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Learning High-Order Control Barrier Functions for Safety-Critical Control with Gaussian Processes

Mohammad Aali, Jun Liu

TL;DR

The paper tackles safety guarantees for high-order control barrier functions (HOCBFs) under model uncertainty by introducing a Gaussian process (GP) framework to learn the residual impact of uncertainty on the safety certificate. By enforcing linear class-$\mathcal{K}$ functions, the residual becomes affine in the disturbance, allowing the uncertain safety constraint to be reformulated as a convex second-order cone program (SOCP) that can serve as a real-time safety filter. The authors derive probabilistic confidence bounds for the GP estimate and show how to convert the resulting uncertainty-aware constraint into an SOCP with tractable feasibility conditions. Two numerical studies—adaptive cruise control with collision avoidance and a quarter-car active suspension system—demonstrate that the GP-based SOCP-HOCBF maintains safety under significant model mismatch, outperforming a nominal QP-HOCBF and approaching the performance of a true-model oracle. The work provides a practical, data-driven approach to robust safety in control-affine systems, with future directions including relaxing global relative-degree assumptions.

Abstract

Control barrier functions (CBFs) have recently introduced a systematic tool to ensure system safety by establishing set invariance. When combined with a nominal control strategy, they form a safety-critical control mechanism. However, the effectiveness of CBFs is closely tied to the system model. In practice, model uncertainty can compromise safety guarantees and may lead to conservative safety constraints, or conversely, allow the system to operate in unsafe regions. In this paper, we use Gaussian processes to mitigate the adverse effects of uncertainty on high-order CBFs (HOCBFs). A properly structured covariance function enables us to convert the chance constraints of HOCBFs into a second-order cone constraint. This results in a convex constrained optimization as a safety filter. We analyze the feasibility of the resulting optimization and provide the necessary and sufficient conditions for feasibility. The effectiveness of the proposed strategy is validated through two numerical results.

Learning High-Order Control Barrier Functions for Safety-Critical Control with Gaussian Processes

TL;DR

The paper tackles safety guarantees for high-order control barrier functions (HOCBFs) under model uncertainty by introducing a Gaussian process (GP) framework to learn the residual impact of uncertainty on the safety certificate. By enforcing linear class- functions, the residual becomes affine in the disturbance, allowing the uncertain safety constraint to be reformulated as a convex second-order cone program (SOCP) that can serve as a real-time safety filter. The authors derive probabilistic confidence bounds for the GP estimate and show how to convert the resulting uncertainty-aware constraint into an SOCP with tractable feasibility conditions. Two numerical studies—adaptive cruise control with collision avoidance and a quarter-car active suspension system—demonstrate that the GP-based SOCP-HOCBF maintains safety under significant model mismatch, outperforming a nominal QP-HOCBF and approaching the performance of a true-model oracle. The work provides a practical, data-driven approach to robust safety in control-affine systems, with future directions including relaxing global relative-degree assumptions.

Abstract

Control barrier functions (CBFs) have recently introduced a systematic tool to ensure system safety by establishing set invariance. When combined with a nominal control strategy, they form a safety-critical control mechanism. However, the effectiveness of CBFs is closely tied to the system model. In practice, model uncertainty can compromise safety guarantees and may lead to conservative safety constraints, or conversely, allow the system to operate in unsafe regions. In this paper, we use Gaussian processes to mitigate the adverse effects of uncertainty on high-order CBFs (HOCBFs). A properly structured covariance function enables us to convert the chance constraints of HOCBFs into a second-order cone constraint. This results in a convex constrained optimization as a safety filter. We analyze the feasibility of the resulting optimization and provide the necessary and sufficient conditions for feasibility. The effectiveness of the proposed strategy is validated through two numerical results.
Paper Structure (13 sections, 9 theorems, 50 equations, 3 figures)

This paper contains 13 sections, 9 theorems, 50 equations, 3 figures.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND AND PRELIMINARIES
  3. High-order control barrier functions
  4. Impact of model uncertainty on hocbfs
  5. PROPOSED GP-BASED HOCBF DESIGN
  6. GP for real-valued functions
  7. GP for uncertainty-aware HOCBF
  8. Confidence bounds for the estimation of the effect of uncertainty
  9. Feasibility analysis
  10. Simulation results
  11. Adaptive cruise control with collision avoidance
  12. Active suspension system
  13. Conclusion

Key Result

Corollary 1

Given CBF $h:\mathbb{R}^n \rightarrow \mathbb{R}$ with the associated set $\mathcal{C}$ in (eq3), if $\nabla h(\bm x) \neq 0$ for all $\bm x \in \partial \mathcal{C}$, any Lipschitz continuous controller $\boldsymbol{u}(\boldsymbol{x})$ satisfying (eq4) guarantees that $\mathcal{C}$ is forward invar

Figures (3)

  • Figure 1: The state variables and control input in the presence of uncertainty when applying the GP-based SOCP-HOCBF (yellow) compared to the nominal QP-HOCBF (blue) and the QP-HOCBF design based on the true system (red dashed).
  • Figure 2: Active suspension system.
  • Figure 3: Comparison of the GP-based SOCP-HOCBF (yellow), nominal QP-HOCBF (blue), and true QP-HOCBF (dashed red). States $x_1$ and $x_3$ of the active suspension system (left column). Control input and the road disturbance profile (right column).

Theorems & Definitions (20)

  • Definition 1: Class $\mathcal{K}$ function khalil2002nonlinear
  • Definition 2: Control barrier function ames2019control
  • Corollary 1
  • Definition 3: Relative degree
  • Definition 4
  • Proposition 1
  • Theorem 1
  • proof
  • Remark 1
  • Proposition 2
  • ...and 10 more