Why do we need pole in the lead compensator?

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What if I use a controller (s+a) instead of a lead compensator (s+a)/(s+b), since (s+a) can also shift the phase. (There seems to be no need to bring the phase back by adding a pole.)

You can certainly use a controller of the form (s+a). In fact, this is a proportional derivative controller. However, this controller adds increased magnitude to the loop transfer functions at high frequency. High frequency inputs are generally noise, which we'd like to cancel, so it is typically bad to have the magnitude of the loop transfer function be high at large frequencies. If the system already has several poles then a single zero may not matter, but if not then the PD controller could amplify high frequency error.

--Mary Dunlop 17:30, 10 November 2006 (PST)