Editing Step response (section)

=== Linear dynamical system ===
For a [[linear system|linear]] [[Time-invariant system|time-invariant]] (LTI) black box, let <math>\mathfrak{S} \equiv S</math> 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

:<math>a(t) = (h*H)(t) = \int_{-\infty }^{+\infty} h(\tau) H(t - \tau)\,d\tau = \int_{-\infty}^t h(\tau)\,d\tau.</math>

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:

:<math>h(t) = \frac{d}{dt}\,a(t).</math>

However, these simple relations are not true for a non-linear or [[time-variant system]].<ref name="Shmaliy2007">{{cite book|author=Yuriy Shmaliy|title=Continuous-Time Systems|url=https://archive.org/details/continuoustimesy00shma|url-access=limited|year=2007|publisher=Springer Science & Business Media|isbn=978-1-4020-6272-8|page=[https://archive.org/details/continuoustimesy00shma/page/n61 46]}}</ref>