Closed-loop transfer function: Difference between revisions

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

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Using this figure we can write

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 (information theory)|signal]] to the [[plant (control theory)|plant]] under control.

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

== Overview ==

: <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>

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

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

: <math>\Rightarrow \dfrac{Y(s)}{X(s)} = \dfrac{G(s)}{1 + G(s) H(s)}</math>

[[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>\Rightarrow \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==

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

Using this figure we can 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) = Z+G(s)GH(s) \Rightarrow ZY(s) = \dfrac{YG(s)}{GX(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]]

[[Category:Cybernetics]]

{{mathapplied-stub}}