We propose a lightweight extension to Control Barrier Function (CBF) safety filters that introduces a stateful relaxation variable to encode temporal feasibility stress. Unlike standard approaches where slack variables are recomputed independently at each optimisation step, the proposed method evolves relaxation as a dynamic state within the closed-loop system.This allows the controller to retain memory of constraint pressure and adapt projection aggressiveness over time. The result is a hybrid safety layer compatible with existing CBF-QP and MPC architectures.We establish: boundedness of the relaxation state under standard assumptions sufficient conditions for recursive feasibility practical stability interpretation under bounded disturbance The contribution is primarily structural: a minimal modification to existing constrained control pipelines that introduces temporal coupling in feasibility handling. 1. Introduction Constrained control methods such as: Model Predictive Control (MPC) Control Barrier Function (CBF) filters Reference governors enforce safety by solving optimisation problems at each timestep. These typically introduce instantaneous slack variables or constraint softening terms. However, such slack variables are: memoryless (do not persist across time) independently at each step unable to encode persistent feasibility stress Key ideaWe introduce a stateful relaxation variable ρₜ that evolves dynamically and influences constraint enforcement strength over time. This creates a lightweight memory mechanism for feasibility pressure.
Jason Crowe (Wed,) studied this question.