An LTI system has a unit step response y(t) = (2 − e−t)u(t). What is the transfer function of this system?
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Answer & Explanation
Correct AnswerC. (s + 2)/(s + 1)
Quick Explanation
For a unit-step input, U(s) = 1/s and G(s) = Y(s)/U(s) = sY(s). Transforming the given response gives Y(s) = (s + 2)/[s(s + 1)]; therefore G(s) = (s + 2)/(s + 1).
Formula / Key Relation
Y(s) = 2/s − 1/(s + 1)
Y(s) = (s + 2)/[s(s + 1)]
U(s) = 1/s
G(s) = Y(s)/U(s) = sY(s)
G(s) = (s + 2)/(s + 1)
Detailed Explanation
Correct Answer
C — (s + 2)/(s + 1)
The given expression is the output to a unit-step input, so its Laplace transform must be divided by U(s)=1/s to obtain the transfer function.
What This Question Is Really Testing
The key distinction is between a step response and a transfer function. A step response already includes the dynamics of the system and the 1/s factor contributed by the step input.
Concept Foundation
For an LTI system with zero initial conditions:
G(s) = Y(s)/U(s)
If the input is a unit step, then U(s)=1/s, so the useful shortcut is G(s)=sY(s).
Step 1 — Take the Laplace Transform of the Step Response
y(t) = 2u(t) − e−tu(t)
Y(s) = 2/s − 1/(s + 1)
Step 2 — Combine the Two Terms
Y(s) = [2(s + 1) − s]/[s(s + 1)]
Y(s) = (s + 2)/[s(s + 1)]
Step 3 — Divide by the Unit-Step Input
U(s) = 1/s
G(s) = Y(s)/(1/s)
G(s) = sY(s)
G(s) = (s + 2)/(s + 1)
Hence the correct option is C.
Why This Method Works
The step input contributes one factor 1/s to the output transform. Removing that input factor leaves only the system transfer function.
Independent Verification / Cross-Check
Final step-response value = 2
G(0) = (0 + 2)/(0 + 1) = 2
The DC gain of the derived transfer function equals the final value of the stable unit-step response, so the answer is consistent.
Concept Extension
Impulse-response transform: H(s)=G(s).
Unit-step-response transform: Y(s)=G(s)/s.
Exam Strategy
When a unit-step response is given, first transform it and then use G(s)=sY(s) under zero initial conditions. Use the final value as a fast DC-gain check.
Quick Trick
Unit-step response given? Transform it and multiply by s.
Why Other Options Are Wrong
A and B do not reproduce the given step response. D has G(0)=1/2, whereas the given step response settles to 2.
Common Mistake
Using Y(s) directly as G(s) and forgetting that the unit-step input itself has U(s)=1/s.
Exam Tip
After finding G(s), check G(0) against the final value of a stable unit-step response.
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