Definitions

# 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:

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

## Derivation

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

Using this figure we can write

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

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

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