User:Stephen Burnett/Maths

Implementation considerations
It is often the case that the available computing power is insufficient to process the filter equations. There are various techniques available to the designer to reduce the complexity, at the cost of a sub-optimal filter.

Deleting states
If states have small rms values or do not significantly influence states of interest, they may be deleted and the dimension of the filter reduced. Performance of the reduced-order filter can be evaluated by running it against a full-order system model.

Decoupling states
Gelb describes a model where the equations are:


 * $$ \begin{bmatrix} \dot{x_{1}} \\ \dot{x_{2}} \end{bmatrix} = \begin{bmatrix} F_{11} & F_{12} \\ F_{21} & F_{22} \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2}  \end{bmatrix} + \begin{bmatrix} w_{1} \\ w_{2}  \end{bmatrix}$$

If the elements of $$F_{12}$$ and $$F_{21}$$ are small, and if $$w_{1}$$ and $$w_{2}$$ are uncorrelated, a system of dimension N can be reduced to two systems of dimension p and q, where p + q = N:


 * $$\dot{x_{1}} = F_{11} x_{1} + w_{1}$$
 * $$\dot{x_{2}} = F_{22} x_{2} + w_{2}$$

The performance improvement arises from the fact that processing a system of size N requires time $$O(N^2)$$, and $$p^2 + q^2 << N^2$$ for any p and q subject to p + q = N.

Measurement averaging
The measurement update of the filter involves calculation of the Kalman gain matrix '''Kk = Pk HkT Sk-1

The inverse involved can make the time to process every measurement excessive, but to discard too many of them might severely affect the filter performance. Gelb presents a simplified technique of processing a batch of scalar measurements into a single equivalent which can be used in the filter update, and provides references for more elaborate techniques.