A Trajectory-based Approach to the Computation of Controlled Invariants with application to MPC

Abstract: In this paper, we revisit the computation of controlled invariant sets for linear discrete-time systems through a trajectory-based viewpoint. We begin by introducing the notion of convex feasible points, which provides a new characterization of controlled invariance using finitely long state trajectories. We further show that combining this notion with the classical backward fixed-point algorithm allows us to compute the maximal controlled invariant set. Building on these results, we propose two MPC schemes that guarantee recursive feasibility without relying on precomputed terminal sets. Finally, we formulate the search for convex feasible points as an optimization problem, yielding a practical computational method for constructing controlled invariant sets. The effectiveness of the approach is illustrated through numerical examples.