Control Barrier Function Approaches

• Control barrier function approaches are a set of methods that define safe sets for nonlinear systems using barrier functions to ensure forward invariance and prevent unsafe trajectories.
• They integrate quadratic program-based safety filters in real-time applications such as automotive control, robotics, and systems with input constraints.
• Recent extensions include high-order formulations, robust and input-aware designs, and learning-based techniques to adapt safe sets amid uncertainties and dynamic constraints.

Control barrier function (CBF) approaches constitute a central methodology for certifying and enforcing safety of nonlinear control systems via constructive, optimization-based safety filters. These methods synthesize supervisory constraints that guarantee forward invariance of a prescribed "safe set," typically formulated as a superlevel set of one or several barrier functions, thereby preventing trajectories from entering undesirable or unrecoverable regions of state space. CBFs are employed in safety-critical domains including automotive control at the handling limit, robotics, and input-constrained or uncertain systems. Contemporary CBF approaches encompass extended high-order barrier formulations, robust and input-aware designs, optimization-based learning of the safe set itself, and scalable shared-control integrations.

1. Foundational Control Barrier Function Frameworks

The canonical CBF framework considers a locally Lipschitz control-affine system

$$\dot{x} = f(x) + g(x)u, \quad x \in \mathbb{R}^n, \; u \in \mathcal{U} \subseteq \mathbb{R}^m$$

and a continuously differentiable barrier function $h : \mathbb{R}^n \to \mathbb{R}$ defining a safe set $\mathcal{C} = \{x : h(x) \geq 0\}$. The CBF condition requires existence of a class-K function $\alpha$ such that for all $x$ in a suitable superlevel set,

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

which guarantees via Nagumo's Theorem that any controller satisfying this constraint renders $\mathcal{C}$ forward invariant (Panja, 2024, Cortez et al., 2022). In closed-loop, such constraints are commonly enforced at runtime by embedding the CBF inequality as a single affine constraint within a quadratic program (CBF-QP), minimally altering a nominal control input while preserving safety.

CBFs have been extended to high-order (relative degree $r>1$) settings via recursive formulations (HOCBFs) and polynomial Lie-derivative chains. The exponential CBF of relative degree 2, for example, considers a sequence of intermediate barrier variables,

$$\nu_0(x) = h(x), \quad \nu_1(x) = \dot{\nu}_0(x) + \alpha_0 \nu_0(x), \quad \nu_2(x) = \dot{\nu}_1(x) + \alpha_1 \nu_1(x)$$

and enforces a second-order inequality involving both state and input, as detailed in shared-control vehicle safety (Dallas et al., 25 Mar 2025).

Zeroing-CBFs (ZCBFs) and their generalizations to Type-II ZCBF (i.e., only requiring the barrier condition within an annulus near the safe set boundary) address robustness and continuity properties, and allow the application of LaSalle’s Invariance Principle for convergence guarantees (Cortez et al., 2022). For input and state constrained systems, new zero-order CBF (ZOCBF) approaches enforce barrier difference inequalities sample-to-sample, thus circumventing derivative-based relative-degree notions and enabling input-dependent and high-relative-degree constraints in sampled-data settings (Tan et al., 2024).

2. Quadratic Program–Based Shared Control and Real-Time Implementation

Real-time CBF deployment is typically realized by synthesizing a QP-based "safety filter" that directly modifies the operator or base controller's command to ensure forward invariance of the safe set. For example, in the context of shared control for high-performance vehicle drifting, the CBF is defined on the maximal phase-recoverable ellipse (MPREl) in the sideslip-yaw rate plane: $$h(x) = d - \left( a \beta^2 + b \beta r + c r^2 \right) \geq 0, \qquad x = [ r, \beta, V, \delta, \tau ]^\top$$ where the coefficients $a, b, c, d$ are fitted to the controllable envelope by sweeping input extremes. Due to the relative-degree two of $h(x)$ (inputs appear only after two derivatives), an exponential CBF is constructed, and the safety constraint on the control input reads (Dallas et al., 25 Mar 2025): $$L_f^2 h(x) + L_g L_f h(x) u + p_1 L_f h(x) + p_0 h(x) \geq 0$$ with $p_0, p_1$ constructed from class-K gain parameters. This constraint is embedded in a QP together with a tracking cost relative to the operator’s command, and includes a slack to accommodate model mismatch.

Validated both