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

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.

💡 Research Summary

The paper addresses a critical gap in safety‑critical control: while High‑Order Control Barrier Functions (HOCBFs) guarantee forward invariance of safe sets, they do not ensure that the resulting control input varies smoothly in a bounded manner. Abrupt changes in the input can cause actuator saturation, wear, and even jeopardize the safety guarantees that rely on Lipschitz continuity of the closed‑loop dynamics. To close this gap, the authors introduce Filtered Control Barrier Functions (FCBFs), a novel construct that augments the standard HOCBF framework with an auxiliary dynamical system acting as an input regularization filter.

The auxiliary filter is defined by (\dot \pi = F(\pi) + G(\pi)\nu), where the filter state (\pi) stacks the control input and its successive time derivatives: (\pi_1 = u_f), (\pi_2 = \dot u_f), …, (\pi_{m_a}=u_f^{(m_a-1)}). The final component of the filter, (\nu), is the new decision variable supplied by the quadratic program (QP). By embedding the original HOCBF safety condition (\psi_m(x,u)\ge 0) into the filter’s dynamics as (\psi_{0,f}(x,\pi)=\psi_m(x,u_f)), the authors construct a hierarchy of functions (\psi_{i,f}) (i = 1 … (m_a)) using the same class‑(\kappa) functions as in standard HOCBFs. Each (\psi_{i,f}) yields a set (C_{i,f}={(x,\pi)\mid \psi_{i,f}\ge0}). The collection ({C_{0,f},\dots,C_{m_a-1,f}}) defines the filtered barrier constraints.

The key insight is that enforcing (\psi_{i,f}\ge0) for all i guarantees that the filtered input (u_f) is Lipschitz continuous, because the filter dynamics are locally Lipschitz and the class‑(\kappa) functions bound the growth of each (\psi_{i,f}). Consequently, the closed‑loop system satisfies the existence‑and‑uniqueness conditions required for safety proofs. Moreover, the Lipschitz constant can be tuned by adjusting the gains of the class‑(\kappa) functions, providing a design knob to trade off smoothness against conservatism.

All safety, control‑limit, and performance objectives are merged into a single QP:

• Objective: minimize a quadratic cost on the (unfiltered) control effort or energy.
• Soft constraints: a Control Lyapunov Function (CLF) for convergence to a desired equilibrium.
• Hard constraints: the original HOCBF inequality for the safety set, and the FCBF inequalities (\psi_{i,f}\ge0) that enforce input bounds and Lipschitz continuity.

Because the filter state (\pi) is treated as part of the augmented system, the QP remains convex and can be solved at each sampling instant with the same computational burden as a standard CBF‑CLF QP. The authors provide a formal theorem stating that any solution of the QP yields a control input that is (i) admissible (within the prescribed bounds), (ii) guarantees forward invariance of the safe set, and (iii) is Lipschitz continuous with a known bound.

The theoretical development is illustrated on a unicycle model with constant forward speed and a steering input (u). The safety requirement is to avoid a circular obstacle, which has relative degree two with respect to the dynamics. A conventional HOCBF leads to a control law that can exhibit large spikes when the vehicle passes near the obstacle. By applying the FCBF approach with a second‑order filter ((m_a=2)), the authors obtain a filtered steering command whose variation is dramatically reduced (approximately 70 % lower peak rate) while still satisfying the obstacle avoidance constraint. Comparative simulations also include a “smoothness‑penalized” HOCBF variant; the FCBF method outperforms it both in terms of reduced input variation and in preserving safety.

In summary, the paper makes three substantive contributions:

1. It identifies the lack of Lipschitz continuity guarantees in existing CBF‑based safety controllers as a practical and theoretical limitation.
2. It proposes a systematic method—Filtered Control Barrier Functions—that embeds an input regularization filter into the barrier function framework, yielding provable Lipschitz‑continuous control inputs.
3. It demonstrates, both analytically and via simulation, that the method retains safety guarantees, respects input constraints, and produces significantly smoother control actions without sacrificing performance.

The approach is compatible with any system for which standard CBFs are applicable, making it a versatile addition to the safety‑critical control toolbox. Future work suggested includes automatic tuning of filter parameters, extension to stochastic disturbances, and experimental validation on real robotic platforms.