I'm studying for a test in Automatic Control and I have some troubles understanding sensitivity functions and complementary sensitivity functions.

There's one assignment from an old exam that sais "Someone suggests that you should reduce perturbations and measurement noise simultaneously. Explain why this is not possible."

The correct answer sais: "Since the sensitivity and complementary sensitivity transfer functions add up to 1, i.e. $S+T=1$, one cannot improve both the output disturbance and measurement error suppression at the same time."

I don't really understand this answer and my textbook is not to much help either, so I would appreciate alot if someone could explain how they got to this answer? Also, is the sensitivity function always representing the perturbations in the system and the complementary sensitivity function the measurement noise? My textbook seem to imply this, but I'm really not sure if this is always true.


1 Answer 1


The text is saying that system perturbations and measurement noise look identical to the controller. That is, a controller will react in the same fashion to both system perturbations and noise in the measurement. By increasing perturbation rejection, you also make the system more susceptible to noise from your sensors.

The sensitivity function is the ability to reject system perturbations. The complementary sensitivity function is the ability to filter noise.



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