Approximate Stochastic Optimal Control for Linear Time Invariant Systems with Heavy-tailed Disturbances

TL;DR

This paper introduces a convex optimization-based open loop control method for linear systems affected by heavy-tailed disturbances, enabling fast, safe control in multi-vehicle scenarios with heavy-tailed noise.

Contribution

It develops a novel quantile reformulation approach for stochastic control under heavy-tailed disturbances using convex optimization and difference-of-convex programming.

Findings

Successfully applied to satellite rendezvous scenarios

Achieves computational efficiency with convex reformulations

Provides safety guarantees despite heavy-tailed noise

Abstract

We propose an open loop control scheme for linear time invariant systems perturbed by multivariate $t$ disturbances through the use of quantile reformulations. The multivariate $t$ disturbance is motivated by heavy tailed phenomena that arise in multi-vehicle planning planning problems through unmodeled perturbation forces, linearization effects, or faulty actuators. Our approach relies on convex quantile reformulations of the polytopic target sets and norm based collision avoidance constraints to enable fast computation. We embed quantile approximations of the Student's $t$ distribution and the beta prime distribution in a difference-of-convex function framework to compute provably safe but likely suboptimal controllers. We demonstrate our method with three satellite rendezvous examples and provide a comparison with particle control.