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Different PID Equations

The equation used to describe PID control so far in this chapter is the simplest form, sometimes called the parallel equation, because each action (P, I, and D) occurs in separate terms of the equation, with the combined 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 to be 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, and D).

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


Here, the gain constant (Kp) is distributed to all terms within the parentheses, equally affecting all three control actions. 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 or Interacting equation:


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

This “interacting” equation was an artifact of certain pneumatic and electronic controller designs. Back when these were the dominant technologies, and PID controllers were modularly designed such that integral and derivative actions were separate hardware modules included in a controller at additional cost beyond proportional-only action, the easiest way to implement the integral and derivative actions was in a way that just happened to have an interactive effect on controller gain. In other words, this odd equation form was a sort of compromise 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 is no longer a necessary artifact of controller design.

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