Forward Reachability for Discrete-Time Nonlinear Stochastic Systems via Mixed-Monotonicity and Stochastic Order

TL;DR

This paper introduces a novel method to overapproximate the forward stochastic reach sets of discrete-time nonlinear systems using mixed-monotonicity and stochastic order theory, enabling probabilistic safety analysis.

Contribution

It extends mixed-monotone system theory to stochastic orders and develops an algorithm for overapproximating reach sets with probabilistic bounds.

Findings

Effective overapproximation of stochastic reach sets demonstrated on aerospace examples.

The method provides probabilistic guarantees for system states within computed bounds.

Algorithm shows computational efficiency and accuracy in complex nonlinear stochastic systems.

Abstract

We present a method to overapproximate forward stochastic reach sets of discrete-time, stochastic nonlinear systems with interval geometry. This is made possible by extending the theory of mixed-monotone systems to incorporate stochastic orders, and a concentration inequality result that lower-bounds the probability the state resides within an interval through a monotone mapping. Then, we present an algorithm to compute the overapproximations of forward reachable set and the probability the state resides within it. We present our approach on two aerospace examples to show its efficacy.