Analysis and applications of time-triggered stochastic hybrid system

Abstract

In this paper, we provide a new method to derive the exact analytical solutions of the moments for a general class of stochastic hybrid systems. We identified a sub-class of stochastic hybrid systems where stochastic resets change the states of the system, and extend our analysis to the systems in which random resets can change both dynamics and the states of the system. We provide the exact solutions of first and second order moments. Further, we analyze a class of time-triggered stochastic hybrid systems where the state-space evolves as per a linear time-invariant dynamical system. This continuous time evolution is interspersed with two kinds of stochastic resets. The first reset occurs based on an internal timer that measures the time elapsed since it last occurred. Whenever the first reset occurs the states-space undergoes a random jump and the timer is reset to zero. The second reset occurs based on an arbitrary timer-depended rate, and whenever this reset fires, the state-space is changed based on a given random map. For this class of systems, we provide exact conditions that lead to finite statistic moments, and the corresponding exact analytical expressions for the first two moments.