when starting out tuning, you first need to know the boundaries of your control.
at one extreme, there is no control and the process runs to whatever balance it has.
for instance, a pump will lift water, with a small lift, it will push a lot of water
with a high lift, almost no water.
as an example, you have a pump that does 200lmp with no lift, but with your lift, it does 97 lpm at full speed.
so, with whatever lift you have, say you want your setpoint to be half. you set proportional so it is swinging wildly at setpoint, every movement is exaggerated and it over-corrects continuously. this should result in a specific frequency that is dependant on the process.
think of it this way, the pump can deliver 100 lpm, you want 50
you set the P for 49.9999 and 50.001
it is not possible, so the control over corrects no matter what.
but, there will be a magnitude and a frequency. both measurable.
you de-tune, that to say, 47 and 53 and the magnitude gets smaller and the frequency of correction lengthens.
you de-tune it more and say at 45/55 it gets to 54 and does not change. you have set the window wider than the controller needs for a correction and the frequency goes to infinity.
a note here is that from this point, 44/56 or 25/75 will act the same way, no attempt to change as you are within the window. you want to tighten up your P so as to find that point where there is instability. note that, then back off slightly.
from that point, you can set your integral and that will look at that 4 lpm offset and just add itself up
0 + 4 = 4
4 + 4 = 8
8 + 4 = 12......
in short order you hit 100 (your integral setting) and nudge it back towards your setpoint.
after the first nudge, the process goes to 53.5
old_error = setpoint (50) - error (53.5)
0 + 3.5 = 3.5
3.5 + 3.5 = 7
it will take longer before the next nudge
and each nudge will take longer and longer as you get closer to your setpoint
once you are near to setpoint, say 50.5, AND you have some noise on your readings.
0 + 0.5 = 0.5
0.5 + 0 = 0.5 // noise has reading at 50 , so there was no error
as you can see, this could take a long time between nudges.
if you over shoot, it will be by a count of 1 as your nudge is just changing your output by a small count.
the, your error will be just under yoursetpoint and close, so the frequency of correction will be very long, but control will be very close to setpoint.
so, your question about where to start is, anywhere. then make changes, double or half, to make an an upset.
response time is dependant on the process being controlled.
air pressure moves at the speed of sound.
temperature moves slightly faster than paint dries.
there are ways to calculate based on the process and the reaction of your final control element, but in the end, you will tweak your numbers based on how the control works.
as for derivative, that only comes into use with highly dynamic processes. pressure, flow, balance, etc.
and as the process becomes so dynamic is is almost unstable, then inverse derivative comes into use.
my background has been HVAC ; pressure, flow and temperature.
an industrial engineer calls HVAC 'casual' engineering where 'close enough' is the goal.
the Segway and other balance devices are like magic in comparison.