Step response

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Black box representation of a dynamical system, its input and its step response

The step response of a dynamical system consists of the time (or more generally the evolution parameter) behavior of its outputs when its control inputs are Heaviside step functions, for a given initial state. In electronic engineering and control theory, step response is the time behaviour of the outputs of a general system when its inputs change from 0 to unity value in a very short time. Knowing the step response of a dynamical system gives information on the stability of such a system, and on its ability to reach a stationary state starting from another when required.

1 Mathematical description
- 1.1 Nonlinear dynamical system
- 1.2 Linear dynamical system
2 Qualitative characterization of the system performance
3 See also
4 References

The aim of this section is to define the step response for a general dynamical system $\scriptstyle\mathfrak{S}$ : all notations and assumptions required for the following analysis are listed here.

$\scriptstyle t\in T$ is the evolution parameter of the system, called "time" for the sake of simplicity,
$\scriptstyle\boldsymbol{x}|_t\in M$ is the state of the system at time $t\,$ , called "output" for the sake of simplicity,
$\scriptstyle\Phi:T\times M\longrightarrow M$ is the dynamical system evolution function,
$\scriptstyle\Phi(0,\boldsymbol{x})=\boldsymbol{x}_0\in M$ is the dynamical system initial state,
$\scriptstyle H(t)\,$ is the Heaviside step function

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

$\boldsymbol{x}|_t={\Phi_{\{H(t)\}}{\left(t,{\boldsymbol{x}_0}\right)}}$

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 system $\scriptstyle\mathfrak{{S}}\equiv S$ for notation convenience: the step response can be obtained by convolution of the Heaviside step function control and the impulse response of the system itself

$a(t) = {h*H}(t) = {H*h}(t) = \int\limits_{-\infty }^{+\infty}\!\!{h(\tau )H(t - \tau )} d\tau = \int\limits_{-\infty}^t\!\!{h(\tau)}d\tau$

The step response could be characterized by the following quantities related to its time behavior,

In the case of linear dynamic systems, a great deal can be inferred about the system from these characteristics. Depending on the application, system performance may be specified in terms of these characteristics instead of bandwidth.

Vladimir Igorevic Arnol'd "Ordinary differential equations", various editions from MIT Press and from Springer Verlag, chapter 1 "Fundamental concepts"

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Step response

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