Control Barrier Function Augmentation

• Control Barrier Function augmentation is a method to enhance safety certificates by systematically relaxing invariance properties to account for disturbances in dynamical systems.
• It establishes robust forward invariance and ISS-based stability, ensuring that the relaxed safe sets remain invariant even in the presence of bounded model errors.
• Integrating CLF-CBF-QP synthesis yields locally Lipschitz feedback controllers, guaranteeing safe and performance-oriented control across various applications.

Control Barrier Function (CBF) augmentation concerns the systematic enhancement of conventional barrier function–based safety certificates for controlled dynamical systems. Augmentation methods address robustness, feasibility, and compositionality in the enforced invariance property that underpins safety-critical control. The concept draws on the analogy with Lyapunov function augmentation in control Lyapunov function (CLF) theory, enabling synthesis of feedback controllers that uphold set invariance—here, safety requirements—even in the presence of perturbations, modeling errors, or hard constraints. This article rigorously formulates the main augmentation results, their mathematical underpinnings, and their implications for controller design.

1. Zeroing Control Barrier Function: Definition and Forward Invariance

Let the uncontrolled dynamical system be

$$\dot x = f(x), \quad f:\mathbb{R}^n \rightarrow \mathbb{R}^n$$

with $f$ locally Lipschitz. Define a continuously differentiable scalar function $h: \mathbb{R}^n \rightarrow \mathbb{R}$ and the safe set

$C = \{ x \in \mathbb{R}^n : h(x) \geq 0 \}.$

A Zeroing Barrier Function (ZBF) is a function $h$ for which there exists an extended-class $\mathcal{K}$ function $\alpha$ (strictly increasing, $\alpha(0) = 0$) such that

$$L_f h(x) = \nabla h(x) \cdot f(x) \geq -\alpha(h(x)), \quad \forall x \in D$$

for some open set $D \supset C$. By Nagumo’s theorem, this implies forward invariance of $C$ under $\dot x = f(x)$; i.e., if $x(0) \in C$, then $x(t) \in C$ for all $t \geq 0$.

2. Augmentation under Disturbances: Input-to-State Stability Relaxation

Consider now the disturbed system

$$\dot x = f(x) + d(t), \qquad \|d\|_\infty \leq \bar d,$$

and define the distance to $C$ as

$\|x\|_C = \inf_{y \in C} \|x - y\|.$

Let $V_C(x) = 0$ for $x \in C$ and $V_C(x) = -h(x)$ for $x \notin C$. For $x \notin C$, one obtains

$$\dot V_C(x, d) = -L_f h(x) + \nabla_x V_C(x) \cdot d \leq -\alpha(V_C(x)) + \|\nabla V_C(x)\| \|d\| .$$

By ISS-Lyapunov theory, there exist class $\mathcal{K}$ and $\mathcal{KL}$ functions $\gamma$, $\beta$ so that

$$\|x(t)\|_C \leq \beta(\|x(0)\|_C, t) + \gamma(\|d\|_\infty) .$$

Consequently, trajectories starting inside or near $C$ cannot exit the relaxed safe set

$$C_\varepsilon = \{ x : h(x) \geq -\varepsilon \}, \qquad \varepsilon = \gamma(\bar d) .$$

This "tube" augmentation establishes that the level set shifted by $-\varepsilon$ is invariant under bounded disturbances.

3. Robust Forward Invariance and Asymptotic Stability of Relaxed Sets

Let $h$ be a ZBF on open $D \supset C$. For a disturbance magnitude $\bar d \geq 0$, define $\varepsilon = \gamma(\bar d)$ as before, and

$C_\varepsilon = \{ x : h(x) \geq -\varepsilon \}.$

Robust Forward Invariance Theorem: The set $C_\varepsilon$ is forward invariant under

$$\dot x = f(x) + d(t), \qquad \|d\|_\infty \leq \bar d .$$

Further, for sufficiently small $\bar d$, $C_\varepsilon$ is asymptotically stable in the sense of sets: solutions converge to $C_\varepsilon$ from nearby initial states and remain therein, with the residual error determined by $\varepsilon$.

