Control and Feedback in electronic circuits

[Rushers]

I'm trying to understand the concept of a control system in electronics.
In a control system, we have a substractor, forward path and feedback path.

The forward path is described by a function g ( a gain function ) and the feedback by another function ( gain again ).
The input is Xi and the output is Xo.

Let's suppose we are controlling the voltage so X is a voltage.
Our sensor in the feedback path reads the value of the voltage. This value is multiplied by B. My question is .Why?

If we want to substract from the reference value at the substractor, why would we multiply by a gain and modify the real value of voltage?

Then we substruct so we get (Xi -BXo) and then multiply by A to get A(Xi - BXo).
This new value is supposed to be the new value of voltage.
Xo = A(Xi -BXo) with TWO values of Xo. We have Xo previous and Xo after the operation of modification.
But in the textbook they say these values are the same and deduce a "gain" of the circuit (A/ (1 + AB)).
Why, if we have two values of Xo, are they taken to be similar?
AND what does this "gain" represent?

 

[es1]

This is a really insightful question.

Basically for now, to keep things simple, the textbook is assuming that all the amplifiers are ideal. Or more specifically, that it takes zero time for the signal to pass through the amplifier. If this it true then the two Xo's in the equation are the same. This assumption keeps the math a lot easier.

However, as you mention, this is not true for real amplifiers. It takes some time for the signal to pass through them. The measure of this time is called the phase of the system and it has huge implications on the stability of the control system.

Even more confusing is that for different d(Xi)/dt's the delay (and gain) through the system will be different. Determining this is called finding the frequency response of the system.
http://en.wikipedia.org/wiki/Frequency_response

And I'll bet it is something that your textbook will cover in future chapters as it is hugely important.

For all the why do we do this questions...
There are many good reasons. I recommend reading this introductory chapter. I think it is really good.
http://www.cds.caltech.edu/~murray/amwiki/index.php/Introduction

 

[es1]

Oh ya, that gain represents the fraction Xo/Xi.

So: Xo/Xi = (A/ (1 + AB))