Forward Stochastic Reachability Analysis for Uncontrolled Linear Systems using Fourier Transforms

TL;DR

This paper introduces a scalable Fourier transform-based approach for exact forward stochastic reachability analysis of uncontrolled linear systems with affine disturbances, applicable to bounded or unbounded disturbances, and demonstrates its effectiveness through examples.

Contribution

It presents a novel Fourier transform-based method for exact stochastic reachability analysis, improving scalability and accuracy over traditional approximation methods.

Findings

Provides exact analytical expressions for reachability densities.

Applicable to systems with bounded and unbounded disturbances.

Demonstrates effectiveness on example scenarios.

Abstract

We propose a scalable method for forward stochastic reachability analysis for uncontrolled linear systems with affine disturbance. Our method uses Fourier transforms to efficiently compute the forward stochastic reach probability measure (density) and the forward stochastic reach set. This method is applicable to systems with bounded or unbounded disturbance sets. We also examine the convexity properties of the forward stochastic reach set and its probability density. Motivated by the problem of a robot attempting to capture a stochastically moving, non-adversarial target, we demonstrate our method on two simple examples. Where traditional approaches provide approximations, our method provides exact analytical expressions for the densities and probability of capture.