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 X _K governed by an equation of the form:

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

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 Y _K governed by an equation of the form:

Y _K is called the observation process.

N _K and W _K are the system noise and the measurement noise respectively and will be explained in more detail in what follows.

The Kalman filter, with its creation, brought into widespread use the optimal filter for non-stationary systems.

It is also recursive: the predicted is obtained starting from the filtration at the preceding instant and the filtration from its predicted and from the measurement of the process Y _K+1 at the instant that we are making our estimation.

Moreover, if the observable system is known and linear, the objective consists of, starting from measurements of the system, determining the best possible estimate in the sense of the criteria specified above.

If the observable system is known but non-linear an approximate solution can be given by effecting a linearization of the equations of state and of the observations...