$4.($a$)$ Consider the control system shown in Figure $5,$ where $\backslash ($ K $\backslash )$ and $\backslash ($ T $\backslash )$ are compensator parameters to be determined.

Figure $5$

Determine $\backslash ($ K $\backslash )$ and $\backslash ($ T $\backslash )$ so that the gain crossover frequency and phase margin of the system are $\backslash (1\backslash$mathrm$\{$rad$\}/\backslash$mathrm$\{$s$\}\backslash )$ and $\backslash (45^\{\backslash$circ$\}\backslash ),$ respectively.

$(10$ Marks$)$