Stochastic analysis of linear time-invariant systems with renewal transitions

Stochastic hybrid systems contain both continuous dynamics and discrete random events. Here we consider a subclass of these systems in which the continuous dynamics is governed via linear time-invariant ordinary differential equations. The discrete events are assumed to occur at random times, wherein the time intervals between events are independent and identically distributed random variables. Moreover, whenever the event occurs, the state is reset to a random value, whose statistics is dependent on its value just before the event. For this class of stochastic hybrid systems we give the necessary and sufficient conditions for having finite moments. Further we derive exact analytical expressions for the steady-state moments. Finally, we use our results to study the effect of noise in timing of the events on a linear scalar system. Surprisingly our derivation shows that high level of randomness in timing of events can be beneficial to a system, in the sense that it reduces the fluctuations in the state of the system.