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A '''closed-loop transfer function''' in [[control theory]] is a mathematical expression ([[algorithm]]) describing the net result of the effects of a closed ([[feedback]]) [[loop (telecommunication)|loop]] on the input [[signal (information theory)|signal]] to the circuits enclosed by the loop.


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== Overview ==
 
The closed-loop [[transfer function]] is measured at the [[output]]. The output signal [[waveform]] can be calculated from the closed-loop transfer function and the input signal waveform.
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An example of a closed-loop transfer function is shown below:
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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:
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: <math>\dfrac{Y(s)}{X(s)} = \dfrac{G(s)}{1 + G(s) H(s)}</math>
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==Derivation==
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Let's define an intermediate signal Z shown as follows:
 
 
</ul>
[[Image:Closed Loop Block Deriv.png]]
 
Using this figure we can write
 
: <math>Y(s) = Z(s)G(s) \Rightarrow Z(s) = \dfrac{Y(s)}{G(s)} </math>
 
: <math>X(s)-Y(s)H(s) = Z(s) = \dfrac{Y(s)}{G(s)} \Rightarrow X(s) = Y(s) \left[{1+G(s)H(s)} \right]/G(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]]
 
== References ==
*{{FS1037C}}
 
[[Category:Control theory]]
[[Category:Cybernetics]]
 
 
{{mathapplied-stub}}

Revision as of 21:20, 3 February 2014

A closed-loop transfer function in control theory is a mathematical expression (algorithm) describing the net result of the effects of a closed (feedback) loop on the input signal to the circuits enclosed by the loop.

Overview

The closed-loop transfer function is measured at the output. The output signal waveform can be calculated from the closed-loop transfer function and the input signal waveform.

An example of a closed-loop transfer function 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:

Derivation

Let's define an intermediate signal Z shown as follows:

Using this figure we can write

See also

References


Template:Mathapplied-stub