速溶咖啡 品牌 高端排行榜前十名 Equation of closed and open loop systems

Equation of closed and open loop systems Introduction

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.

Open loop

Let鈥檚 consider the following open loop system:

The transfert function of the system is given by:

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

Closed loop

Let鈥檚 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:

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

See also Ball and beam model Dynamic model of an inverted pendulum (part 1) Dynamic model of an inverted pendulum (part 2) Dynamic model of an inverted pendulum (part 3) Dynamic model of an inverted pendulum (part 4) Dynamic model of an inverted pendulum (part 5) Dynamic model of an inverted pendulum (part 6) Modelling of a simple pendulum PI-based first-order controller Last update : 01/23/2021