A Trajectory-Based Approach to Controlled Invariance and Recursively Feasible MPC

TL;DR

This paper introduces a trajectory-based method for computing controlled invariant sets in linear systems, enabling recursive feasible MPC without precomputed terminal sets.

Contribution

It presents a new characterization of controlled invariance using convex feasible points and integrates this with a fixed-point algorithm for invariant set computation.

Findings

Successfully computes maximal controlled invariant sets using the proposed method.

Guarantees recursive feasibility in MPC without precomputed terminal sets.

Provides a practical optimization-based approach for constructing invariant sets.

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 for the computation of the maximal controlled invariant set. Building on these results, we propose a model predictive control (MPC) scheme that guarantees 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.