{{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.
== 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:
Image: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:
: <math>\dfrac{Y(s)}{X(s)} = \dfrac{G(s)}{1 + G(s) H(s)}</math>
<math>G(s) </math> is called the feed forward transfer function, <math>H(s) </math> is called the feedback transfer function, and their product <math>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>Y(s) = G(s)Z(s) </math>
: <math>Z(s) =X(s)-H(s)Y(s) </math>
Now, plug the second equation into the first to eliminate Z(s):
:<math>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>Y(s)+G(s)H(s)Y(s) = G(s)X(s)</math>
Therefore,
:<math>Y(s)(1+G(s)H(s)) = G(s)X(s)</math>
:<math>\Rightarrow \dfrac{Y(s)}{X(s)} = \dfrac{G(s)}{1+G(s)H(s)}</math>
==See also== *Federal Standard 1037C *Open-loop controller * {{section link|Control theory|Open-loop and closed-loop (feedback) control}}
== References == *{{FS1037C}}
Category:Classical control theory Category:Cybernetics