transfer function

Abstract

The use of a control system automatically implies that the system has an input. This implies, that we are looking for the FORCED RESPONSE.

Our actual transfer function should be the ratio of output to input in the Laplace domain

FREE RESPONSE - This is the response purely based on the initial conditions and no other inputs FORCED RESPONSE - This is the response based on the additional conditions added by an input

Any General TF can be expressed as:

$$\frac{X(s)}{u(s)}=\frac{p(s)}{q(s)}$$

Where $q(s)=0$ is called the characteristic equation This helps us determine the stability of the system.

Examples

FIRST ORDER SYSTEM

Let us assume our initial conditions are 0:

$$x(0)=\dot{x}(0)=0$$

For our First Order System we have: General Form

$$\dot{x}(t)+\frac{1}{\tau }x(t)=Ku(t)$$

We are then going to take the Laplace Transform:

$$sX(s)-[x(0){]}^{=0}+\frac{1}{\tau }X(s)=Ku(s)$$

Which can be simplified as:

$$X(s)[s+\frac{1}{\tau }]=Ku(s)$$ $$\frac{X(s)}{u(s)}=\frac{K}{s+\frac{1}{\tau }}$$

This $\frac{X(s)}{u(s)}$ is our transfer function

SECOND ORDER SYSTEM

General Form

$$\ddot{x}(t)+2\xi {\omega }_n\dot{x}(t)+({\omega }_n{)}^{2}x(t)=Ku(t)$$

Taking the Laplace of this:

$$s^{2}X(s)+2\xi {\omega }_{n}sX(s)+{\omega }_n^{2}x(s)=Ku(s)$$ $$X(s)[s^2+2s\xi {\omega }_n+{\omega }_n^2]=Ku(s)$$ $$\frac{X(s)}{u(s)}=\frac{K}{[s^2+2s\xi {\omega }_n+{\omega }_n^2]}$$

Error Of Control System

$$\frac{E(s)}{R(s)}=1-\frac{G_1(s)G_2(s)}{1+G_1(s)G_2(s)}=\frac{1}{1+G_1(s)G_2(s)}$$

Where:

Closed Loop TF of Controlled System

$$\frac{C(s)}{R(s)}=\frac{G_1(s)G_2(s)}{1+G_1(s)G_2(s)}$$

Where: