Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises, 4th Edition

6

Smoothing and Further Intermediate Topics

Chapters 4 and 5 were devoted entirely to Kalman filtering and prediction. We will now look at the smoothing problem. This is just the opposite of prediction, because in smoothing we are concerned with estimating the random process x in the past, rather than out into the future as was the case with prediction. To be more exact, we will be seeking the minimum-mean-square-error (MMSE) estimate of x (t + α) where a is negative. (Note that α = 0 is filtering and α > 0 is prediction.) The three classifications of smoothing will be discussed in detail in Sections 6.1 through 6.4. Then, in Sections 6.5 through 6.9 we will take a brief look at five other intermediate topics which have relevance in applications. There are no special connections among these topics, so they may be studied in any desired order.

6.1 CLASSIFICATION OF SMOOTHING PROBLEMS

A great deal has been written about the smoothing problem, especially in the early years of Kalman filtering. An older text by Meditch (1) is still a fine reference on the subject. A more recent book by Grewal and Andrews (2) is also an excellent reference. The early researchers on the subject were generally searching for efficient algorithms for the various types of smoothing problems. Computational efficiency was especially important with the computer facilities of the 1960s, but perhaps efficiency is not so much of an issue today. Precision and algorithm stability in processing large amounts of data ...