# Open and closed loop

## Open loop

Let’s consider the following system in open loop:

The transfert function of the system is given by:

$\frac {y}{u} = G$

## Closed loop

Let’s now consider the same system in closed loop:

The error $\epsilon$ is defined by the difference between the reference (expected value) and the output of the system (the real value):

$\epsilon = y_c - y$

The output of the system is given by:

$y=G.u=G.\epsilon$

By replacing $\epsilon$ in the previous equation we get:

$y=G.(y_c - y) = G.y_c - G.y$

This equation can be rewritten to get the transfert function:

$\frac{y}{y_c} = \frac {G}{1+G}$

## Closed loop with controller

Let’s now assume that a controller is added:

We can deduce the new transfert function:

$\frac{y}{y_c} = \frac {CG}{1+CG}$