Solution of the linear-estimation problem in the s-domain

The finite-time optimal linear-estimation problem is considered where the system is assumed constant and the noise is stationary. A transfer function form of estimator is defined which will give an optimal state estimate at some chosen time T. It is shown in general that this time-invariant estimator cannot be realised by simply using a constant gain Kalman estimator. The time-invariant estimator will, for a given observation signal, give the same state estimate as that from the Kalman estimator in both the fixed-point filtering problem and in the fixed-interval prediction problem. The s-domain solution for the time-invariant estimator contains the solution to the Wiener estimation problem and enables the Kalman estimator to be determined. The time-varying Kalman gain matrix can be calculated directly from the transfer function matrix for the time-invariant estimator. A suboptimal Kalman estimator is also defined from the s-domain results. This has a time-varying gain matrix which can be calculated relatively easily. The suboptimal and time-invariant estimators are both simpler to implement than the continuous-time Kalman estimator.