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This paper introduces control barrier functions for discrete-time systems,which can be shown to be necessary and sufficient for controlled invariance ofa given set. Moreover, we propose nonlinear discrete-time control barrierfunctions for partially control affine systems that lead to controlledinvariance conditions that are affine in the control input, leading to atractable formulation that enables us to handle the safety optimal controlproblem for a broader range of applications with more complicated safetyconditions than existing approaches. In addition, we develop mixed-integerformulations for basic and secondary Boolean compositions of multiple controlbarrier functions and further provide mixed-integer constraints for piecewisecontrol barrier functions. Finally, we apply these discrete-time controlbarrier function tools to automotive safety problems of lane keeping andobstacle avoidance, which are shown to be effective in simulation.

Tractable Compositions of Discrete-Time Control Barrier Functions with Application to Lane Keeping and Obstacle Avoidance