Closed-loop transfer function

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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 on the input signal to the circuits enclosed by the loop.

1 Overview
2 Derivation
3 See also
4 References

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:

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

Derivation

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

Using this figure we can write

$Y(s) = Z(s)G(s) \Rightarrow Z(s) = \dfrac{Y(s)}{G(s)}$

$X(s)-Y(s)H(s) = Z(s) = \dfrac{Y(s)}{G(s)} \Rightarrow X(s) = Y(s) \left[{1+G(s)H(s)} \right]/G(s)$

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

References

This article does not cite any references or sources.
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