Control SystemsPreliminary engineering
PID Performance Calculator
The complete closed-loop performance report for a tuned PID loop — rise/peak/settling time, overshoot, steady-state error, bandwidth, and margins — exact from the model, or from the second-order approximation, with the validity of that approximation checked.
About this calculator
After tuning comes the specification check: does the loop meet its numbers? This calculator produces the full performance sheet two ways and tells you when the shortcut is trustworthy.
EXACT mode takes the plant and the PID gains and computes everything from the true closed loop by the shared engine — metrics measured on the simulated response against the analytic final value, the −3 dB closed-loop bandwidth found on the exact frequency response, and the open-loop gain and phase margins by bisection. This is the number set to put in a design report.
SECOND-ORDER mode evaluates the classical ζ/ωn formulas (overshoot, peak time, the 4/(ζωn) settling estimate, rise time solved on the exact canonical response, resonant peak, bandwidth) — the hand-calculation vocabulary every specification is written in.
The bridge between them is the dominance question, and this tool answers it instead of assuming it: in exact mode it locates the dominant closed-loop pole pair, reports its effective ζ and ωn, and states plainly whether the second-order picture applies — a nearby third pole or a closed-loop zero (PID controllers always introduce zeros) can make the canonical formulas wrong by a factor of two, and the warning fires precisely when that risk exists.
Every output states its definition (2% band, 10–90% rise, amplitude-convention dB), because a performance number without its convention is a future argument with a client.
Design notes & common mistakes
- PID closed loops always carry numerator zeros (from Kd·s² + Kp·s + Ki) — they add overshoot beyond the %OS(ζ) prediction. Quoting hand-formula overshoot for a PID loop without checking is the most common spec-sheet error.
- State the band with every settling time: 2% here; contracts often mean 5% or 1%. The same response differs by tens of percent between bands.
- Bandwidth here is CLOSED-loop −3 dB (relative to T(0)); do not compare it against an open-loop crossover spec — they are related (ωBW ≈ 1–1.5·ω_gc) but not equal.
- A performance sheet without margins is half a sheet: two loops with identical step metrics can differ wildly in robustness.
Assumptions
- Exact mode: parallel-form ideal-derivative PID on the entered plant, negative unity feedback, zero initial conditions — all computation by the shared tested engine.
- Second-order mode: the canonical two-pole system with no finite zeros; validity for a real loop is exactly what exact mode checks.
- Metric definitions as stated in the convention note — always quote them alongside the numbers.
When to use this calculator
Appropriate for
- Producing a full closed-loop performance sheet from a plant and PID gains, or from a second-order approximation
- Checking whether the second-order approximation is trustworthy for a given loop before quoting hand-formula numbers
- Documenting rise/peak/settling time, overshoot, steady-state error, bandwidth, and margins for a report draft
Not suitable for
- Quoting second-order hand-formula metrics for a PID loop without the validity check — PID zeros add overshoot the pole picture misses
- Systems with saturation or rate limits, where the linear metrics do not hold for large steps
- Final specification without independent verification against the real closed loop
What this calculator does not cover
- Setpoint-step performance only; disturbance-response and noise metrics (IAE/ISE/ITAE) are planned with the discrete-PID work.
- The second-order mode is deliberately the textbook idealization — its entire purpose is to be compared against exact mode's validity check.
- Ideal derivative; a filtered derivative slightly changes high-frequency margins.
- No saturation or rate limits — large setpoint steps in the real loop will differ from this linear report.
- 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 do my hand-calculated numbers disagree with the exact ones?
Usually the closed-loop zeros: a PID controller's numerator (Kd·s² + Kp·s + Ki) becomes zeros of the closed loop, and zeros add overshoot and speed that the ζ/ωn pole picture doesn't capture. This tool measures the disagreement and tells you when the hand formulas are safe.
Which numbers should go in a design report?
Exact mode's, with their conventions attached: 2% settling band, 10–90% rise, overshoot relative to final value, −3 dB closed-loop bandwidth, and both margins. The second-order numbers are for communication and sanity checks, not for the record.
What is 'effective ζ' for a higher-order loop?
The damping ratio of the dominant closed-loop pole pair — the pair nearest the imaginary axis. It is the honest bridge to second-order intuition, but it summarizes the loop faithfully only when other poles are well separated and no zero sits nearby; the tool states whether that holds.
References
- Nise, N. S., Control Systems Engineering, 8th ed., Ch. 4 (metric definitions) and Ch. 9 (PID zeros and their effect)
- Franklin, G., Powell, J. & Emami-Naeini, A., Feedback Control of Dynamic Systems, 8th ed. (bandwidth-rise-time relations, margins)
- Åström, K. J. & Hägglund, T., PID Controllers: Theory, Design, and Tuning, 2nd ed.
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