The answer to your question is more of a PID/math theory question .. you'll have to look more at the various PID "forms".
Yeah, I have found this pattern in all binarys I have so it must be done this way for a good reason
Thing is I cant find myself comfortable to tune a function whos logic I dont understand.
The basic form "Kp(P) + Ki(I) + Kd(D)" isn't the only form.
Do you think it can be applied though? For simplicitys sake?