On optimal estimation in linear systems with time delay

Optimal state estimation in linear systems with time delay is investigated. The application of optimum control theory leads to a split boundary value problem. The existence of the solution to the two-point boundary value problem is investigated. It is shown that the adjoint system (costate equations) must be completely controllable to a function (complete observability) with respect to the initial function in order to solve the two-point boundary value problem. Necessary and sufficient conditions are presented. Equations for the optimal estimator, which can be solved on-line, are derived. These equations are applied to an example to illustrate the applicability of the approach.