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: 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: G(s) is called the feed forward transfer function, H(s) is called the feedback transfer function, and their product G(s)H(s) is called the open-loop transfer function.

Derivation

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: Therefore,