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Complementary sensitivity is the closed-loop transfer contribution of loop gain.
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Formula
T dimensionless for compatible input/output quantities; s in s−1
| Symbol | Meaning | Unit |
|---|---|---|
| Complementary sensitivity function | dimensionless | |
| Forward transfer function | dimension-dependent | |
| Feedback transfer function | dimension-dependent |
T(s)=\frac{G(s)H(s)}{1+G(s)H(s)}For loop gain L=GH and negative feedback, T=L/(1+L), while S=1/(1+L) and S+T=1. T often describes noise transmission or model tracking at particular locations, depending on the block diagram. Small S and small T cannot generally be achieved at all frequencies simultaneously.
For loop gain L=GH and negative feedback, T=L/(1+L), while S=1/(1+L) and S+T=1. T often describes noise transmission or model tracking at particular locations, depending on the block diagram. Small S and small T cannot generally be achieved at all frequencies simultaneously.
Complementary sensitivity is the closed-loop transfer contribution of loop gain.
Base equation for the target quantity
Preserve signs, boundary conditions and units when rearranging.
Work through the numerical example for “Complementary sensitivity function” and check its unit and magnitude.
At L=9, T=0.9 and S=0.1; the functions sum to one.
Substituting a related quantity that shares the same symbol.
Use variable definitions and check: T dimensionless for compatible input/output quantities; s in s−1.
Formula cluster
Complementary sensitivity is the closed-loop transfer contribution of loop gain.
At L=9, T=0.9 and S=0.1; the functions sum to one.
Analyze noise transmission and robustness trade-offs in feedback systems.
Here is how to work through a typical Complementary sensitivity function (T = GH/(1+GH)) task step by step:
Task
Work through the numerical example for “Complementary sensitivity function” and check its unit and magnitude.
Solution path
At L=9, T=0.9 and S=0.1; the functions sum to one.
S=1/(1+L) describes sensitivity to certain disturbances; T=L/(1+L) is complementary and often relevant to reference tracking or measurement noise. In the ideal single loop S+T=1, but exact disturbance paths depend on injection point. T and S describe different closed-loop paths; identify disturbance location. Application: Analyze noise transmission and robustness trade-offs in feedback systems. The numerical example At L=9, T=0.9 and S=0.1; the functions sum to one anchors the idea in a concrete measurement case. The corresponding units are T dimensionless for compatible input/output quantities; s in s−1. At L=9, T=0.9 and S=0.1; the functions sum to one. This checks that the reference quantities are compatible.
With high loop gain, T is often near one. This can mean good reference tracking, but sensor measurement noise may also reach the output. Assess T with S, bandwidth and stability margins. T and S describe different closed-loop paths; identify disturbance location. The corresponding units are T dimensionless for compatible input/output quantities; s in s−1. At L=9, T=0.9 and S=0.1; the functions sum to one. This checks that the reference quantities are compatible. Scope: For loop gain L=GH and negative feedback, T=L/(1+L), while S=1/(1+L) and S+T=1. T often describes noise transmission or model tracking at particular locations, depending on the block diagram. Small S and small T cannot generally be achieved at all frequencies simultaneously. A practical application is Analyze noise transmission and robustness trade-offs in feedback systems.
It complements sensitivity to unity: T+S=1 for the same loop gain and feedback convention. Together they describe key properties of a simple closed loop. T and S describe different closed-loop paths; identify disturbance location. Scope: For loop gain L=GH and negative feedback, T=L/(1+L), while S=1/(1+L) and S+T=1. T often describes noise transmission or model tracking at particular locations, depending on the block diagram. Small S and small T cannot generally be achieved at all frequencies simultaneously. A practical application is Analyze noise transmission and robustness trade-offs in feedback systems.
Write the quantity and unit from the variable list next to every value: T dimensionless for compatible input/output quantities; s in s−1. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example At L=9, T=0.9 and S=0.1; the functions sum to one. 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. T and S describe different closed-loop paths; identify disturbance location.
The background states this starting condition: For loop gain L=GH and negative feedback, T=L/(1+L), while S=1/(1+L) and S+T=1. T often describes noise transmission or. 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. T and S describe different closed-loop paths; identify disturbance location.
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Which formula describes Complementary sensitivity function?
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How do you rearrange Complementary sensitivity function for Base equation for the target quantity?
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Which common mistake happens with Complementary sensitivity function?
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