Discussion
The response of a system is obtained as \(C(t) = 1 + 0.2\exp(-60t) - 1.2\exp(-10t)\), when subjected to a unit-step input. The close loop transfer function is:
Answer: C
Take the Laplace transform of the step response and divide by the transform of the unit step, \(1/s\).
\[ \begin{aligned} C(s) \\ &= \frac{1}{s} + \frac{0.2}{s + 60} - \frac{1.2}{s + 10} \end{aligned} \]
The common denominator is \(s(s + 10)(s + 60)\), and the numerator works out to \(600\):
\[ \begin{aligned} \frac{(s+10)(s+60) + 0.2s(s+10) - 1.2s(s+60)}{s(s+10)(s+60)} \\ &= \frac{600}{s(s+10)(s+60)} \end{aligned} \]
\[ \begin{aligned} \frac{C(s)}{R(s)} = sC(s) \\ &= \frac{600}{(s + 10)(s + 60)} \\ &= \frac{600}{s^2 + 70s + 600} \end{aligned} \]
The dc gain is \(600/600 = 1\), matching the final value 1. The answer is option C.
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