Closed-loop transfer function: Difference between revisions

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{{short description|Function describing the effects of feedback on a control system}}

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.

AIn [[control theory]], a '''closed-loop transfer function''' in [[control theory]] is a [[mathematical expression ([[algorithmfunction]]) describing the net result of the effects of a closed ([[feedback]]) [[loopcontrol (telecommunication)|loop]] on the input [[signal (information theory)|signal]] to the circuits[[plant enclosed(control bytheory)|plant]] theunder loopcontrol.

== 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. Signals may be [[waveform|waveforms]], [[image|images]], or other types of [[data stream|data streams]].

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An example of a closed-loop transfer function is shown below:

An example of a closed-loop block diagram, from which a transfer function may be computed, is shown below:

[[Image:Closed_Loop_Block_Diagram.png]]

[[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 (control)|feed forward]] transfer function, <math>H(s) </math> is called the [[Feedback#Control theory|feedback]] transfer function, and their product <math>G(s)H(s) </math> is called the '''open-loop transfer function'''.

==Derivation==

Let'sWe define an intermediate signal Z (also known as [[error signal]]) shown as follows:

Using this figure we can write:

[[Image:Closed_Loop_Block_Deriv.png]]

: <math>Y(s) = G(s)Z(s) </math>

Using this figure we can write

: <math>YZ(s) = ZX(s)G-H(s) \Rightarrow Z(s) = \dfrac{Y(s)}{G(s)} </math>

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

: <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[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>X(s)-Y(s)H(s) = Z(s) =\Rightarrow \dfrac{Y(s)}{GX(s)} \Rightarrow X(s) = Y\dfrac{G(s) \left[}{1+G(s)H(s)} \right]/G(s)</math>

==See also==

*[[Federal Standard 1037C]]

*[[Open-loop controller]]

* {{section link|Control theory|Open-loop and closed-loop (feedback) control}}

== References ==

*{{FS1037C}}

[[Category:ControlClassical control theory]]

{{mathapplied-stub}}

[[Category:Control theory]]

[[Category:Cybernetics]]

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