Stochastic Optimal Control For Gaussian Disturbances with Unknown Mean and Variance Based on Sample Statistics

We propose an open loop methodology based on sample statistics to solve chance constrained stochastic optimal control problems with probabilistic safety guarantees for linear systems where the additive Gaussian noise has unknown mean and covariance. We consider a joint chance constraint for time- varying polytopic target sets under assumptions that the disturbance has been sufficiently sampled. We derive two theorems that allow us to bound the probability of the state being more than some number of sample standard deviations away from the sample mean. We use these theorems to reformulate the chance constraint into a series of convex and linear constraints. Here, solutions guarantee chance constraint satisfaction almost surely. We demonstrate our method on a satellite rendezvous maneuver and provide comparisons with the scenario approach.