I am trying to design a P controller (proportional constant only) to satisfy certain overshoot and settling time requirements for my plant. My values for \$\zeta\$ and \$w_n\$ which would satisfy the requirements are 0.86 and 0.52 respectively. So I went ahead and made a root locus of the plant's transfer function showing the lines for \$\zeta\$ and \$w_n\$ using the sgrid command. This would help me find the value of K.

Desired closed-loop poles

As seen in the plot, the root locus does not lie within the desired location (in between the two dotted lines and outside the semi-circle). I am not sure how I can select the desired pole locations in this case. I might be doing something wrong, or not fully understanding what needs to be done. Could someone please guide me here?

EDIT: My plant transfer function is \$\frac{8.25}{6.6s^2+s}\$
I need a proportional controller for my plant which would meet my requirements of \$<0.5\%\$ overshoot and \$<8s\$ settling time


2 Answers 2


The issue you have is you have plotted the natural frequency \$\omega_n\$ of the system not a design requirement for settling time. You want to plot a vertical line at $$ s \simeq - \dfrac{3.9}{t_s} $$

If you have Matlab you can also accomplish this with the SISO tool. Add a design requirement for settling time and you will get a vertical line. If the complex pole-pair exists towards neg-infinity in the left-hand plane, you have satisfied the settling time requirement.

Sample root locus plot below. enter image description here

Here any complex-pole pair that exists in the white space satisfies design requirements for overshoot and settling time.

  • \$\begingroup\$ I tried using SISO tool beforehand, and this is what I am getting for my design requirements: i.imgur.com/h1v1uAP.png It seems impossible to place a closed-loop pole anywhere in the white area since it does not lie on the root locus. Also, the natural frequency and damping coefficient I plotted earlier are in fact obtained from the settling time and overshoot. This is the guide I followed: ctms.engin.umich.edu/CTMS/… Any ideas how I can add a compensator to my system with the same plant transfer function/requirements? \$\endgroup\$
    – Jspake
    Commented Jun 13, 2017 at 16:09
  • \$\begingroup\$ Sure, can you update your Question to include the transfer-function of your plant \$\endgroup\$
    – sstobbe
    Commented Jun 13, 2017 at 17:21

The root-locus results indicate the desired requirements cannot be met with just a proportional controller. So, you are left with two options: change the controller, or change the requirements.

A controller with a zero in -1, a pole in -10 and gain 25 meets the requirements just fine:

enter image description here

$$C(s) = 25\frac{s+1}{s+10}$$


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