Robust Zero-Order Control Barrier Function

• The paper introduces a robust framework that guarantees safety by ensuring forward invariance of a safe set using Lyapunov-like zeroing barrier functions.
• The methodology integrates safety and performance by synthesizing control inputs via a convex quadratic program that couples control Lyapunov and barrier functions.
• The framework explicitly accounts for bounded disturbances by inflating the safe set and ensuring locally Lipschitz continuous controllers for practical safety-critical implementations.

A Robust Zero-Order Control Barrier Function (R-ZOCBF) is a mathematically rigorous framework for enforcing safety in control systems by guaranteeing the forward invariance of a constraint set, even in the presence of bounded model perturbations or exogenous disturbances. The R-ZOCBF architecture formalizes how safety requirements and robustness guarantees can be achieved simultaneously, typically as a hard constraint within a quadratic program (QP) that synthesizes the control input. The approach is grounded in a Lyapunov-like zeroing barrier function (ZBF), with explicit characterizations of robustness properties, stabilizability, and regularity (Lipschitz continuity) of the feedback law (Xu et al., 2016).

1. Mathematical Foundations: Zeroing Barrier Functions and Forward Invariance

A zeroing barrier function (ZBF) is a continuously differentiable function $h: \mathbb{R}^n \rightarrow \mathbb{R}$ defined on an open set $\mathcal{D}$, such that the safe set is $$\mathcal{C} = \{ x \in \mathbb{R}^n : h(x) \geq 0 \}$$. The ZBF property requires the existence of an extended class-$\mathcal{K}$ function $\alpha$ for which

$$L_f h(x) \geq -\alpha(h(x)), \quad \forall x \in \mathcal{D}$$

where $L_f h(x)$ denotes the Lie derivative along the drift vector field $f(x)$. This condition, inspired by the Lyapunov approach to stability, ensures that $h(x)$ cannot decrease too rapidly and, consequently, that the safe set $\mathcal{C}$ is forward invariant: for any initial condition $x_0 \in \mathcal{C}$, the state remains in $\mathcal{C}$ for all future time under the nominal dynamics.

A Lyapunov candidate for the safe set is given by

$$V_\mathcal{C}(x) = \begin{cases} 0 & \text{if}~x \in \mathcal{C} \ -h(x) & \text{if}~x \in \mathcal{D} \setminus \mathcal{C} \end{cases}$$

which is positive outside $\mathcal{C}$ and decreases along system trajectories if the ZBF property holds.

2. Robustness to Disturbances and the Role of the R-ZOCBF

The robustness analysis is central to R-ZOCBFs. The controlled system is considered in the disturbed form: $\dot{x} = f(x) + d(x, t)$ where $d(\cdot, \cdot)$ is an unknown disturbance term, possibly state-dependent and time-varying. The R-ZOCBF theory asserts that if the disturbance vanishes sufficiently rapidly as one approaches the safe set, or is bounded, the set $\mathcal{C}$ remains robustly forward invariant or, more generally, an “inflated” safe set remains invariant:

• If $\|d(x, t)\| \leq \sigma(\|x\|_\mathcal{C})$ for a class-$\mathcal{K}$ function $\sigma$, then $\mathcal{C}$ is asymptotically stable (i.e., robust to vanishing disturbances).
• For bounded, nonvanishing disturbances $\|d(\cdot, t)\|_\infty \leq k$, there exists a class-$\mathcal{K}$ function $\gamma$ such that

$$\mathcal{C}_{\gamma(k)} = \{ x \in \mathbb{R}^n : h(x) \geq -\gamma(k) \}$$

is asymptotically stable. The safe set “inflates” by $\gamma(k)$, yielding a robust extension of the traditional ZBF, i.e., the R-ZOCBF.

This shows that the R-ZOCBF provides a quantitative mechanism for bounding how much violation of the nominal safety set must be tolerated under given levels of disturbance and that forward invariance or asymptotic stability of the safe set (or an enlargement) can be guaranteed.

3. Control Synthesis via QP: Integration of Safety and Performance

To reconcile safety with control objectives (such as stabilization), the R-ZOCBF is combined with a Control Lyapunov Function (CLF) in a convex quadratic program: $$\begin{aligned} \min_{u, \delta} \quad & \frac{1}{2} u^\top u + k^2 \delta^2 \ \text{subject to}\quad & L_f V(x) + L_g V(x) u + c V(x) - \delta \leq 0 \quad \text{(CLF soft constraint)} \ & L_f h(x) + L_g h(x) u + \alpha(h(x)) \geq 0 \quad \text{(CBF hard constraint)} \end{aligned}$$ where $V(x)$ is a CLF, $h(x)$ is the R-ZOCBF, $u$ is the control input, $\delta$ is a slack variable for the CLF constraint, and $k$, $c$ are weights. The CLF constraint is soft (relaxed by $\delta$) to ensure feasibility, while the R-ZOCBF constraint is hard and ensures safety as a strict priority.

