Safe and Performant Control via Efficient Overapproximation of the Reachable Set Probability Distribution

We consider a nonlinear system subject to an unknown state-dependent disturbance input and assume availability of state-dependent upper and lower bounds on the disturbance that hold with any user-prescribed probability available from, e.g., Gaussian Process estimation. Using methods from mixed monotone systems theory, we then propose an efficient technique for overbounding the probabilistic reachable set of the system for any prescribed probability. Next, we consider a reach-avoid control synthesis problem and propose using a weighted sum of reachability quantiles as the control objective to balance safety and performance. We show via a case study of a kinematic bicycle vehicle model that this approach generally outperforms using a single fixed probability bound.