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Control engineering
Damping ratio from measured overshoot
Inverse of the overshoot formula: the damping ratio that gives a measured percentage overshoot in a standard second-order step response.
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- CTL-06
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- Result
- ζ
This is the overshoot formula Mp = e−πζ/√(1−ζ²) solved for ζ. Enter the overshoot as a percentage of the final value, read from a step response. The result is the damping ratio of the second-order model that would overshoot by that much.
Inverse of the overshoot formula: the damping ratio that gives a measured percentage overshoot in a standard second-order step response.
Calculator
- Natural logarithm of the overshoot fraction, ln(Mp/100)−1.814
- Pole angle from the negative real axis, cos⁻¹ ζ60 °
- Rule-of-thumb phase margin, 100ζ (reasonable below about ζ = 0.6)50 °
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.
- An overshoot of 16.3 % corresponds to ζ ≈ 0.5. Overshoots of 5 % and 25 % correspond to ζ ≈ 0.69 and 0.404.
- Halving the overshoot to 8.15 % raises the required damping from 0.5 to 0.624. The overshoot is very sensitive to damping, so a small change in ζ makes a large visible difference.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Mp Percentage overshoot | 1 to 60 % | 0.1605 | 0.8261 | 0.6656 | +10% | −4.02 % |
Sweep of Percentage overshoot
| Percentage overshoot (%) | Damping ratio |
|---|---|
| 1 | 0.8261 |
| 8.375 | 0.6196 |
| 15.75 | 0.5071 |
| 23.13 | 0.4225 |
| 30.5 | 0.3536 |
| 37.88 | 0.2953 |
| 45.25 | 0.2447 |
| 52.63 | 0.2002 |
| 60 | 0.1605 |
Sweep one variable
Variables and how each one contributes
Damping ratio
What the equation gives you. Everything else feeds it.
Percentage overshoot
The percentage overshoot: (peak − final value) / final value × 100. More overshoot means less damping, but ζ changes more slowly with overshoot at the high end. Raising it by 10 % (to 17.93) moves the result by −4.02 %, from 0.5 to 0.4799.
Worked example with the values above
- Logarithm of the fraction: ln(0.163) = −1.814.
- Denominator: √(π² + ln²) = √(9.87 + 3.291) = 3.628.
- ζ = −(−1.814) / 3.628 = 0.5.
Assumptions and limits
- The system behaves like a standard second-order system with no zeros, so that overshoot depends on ζ only.
- A step from rest, measured from a record with the final value reached.
Common mistakes
- Entering the overshoot as a fraction (0.163) when the input is a percentage (16.3).
- Applying it to a loop with a zero or a slow extra pole, where the overshoot says little about the true damping.
Statistics and uncertainty
A step response with noise gives an overshoot that is biased high, so it should be measured on a smoothed record and ideally averaged over several steps. Because ζ comes from a logarithm of the overshoot, an error in a small overshoot (a few per cent) matters much more than the same absolute error in a large one.
Where it is used
- Identifying the damping of a servo, vehicle or actuator from a recorded step.
- Converting an overshoot specification into a damping ratio for design.
Maths you need
- Natural logarithm: the inverse of the exponential
- Step response of a second-order system: peak overshoot Mp