Closed-loop transfer function


A closed-loop transfer function in control theory is a mathematical expression describing the net result of the effects of a closed loop on the input signal to the circuits enclosed by the loop.

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 and H 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.

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,