Complementary sensitivity function

The complementary sensitivity function \(T(s)\) is the frequency-domain measure of how well a feedback control system tracks its reference input. Defined as \(T(s) = L(s)/(1+L(s)) = 1 - S(s)\), where \(S(s)\) is the sensitivity function and \(L(s)\) is the loop gain, it quantifies the closed-loop transfer function from reference to output. Large \(|T(j\omega)|\) means the output faithfully reproduces the reference at that frequency; small \(|T(j\omega)|\) means the reference is attenuated. Because \(S(s) + T(s) = 1\), the complementary sensitivity function embodies the fundamental tradeoff of control design: good tracking and good disturbance rejection cannot coexist at the same frequency. The complementary sensitivity function also determines the system's sensitivity to sensor noise and its bandwidth — the range of frequencies it can track before attenuation sets in.