control//frequency response//sensitivity function

The sensitivity function of a feedback loop is the fraction of a disturbance acting on the output that still reaches the output once the loop has done its work, given at each frequency by \(S=1/(1+L)\) where \(L\) is the loop gain, and engineers use it with its twin, the complementary sensitivity \(T=L/(1+L)\), to see where in frequency a loop rejects disturbances and where it simply passes on its sensor's errors. With \(d\) a disturbance at the output and \(n\) the sensor's noise,

The sensitivity function of a feedback loop is the fraction of a disturbance acting on the output that still reaches the output once the loop has done its work, given at each frequency by S=1/(1+L)S=1/(1+L)S=1/(1+L) where LLL is the loop gain, and engineers use it with its twin, the complementary sensitivity T=L/(1+L)T=L/(1+L)T=L/(1+L), to see where in frequency a loop rejects disturbances and where it simply passes on its sensor's errors. With ddd a disturbance at the output and nnn the sensor's noise,

y=T (r−n)+S d,S+T=1.y=T\,(r-n)+S\,d,\qquad S+T=1 .y=T(r−n)+Sd,S+T=1.

TTT is the share of the setpoint, and of the sensor noise, that reaches the output. The two add up to one at every frequency. Where the loop has high gain (low frequencies) S≈0S\approx0S≈0 and disturbances are rejected, but T≈1T\approx1T≈1 and the sensor's noise and bias go straight through; where the gain is low the noise is ignored and the disturbances pass.

That second half has a blunt consequence: at low frequencies feedback is exactly as good as its sensor. A barometer with a 0.5 m bias gives a drone that hovers 0.5 m off its target and is perfectly regulated there; nothing in the loop can see the error, since the loop corrects towards what the sensor says. That is why estimation comes before control: the state estimation that cleans the measurement sets the floor that the controller can only reach.

S + T = 1 means the error moves; it is never removed.

At each frequency the designer chooses whether to reject disturbances or to ignore sensor noise, and choosing where the loop gain crosses one is choosing at which frequencies each kind of error lives (error relocation).

The peak of ∣S∣|S|∣S∣, written MsM_sMs​, is the single best number for robustness. It is the inverse of the loop's closest distance to the critical point −1-1−1, and the usual target is 1.2 to 2. Ms=2M_s=2Ms​=2 guarantees at once a gain margin of 2 and a phase margin of 29 degrees, so it covers the cases where gain and phase degrade together, which the separate stability margins miss. It is checked at every point where the loop can break: each actuator and each sensor.

The trade cannot be dodged by clever tuning. Bode proved in 1945 that for a loop stable in open loop, with enough roll-off at high frequency, ∫0∞ln⁡∣S(jω)∣ dω=0\int_0^\infty\ln|S(j\omega)|,d\omega=0∫0∞​ln∣S(jω)∣dω=0: push SSS down in one band and it rises in another (the waterbed effect). With an unstable plant the integral is positive, a tax paid for the instability.

The same split is the one estimators make. A complementary filter is built from a low-pass and a high-pass that sum to one, trusting one sensor below a crossover and another above it.

The plot of ∣S∣|S|∣S∣ and ∣T∣|T|∣T∣ against frequency is read beside the Bode plot of LLL; robust design methods such as H-infinity control shape them directly with frequency weights.