The unit step response of a second order system is = 1 − e^(−5t) − 5t e^(−5t). Consider the following statements:1.The under damped natural frequency is 5 rad/s.2.The damping ratio is1.3.The impulse response is 25t e^(−5t).Which of the statements given above are correct?
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The response 1−e⁻⁵ᵗ−5te⁻⁵ᵗ is the standard unit-step response of a critically damped second-order system with repeated pole at −5. Hence ζ=1, and differentiating the step response gives impulse response h(t)=25te⁻⁵ᵗ. Statement 1 is false because there is no underdamped natural frequency.
Formula / Key Relation
c(t)=1−(1+5t)e⁻⁵ᵗ
ζ = 1
ωn = 5 rad/s
ωd = ωn√(1−ζ²)=0
h(t)=dc(t)/dt
=25t e⁻⁵ᵗ
Detailed Explanation
Concept & ReasoningFor a critically damped system G(s)=ω_n²/(s+ω_n)² with ζ=1. Its unit-step response is 1−e^{-ω_n t}(1+ω_n t). Comparing with the given expression gives ω_n=5 rad/s and ζ=1. The damped oscillation frequency ω_d=ω_n√(1−ζ²)=0, so calling 5 rad/s an 'underdamped natural frequency' is incorrect. The impulse response is the derivative of the unit-step response.How to Solve It in the ExamSpot (1+ω_nt)e^{-ω_nt} → critically damped repeated pole.Important Exam PointImpulse response = derivative of unit-step response for an LTI system.Editorial / Answer-Key NoteThe official GPSC final answer key revised the provisional outcome for this question from D to B.Related Revision PathThis question sits in…
Concept & Reasoning
For a critically damped system G(s)=ω_n²/(s+ω_n)² with ζ=1. Its unit-step response is 1−e^{-ω_n t}(1+ω_n t). Comparing with the given expression gives ω_n=5 rad/s and ζ=1. The damped oscillation frequency ω_d=ω_n√(1−ζ²)=0, so calling 5 rad/s an 'underdamped natural frequency' is incorrect. The impulse response is the derivative of the unit-step response.
How to Solve It in the Exam
Spot (1+ω_nt)e^{-ω_nt} → critically damped repeated pole.
Important Exam Point
Impulse response = derivative of unit-step response for an LTI system.
Editorial / Answer-Key Note
The official GPSC final answer key revised the provisional outcome for this question from D to B.
Related Revision Path
This question sits in the revision path Control Systems → Second-Order Systems → Time Response .
Quick Trick
Spot (1+ω_nt)e^{-ω_nt} → critically damped repeated pole.
Common Mistake
Calling ω_n=5 an underdamped frequency when ζ=1.
Exam Tip
Impulse response = derivative of unit-step response for an LTI system.
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