Linear system

A linear system is a model of system which applies a linear operator to an entry signal. A linear system typically posts characteristics and properties much simpler than the non-linear general case.

It is mathematics a very useful abstraction in Automatique, Treatment of the signal, Mécanique and Télécommunications. The linear systems are thus frequently used to describe a nonlinear system, either by being unaware of small non-linearities on the assumption of the small movements (see oscillating Systèmes with a degree of freedom), or while proceeding to a linearization optimized in the contrary case.

If the system is governed by the principle of superposition , one speaks about linear system . Whatever the mathematical nature of the equations which describe it, it can be characterized by its Impulse response or its Transfer function transfer .

If the system is in more invariant, then one speaks about a SLI (linear System invariant), which is at the base of the methods of the Impulse response and the frequential Réponse. The differential equations of the linear systems invariants lend themselves well to the analysis by using the Transformée of Laplace in the continuous case, and the Transformée into Z in the discrete case

Principle of superposition

A deterministic system can generally be described by an operator $H$ who associates the entry $x (T)$ function of $t$ with the exit $y (T)$ . The linear systems check the Principe of superposition:

That is to say two valid entries $x_1 (T) \$ and $x_2 (T) \$ and corresponding exits:

y_1 (T) = H \ left (x_1 (T) \ right)

y_2 (T) = H \ left (x_2 (T) \ right)

then a linear system must check:

H \ left (\ alpha x_1 (T) + \ beta x_2 (T) \ right) = \ alpha H \ left (x_1 (T) \ right) + \ beta H \ left (x_2 (T) \ right) = \ alpha y_1 (T) + \ beta y_2 (T) \ \ forall \ alpha, \ beta \ in \ mathbb {R}

This result spreads then with an unspecified number of excitations. In other words, if one can break up an excitation into a sum of simple functions, it will be possibly possible to explicitly calculate the corresponding answer while adding with the calculable individual answers. This mathematical property returns the resolution of the equations of modeling simpler than of many nonlinear systems.

Transfer transfer function

General information

Instead of explicitly calculating the answer of the system in time, it is often more interesting to determine its contents in frequencies, the passage from one field to another being done using the transformation of Fourier. One shows in mathematics that the transform of a convolution is simply the product of the transforms. By using the corresponding capital letters for these last, one obtains the following equation in which $H (\ Omega)$ is called Transfer function transfer system:

$Y (\ Omega) = H (\ Omega) X (\ Omega) \,$

Case of a sinewave excitation

The energy of a sinusoid is concentrated on only one frequency. In terms of transform of Fourier, it is represented by a delta positioned on this frequency (a more rigorous analysis results in considering two complex deltas). The preceding formula transforms the delta of entry into another delta corresponding to another of the same sinusoid frequency, which gives the physical significance of the transfer transfer function.

According to the linearity, this one thus made correspond to a sum of sinusoids another sum of sinusoids which have the same frequencies (on the contrary, a non-linear system creates new frequencies). In the case of a periodic signal, it acts of sinusoids of finished amplitudes. One will consider below two cases in which intervene of the infinitely small sinusoids (see on this subject spectral Analyze).

Case of a transitory excitation

The preceding formula applies directly to such an excitation, often called to finished total energy, provided with a transform of Fourier.

Case of an excitation with finished variance

The concept of transfer transfer function also applies, at the price of some modifications, an excitation by a signal with finished variance or finished average power, having a spectral concentration. The stochastic concept of Processus then makes it possible to determine in a more or less precise way the characteristics of the answer. If one can suppose that the excitation is Gaussian, the linearity of the system involves the same property for the answer, which provides tools for a precise statistical description.

Another expression of the transfer transfer function

In certain fields, one is interested less in the answer to an excitation given than to the stability of the system. In this case, one uses an expression slightly different deduced from the transformation from Laplace.

See too

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