A finite-time linear filter for discrete-time systems
Michael John Grimble · International Journal of Control · 1980
The discrete-time linear filtering problem is considered where the system is constant and the noise is stationary. An optimal time-invariant linear filter is obtained in z-transfer function matrix form. The filter gives an unbiased minimum variance state estimate at the end of a given fixed filtering interval [ 0, T] This state estimate is the same as that which would be obtained using a time-varying Kalman filter, The estimate is therefore better than that which would be obtained from a Wiener filter at this time. The state estimates within the finite filtering interval can also be an improvement over those from a Wiener filter. The major advantage of this filter is that it may be easily implemented because it is time-invariant. The weighting sequence for the proposed filter is zero for times greater than the fixed filtering interval. The filter, at times t⩽ T, therefore uses the observations in a moving window of length T. The fixed memory length property of the filter has well known advantages in reducing divergence problems. The filter may be used in a multiple interval mode to give optimal state estimates ( same as Kalman estimates) at a number of fixed times. The Kalman filter solution to the above problem may also be calculated directly from the results for the time-invariant filter. As the filtering interval tonds to infinity the time-invariant filter becomes identical to a discrete form of the Wierier filter.