Library / Control engineering / CTL-03
Control engineering
Percentage overshoot of an underdamped second-order system
Peak overshoot of the step response of a standard second-order system with no zeros.
- Ref
- CTL-03
- Rev
- v1
- Field
- Control engineering
- Result
- Mp [%]
ζ is the damping ratio of the standard second-order system ωn²/(s² + 2ζωns + ωn²). The unit-step response overshoots its final value by Mp at its first peak, given here as a percentage. It depends on ζ only: a faster or slower system with the same ζ overshoots by the same amount.
Peak overshoot of the step response of a standard second-order system with no zeros. It depends on the damping ratio alone, not on the natural frequency.
Calculator
- Ratio of successive peaks, Mp²0.02658
- 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.
- At ζ = 0.5 the overshoot is 16.3 %. A more heavily damped ζ = 0.7 gives 4.6 %, and a lightly damped ζ = 0.2 gives 52.7 %.
- Overshoot is set by the angle of the poles, not their speed. Poles on a line through the origin at the same angle all give the same overshoot, and the natural frequency then only sets the time scale of the response.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (%) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ζ Damping ratio (0 < ζ < 1) | 0.1 to 0.9 | 0.1524 | 72.92 | 72.77 | +10% | −22.5 % |
Sweep of Damping ratio (0 < ζ < 1)
| Damping ratio (0 < ζ < 1) | Percentage overshoot (%) |
|---|---|
| 0.1 | 72.92 |
| 0.2 | 52.66 |
| 0.3 | 37.23 |
| 0.4 | 25.38 |
| 0.5 | 16.3 |
| 0.6 | 9.478 |
| 0.7 | 4.599 |
| 0.8 | 1.516 |
| 0.9 | 0.1524 |
Sweep one variable
Variables and how each one contributes
Percentage overshoot
What the equation gives you. Everything else feeds it.
Damping ratio (0 < ζ < 1)
The damping ratio. Below 1 the response oscillates and overshoots. At 1 the system is critically damped, with no overshoot. Overshoot falls steeply as damping increases. Raising it by 0.1 (to 0.6) moves the result by −41.9 %, from 16.3 to 9.478 %.
Worked example with the values above
- Exponent: πζ / √(1 − ζ²) = π × 0.5 / √(1 − 0.25) = 1.814.
- Overshoot as a fraction: e−1.814 = 0.163.
- Mp = 100 × 0.163 = 16.3 %.
Assumptions and limits
- A standard second-order system with no zeros, so the numerator is fixed at ωn². A zero or a third slow pole changes the overshoot.
- A unit step from rest, with 0 < ζ < 1.
Common mistakes
- Applying it to a loop that has a zero, such as one with a PI controller. The overshoot can then be much larger than the formula says.
- Mixing the fraction with the percentage: 0.163 and 16.3 % are the same overshoot.
Statistics and uncertainty
Overshoot is read at a peak, so measurement noise inflates it, and a record should be smoothed before the peak is read. Near the default the overshoot is sensitive to ζ: a ±10 % error in ζ (0.45 to 0.55) gives an overshoot from 12.6 % to 20.5 %.
Where it is used
- Choosing a damping ratio to meet an overshoot limit, for example under 10 %.
- Checking a measured step response against the damping a design was meant to have.
Maths you need
- Complex poles: s = −ζωn ± jωd for 0 < ζ < 1
- The exponential function: e^(−x) for the decay of the oscillation