# Block Diagram Simplification and Routh-Hurwitz

I'm trying to solve an exercise of block diagrams and Routh-Hurwitz, but I stuck in the simplification part. Could you help me?

1. Reduce to an equivalent single block: 1. For what values of K the system will be stable?

It's my first time trying to solve this, so I'm sorry if this is a "too easy question".

My effort:

I initially simplified the block diagram to this: Then this: Anyone can confirm if I'm right?

• What have you tried? Have you worked out the total transfer function $\frac{y(s)}{u(s)}$? Do you know what defines a stable system from an unstable one? We are happy to help but we want to see your efforts too. – Warren Hill Sep 5 '17 at 8:38
• Hi! I tried to simplify the block diagram and I stuck in this part (after some steps): imgur.com/B4Y0Zau. But now I don't know what to do with this H1 block... Regarding HR, I think I can solve by my own... – misc Sep 5 '17 at 8:42
• I've added this detail to your question for you. Please only use comments for comments additional detail for the question, including your efforts should be part of the question not comments. I appreciate a a new user you can only add one image. I've also up-voted for you. – Warren Hill Sep 5 '17 at 8:56

Lets start by labelling your first effort

We will call the bottom block $G_3 = \frac{G_1}{1+ G_1 \cdot H_1}$

The top block $G_4 = \frac{G_2}{1+ G_2 \cdot H_2}$

For convenience I've dropped the (s) notation from $y$ and $u$ since we know they are functions of s.

$w$ is the input to $G_4$,
$x$ is the output of $G_3$

Now lets simplify further

$y = x + w \cdot G_4$

$w = u - x \cdot H_1$

$x = u \cdot G_3$

We want $\frac{y}{u}$

$y = u \cdot G_3 + w \cdot G_4$

$w = u - u \cdot G_3 \cdot H_1$

$y = u \cdot G_3 + (u - u \cdot G_3 \cdot H_1) \cdot G_4$

$y = u \cdot \left( G_3 + G_4 - G_3 \cdot G_4 \cdot H_1 \right)$

If we now replace $G_3$ and $G_4$ with their definitions and these by their transfer functions (in $s$) we have the transfer function for the entire system. Can you take it from here?

• Yes! I tried a little more after my first effort (edited the OP with my answer) and I get the same result :D Actually, I think your answer has a little typo mistake in the definition of G4. It should be G2 in the numerator, right? – misc Sep 5 '17 at 9:44
• Edited to correct, Also put your original effort back in so this answer made sense, still has your latest answer. – Warren Hill Sep 5 '17 at 9:50