Modeling, Estimation and Optimal Filtration in Signal Processing

Chapter 5

Kalman Filtering

5.1. Introduction

In the previous chapters, we presented different techniques for the estimation of the parameters of linear models. We noted the non-recursive character of the Wiener filter which was applied to identification. This chapter will present the Kalman filter, a recursive alternative to the Wiener filter. It was first introduced in the 1960s [12] [13]. Kalman transformed the integral equation of the continuous-time Wiener filter to differential equations [27]. Other approaches were also used to obtain the Kalman filter:

– Sage and Masters' technique based on the least squares method [23];

– Athans and Tse's technique based on the Pontriaguin maximum principle [3].

We hope that this chapter will serve as a reader-friendly introduction to the Kalman filter. We present this filter using an algebraic approach which, though it may not be the most elegant, has the advantage of not requiring any prior knowledge. This algebraic description can also be the starting point for more formal descriptions, such as that presented by T. Kailath, A. Sayed and B. Hassibi in [11].

In this description, the Kalman filter is presented with a special emphasis on its utilization in the estimation of model parameters. We also present the so-called “extended” Kalman filter for nonlinear estimation.

5.2. Derivation of the Kalman filter

5.2.1. Statement of problem

Let there be a system which is represented in the state space domain as follows:

The driving process