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.

Overview

The closed-loop transfer function is measured at the output. The output signal waveform can be calculated from the closed-loop transfer function and the input signal waveform.

An example of a closed-loop transfer function is shown below:

The summing node and the G(s) and H(s) blocks can all be combined into one block, which would have the following transfer function:

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

Derivation

We define an intermediate signal Z shown as follows:

Using this figure we write:

$Y(s)=Z(s)G(s)$

$Z(s)=X(s)-Y(s)H(s)$

$X(s)=Z(s)+Y(s)H(s)$

$X(s)=Z(s)+Z(s)G(s)H(s)$

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

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

See also

• Federal Standard 1037C
• Open-loop controller

References

•  This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".