The characteristic equation is \(1 + {\rm{G}}\left( {\rm{s}} \right){\rm{H}}\left( {\rm{s}} \right) = 0\)
\(\begin{array}{l}
\Rightarrow 1 + \frac{{\rm{k}}}{{{\rm{s}}\left( {{\rm{s}} + 1} \right)\left( {{\rm{s}} + 3} \right)}} = 0\\
\Rightarrow {{\rm{s}}^3} + 4{{\rm{s}}^2} + 3{\rm{s}} + {\rm{k}} = 0
\end{array}\)
Solving the Routh’s table,
\({{\rm{s}}^3}\)

1

3

\({{\rm{s}}^2}\)

4

k

\({{\rm{s}}^1}\)

\(\frac{{12  {\rm{k}}}}{4}\)


\({{\rm{s}}^0}\)

k


For finding imaginary axis poles, we set \(\frac{{12  {\rm{k}}}}{4} = 0\)
Thus,
\({\rm{k}} = 12\)