Moment Analysis of Stochastic Hybrid Systems Using Semidefinite Programming

TL;DR

This paper introduces a semidefinite programming approach to estimate bounds on moments of stochastic hybrid systems, providing a systematic way to analyze complex systems with polynomial dynamics and extending to certain non-polynomial cases.

Contribution

It develops a novel semidefinite programming method for bounding moments in stochastic hybrid systems, including a reduction technique for multi-mode systems and extensions to non-polynomial systems.

Findings

Bounds improve with larger semidefinite programs

Method applies to diverse disciplines' systems

Extension to non-polynomial systems via state augmentation

Abstract

This paper proposes a semidefinite programming based method for estimating moments of a stochastic hybrid system (SHS). For polynomial SHSs -- which consist of polynomial continuous vector fields, reset maps, and transition intensities -- the dynamics of moments evolve according to a system of linear ordinary differential equations. However, it is generally not possible to solve the system exactly since time evolution of a specific moment may depend upon moments of order higher than it. One way to overcome this problem is to employ so-called moment closure methods that give point approximations to moments, but these are limited in that accuracy of the estimations is unknown. We find lower and upper bounds on a moment of interest via a semidefinite program that includes linear constraints obtained from moment dynamics, along with semidefinite constraints that arise from the non-negativity of moment matrices. These bounds are further shown to improve as the size of semidefinite program is increased. The key insight in the method is a reduction from stochastic hybrid systems with multiple discrete modes to a single-mode hybrid system with algebraic constraints. We further extend the scope of the proposed method to a class of non-polynomial SHSs which can be recast to polynomial SHSs via augmentation of additional states. Finally, we illustrate the applicability of results via examples of SHSs drawn from different disciplines.