Sensitivity and Complementary Sensitivity
The sensitivity and complementary sensitivity functions summarize disturbance rejection, tracking, robustness, and noise attenuation in one algebraic identity.
Two functions, one constraint
For a unity-feedback loop with loop gain L equal to the plant times the controller, the sensitivity is S equal to 1 over 1 plus L, and the complementary sensitivity is T equal to L over 1 plus L. They satisfy the algebraic identity S plus T equal to 1 at every frequency. This single equation encodes the central trade-off of feedback: you cannot make both small at the same frequency.
What each shapes
S is the transfer function from output disturbance to output and also governs tracking error, so small S at low frequency means good regulation and reference following. T is the transfer function from reference and from measurement noise to output, so small T at high frequency means good noise rejection and robustness against multiplicative uncertainty. The identity forces a crossover region where neither is negligible.
Bode's integral
The waterbed effect makes the trade-off unavoidable. Bode's sensitivity integral states that for a stable open loop with sufficient roll-off, the integral of the logarithm of the magnitude of S over frequency is zero. Pushing S below one in some band forces it above one elsewhere; attenuation is conserved, only redistributed. Right-half-plane poles make the integral positive, worsening the penalty.
- S plus T equals 1 identically
- Small S: disturbance rejection and tracking
- Small T: noise rejection and robustness
- Bode integral conserves log-sensitivity area
Design specifications are naturally weights on S and T: a low-frequency bound on S, a high-frequency bound on T. Stacking weighted S, T, and control sensitivity is exactly the mixed-sensitivity H-infinity problem.
For a plasma-shape control loop in a design-stage machine, the sensitivity function quantifies how well disturbances such as fueling transients are suppressed, and reading its peak reveals the robustness margin at a glance. These functions are the vocabulary of loop shaping.