Step / Impulse Response of an LTI System

The impulse response is the complete time-domain fingerprint of an LTI system, and the step response is what most measurements actually show. This tool simulates either one for a transfer function H(s) you provide, using a fixed-step RK4 integration of the controllable-canonical state-space realization — the same tested engine the Control Systems toolkit uses — and reports the standard step-response metrics: steady-state value, percent overshoot, rise time, peak time, and 2% settling time.

As with the frequency-response tool, the framing is analysis of a GIVEN system, not the design of a controller or a physical plant. Enter H(s) as numerator and denominator coefficients in descending powers of s; the simulation window is chosen automatically from the pole locations (a few settling constants for stable systems, a bounded window for unstable ones, with the divergence flagged honestly rather than plotted as garbage).

The step metrics are measured against the analytic final value (the DC gain), not the last simulated sample, so a window that ends mid-ring cannot bias the overshoot. For an unstable or marginally-stable system the metrics are reported as not meaningful — the tool says so instead of printing numbers that look authoritative. Use it to see how poles and zeros shape the transient, or to confirm a hand analysis of a second-order system.