The Kalman filter between statistics and ordinary differential equations

Jean-Marc Blanc

Ecole d'Ingénieurs de Fribourg, Bd de Pérolles 80 - CP 32, CH-1705 Fribourg, Switzerland

jmarc.blanc@eif.ch

Contributed talk

The Kalman filter is an algorithm that provides an efficient solution of the general problem to estimate the state of a dynamic process governed by the state equation

$${d \vec x\over dt} \, = \, A\cdot \vec x + B\cdot \vec u + B \cdot \vec v$$

and observed by measurements

$$\vec z \, = \, H\cdot \vec x + D\cdot \vec u + E\cdot\vec w$$

where $\vec x$ is the state vector and $\vec u$ the control input.

Both the process and the measurement $\vec z$ are contaminated by independent white noises $\vec v$ and $\vec w$ with normal probability distributions and given covariances.

The Kalman filter is a combination of an iterative algorithm to integrate the state equation with a feedback in form of noisy measurements. In its original form the integration scheme was based on the the first order point slope formula

$$\vec x_{k+1} \, = \, \vec x_k + \Delta t \, \bigl( A\cdot \vec x_k + B\cdot \vec u_k + B \cdot \vec v_k \bigr) .$$

A first example will show that the filter can only partially correct the error due to a poor integration scheme. Using a better algorithm (fourth order Runge-Kutta) brought a significant improvement of the accuracy.

A second example shows the result of the filtering applied to true experimental data.