Control Loop Lab

1. 1

   Identify the damping regime

   G(s) = ωn² / (s² + 2ζωn·s + ωn²)

   ζ = 0.5 < 1 → underdamped (complex poles): it overshoots and rings

   ζ alone decides the regime; ωn only sets how fast everything happens.

2. 2

   Damped natural frequency

   ω_d = ωn·√(1 − ζ²)

   ω_d = 2·√(1 − 0.5²) = 1.73 rad/s

   The ringing oscillates at ω_d — always a little slower than ωn.

3. 3

   Peak overshoot

   Mp = e^(−ζπ / √(1 − ζ²))

   Mp = e^(−0.5·π / √(1 − 0.5²)) = 16.3 %

   Overshoot depends on ζ only — drag ωn and watch this number stay put. Try ζ ≈ 0.7, the classic design point.

4. 4

   Time to the first peak

   t_p = π / ω_d

   t_p = π / 1.73 = 1.81 s

5. 5

   Rise time (0→100%)

   t_r = (π − cos⁻¹ζ) / ω_d

   t_r = (π − cos⁻¹ 0.5) / 1.73 = 1.21 s

6. 6

   Settling time (±2% band)

   t_s ≈ 4 / (ζ·ωn)

   t_s ≈ 4 / (0.5·2) = 4 s

   After t_s the response stays inside the shaded ±2% band around the setpoint for good.

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