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.

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:

Y ( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s ) {\displaystyle {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}}

G ( s ) {\displaystyle G(s)} is called the feed forward transfer function, H ( s ) {\displaystyle H(s)} is called the feedback transfer function, and their product G ( s ) H ( s ) {\displaystyle G(s)H(s)} 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:

Y ( s ) = G ( s ) Z ( s ) {\displaystyle Y(s)=G(s)Z(s)}

Z ( s ) = X ( s ) − H ( s ) Y ( s ) {\displaystyle Z(s)=X(s)-H(s)Y(s)}

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

Y ( s ) = G ( s ) [ X ( s ) − H ( s ) Y ( s ) ] {\displaystyle Y(s)=G(s)[X(s)-H(s)Y(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:

Y ( s ) + G ( s ) H ( s ) Y ( s ) = G ( s ) X ( s ) {\displaystyle Y(s)+G(s)H(s)Y(s)=G(s)X(s)}

Therefore,

Y ( s ) ( 1 + G ( s ) H ( s ) ) = G ( s ) X ( s ) {\displaystyle Y(s)(1+G(s)H(s))=G(s)X(s)}

⇒ Y ( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s ) {\displaystyle \Rightarrow {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}}

• Federal Standard 1037C
• Open-loop controller
• Control theory § Open-loop and closed-loop (feedback) control

•  This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22.