# Finding gain on root-locus

I have to design a compensator using root locus for the system $$G(s) = \frac{1}{s\cdot(s-4)}$$ that follow the following criteria:

$$op\% \le 5\%$$

$$t_{2\%} \le 4s$$

So I started computing $$\zeta$$ and $$\omega_n$$ this way:

$$\zeta = \sqrt{\frac{ln(op\%)^2}{\pi^2 + ln(op\%)^2}} = \sqrt{\frac{ln(0.05)^2}{\pi^2 + ln(0.05)^2}} = 0.6901$$

$$t_{2\%} = \frac{4}{\zeta \omega_n}\Rightarrow \omega_n = \frac{4}{4\cdot0.6901} = 1.4491$$

I've added a compensator $$C(s) = (s+1)$$ and plotted the root-locus, a horizontal line at $$-\omega_n\zeta$$ and a line with angle $$\theta = \cos^{-1}(\zeta)$$ which looks like this:

Now, from my understanding of the root-locus method picking a point in the same region this point on the next picture was chosen should fulfill my design criteria:

But if I use 9.35 as my gain on the feedback loop this is the step response I get:

The timing is alright but the overshoot is gigantic and I can't figure out why. What am I doing wrong here and why is my system not following the system criteria?

I define the system as such:

Kp = 9.35;
s = tf('s');
G = 1/(s*(s-4));
C = (s+1);


The root-locus is drawn like that:

rlocus(C*G)
line([0 -30], [0 30*tan(theta)])
line([0 -30], [0 -30*tan(theta)])


and the system is defined as such:

sys = feedback(Kp*C*G, 1);

• I think this is due to your zero at -1. Perhaps you could try placing your zero further left. – Ben May 23 at 23:58