Control Barrier Functions for Safety Control

• Control Barrier Functions (CBFs) are mathematical constructs that ensure forward invariance of safe sets by encoding safety constraints.
• CBFs integrate with Control Lyapunov Functions via quadratic programming to mediate safety and performance in real-time control decisions.
• CBFs have been validated in automotive and robotics, providing less conservative controllers under actuator and input constraints.

A Control Barrier Function (CBF) is a mathematical construct for formally encoding safety requirements as controlled invariance of a designated set in the system’s state space. The fundamental objective is to guarantee that, through suitable control action, the system state remains perpetually within a “safe set,” even in the presence of potentially conflicting performance (stabilization or tracking) requirements. CBFs facilitate the rigorous mediation of safety and performance in safety-critical systems by embedding both objectives as constraints within real-time, optimization-based controllers. The methodology has been influential in areas such as automotive adaptive cruise control and lane keeping, where safety and performance objectives must be satisfied simultaneously, frequently under actuator or input constraints.

1. Control Barrier Function Concepts and Forward Invariance

CBFs formalize safety constraints as forward invariance of a set: $C = \{ x \in \mathbb{R}^n : h(x) \geq 0 \}$ where $h$ is a continuously differentiable function. Two barrier function types are introduced:

• Reciprocal Barrier Function (RBF):

$B(x) = -\log\left(\frac{h(x)}{1 + h(x)}\right)$

$B(x)$ tends to infinity at the set boundary ($h(x) \to 0^+$) and satisfies comparison inequalities through class-$\mathcal{K}$ functions $\alpha_1, \alpha_2$:

$$\frac{1}{\alpha_1(h(x))} \leq B(x) \leq \frac{1}{\alpha_2(h(x))}$$

• Zeroing Barrier Function (ZBF):

Here, $h(x)$ itself serves as the barrier, and invariance requires a class-$\mathcal{K}$ function $\alpha$:

$L_f h(x) \geq -\alpha(h(x))$

ensuring $h(x(t))$ remains nonnegative for all $t\geq0$.

Both forms generalize Lyapunov stability—while Lyapunov functions drive states towards an equilibrium, CBFs ensure that trajectories are blocked from entering unsafe regions by controlling the sign and growth rates of barrier derivatives.

2. Unification of Safety and Performance via Quadratic Programming

CBFs are naturally unified with Control Lyapunov Functions (CLFs), which enforce stabilization or performance, through quadratic programming: $$\begin{align*} &\min_{u, \delta}~ \frac{1}{2}\|(u, \delta)\|^2 \ &\text{subject to:}~ L_f V(x) + L_g V(x)u + c_3 V(x) - \delta \leq 0 \ &\hspace{57pt}L_f B(x) + L_g B(x)u - \alpha_3(h(x)) \leq 0 \end{align*}$$

• CLF constraint: Drives the state toward a reference or equilibrium (with slack variable $\delta$ for soft enforcement).
• CBF constraint: Ensures invariance of the safe set (enforced as a hard constraint).

In the context of Adaptive Cruise Control (ACC), for instance, performance is encoded as convergence to desired velocity, while safety (e.g., time headway) is encoded as a function of relative distance. The quadratic program generates real-time control actions that prioritize safety if the objectives conflict and relax the stabilization requirement only as necessary.

3. Mathematical Formalism

The evolution of the system under input-affine dynamics: $\dot{x} = f(x) + g(x)u$ extends the CBF conditions:

• Reciprocal Control Barrier Function (RCBF):

$$\inf_{u \in U} \left\{ L_f B(x) + L_g B(x)u - \alpha_3(h(x)) \right\} \leq 0$$

• Zeroing Control Barrier Function (ZCBF):

$$\sup_{u \in U} \left\{ L_f h(x) + L_g h(x)u + \alpha(h(x)) \right\} \geq 0$$

These conditions are affine in $u$, enabling embedding as linear constraints within optimization problems.

4. Safety-Critical Applications

Adaptive Cruise Control (ACC)

• Objective: Track a desired speed $v_d$ while maintaining a minimum following distance.
• CLF: $V(x) = (v_f - v_d)^2$ (where $v_f$ is ego vehicle speed).
• CBF: $h(x) = D - \tau_d v_f \geq 0$ (with $D$ as inter-vehicle distance, $\tau_d$ the desired headway).
• QP constraints: Safety takes priority if a faster speed would reduce headway below threshold.

Lane Keeping (LK)

• Objective: Stay within lane and limit lateral displacement $|y| \leq y_{max}$.
• CBF: Designed using position and velocity to restrict lateral deviation and maintain comfort constraints (e.g., bounded lateral acceleration).
• QP execution: Control actions computed such that deviation from reference (tracking) is penalized but never at the cost of violating lane or actuator constraints.

Simulations show that the dual-objective QP prevents violations of safety constraints (e.g., crashes, lane departures) and only sacrifices performance when strictly necessary due to conflict.

5. Novel Methodological Contributions

• The introduction of RBFs and ZBFs in this context relaxes the classical requirement for monotonic nonincreasing barrier functions: the constructed CBF is allowed to grow in the safe set’s interior and is only tightly constrained near the boundary.
• This leads to less conservative controllers, maximizing the available set of feasible control actions.
• The mediation of safety and performance within a QP directly resolves trade-offs in real time, ensuring that stabilization can be “sacrificed” to maintain safety but safety is never compromised.
• Treatment of higher relative degree barriers (i.e., those requiring backstepping) and preservation of local Lipschitz continuity for well-posedness and implementation robustness.

6. Implications for Safety-Critical Systems

The formal synthesis of CBF-based controllers provides explicit guarantees for forward invariance and safety in complex, actuator-constrained systems, as demonstrated in automotive scenarios but broadly applicable to robotics, aerospace, and networked cyber–physical systems. The approach’s flexibility and modularity (via QP-based mediation of constraints) make it suitable for scalable, real-time embedded control, and the foundational generalizations introduced by reciprocal and zeroing barrier functions address both computational tractability and reduction in conservativeness.

7. Conclusion

This framework establishes a rigorous method for synthesizing controllers that guarantee forward invariance of a safe set through generalized CBFs, and unify safety with performance via quadratic programming. The resulting optimization-based controllers are theoretically grounded, less conservative, and practically validated in safety-critical automotive applications, with future potential for broad adoption in complex engineered systems requiring both provable safety and agile performance.