  Inspired by the success of imitation and inverse reinforcement learning in
replicating expert behavior through optimal control, we propose a learning
based approach to safe controller synthesis based on control barrier functions
(CBFs). We consider the setting of a known nonlinear control affine dynamical
system and assume that we have access to safe trajectories generated by an
expert - a practical example of such a setting would be a kinematic model of a
self-driving vehicle with safe trajectories (e.g., trajectories that avoid
collisions with obstacles in the environment) generated by a human driver. We
then propose and analyze an optimization-based approach to learning a CBF that
enjoys provable safety guarantees under suitable Lipschitz smoothness
assumptions on the underlying dynamical system. A strength of our approach is
that it is agnostic to the parameterization used to represent the CBF, assuming
only that the Lipschitz constant of such functions can be efficiently bounded.
Furthermore, if the CBF parameterization is convex, then under mild
assumptions, so is our learning process. We end with extensive numerical
evaluations of our results on both planar and realistic examples, using both
random feature and deep neural network parameterizations of the CBF. To the
best of our knowledge, these are the first results that learn provably safe
control barrier functions from data.
