Editing Step response (section)

== Formal mathematical description ==
[[Image:Step response.jpg|right|frame|Figure 4: Black box representation of a dynamical system, its input and its step response.]]
This section provides a formal mathematical definition of  step response in terms of the abstract mathematical concept of a [[dynamical system (definition)|dynamical system]] <math>\mathfrak{S}</math>: all notations and assumptions required for the following description are listed here. 
*<math> t\in T</math> is the [[dynamical system (definition)|evolution parameter]] of the system,  called "[[time]]" for the sake of simplicity,
*<math>\boldsymbol{x}|_t\in M</math> is the [[dynamical system (definition)|state]] of the system at time <math>t</math>, called "output" for the sake of simplicity,
*<math>\Phi:T \times M \to M</math> is the dynamical system [[dynamical system (definition)|evolution function]],
*<math>\Phi(0,\boldsymbol{x}) = \boldsymbol{x}_0 \in M</math> is the dynamical system [[dynamical system (definition)|initial state]],
*<math> H(t)</math> is the [[Heaviside step function]]

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

:<math> \boldsymbol{x}|_t = \Phi_{\{H(t)\}} \left(t,{\boldsymbol{x}_0} \right). </math>

It is the [[dynamical system (definition)|evolution function]] when the control inputs (or [[linear differential equation|source term]], or [[forcing input]]s) are Heaviside functions: the notation emphasizes this concept showing ''H''(''t'') as a subscript.

=== 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>