Kalman Filters

Abstract

The Gaussian concept of estimation by least squares, originally stimulated by astronomical studies, has provided the basis for a number of estimation theories and techniques during the ensuing 170 years-probably none as useful in terms of today's requirements as the Kalman filter.
	H. W. Sorenson University of California, San Diego IEEE Spectrum, vol. 7, July 1970

In 1960, R.E. Kalman published his famous paper describing a recursive solution to the discrete-data linear filtering problem [Kalman60]. Since that time, due in large part to advances in digital computing, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation.

The Kalman filter is a set of mathematical equations that provides an efficient computational (recursive) solution of the least-squares method. The filter is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown.

For this STC lecture our goal is to provide a practical one-hour introduction to the Kalman filter. This includes the following:

some intuition about the filter operation,
the introduction of the basic discrete Kalman filter equations,
a relatively simple (tangible) example with real numbers & results,
highlights of some applications, and
a Kalman filter handout and pointers to more detailed reading.

A very "friendly" introduction to the general idea of the Kalman filter can be found in Chapter 1 of [Maybeck79], while a more complete introductory discussion can be found in [Sorenson70], which also contains some interesting historical narrative. More extensive references include [Gelb74], [Maybeck79], [Lewis86], [Brown92], and [Jacobs93].

References

Kalman60: Kalman, R. E. 1960. "A New Approach to Linear Filtering and Prediction Problems," Transaction of the ASME-Journal of Basic Engineering, pp. 35-45 (March 1960).

Maybeck79: Maybeck, Peter S. 1979. Stochastic Models, Estimation, and Control, Volume 1, Academic Press, Inc.

Sorenson70: Sorenson, H. W. 1970. "Least-Squares estimation: from Gauss to Kalman," IEEE Spectrum, vol. 7, pp. 63-68, July 1970.

Gelb74: Gelb, A. 1974. Applied Optimal Estimation, MIT Press, Cambridge, MA.

Lewis86: Lewis, Richard. 1986. Optimal Estimation with an Introduction to Stochastic Control Theory, John Wiley & Sons, Inc.

Brown92: Brown, R. G. and P. Y. C. Hwang. 1992. Introduction to Random Signals and Applied Kalman Filtering, Second Edition, John Wiley & Sons, Inc.

You can read the paper or print it, or print the slides in thumbnail form.

Brook Conner

Kalman Filters

Gary Bishop and Greg Welch

UNC

Abstract

References