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Chance Constrained Stochastic Optimal Control Based on Sample Statistics With Almost Surely Probabilistic Guarantees

While techniques have been developed for chance constrained stochastic optimal control using sample disturbance data that provide a probabilistic confidence bound for chance constraint satisfaction, far less is known about how to use sample data in a manner that can provide almost surely guarantees of chance constraint satisfaction. In this paper, we develop a method for solving chance constrained stochastic optimal control problems for linear time-invariant systems with unknown but sampled additive disturbances. We propose an open-loop control scheme for multi-vehicle planning, with both target sets and collision avoidance constraints. To this end, we derive a concentration inequality to bound the tail probability of a random variable based on sample statistics. Using the derived concentration inequality, we reformulate each chance constraint in terms of the sample mean and sample standard deviation. While the reformulated bounds are conservative with respect to the original bounds, they have a simple and closed form, guarantee chance constraint satisfaction almost surely, and are amenable to difference of convex optimization techniques. We demonstrate our approach on two multi-satellite rendezvous problems.

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