Chance Constrained Stochastic Optimal Control for Arbitrarily Disturbed LTI Systems Via the One-Sided Vysochanskij-Petunin Inequality

TL;DR

This paper introduces a computationally efficient method for chance constrained stochastic optimal control of LTI systems with non-Gaussian disturbances, utilizing the Vysochanskij-Petunin inequality for tail probability bounds.

Contribution

It develops a novel control approach for non-Gaussian disturbances using the Vysochanskij-Petunin inequality, enabling closed-form reformulation of chance constraints.

Findings

Effective multi-vehicle planning with target and collision constraints

Reformulated chance constraints are conservative but simple and closed-form

Validated approach on a multi-satellite rendezvous problem

Abstract

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 paper, 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 multi-vehicle 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 multi-satellite rendezvous problem.