Constraint Horizon in Model Predictive Control

TL;DR

This paper introduces a two-horizon Model Predictive Control (MPC) framework that decouples the prediction horizon from the constraint horizon, enabling constraints to be enforced over only a portion of the horizon while preserving recursive feasibility and constraint satisfaction. It develops a suboptimality bound by relating the infinite-horizon cost $J_\infty^{N,\tilde{N}}(x)$ to the finite-horizon value function $V_N^{\tilde{N}}(x)$ through a relaxed dynamic programming argument, yielding an explicit bound parameter $\alpha = 1 - \frac{\beta^{N-\tilde{N}+1}}{(\beta+1)^{N-\tilde{N}-1}}$ that improves as the horizons widen or the constraint horizon tightens. The framework is extended to safety-critical control by incorporating Control Barrier Function (CBF) constraints into the horizon-based MPC, mitigating myopic safety behavior and resolving CLF-CBF compatibility issues via receding-horizon optimization. Numerical simulations on a safety-critical system (e.g., a double-integrator) validate the theoretical bounds and demonstrate that longer horizons or more extensive constraint enforcement yield costs closer to the open-loop bound and improved safety performance. Overall, the work provides a rigorous, horizon-aware methodology for analyzing and designing safe, suboptimal MPC controllers with explicit performance and stability guarantees.

Abstract

In this work, we propose a Model Predictive Control (MPC) formulation incorporating two distinct horizons: a prediction horizon and a constraint horizon. This approach enables a deeper understanding of how constraints influence key system properties such as suboptimality, without compromising recursive feasibility and constraint satisfaction. In this direction, our contributions are twofold. First, we provide a framework to estimate closed-loop optimality as a function of the number of enforced constraints. This is a generalization of existing results by considering partial constraint enforcement over the prediction horizon. Second, when adopting this general framework under the lens of safety-critical applications, our method improves conventional Control Barrier Function (CBF) based approaches. It mitigates myopic behaviour in Quadratic Programming (QP)-CBF schemes, and resolves compatibility issues between Control Lyapunov Function (CLF) and CBF constraints via the prediction horizon used in the optimization. We show the efficacy of the method via numerical simulations for a safety critical application.

Figures (3)

• Figure 1: Distinct horizons on MPC. Line coding: Single lines - system trajectories, double lines - system inputs. Color coding: Black - past actions, Blue - safe actions (within constraint horizon), Yellow - possibly unsafe actions (within prediction horizon but no constraints are enforced), Red - unsafe set, Green - safe set.
• Figure 2: Comparison between calculated and estimated closed loop costs according to \ref{['eq:optim_alpha']}.
• Figure 3: System trajectories for $N=6$ and different values of $N-\tilde{N}$. Circle and cross are start and endpoint respectively.

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Lemma 1
• Theorem 1
• Lemma 2
• proof
• proof
• Definition 1