The hard constraint guarantees forward invariance of the safe set; in the presence of disturbances, the robustness theory ensures that either the nominal or an “inflated” safe set is rendered invariant. The method is compatible with disturbances in both the underlying dynamics and the input channel.

4. Regularity and Well-Posedness of the Feedback Law

It is essential for the controller synthesized by the QP (either for the ZCBF constraint alone or the coupled R-ZOCBF–CLF system) to be locally Lipschitz continuous. This condition ensures:

• The closed-loop vector field is locally Lipschitz.
• Existence and uniqueness of solutions to the closed-loop ODE (by Picard–Lindelöf).
• Absence of discontinuities or “jumps” in the control input (i.e., chattering is avoided).

For relative degree one barrier functions ($L_g h(x) \neq 0$ everywhere on $\mathcal{D}$), the explicit solution to the minimum-norm CBF QP is given as

$$u^*(x) = \begin{cases} 0 & \text{if}~L_f h(x) + \alpha(h(x)) > 0 \ - \frac{L_f h(x) + \alpha(h(x))}{L_g h(x) L_g h(x)^\top} L_g h(x)^\top & \text{otherwise} \end{cases}$$

which is locally Lipschitz continuous if $h$ is smooth and $L_g h(x)$ does not vanish. This property extends to more elaborate output-coupled QPs featuring both CLF and CBF constraints.

5. Practical Considerations and Applications

The R-ZOCBF paradigm offers a robust, scalable method for synthesizing safety-critical controllers in a broad spectrum of applications:

• Safety filters for autonomous robots and vehicles, providing guarantees even under model mismatch or perturbations.
• Integration with performance controllers (e.g., trajectory tracking) through the QP framework, enabling coexistence of control and safety objectives.
• Feasibility in hard-constrained actuation settings, since the QP can accommodate input bounds and ensures locally smooth feedback.

The modular structure allows for augmenting with higher-order barrier conditions if the safety set has higher relative degree, and the specific characterization of robustness enables explicit computation (offline or online) of the minimum “buffer” needed to guarantee safety under anticipated disturbance magnitudes.

6. Limitations and Extensions

While the R-ZOCBF approach is robust to “small” additive disturbances, arbitrarily large or rapidly varying uncertainties may require a (possibly significant) inflation of the safe set, potentially compromising control objectives. The method presumes knowledge of reasonable upper bounds on the disturbance for its invariance inflation parameter, and the geometry of the safe set defined by $h(x)$ must be compatible with system dynamics (e.g., must avoid unbounded growth of $\alpha(h(x))$ or singularities in $L_g h(x)$). For systems of higher relative degree, extensions to high-order barrier functions and more elaborate “lifted” QP constraints are needed.

Additionally, computational complexity is largely dictated by the dimension of the QP (number of outputs, input constraints, and performance objectives), but closed-form feedback expressions can be obtained in simple cases (see explicit formulas in (Xu et al., 2016)).

7. Summary Table: Core Components of R-ZOCBF

Component	Description	Robustness Mechanism
Safety Function $h(x)$	Continuously differentiable, vanishing on boundary of safe set	Buffer via class-$\mathcal{K}$ function in presence of disturbance
ZOCBF Inequality	$L_f h(x) \geq -\alpha(h(x))$, or $L_f h(x) + L_g h(x) u + \alpha(h(x)) \geq 0$	Extended by QP for robust control input
Disturbance Robustness	Asymptotic stability of an inflated safe set under bounded exogenous perturbations	Explicitly via class-$\mathcal{K}$ function $\gamma$
QP Control Synthesis	Minimizes performance cost subject to safety (hard) and performance (soft) constraints	Hard safety constraint guarantees invariance
Lipschitz Continuity	Constructed for relative degree one via explicit QP solution; crucial for unique closed-loop solutions	Avoids chattering and ill-posed ODEs

8. Concluding Perspective

Robust Zero-Order Control Barrier Functions enable constraint satisfaction and forward invariance for control-affine systems in the presence of bounded, possibly state-dependent disturbances and model uncertainties. The framework yields explicit quantitative guarantees on safety via set “inflation” and is compatible with existing optimization-based synthesis methods (e.g., convex QPs mixing control Lyapunov and barrier functions). Lipschitz continuity of the resulting feedback law ensures well-posedness and implementability in safety-critical cyberphysical systems (Xu et al., 2016).