Sensitivity and Complementary Sensitivity
The sensitivity and complementary sensitivity functions quantify disturbance rejection and noise sensitivity, bound by the constraint that they sum to one.
Two Functions That Describe a Loop
For a feedback loop with loop gain L(s), two transfer functions capture almost everything about its behavior. The sensitivity S = 1/(1 + L) is the transfer function from disturbance to output, and the complementary sensitivity T = L/(1 + L) is the transfer function from reference to output and from sensor noise to output.
The fundamental identity
At every frequency, S + T = 1. This simple algebraic fact is a hard constraint on design: you cannot make both small at the same frequency. Good disturbance rejection (small S) and good noise rejection (small T) cannot coexist at one frequency, so they must be assigned to different frequency bands.
The standard assignment
- Low frequency: make S small for tracking and disturbance rejection, accepting T near one.
- High frequency: make T small for noise rejection and robustness to unmodeled dynamics, accepting S near one.
- The crossover region between them is where stability margins are set and must be handled carefully.
The waterbed effect
There is a deeper limit. For many plants the Bode sensitivity integral requires that the area of log-sensitivity above and below one balance out: pushing S down in one band forces it up in another, like pressing on a waterbed. Non-minimum-phase zeros and time delays worsen this, capping how much disturbance rejection is achievable without a robustness penalty elsewhere.
Use in robust design
Sensitivity functions are the language of loop shaping and H-infinity design. Performance specifications are stated as frequency-dependent bounds on S and T, and the controller is synthesized to meet them. The peak value of S, in particular, is a robustness measure: a large sensitivity peak means the Nyquist curve passes close to the critical point and margins are thin.
Reading S and T across frequency gives a complete picture of a loop's trade-offs in one view, which is why they are central to both classical loop shaping and modern robust control.