Root Locus

Here 5 Root Locus branches are diverging from same point, this can possible only when if we have 5 poles in the system at the same point because Root Locus branch departs from open loop pole and
Number of Root Locus branches = Number of open loop poles or Number of zero (Whichever is greater).
Here, there are 5 multiple Real poles, and matching with option (A).

Q(s)=1+G(s)H(s)=0
s^{3}+3s^{2}+2s+ks+3k=0
-k=\frac{s^{3}+3s^{2}+2s}{s+3}
-\frac{\mathrm{d} k}{\mathrm{d} s}=\frac{(s+3)(3s^{2}+6s+2)-(s^{3}+3s^{2}+2s)}{(s+3)^{2}}=0
3s^{3}+6s^{2}+2s+9s^{2}+18s+6-s^{3}-3s^{2}-2s=0
2s^{3}+12s^{2}+18s+6=0
s=-0.46,-3.87,-1.65

\therefore Break-away point lies between (0, -1), i.e. (-1, 0)

Given that,
G(s)=\frac{K\left(s^{2}+2 s+2\right)}{s^{2}-3 s+2}
The root locus plot of the given system is follows:


\therefore System is stable for K \gt 1.5.

\begin{aligned} G(s) &=\frac{K(s+2)}{s^{2}+2 s+2} \\ H(s)&=1 \\ 1+G(s) H(s) &=0 \\ 1+\frac{K}{\left(s^{2}\right)+2 s+2}(s+2) &=0\\ K&=-\frac{\left(s^{2}+2 s+2\right)}{(s+2)} \end{aligned}


\frac{d K}{d s}=-\left(\frac{(2 s+2)(s+2)-\left(s^{2}+2 s+2\right)}{(s+2)^{2}}\right)
For break away points, \frac{d K}{d s}=0
\begin{array}{l} (2 s+2)(s+2)-\left(s^{2}+2 s+2\right)=0 \\ 2 s^{2}+6 s+4-s^{2}-2 s-2=0 \\ s^{2}+4 s+2=0 \end{array}
\begin{aligned} s&=\frac{-4 \pm \sqrt{16-8}}{2} \\ &=\frac{-4 \pm 2 \sqrt{2}}{2}=-2 \pm \sqrt{2} \\ s&=-0.58 ; s=-3.41 \\ \text{But,}\quad s&=-3.41 \text{ lies on root locus }\\ \text{Hence}\quad s&=-3.41 \end{aligned}

Characteristic equation 1+G(s)H(s)=0
1+\frac{K}{s^2+5s+5}=0
K=-s^2-5s-5
For break away point \frac{dK}{ds}=0
\frac{dK}{ds}=-2s+5=0\;\Rightarrow \; s=-2.5
According to magnitude condition,
\begin{aligned} |G(s)H(s)|_{s=-2.5} &=1 \\ |G(s)H(s)|_{s=-2.5} &=\frac{|K|}{|(2.5)^2 +5 \times -2.5+5|}=1 \\ |K|&=|(6.25+5-12.5)|=1.25 \\ K&=\pm 1.25 \end{aligned}

The open loop transfer function of the system:
G(s) H(s)=\frac{K(s+3) \times 10}{(s+2)}
In order to lie a point s=-2.75, the angle condition must be satisfy and therefore the magnitude condition is
\begin{aligned} &\mid G(s) H(s) \|_{s=-275}=1 \\ &=\left.\frac{10 K \sqrt{s^{2}+3^{2}}}{\sqrt{s^{2}+2^{2}}}\right|_{s=-2.75}=1 \\ &=\frac{10 K \sqrt{(3-2.75)^{2}}}{\sqrt{(2-2.75)^{2}}}=1 \\ &=\frac{0.25 \times 10 K}{0.75} or K=0.3 \end{aligned}

\begin{aligned} G(s) &=\frac{K(s+4)}{(s+8)\left(s^{2}-9\right)} \\ &=\frac{K(s+4)}{(s+8)(s+3)(s-3)} \end{aligned}
For the point (-1+2 j) to lie on the root locus the angle condition must be satisfy
i.e. \angle G(s) H(s)=\pm(2 Q+1) 180^{\circ}
Taking LHS:
\begin{array}{l} \left.\angle G(s) H(s)\right|_{s=-1+2 j} \\ =\frac{\angle K+\angle(s+4)}{\angle(s+8)+\angle(s+3)+\angle(s-3)} \\ =\frac{\angle K+\angle(-1+2 j+4)}{\angle(-1+2 j+8)+\angle(-1+2 j+3)+\angle(-1+2 j-3)} \\ =\frac{0+\tan ^{-1}\left(\frac{2}{3}\right)}{\tan ^{-1}\left(\frac{2}{7}\right)+\tan ^{-1}(1)+180^{\circ}-\tan ^{-1}\left(\frac{1}{2}\right)} \\ =33.69^{\circ}-15.945^{\circ}-45^{\circ}-180^{\circ}+26.565^{\circ} \\ \simeq 180^{\circ} \end{array}
As angle condition is satisfied the value of system gain K can be obtained by using magnitude
condition
\begin{array}{l} i.e. |G(s) H(s)|_{s=-1+2 j}=1 \\ =\frac{K \sqrt{(-1+4)^{2}+2^{2}}}{\sqrt{(-1+8)^{2}+2^{2}} \cdot \sqrt{(-1+3)^{2}+2^{2}} \cdot \sqrt{(-1-3)^{2}+2^{2}}} \\ =1 \\ =\frac{K \sqrt{9+4}}{\sqrt{49+4} \sqrt{4+4} \sqrt{16+4}}=1 \\ =\frac{K \sqrt{13}}{\sqrt{53} \sqrt{8} \sqrt{20}}=1 \\ K=\frac{\sqrt{53 \times 8 \times 20}}{\sqrt{13}}=25.54 \end{array}

G(s)=\frac{K}{s(s+1)(s+3)}
The characteristic equation
\begin{array}{l} =1+G(s) H(s)=0 \\ =s(s+1)(s+3)+K=0 \\ =s^{3}+4 s^{2}+3 s+K=0 \end{array}
Using Routh's tabular form
\begin{array}{c|cc} s^{3} & 1 & 3 \\ s^{2} & 4 & K \\ s^{1} & \frac{12-K}{4} & 0 \\ s^{0} & K & \end{array}
In order to cross the imaginary axis, system
should be marginally stable
\begin{aligned} \frac{12-K}{4}&=0\\ \text{or }\quad K&=12 \end{aligned}

Since the root locus always emerges from the break-away points i.e. when two poles are at a part of same root locus lie. So \sigma =-1 and -2 will surely be poles.
Also we know that locus emerges from poles and terminates at zero. So, s=-4 is a zero and s=-7 is a pole
\therefore T(s)=\frac{(s+4)}{(s+1)(s+2)(s+7)}

From plot we can observe that one pole terminates at one zero at position -1 and three poles terminates to \infty. It means there are four poles and 1 zero. Pole at -3 goes on both sides. It means there are two poles at -3.