Approximation and decomposition of singularly perturbed stochastic hybrid systems

This paper considers a singularly perturbed stochastic hybrid system whose state equations are governed by a stochastic switching process, which is modelled as a nearly completely decomposable continuous time finite state Markov chain. Approximate models for the slow mode subsystem are derived over the intervals of rapid switching and their accuracy (the mean-squared errors between the approximate models and the original systems) is quantified. Aggregate models for the subsystem are obtained over a longer period. The almost surely exponential stability of the subsystem is studied. The behaviour of the fast subsystem which depends on the relative size of the perturbation parameters is analysed. The decomposition of the overall system and the switching process together into slow and fast subsystems is investigated. The results derived in the paper are shown to hold when the switching process is stationary and irreducible, and each group of strongly interacting states is irreducible and time reversible. Finally a simple example is detailed to illustrate the aforementioned techniques