This page explains how to calculate the equation of a closed loop system. We first present the transfer function of an open loop system, then a closed loop system and finally a closed loop system with a controller.

Let’s consider the following open loop system:

The transfert function of the system is given by:

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

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:

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

Let's now assume that a controller is added:

We can deduce the new transfert function:

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

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Last update : 01/23/2021