Control Theory: Closed loop system math

Question

I'm currently studying for control theory and I'm stuck on a math issue.

The following picture is showing a block diagram of the system I'm calculating.

Negative feedback closed loop system

The total transfer function of this system is:

$H=\frac{Y}{X}=\frac{H1}{1+H_1H_2}$

To come to this answer I start with $E=X-H_2Y$ and $Y=H_1E$

What I'm doing to try to come to this answer is the following:

$Y=H_1(X-H_2Y)$

$Y=H_1X+H_1H_2Y$

$Y-YH_1H_2= H_1X$

$?$

But after the last transformation I'm lost and I have no idea what to do. I think it is a simple math rule that I do not know.

Going back from the answer to the beginning I'm doing the following steps:

$\frac{Y}{X}=\frac{H_1}{1+H_1H_2} * \frac{H_2}{H_2}$

$\frac{Y}{X}=\frac{H_1H_2}{H2+H_1H_2^2}$

$Y=\frac{H_1H_2}{H2+H_1H_2^2}X$

$Y(H2+H_1H_2^2)=H_1H_2X$

$H2Y+H_1H_2^2Y=H_1H_2X$ (this transformation is the problem comming from the other way around.)

$Y+H_1H_2Y=H_1X$

$Y=H_1X-H_1H_2Y$

$Y=H_1(X-H_2Y)$

As it is probably clear to notice, math is not my strongest point. So a answer with a few more steps clarifying the math is highly appreciated.

Answer 1

You have a sign error when you multiply into your parenthesis. The next step you are missing is to take Y outside a parenthesis. I have put down all the steps for your clarification:

$$\begin{array}{l} E = X - {H_2}Y\\ Y = {H_1}E\\ \\ Y = {H_1}\left( {X - {H_2}Y} \right)\\ Y = {H_1}X - {H_1}{H_2}Y\\ Y + {H_1}{H_2}Y = {H_1}X\\ Y(1 + {H_1}{H_2}) = {H_1}X\\ Y = \frac{{{H_1}X}}{{(1 + {H_1}{H_2})}}\\ \frac{Y}{X} = \frac{{{H_1}}}{{(1 + {H_1}{H_2})}} \end{array}$$

Answer 2

Just a sign mistake :

$Y = H_{1}(X - H_{2}Y)$

$Y = H_{1}X - H_{1}H_{2}Y$ instead of your $+$

$Y(1+H_{1}H_{2})=H_{1}X$

Then you have your whole transfer