Relationship between root locus pole and percent overshoot and gain in MATLAB

Question

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?

Answer 1

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.

The features of a dipole in the transient response are: a relatively large initial overshoot, and a long-tail (i.e. the settling time is longer than that promised by the initial response characteristic) . Both of these are apparent in your step response plot.

Answer 2

In the root locus plot, MATLAB computes the percentage overshoot according to the lines of constant damping ratio (ζ). The problem is that the damping ratio only makes sense for a second order system and the transfer function used (gproc) is not a second-order system since it has 2 zeros and 3 poles.