# Relationship Between Root Locus Pole and Percent Overshoot and Gain in MATLAB

I have a peculiar confusion regarding the relationship between a closed-loop pole on a root locus plot and its gain and percent overshoot. I have found a point on the root locus plot that MATLAB claims to have a percentage overshoot of 5%: using the following code:

    s = tf('s');
gproc = ((s+3)*(s+6))/(s^2*(s+2));
rlocus(gproc); % plot root locus


However, when I plot the step response, the percent overshoot seems to be 22%: using the following code:

    gopen = 13.71*gproc;
step(feedback(gopen,1));


What is the reason for this discrepancy? Have I misunderstood the MATLAB code?

From the root locus, the dominant closed loop pole will be the real pole between $$\\small s=-2\$$ and $$\\small s=-3\$$, since it's approximately three times further from the origin than the 2nd order complex poles.
Now, there's also a closed loop zero at $$\\small s=-3\$$, so you have a closed pole and a closed loop zero quite close together. In control engineering, this is called a dipole.
Note, a dipole is often created when attempting to cancel a troublesome, perhaps slow, pole by plonking a zero on top of it - in practice, there's always an error between the pole and zero values, hence a dipole is born. Essentially, that's what you've done here - by choosing the value of $$\\small K=13.7\$$, the first order pole is dominant. Choosing a smaller value for $$\\small K\$$ would have given dominance to the 2nd order roots, which would then make the system more oscillatory. But there's a limit to what can be done when there's only one control parameter ($$\\small K\$$) to play with.