... control system is: c(t) = 1 + 0.25e^-50t - 1.25e^-10t

Second Order Control System

Question:

What are the damping ratio (ζ) and undamped natural frequency (ωₙ) if the output of a control system is:

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

1. Identify the Exponential Terms

The response contains two exponential terms:

• e^-50t
• e^-10t

These correspond to the system poles.

s₁ = -50

s₂ = -10

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2. Form the Characteristic Equation

For poles s₁ and s₂:

(s - s₁)(s - s₂) = 0

Substitute the values:

(s + 50)(s + 10) = 0

Expanding:

s² + 60s + 500 = 0

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3. Compare with Standard Second-Order Form

s² + 2ζωₙ s + ωₙ²

Comparing with:

s² + 60s + 500

We get:

2ζωₙ = 60

ωₙ² = 500

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4. Calculate Undamped Natural Frequency

ωₙ = √500

ωₙ = 22.36 rad/s

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5. Calculate Damping Ratio

ζ = 60 / (2 × 22.36)

ζ ≈ 1.34

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Final Results

• Undamped Natural Frequency (ωₙ) = 22.36 rad/s
• Damping Ratio (ζ) = 1.34

Conclusion: Since ζ > 1, the system is overdamped (non-oscillatory response).