ISRO EC 2023 Q.63: solution and answer

Official answer: CSource: ISRO's revised and final answer key of 16 February 2024

1. A PID controller acts on the error $E(s)$ with three parts: proportional, integral and derivative. Its output is the sum of the three.

2. Proportional action: $u_p(t)=K_{p}e(t)$, so its transfer function is $K_p$.

3. Integral action: $u_I(t)=K_I\int e(t)dt$. Integration in time corresponds to division by $s$, so this part is $\frac{K_I}{s}$.

4. Derivative action: $u_d(t)=K_d\frac{de(t)}{dt}$. Differentiation in time (zero initial conditions) corresponds to multiplication by $s$, so this part is $K_{d}s$.

5. Adding the three parts: $$G_c(s)=K_p+K_{d}s+\frac{K_I}{s}$$ This is option (c).

6. Option (a) has $K_I{s}^2$: multiplying by $s^2$ means differentiating twice, not integrating. Option (b) has $K_d/s$ and $K_I/s^2$: dividing by $s$ means integrating, so the derivative term is wrong and the integral term is a double integral. Option (d) equals $s\left(K_p+K_{d}s+K_I\right)$, which has no $1/s$ integral term and no constant proportional term.