Second-order response and steady-state error

Drag a pole in the s-plane and watch overshoot, peak time, rise time and settling time change. Formula values sit beside measured ones, and you can add a third pole to test dominance. A second tab shows type 0/1/2 loops tracking step, ramp and parabola inputs.

s-plane — drag the pole. Rays are constant ζ, the circle is constant ω_n, the vertical line is constant settling time.

\zeta \text{ (damping ratio)}

0.35

\omega_n

4 rad/sAdd a third (real) pole — test the dominant-pole approximation

	formula	measured
$\text{peak overshoot } M_p$	$30.9\%$	30.9%
$\text{peak time } t_p = \pi/\omega_d$	$0.838\,\text{s}$	0.838 s
$\text{rise time } t_r = (\pi-\cos^{-1}\zeta)/\omega_d$	$0.515\,\text{s}$	0.515 s
$\text{settling } t_s \approx 4/\zeta\omega_n$	$2.857\,\text{s}$	2.746 s (exact 2%)

What to notice

Overshoot depends on $\zeta$ only: $M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}}$ . Change $\omega_n$ and the curve only stretches in time; the peak height stays put.
Poles on the same ray from the origin share $\zeta$ , hence the same overshoot. Poles on the same vertical line share $\sigma = \zeta\omega_n$ , hence the same settling time.
Here $\omega_d = \omega_n\sqrt{1-\zeta^2}$ $= 3.747$ rad/s.

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