Parametric Adaptive HOCBFs (PACBFs)

• PACBFs are a general framework for synthesizing safety-critical controllers for nonlinear affine systems using high-order barrier constructions and adaptive parameter estimation.
• The approach ensures robust forward invariance of safety sets by dynamically solving quadratic programs that incorporate real-time parameter updates.
• Simulation studies in applications like adaptive cruise control and robotic navigation demonstrate PACBFs’ reduced conservatism and enhanced safety performance.

Parametric Adaptive High-Order Control Barrier Functions (PACBFs) are a general framework for synthesizing safety-critical controllers for nonlinear affine systems under parametric uncertainty and arbitrary relative-degree state constraints. PACBFs combine high-order barrier constructions, adaptive parameter estimation, and optimization-based control policies to ensure robust forward invariance of safety sets despite uncertainty in the system model.

1. System Class and Problem Formulation

PACBF methodology applies to control-affine systems of the form

$\dot x = f(x) + g(x)\,u + Y(x)\,\theta^*, \quad x\in\R^n,\,u\in\U \subset \R^m,$

where $f$ and $g$ are known, locally Lipschitz, $Y(x) \in \R^{n\times p}$ is a known regressor, and $\theta^*\in\Theta\subset\R^p$ is an unknown (but constant) parameter vector within a known convex compact set. The principal safety objective is to guarantee forward invariance of

$\mathcal{C} := \{\,x \in \R^n \mid h(x)\ge 0\,\},$

for a $C^r$ constraint function $h$ with (nominal) relative degree $r$—that is, for all $x$ in a region $R$,

$$L_g L_f^{i-1}h(x)\equiv 0,\;\forall i=1,\dots,r-1,\quad L_g L_f^{r-1}h(x)\not=0.$$

PACBFs address the challenge of enforcing $h(x)\ge0$ for all $t$, despite parametric uncertainty in the dynamics (Cohen et al., 2022).

2. High-Order Robust Adaptive Barrier Construction

PACBFs leverage a high-order cascade of barrier functions,

$$\psi_0(x) = h(x), \quad \psi_i(x) = \dot \psi_{i-1}(x) + \alpha_i(\psi_{i-1}(x)),\; i=1,\dots, r-1,$$

with $\alpha_i$ extended class-$\mathcal{K}$ functions, such that recursively applying Lie derivatives yields control-affine inequality constraints at each relative degree. For systems with parametric uncertainty, the top-level condition becomes \begin{align*} \psi_r(x,u) &= L_f\psi_{r-1}(x) + L_Y\psi_{r-1}(x)\,\theta^* + L_g\psi_{r-1}(x)\,u \ &\quad + \alpha_r(\psi_{r-1}(x)). \end{align*} Online parameter estimation provides an adaptive estimate $\hat\theta(t)$ and an uncertainty radius $\nu(t)$ such that

$$L_Y\psi_{r-1}(x)\,\theta^* \ge L_Y\psi_{r-1}(x)\,\hat\theta - \|L_Y\psi_{r-1}(x)\|\,\nu(t).$$

The PACBF (or HO-RaCBF) constraint is then

$$L_f\psi_{r-1}(x) + L_Y\psi_{r-1}(x)\,\hat\theta + L_g\psi_{r-1}(x)\,u + \alpha_r(\psi_{r-1}(x)) - \|L_Y\psi_{r-1}(x)\|\,\nu(t) \ge 0,$$

ensuring robust forward invariance of the safe set for all admissible $\theta^*$ and $u\in U$ (Cohen et al., 2022).

3. Parameter Estimation and Adaptation Laws

PACBF frameworks incorporate concurrent-learning adaptation or event-triggered estimation. In the concurrent-learning approach, an integral over a sliding window provides sufficient excitation for parameter convergence: $$\dot{\hat\theta} = \gamma\sum_{j=1}^M Y_j^\top [x_j-x_j^--F_j-Y_j\hat\theta-G_j], \quad \gamma>0,$$ with $Y_j$ constructed from state/parameter regressor data over the interval, and corresponding adaptation Lyapunov arguments establishing that the uncertainty set shrinks exponentially in the presence of sufficient persistent excitation (Cohen et al., 2022).

Alternatively, event-triggered approaches update $\hat\theta$ and enforce safety constraints at discrete times based on state-prediction error and its derivatives. Between events, PACBF constraints are guaranteed via a priori tube-based bounds (Xiao et al., 2021).

