Problem #$3(20$ marks$--$ a$)-12$marks$-($b$)4$marks, $($c$)4$marks$)$

The open$-$loop transfer function for a unity feedback system is:

$G(s)=\frac{K}{s(s+4)(s^2+8s+32)}$

$($a$)$ Draw the root$-$locus for the closed$-$loop system as K varies from $Ks=-1\mathrm{.}33+-j1\mathrm{.}33G(s)=\frac{0\mathrm{.}0065(s^2+9s+49)}{s^2(s+1)}G(s)0,$ showing all relevant steps and details.

$(b)$ Determine the maximum allowable gain $Kof$ the amplifier for a stable system.

$(c)$ Show that the closed$-$loop poles $s=-1\mathrm{.}33+-j1\mathrm{.}33$ lie $on$ the locus

Problem #$4(20$ marks$-(a)-10$marks, $(b)-6$marks, $(c)-4$marks$)$

A unity feedback system has a plant transfer function given $by$

$G(s)=\frac{0\mathrm{.}0065(s^2+9s+49)}{s^2(s+1)}$

$a$

Please ansewr problems $3$ and $4$ as if they where an exam.

Thank you