Hi everyone , Hope u fine.
I have a negative feedback system that consists of an integral controller and a process.

The task is to sketch the root locus(no the issue) , and the system should the following specifications which are settling time Ts<=10 seconds and percent overshoot PO<=10%.

OK, I belive that Wn (nature frequency) will be in terms of K and here where i should manipulate numbers in order to meet the specifications the problem is how to get zeta and w out of 3rd order characteristic equation ???

Please see my attempt and the question in the attached file

 

Are you familiar with root locus or other tools that assist with the design process?
Given your target overshoot and settling time limits, you will be able to determine the required or target "dominant" second order complex conjugate closed loop poles, since a specific damping factor & natural frequency would equate to the overshoot and settling time criteria. Since we have a third (not second) order system, the only concern will be the third real valued closed loop pole and how it impacts the overall response. Some final trial and error tweaking of the gain term K might be required.
If your target closed loop pole pair don't match the actual system dominant poles as the gain K varies, you might focus initially on the damping factor \(\zeta\).
For a given percentage overshoot (PO) you would anticipate the approximating second order response to have a minimum damping factor (<1) given by

\(\text{\zeta_{min}=sqrt{\frac{\( ln(\frac{PO}{100})\)^2}{\pi^2+\( ln(\frac{PO}{100})\)^2}}\)

The "dominant" closed loop poles as gain K varies would have some form

\(\text{\(a \pm jb \)}\)

Provided the equality

\(\text{\frac{b}{a} \gt = {\frac{sqrt{\( 1- \zeta_{min}^2\)}}{\zeta_{min}}}}\)

is met, then the overshoot should be within spec. It remains then to check whether the settling time constraint is met for the explicit values of \(\text{\( a \pm jb \)}\)

Keep in mind that the settling time Ts only has meaning when one specifies the allowable variation in response value from time Ts onwards. This might be +/- 5% of the final steady state value - as the benchmark.