GATE 2022 CH – Question 60
Consider a single-input-single-output (SISO) system with the transfer function $G_p(s) = \frac{2(s + 1)}{\left(\frac{1}{2}s + 1\right)\left(\frac{1}{4}s + 1\right)}$ where the time constants are in minutes. The system is forced by a unit step input at time $t = 0$. The time at which the output response reaches the maximum is __________ minutes (rounded off to two decimal places).
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Correct answer: 0.54 to 0.56
Explanation
Rewrite $G_p = \frac{16(s + 1)}{(s + 2)(s + 4)}$. The step response is $\frac{16(s + 1)}{s(s + 2)(s + 4)} = \frac{2}{s} + \frac{4}{s + 2} - \frac{6}{s + 4}$, so $y(t) = 2 + 4e^{-2t} - 6e^{-4t}$. Setting $\frac{dy}{dt} = -8e^{-2t} + 24e^{-4t} = 0$ gives $e^{2t} = 3$ and $t = \frac{\ln 3}{2} = 0.55$ min.