Explicit Control Barrier Function-based Safety Filters and their Resource-Aware Computation

This paper studies the efficient implementation of safety filters that are designed using control barrier functions (CBFs), which minimally modify a nominal controller to render it safe with respect to a prescribed set of states. Although CBF-based safety filters are often implemented by solving a quadratic program (QP) in real time, the use of off-the-shelf solvers for such optimization problems poses a challenge in applications where control actions need to be computed efficiently at very high frequencies. In this paper, we introduce a closed-form expression for controllers obtained through CBF-based safety filters. This expression is obtained by partitioning the state-space into different regions, with a different closed-form solution in each region. We leverage this formula to introduce a resource-aware implementation of CBF-based safety filters that detects changes in the partition region and uses the closed-form expression between changes. We showcase the applicability of our approach in examples ranging from aerospace control to safe reinforcement learning.

💡 Research Summary

This paper addresses a fundamental bottleneck in the deployment of control‑barrier‑function (CBF) safety filters: the need to solve a quadratic program (QP) at every control update. While CBF‑based safety filters are widely used because they minimally modify a nominal controller to guarantee forward invariance of a safe set, the real‑time solution of the associated QP can be computationally prohibitive in high‑frequency control loops, safety‑critical aerospace software, or large‑scale reinforcement‑learning (RL) training that runs on GPUs.

The authors propose an explicit, closed‑form representation of the optimal control law that results from the CBF‑QP. By introducing the notation (b_i(x) = -L_g h_i(x)^\top) and (a_i(x) = -L_f h_i(x) - \alpha(h_i(x))), each CBF constraint becomes an affine inequality in the control input: (b_i(x)^\top u + a_i(x) \le 0). For any candidate active‑set (I \subseteq {1,\dots,p}) they derive the Lagrange multipliers
\