Moment Propagation of Discrete-Time Stochastic Polynomial Systems using Truncated Carleman Linearization

TL;DR

This paper introduces a novel method using Carleman linearization to approximate moments of discrete-time stochastic polynomial systems, enabling effective safety analysis and moment computation.

Contribution

It develops a truncated Carleman linearization approach to approximate moments of stochastic polynomial systems with error bounds, enhancing safety analysis capabilities.

Findings

Accurately computes moments for small degrees and time steps

Provides explicit error bounds for moment approximations

Demonstrates effectiveness on logistic map and vehicle dynamics examples

Abstract

We propose a method to compute an approximation of the moments of a discrete-time stochastic polynomial system. We use the Carleman linearization technique to transform this finite-dimensional polynomial system into an infinite-dimensional linear one. After taking expectation and truncating the induced deterministic dynamics, we obtain a finite-dimensional linear deterministic system, which we then use to iteratively compute approximations of the moments of the original polynomial system at different time steps. We provide upper bounds on the approximation error for each moment and show that, for large enough truncation limits, the proposed method precisely computes moments for sufficiently small degrees and numbers of time steps. We use our proposed method for safety analysis to compute bounds on the probability of the system state being outside a given safety region. Finally, we illustrate our results on two concrete examples, a stochastic logistic map and a vehicle dynamics under stochastic disturbance.