given. A numerical example demonstrates that pIC provides similar performance compared to more expensive optimization-based schemes previously proposed in the literature, though it employs a naive representation of the controllable invariant set.