Gate 2016 EE - Set 2 - Question 39

For closed loop system, being stable, the characteristic equation is, 1+G(S)H(S)=0

1+\frac{k(S+1)}{S(1+Ts)(1+2S)}=0

S(1+TS)(1+2S)+k(S+1)=0

2TS^{3}+2S^{2}+TS^{2}+S+Sk+k=0

2TS^{3}+S^{2}(2+T)+S(1+k)+k=0

Using Routh's criteria, we arrive at the stable condition below, k> 0

(2+T)(1+k)-2Tk> 0

k(2+T-2T)+(2+T)> 0

k(2-T)+(2+T)> 0

k< \frac{(T+2)}{(T-2)}

0< k< \left ( \frac{T+2}{T-2} \right )