I was looking over practice problems for an upcoming exam and I am struggling with root locus. I understand how to plot the basic root locus but for some of the aspects when gain is asked to be calculated for plots I'm stumped.

Practice problem from exam study guide

I was wondering if anyone knew a good step by step method to solving something like this?


First determine the damping ratio $\zeta$ and natural frequency $\omega$ of the closed loop poles.

The general characteristic equation is $s^2+2 \zeta s \omega +\omega ^2$. For the desired pole locations the characteristic equation is $(s+10 -8.83 i) (s+10 +8.83 i)$. Equate the coefficients and solve for $\zeta$ and $\omega$.

Now draw lines from the origin to the desired closed-loop poles at $-\zeta \omega \pm i \sqrt{1-\zeta ^2} \omega $. The lines must intersect with the root-locus plot to get a feasible $K$. The $K$ value at which the intersection occurs is the value you are looking for.

You did not provide the transfer function, so I made one up. Your analysis will be something like this.

enter image description here


The solution according to your professor's logic:

The characteristic equation is $$1 + K G(s) = 0$$

This implies $$K = \left| \frac{1}{G(s)}\right|$$

From the figure we know that $G(s)$ has two finite poles and one finite zero. So write $$G(s) = \frac{s-z}{(s-p_1)(s-p_2)}$$

The closed loop pole needs to be at $s = -10 +8.83 i$ (the conjugate will give the same result). Thus:

$$K = \left|\frac{\left(-10+8.83 i-p_1\right) \left(-10+8.83 i-p_2\right)}{-10+8.83i-z}\right|$$

But you again need to read off the values of the open-loop zero and open-loop poles from the figure to get $z$, $p_1$, and $p_2$.

  • $\begingroup$ There is not transfer function for this problem. I found time to speak with the professor. K is the magnitude of 1/G(s). 1/G(s) can be flipped so you have the Poles in the numerator and the zeros in the denominator. The magnitude is the distances from the point of interest to the poles and zeros. So you draw lines from the point of interest to the zeros and poles use geometry to find the distances and then you can multiply the two pole distances and divide by the radius distance to get the magnitude of 1/G(s). I'm not the best at explaining but that's the general way he explained it. $\endgroup$ Nov 6 '15 at 20:06
  • $\begingroup$ Ok. Then what are the poles and zeros. The zero location is 10 (where one of the roots end). I approximated the pole locations. Can you tell me the exact location where the poles start, i.e. the location of the x's. They will be complex conjugates. If you figure out the poles and zeros, you figure out the transfer function with K=1! $\endgroup$ Nov 6 '15 at 21:33

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