Open-Loop Chance Constrained Stochastic Optimal Control via the One-Sided Vysochanskij–Petunin Inequality

While many techniques have been developed for chance constrained stochastic optimal control with Gaussian disturbance processes, far less is known about computationally efficient methods to handle non-Gaussian processes. In this article, we develop a method for solving chance constrained stochastic optimal control problems for linear time-invariant systems with general additive disturbances with finite moments and unimodal chance constraints. We propose an open-loop control scheme for multivehicle planning, with both target sets and collision avoidance constraints. Our method relies on the one-sided Vysochanskij–Petunin inequality, a tool from statistics used to bound tail probabilities of unimodal random variables. Using the one-sided Vysochanskij–Petunin inequality, we reformulate each chance constraint in terms of the expectation and standard deviation. While the reformulated bounds are conservative with respect to the original bounds, they have a simple and closed form, and are amenable to difference of convex optimization techniques. We demonstrate our approach on a multisatellite rendezvous problem.