Solution of the Kalman filtering problem for stationary noise and finite data records
Michael John Grimble · International Journal of Systems Science · 1979
A solution is obtained in the s-domain to the finite-time optimal filtering problem where the system is constant and the noise is stationary. This enables the time-varying Kalman filter gain matrix to be calculated using transform techniques. In the solution of this problem, to find the optimal time-varying linear filter, a new fixed point filtering problem is posed and solved. That is, the optimal time-invariant filter is found which will give the best state estimate after some fixed time interval T. This state estimate is the same as would be obtained from the equivalent Kalman filter at this time. It follows that this time-invariant filter can give a better state estimate than the Wiener filter, when the filtering time is finite Sub-optimal versions of both the time-invariant filter and the Kalman filter are defined from a simple approximation which can be made in the s-domain solution. This considerably simplifies the calculation of the gain matrix for the continuous-time Kalman filter. There are important advantages over the Kalman filter in the implementation of both the time-invariant filter and the sub-optimal filters.