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GATE 2025 CH – Question 25

Instrumentation and Process Control · Process modelling, transfer functions and dynamic response · 1 mark · Multiple choice

Choose the transfer function that best fits the output response to a unit step input change shown in the figure

the output response to a unit step in the input. The output stays flat for a short dead time, then rises in an S-shaped curve, overshoots the final value slightly and settles back to it.
  1. $\frac{(\alpha s + 1)e^{-\theta s}}{(\tau_1 s + 1)(\tau_2 s + 1)^2}$
  2. $\frac{(\alpha s + 1)e^{-\theta s}}{(\tau_1 s + 1)(\tau_2 s + 1)}$
  3. $\frac{(\alpha s + 1)}{(\tau_1 s + 1)(\tau_2 s + 1)^2}$
  4. $\frac{(\alpha s + 1)^2e^{-\theta s}}{(\tau_1 s + 1)(\tau_2 s + 1)^2}$

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Correct answer: (A) $\frac{(\alpha s + 1)e^{-\theta s}}{(\tau_1 s + 1)(\tau_2 s + 1)^2}$

Explanation

Three features can be read from the curve. First, there is a flat start, which needs a dead time $e^{-\theta s}$, so option C (no delay) is out. Second, after the dead time the response starts with zero slope and then bends upward (an S-shape), which needs a transfer function with the denominator at least two orders higher than the numerator. Option B has a difference of only one, so its response would start with a non-zero slope. Third, the output overshoots the final value, which needs a lead term $(\alpha s + 1)$ in the numerator. Option D has two numerator zeros that make the difference one again. Option A, with one lead term, a third-order denominator and a dead time, matches all three features.