A: Ziegler and Nichols used an interacting PID controller in their studies, but it was even more interacting than the current interacting model, whose proportional gain is multiplied by the factor (1 + D/I), where D and I are derivative and integral time constants respectively. Its positive and negative feedback loops (providing integral and derivative action respectively) around the amplifier were in parallel. As a result, the controller proportional gain was multiplied by the factor (1 + D/I)/(1 - D/I), This causes the gain to reverse signs as D > I, which was carefully avoided. As a result, they kept the D/I ratio at 1:4, rather than my choice of 1:2.7.
For more detail, see Shinskey, Feedback Controllers for the Process Industrues, McGraw-Hill, 1994, p.71-73, regarding the Taylor Fulscope controller.
A: The PID equation typically used for the digital feedback loop control is designed to approximate traditional pneumatic controllers that were in use when Ziegler and Nichols created their loop-tuning methods. This form of the equation is called the non-interacting or standard form. Here it is described in a clip from the Wikipedia page on PID loop control: http://en.wikipedia.org/wiki/PID_controller#Alternative_nomenclature_and_PID_forms.
The form of the PID controller most often encountered in industry, and the one most relevant to tuning algorithms is the standard form. In this form the Kp gain is applied to the Ioutand Dout terms, yielding:
Ti is the integral time
Td is the derivative time.
In this standard form, the parameters have a clear physical meaning. In particular, the inner summation produces a new single error value, which is compensated for future and past errors. The addition of the proportional and derivative components effectively predicts the error value at Ti seconds (or samples) in the future, assuming that the loop control remains unchanged. The integral component adjusts the error value to compensate for the sum of all past errors, with the intention of completely eliminating them in seconds (or samples). The resulting compensated single error value is scaled by the single gain Kp.
In the ideal parallel form, shown in the controller theory section:
Clearly, adjustment of the Kp term affects the gain of the integral and derivative terms, Ki = Kp/Ti and Kd = KpTd, but this is still referred to as the non-interactive form by tradition. Wikipedia goes on to show the parallel form of the PID equation typically used outside process control in which each of the three terms is independent.
A: The best answer is "neither." The Ziegler-Nichols tuning methods were developed for a Taylor Fulscope pneumatic controller. (Both Ziegler and Nichols were employees of Taylor Instrument Co., Rochester, NY.) Taylor was an early leader in pneumatic instruments. Over the years, the company was acquired by a number of companies, and now is a part of ABB. They developed these rules before control theory, transfer functions, etc. became common knowledge for control engineers. I've done a detailed analysis of the Fulscope, and though I don't recall the exact transfer function I came up with, I do recall that it was similar to, but did not exactly match, either the interacting nor the non-interacting form of PID.
Perhaps a more appropriate question to ask would be "which form of PID are these tuning relations best suited for?" Suppose there had not have been a Ziegler and Nichols, and the relations were suddenly discovered, say, under a rock or floating in the sea in a bottle, with no indication of source nor intended use. Then, the only question that could be asked is, "What should they be used for?"
In my opinion, based on numerous simulation studies using a variety of process models, they are best suited to the non-interactive form of PID. Then, if one has an interactive PID, there are widely published conversion relations to go from one form to the other.