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 plant under control.


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 transfer function is shown below:

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:

is called feedforward transfer function, is called feedback transfer function, and their product 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:

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


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  • Public Domain This article incorporates public domain material from the General Services Administration document: "Federal Standard 1037C".