The Extended Kalman Filter: An Interactive Tutorial for Non-Experts

Part 9: Modifying the Estimates

Here again are our more realistic / more general equations for the state and observation variables of our system: \[ x_k = a x_{k-1} + b u_k\] \[ z_k = c x_k + v_k\] As we might expect, the introduction of these new components into our model requires a corresponding modification to the prediction and update equations:

Predict:

$\hat{x}_k = a\hat{x}_{k-1}$ $ + b u_k $
\[p_k = ap_{k-1}a \] Update:
$g_k = p_k$ $c$ $/ ($$c$ $p_k$ $c$$~+~r)$

$\hat{x}_k \leftarrow \hat{x}_k + g_k(z_k - $$c$ $\hat{x}_k)$

$p_k \leftarrow (1 - g_k $$c$) $p_k$

Here is an extension of our airplane demo, showing a longer duration of time and adding in a control signal representing the pilot steadily pulling back on the control column to raise the altitude of the plane. Try moving around the sliders to adjust the values of the different constants. As in the previous demo, the original signal is shown blue, the observed signal in red, and the Kalman-filtered signal in green.

$a =$ 0.95 $r = $ 100
$b =$ 0.5
$c =$ 1.0    

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