$9.$ Consider the system is given as in figure below with $\backslash (\backslash$mathrm$\{$T$\}=1\backslash$mathrm$\{$~s$\}\backslash ).$ Let the digital controller be a variable gain $\backslash ($ K $\backslash )$ such that $\backslash ($ D$($z$)=$K $\backslash ).$ Hence $\backslash ($ m$($k T$)=\backslash$operatorname$\{$Ke$\}($k T$)\backslash ).$

$($i$)$ Write the closed$-$loop system characteristic equation

$($ii$)$ Determine the range of K for which the system is stable.

$($iii$)$ Determine the range of K for which the system is stable using Routh$-$Hurtwiz criterion.

$($iv$)$ Plot the z$-$plane root locus.

$($v$)$ Plot the w$-$plane root locus.

$($vi$)$ Determine the range of K for stability using results of part $($iv$).$

$($vii$)$ Determine the range of K for stability using results of part $($v$).$