Range of K - Problem on Stability1.3

Ask

For the characteristic equation:$s^4+Ks^3+s^2+s+1=0$ the range of K for stability is

(a)K&gt0

(b)K&gt1

(c)K&lt1

(d)none of above

Answer is ( d )

Concept

Explanation : Construct Roth's array:

$$ \begin{array}{c|lcr} n & \text{Left} & \text{Center} & \text{Right} \\ \hline s^{4} & 1 & 1 & 1 \\ s^{3} & K & 1 & \\ s^{2} & {K-1 \over K} & 1 & \\ s^{1} & {K-1 - K^{2} \over K} & & \\ s^{0} & 1 \end{array} $$ For stability of system $ \left. \begin{array}{l} \ K &gt0 & \\ \frac{K}{K-1} &gt 0 & \\ \end{array} \right\} $ = K&gt1 also , by substituting K &gt 1 in equation $ frac{K-1-K^{2}}{K-1} &gt 0 $, the equation gives negative value.Since all three conditions cannot be satisfied simultaneously.Hence (d).