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
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)