Problem $3.(8$pts$)$ A transfer function of a DC motor $G(s)$ that relates the motor spped to the input voltage is given by

$G_p(s)=\frac{({}^{})(s)}{V_{in}(s)}=\frac{10}{s^2+2s+10}$

The system $G(s)$ is placed in a negative unity feedback loop as shown in figure below.

Design a PID$-$controller, $C(s)=\frac{K_D{s}^2+K_{P}s+K_{\text{I}}}{s},$ by assuming second$-$order dominance to give a rise time $t_r$ of $0\mathrm{.}18$ s and $2\%$ settling time $t_s$ of $4$ s $.$ Place the non$-$dominant pole at a distance $10$ times the distance from the dominant second$-$order complex$-$conjugate poles th the imaginary axis.

Hints: $M_p=e^{-\frac{}{\sqrt[\phantom{2}]{1-{}^2}}},t_r=\frac{1\mathrm{.}8}{{}_n},t_s=\frac{4}{{}_n},t_p=\frac{}{{}_d}$