Improving the Feasibility of Moment-Based Safety Analysis for Stochastic Dynamics

TL;DR

This paper enhances a moment-based safety analysis method for stochastic differential equations, enabling it to handle higher-dimensional and more complex nonlinear systems by reformulating constraints and augmenting states.

Contribution

The authors introduce a state augmentation technique and reformulate optimization constraints to extend moment-based safety analysis to higher-dimensional, nonlinear stochastic systems.

Findings

Successfully applied to multi-dimensional physical systems

Improved computational scalability for higher dimensions

Accurately characterized safety via expected exit times

Abstract

Given a stochastic dynamical system modelled via stochastic differential equations (SDEs), we evaluate the safety of the system through characterizations of its exit time moments. We lift the (possibly nonlinear) dynamics into the space of the occupation and exit measures to obtain a set of linear evolution equations which depend on the infinitesimal generator of the SDE. Coupled with appropriate semidefinite positive matrix constraints, this yields a moment-based approach for the computation of exit time moments of SDEs with polynomial drift and diffusion dynamics. However, the existing moment approach suffers from drawbacks which impede its applicability to the analysis of higher dimensional physical systems. To apply the existing approach, the dynamics of the systems are limited to polynomials of the state - excluding a large majority of real world examples. Computational scalability is also poor as the dimensionality of the state increases, largely due to the combinatorial growth of the optimization program. In this paper, we propose changes to the existing moment method to make feasible the safety analysis of higher dimensional physical systems. The restriction to polynomial dynamics is lifted by using a state augmentation method which allows one to generate the evolution equations for a broader class of nonlinear stochastic systems. We then reformulate the constraints of the optimization to mitigate the computational limitations associated with an increase in state dimensionality. We employ our methodology on two example processes to characterize their safety via expected exit times and demonstrate the ability to handle multi-dimensional physical systems that were previously unsupported by the existing SDP method of moments.