Concept:

The characteristic equation for a given open loop transfer function G(s) is

1 + G(s) H(s) = 0

According to the Routh tabulation method,

The system is said to be stable if there are no sign changes in the first column of Routh array.

For marginally stable, s¹ row must be zero.

The number of poles lie on the right half of s plane = number of sign changes

Calculation:

Given that, \(G\left( s \right) = \frac{K}{{s\left( {s + 2} \right)\left( {{s^2} + 2s + 2} \right)}}\)

Feedback gain, H(s) = 1

Characteristic equation: \(1 + \frac{K}{{s\left( {s + 2} \right)\left( {{s^2} + 2s + 2} \right)}} = 0\)

⇒ s (s + 2) (s² + 2s + 2) + K = 0

⇒ s⁴ + 4s³ + 6s² + 4s + K = 0

By applying Routh tabulation method,

\(\begin{array}{*{20}{c}} {{s^4}}\\ {{s^3}}\\ {{s^2}}\\ {{s^1}}\\ {{s^0}} \end{array}\left| {\begin{array}{*{20}{c}} 1&6&K\\ 4&4&0\\ 5&K&{}\\ {\frac{{20 - 4K}}{5}}&0&{}\\ K&{}&{} \end{array}} \right.\)

For marginally stable, s¹ row must be zero.

\( \Rightarrow \frac{{20 - 4K}}{5} = 0\)