Safety-critical control of discontinuous systems with nonsmooth safe sets

This paper studies the design of controllers for discontinuous dynamics that ensure the safety of non-smooth sets. The safe set is represented by arbitrarily nested unions and intersections of 0-superlevel sets of differentiable functions. We show that the satisfaction of the point-wise active safety constraints only does not necessarily imply safety. This rules out the standard techniques developed for safety of continuous dynamics. This motivates the introduction of the notion of transition functions, which allow us to incorporate even the inactive safety constraints without falling into unnecessary conservatism. These functions allow system trajectories to leave a component of the nonsmooth safe set to transition to a different one. The resulting controller is then defined as the solution to a convex optimization problem, which we show is feasible and continuous wherever the system dynamics is continuous. We illustrate the effectiveness of the proposed design approach in a multi-agent coverage control problem.