Note the parenthesis. Which is correct? Is there a standard PID algorithm? Wouldn’t use of the second form (with parenthesis) cause the gain (Kp) to be interactive with the other terms?

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Answer:

There is no “standard” or correct PID algorithm. However, the second (Kp outside of parenthesis) form is the most common. Also, most standard tuning methods, such as Ziegler-Nichols and others, are based on that form.

As to the interaction, the parenthesis form of the equation actually reduces interaction. Using that form the reset (Ki) and derivative (Kd) parameters. The reason is that this form closely matches the process. Using this form, the Ki and Kd parameters relate only to dynamics of the process, and the Kp (proportional gain) relates to the gain of the process.

This is demonstrated by the Closed Loop Ziegler-Nichols tuning method. A controller is tuned with no or minimum reset rate (longest possible reset time) and no or minimum derivative. The gain is increased while the process is disturbed to find the gain at which the controller oscillates with constant oscillations-neither shrinking nor growing. That gain and the period of the oscillation is noted. The desired gain of the controller is calculated from the gain producing the constant oscillation. Ki and Kd are calculated from the period. In other words, if the dynamics of the process changes, using the standard algorithm (Kp affects all terms), Ki and Kd are changed. If the gain of the process changes, only Kp is changed.