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Q#1 Stability and Frequency Analysis of Control Systems GATE CH 2024 NAT +2 marks -0 marks

Consider the surge drum in the figure. Initially the system is at steady-state with a hold-up \(\begin{align*} \bar{V} = 5 \text{ m}^3, \end{align*}\)
, which is 50% of full tank capacity, \(𝑉_{𝑓𝑒𝑙𝑙}\) , and volumetric flow rates \(𝐹̅_{𝑖𝑛} = 𝐹̅_{π‘œπ‘’π‘‘} = 1 m^3 h^{−1}\). The high hold-up alarm limit \(V_{high} = 0.8 V_{full}\) while the low hold-up alarm limit \(𝑉_{π‘™π‘œπ‘€} = 0.2 𝑉_{𝑓𝑒𝑙𝑙}\). A proportional (P-only) controller manipulates the outflow to regulate the hold-up 𝑉 as 

\(\begin{align*} F_{\text{out}} = K_c (V - \bar{V}) + \bar{F}_{\text{out}} \end{align*}\)

At 𝑑 = 0, 𝐹𝑖𝑛 increases as a step from\(1 m^3 h^{−1}\) to \(2 m^3 h^{−1}\).  Assume linear control valves and instantaneous valve dynamics. Let \(𝐾_𝑐 ^{π‘šπ‘–π‘›}\) be the minimum controller gain that ensures 𝑉 never exceeds \(V_{high}\). The value of \(𝐾_𝑐^{π‘šπ‘–π‘›}\) , in \(β„Ž^{−1}\) , rounded off to 2 decimal places, is _________

.

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