Auxiliary-Variable Adaptive Control Barrier Functions for Safety Critical Systems

TL;DR

This work tackles safety guarantees for systems with time-varying control bounds by improving the feasibility of the CBF-QP framework. It introduces Auxiliary-Variable Adaptive CBFs (AVCBFs), which place auxiliary variables in front of each CBF constraint and propagate their dynamics through auxiliary systems to maintain positive gains, preserving classical CBF structure while expanding feasible control space. The approach retains adaptivity without extensive hyperparameter tuning and reduces overshoot near safety boundaries, demonstrated on an Adaptive Cruise Control problem with time-varying bounds and compared to PACBFs. The results indicate AVCBFs provide safer, smoother, and more adaptable control under varying road conditions, with potential broad applicability to safety-critical systems beyond ACC.

Abstract

This paper studies safety guarantees for systems with time-varying control bounds. It has been shown that optimizing quadratic costs subject to state and control constraints can be reduced to a sequence of Quadratic Programs (QPs) using Control Barrier Functions (CBFs). One of the main challenges in this method is that the CBF-based QP could easily become infeasible under tight control bounds, especially when the control bounds are time-varying. The recently proposed adaptive CBFs have addressed such infeasibility issues, but require extensive and non-trivial hyperparameter tuning for the CBF-based QP and may introduce overshooting control near the boundaries of safe sets. To address these issues, we propose a new type of adaptive CBFs called Auxiliary-Variable Adaptive CBFs (AVCBFs). Specifically, we introduce an auxiliary variable that multiplies each CBF itself, and define dynamics for the auxiliary variable to adapt it in constructing the corresponding CBF constraint. In this way, we can improve the feasibility of the CBF-based QP while avoiding extensive parameter tuning with non-overshooting control since the formulation is identical to classical CBF methods. We demonstrate the advantages of using AVCBFs and compare them with existing techniques on an Adaptive Cruise Control (ACC) problem with time-varying control bounds.

Figures (4)

• Figure 1: Control input $u(t)$ varies as $b(\boldsymbol{x(t)})$ goes to 0 under different lower control bounds. The arrows denote the changing trend for $b(\boldsymbol{x(t)})$ and $c_{d}(t)$ over time. $b(\boldsymbol{x(0)})=90$ and $b(\boldsymbol{x(t)})\ge 0$ implies safety. Hyperparameters are set as $k_{1}=k_{2}=l_{1}=l_{2}=0.1, a_{1,w}=1,T=50s.$
• Figure 2: Control input $u(t)$ varies as $b(\boldsymbol{x(t)})$ goes to 0 under different lower control bounds. The arrows denote the changing trend for $b(\boldsymbol{x(t)})$ and $c_{d}(t)$ over 50 seconds. $b(\boldsymbol{x(0)})=90$ and $b(\boldsymbol{x(t)})\ge 0$ implies safety. Different sets of hyperparameters for class $\kappa$ functions are tested.
• Figure 3: Control input $u(t)$ varies as $b(\boldsymbol{x(t)})$ goes to 0 under different lower control bounds. The arrows denote the changing trend for $b(\boldsymbol{x(t)})$ and $c_{d}(t)$ over 50 seconds. $b(\boldsymbol{x(0)})=90$ and $b(\boldsymbol{x(t)})\ge 0$ implies safety. Solid curves denote AVCBFs and dashed curves denote PACBFs.
• Figure 4: Control input $u(t)$, velocity $v(t)$, time-varying $p_{1}(t)$ and distance between two vehicles $b(\boldsymbol{x(t)})$ over 30 seconds for AVCBFs and PACBFs. $b(\boldsymbol{x(t)})\ge 0$ implies safety. Solid curves denote AVCBFs and dashed curve denotes PACBFs.

Theorems & Definitions (13)

• Definition 1: Class $\kappa$ function Khalil:1173048
• Definition 2
• Definition 3
• Definition 4: HOCBF xiao2021high
• Theorem 1: Safety Guarantee xiao2021high
• Definition 5: CLF ames2012control
• Remark 1
• Definition 6: AVCBF
• Theorem 2
• proof