The Kalman filter: an introduction to the mathematics of linear least mean square recursive estimation

Abstract

This paper introduces the fundamental ideas of Kalman filtering, a recursive estimation technique widely used for continuous estimation of the state of a dynamic system. The estimation problem is posed within the well known Hilbert space framework of classical linear analysis. This permits an easily grasped geometric interpretation which is stripped of cumbersome details that tend to obscure the essential notions. Considerable emphasis is placed on the development of the mathematical models for the state and measurement equations. A practical example of a real‐world dynamic system (the motion of a ship) is used to motivate the form of the state equation required in Kalman filtering, as well as the measurement equation. The recursive estimator and error covariance equations are derived for a one‐dimensional dynamic system using a sequence of geometric visualizations as the derivation proceeds. The more tedious algebraic manipulations, which are not needed for an essential understanding of the derivation, are relegated to Appendices. This approach removes distracting details which often accompany derivations found in the engineering literature, and makes it apparent that Kalman filtering is based on elegant classical mathematics.