Control Barrier Function Techniques

• Control barrier functions are mathematical tools that guarantee forward invariance of safe sets in dynamical systems.
• They are implemented via optimization-based controllers, such as QPs, to balance safety constraints with performance objectives.
• Recent advances extend CBF techniques to high-order, discrete-time, and robust settings, enhancing safety in automotive, aerospace, and robotics.

A control barrier function (CBF) is a mathematical construct used to enforce safety constraints in the control of dynamical systems by ensuring forward invariance of a designated safe set within the state space. CBF techniques are fundamentally based on Lyapunov-like conditions and measure-theoretic invariance principles, and they provide tractable certificates and synthesis procedures that guarantee system trajectories do not leave prescribed safe regions, even under actuation limits, external disturbances, or model uncertainties. Modern CBF methodologies integrate seamlessly with optimization-based controllers, especially quadratic programs (QPs), and extend classical state-constraint tools to highly nonlinear, high-relative-degree, and even discrete-time or learning-based systems.

1. Fundamental Principles and Variants of Control Barrier Functions

The classical CBF framework is defined for a control-affine system

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

with $x \in \mathbb R^n$, $u \in \mathbb R^m$, and a twice continuously differentiable function $h: \mathbb R^n \to \mathbb R$ specifying the safe set $C := \{x \mid h(x) \ge 0\}$. The CBF is constructed such that, for some extended class-$\mathcal K$ function $\alpha$ (i.e., $\alpha$ strictly increasing and $\alpha(0) = 0$), the following inequality is satisfied for all $x \in C$,

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

where $L_f h$ and $L_g h$ are the Lie derivatives of $h$ along $f$ and $g$ respectively. Any Lipschitz continuous controller $u(x)$ that enforces

$L_f h(x) + L_g h(x) u(x) + \alpha(h(x)) \ge 0$

ensures forward invariance of $C$; that is, $x(0) \in C$ implies $x(t) \in C$ for all $t \ge 0$ (Ames et al., 2016).

Two principal forms are employed: zeroing barrier functions (ZBFs), which vanish on the boundary of $C$ and satisfy Lyapunov-like decrease conditions, and reciprocal barrier functions (RBFs), which diverge as the system state approaches the boundary of $C$ but remain finite in the interior. ZBFs are generally favored for constructive control synthesis due to their affinity with the input $u$.

Substantial generalizations have been introduced:

• High-Order CBFs (HOCBFs): Handle constraints of relative degree $r > 1$ (i.e., input appears after $r$ derivatives), using a recursive sequence of class-$\mathcal K$ functions and auxiliary variables to generate a chain of inequalities guaranteeing forward invariance on a funnel of progressively tighter sets (Xiao et al., 2022).
• Exponential CBFs (ECBFs): Use linear or higher-order feedback to assign the decay rate of $h$ and its derivatives, particularly for systems with higher relative degree, resulting in exponential decrease toward the interior of $C$ (Dallas et al., 25 Mar 2025).
• Discrete-Time CBFs: The extension to sampled-data or discrete-time systems is formulated such that invariance is maintained by ensuring that the safe set is mapped into itself at each time step, leading to a necessary and sufficient condition for controlled invariance in the form $\sup_{u\in U} h(f(x) + \tilde g(x)u) \ge 0$ (Cavorsi et al., 2020).

CBFs can be relaxed via pointwise optimization of the decay rate (optimal-decay CBFs) to guarantee that the relevant safety QP is always feasible in the interior of $C$, especially in the presence of input constraints (Zeng et al., 2021).

2. Optimization-Based Controller Synthesis: QP/SDP Realizations

CBFs are operationalized via optimization-based safety filters, commonly by formulating a quadratic program (QP) that mediates between safety and performance objectives (the latter often specified via a control Lyapunov function, CLF): $$\begin{aligned} &\min_{u\in U} \|u - u_{\mathrm{nom}}(x)\|^2 + p \delta^2 \ &\text{subject to } L_f h(x)+L_g h(x)u + \alpha(h(x)) \ge 0 \qquad \text{(CBF, hard constraint)}\ &\hspace{40pt} L_f V(x) + L_g V(x)u + c V(x) \le \delta \qquad \quad \text{(CLF, soft constraint)}\ &\hspace{40pt} u \in U. \end{aligned}$$ Here $u_{\mathrm{nom}}(x)$ is a nominal performance control, $V(x)$ a CLF, and $\delta \ge 0$ a slack variable. The hard inequality on the CBF maintains safety, while the CLF objective is achieved as well as possible without violating safety (Ames et al., 2016, Zeng et al., 2021). Under regularity conditions, the QP admits a unique, piecewise affine, and locally Lipschitz continuous solution (Wang et al., 2022).

