Identification of system

Principles and Objectives

The identification consists in applying signals of disturbance to the entry of a system (for example for an electronic system, those can be of binary type random or pseudo-random, Welsh, sine at multiple frequencies…) and to analyze the exit with an aim of it of obtaining a purely mathematical model. The various parameters of the model do not correspond to any physical reality in this case. The identification can be done is in time (temporal space) or in frequency (space of Laplace). To avoid the purely theoretical models starting from the physical equations (in general of the differential equations), which is long to obtain and often too complex for the time of development given, is thus possible with this technique.

Note: It could be possible to find a model effective of the universe since we know of it the entry and the exit from the point of view of the entropy.

Various types of models

The principle of a parametric Identification is to extract a mathematical model starting from observations. The model must make it possible to calculate the exit of the process there at any moment T if the initial conditions of the system are known. For that one can make use of the values of the entries at the moments present and preceding ( U (T), U (T-1),… ) and preceding values of the exit ( there (T-1), there (t-2),… ) in the case of a regressive model.

It is all the same important to have basic knowledge of the system to choose a type of adapted model

Model having an entry/an exit (SISO) or several entries and several exits (MIMO)
linear Model or non-linear (in this case, what is non-linear according to what)
continuous Modèle or discrete
regressive or independent Modèle: for a regressive model, the exit at one moment T , there (T), depends on the previous moments (there (Ti)).
stochastic Model or deterministic

In general, the model is represented in the form of Transfer function transfer using the Transformée into Z. The identification requires a structure of model known a priori to come to identify in this structure various parameters. Here 3 the most used structures of model:

Model ARX

Model ARX ( Regressive Car model with external inputs ) is a regressive model car which included entries U (T) and a white Bruit

\ zeta (T)

of null average. Moreover, the model included a pure delay of K blows of clock. If the system is sampled at one period of sampling T , then the delay will be of k*T .

In temporal form:

Y (T) = B \ cdot ^T - has \ cdot ^T + has \ cdot

In a discrete space using the Transformed into Z:

y (Z) =z^ {- K} \ cdot \ frac {B (z^ {- 1})}{Has (z^ {- 1})}\ cdot U (Z) + \ zeta (Z)

Model ARMAX

Model ARMAX ( Regressive Car Moving Average with external inputs ) takes again the attributes of model ARX but included a Transfer function transfer with an adjustable average on the white Bruit. In general the white vibration makes it possible to model not-measurable disturbances in the model. However, these not-measurable disturbances (thermal fluctuations, vibrations of the ground…) are seldom of null average and also can répondrent with a model.

y (Z) =z^ {- K} \ cdot \ frac {B (z^ {- 1})}{Has (z^ {- 1})}\ cdot U (Z) + \ frac {T (z^ {- 1})}{Has (z^ {- 1})} \ cdot \ zeta (T)

Model ARIMAX (or CARIMA)

In model ARIMAX ( Regressive Car Integrated Moving Average with external inputs ) the model of the noise is directly integrated:

y (Z) =z^ {- K} \ cdot \ frac {B (q^ {- Z})}{Has (q^ {- Z})}\ cdot U (Z) + \ frac {T (z^ {- 1})}{(1-z^ {- 1}) \ cdot has (z^ {- 1})} \ cdot \ zeta (Z)

Procedure of identification

To obtain a consistent model, it is important to excite the process with all the frequencies of its operating range. The entry signal applied must thus be rich in frequencies (to have a broad spectrum). In general a pseudo-random periodic signal is applied (PRBS).

When the system has several entries/several exits, it is important to apply décorrélés signals not to introduce skew of identification. A common idea consisting in exciting one after the other the entries is a bad method because it introduces a skew of identification and does not give an account of the normal functioning of the system. It is important to observe a rigorous procedure to identify a process:

Determination of a protocol of test: statistical properties of the entry signals to sweep all the interesting frequencies, the signal-to-noise ratio must be sufficiently important and the number of points of measurements must be significant for the test (>1000)
Détermination of the structure of the model: type of model, order and delay
Identification: choice of a algorithm to find the model by minimizing the errors between measurements and the model, in general algorithm based on the Method of least squares (LS, RLS, RELS).
Validation of the model: Realization of several tests of checking. It is necessary for this stage to use measurements different from those used during the identification.

Other approaches are also possible, in particular by looking at the matrices of subspace of a system (but less effective than above for the non-linear systems).

This can thus easily give a model less " théorique" and to help with the improvement of the output, control or the prediction (for values of action in an economic system for example).

toolbox Matlab and Scilab exist for the resolution of the algorithms (of type ARMAX for example). Those for Octave are to be created.

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