Controlled invariant sets: implicit closed-form representations and applications

In this paper we revisit the problem of computing robust controlled invariant sets for discrete-time linear systems. The key idea is that by considering controllers that exhibit eventually periodic behavior, we obtain a closed-form expression for an implicit representation of a robust controlled invariant set in the space of states and finite input sequences. Due to the derived closed-form expression, our method is suitable for high dimensional systems. Optionally, one obtains an explicit robust controlled invariant set by projecting the implicit representation to the original state space. The proposed method is complete in the absence of disturbances, with a weak completeness result established when disturbances are present. Moreover, we show that a specific controller choice yields a hierarchy of robust controlled invariant sets. To validate the proposed method, we present thorough case studies illustrating that in safety-critical scenarios the implicit representation suffices in place of the explicit invariant set.