Peak Overshoot⚠ unverified
Physics / Controls · Percent overshoot of a second-order system step response
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| zeta | ζ | — | 0.5 | Damping ratio |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| overshoot | %OS | — | Percent overshoot (%) |
The science & history
Understanding the Parameters
-
$\zeta$ — larger damping → smaller overshoot. $\zeta = 0$ → 100% theoretical infinite ring (formula $\to 100$); $\zeta \to 1^-$ → overshoot $\to 0$. Critically/overdamped systems have no overshoot (formula not applicable).
-
$\%OS$ — $(c_{\max} - c_{\infty})/c_{\infty}\times 100\%$ for a unit-step final value of 1.
Derivation (Approaching a Proof)
The underdamped step response peaks at $t_p = \pi/(\omega_n\sqrt{1-\zeta^2})$. Substituting into $c(t) = 1 - e^{-\zeta\omega_n t}/\sqrt{1-\zeta^2}\,\sin(\omega_d t+\phi)$ yields the first peak value $1 + e^{-\pi\zeta/\sqrt{1-\zeta^2}}$, hence the percent overshoot above.
History
Percent overshoot vs $\zeta$ is a standard second-order design chart in classical control (Nise, Ogata, Dorf & Bishop).
Related Concepts: Damping Ratio From Overshoot, Settling Time 2%, Rise Time, Phase Margin From Damping
Notes: Registry calculator peak-overshoot (unverified). Underdamped second-order idealisation only.