$(20$ pts$)$ Consider the following closed$-$loop PD control system,

Design the controller to meet the specifications: $Ts=0\mathrm{.}5sec,\%OS=5\%.$

a$.$ Determine the target values for $$ and ${}_n.$

$5=\mathrm{.}69$

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

$\frac{4}{20.j{x}_n}=\mathrm{.}5$

$w_n=\frac{4}{(\mathrm{.}69)(\mathrm{.}5)}=11\mathrm{.}59$

$T=\frac{(k_1+k_{2}s)}{s^2+10s+20+k_1+k_{2}s}$

$=\frac{k_1+k_{2}s}{s^2+s(40+k_2)+(k_1+70)}$

b$.$ Solve for K$1$ and K$2$ to meet the design spec

$x_1+20={}_{a^2}=11\mathrm{.}5a^2=134\mathrm{.}3$

$k_1=114\mathrm{.}3$