title: Closed-loop transfer function
text: # 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 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:
  
      ${\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}$
  
  $G(s)$ 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):
  
      $Y(s)=G(s)[X(s)-H(s)Y(s)]$
  
  Move all the terms with Y(s) to the left hand side, and keep the term with X(s) on the right hand side:
  
      $Y(s)+G(s)H(s)Y(s)=G(s)X(s)$
  
  Therefore,
  
      $Y(s)(1+G(s)H(s))=G(s)X(s)$
      $\Rightarrow {\dfrac {Y(s)}{X(s)}}={\dfrac {G(s)}{1+G(s)H(s)}}$
  