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I have to draw a root locus with respect to K and find for which values of K the system is stable, then draw conclusions on the principal shape of the step response.

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$$ G_{0}(s)=\frac{K(s+2)}{s(s+1)(s+3)} $$

  1. Isn't the step response just $\displaystyle \frac{1}{s}$?

  2. I can get the characteristic polynomial of $G_c$ which is $s(s + 1)(s + 3) + K(s + 2)=0$, and I know it is in the form of $Q + KP$ wher $Q$ stands for $G_0$ numerator and $P$ for $G_0$ denominator, so the function is supposed to move between the zeroes to the poles. I checked the solution and I still don't understand how to do the asymptotes, how do they come up with the intersection with the real axis, how do they draw the shape in form of a bow and 2 bars on the real axis. The intersection with the imaginary axis part makes much more sense.

What's the reasoning behind the asymptotes and the root locus?

The solution:

enter image description here enter image description here

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