## Open loop

Let’s consider the following system in open loop:

The transfert function of the system is given by:

## Closed loop

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

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

The output of the system is given by:

By replacing in the previous equation we get:

This equation can be rewritten to get the transfert function:

## Closed loop with controller

Let’s now assume that a controller is added:

We can deduce the new transfert function: