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GATE 2023 CH – Question 45

Instrumentation and Process Control · Closed-loop stability and frequency response · 2 marks · Multiple choice

Level ($h$) in a steam boiler is controlled by manipulating the flow rate ($F$) of the make-up (fresh) water using a proportional (P) controller. The transfer function between the output and the manipulated input is $\frac{h(s)}{F(s)} = \frac{0.25(1 - s)}{s(2s + 1)}$. The measurement and valve transfer functions are both equal to 1. A process engineer wants to tune the controller so that the closed-loop response gives decaying oscillations under servo mode. Which one of the following is the CORRECT value of the controller gain to be used by the engineer?

  1. 0.25
  2. 2
  3. 4
  4. 6

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Correct answer: (B) 2

Explanation

The closed-loop characteristic equation is $1 + K_c\frac{0.25(1 - s)}{s(2s + 1)} = 0$, that is $2s^2 + (1 - 0.25K_c)s + 0.25K_c = 0$. For stability, $K_c < 4$. For complex roots (oscillation), the discriminant $(1 - 0.25K_c)^2 - 2K_c < 0$, which gives $K_c^2 - 40K_c + 16 < 0$ and so $0.4 < K_c < 39.6$. So decaying oscillations need $0.4 < K_c < 4$, and only $K_c = 2$ is in this range. ($K_c = 4$ gives sustained oscillation, 6 is unstable and 0.25 is overdamped.)