GATE 2023 EC Q.41: solution and answer

Official answer: ASource: Official answer key, GATE 2023 EC (IIT Kanpur)

1. The loop has unity feedback and $G(s)=\frac{k}{s(s+2)}$. $G(s)$ has one pole at the origin, so the system is type 1. A type-1 system follows a ramp with a finite, nonzero steady-state error.

2. Check stability first, because the final value theorem only applies to a stable loop. The characteristic equation is $1+G(s)=0\Rightarrow s^2+2s+k=0$. For a second-order polynomial, all coefficients positive is enough for stability, and that holds for every $k>0$.

3. Velocity error constant: $$K_v=\underset{s\to 0}{\lim }sG(s)=\underset{s\to 0}{\lim }\frac{k}{s+2}=\frac{k}{2}$$

4. Apply the final value theorem to $E(s)=\frac{R(s)}{1+G(s)}$: $$e_{ss}=\underset{s\to 0}{\lim }s\cdot \frac{\alpha /s^2}{1+\frac{k}{s(s+2)}}=\underset{s\to 0}{\lim }\frac{\alpha (s+2)}{s(s+2)+k}=\frac{2\alpha }{k}$$

5. Cross-check with the shortcut for a ramp of slope $\alpha$: $e_{ss}=\frac{\alpha }{K_v}=\frac{\alpha }{k/2}=\frac{2\alpha }{k}$. The answer is option (A).