Optimal Filtering for Linear Systems with Multiple State and Observation Delays

In this paper, the optimal filtering problem for linear systems with multiple state and observation delays is treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate, error variance, and various error covariances. The resulting system of equations for determining the filter gain matrix consists, in the general case, of an infinite set of equations. It is however demonstrated that a finite set of the filtering equations can be obtained in the particular case of equal or commensurable (tau_j =q_jh, q_j are natural) delays in the observation and state equations. In the example, performance of the designed optimal filter for linear systems with state and observation delays is verified against the best Kalman-Bucy filter available for linear systems without delays