  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.
