Closed-loop transfer function

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.

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:

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:

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

${\displaystyle G(s)}${\displaystyle G(s)} is called the feed forward transfer function, ${\displaystyle H(s)}${\displaystyle H(s)} is called the feedback transfer function, and their product ${\displaystyle G(s)H(s)}${\displaystyle G(s)H(s)} is called the open-loop transfer function.

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

Using this figure we write:

${\displaystyle Y(s)=G(s)Z(s)}${\displaystyle Y(s)=G(s)Z(s)}

${\displaystyle Z(s)=X(s)-H(s)Y(s)}${\displaystyle Z(s)=X(s)-H(s)Y(s)}

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

${\displaystyle Y(s)=G(s)[X(s)-H(s)Y(s)]}${\displaystyle 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:

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

Therefore,

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

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

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

• Public Domain This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022年01月22日.

• Classical control theory
• Cybernetics

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• Short description is different from Wikidata
• Wikipedia articles incorporating text from the Federal Standard 1037C