EE$3011$

$($b$)$ The control system as shown in Figure $2$ has a proportional controller $K>0$ which can be adjusted to yield certain response, and that

$G(s)=\frac{10}{(s+1)(s+3{)}^2}$

$($i$)$ Determine the range of $K$ for which the control system is stable. Compute also the value of $K$ that will produce an undamped oscillation and determine its oscillation frequency, if any.

$($ii$)$ Determine the range of $K,$ if any, for which the closed$-$loop system will have all its poles lying to the left of $s=-1\mathrm{.}0$ in the complex plane.