4. Quadratic Program-Based Control Synthesis

PACBF enforcement is realized by solving a convex quadratic program (QP) at each time step (or event trigger). The general QP structure is: $$\begin{aligned} \min_{u} &\quad \frac{1}{2}\|u-u_{\text{nom}}(x)\|^2 \ \text{s.t.} &\quad L_f\psi_{r-1}(x) + L_Y\psi_{r-1}(x)\,\hat\theta \ &\quad + L_g\psi_{r-1}(x)\,u + \alpha_r(\psi_{r-1}(x)) - \|L_Y\psi_{r-1}(x)\|\,\nu(t) \ge 0 \ &\quad u\in U, \end{aligned}$$ where $u_{\text{nom}}(x)$ is a nominal controller. All terms are computable from the state and parameter estimate (Cohen et al., 2022). Extensions support simultaneously handling adaptive penalties (as in AdaCBF), parameterized barriers (as in PCBF), and composite Lyapunov-barrier approaches (Xiao et al., 2020, Jang et al., 17 Jul 2025, Kamaldar, 25 Jan 2026).

5. Theoretical Guarantees and Robustness

PACBFs (HO-RaCBFs) guarantee forward invariance of the intersection of all barrier levels $\cap_{i=1}^r\{\psi_{i-1}\ge0\}$ for the true uncertain system, provided the QP is feasible. The main robustness property is that the margin $\|L_Y\psi_{r-1}(x)\|\,\nu(t)$ in the barrier constraint shrinks as parameter uncertainty $\nu(t)\to0$, reducing conservatism online. Unlike standard adaptive CBFs limited to relative degree one, PACBFs extend to arbitrary $r\ge1$ under a mild matching condition. Simulation studies demonstrate that PACBFs significantly reduce unnecessary conservatism compared to worst-case robust CBFs and maintain strict invariance even during adaptation transients (Cohen et al., 2022, Kamaldar, 25 Jan 2026).

6. Variants and Connections

PACBF encompasses several formulations:

• HO-RaCBF: High-order, robust adaptive CBFs for control-affine, linearly parameterized models (Cohen et al., 2022).
• AdaCBF: Adaptive penalties for constraint relaxation via auxiliary HOCBFs and auxiliary CLFs to ensure feasibility and convergence (Xiao et al., 2020).
• PCBF (Parametrized CBF): Continuously parameterized barrier functions to dynamically reshape invariant sets using parameter dynamics, with corresponding QP enforcement and high-order extension (Jang et al., 17 Jul 2025).
• Composite Adaptive CBF: Unifies Lyapunov, barrier, and parameter error terms in a single composite energy function, with adaptation laws ensuring safety and boundedness simultaneously, even without parameter convergence (Kamaldar, 25 Jan 2026).
• Event-Triggered PACBF: Event-triggered adaptation and prediction-error tube-based constraint bounding for systems with limited computation or online data (Xiao et al., 2021).

7. Representative Applications

PACBF techniques have been validated numerically in several safety-critical settings:

• Adaptive Cruise Control with Unknown Drag: PACBF achieves safety with tighter margins (reducing safety buffer from $\sim6$ m to $0.5$ m) compared to robust CBFs while tracking speed references (Kamaldar, 25 Jan 2026).
• Robotic Navigation under Uncertain Disturbances: PACBF learns disturbances such as drift or wind, enabling navigation close to obstacles while maintaining safety, outperforming robust and modular decoupled designs (Kamaldar, 25 Jan 2026).
• Planar Drone Navigation with Unknown Cross-Wind: Robust CBFs may become infeasible due to overlapping margins; PACBF adaptively reduces uncertainty online and completes the navigation task with minimal clearance (Kamaldar, 25 Jan 2026).
• Adaptive Cruise Control under Braking and Slip: AdaCBF variants maintain feasibility and safety even under sudden loss of braking ability or dynamics noise by adaptively increasing penalty terms (Xiao et al., 2020).

These results confirm that PACBF-style methods provide strict, non-conservative safety guarantees in the presence of unknown constant parameters and time-varying model or constraint uncertainty. The framework generalizes to a variety of parametric structures and implementation paradigms (Cohen et al., 2022, Jang et al., 17 Jul 2025, Kamaldar, 25 Jan 2026, Xiao et al., 2021, Xiao et al., 2020).