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:


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}

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