Discrete System

A discrete system is a system which brings into play information which is taken into account only at exact moments. In general these moments are spaced one duration constant called period of sampling. The discrete term comes from the Mathématiques. In the majority of the cases, the value of this information is it also sampled. There is an approximation due to the use of numerical memories to store information.

The Automatique puts increasingly concerned sampled systems. That is due to the computerization which makes it possible to carry out many Calcul S by a Ordinateur, a Microcontrôleur or very other digital Computer. A computer, from its principle, functions in a discontinuous way. The duration between each operation depends on the components of the computer (a given rhythm processor with 1GHz corresponds to one duration of 1ns). Information cannot thus be treated uninterrupted (contrary to an assembly Analogique).

Therefore it was necessary to determine methods of analysis for these systems.

Study

What it is necessary to include/understand, it is that sampling implies a loss of information on the entry. One does not know what it occurred between the two moments when the entry was observed.
In general, one knows the signal roughly to be observed. One of deduced the acceptable maximum duration enters the samples in order not to lose information.

For periodic signals, one uses the Théorème of sampling of Nyquist-Shannon: One takes the most important Fréquence of his Fourier series and one arranges oneself so that the sampling rate is at least twice higher.

If one is sure not to lose information, one can use the same principle of study as for continuous systems.
La principal difference is that one cannot in the case of use the Transformée of Laplace a sampled signal. One replaces the transforms of Laplace by transformed into Z.

The study is then overall the same one.

Discrete-continuous mixture

In reality, the systems are often a coupling between the discrete one (the order, often computerized) and the continuous one (physical phenomena). It is thus necessary to be able to pass from discrete to continuous and vice versa.

When a signal leaves a module of order, it is discretized. However a physical process will require a continuous signal. One thus places a Bloqueur between them. There exist several kinds of blockers. Their principle is “to recreate” signal between the discrete values. Simplest is the blocker of order 0 whose exit keeps the same value until the arrival of a new value in entry.

In the other direction, the problem does not arise truly since the entry will be taken into account only at determined times, it does not matter if it is continuous or discrete.

The difficulty lies in the general study of the system. One can model the system by a continuous system by checking that the entries and the exits are quite continuous and while passing from the transforms in Z characterizing the discrete zones of the system to the transforms of Laplace by using for example tables of transformation.
Inversement, one can also compare the system to an entirely discrete system and pass from the transforms of Laplace characterizing the continuous zones of the system to the transforms in Z, provided that the entries and the exits are taken into account in their discrete form.

The passages discrete-continuous are modelled by blockers of which it is necessary to take into account the transfer transfer function at the time of the study.

See too

discrete Topology

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