Sensitivity Function

In a standard feedback loop with plant G(s) and controller C(s), the open-loop transfer function is L(s) = C(s) G(s). The sensitivity function S(s) is defined as:

It is the closed-loop transfer function from a disturbance entering at the plant input (or from the reference) to the tracking error e(s) = r(s) - y(s). A small value of |S(j\omega)| at a given frequency means the loop effectively rejects disturbances at that frequency — the error produced by a disturbance of unit amplitude is small. A large value means the loop amplifies disturbances at that frequency.

The complementary sensitivity function

The complementary sensitivity function T(s) is defined as:

It is the closed-loop transfer function from the reference to the output — the transfer function a designer would ideally want to be unity (perfect tracking). It also describes how sensor noise enters the output: high-frequency sensor noise, which passes through T(s), is amplified wherever |T(j\omega)| is large. Because S(s) + T(s) = 1 identically, the two functions cannot both be made small at the same frequency: making S small (good disturbance rejection) makes T large (poor noise attenuation), and vice versa.

Frequency-domain interpretation

At low frequencies, where the loop gain |L(j\omega)| is large, |S(j\omega)| \approx 1/|L(j\omega)| is small — the loop rejects low-frequency disturbances effectively, as expected from integral action in the controller. At high frequencies, where the plant rolls off and |L(j\omega)| \ll 1, |S(j\omega)| \approx 1 — the loop provides no disturbance rejection and the output disturbance passes through unattenuated. The gain crossover frequency \omega_{gc} marks the approximate boundary between the disturbance-rejecting and non-rejecting regimes.

Near \omega_{gc}, the sensitivity function often exhibits a peak \|S\|_\infty > 1, meaning the loop amplifies disturbances in that frequency band. The peak value M_s = \max_\omega |S(j\omega)| is a robustness measure: it is the inverse of the shortest distance from the Nyquist curve of L(j\omega) to the critical point (-1, 0). A large M_s (small distance to -1) indicates low robustness margins. Typical design requirements specify M_s \leq 2 (6 dB), corresponding to minimum gain and phase margins of approximately 6 dB and 29° respectively.

Bode’s sensitivity integral

A fundamental constraint on the sensitivity function is given by Bode’s integral theorem. For a stable open-loop system with at least two more poles than zeros, the integral of the log-sensitivity over all frequencies is fixed:

This conservation law states that the total area under the log-sensitivity curve is zero. Any reduction of |S(j\omega)| below unity in one frequency band — corresponding to improved disturbance rejection — must be compensated by an equal increase above unity in another band. Feedback cannot reduce disturbances at all frequencies simultaneously: it can only redistribute them. Unstable open-loop poles make the integral positive, meaning that the mandatory water-bed effect is even more severe, and a greater penalty in sensitivity amplification must be paid somewhere in the spectrum.

Design use

Sensitivity functions are central to H_\infty robust control design, where the controller is synthesised to minimise the peak of a weighted sensitivity function subject to stability constraints. They also provide a frequency-domain language for specifying performance requirements: a performance weight W_1(s) is chosen such that the design requirement |S(j\omega)| \leq 1/|W_1(j\omega)| captures the desired disturbance rejection profile across frequency, and the controller is designed to satisfy this bound.

Common mistakes

A common mistake is reading a low value of S at low frequency as proof that the whole loop is robust. The peak sensitivity, complementary sensitivity, actuator limits, sensor noise, delays, unmodelled modes, and saturation must also be checked. Another mistake is treating disturbance rejection and noise attenuation as independent goals, even though S(s)+T(s)=1 creates a direct tradeoff. A strong review states the loop transfer function, disturbance entry point, frequency weighting, peak sensitivity, stability margins, uncertainty model, and validation data.