The Ziegler-Nichols method of tuning PID control loops has been around since the 1940's, and has been widely applied in many areas of engineering. I understand that it is really only a heuristic method of finding the right control parameters, and many others have been created over the years. However, when it was originally created, was there any mathematical reason for how the method worked. I am talking specifically about the open-loop step response method. I cannot find any reason for why they decided upon the method they did. Is there any mathematical reason behind this?
This is supposed to be a comment rather than an answer, because I only reference I understand that it is really only a heuristic method:
You might also want to consider Ackermann's formula for determining the control parameters, which would be a mathematical way with which you might find parallels to Ziegler-Nichols method. If and when you do find parallels, please consider posting your findings here as I am very interested in the results. Thanks. I hope it helps.
EDIT (01/04/20): Note that Ackermann's formula is only valid for invertible controllability matrix (full rank), i.e. when your system is controllable in Kalman's sense.