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.


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)}


We define an intermediate signal Z shown as follows:

Closed Loop Block Deriv.png

Using this figure we 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)}

See also[edit]