A high$-$performance aircraft shown below uses the ailerons, rudder, and elevator to steer through a three$-$dimensional flight path. The pitch rate control system for a fighter mireraft at $11000$ m and Nach I. I cun be represented by the system in Iigure $($b$).$

$G(s)=\frac{-20(s+0\mathrm{.}05)(s+0\mathrm{.}4)}{(s^2+1\mathrm{.}5s+17)(s^2+0\mathrm{.}06s+0\mathrm{.}005)}$

$($a$)$ Sketch the root locus when the controller is a simple gain, so that G $.($s$)=K,$ and determine K when $$ for the roots with $0_m>3\mathrm{.}5$ is larger than $0\mathrm{.}35($seck a maximum $4).($b$)$ Plot the response $q(t)$ for a step input $r(t)$ with $K$ as in $($a$).($c$)$ A designer suggests an anticipatory controller with $G_d(s)=K_1+K_{2}s=K(s+3).$ Sketch the root locus for this system as $K$ varies and determine a K so that Gof all the closed$-$loop roots is $$ U$.$S$.($d$)$ Plot the response q$($t$)$ for a step input $r(t)$ with $K$ as in $(c).$ You may use MATI.AB $IMILINK for this problem.