Problem #$1(20$ marks$-($a$)-5$marks, $($b$)-10$marks, $($c$)-5$marks$)$

The system block diagram is shown in the following figure. $($a$).$ Draw the signal flow graph, $($b$)$ Determine the transfer function $T(s)=Y\frac{s}{R}(s).($c$)$ Determine the transfer function $Y\frac{s}{D}(s)$

$D((s){)}_{\text{n}\text{i}\text{s}\text{t}\text{u}\text{r}\text{h}\text{m}\text{e}\text{s}\text{.}}$

Problem #$2(20$mark$-($a$)-12$marks, $($b$)-4$marks, $($c$)-4$marks$)$

a$)$ Plot the Nyquist diagram of the following system for $k=1.$

$GH(s)=\frac{0\mathrm{.}5k(s+1+j)(s+1-j)}{s(s-2)}$

b$)$ Investigate the stability of the system based on the Nyquist diagram for $k=1.$

c$)$ Using the Nyquist diagram determine the range of k where the closed$-$loop system is stable.

Please answer problems $1$ and $2$ as if this was an exam.