Step response

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

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[edit] Mathematical description

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.

[edit] Nonlinear dynamical system

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.

[edit] Linear dynamical system

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

[edit] Qualitative characterization of the system performance

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.

[edit] See also

[edit] References

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


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