View source for 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 (control theory)|bandwidth]] — the range of frequencies it can track before attenuation sets in.

See also: [[Sensitivity function]], [[Control theory]], [[Feedback]], [[Reference tracking]], [[Bandwidth (control theory)]], [[Noise sensitivity]]

[[Category:Systems]]
[[Category:Control Theory]]
[[Category:Mathematics]]