I do apologise, I should really have explained my iterative tuning approach in a bit more detail.
When I tune a process like the one given, I would typically (but not always) use an iterative method to tune it. If I know that the process includes a pure integrator, I would use Proportional Control only, because I know that then you don't need Integral Control. My iterative tuning would be: with the controller in AUTO, start with very weak Proportional Control action and observe the closed response, you can either rely on natural process disturbances or make setpoint changes. If you observe that there are no sign of closed loop oscillatory behaviour, you increase the proportional control action and repeat this action iteratively, until you observe the initial signs of oscillatory response. If your objective is to find critical damped closed loop response, you end your iterative tuning process the moment you have found the strongest Proportional Control Action before any overshoot on setpoint changes.
Let me just qualify this: I do not say this is THE way to tune controllers. There are obviously many ways, and I will not even try to argue that this way is better than any other way; it depends on too many factors. All I claim is that it does work for me; not always, I can also give you many examples of where this method does not work. Normally my focus would be to get the loop on the real plant to work well, with the minimum of effort. I love maths too, earlier in my career I used to apply control theory to solve practical problems. It did work and I did enjoy it. There is an intellectual satisfaction you get from model a process and apply the nice maths based control theory to design a control system and the to implement it successfully. But, especially if you are pressurised for time, I find that there is a place (again, not always and not for all processes) in the world for the practical iterative tuning method explained above.
Pieter Steenekamp