The output of a control system is c(t) = 1 + 0.25e ...

22. The output of a control system is

c(t) = 1 + 0.25e^-50t - 1.25e^-10t

For an unit step input, its undamped natural frequency and damping ratio will be:

• A. 50 rad/s & 1.34
• B. 500 rad/s & 0.5
• C. 22.36 rad/s & 0.35
• D. 22.36 rad/s & 1.34

Answer: Option D Solution:

Solution:

Given output response:

c(t) = 1 + 0.25e^-50t - 1.25e^-10t

Comparing with the standard overdamped second-order system response:

c(t) = 1 + A e^s₁t + B e^s₂t

The poles are:

• s₁ = -50
• s₂ = -10

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For a second-order system:

Characteristic equation:

s² + 2ζω_ns + ω_n² = 0

Sum of roots:

s₁ + s₂ = -2ζω_n

(-50) + (-10) = -2ζω_n

-60 = -2ζω_n

ζω_n = 30

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Product of roots:

s₁s₂ = ω_n²

(-50)(-10) = ω_n²

500 = ω_n²

ω_n = √500

ω_n = 22.36 rad/s

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Now,

ζ = 30 / 22.36

ζ = 1.34

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Correct Answer: D. 22.36 rad/s & 1.34

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