For multi-input or multi-constraint systems, joint or compositional CBF conditions are enforced via semidefinite programming (SDP) extensions (matrix-valued CBFs), which allow the safe set to be specified by semidefinite or indefinite matrix inequalities. These SDPs enable provable continuity of the safety filter, admit rich Boolean compositions (AND, OR), and efficiently enforce simultaneous or alternative safety requirements (Ong et al., 15 Aug 2025).

3. Shared Control, High Relative-Degree Safety, and Real-Time Implementation

Modern CBF methodologies can be deployed for shared control scenarios where a human or autonomous driver provides nominal inputs that are minimally modified to enforce safety. One prominent application is in vehicle dynamics at the handling limits, such as controlled drifting maneuvers. In this context:

• The maximal phase recoverable ellipse (MPREl) is constructed in the sideslip–yaw rate plane $(\beta, r)$, by simulating boundary trajectories using extremal steering inputs, and fitting an ellipse that is entirely contained in the recoverable region. The safe set is parameterized by $a \beta^2 + b \beta r + c r^2 \le d$.
• An order-2 ECBF is defined for this set, exploiting the fact that $h(x)$ (as a function of state $x = [r, \beta, V, \delta, \tau]^\top$) has relative degree two. The ECBF is enforced via the condition

$$L_f^2 h(x) + L_g L_f h(x) u \ge - [p_1 L_f h(x) + p_0 h(x)],$$

with appropriately chosen gains $p_1, p_0$.

• A real-time QP at each sample selects actuator rate commands $u = [\dot\delta, \dot\tau]$, penalizing deviations from the driver's request while enforcing the ECBF with a slack variable $\epsilon$ for feasibility under model mismatch.
• On physical hardware (e.g., a GRIP test vehicle), this filter preserves system safety at the handling limit with mean QP solution times $<50\ \mu$s at a 200 Hz update rate. The barrier filter intervenes only near the boundary of the MPREl, maintaining minimally invasive shared control and preventing spin-out in aggressive maneuvers (Dallas et al., 25 Mar 2025).

Applications in high relative-degree safety (e.g., output constraints that appear only after several time derivatives) are tackled using ECBFs and, for robustness to model uncertainty and high-performance set tracking, via sliding-mode CBF schemes that enhance rejection of bounded uncertainties at the boundary (Chinelato et al., 2020).

4. Robustness to Model Uncertainty and Sensor Error

A key limitation of basic CBF approaches is their reliance on accurate models of $f(x), g(x)$ and state measurements. To address this, robust and adaptive CBF methodologies have been developed:

• Measurement-Robust Incremental CBFs (MRICBF): MRICBFs leverage high-rate sensor data and incremental linearization to compensate for parametric/model error and sensor biases. Explicit error compensation terms quantify the worst-case impact of both model approximation and measurement uncertainty on the time derivative of $h$. The MRICBF QP modifies the incremental control to directly account for real-time observation, ensuring immediate safety despite time-varying or biased sensors, with formally quantified conservatism (Autenrieb et al., 10 Oct 2024).
• Set-Membership and Adaptive CBFs: For time-varying or uncertain parameters, set-membership identification of the uncertainty set (parameter bounding in real time) is integrated with robust CBF design to achieve reduced conservatism and input-to-state safety in the presence of disturbances. This schema ensures that the tightened CBF constraint, adjusted for the estimated parameter set and disturbance bound, maintains forward invariance for all admissible states (Kim et al., 17 Jun 2025).
• Reciprocal Resistance CBFs: Robustness to disturbances without a priori knowledge of their bound is achieved via reciprocal-resistance terms $\beta(1/h)$ in the CBF condition. As $h \to 0^+$, the reciprocal-resistance-term dominates, dynamically generating a non-vanishing buffer zone at the safe set boundary and ensuring invariance under arbitrarily large but bounded disturbances. This can be augmented with disturbance observers for further reduction of conservatism (Wang et al., 25 Jul 2025).

Conservatism (shrinkage of the effective safe set due to robustness margins) is a recurring theme. Conservative bounds cause the system to operate deeper within $C$ than nominally required, but as sensor/model fidelity improves, robust CBFs converge to the nominal case.

5. Compositional, Discrete-Time, and Predictive CBF Techniques

CBF theory has been developed to support a spectrum of compositional and discrete-time safety scenarios:

• Compositional and Boolean-encoded CBFs: Mixed-integer programming formulations allow conjunctive (AND), disjunctive (OR), and other Boolean compositions of multiple barrier constraints, yielding piecewise- or region-dependent constraints tractable via MIP/MIQP solvers (Cavorsi et al., 2020). This enables safety with respect to multiple, possibly nonconvex, environmental features (e.g., obstacle avoidance with lane splitting).
• Matrix CBFs: Matrix-valued CBFs seamlessly encode both conjunctions (via semidefinite LMIs) and disjunctions (via indefinite matrix conditions), eliminating heuristic soft relaxations and ensuring continuity of the safety filter by design (Ong et al., 15 Aug 2025).
• Discrete-Time CBFs: Discrete-time control systems admit a directly analogous barrier framework: $h(f(x_k,u_k)) \ge h(x_k) + \alpha(h(x_k))$. Necessary and sufficient conditions exist for controlled invariance in the discrete-time sense. Robustness is enforced via Lipschitz bounds on $h$ and additive disturbance margins (Shakhesi et al., 16 Jun 2025). Synthesis can be performed using learning-based counterexample-guided refinement (adding sampled states where constraints fail), guaranteeing global validity on the safe set.

