Editing Kalman filter (section)

=== Update ===
{|
|-
| style="width:180pt;" | [[Innovation (signal processing)|Innovation]] or measurement pre-fit residual
| <math>\tilde{\mathbf{y}}_k = \mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k\mid k-1}</math>
|-
| Innovation (or pre-fit residual) covariance
| <math>\mathbf{S}_k = \mathbf{H}_k {\mathbf{P}}_{k\mid k-1} \mathbf{H}_k^\textsf{T} + \mathbf{R}_k</math>
|-
| ''Optimal'' Kalman gain
| <math>\mathbf{K}_k = {\mathbf{P}}_{k\mid k-1}\mathbf{H}_k^\textsf{T} \mathbf{S}_k^{-1}</math>
|-
| Updated (''a posteriori'') state estimate
| <math>\hat{\mathbf{x}}_{k\mid k} = \hat{\mathbf{x}}_{k\mid k-1} + \mathbf{K}_k\tilde{\mathbf{y}}_k</math>
|-
| Updated (''a posteriori'') estimate covariance
| <math>\mathbf{P}_{k|k} = \left(\mathbf{I} - \mathbf{K}_k \mathbf{H}_k\right) {\mathbf{P}}_{k|k-1} </math>
|-
| Measurement post-fit [[residuals (statistics)|residual]]
| <math>\tilde{\mathbf{y}}_{k\mid k} = \mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k\mid k}</math>
|}

The formula for the updated (''a posteriori'') estimate covariance above is valid for the optimal '''K'''<sub>k</sub> gain that minimizes the residual error, in which form it is most widely used in applications. Proof of the formulae is found in the ''[[#Derivations|derivations]]'' section, where the formula valid for any '''K'''<sub>k</sub> is also shown.

A more intuitive way to express the updated state estimate (<math>\hat{\mathbf{x}}_{k\mid k}</math>) is:

:<math>\hat{\mathbf{x}}_{k\mid k} = (\mathbf{I} - \mathbf{K}_k \mathbf{H}_k) \hat{\mathbf{x}}_{k\mid k-1} + \mathbf{K}_k \mathbf{z}_k</math>

This expression reminds us of a linear interpolation, <math>x = (1-t)(a) + t(b)</math> for <math>t</math> between [0,1]. 
In our case:
* <math>t</math> is the matrix <math>\mathbf{K}_k \mathbf{H}_k</math> that takes values from <math>0</math> (high error in the sensor) to <math>I</math> or a projection (low error).
* <math>a</math> is the internal state <math>\hat{\mathbf{x}}_{k\mid k-1}</math> estimated from the model.
* <math>b</math> is the internal state <math>\mathbf{K}_k \mathbf{z}_k</math> estimated from the measurement, assuming <math>\mathbf{K}_k</math> is nonsingular.
This expression also resembles the [[alpha beta filter]] update step.