Second-Order Step Response
Overshoot, peak and settling time from the damping ratio and natural frequency.
Everything from ζ and ωₙ
A second-order system ωₙ² / (s² + 2ζωₙs + ωₙ²) hides its entire step response in
two numbers: the damping ratio ζ sets the shape (how much it overshoots and
rings), and the natural frequency ωₙ sets the speed. For an underdamped system
(ζ < 1):
where ω_d = ωₙ√(1−ζ²) is the damped (ringing) frequency and σ = ζωₙ is
the decay rate. The overshoot depends only on ζ — ζ = 0.5 always gives ~16 %, ζ = 0.707
gives ~4 %, and ζ ≥ 1 gives none. Settling time falls as ζωₙ rises, which is exactly
the trade-off a controller tunes: more damping kills overshoot but a higher ωₙ is what makes it
fast. Map these targets back to a controller with the PID
simulator, or convert between ζ, overshoot and phase margin with the
damping converter.