Explicit Solutions for Safety Problems Using Control Barrier Functions

The control barrier function approach has been widely used for safe controller synthesis. By solving an online convex quadratic programming problem, an optimal safe controller can be synthesized implicitly. Since the solution is unique, the mapping from the state-space to the control inputs is injective, thus enabling us to evaluate the underlying relationship. In this paper we aim at explicitly synthesizing a safe control law as a function of the state for nonlinear control-affine systems with limited control ability. We transform the online quadratic programming problem into an offline parameterized optimisation problem which considers states as parameters. The obtained explicit safe controller is shown to be a piece-wise Lipschitz continuous function over the partitioned state space if the program is feasible. We address the infeasible cases by solving a parameterized adaptive control barrier function-based quadratic programming problem. Extensive simulation results show the state-space partition and the controller properties.