Moment dynamics for a class of time-triggered stochastic hybrid systems

TL;DR

This paper introduces a class of stochastic hybrid systems with closed-form moment dynamics, enabling exact computation of moments for systems with random event timings and state changes, relevant to control and biological systems.

Contribution

It identifies linear time-triggered SHS with closed moment dynamics and models event timing using phase-type processes, providing exact moment calculations.

Findings

Closed-form differential equations for moments in TTSHS

First and second moments depend only on mean event interval

Applicable to network control and systems biology

Abstract

Stochastic Hybrid Systems (SHS) constitute an important class of mathematical models that integrate discrete stochastic events with continuous dynamics. The time evolution of statistical moments is generally not closed for SHS, in the sense that the time derivative of the lower-order moments depends on higher-order moments. Here, we identify an important class of SHS where moment dynamics is automatically closed, and hence moments can be computed exactly by solving a system of coupled differential equations. This class is referred to as linear time-triggered SHS (TTSHS), where the state evolves according to a linear dynamical system. Stochastic events occur at discrete times and the intervals between them are independent random variables that follow a general class of probability distributions. Moreover, whenever the event occurs, the state of the SHS changes randomly based on a probability distribution. Our approach relies on embedding a Markov chain based on phase-type processes to model timing of events, and showing that the resulting system has closed moment dynamics. Interestingly, we identify a subclass of linear TTSHS, where the first and second-order moments depend only on the mean time interval between events and invariant of their higher-order statistics. Finally, we discuss applicability of our results to different application areas such as network control systems and systems biology.