Composite Adaptive Control Barrier Functions for Safety-Critical Systems with Parametric Uncertainty

TL;DR

CaCBF integrates safety certification and parameter adaptation for nonlinear control-affine systems with linear parametric uncertainty. By stacking a logarithmic barrier, a CLF term, and a parameter error term into a single energy function $V_c$, and deriving a projection-based adaptation law, the method guarantees forward invariance of the safe set without requiring parameter convergence. The CLF-CBF-QP control law simultaneously enforces safety and stabilization, while a feasibility analysis shows universal boundedness of all signals under mild assumptions. Compared with robust CBFs, CaCBF expands the admissible safety set and recovers the exact safe region as parameter estimates converge, demonstrated across adaptive cruise control, a drifted omnidirectional robot, and a crosswind-plagued planar drone. This approach yields substantial reductions in conservatism while maintaining strict safety in the presence of uncertainty, with potential extensions to unstructured dynamics and partial-state scenarios.

Abstract

Control barrier functions guarantee safety but typically require accurate system models. Parametric uncertainty invalidates these guarantees. Existing robust methods maintain safety via worst-case bounds, limiting performance, while modular learning schemes decouple estimation from safety, permitting state violations during training. This paper presents the composite adaptive control barrier function (CaCBF) algorithm for nonlinear control-affine systems subject to linear parametric uncertainty. We derive adaptation laws from a composite energy function comprising a logarithmic safety barrier, a control Lyapunov function, and a parameter error term. We prove that CaCBF guarantees the forward invariance of the safe set and the uniform boundedness of the closed-loop system. This safety guarantee holds without requiring parameter convergence. Simulations of adaptive cruise control, an omnidirectional robot, and a planar drone demonstrate the efficacy of the CaCBF algorithm.

Figures (4)

• Figure 1: Geometric interpretation of Theorem \ref{['prop:reduced_conservatism']} at each $x\in{\mathcal{C}}$, $\theta_{\rm e}\in{{\mathbb R}^p}$, and $\hat{\theta}\in\Theta$. The conservative robust safe control set ${\mathcal{U}}_{\rm rob}(x, \theta_{\rm e})$ (inner, blue) is nested within the adaptive safe control set ${\mathcal{U}}_{\rm adp}(x, \hat{\theta})$ (middle, red, dashed). The adaptive set expands toward the true safe control set ${\mathcal{U}}_{\rm cbf}(x)$ (outer, solid) as $\hat{\theta}$ converges to the true parameter $\theta_*$.
• Figure 2: Example \ref{['ex_ACC']}: Simulation results for the adaptive cruise control with unknown resistance force. The top subplot shows that the proposed CaCBF (red) tracks the desired speed more effectively than the R-CBF (blue), which exhibits spurious braking. The second subplot tracks the inter-vehicle distance $d$, where the green shaded region visualizes the conservatism gap: the robust controller maintains an unnecessary buffer due to worst-case assumptions, whereas the adaptive controller operates closer to the effective safety limit $1.8v$. The third subplot plots the control effort $u$, revealing that while both controllers strike the actuation boundaries (dashed black lines), they do so for distinct reasons. The R-CBF is forced into saturation by its conservative worst-case margins, whereas the CaCBF leverages braking authority to maximize tracking performance while remaining strictly safe. The bottom subplot confirms that the parameter estimates $\hat{\theta}$ (solid) need not converge perfectly to the true values $\theta_*$ (dashed) to guarantee safety.
• Figure 3: Example \ref{['ex_Omni']}: Simulation results for the omnidirectional robot navigation with unknown drift. The top subplot shows the 2D trajectories: while both controllers reach the target (star), the R-CBF (blue) takes a longer route to maintain a robust margin against worst-case drift, whereas the CaCBF (red) adapts to the true drift and hugs the obstacle boundary efficiently. The second subplot tracks the safety margin $h(x)$, where the green shaded region visualizes the conservatism gap: R-CBF maintains a larger buffer $h_{\min}=0.42$ m than the adaptive controller buffer $h_{\min}=0.04$ m. The third subplot confirms that the control inputs remain within the saturation limit $\|u\| \le 2$ m/s. The bottom subplot shows the parameter estimates $\hat{\theta}$ (solid) converging toward the true drift values $\theta_*$ (dashed), which enables the CaCBF to safely reduce the conservatism gap.
• Figure 4: Example \ref{['ex_Drone_Gate']}: Simulation results for the planar drone subject to unknown crosswind navigating a passage. The top subplot shows the trajectories, where the R-CBF (blue) is blocked because the robust margins (dotted) of the two obstacles overlap, effectively closing the gate. The CaCBF (red) learns the wind parameters, shrinks these margins, and successfully passes through the gap to reach the target (star). The second subplot tracks the minimum distance to the nearest wall, confirming that the CaCBF safely utilizes the available space ($h>0$). The third subplot shows the control effort $\|u\|$, illustrating the maneuvering required to pass the gate. The bottom subplot confirms the convergence of the parameter estimates $\hat{\theta}$ (dotted) toward the true wind values $\theta_*$ (dashed).

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Theorem 1
• Theorem 2
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3