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GATE 2025 EC – Question 36

Control Systems · Compensators and PID Controller · 2 marks · Multiple choice

Let $G(s)=\dfrac{1}{10s^2}$ be the transfer function of a second-order system. A controller $M(s)$ is connected to the system $G(s)$ in the configuration shown below.

Consider the following statements.

(i) There exists no controller of the form $M(s)=\dfrac{K_I}{s}$, where $K_I$ is a positive real number, such that the closed loop system is stable.

(ii) There exists at least one controller of the form $M(s)=K_P+sK_D$, where $K_P$ and $K_D$ are positive real numbers, such that the closed loop system is stable.

Which one of the following options is correct?

Diagram for GATE 2025 EC question 36
  1. (i) is TRUE and (ii) is FALSE
  2. (i) is FALSE and (ii) is TRUE
  3. Both (i) and (ii) are FALSE
  4. Both (i) and (ii) are TRUE

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Correct answer: (D) Both (i) and (ii) are TRUE

Explanation

(i) With $M=K_I/s$ the characteristic equation is $1+\dfrac{K_I}{10s^3}=0$, i.e. $10s^3+K_I=0$. It has a missing $s^2$ and $s$ term, so it has roots in the right half-plane for every $K_I>0$: no stabilising controller, statement TRUE. (ii) With $M=K_P+sK_D$ the characteristic equation is $10s^2+K_Ds+K_P=0$. For a second-order polynomial all coefficients positive means stable, so any $K_P,K_D>0$ works: statement TRUE.