In this article we will show how to do a third order system identification using excel. The identification takes place in the Laplace domain (
) where the optimal system parameters are found by using the gradient descend method.
Consider the following second order system, a DC motor for example:
![\[ Y(s) = \frac{1}{a_3 s^3 + a_2 s^2 + a_1 s} V(s)\]](https://www.johsac.com/wp-content/ql-cache/quicklatex.com-f00a574278265cea23fdffd52c29bc79_l3.png "Rendered by QuickLaTeX.com")
Where 
, 
, 
are the optimal parameters to be found by minimizing the cost function:
Transforming the system to the Z-domain by using Euler’s forward approximation:
![\[ \frac{Y(z)}{V(z)} = \frac{a z^{-3}}{-z^{-3} b - c z^{-2} - d z^{-1} + 1 } \]](https://www.johsac.com/wp-content/ql-cache/quicklatex.com-592880c758646ae2655152f0f46a0b50_l3.png "Rendered by QuickLaTeX.com")
Which leads to the following difference equations:
The cost function to be minimized is :
The validation shows a satisfactory match between the estimated and the real model:
$ls postfix.txt > Johnny