Control SystemsDesign workbench
Second-Order System Calculator
Sweep ζ and ωn with live sliders and watch the poles move and the step response reshape — with every time- and frequency-domain characteristic computed in closed form.
About this calculator
The canonical second-order system G(s) = Kωn²/(s² + 2ζωn·s + ωn²) is the most important two-parameter object in control engineering: most well-designed loops are deliberately shaped so a single complex pole pair dominates, which makes second-order intuition transferable to systems of any order. Two numbers tell the whole story — the damping ratio ζ sets the SHAPE of the response, the natural frequency ωn sets its SPEED — and they separate cleanly: overshoot depends on ζ alone, while every time metric scales as 1/ωn.
Drag the sliders and the connection becomes physical: lowering ζ swings the pole pair toward the imaginary axis along the circle of radius ωn and the step response rings harder; raising ωn pushes the pair radially outward and the same-shaped response simply plays faster. This single interaction — poles moving, response reshaping, numbers updating together — is the mental model every later design tool builds on.
All characteristics come from the classical closed-form results: percent overshoot, peak time, the 4/(ζωn) settling approximation, damped frequency, and the frequency-domain trio (resonant peak, resonant frequency, bandwidth). Rise time is solved numerically from the exact step response rather than a curve-fit constant, so the number is defensible rather than memorized. Overdamped systems (ζ ≥ 1) have no peak and no closed-form rise time; there the tool reports simulated metrics and says which is which.
The worked solution derives each quantity symbol → substitution → result, exactly as the derivation runs in Nise or Ogata.
Design notes & common mistakes
- ζ and ωn are independent knobs: overshoot is pure ζ, speed is pure ωn. If a spec gives %OS and Ts, solve ζ from the overshoot first, then ωn = 4/(ζTs) — in that order.
- The 4/(ζωn) settling formula is an envelope approximation: the true 2% time steps discontinuously as ring peaks cross the band, so simulated Ts can differ from 4/(ζωn) by up to one half-period. Both are 'right'; state which you used.
- Adding a zero near the pole pair raises overshoot above the %OS(ζ) value; adding a far pole slows the rise. The canonical formulas describe the PURE two-pole system — check dominance before quoting them for a bigger model.
- Critical damping is rarely the practical optimum: ζ ≈ 0.7 settles faster to a band than ζ = 1 for the same ωn because it tolerates one small undershoot of the band.
Assumptions
- Pure canonical two-pole system: no finite zeros, no additional poles — the closed forms are exact only for this structure.
- Unit-step input, zero initial conditions; %OS and the settling band are relative to the final value K.
- Ts = 4/(ζωn) is the standard 2% envelope approximation (its assumption is stated in the worked solution).
- Rise time for ζ < 1 is solved numerically on the exact response (bisection), not from a fitted formula.
When to use this calculator
Appropriate for
- Exploring how damping ratio and natural frequency shape the step response of the canonical second-order system
- Computing overshoot, peak/rise/settling time, resonant peak, and bandwidth from ζ and ωn
- Building intuition that transfers to any loop with a dominant pole pair
Not suitable for
- Systems with a nearby zero or third pole, which shift every metric away from the pure two-pole prediction
- Real plants with saturation or rate limits, where the linear response does not hold for large steps
- Any use of these formulas on a higher-order system without first confirming a dominant pair actually governs it
What this calculator does not cover
- Describes the pure canonical system only — a zero or third pole in a real plant shifts every metric (the pole-zero analyzer checks dominance for such systems).
- ζ ≤ 0 is rejected: an undamped or negatively damped model has no settling behavior to characterize.
- The frequency-domain characteristics (Mr, ωr, ωBW) are for this G(s) as written, which equals the closed-loop shape only when your loop actually reduces to it.
- No saturation or rate limiting — real actuators bend large steps away from the linear prediction.
- As with every calculator on this site: results are preliminary and educational, are not verified for any specific installation, and must be reviewed against the applicable code edition and stamped by a licensed Professional Engineer before real-world use.
Frequently asked questions
Why does percent overshoot depend only on ζ?
Normalize time by ωn and the step response of the canonical system becomes a one-parameter family in ζ — ωn only stretches the time axis. The peak height is therefore a pure function of ζ: %OS = 100·exp(−ζπ/√(1−ζ²)).
What damping ratio should I aim for?
ζ ≈ 0.6–0.8 is the classic pocket: about 2–10% overshoot, near-minimum settling for a given ωn, and no significant frequency-domain resonance (the resonant peak disappears entirely at ζ = 0.707). Position servos often accept less damping; process loops often want more.
How is this different from the RLC second-order filter calculator?
Same mathematics, different vocabulary and outputs: the filter tool thinks in f₀ and Q with frequency-response outputs, this tool thinks in ζ = 1/(2Q) and ωn with step-response metrics. Use whichever matches the design conversation.
Why does my simulated settling time differ slightly from 4/(ζωn)?
4/(ζωn) tracks the exponential envelope, but the true response leaves the 2% band at a discrete ring peak, so the measured value can jump by up to half a damped period as ζ changes. Both numbers are reported honestly — the formula as the classical approximation, the simulation as the measurement.
References
- Nise, N. S., Control Systems Engineering, 8th ed., Ch. 4 (underdamped second-order response: %OS, Tp, Ts derivations)
- Ogata, K., Modern Control Engineering, 5th ed., Ch. 5 (second-order transient response)
- Franklin, G., Powell, J. & Emami-Naeini, A., Feedback Control of Dynamic Systems, 8th ed., Ch. 3 (time-domain specifications) and Ch. 6 (Mr, ωr, bandwidth relations)
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