Closed-loop transfer function

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 to the plant under control.

Overview

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 and H blocks can all be combined into one block, which would have the following transfer function:
is called the feed forward transfer function, is called the feedback transfer function, and their product is called the open-loop transfer function.

Derivation

We define an intermediate signal Z shown as follows:
Using this figure we write:
Now, plug the second equation into the first to eliminate Z:
Move all the terms with Y to the left hand side, and keep the term with X on the right hand side:
Therefore,