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