Wasserstein Distributionally Robust Control Barrier Function using Conditional Value-at-Risk with Differentiable Convex Programming

TL;DR

This paper introduces a distributionally robust control barrier function (DR-CBF) that uses Wasserstein metrics and CVaR to ensure safety under environmental distributional shifts, employing differentiable convex programming for efficient computation.

Contribution

The paper proposes a novel single-level convex reformulation for CVaR estimation under Wasserstein distributional shift and applies differentiable convex programming to enable gradient-based optimization for safety guarantees.

Findings

Validated safety guarantees under distributional shift in simulations

Demonstrated effectiveness for first and second-order systems

Provided an approximate variant for higher-order systems

Abstract

Control Barrier functions (CBFs) have attracted extensive attention for designing safe controllers for their deployment in real-world safety-critical systems. However, the perception of the surrounding environment is often subject to stochasticity and further distributional shift from the nominal one. In this paper, we present distributional robust CBF (DR-CBF) to achieve resilience under distributional shift while keeping the advantages of CBF, such as computational efficacy and forward invariance. To achieve this goal, we first propose a single-level convex reformulation to estimate the conditional value at risk (CVaR) of the safety constraints under distributional shift measured by a Wasserstein metric, which is by nature tri-level programming. Moreover, to construct a control barrier condition to enforce the forward invariance of the CVaR, the technique of differentiable convex programming is applied to enable differentiation through the optimization layer of CVaR estimation. We also provide an approximate variant of DR-CBF for higher-order systems. Simulation results are presented to validate the chance-constrained safety guarantee under the distributional shift in both first and second-order systems.