State estimation in stochastic hybrid Systems with sparse observations

In this note we study the problem of state estimation for a class of sampled-measurement stochastic hybrid systems, where the continuous state x satisfies a linear stochastic differential equation, and noisy measurements y are taken at assigned discrete-time instants. The parameters of both the state and measurement equation depend on the discrete state q of a continuous-time finite Markov chain. Even in the fault detection setting we consider-at most one transition for q is admissible-the switch may occur between two observations, whence it turns out that the optimal estimates cannot be expressed in parametric form and time integrations are unavoidable, so that the known estimation techniques cannot be applied. We derive and implement an algorithm for the estimation of the states x, q and of the discrete-state switching time that is convenient for both recursive update and the eventual numerical quadrature. Numerical simulations are illustrated