Formula used
The calculator forms an open-loop transfer function: L(s)=C(s)G(s). For unity feedback, the closed-loop transfer is: T(s)=L(s)/(1+L(s)).
Poles are roots of the characteristic equation: D(s)+N(s)=0, where L(s)=N(s)/D(s). Stability is checked by verifying that all pole real parts are negative.
When a dominant complex pair exists, estimates use: ωn=√(σ²+ωd²), ζ=σ/ωn, Mp≈e^{-πζ/√(1-ζ²)}, and Ts≈4/(ζωn).
How to use this calculator
- Select a plant model that matches your process.
- Choose a controller family and enter its parameters.
- Set the step amplitude and simulation settings if needed.
- Press Submit to view poles, stability, and metrics.
- Use CSV or PDF exports for review and documentation.
Example data table
This example demonstrates a first-order plant with a PID controller.
| Item | Value | Notes |
|---|---|---|
| Plant model | First Order | K/(τs+1) |
| K | 1.0 | Plant gain |
| τ | 2.0 | Time constant |
| Controller | PID | Kp=2.0, Ki=1.0, Kd=0.2 |
| Step | 1.0 | Unit reference |
| What to check | Poles and Ts | Confirm stable and responsive |
Run the same values above to compare your results.
Plant and controller coverage
The calculator supports gain-only, first-order, second-order, integrator, and FOPDT plants. For FOPDT, dead time is approximated using a first-order Padé term. Controller choices include P, PI, PD, PID, lead, and lag, so you can compare structures without changing pages easily. Use K > 0 and τ > 0 for physical processes, set ωn in rad/s, and keep ζ between 0 and 1 for stable second-order behavior.
Closed-loop math and pole location
Open-loop dynamics are built as L(s)=C(s)G(s). With unity feedback, the closed-loop transfer is T(s)=L(s)/(1+L(s)). The characteristic polynomial is D(s)+N(s)=0 when L(s)=N(s)/D(s). Polynomials are formed by coefficient multiplication and addition, and the resulting numerator and denominator are displayed as coefficient arrays for traceability. Poles are calculated numerically for orders up to four, then classified by real-part sign for stability.
Performance indicators used for engineering decisions
When a dominant complex pair exists, the tool estimates natural frequency and damping ratio from pole location: ωn=√(σ2+ωd2) and ζ=σ/ωn. It reports overshoot, rise time, peak time, and 2% settling time using standard second-order relationships. A practical tuning target is ζ between 0.5 and 0.8 for balanced speed and overshoot; for example, ζ=0.6 gives about 9.5% overshoot, while 0.7 gives about 4.6%.
Simulation checks and error-based scoring
An optional step simulation validates the computed transfer using a state-space realization and explicit Euler integration. The time-step is clamped between 0.0005 and 0.2 seconds to reduce numerical blowup. In addition to sampled response values, the tool computes IAE, the integral of absolute error, as a single score for tracking improvements. Smaller dt typically improves accuracy but increases computation and table density.
Reporting, traceability, and limits
Results can be exported as CSV for spreadsheets and as a PDF report for reviews. Stored inputs and outputs improve traceability during audits, design reports, and commissioning. For a Type 0 loop, the step steady-state error is 1/(1+Kp) using the position error constant; Type 1 or higher drives step error toward zero. If your model produces a closed-loop order above four, simplify the model or reduce controller order. Always validate final settings on the real process.
FAQs
1) What does “system type” mean here?
It is the count of open-loop poles at the origin. Type 0 has finite step error; Type 1 or higher drives step steady-state error toward zero under unity feedback.
2) Why is dead time approximated for FOPDT?
Pure delay introduces an exponential term, not a polynomial. A first-order Padé approximation converts it into a rational form so poles, stability, and time response can be computed.
3) Are the time metrics always reliable?
They are most reliable when a dominant complex pole pair governs the response. If real poles dominate or multiple modes compete, treat rise and settling times as engineering estimates.
4) What does IAE tell me?
IAE integrates absolute tracking error over the simulation horizon. Lower IAE usually indicates faster correction with less sustained deviation, helping you compare tunings objectively.
5) Why can simulation differ from estimates?
Estimates assume a dominant second-order form. Simulation uses the full computed transfer, including added poles, zeros, and Padé terms, revealing behaviors the approximations may miss.
6) How should I choose a starting controller?
Start with PI for many first-order processes to remove step error. Add derivative action with PID or lead when you need faster response but must manage overshoot and noise sensitivity.