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I have this block diagram, and I just can't arrive at the transfer function \$H(s) = \frac{y(s)}{u(s)} \$ enter image description here

Starting from the end, we see that $$y(s) = xC $$ Let's find out what x is. $$x=\Bigg[\bigg(u(s)-x\bigg)A-y(s) \Bigg]B $$ But as you can see from the equation, it is recursive. I also can't see how the block diagram can be simplified, since nothing is in series or parallel. I hope someone can help me out.

Carl
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    It's just algebra, Carl, to solve for \$x\$. Different people will have slightly different preferences. But they all learned to do it. The equation is recursive only if you substitute over and over for \$x\$. But for solving for \$x\$ you can just expand all of factors on the right hand side into individual terms, then collect terms that include \$x\$ as a factor onto the left hand side, factor out \$x\$, and then divide both sides with remaining cofactor of \$x\$. You did well in stating the issue, though! – jonk May 14 '21 at 16:42
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    Thank you @jonk , I solved it using the block diagram reduction method suggested by AJN and it worked out. – Carl May 14 '21 at 18:04

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Second equation contains a term \$ -x\cdot A \cdot B\$ in the RHS. Take the \$ -x\cdot A \cdot B\$ from the second equation to its LHS. Then, substitute for x in the first equation with x from the (now re arranged) second equation. This eliminates the variable x in the final equation.

Other methods

Apart from the above manual elimination of variables, there are two more methods. Block diagram reduction techniques and Mason Gain Formula. If you are familiar with either technique, use one which is convenient.

Block diagram reduction

Shift the junction labelled x to the right of the C block by using an additional 1/C block in the feedback path. Now you have nested feedbacks. Use the G/(1+GH) feedback formula on the innermost loop and repeat until no more loops remain.

AJN
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