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GATE 2024 Chemical Engineering, Q64 · Process Control

The correct answer is 22.1 to 22.3.

Question

A PD controller with transfer function Gc=Kc(τDs+1)/[(τD/20)s+1]G_{c} = K_{c}(τ_{Ds}+1)/[(τ_{D}/20)s+1] is used to stabilize an open-loop unstable process with transfer function Gp=G_{p} = 1/[(s−1)(10s+1)], shown in the figure, and time is in minutes. From the necessary conditions for closed-loop stability, the maximum feasible value of τD,τ_{D}, in minutes, rounded off to 1 decimal place, is ____________

Figure for Q64 (Process Control, PD Control): A PD controller with transfer function G_c = K_c(τ_Ds+1)/[(τ_D/20)s+1] is used to stabilize an open-…

Official answer: 22.1 to 22.3

Why

GATE 2024 official key: 22.1 to 22.3. The closed-loop characteristic equation 1+GcGp=01+G_{c}G_{p}=0 expands to (τD/2)s3+[10−9τD/20]s2+[−9−τD/20+KcτD]s+(Kc−1)=0.(τ_{D}/2)s³ + [10−9τ_{D}/20]s² + [−9−τ_{D}/20+K_{c}τ_{D}]s + (K_{c}−1) = 0. A necessary Routh-Hurwitz condition (independent of Kc)K_{c}) requires all same-sign coefficients, so the s2s² coefficient must stay positive: 10−9τD/20>0⇒τD<200/9≈22.210−9τ_{D}/20 > 0 \Rightarrow τ_{D} < 200/9 \approx 22.2 min.

Explanation cross-checked for consistency. Final answer matches the official IIT answer key.

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