Pump Transfer Function


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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 at control system and some help would be greatly appreciated.

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.

The best wasy to do this is to linearize the pump characteristic. By this I mean that you create a curve that is the opposite of the pump characteristic. So if you add the characterization curve to the pump curve you would get a straight line - the characterization curve would essentially be the opposite.