# Tuning Interacting Controllers

## Is the Famous Ziegler-Nichols (ZN) Open-Loop Tuning/Closed-Loop Tuning Parameter Calculation for Interacting or Non-Interacting PID?

*This column is moderated by Béla Lipták, automation and safety consultant, former Chief Instrument Engineer of C&R, former Yale professor of process control and the editor of the Instrument Engineer's Handbook. If you have automation-related questions for this column, write to liptakbela@aol.com.*

**Q:** I was reading your answer on Controlglobal.com regarding "Using Feed-Forward PID for External Reset." My question is whether the famous Ziegler-Nichols (ZN) open-loop tuning/closed-loop tuning parameter calculation is for interacting or non-interacting PID?

*Kumar Chennai**kumaralstom@hotmail.com*

**A:** The design of the old pneumatic controllers made their interacting behavior unavoidable because their three settings (gain spring, integral and derivative restriction) were physically interconnected (Figure 1) and, therefore, if one setting changed, it affected all.

One of the advantages of this design was that the actual working derivative (D) could never become more than one quarter of the integral (I). This feature provided safety, because if D > I/4, the controller action reverses, which can cause accidents.

The tuning parameters in those early days were named differently than today: Gain (G) was named proportional band (PB = 100/G), integral (I) was usually given in units of repeats per minute, while today the units of both integral (I) and derivative (D) are in minutes.

Now, turning to the subject of tuning: any controller (interacting, non-interacting or any other), can be tuned by either open- or closed-loop methods, and the dynamic parameters obtained can be converted into PID settings by any of the algorithms developed over the years (ZN, Shinskey, 3 C, Cohen-Coon, etc.).

Open-loop tuning means that we evaluate only the dynamics of the process, while closed-loop means we view the dynamics of the total loop. Open-loop tuning considers the response of the controlled variable only to load change (not setpoint change). The load is "stepped" after bringing the process to a steady state, and then applying a step change (suddenly opening or closing the control valve by some percentage). This step change causes the controlled variable to react. For example, in case of a heat transfer process, increasing the steam valve opening causes the temperature to rise.

After the step change, it takes some time for the controlled variable to react, and this we call the "dead time" (T_{d}). After Td has passed, the process starts to respond. The maximum rate of rise is called the reaction rate, and the process time constant (T) is the time it takes for the controlled variable to reach 63% of the rise (or decay). The process gain is the ratio of the total percent rise (or fall) divided by the size (in percent) of the step change that caused it. A process is easy to control if its Td is short, and T is large (the process response is slow).

In the closed-loop method, the controller is in automatic. The method applied can either be the "ultimate" or the "damped oscillation" method. Seventy years ago, in 1942, Ziegler and Nichols developed the ultimate method. It is applied by determining the ultimate gain (K_{u}) and the ultimate period (P_{u}). K_{u} is the ultimate gain that causes continuous cycling. When the loop is cycling in (un-dampened oscillation), the loop gain 1.0 and the amplitude of the sinusoidal is constant. ZN recommends tuning for quarter-amplitude damping (the amplitude of each cycle is one-fourth of the previous), which occurs when the loop gain is 0.5, meaning that the product of the gains in the loop components (process, sensor, transmitter, controller and valve) is 0.5.

For proportional (P) controllers, the period of oscillation is 2 to 5 dead times; for PI it is 3 to 5; and for PID around 3 periods. This in flow loops results in oscillation periods of 1 to 3 seconds; in level loops, 3 to 30 seconds; in pressure loops, 5 to 100 seconds; in temperature loops, 30 to 20 minutes; and in analytical loops, minutes to hours. For non-interacting PID loops with no dead time, one would set integral (I) in minutes/repeat to a value equaling 50% of the period of oscillation and the derivative time (D) to about 18 % of that period.

The advantages of open-loop tuning include its speed (you do not need to wait for several cycles), the fact that size of the amplitude of upset is predictable, and the test can be performed before the control loop is installed. The disadvantages include that only the process dynamics are determined (the dynamic contributions of the loop components are not) and that, on noisy processes, the inflection point of the reaction curve is hard to determine. What I normally do is to select the preliminary settings by the open-loop method and refine them later with the closed-loop one.

*Béla Lipták**liptakbela@aol.com*

**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.

*Greg Shinskey**shinskey@metrocast.net*

**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:

where

T_{i} is the integral time

T_{d} 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 T_{i} 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 K_{p}.

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, K_{i} = K_{p}/T_{i} and K_{d} = K_{p}T_{d}, 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.

*Dick Caro**Dick@Caro.us*

**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.

*Harold Wade**Hlwade@aol.com*