Controlled invariant sets: implicit closed-form representations and applications

TL;DR

This paper introduces a novel implicit, closed-form approach for computing controlled invariant sets in discrete-time linear systems, enabling efficient handling of high-dimensional systems and applications in safety-critical scenarios.

Contribution

It proposes an implicit, closed-form representation for controlled invariant sets, allowing single-step computation and improved scalability over traditional iterative methods.

Findings

Efficient computation of invariant sets in high-dimensional systems.

Implicit representation suffices for safety-critical applications.

Method is complete without disturbances, with partial guarantees under disturbances.

Abstract

We revisit the problem of computing (robust) controlled invariant sets for discrete-time linear systems. Departing from previous approaches, we consider implicit, rather than explicit, representations for controlled invariant sets. Moreover, by considering such representations in the space of states and finite input sequences we obtain closed-form expressions for controlled invariant sets. An immediate advantage is the ability to handle high-dimensional systems since the closed-form expression is computed in a single step rather than iteratively. 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. The proposed method is complete in the absence of disturbances, and we provide a weak completeness result when disturbances are present.