How to estimate transfer function from real measurements data (Partially known)

[WCR · Accepted solution]

A very simple model with magnitude and phase plots similar to the ones you posted, is a L-R-C band pass filter, plus a delay.  It has only three adjustable parameters, whereas the fitted transfer function you tried has 11 adjustable parameters (4 numerator coefficients and 7 denominator coefficients). One chooses L, and sets C equal to L.  By choosing L and C you have chosen the undamped natural frequency, w0 = 1/sqrt(LC).  Then one chooses R, which sets the sharpness or "Q" of the filter.  Then one chooses the delay, to get the slope of the phase about right at high frequencies.  I chose L=C=0.06, R=0.09, and Tdelay=0.07 s.  This was done in a few trials by eye-balling the resulting magnitude and phase curves, all in an Excel spreadsheet.  One figure shows the LRC circuit.  The delay element is not shown, but can be cascaded upstream or downstream of the filter circuit.  The other figure shows the magnitude and phase response.

[Nemesis]

WCR it was brilliant !!! I did not take into consideration delay in my system. I do not know why:/ I was that deeply confident what I was doing that I thought that it is the best way...but I was wrong. Sometimes a fresh look from another person is the best solution. Now my transfer function reduce a lot. I will do simulation tonight and tomorrow I will give a shoot in real-time. I will let you know. What is the best I did it manually and I found the same delay that you have just written Tdelay=0.07.

Kamil

[Nemesis]

As I said before a time delay help a lot and thank you WCR to point it. I did experiments with a new found transfer function and it is much better than I had before. I put in the attachment a gain for the found transfer function. Now it is sufficient to have 3 numerators and 5 denominators plus delay. However I have to still improve the way to measure of the excitation signal. I notice (in the attachment) that from some frequency I cannot keep constant mean flow (did you have similar problem or your system is completely different? ). This really disturbes my real measured signal. I cannot have a nice signal for lower mean flow (not shown). Normally I should have different responses of the system for different mean flow rate (It can explain equations that I put above). I will try to use different excitation signal and try to fit the transfer function. 

Thank you again for your help.

Kamil