Q$2.$ Consider the unity negative feedback system with the open$-$loop transfer function given as

$\backslash [$

G$($s$)=\backslash$frac$\{2($s$+$K$)\}\{\backslash$left$($s$^\{2\}+3$ s$+$K$\backslash$right$)($s$+1)\}$

$\backslash ]$

a$.$ Sketch the root locus for $\backslash ($ K $\backslash$geq $0\backslash )$ with all the detalls like break$-$away$/$in points and imaginary axis crossings.

b$.$ Based on the root locus, what can you say about the behavior of the frequency of oscillations observed in the step$-$response of the closed$-$loop system, i$.$e$.,$ does it increase$/$decrease$/$remain constant? Explain you answer.

c$.$ Based on your root locus, find the minimum settling$-$time $(2\backslash \%)$ the closed$-$loop system can have. Hint: For this system, the minimum settling$-$time is reached when all three closed$-$loop poles have the same real part.