Predictive control barrier functions (PCBFs) integrate a look-ahead (model-predictive) safety assessment into the CBF framework. By leveraging the value function of a constrained model predictive control (MPC) problem as a CBF—either as a terminal ingredient or directly as a safety certificate—forward invariance of the associated level set is achieved via a single-stage QP at the current time, exploiting the horizon’s look-ahead to reduce conservatism and resolve infeasibility (Huang et al., 12 Feb 2025, Wabersich et al., 2021, Breeden et al., 2022). The predictive framework enhances recovery properties (steering back into the safe set after perturbation) and is suited to learning-based or uncertain settings where constraint violations are otherwise not precluded.

6. Applications and Performance in Complex Environments

CBF methods have been demonstrated across a broad range of safety-critical domains:

• Automotive systems: Adaptive cruise control, lane keeping, lane changing, and aggressive maneuvering (e.g., drifting) all leverage real-time CBF-QP formulations that enforce time headway, lane boundary, and performance constraints under actuation limits (Ames et al., 2016, Dallas et al., 25 Mar 2025, Cavorsi et al., 2020).
• Aerospace and Flight Control: Overactuated flight vehicles, hypersonic gliders, and robotic spacecraft employ CBFs to maintain attitude, prevent escape from flight envelopes, and enforce multiple simultaneous constraints, with robustification via MRICBF and high-order CBFs (Autenrieb et al., 10 Oct 2024).
• Robotics: Obstacle avoidance, surface treatment with force constraints, and collision avoidance for multi-body agents utilize both scalar and matrix CBFs, as well as compositional and time-varying extensions (Molnar, 22 May 2025, Aali et al., 2022, Kim et al., 17 Jun 2025).
• Learning and Imitation: Safety-augmented learning (e.g., CBFIRL) combines neural parameterization of barrier functions with imitation learning, adding CBF-inspired loss terms to minimize policy violations, thereby achieving a measurable reduction in collision rates compared to unconstrained learning (up to $50\%$ in complex drone domains) (Yang et al., 2022).
• Multi-objective and input-constrained control: Matrix CBFs provide natural continuity of the safety filter in the presence of AND/OR constraints. The barrier-state (BaS) theory reinterprets CBF-based constraints as dynamical states embedded in the system, enabling unified stability and safety via a single augmented feedback law (Almubarak et al., 2023).

7. Theoretical Guarantees, Scalability, and Limitations

CBF controllers offer rigorous guarantees:

• Forward invariance: Provided regularity of $h$, $f$, $g$ and Lipschitz continuity of the realized feedback, the safe set $C$ remains invariant for all time under the closed-loop dynamics.
• Online feasibility: QP and SDP-based safety filters are real-time feasible on embedded hardware for modest state/input dimensions (e.g., $< 50\mu$s for 3-variable QPs at 200Hz in vehicle examples, $<2$ms for moderate-sized SDPs on quadrotor networks) (Dallas et al., 25 Mar 2025, Ong et al., 15 Aug 2025).
• Robustness: Explicit compensation for model error, bounded disturbances, and sensor bias yields quantifiable safety margins, at the expense of conservativeness if large margins are required (Autenrieb et al., 10 Oct 2024, Wang et al., 25 Jul 2025).
• Continuity and regularity: Active-set and semidefinite formulations guarantee piecewise Lipschitz continuity of the feedback law in the absence of degeneracy. Boolean compositions and high-order CBFs maintain regularity through judicious construction (Ong et al., 15 Aug 2025).

Limitations and open challenges include:

• Conservatism: Robust and measurement-robust CBFs can shrink the effective safe set. As parameter uncertainty and sensor accuracy improve, this effect diminishes, and the robust CBF converges to its nominal case (Autenrieb et al., 10 Oct 2024, Kim et al., 17 Jun 2025).
• Scalability: While explicit and semi-separable QPs/SDPs scale modestly with problem size, large-scale nonlinear or high-dimensional systems may require approximation or tailored solvers.
• Model Uncertainty: Current methods are sensitive to unmodeled dynamics and large parametric variations unless explicitly accounted for, motivating integration with learning- and adaptive-barrier constructions (Yang et al., 2022).
• Discrete-time and sampled-data effects: Robustness under discretization and time-delay requires additional margins or predictive barrier constructions (Cavorsi et al., 2020).

CBF techniques persist as a central paradigm for provably safe real-time control under performance constraints, with continued advancements extending their applicability to learning-enabled, high-dimensional, and uncertain systems.