Characterising a system from its input and output
Use the matlab cpsd (cross power spectral density) function on your two signals.
That will give you the frequency response.
Do an xcorr (cross correlation) to find the time delay between the two signals.
Do this for several pairs of input and output files and see if the results match between different pairs.
If they are (reasonably) similar then you could create an FIR filter that reproduces the frequency response and delay.

You could then apply the filter to a set of input data to approximate the expected output.

If there is a PID controller in the middle, however,  I don't know if this procedure will be enough.