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Different PID EquationsThe equation used to describe PID control so far in this chapter is the simplest form, sometimes called theparallel equation, because each action (P, I, and D) occurs in separate terms of the equation, with thecombined effect being a simple sum:

In the parallel equation, each action parameter (Kp, τi, τd) is independent of the others. At first, this may seem tobe an advantage, for it means each adjustment made to the controller should only affect one aspect of its action.However, there are times when it is better to have the gain parameter affect all three control actions (P, I, andD).

An alternate version of the PID equation exists to provide this very functionality. This version is called the Ideal orISA equation:

Here, the gain constant (Kp) is distributed to all terms within the parentheses, equally affecting all three controlactions. Increasing Kp in this style of PID controller makes the P, the I, and the D actions equally more aggressive.

A third version, with origins in the peculiarities of pneumatic and analog electronic circuits, is called the Series orInteracting equation:

Here, the gain constant (Kp) affects all three actions (P, I, and D) just as with the “ideal” equation. Thedifference, though, is the fact that both the integral and derivative constants have an effect on proportionalaction as well! That is to say, adjusting either τi or τd does not merely adjust those actions, but also influencesthe aggressiveness of proportional action.

This “interacting” equation was an artifact of certain pneumatic and electronic controller designs. Back whenthese were the dominant technologies, and PID controllers were modularly designed such that integral andderivative actions were separate hardware modules included in a controller at additional cost beyondproportional-only action, the easiest way to implement the integral and derivative actions was in a way that justhappened to have an interactive effect on controller gain. In other words, this odd equation form was a sort ofcompromise made for the purpose of simplifying the design of the controller hardware.

Interestingly enough, some digital PID controllers still implement the “interacting” PID equation even though it isno longer a necessary artifact of controller design.

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