Chapter 7

The Kalman Filter

7.1. Position of problem

The aim of the filtering that we are going to study consists of “best estimating”, in the sense of the classic criteria of least mean squares, a discrete process XK governed by an equation of the form:

image

This process (physical, biological, etc.), called the state process, is what interests users.

It represents, for example, the position, speed and acceleration of a moving object.

This process is inaccessible directly and it is studied by means of a process YK governed by an equation of the form:

image

where YK is called the observation process.

NK and WK are the system noise and the measurement noise respectively and will be explained in more detail in what follows.

The Kalman filter, by its conception, generalizes the optimal filter for non-stationary signals.

It is also recursive: the predicted imageK+1|K is obtained starting from the filtration at the preceding instant imageK|K and the filtration imageK+1|K+1 from its predicted K+1|K and from the measurement ...

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