Pump transfer function

[COLOR=#000120][FONT=Times]Hello,[/FONT][/COLOR]
[COLOR=#000120][FONT=Times]I'm working on controlling the flow of a pump. The pump in question has non-linear characteristic and I'm having trouble finding a transfer function so that I can tune the PID controller. Is such a non-linear system even possible to control with a PID-controller? I'm new to control systems and some help would be greatly appreciated.
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[COLOR=#000120][FONT=Times]The attached graphs show how the pump reacts when I make it go from 0-x % of the max flow. The x-axis displays time and the y-axis displays the signal from a flowmeter with a 4-20mA signal. Also please note that we can't read the Dead Time from the graphs since we didn't start the measuring exactly when we gave the pump a signal.

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