GATE 2024 IN Q.44: solution and answer

Official answer: ASource: Official answer key, GATE 2024 IN (IISc Bengaluru)

1. Closed-loop transfer function, negative feedback with $G(s)=\frac{1}{s(s+1)}$ and $H(s)=K$: $$\frac{Y(s)}{R(s)}=\frac{G}{1+GH}=\frac{\frac{1}{s(s+1)}}{1+\frac{K}{s(s+1)}}=\frac{1}{s^2+s+K}$$

2. Stability check: the characteristic equation $s^2+s+K=0$ is second order, so it is stable exactly when all coefficients are positive, that is for $K>0$. Under that condition the final value theorem can be used.

3. The question defines the error as $e=r-y$. In a non-unity feedback loop that is not the actuating signal $r-Ky$, so the usual $\frac{1}{1+K_p}$ formula does not apply. Work it out directly: $$E(s)=R(s)-Y(s)=R(s)\left[1-\frac{1}{s^2+s+K}\right]=R(s)\frac{s^2+s+K-1}{s^2+s+K}$$

4. Final value theorem with $R(s)=\frac{1}{s}$: $$e_{ss}=\underset{s\to 0}{\lim }s\cdot \frac{1}{s}\cdot \frac{s^2+s+K-1}{s^2+s+K}=\frac{K-1}{K}$$

5. Check: the closed-loop DC gain is $\frac{1}{K}$, so $y(\infty )=\frac{1}{K}$ and $e_{ss}=1-\frac{1}{K}=\frac{K-1}{K}$. The error is zero only when $K=1$ (unity feedback). This is option (A).