Safety-Critical Control with Guaranteed Lipschitz Continuity via Filtered Control Barrier Functions

TL;DR

Filtered Control Barrier Functions (FCBFs) are introduced, which extend HOCBFs by incorporating an auxiliary dynamic system-referred to as an input regularization filter-to produce Lipschitz continuous control inputs and simulations on a unicycle model demonstrate the effectiveness of the proposed method compared to standard and smoothness-penalized HOCBF approaches.

Abstract

In safety-critical control systems, ensuring both system safety and smooth control input is essential for practical deployment. Existing Control Barrier Function (CBF) frameworks, especially High-Order CBFs (HOCBFs), effectively enforce safety constraints, but also raise concerns about the smoothness of the resulting control inputs. While smoothness typically refers to continuity and differentiability, it does not by itself ensure bounded input variation. In contrast, Lipschitz continuity is a stronger form of continuity that not only is necessary for the theoretical guarantee of safety, but also bounds the rate of variation and eliminates abrupt changes in the control input. Such abrupt changes can degrade system performance or even violate actuator limitations, yet current CBF-based methods do not provide Lipschitz continuity guarantees. This paper introduces Filtered Control Barrier Functions (FCBFs), which extend HOCBFs by incorporating an auxiliary dynamic system-referred to as an input regularization filter-to produce Lipschitz continuous control inputs. The proposed framework ensures safety, control bounds, and Lipschitz continuity of the control inputs simultaneously by integrating FCBFs and HOCBFs within a unified quadratic program (QP). Theoretical guarantees are provided and simulations on a unicycle model demonstrate the effectiveness of the proposed method compared to standard and smoothness-penalized HOCBF approaches.

Figures (4)

• Figure 1: Closed-loop trajectories with controllers derived using FCBF (magenta), HOCBF (blue) and sp-HOCBF (red). FCBF ($\alpha=1$) and HOCBF perform well in safety-critical navigation when the initial heading angle is small. FCBF produces smoother trajectories, demonstrates strong adaptability, and maintains feasibility across a range of hyperparameters.
• Figure 2: Control input $u_{1}$ (angular velocity) over time with different controllers. FCBF ($k_{3}=\alpha=1$) ensures smoother transitions of $u_{1}$ compared to HOCBF and sp-HOCBF.
• Figure 3: Control input $u_{2}$ (driven force) over time with different controllers. FCBF ($k_{3}=\alpha=1$) ensures smoother transitions of $u_{2}$ compared to HOCBF and sp-HOCBF.
• Figure 4: FCBF ($k_{3}=1$, $\tau=2\cdot 10^{-3}$) with different class $\kappa$ function hyperparameter ($\alpha$) is evaluated. Smaller $\alpha$ tends to promote smoother variations in $\boldsymbol{u}$.

Theorems & Definitions (19)

• Definition 1: Lipschitz continuity rockafellar1998variational
• Definition 2: Class $\kappa$ function Khalil:1173048
• Definition 3
• Definition 4
• Lemma 1: Comparison Lemma, e.g., Khalil:1173048
• Definition 5: HOCBF xiao2021high
• Theorem 1: Safety Guarantee xiao2021high
• Definition 6: CLF ames2012control
• Definition 7: FCBF
• Theorem 2: Safety Guarantee