Probabilistic Reachability of Discrete-Time Nonlinear Stochastic Systems

TL;DR

This paper introduces a unified probabilistic framework for computing reachable sets in discrete-time nonlinear stochastic systems, effectively separating deterministic and stochastic effects using a novel energy function, and validating the approach through case studies and experiments.

Contribution

It presents a new energy function-based approach to tightly bound stochastic trajectories, enabling efficient probabilistic reachability analysis for nonlinear systems.

Findings

The probabilistic bounds are tight for nonlinear systems and exact for linear systems.

The framework effectively combines deterministic and stochastic reachability methods.

Numerical experiments validate the theoretical bounds and the framework's efficiency.

Abstract

In this paper we study the reachability problem for discrete-time nonlinear stochastic systems. Our goal is to present a unified framework for calculating the probabilistic reachable set of discrete-time systems in the presence of both deterministic input and stochastic noise. By adopting a suitable separation strategy, the probabilistic reachable set is decoupled into a deterministic reachable set and the effect of the stochastic noise. To capture the effect of the stochastic noise, in particular sub-Gaussian noise, we provide a probabilistic bound on the distance between a stochastic trajectory and its deterministic counterpart. The key to our approach is a novel energy function called the Averaged Moment Generating Function, which we leverage to provide a high probability bound on this distance. We show that this probabilistic bound is tight for a large class of discrete-time nonlinear stochastic systems and is exact for linear stochastic dynamics. By combining this tight probabilistic bound with the existing methods for deterministic reachability analysis, we propose a flexible framework that can efficiently compute probabilistic reachable sets of stochastic systems. We also provide two case studies for applying our framework to Lipschitz bound reachability and interval-based reachability. Three numerical experiments are conducted to validate the theoretical results.