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, $$k> 0$$

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

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

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

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

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