I read this in a magazine years ago, super helpful. I learned the most when I sat down and worked though it by hand. In practice, I've never needed D, PI has been enough to do the job.
In the typical pedagogy, D is only necessary if you feel the PI isn’t responding fast enough (for rapid control). D, being very sensitive to high frequencies, helps with that.
And the reason you’d leave it out is because a PI controller is guaranteed to be stable with positive gains. Throw the D term in there and you have to start checking pole positions to make sure your system doesn’t blow up.
