Model Predictive Control with Multiple Constraint Horizons

Abstract

We propose a Model Predictive Control (MPC) formulation for nonlinear systems without terminal penalty or dedicated stabilizing terminal set, in which state constraints are enforced heterogeneously along the prediction horizon. In this setting a control-invariant set certifies near-term safety, while a less restrictive set constrains later predictions. This structure is motivated by safety-critical formulations, such as Control Barrier Function (CBF) based MPC, collision avoidance, and robotic receding-horizon planning, where near-term predictions must be certified safe and later predictions can be re-certified in future updates. We develop a value-function-difference analysis separating the effects of constraint-set selection and prediction-horizon length on closed-loop performance bounds, yielding implicit suboptimality certificates. Assuming cost-controllability, we further propose an upper-bound certificate which accounts explicitly for distinct constraint sets, their associated decay rates, and horizons. We also provide a lower-bound certificate for the closed-loop cost beyond the finite-horizon open-loop cost, which, for the admissible parameter range, is not weaker than the standard finite-horizon bound. Simulations on linear and nonlinear safety-critical systems demonstrate the proposed certificates \textit{a priori} and \textit{a posteriori}.