This paper introduces a class of stochastic hybrid models for the analysis of closed-loop control systems implemented with NASA's Recoverable Computer System (RCS). Such systems have been proposed to ensure reliable control performance in harsh environments. The stochastic hybrid model consists of a stochastic finite-state automaton driven by a Markov input process, which in turn drives a switched linear discrete-time dynamical system. Stability and output tracking performance are analyzed using an extension of the existing theory for Markov jump-linear systems. The theory is then applied to predict the tracking error performance of a Boeing 737 at cruising altitude and in closed-loop with an RCS subject to neutron-induced single-event upsets. The results are validated using experimental data obtained from a simulated neutron environment in NASA's SAFETI Laboratory.