Find all the time domain specifications for a unity feedback control system whose open loop transfer function is given by G(S) 25

🧮 Question

Find all the time domain specifications for a unity feedback control system whose open-loop transfer function is given by:

✅ Answer:

The open-loop transfer function is:

For a unity feedback system, the closed-loop transfer function T(s) is:

Substituting G(s):

This is a standard second-order system of the form:

Comparing:

We identify:

• 2ζω_n = 4
• ω_n² = 25 ⇒ ω_n = √25 = 5 rad/sec
• ζ = 4 / (2 × 5) = 0.4

📊 Time Domain Specifications

• Natural Frequency (ω_n) = 5 rad/sec
• Damping Ratio (ζ) = 0.4
• Damped Natural Frequency (ω_d) = ω_n√(1 – ζ²) = 5√(1 – 0.16) = 5√0.84 ≈ 4.58 rad/sec
• Peak Time (T_p) = π / ω_d ≈ π / 4.58 ≈ 0.686 sec
• Rise Time (T_r) ≈ (π – cos⁻¹(ζ)) / ω_d ≈ (π – 1.159) / 4.58 ≈ 0.434 sec
• Settling Time (T_s) ≈ 4 / (ζ × ω_n) = 4 / (0.4 × 5) = 2 sec
• Maximum Overshoot (M_p) = e^{-πζ / √(1 – ζ²)} × 100
  ⇒ e^{-π×0.4 / √(1 – 0.16)} × 100 ≈ 25.4%

🎯 Final Summary:

• ω_n = 5 rad/s
• ζ = 0.4
• ω_d ≈ 4.58 rad/s
• Rise Time (T_r) ≈ 0.434 sec
• Peak Time (T_p) ≈ 0.686 sec
• Settling Time (T_s) ≈ 2 sec
• Maximum Overshoot (M_p) ≈ 25.4%