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.
Best Answer
To find the \$K_p\$ value that leads to less than a 15% Overshoot for the system. we can use the root locus plot, by looking at the position of the dominant poles (the two closer to the imaginary line). I have also added the
sgrid
lines showing the \$\zeta\$ and \$\omega\$ positions. Using the formula that $$ \zeta \geq \frac{-\ln(PO/100)}{\sqrt{ \pi^2 + \ln^2(PO/100) }},$$ we have that $$ \zeta \geq \frac{-\ln(15/100)}{\sqrt{ \pi^2 + \ln^2(15/100) }}= 0.517.$$ So, by positioning the poles below that 0.55 damping line should get you the overshoot to meet your requirement. Therefore, for a gain in \$K_p \in [-1,2]\$ you would meet that requirement. (the image looks horrible, but if you click on it it gets slightly better)