$2.$ Given the system in Fig. $1$ with $\backslash ($ G$($s$)=\backslash$frac$\{10\}\{$s$($s$+2)($s$+6)\}\backslash ).$

$(1)$ Design the simplest controller $\backslash (\backslash$mathrm$\{$G$\}\_\{\backslash$mathrm$\{$C$\}\}(\backslash$mathrm$\{$s$\})\backslash )$ in Fig. $1$ to yield a $20\backslash \%$ overshoot, two$-$second settling time and zero steady$-$state error for the closed$-$loop system with a unit step input using the Bode plots approach. $(10$ pts$.)$

$(2)$ Obtain the state$-$space model for $\backslash (\backslash$mathrm$\{$G$\}(\backslash$mathrm$\{$s$\})\backslash )$ in the phase variable canonical form by inspection. $(5$ pts$.)$

$(3)$ If all the state variables in part $(2)$ are accessible, design a state feedback controller to yield a $\backslash (20\backslash \%\backslash )$ overshoot and two$-$second settling time for the closed$-$loop system with a unit step input. $(10$ pts$.)$