Conditional densities for continuous-time nonlinear hybrid systems with applications to fault detection

Continuous-time nonlinear stochastic differential state and measurement equ ations, all of which have coefficients capable of abrupt changes at a rando m time, are considered; finite-state jump Markov chains are used to model t he changes. Conditional probability densities, which are essential in obtai ning filtered estimates for these hybrid systems, are then derived. They ar e governed by a coupled system of stochastic partial differential equations . When the Q matrix of the Markov chain is either lower or upper diagonal, it is shown that the system of conditional density equations is finite-dime nsional computable. These findings are then applied to a fault detection pr oblem to compute state estimates that include the failure time.