Tractable Compositions of Discrete-Time Control Barrier Functions with Application to Driving Safety Control

This paper introduces control barrier functions for discrete-time systems, which can be shown to be necessary and sufficient for controlled invariance of a given set. In particular, we propose nonlinear discrete-time control barrier functions for control affine systems with an additional structure that lead to controlled invariance conditions that are affine in the control input, resulting in a tractable formulation that enables us to handle the safety optimal control problem for a broader range of applications with more complicated safety conditions than existing approaches. Moreover, we develop alternative mixed-integer formulations for basic and secondary Boolean compositions of multiple control barrier functions and further provide mixed-integer constraints for piecewise control barrier functions. Finally, we apply these proposed tools to driving safety problems of lane keeping and obstacle avoidance, which are shown to be effective in simulation.