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Concept
Réponse indicielle
Résumé
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 : all notations and assumptions required for the following description are listed here.
is the evolution parameter of the system, called "time" for the sake of simplicity,
is the state of the system at time , called "output" for the sake of simplicity,
is the dynamical system evolution function,
is the dynamical system initial state,
is the Heaviside step function
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 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.
Source officielle
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Concepts associés (17)
Réponse indicielle
En automatique la réponse indicielle est la réponse d'un système dynamique à une fonction marche de Heaviside communément appelée échelon. Si le système est un système linéaire invariant (SLI) à temps continu ou discret, alors la réponse indicielle est définie par les relations respectives suivantes : Lorsque le système est asymptotiquement stable, la réponse indicielle converge vers une valeur limite (asymptote horizontale) appelée valeur stationnaire ou finale.
Overshoot (signal)
In signal processing, control theory, electronics, and mathematics, overshoot is the occurrence of a signal or function exceeding its target. Undershoot is the same phenomenon in the opposite direction. It arises especially in the step response of bandlimited systems such as low-pass filters. It is often followed by ringing, and at times conflated with the latter. Maximum overshoot is defined in Katsuhiko Ogata's Discrete-time control systems as "the maximum peak value of the response curve measured from the desired response of the system.
Rise time
In electronics, when describing a voltage or current step function, rise time is the time taken by a signal to change from a specified low value to a specified high value. These values may be expressed as ratios or, equivalently, as percentages with respect to a given reference value. In analog electronics and digital electronics, these percentages are commonly the 10% and 90% (or equivalently 0.1 and 0.9) of the output step height: however, other values are commonly used.