Here is an excerpt from "G. Ellis, Control System Design Guide":
"Compensators are specialized filters ...designed to provide a specific gain and phase shift, usually at one frequency. The effects on gain and phase either above or below that frequency are secondary."
My opinion: There is no clear threshold between "compensators" and "controllers".
Some authors use one or the other term (or mixed). Am I wrong?
To be more precise: I think, each compensator (P, I, PI, PID) can be regarded as a controller - but not vice versa. The bang-bang controller certainly is not a compensator.
