Finding desired closed-loop poles from root locus

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.

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

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.

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

• 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? Commented Jun 13, 2017 at 16:09
• Sure, can you update your Question to include the transfer-function of your plant 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:

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