GATE 2020 IN Q.37: solution and answer

GATE 2020 · INQ.37 · 2 marks · NAT

The system shown in Fig. (a) has a time response y(t) to an input $r (t) = 10 u (t)$ as shown in Fig. (b), $u (t)$ being the unit step input. Both $K$ , $τ$ are positive. The gain $K$ of the system is ____

GATE 2020 IN question 37: Fig. — Figure from the GATE 2020 IN paper. It is part of the question.

Answer and solution

ANSWER

The gain is $K = 4$ .

Official answer: 4Source: Official answer key, GATE 2020 IN (IIT Delhi)

Closed-loop transfer function with unity negative feedback: $\frac{Y ( s )}{R ( s )} = \frac{\frac{K}{τ s + 1}}{1 + \frac{K}{τ s + 1}} = \frac{K}{τ s + 1 + K}$
The input is $r (t) = 10 u (t)$ , so $R (s) = \frac{10}{s}$ .
Final value theorem: $y (\infty) = s \to 0 lim s \cdot \frac{K}{τ s + 1 + K} \cdot \frac{10}{s} = \frac{10 K}{1 + K}$ The theorem applies because the only closed-loop pole, $s = - \frac{1 + K}{τ}$ , lies in the left half-plane when $K, τ > 0$ .
Fig. (b) shows a steady-state value of $8$ : $\frac{10 K}{1 + K} = 8 \Rightarrow 10 K = 8 + 8 K \Rightarrow K = 4$
Check: the loop is type 0 with $K_{p} = lim_{s \to 0} \frac{K}{τ s + 1} = K = 4$ . So $e_{ss} = \frac{10}{1 + K _{p}} = 2$ and $y (\infty) = 10 - 2 = 8$ , which agrees with Fig. (b). The steady state does not depend on $τ$ , so the figure gives no information about it and none is needed.

KEY IDEA

With unity feedback around a type-0 first-order plant, the closed-loop DC gain is $\frac{K}{1 + K}$ . The step response settles at $10 \cdot \frac{K}{1 + K}$ , not at $10 K$ . Set this equal to the asymptote in the plot, 8, and solve for $K$ . The time constant $τ$ only changes how fast the response rises, not where it settles.

WATCH OUT FOR

Using the open-loop gain as though the loop were open, $10 K = 8$ giving $K = 0.8$ , which ignores the feedback path. Another error is reading the 8 as the steady-state error rather than the output, which gives $\frac{10}{1 + K} = 8$ and $K = 0.25$ . Some candidates also try to find $τ$ from the curve, but the plot has no time scale and $τ$ is not asked for.

From the GATE 2020 IN paper. Every question of this paper · Download the full paper · All 22 Time response questions, with notes

GATE 2020 IN Q.37: solution and answer

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