Closed-loop transfer function

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title: "Closed-loop transfer function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["classical-control-theory", "cybernetics"] description: "Function describing the effects of feedback on a control system" topic_path: "general/classical-control-theory" source: "https://en.wikipedia.org/wiki/Closed-loop_transfer_function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary 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.

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 block diagram, from which a transfer function may be computed, is shown below:

::figure[src="https://upload.wikimedia.org/wikipedia/commons/b/b8/Closed_Loop_Block_Deriv.png"] ::

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)}

G(s) is called the feed forward transfer function, H(s) is called the feedback transfer function, and their product G(s)H(s) 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:

: Y(s) = G(s)Z(s)

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

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

:Y(s) = G(s)[X(s)-H(s)Y(s)]

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

:Y(s)+G(s)H(s)Y(s) = G(s)X(s)

Therefore,

:Y(s)(1+G(s)H(s)) = G(s)X(s)

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

References

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::