Library / Control engineering / CTL-05
Control engineering
Approximate 2 % settling time of a second-order system
Common estimate of the time for the step response to stay within 2 % of its final value.
- Ref
- CTL-05
- Rev
- v1
- Field
- Control engineering
- Result
- ts [s]
The oscillation of an underdamped step response is wrapped in an envelope that decays as e−ζωnt. The response has settled within 2 % when that envelope has fallen to 2 %, i.e. when ζωnt = 4 (since e−4 ≈ 0.018). For a 5 % band use 3/(ζωn) instead.
Common estimate of the time for the step response to stay within 2 % of its final value. It is set by how fast the oscillation envelope decays, which is the real part of the poles.
Calculator
- Real part of the poles, ζωn2 1/s
- Envelope time constant, 1/(ζωn)0.5 s
- Damped cycles before settling, ts ωd / 2π1.103
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The settling time depends on the product ζωn only, the distance of the poles from the imaginary axis. At the defaults it is 2 s. Poles with the same real part settle in about the same time, however oscillatory they are.
- Raising the damping to ζ = 0.7 at the same ωn shortens the settling time to 1.429 s and cuts the overshoot from 16.3 % to 4.6 %.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ζ Damping ratio (below 1) | 0.1 to 0.95 | 1.053 | 10 | 8.947 | +10% | −9.09 % |
| ωn Natural frequency | 1 to 20 rad/s | 0.4 | 8 | 7.6 | +10% | −9.09 % |
Sweep of Damping ratio (below 1)
| Damping ratio (below 1) | Settling time (2 %) (s) |
|---|---|
| 0.1 | 10 |
| 0.2062 | 4.848 |
| 0.3125 | 3.2 |
| 0.4187 | 2.388 |
| 0.525 | 1.905 |
| 0.6312 | 1.584 |
| 0.7375 | 1.356 |
| 0.8438 | 1.185 |
| 0.95 | 1.053 |
Sweep one variable
Variables and how each one contributes
Settling time (2 %)
What the equation gives you. Everything else feeds it.
Damping ratio (below 1)
The damping ratio, below 1. Inversely proportional: more damping at the same ωn moves the poles further from the imaginary axis and shortens the settling time. Raising it by 10 % (to 0.55) moves the result by −9.09 %, from 2 to 1.818 s.
Natural frequency
The natural frequency in rad/s. Inversely proportional: a faster system settles sooner by the same percentage. Raising it by 10 % (to 4.4) moves the result by −9.09 %, from 2 to 1.818 s.
Worked example with the values above
- Real part of the poles: ζωn = 0.5 × 4 = 2 1/s.
- Four envelope time constants: ts ≈ 4 / 2.
- ts ≈ 2 s.
Assumptions and limits
- A standard second-order system with no zeros, 0 < ζ < 1, and moderate damping.
- A 2 % band around the final value.
Common mistakes
- Using 4/(ζωn) for an overdamped or critically damped system, where it underestimates the time.
- Quoting it for a 5 % band, which needs 3 envelope time constants instead of 4.
Statistics and uncertainty
The 2 % rule is an approximation of a stepped quantity: the exact settling time jumps when an oscillation peak crosses the 2 % band, and can differ from 4/(ζωn) by roughly a quarter of the damped period. A measured settling time also depends on the noise and on the band chosen, so state both.
Where it is used
- Turning a settling-time requirement into a limit on the real part of the closed-loop poles.
- Checking whether a design settles within the time available between commands.
Maths you need
- Pole location: poles at −ζωn ± jωd
- Exponential envelope: the oscillation decays as e^(−ζωn t)