Learning Performance-Oriented Control Barrier Functions Under Complex
  Safety Constraints and Limited Actuation
Shaoru Chen Mahyar Fazlyab
Control Barrier Functions (CBFs) provide an elegant framework for designing
safety filters for nonlinear control systems by constraining their trajectories
to an invariant subset of a prespecified safe set. However, the task of finding
a CBF that concurrently maximizes the volume of the resulting control invariant
set while accommodating complex safety constraints, particularly in high
relative degree systems with actuation constraints, continues to pose a
substantial challenge. In this work, we propose a novel self-supervised
learning framework that holistically addresses these hurdles. Given a Boolean
composition of multiple state constraints that define the safe set, our
approach starts with building a single continuously differentiable function
whose 0-superlevel set provides an inner approximation of the safe set. We then
use this function together with a smooth neural network to parameterize the CBF
candidate. Finally, we design a training loss function based on a
Hamilton-Jacobi partial differential equation to train the CBF while enlarging
the volume of the induced control invariant set. We demonstrate the
effectiveness of our approach via numerical experiments.