Closed-loop transfer function

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.

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 transfer function 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:

: $\dfrac = \dfrac$

$G(s)$ is called feedforward transfer function, $H(s)$ is called 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:

: $Y(s) = G(s)Z(s)$

: $Z(s) =X(s)-H(s)Y(s)$

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

:Y(s) = G(s) (s)-H(s)Y(s)/math>

Move all the terms with Y(s) to the left hand side, and keep the term with X(s) on the right hand side:

:$Y(s)+G(s)H(s)Y(s) = G(s)X(s)$

Therefore,

:$Y(s)(1+G(s)H(s)) = G(s)X(s)$

:$\Rightarrow \dfrac = \dfrac$

See also

* Federal Standard 1037C
* Open-loop controller
* Open-Loop Transfer Function
* Control_theory#Closed-loop_transfer_function

References

*{{FS1037C

Classical control theory
Cybernetics