• Source: 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.




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:

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:








Y
(
s
)


X
(
s
)




=




G
(
s
)


1
+
G
(
s
)
H
(
s
)






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





G
(
s
)


{\displaystyle G(s)}

is called the feed forward transfer function,



H
(
s
)


{\displaystyle H(s)}

is called the feedback transfer function, and their product



G
(
s
)
H
(
s
)


{\displaystyle 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
)


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





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):




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:




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,




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


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





⇒




Y
(
s
)


X
(
s
)




=




G
(
s
)


1
+
G
(
s
)
H
(
s
)






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





See also


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




References


This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.

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