Wasserstein Distributionally Robust Control Barrier Function using
  Conditional Value-at-Risk with Differentiable Convex Programming
Alaa Eddine Chriat Chuangchuang Sun
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.