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A compensator is a controller meant to improve characteristics of the open-loop plant so that it can safely be used with feedback control.

For example, there is a (parasitic) feedback system in a commercial audio system (e.g., in a theater) formed from a microphone, the soundboard connected to it, and the speakers in the theater connected to the soundboard. If the microphone is positioned in certain locations near the speakers and there is enough gain in the system, a single tone (and possibly its higher harmonics) can be reinforced by the feedback system causing it to grow and grow until it is deafeningly loud.

It is then the sound engineers' job to "compensate" this system at the soundboard by adjusting the system's gain and phase characteristics around that particular frequency. So here, the soundboard is certainly a controller (compensation is a control topic), but the controller is not responsible for tracking or regulation. Instead, it is enforcing stability of the closed-loop system at all frequencies.

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Q: Difference between controllers and compensators in control system?
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