Closed-loop transfer function

A closed-loop transfer function in control theory is a mathematical expression (algorithm) describing the net result of the effects of a closed (feedback) loop on the input signal to the circuits enclosed by the loop.

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Derivation

Let's define an intermediate signal Z shown as follows:

Using this figure we can write

$Y(s)=Z(s)G(s)\Rightarrow Z(s)={\dfrac {Y(s)}{G(s)}}$

$X(s)-Y(s)H(s)=Z(s)={\dfrac {Y(s)}{G(s)}}\Rightarrow X(s)=Y(s)\left[{1+G(s)H(s)}\right]/G(s)$

$\Rightarrow {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}$

See also

• Federal Standard 1037C
• Open-loop controller

References

•  This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.