Control SystemsDesign workbenchPreliminary engineering
Step / Impulse / Ramp Response
Simulate the step, impulse, or ramp response of H(s) and read the performance metrics — rise time, peak time, overshoot, settling time — with markers on the curve.
About this calculator
A transfer function answers 'what will it do' only after you pick an input, and three inputs cover most engineering questions. The step is the workhorse — a setpoint change, a load switching in — and carries the standard performance vocabulary: 10–90% rise time, peak time, percent overshoot, and the 2%-band settling time. The impulse response is the system's fingerprint (its inverse Laplace transform directly), and the ramp exposes tracking behavior that steps hide.
This tool simulates all three numerically: the transfer function is realized in controllable canonical state-space form and integrated with a fixed-step fourth-order Runge–Kutta method, with the step size chosen from the fastest pole so integration error stays far below display precision. The method, its accuracy reasoning, and its verification against exact first- and second-order solutions are documented and tested in the engine.
Metrics are measured on the simulated trace against the analytic final value (the DC gain), so a window that ends mid-ring cannot bias the overshoot. When the metrics don't mean anything — an unstable system, an integrator with no finite final value, a marginal oscillator that never settles — the tool says so explicitly instead of printing numbers that look authoritative and are meaningless.
For underdamped second-order systems the classic ζ/ωn formulas apply exactly; the second-order calculator computes them analytically, and this tool's simulation is tested to agree with it.
Design notes & common mistakes
- Rise time and bandwidth trade directly (Tr ≈ 1.8/ωn for ζ ≈ 0.5): asking for a faster rise IS asking for more bandwidth and more actuator effort.
- The 2% settling band is a convention — contracts sometimes specify 5% or 1%. State the band; a bare 'settling time' number is ambiguous.
- Impulse-response 'overshoot' is not the step %OS. The standard metric set belongs to the step response; this tool labels it accordingly.
- A long simulation window hides fast detail and a short one clips settling — this tool sizes the window from the slowest pole, but always sanity-check that the trace has actually flattened.
Assumptions
- Zero initial conditions; the input is applied at t = 0.
- Numerical integration: controllable-canonical state-space + fixed-step RK4 with h·|p|max ≤ 0.05 — global error orders of magnitude below display precision for the supported systems (verified against exact first-/second-order solutions in the engine tests).
- The simulation window is sized automatically from the slowest pole (≈ 6 decay constants) or, for unstable systems, a few divergence constants.
- Step metrics reference the analytic DC gain as the final value.
When to use this calculator
Appropriate for
- Simulating step, impulse, or ramp response of an LTI transfer function and reading rise/peak/settling time and overshoot
- Checking whether a system meets time-domain specifications under zero initial conditions
- Teaching how pole locations translate into transient behavior
Not suitable for
- Systems with saturation, rate limits, or other nonlinearities, which this linear simulation does not capture
- Unstable or marginally stable systems, where the standard transient metrics are not defined (the tool refuses them)
- Predicting a real plant's response without validating the model and initial conditions
What this calculator does not cover
- Zero initial conditions only — no initial-state response or disturbance-plus-input superposition.
- The standard metric set is defined for stable step responses; for impulse and ramp inputs the chart is exact but only the step metrics are reported.
- Biproper systems: the impulse plot omits the δ(t) feedthrough term (noted on the chart).
- No transport delay e^(−sT); a delayed response must be approximated (e.g. Padé) before entry, which changes high-frequency behavior.
- 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
How are rise time, overshoot, and settling time defined here?
Rise time is 10% to 90% of the final value; percent overshoot is (y_peak − y_final)/y_final × 100; settling time is when the response last leaves the ±2% band around the final value. The final value is the analytic DC gain H(0), not the last simulated point.
Why does the tool refuse to give metrics for my system?
Because they would be meaningless: an unstable response has no peak-relative-to-final or settling time, a marginal oscillator never settles, and an integrator has no finite final value. The chart still shows the honest trace.
How accurate is the numerical simulation?
The solver is classical fixed-step RK4 with the step chosen so h·|p|max ≤ 0.05 for the fastest pole. Against exact first- and second-order solutions the trace agrees to at least 5 decimal places — far below anything visible in the metrics.
What does the ramp response tell me that the step doesn't?
Tracking. A type-0 system falls behind a ramp with ever-growing error; a type-1 system follows with a constant lag equal to 1/Kv. The plot shows the input ramp alongside the output so the lag is visible directly.
References
- Nise, N. S., Control Systems Engineering, 8th ed., Ch. 4 (time response; Tr/Tp/%OS/Ts definitions)
- Ogata, K., Modern Control Engineering, 5th ed., Ch. 5 (transient and steady-state response analysis)
- Press, W. et al., Numerical Recipes, 3rd ed., Ch. 17 (Runge–Kutta integration of ODEs)
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