Mathematical and Simulation models for control valves

Valve position shoule be a linear function of water flow through control valve, which is virtually the same thing. Water flow into system apparently provides input to proportion flow through valve -- is that correct? Need more info.

Flow through a control valve is proportional to the pressure drop across it. The missing data from the post is "what happens downstream?", which confounds issues, as increasing flow may cause reduced pressure drop at the valve depending on the process equipment configuration (which is why equal percentage characteristics, rather than linear, are frequently specified for a valve working on liquids).

Flow control can be achieved by proportional-only action at the controller, though the loop will always have an offset from the set point. To eliminate the offset, introduce a little integral action. Increasing integral action will result in a faster response to disturbances and more overshoot. Derivative action will tend to reduce the overshoot. Far too much integral action will result in oscillation that may be potentially damaging to the process or the components of the system.

If the selected controller has an "auto-tune" facility, like many off-the-shelf products these days, use that to determine the correct controller settings. If it hasn't then the only solution is to do it the time-consuming way, which is to disturb the loop in proportional-only, see what happens, and turn up the gain repeatedly until it just begins to oscillate. Once the period of oscillation is known, the controller parameters are then set to the values given in the Ziegler-Nicholls formulae either for fastest response or minimum overshoot, depending on circumstance. Currently Wikipaedia knows nothing about Zeigler-Nicholls, the textbook containing it is not to hand and it's a difficult thing to remember unless one is using it regularly. There may be a colleague nearby who can advise?