Closed-loop transfer function

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:

: $\backslash dfrac\{Y(s)\}\{X(s)\}\; =\; \backslash dfrac\{G(s)\}\{1\; +\; G(s)\; H(s)\}$

$G(s)$ is called the feed forward transfer function, $H(s)$ is called the feedback transfer function, and their product $G(s)H(s)$ is called the open-loop transfer function.

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,

:$Y(s)(1+G(s)H(s))\; =\; G(s)X(s)$

:$\backslash Rightarrow\; \backslash dfrac\{Y(s)\}\{X(s)\}\; =\; \backslash dfrac\{G(s)\}\{1+G(s)H(s)\}$

• Federal Standard 1037C
• Open-loop controller
• {{section link|Control theory|Open-loop and closed-loop (feedback) control}}

• {{FS1037C}}

Category:Classical control theory

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