Question $1(25$ points$)$

Design a PD Control Given the Percent Overshoot as$,PO=15\%$ and the Peak time as $t_p=0\mathrm{.}1$ Seconds. Show calculated damping ratio $$ and the natural frequency ${}_n.$

Obtain the Proportional gain $k_p$ and the derivative gain $k_d$ for the above parameters for the PD control.

$K$ Given is $23\mathrm{.}2$ and $$ is $0\mathrm{.}13$ for the system.

Damping ratio $()$:

$=-ln(\frac{PO}{100})**\sqrt[\phantom{2}]{\frac{1}{ln(\frac{PO}{100}{)}^2+{}^2}}$

Natural Frequency $({}_n.)$:

${}_n=\frac{}{t_p\sqrt[\phantom{2}]{1-{}^2}}$

Standard Second Order Equation:

$\frac{Y(s)}{R(s)}=\frac{{}_n^2}{s^2+2{}_{n}s+{}_n^2}$

Closed loop solution for Qube Servo Motor Model:

$\frac{Y(s)}{R(s)}=\frac{K{K}_p}{s^2+(1+K{K}_d)s+K{K}_p}$