Calculation
tₛ = −(T₀/D)·ln[(p/100)·√(1−D²)]
Intersecting the exponential oscillation envelope with the permitted final-value error gives a traceable upper estimate of settling time.
Zacher/Reuter 2024, Abschnitte 3.5 und 8.1: PT2-Hüllkurve und Ausregelzeit
Second-order settling time for a tolerance band
Intersecting the exponential oscillation envelope with the permitted final-value error gives a traceable upper estimate of settling time.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Conservative settling time tₛ — Time from the input step after which the second-order envelope lies inside the selected tolerance band. Use it for observation duration, test time or a cautious wait before steady-state evaluation.
- Damping ratio D — Dimensionless decay parameter of the second-order model. Obtain it from model parameters or a decay curve. This calculator applies only to the oscillatory range 0<D<1.
- Natural time T₀ — Reciprocal of undamped natural angular frequency: T₀=1/ω₀. Take it from the identified second-order model; together with D it sets decay speed.
- Relative tolerance band p [%] — Permitted absolute deviation from final value as a percentage of the total step change. Common choices are 2% or 5%; use the value required by your test or process.
Example
For D=0.5, T₀=1 s and a ±2% band, the envelope gives tₛ≈7.801 s. From then on, the modelled error magnitude is guaranteed to be below 2% of the step change.
Assumptions and limits
Normalised linear time-invariant second-order system without dead time, zeros or additional dominant poles, underdamped case 0<D<1. The result is envelope-based and therefore conservative; the actual last band crossing may occur earlier.
Understand Second-order settling time for a tolerance band
This calculator answers a practical question: how long should one wait after a step until an oscillatory second-order response is guaranteed to remain close enough to its final value? It provides a conservative estimate from the exponential envelope.
What does this quantity describe?
Settling time is the instant after which the response remains permanently inside a specified band around final value. A second-order process has two energy-storage elements. For 0<D<1 it oscillates; D is damping ratio and T₀ is natural time. The calculator equates the maximum envelope error with band p.
Formula and variables
tₛ = −(T₀/D)·ln[(p/100)·√(1−D²)]
Relative envelope: h(t)=exp(−D·t/T₀)/√(1−D²)Condition: h(tₛ)=p/100tₛ=−(T₀/D)·ln[(p/100)·√(1−D²)]
| Symbol / input | Meaning |
|---|---|
| Conservative settling time tₛ | Time from the input step after which the second-order envelope lies inside the selected tolerance band. Use it for observation duration, test time or a cautious wait before steady-state evaluation. |
| Damping ratio D | Dimensionless decay parameter of the second-order model. Obtain it from model parameters or a decay curve. This calculator applies only to the oscillatory range 0<D<1. |
| Natural time T₀ | Reciprocal of undamped natural angular frequency: T₀=1/ω₀. Take it from the identified second-order model; together with D it sets decay speed. |
| Relative tolerance band p [%] | Permitted absolute deviation from final value as a percentage of the total step change. Common choices are 2% or 5%; use the value required by your test or process. |
Choose the inputs correctly
D comes from the process model or a measured decay curve. T₀=1/ω₀ is the reciprocal of undamped natural angular frequency and belongs to the second-order model. p is permitted error as a percentage of the complete step change, not of the instantaneous reading. Obtain it from a test plan, process requirement or justified accuracy choice.
How to use the calculator
First identify a second-order model and verify 0<D<1. Choose a justified tolerance band, commonly ±2% for strict and ±5% for coarser assessment. Use the result as a minimum evaluation or simulation duration; add any separate process dead time.
Worked example
With D=0.5, T₀=1 s and p=2%, tₛ≈7.801 s. For the same system at 5%, conservative time decreases to about 5.966 s. Doubling T₀ doubles both times.
Understand the result and units
After tₛ, even the mathematical upper envelope lies within ±p. The real oscillation may make its final band crossing earlier; the calculator deliberately takes the safe side. A tighter band or weaker damping increases waiting time.
T₀ and tₛ are times; seconds, minutes and hours convert internally through SI seconds. D is dimensionless. p is entered as a percentage and divided by 100 internally.
Typical applications
Useful for test duration, waiting before accepting steady readings, simulation horizons, comparing damping cases and explaining a control-loop response.
Assumptions, limits and common mistakes
The result applies to an ideal underdamped second-order model without dead time, zeros or additional dominant dynamics. It is not an exact numerical search for the last curve crossing but a guaranteeing envelope estimate. Measurement noise or steady-state error needs a separate error allowance.
Common mistake: p=2 means 2%, not 0.02%. Do not confuse T₀ with damped oscillation period. Process dead time does not disappear and must be added to the calculated dynamic settling time.
Frequently asked questions
What is “Second-order settling time with a selectable tolerance band” used for?
Useful for test duration, waiting before accepting steady readings, simulation horizons, comparing damping cases and explaining a control-loop response.
Where do the input values come from?
D comes from the process model or a measured decay curve. T₀=1/ω₀ is the reciprocal of undamped natural angular frequency and belongs to the second-order model. p is permitted error as a percentage of the complete step change, not of the instantaneous reading. Obtain it from a test plan, process requirement or justified accuracy choice.
What does the result not cover?
The result applies to an ideal underdamped second-order model without dead time, zeros or additional dominant dynamics. It is not an exact numerical search for the last curve crossing but a guaranteeing envelope estimate. Measurement noise or steady-state error needs a separate error allowance.