GATE 2022 CH – Question 59
Consider the tank level control system shown in the figure, where the cross-section area of the tank is $A$. Assume perfect flow controllers (FC). The level controller (LC) is proportional-integral (PI). For an integral time, $\tau_I$, the level controller gain, $K_c$, is tuned for critical damping. The value of $\frac{K_c\tau_I}{A}$ is ________ (rounded off to the nearest integer).
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Correct answer: 4
Explanation
The tank is an integrator, $A\frac{dh}{dt} = F_{in} - F_{out}$, and the PI controller sets $F_{out} = K_c\left(e + \frac{1}{\tau_I}\int e\,dt\right)$ with $e = h - h_{sp}$. The closed loop is $A\tau_Is^2 + K_c\tau_Is + K_c = 0$, that is $s^2 + \frac{K_c}{A}s + \frac{K_c}{A\tau_I} = 0$. Critical damping needs a zero discriminant: $\left(\frac{K_c}{A}\right)^2 = \frac{4K_c}{A\tau_I}$, so $\frac{K_c\tau_I}{A} = 4$.