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The sensitivity function describes how feedback attenuates disturbances and model changes in the loop.
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Formula
S dimensionless; G and H are loop transfer functions
| Symbol | Meaning | Unit |
|---|---|---|
| Sensitivity function | dimensionless | |
| Forward-path transfer function | dimension-dependent | |
| Feedback-path transfer function | dimension-dependent |
S(s)=\frac{1}{1+G(s)H(s)}For negative feedback, S=1/(1+L) with loop transfer L=GH. Small |S| can attenuate certain disturbances, but frequency range and stability margins constrain the trade-off. Complementary sensitivity is T=L/(1+L), with S+T=1.
For negative feedback, S=1/(1+L) with loop transfer L=GH. Small |S| can attenuate certain disturbances, but frequency range and stability margins constrain the trade-off. Complementary sensitivity is T=L/(1+L), with S+T=1.
The sensitivity function describes how feedback attenuates disturbances and model changes in the loop.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Feedback sensitivity function” and check its unit and magnitude.
At a frequency where L=9, S=0.1 in the ideal model.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: S dimensionless; G and H are loop transfer functions.
Formula cluster
The sensitivity function describes how feedback attenuates disturbances and model changes in the loop.
At a frequency where L=9, S=0.1 in the ideal model.
Assess disturbance rejection and robustness versus frequency.
Here is how to work through a typical Feedback sensitivity function (S = 1/(1+GH)) task step by step:
Task
Work through the numerical example for “Feedback sensitivity function” and check its unit and magnitude.
Solution path
At a frequency where L=9, S=0.1 in the ideal model.
S commonly describes transmission of certain disturbances and relative parameter variations. The exact disturbance depends on where it enters the block diagram. Identify the injection point before interpreting S. Disturbance location matters; sensitivity varies with frequency and does not alone prove stability. Application: Assess disturbance rejection and robustness versus frequency. The numerical example At a frequency where L=9, S=0.1 in the ideal model anchors the idea in a concrete measurement case. The corresponding units are S dimensionless; G and H are loop transfer functions. At a frequency where L=9, S=0.1 in the ideal model. This checks that the reference quantities are compatible.
Feedback trades off stability, bandwidth and noise amplification. Small |S| in one band can require larger response elsewhere. Complementary sensitivity and stability margins are part of a complete assessment. Disturbance location matters; sensitivity varies with frequency and does not alone prove stability. The corresponding units are S dimensionless; G and H are loop transfer functions. At a frequency where L=9, S=0.1 in the ideal model. This checks that the reference quantities are compatible. Scope: For negative feedback, S=1/(1+L) with loop transfer L=GH. Small |S| can attenuate certain disturbances, but frequency range and stability margins constrain the trade-off. Complementary sensitivity is T=L/(1+L), with S+T=1. A practical application is Assess disturbance rejection and robustness versus frequency.
L=GH is the product of forward and feedback paths around the loop. Magnitude and phase determine frequency-dependent feedback action. Units must make the denominator product dimensionless. Disturbance location matters; sensitivity varies with frequency and does not alone prove stability. Scope: For negative feedback, S=1/(1+L) with loop transfer L=GH. Small |S| can attenuate certain disturbances, but frequency range and stability margins constrain the trade-off. Complementary sensitivity is T=L/(1+L), with S+T=1. A practical application is Assess disturbance rejection and robustness versus frequency.
Write the quantity and unit from the variable list next to every value: S dimensionless; G and H are loop transfer functions. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example At a frequency where L=9, S=0.1 in the ideal model. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Disturbance location matters; sensitivity varies with frequency and does not alone prove stability.
The background states this starting condition: For negative feedback, S=1/(1+L) with loop transfer L=GH. Small |S| can attenuate certain disturbances, but frequency range and. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Disturbance location matters; sensitivity varies with frequency and does not alone prove stability.
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Which formula describes Feedback sensitivity function?
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How do you rearrange Feedback sensitivity function for Base equation for the target quantity?
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Which common mistake happens with Feedback sensitivity function?
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