Definitions

# closed shop

Arrangement whereby a company employs only workers who are members in good standing of a specified labour union. It is the most rigid of the various schemes for protecting labour unions (more flexible arrangements include the union shop). Closed shops were declared illegal in the U.S. under the Taft-Hartley Act of 1947, but in practice they continue to exist in some industries, such as construction.

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 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(s) and H(s) blocks can all be combined into one block, which would have the following transfer function:

$dfrac\left\{Y\left(s\right)\right\}\left\{X\left(s\right)\right\} = dfrac\left\{G\left(s\right)\right\}\left\{1 + G\left(s\right) H\left(s\right)\right\}$

## Derivation

Let's define an intermediate signal Z shown as follows:

Using this figure we can write

$Y\left(s\right) = Z\left(s\right)G\left(s\right) Rightarrow Z\left(s\right) = dfrac\left\{Y\left(s\right)\right\}\left\{G\left(s\right)\right\}$

$X\left(s\right)-Y\left(s\right)H\left(s\right) = Z\left(s\right) = dfrac\left\{Y\left(s\right)\right\}\left\{G\left(s\right)\right\} Rightarrow X\left(s\right) = Y\left(s\right) left\left[\left\{1+G\left(s\right)H\left(s\right)\right\} right\right]/G\left(s\right)$

$Rightarrow dfrac\left\{Y\left(s\right)\right\}\left\{X\left(s\right)\right\} = dfrac\left\{G\left(s\right)\right\}\left\{1 + G\left(s\right) H\left(s\right)\right\}$