PID response sytem types desired info\links

I came across an older manual on industrial electronics that described a couple of different reponse systems that apply in different applications such as first order response and quarterly degenerative response (second one it mentions for tank levels) I would like some information or links that describe the various types of desired responses are best for what types of applications.

sorry the second one is quarterly amplitude decay

well after googling endlessly for various types of system dynamics I believe I have a much better understanding in general rather than at the math level been a bit of a tricky thing to search (keeping the search engine to machine related motions) finally think I have enough material to peruse for now

It depends on just a few things

How much over shoot can be tolerated and how quickly does the PV or actual position need to get the SP or command position.
Even people with the same application will have different opinions. Some may take into account available power, acceleration or velocity limits.

My customers would put our product in the trash compactor if we provided a quarter amplitude decay response. Most customers will not tolerate the overshoot. As a motion control guy I don't think much of ZN and quarter amplitude decays. I also know that temperature and other process control applications can be controlled very precisely with some mathematics and time spend modeling the plant.

Once one has a plant model I have equations to calculate the gains for any kind of response I want as long as the control output doesn't saturate and the plant isn't non-minimum phase.

I know this isn't the answer you want but why not always try for a critically damped response with no over shoot and achieves the SP in the time required? Why put up with overshoot and slow response?

That was one of the reasons why I wanted more info on the subject, recalling earlier discussions and threads posted I realized that my understanding of the system dynamics was lacking also its helping me understand what type of tuning may work best in certain scenarios I do agree with the idea of critically damped systems working in most cases. Would you describe a critically damped system as a first order response?

I made it home to look at that manual I stated earlier and from the descriptions in that manual it describes the quarter amplitude decay resulting in PI is that it could be described the PV does not know its exceeded SP until it happens sort of like driving a car and looking out the side windows to see a stop sign hence the derivitive action which adds the anticipation. So after reading this I will have to agree with a critically damped as being preferred.