help An aircraft pitch control system includes a controller and a gyroscope to automatically control the pitch of the aircraft based on its dynamics. A simplified block diagram of the pitch control system for an aircraft is shown in FIGURE Q$3,$ where the gain $\backslash ($ K$\_\{$p$\}\backslash )$ should be as large as possible to reduce the effect of wind disturbances.

a$.$ Formulate the overall transfer function, $\backslash ($ T$($s$)=\backslash$frac$\{$C$($s$)\}\{$R$($s$)\}\backslash ).$

$[5$ marks$]$

b$.$ Construct the root$-$locus for the closed loop system. Show all the necessary steps and calculations.

$[12$ marks$]$

c$.$ If the damping ratio, $\backslash (\backslash$zeta $\backslash ),$ of the closed loop system is to be limited to $0\mathrm{.}5,$ propose a suitable value for $\backslash ($ K$\_\{$p$\}\backslash )$ to meet the requirement.