The proof constructs a Lyapunov function and applies standard ISS theorems: for $d = 0$, $V_C$ is non-increasing. Under disturbance,

$$\dot V_C \leq -\alpha(V_C) + \mu \|d\|, \quad \mu = \sup_{x \in D} \|\nabla V_C(x)\| .$$

This implies exponential decay to a ball of radius $\gamma(\bar d)$, hence the invariance of $C_\varepsilon$.

4. Synthesis with CLF-CBF-QP and Lipschitz Regularity

For the affine-in-control system

$\dot x = f(x) + g(x) u,$

and given a CBF $h$ and a Control Lyapunov Function (CLF) $V$, one synthesizes the controller by the solution to a Quadratic Program (QP): $$\begin{aligned} &\min_{u, \delta} \tfrac{1}{2} u^\top H u + \tfrac{1}{2} p_{sc}\, \delta^2 \ \text{subject to} \quad & L_f V(x) + L_g V(x) u + c V(x) \leq \delta, \ & L_f h(x) + L_g h(x) u + \alpha(h(x)) \geq 0 , \end{aligned}$$ where $\delta \geq 0$ is a relaxation, penalized in the cost function to softly enforce the CLF constraint.

Under the conditions:

• $f$, $g$ locally Lipschitz;
• $V$, $h$ with locally Lipschitz gradients;
• relative degree one: $L_g h(x) \neq 0$ throughout the domain,

it follows via parametric QP theory that the solution map $x \mapsto (u^*(x), \delta^*(x))$ is locally Lipschitz. This guarantees existence and uniqueness of trajectories for the closed-loop feedback system.

5. Adaptive Cruise Control Example under Unmodeled Disturbances

As an illustration, consider adaptive cruise control subject to unknown road grade $d(t) = g \, \Delta \theta(t)$, with

$$\dot v_\ell = a_\ell, \quad \dot v_f = -\frac{F_r(v_f)}{m} + \frac{1}{m} u + g \Delta \theta(t), \quad \dot D = v_\ell - v_f,$$

and safety encoded by

$$h(x) = D - \tau_{\text{des}} v_f, \quad C = \{ h \geq 0 \} .$$

Choosing $\alpha(s) = \kappa s$, the QP above is solved for feedback $u(x)$. The disturbance bound $\|\Delta \theta\|_\infty \leq \overline{\theta}$ implies that the relaxed safe set

$$C_\varepsilon = \{ h \geq -\varepsilon \}, \quad \varepsilon = \frac{\tau_{\text{des}} g}{\kappa} \overline{\theta}$$

remains invariant. Simulation results confirm that:

• The following vehicle never violates $h \geq -\varepsilon$
• The QP-derived feedback is smooth in $x$
• For large $\kappa$ and in the absence of obstacles, the follower speed tracks the desired $v_d$ asymptotically

6. Significance and Broad Implications

Control Barrier Function augmentation, specifically through the robust relaxation of the invariant set and synthesis via QP, enables:

• Safety certification in the presence of bounded but unmeasured disturbances;
• Explicit computation of the margin $\varepsilon$ by which the safety set is relaxed, as a function of the disturbance;
• Composition with performance (e.g., CLF) objectives through convex optimization, preserving both safety and closed-loop well-posedness;
• Robust, locally Lipschitz feedback design, crucial for implementation in physical systems, where continuity is necessary for existence and uniqueness of solutions.

This framework is not restricted to first-order systems: the methodology extends to higher relative degree safety constraints (via high-order barrier functions), input constraints, and can be modularly composed with various performance objectives in more general controller architectures. It is foundational in practical deployments of safety-critical control, particularly in automotive and robotic domains.