Settling Time (2%)⚠ unverified
Physics / Controls · Approximate 2% settling time of a second-order system
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| wn | ωn | rad/s | 10.0 | Natural frequency |
| zeta | ζ | — | 0.5 | Damping ratio |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| ts | ts | s | Settling time |
The science & history
Understanding the Parameters
- $\omega_n$ — higher $\omega_n$ → faster settling at fixed $\zeta$.
- $\zeta$ — more damping → faster envelope decay; very small $\zeta$ → long settling.
- $t_s$ — rule of thumb from $e^{-\zeta\omega_n t_s} \approx 0.02$ ⇒ $\zeta\omega_n t_s \approx 4$.
Derivation (Approaching a Proof)
The transient envelope is $e^{-\zeta\omega_n t}$. Setting $e^{-\zeta\omega_n t_s} = 0.02$ gives $\zeta\omega_n t_s = -\ln 0.02 \approx 3.91 \approx 4$. Hence $t_s \approx 4/(\zeta\omega_n)$. This ignores the oscillatory factor and is invalid for $\zeta \ge 1$ (different forms).
History
The $4/\zeta\omega_n$ estimate is ubiquitous in introductory control design for specifying speed of response.
Related Concepts: Peak Overshoot, Rise Time, Bandwidth, Damping Ratio From Overshoot
Notes: Registry calculator settling-time (unverified). Approximate underdamped second-order form.
Filename uses ASCII 2% as in the bulk ingest.