Closed-loop transfer function

Short description: Function describing the effects of feedback on a control system

In control theory, a closed-loop transfer function is a mathematical function describing the net result of the effects of a feedback control loop on the input signal to the plant under control.

## Contents

* 1 Overview
* 2 Derivation
* 3 See also
* 4 References

## Overview

The closed-loop transfer function is measured at the output. The output signal can be calculated from the closed-loop transfer function and the input signal. Signals may be waveforms, images, or other types of data streams.

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:

[math]\displaystyle{ \dfrac{Y(s)}{X(s)} = \dfrac{G(s)}{1 + G(s) H(s)} }[/math]

[math]\displaystyle{ G(s) }[/math] is called feedforward transfer function, [math]\displaystyle{ H(s) }[/math] is called feedback transfer function, and their product [math]\displaystyle{ G(s)H(s) }[/math] is called the open-loop transfer function.

## Derivation

We define an intermediate signal Z (also known as error signal) shown as follows:

Using this figure we write:

[math]\displaystyle{ Y(s) = G(s)Z(s) }[/math]

[math]\displaystyle{ Z(s) =X(s)-H(s)Y(s) }[/math]

Now, plug the second equation into the first to eliminate Z(s):

[math]\displaystyle{ Y(s) = G(s)[X(s)-H(s)Y(s)] }[/math]

Move all the terms with Y(s) to the left hand side, and keep the term with X(s) on the right hand side:

[math]\displaystyle{ Y(s)+G(s)H(s)Y(s) = G(s)X(s) }[/math]

Therefore,

[math]\displaystyle{ Y(s)(1+G(s)H(s)) = G(s)X(s) }[/math]

[math]\displaystyle{ \Rightarrow \dfrac{Y(s)}{X(s)} = \dfrac{G(s)}{1+G(s)H(s)} }[/math]

## See also

* Federal Standard 1037C
* Open-loop controller
* Control theory § Open-loop and closed-loop (feedback) control

## References

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