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

Abstract

Given a dynamical system modelled via stochastic differential equations (SDEs), we evaluate the safety of the system through its exit time moments. Using appropriate semidefinite positive matrix constraints, an SDP moment-based approach can be used to compute moments of the exit time. However, the approach is impeded when analyzing higher dimensional physical systems as the dynamics are limited to polynomials. Computational scalability is also poor as the dimensionality of the state grows, largely due to the combinatorial growth of the optimization program. We propose methods to make feasible the safety analysis of higher dimensional physical systems. The restriction to polynomial dynamics is lifted by using state augmentation which allows one to generate the optimization for a broader class of nonlinear stochastic systems. We then reformulate the constraints to mitigate the computational limitations associated with an increase in state dimensionality. We employ our methods on two example processes to characterize their safety via exit times and show the ability to handle multi-dimensional systems that were previously unsupported by the existing SDP method of moments.