The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from zero to one in a very short time. The concept can be extended to the abstract mathematical notion of a dynamical system using an evolution parameter.

From a practical standpoint, knowing how the system responds to a sudden input is important because large and possibly fast deviations from the long term steady state may have extreme effects on the component itself and on other portions of the overall system dependent on this component. In addition, the overall system cannot act until the component's output settles down to some vicinity of its final state, delaying the overall system response. Formally, knowing the step response of a dynamical system gives information on the stability of such a system, and on its ability to reach one stationary state when starting from another.

This section provides a formal mathematical definition of step response in terms of the abstract mathematical concept of a dynamical system ${\displaystyle {\mathfrak {S}}}$: all notations and assumptions required for the following description are listed here.

For a general dynamical system, the step response is defined as follows:

It is the evolution function when the control inputs (or source term, or forcing inputs) are Heaviside functions: the notation emphasizes this concept showing H(t) as a subscript.

For a linear time-invariant (LTI) black box, let ${\displaystyle {\mathfrak {S}}\equiv S}$ for notational convenience: the step response can be obtained by convolution of the Heaviside step function control and the impulse response h(t) of the system itself

which for an LTI system is equivalent to just integrating the latter. Conversely, for an LTI system, the derivative of the step response yields the impulse response:

However, these simple relations are not true for a non-linear or time-variant system.