Risk-Aware Adaptive Control Barrier Functions for Safe Control of Nonlinear Systems under Stochastic Uncertainty

TL;DR

The paper tackles safety in stochastic nonlinear systems with high-relativDegree constraints by introducing Risk-Aware Adaptive CBFs (RACBFs), which fuse Discrete-Time Auxiliary-Variable adaptive CBFs (DAVCBFs) with coherent risk measures. DAVCBFs insert auxiliary variables to relax feasibility in DHOCBF-based safety constraints, preserving forward invariance and enabling multi-step uncertainty handling. RACBFs further incorporate risk measures (notably CVaR) to achieve risk-aware safety without excessive conservatism, offering a principled trade-off between safety and performance. A stochastic unicycle case study demonstrates that RACBFs maintain feasibility and safety under disturbances more reliably than risk-aware or risk-agnostic DHOCBFs, with tunable risk levels reducing conservatism when appropriate.

Abstract

This paper addresses the challenge of ensuring safety in stochastic control systems with high-relative-degree constraints, while maintaining feasibility and mitigating conservatism in risk evaluation. Control Barrier Functions (CBFs) provide an effective framework for enforcing safety constraints in nonlinear systems. However, existing methods struggle with feasibility issues and multi-step uncertainties. To address these challenges, we introduce Risk-aware Adaptive CBFs (RACBFs), which integrate Discrete-time Auxiliary-Variable adaptive CBFs (DAVCBFs) with coherent risk measures. DAVCBFs introduce auxiliary variables to improve the feasibility of the optimal control problem, while RACBFs incorporate risk-aware formulations to balance safety and risk evaluation performance. By extending discrete-time high-order CBF constraints over multiple steps, RACBFs effectively handle multi-step uncertainties that propagate through the system dynamics. We demonstrate the effectiveness of our approach on a stochastic unicycle system, showing that RACBFs maintain safety and feasibility while reducing unnecessary conservatism compared to standard robust formulations of discrete-time CBF methods.

Figures (3)

• Figure 1: Closed-loop trajectories with controllers RACBF (blue, orange, cyan), risk-aware DHOCBF (red) and risk-agnostic DHOCBF (pink). RACBF and risk-aware DHOCBF with $\beta=0.1$ work well for safety-critical navigation under state disturbances.
• Figure 2: First-order functions over time with controllers RACBF (blue), risk-aware DHOCBF (red) and risk-agnostic DHOCBF (pink). RACBF and risk-aware DHOCBF with $\beta=0.1$ effectively maintain the first-order function non-negative under state disturbances.
• Figure 3: Control input $u_{1,t}$ (angular velocity) over time with controllers RACBF (blue), risk-aware DHOCBF (red) and risk-agnostic DHOCBF (pink). Only RACBF with $\beta=0.1$ effectively keeps $u_{1,t}$ within the corresponding bounds under state disturbances, ensuring feasibility when strict bounds are imposed. In contrast, for the other two methods, $u_{1,t}$ exceeds the corresponding bounds, making the optimization problem infeasible under strict input bounds.

Theorems & Definitions (15)

• Definition 1: Relative degree sun2003initial
• Definition 2: DHOCBF xiong2022discrete
• Theorem 1: Safety Guarantee xiong2022discrete
• Definition 3: Conditional Risk Measure
• Definition 4: Coherent Risk Measure
• Remark 1
• Definition 5: DAVCBF
• Theorem 2
• proof
• Definition 6: $\rho$-Forward Invariance