staprespons

Staprespons, or step response, is the time-dependent output of a dynamic system when the input changes from zero to a step function (often a unit step) at time t = 0. It is a fundamental tool in control engineering and signal processing for assessing how systems react to sudden, sustained inputs.

For linear time-invariant systems, the step response y(t) is the convolution of the system’s impulse response h(t) with the unit step, equivalently the integral of h(t): y(t) = ∫0^t h(τ) dτ. In the Laplace domain, if the system has transfer function G(s), the step response is Y(s) = G(s)/s, and the final value is y(∞) = lim s→0 G(s). The response reveals dynamic characteristics such as rise time, overshoot, settling time, and steady-state error.

Common cases include:

- First-order system with G(s) = K/(τs+1): y(t) = K(1 − e^(−t/τ)).

- Second-order underdamped system with natural frequency ω_n and damping ratio ζ: y(t) exhibits overshoot and oscillations, with peak time and settling time determined by ζ and ω_n.

The step response is used to identify model parameters, evaluate stability, and guide controller design. It is sensitive to system nonlinearities, time-variance, and non-minimum phase behavior, which can alter the observed transient and final value. See also impulse response, transfer function, and frequency response.