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# 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, $k> 0$

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

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

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

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

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

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