  The backup control barrier function (CBF) was recently proposed as a
tractable formulation that guarantees the feasibility of the CBF quadratic
programming (QP) via an implicitly defined control invariant set. The control
invariant set is based on a fixed backup policy and evaluated online by forward
integrating the dynamics under the backup policy. This paper is intended as a
tutorial of the backup CBF approach and a comparative study to some benchmarks.
First, the backup CBF approach is presented step by step with the underlying
math explained in detail. Second, we prove that the backup CBF always has a
relative degree 1 under mild assumptions. Third, the backup CBF approach is
compared with benchmarks such as Hamilton Jacobi PDE and Sum-of-Squares on the
computation of control invariant sets, which shows that one can obtain a
control invariant set close to the maximum control invariant set under a good
backup policy for many practical problems.
