Question

A system with the open loop transfer function:

$G\left(s\right)=\frac{K}{s(s+2)(s^2+2s+2)}$ is connected in a negative feedback configuration with a feedback gain of unity. For the closed-loop system to be marginally stable, the value of k is

• A. 5

Answer 1

The correct option is A 5
Given,

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


The characteristic equation of the given unity negative feedback control system is given by

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

$or,1+\frac{K}{s(s+2)(s^2+2s+2)}=0$

$or,s(s+2)(s^2+2s+2)+K=0$

or, $s^4+4s^3+6s^2+4s+K=0$

Forming Routh's array as shown below:

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

For stability of the system,

$K>0and\frac{20-4K}{5}>0$

$or,K<5$

$\therefore$ For stability, 0 < K < 5

For the given system to be marginally stable,

K = 5