GATE 2020 EE Q.36: solution and answer

Official answer: BSource: Official answer key, GATE 2020 EE (IIT Delhi)

1. Closed-loop characteristic equation (unity negative feedback): $1+G(s)=0$. Multiplying through by the denominator of $G(s)$ gives $(s^3+2s^2+2s+K)+(s^2+s+1)=0$.

2. Combine terms: $$s^3+3s^2+3s+(K+1)=0$$

3. Routh array: $$\begin{array}{rl}{s}^3& :13\\ s^2& :3K+1\\ s^1& :\frac{3\cdot 3-1\cdot (K+1)}{3}=\frac{8-K}{3}\\ s^0& :K+1\end{array}$$

4. Poles on the imaginary axis appear when a complete row of the Routh array becomes zero. The $s^0$ entry $K+1$ cannot be zero for $K>0$, so the only possibility is the $s^1$ row: $\frac{8-K}{3}=0\Rightarrow K=8$.

5. Auxiliary equation from the $s^2$ row at $K=8$: $3s^2+9=0\Rightarrow s=\pm j\sqrt{3}$, so two closed-loop poles sit on the $j\omega$ axis at $\omega =\sqrt{3}\text{rad/s}$.

6. Check: at $K=8$ the characteristic polynomial is $s^3+3s^2+3s+9=(s+3)(s^2+3)$, with roots $-3$ and $\pm j\sqrt{3}$. Hence $K=8$, option (B).