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We have the open loop transfer function

$$ G(s)H(s) = \frac{k(s+2)}{s^2+2s+3}$$

There were three parts to this question :

a. The value of $k$ for which repetitive roots occur

b. The range of $k$ for which the closed loop system becomes underdamped

c. The value of $k$ for which the system will have dampening ratio of $0.7$

I got the solution of the first part by putting $\frac{dk}{ds}=0$ and got the value to be $2(1+\sqrt3)$ . Because this is the break-in point of the root-loci on the real axis after the all the poles will be real and negative and hence underdamped .

So my question , is my reasoning for the first part and the answer , correct ? And I am not able to do the third part , please help me with that .

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    $\begingroup$ Add an image of the problem's diagram block to be able to calculate the transfer function. $\endgroup$ – CAGT Jan 23 '14 at 13:46
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First of all you need to close the loop, or otherwise roots would not change with $k$. When you do that you will get a characteristic polynomial

$s^2 + (2 + k)s + (3 + 2k) = s^2 + 2 \zeta \omega s + \omega^2$

Now you can put $\zeta = 1$, $\zeta > 1$ and $\zeta = 0.7$ to get the $k$ values respectively.

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