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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:

Block diagram

  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:

enter image description here

Then this:

Simplified BD

Anyone can confirm if I'm right?

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  • \$\begingroup\$ 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. \$\endgroup\$ – Warren Hill Sep 5 '17 at 8:38
  • \$\begingroup\$ 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... \$\endgroup\$ – misc Sep 5 '17 at 8:42
  • \$\begingroup\$ 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. \$\endgroup\$ – Warren Hill Sep 5 '17 at 8:56
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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.

I've also added labels:
\$ 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?

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  • \$\begingroup\$ 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? \$\endgroup\$ – misc Sep 5 '17 at 9:44
  • \$\begingroup\$ Edited to correct, Also put your original effort back in so this answer made sense, still has your latest answer. \$\endgroup\$ – Warren Hill Sep 5 '17 at 9:50

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