Stochastic Hybrid Systems with renewal transitions

We consider Stochastic Hybrid Systems (SHSs) for which the lengths of times that the system stays in each mode are independent random variables with given distributions. We propose an analysis framework based on a set of Volterra renewal-type equations, which allows us to compute any statistical moment of the state of the SHS. Moreover, we provide necessary and sufficient conditions for various stability notions, and determine the exponential decay or increase rate at which the expected value of a quadratic function of the systems´ state converges to zero or to infinity, respectively. The applicability of the results is illustrated in a networked control problem considering independently distributed intervals between data transmissions and delays.