GATE 2024 CH – Question 64
A PD controller with transfer function $G_c$ is used to stabilize an open-loop unstable process with transfer function $G_p$, where $G_c = K_c\frac{\tau_Ds + 1}{\left(\frac{\tau_D}{20}\right)s + 1}$, $G_p = \frac{1}{(s - 1)(10s + 1)}$ and time is in minutes. From the necessary conditions for closed-loop stability, the maximum feasible value of $\tau_D$, in minutes, rounded off to 1 decimal place, is ____________
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Correct answer: 22.09 to 22.31
Explanation
The characteristic equation is $(s - 1)(10s + 1)\left(\frac{\tau_D}{20}s + 1\right) + K_c(\tau_Ds + 1) = 0$. Let $a = \frac{\tau_D}{20}$. Expanding, $10a\,s^3 + (10 - 9a)s^2 + (K_c\tau_D - 9 - a)s + (K_c - 1) = 0$. A necessary condition for stability is that all the coefficients are positive. The $s^2$ coefficient requires $10 - 9a > 0$, that is $a < \frac{10}{9}$, so $\tau_D < 20 \times \frac{10}{9} = 22.2$ min. (The other conditions, $K_c > 1$ and $K_c\tau_D > 9 + a$, can be met by choosing $K_c$ large enough.)