# Linearization of dataset generated from s-function model

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Mohammed Hamza on 22 Oct 2022
Commented: Rajiv Singh on 7 Nov 2022
Hello everyone,
I have a highly coupled MIMO nonlinear system that Initially intended to fit into a linear model using system identification toolbox. When I used the system identification toolbox it gave me a responses with fair fit and dynamics. Unfortunatly, when I test the models later using a step reponse, the model goes crazy.
After that, I thought about linearizing the system first then using the system identification toolbox. I tried the linearization toolbox, however it didn't work because the level 2 s-function doesn't support the jocobians.
Does anyone have an idea about how can I solve this problem? I would really appritiate it if anyone could help me with that.
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### Answers (1)

Rajiv Singh on 7 Nov 2022
Hard to say without looking at the actual nonlinear model, but note that in general a "linearization" is valid only in a small local neighborhood of the operating point (state and input values) at which the linear model is generated. There is no guarantee that the approximation will hold for other signal profiles or at other points in the operating space.
For data based linearization (i.e., collect small-excitation response of the model at a given operating point and use it for fitting a linear model), you can conduct a series of experiments wherein you change the size/sign of your input signal and initial conditions incrementally and study the change in the resulting model for each profile. This way, you can roughly carve out the size of the operating space region over which a linearization would be valid.
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Rajiv Singh on 7 Nov 2022
In practice, another effctive way is to do a series of sinnusoidal experiments over a range of frequencies, and study the resulting frequency response for things like:
• is the response harmonic?
• if so, is the response corresponding to a given input frequency also harmonic with the same frequency?
The frestimate command in Simulink Control Design automated this analysis. Some examples:

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