State Estimation and Prediction in a Class of Stochastic Hybrid Systems

We consider a dynamical system whose state equation evolves continuously in time according to a linear stochastic differential equation; the parameters of such SDE depend on a discrete variable that follows the laws of a continuous-time Markov process. Noisy measurements of the continuous state are made available at discrete deterministic times, by a static linear equation whose parameters depend, again, on the discrete state. Therefore the discrete state may switch between different values between successive measures. We solve the problem of estimating both the continuous and the discrete state, given the measurements up to a certain time, in an on-line manner. Models like the one we analyze arise naturally in industrial applications such as fault detection.