For the following control system, I am trying to find the characteristics of the system.
I have firstly found the closed loop transfer functions \$T(s) =\frac{C(s)}{R(s)}\$ , to be:
\$G(s) =K_p(s+9)*\frac{10}{s+12}*\frac{1}{s^2+6s+15}\$
\$G(s) =\frac{10K_ps+90K_p}{s^3+18s^2+87s+180}\$
Now as feedback is Uniity Feedback, therefore H = 1, I get;
\$T(s) = \frac{G(s)}{1+G(s)H(s)}\$
Which means \$T(s) = \frac{10K_ps+90K_p}{s^3+18s^2+87s+180+10K_ps+90K_p}\$
My first question for this problem is, is this Transfer Function correct, or have I miscalculated during a particular step up to this point.
Continuing on, I am now trying to find the \$K_p\$ value that leads to less than a 15% Overshoot for the system.
Unfortunately, I am unsure what to do next to find the \$K_p\$ value for this problem.
Any working or help as to how I go about this would be greatly appreciated.