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I have an exercise to draw a root locus plot and determine its gain to make sure that it fulfills the performance requirements.

the given system G(s) is $$G(s) = \frac{1}{s^2+s}$$

And the given controller:

$$K(s) = k \frac{s+2}{s+10}$$

I have to draw the root locus plot and determine k so that the rise time is less than 0.25 s and overshoot is less than 20%.

This is my entire matlab code:

    s = tf('s');
    Gs = 1/(s^2+s)
    Ks = (s+2)/(s+10)
    sys = feedback(Gs*Ks,1)
    hold on
    plot([0 -10], [omega_d omega_d],"LineWidth",3)
    plot([0 -10], [-omega_d -omega_d],"LineWidth",3)
    plot([-sigma -sigma],[-30 30],"LineWidth",3)
    plot([0 -10],[0 rad2deg(asin(zeta))])
    plot([0 -10],[0 -rad2deg(asin(zeta))])
    rlocus(sys);
    grid on
    hold off

this is the root locus plot

Out from the calculations, the frequence domain of the performance requirement are:

  • The natural frequency, $\omega_n = \frac{1.8}{0.25} = 7.2$

  • The damping ratio, $\zeta = 0.4559$

  • The real component, $\sigma = -3.2828$

  • And the imaginary component, $\omega_d = \pm 6.4080$

Now the question is:

How do I find the gain k, where the two poles bypasses the real value of -3.2828, despite the unknown imaginary component.

I did try to plug directly into s but the 3rd pole intersect -3.2828, not two other poles. The resulting gain is 39. (although it fulfills the requirements)

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  • $\begingroup$ you have a zero, which will end up in the closed loop expression... so make sure that pole on the left is close to canceling your zero at the gain you calculate (as it travels rightward along the blue line), or else the "dominant complex pole" simplification/analysis might lead you astray. Better yet, just graph the step response in matlab, for the gain you think you need. $\endgroup$
    – Pete W
    Aug 15, 2021 at 23:15
  • $\begingroup$ That zero is occupied by the third pole. The first and second pole will go to infinity. $\endgroup$
    – Zain
    Aug 16, 2021 at 20:41
  • $\begingroup$ That is the tradeoff.... increasing the gain very much results in good cancellation of the zero vs third pole, but letting the complex conjugate pole pair split apart "toward infinity" means that pole pair's damping ratio (related to angle from horizontal) gets worse.... thus there is an optimim gain... $\endgroup$
    – Pete W
    Aug 16, 2021 at 21:05

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