Rectified Control Barrier Functions for High-Order Safety Constraints

TL;DR

This work introduces Rectified Control Barrier Functions (ReCBFs) to synthesize CBF certificates for safety constraints with high or mixed relative degree. By rectifying the high-order terms with activation through a suitable ReLU-based mechanism, ReCBFs enforce safety only when necessary, addressing singularities and robustness issues that challenge traditional HOCBFs. The authors provide rigorous theory for relative degree 2, mixed-input, and arbitrary weak relative degree cases, and compare ReCBFs with HOCBFs and backstepping, illustrating superior well-definedness in a fixed-wing aircraft pitch example. The results offer a practical pathway to safer, more robust safety enforcement in nonlinear control systems with complex constraint structures.

Abstract

This paper presents a novel approach for synthesizing control barrier functions (CBFs) from high relative degree safety constraints: Rectified CBFs (ReCBFs). We begin by discussing the limitations of existing High-Order CBF approaches and how these can be overcome by incorporating an activation function into the CBF construction. We then provide a comparative analysis of our approach with related methods, such as CBF backstepping. Our results are presented first for safety constraints with relative degree two, then for mixed-input relative degree constraints, and finally for higher relative degrees. The theoretical developments are illustrated through simple running examples and an aircraft control problem.

Figures (5)

• Figure 1: Left: Safe set $\mathcal{S}$ induced by the HOCBF candidate from Example \ref{['ex:motivating-example']}, where the dashed black lines denote the boundary of the constraint set $\mathcal{C}$, the solid red curves denote the boundary of the safe set $\mathcal{S}$, the arrows denote the closed-loop vector field under the resulting quadratic programming-based controller (lighter arrows correspond to larger magnitude), the gray curves illustrate example closed-loop trajectories, and the gray dots denote the initial conditions of such trajectories. Right: Input generated by the resulting HOCBF controller for fixed values of $\dot{x}$ as $x$ is varied.
• Figure 2: Left: Safe set induced by ReCBF \ref{['eq:h-rel-deg-2']} for Example \ref{['ex:new-cbf']} (blue curve), where all other plot elements share the same interpretation as those in Fig. \ref{['fig:hocbf']}. Right: Safe set induced by the CBF \ref{['eq:h-Breeden']} for Example \ref{['ex:Breeden-cbf']} (green curve).
• Figure 3: Left: Safe set induced by the CBF \ref{['eq:backstepping-cbf']} for Example \ref{['ex:backstepping']} (purple curve), where all other plot elements share the same interpretation as those in Fig. \ref{['fig:hocbf']}. Right: Safe set induced by the CBF \ref{['eq:backstepping-cbf-new']} for Example \ref{['ex:backstepping']}.
• Figure 4: Evolution of the pitch angles for different controllers. The blue plot is induced by the ReCBF when used as a safety filter on a nominal control signal that seeks to track the pitch angle shown by the unsafe dotted green line. The red plot, induced by the HOCBF approach, does not have a valid solution after approximately 10.9 seconds.
• Figure 5: Input signals for the ReCBF and HOCBF approach for the aircraft example. The HOCBF input goes unbounded as $L_{\mathbf{g}}L_{\bf}\psi(\mathbf{x})\rightarrow0$.

Theorems & Definitions (17)

• Definition 1: AmesTAC17
• Lemma 1: jankovic2018robust
• Definition 2
• Definition 3: TanTAC22
• Definition 4
• Example 1: CohenARC24
• Theorem 1
• proof
• Example 2: Comparison to HOCBFs WeiTAC22
• Example 3: Comparison to BreedenAutomatica23