A New Approach of Estimation of the Aptnesses of the Models of a Base for the Vague Systems

A New Technique of validities' Computation

for Models' base

Talmoudi Samia - Abderrahim Kamel - Ben Abdennour Ridha *

& Ksouri Mekki **

* Laboratoire de Commande Numérique des Procédés Industriels

Ecole Nationale d’Ingénieurs de Gabès, Route de Mednine, 6029 Gabès, Tunisie.

** Institut National des Sciences Appliquées et de Technologie de Tunis, BP 676, 1080 Tunis Cedex, Tunisie

Abstract : the main idea of the multimodel approach is to represent a complex system by several models having each a given validity'domain chiffred by a validity’degree. These validities'degrees are exploited in the fusion step of the base'models in order to obtain the effective multimodel output. the methods of validity calculation existing in the literature pass generally by the residues calculation. Indeed, these methods lack sometimes a precision. To overcome this problem, we propose, in this work, a new approach of validities estimation of the base’models. This last approach is based on minimising an quadratic criterion.

A comparative study with the residues’approach shows the good performances of the proposed technique.

Key-words : multimodel approach - uncertain systems - validity computation - new approach - residues.

1 Introduction

Generally, industrial processes are complex, non linear and/or uncertain. The multimodel approach is a powerful approach developed to resolve the difficulties of modelling and controlling of these processes types.

The multimodel and multicontrol approaches were notably advocated by several researchers [1, 2, 3, 6, 7, 8, 9, 10, 12, 14, 15, 16, 19, 20, 22, 25, 26]. we note, as examples, Takagi and Sugeno [25], Johansen and Foss [11].

The multimodel approach consists in representing the system by several simple models having each a given validity domain [11]. These models form the said models’base. The advantage of this approach is connected to the simplicity of the established models. It is necessary to calculate, afterward, coefficients named validities of the models.

Finally, there is a stage of fusion or switching of information supplied by the models according to their estimated validities.

In spite of their successes in several domains (academic, biomedical, process industries, etc.), the multimodel and multicontrol approaches are confronted with several problems; such as; the choice and the models’number of the base’models, the adequate technique of fusion or switching between these models and the computation of the models’validities. This last problem establishes the subject of the present paper. Indeed, The existing methods of validities require, generally, the calculation of residues [6, 8, 12, 13, 15, 16]. The geometrical approach is an example of validities’computation that requires the calculation of residue. In spite of their simplicity of computation, these methods lack sometimes precision. Besides that, this technique is limited and can not be adopted when dealing with complex non-linear and/or ill-defined systems [13]. In this paper, we propose a new estimation method of validities’degrees. This method, inspired for the classification’algorithm "k-means", minimise an quadratic criterion. This criterion exploits the centres of classesobtained in the first stage of models’base determination. The obtained validities are then used for processes identification and control.

In the following section, we present a systematic determination approach of the models’base by exploiting the Kohonen card. A brief presentation of the residues method is the subject of the second section. The description of the new proposed approach of validities estimation is the subject of the third section. Two simulation examples, showing the efficiency of the new approach of validities’computation for uncertain systems, are proposed in the fourth section. We finish the present work by a conclusion.

2 A systematicdetermination approach of a models'base [24]

This approach exploits Kohonen's networks for the construction of various vectors. These vectors of observations are necessary for the structural and parametric identification of the different models of the elaborated base. Three stages are necessary for the determination of a models’base :

determination of the classes’number.
classification of numerical data.
structural and parametric estimation.

After determination the suitable number of classes, it is a question of classifying a set of outputs measures of a non stationary system. This classification strategy consists in applying the rule of Kohonen [18, 21, 24]. This rule is characterised by an unsupervised competitive learning. Where, a competition takes place before the modification of the network-weights. Only the neurons which gained the competition have the right to change their weight. The Kohonen rule has the property of self-adapting which allows him to group together a set of data, presented to the corresponding network, around a certain number of representative centroïdes of these data clusters. The exploited Kohonen network is able to give a set of vectors witch represent the classes with their respective centres. The obtained vectors are then exploited for the structural and parametric identification of the elaborated base models on the one hand. on the other hand, the centres are exploited by the new approach for validities computation for models’base,.

3 Classical approach for validity computation

Several validities computation methods was proposed in the literature [6, 8, 12, 13, 15, 16]. All these methods are based on the residues computation. We present the principle of the geometrically approach in the next paragraph.

3.1 The geometric approach

This method consists in measuring the distance between the current state of the process and the considered model Mi(fig. 1).

Fig. 1 : Geometric distance.

Where

M : the current state of the process.

di: The distance between the state model M and the state local model Mi.

The residue is taken as : ri = di

The geometric distance can be calculated by several methods; the simplest one is being the distance ri between the process’output y and the base’models outputs yi:

ri = |y-yi| (1)

3.2 Computation of validities

Frequently, we choose the validities such as all the time their sum is equal to the unity. For example:

(2)

Where is a normalised distance given by :

C represents the number of base’models.

4A new approach for validity computation

The proposed method of validities computation is inspired from the fuzzy version of the algorithm " k-means" [18]. This method is based on the minimisation of the following criterion :

(3)

With (4)

Where:

vjirepresent the degree of validity of the model j at the instant i.

cjis the centre of the class j.

It is a first order problem of optimisation with equality constraint. The resolution of this type of problem requires the determination of the Lagrange’equation. In fact, so that vijis a local extremum of the criterion J, it is necessary that there is a real  such as the Lagrangien L of the problem can be written as follows :

L (vji, ) =J +  g(vji) (5)

is stationary with regard to vji and .

This leads to et (6)

with is Lagrange's multiplier associated to the constraint. The relations (6) lead to the following system.

For a given i, we have :

(7)

this problem becomes :

(8)

The relations (8 ) give :

(9)

This relation becomes :

(10)

Then is giving from the relation (10) and replaced in the equation (9). Finally, we can conclude that the expression of validity’degree for a model Mi can be written as follows :

(11)

Where;

Fig. 2: Euclidean distance illustrated by the new approach of validity computation.

5Simulation examples

5.1Example 1

To evaluate the performance of the proposed’method, we consider the next linear second order system with parameters varying as showed in the figure 3 :

(12)

Fig. 3: The variation laws of the considered process parameters

The structural and parametric estimation of the elaborated base’models from three data sets relative to the different classes obtained after Kohonen's network learning [24], give the three following second-order transfer functions H1(z), H2(z) and H3(z) :

The multimodel output is obtained by fusion of three outputs y1(k),y2(k) and y3(k) of base’models as follows:

yf (k)=v1(k).y1(k)+v2(k).y2(k)+v3(k).y3(k) (13)

Where v1(k), v2(k) and v3(k) are the validities of base’models and are calculated by both methods described previously namely ; the residues method and the new proposed method.

5.1.1Classical method of validities’computation :

The application of the following input sequence is the subject of validation step :

u(k)= 1+exp(-1-0.05*k)*sin(k/5); (14)

The results of validation is given in the figure 4 which shows that the multimodel output yfc(k) obtained from calculated validities by using the residues method, follows the real output of the uncertain process with an important prediction relative error.

Fig. 4: Evolution of the relative prediction error between the real and multimodel outputs (Method of residues).

The evolution of the three validities are given in the figure 5. This figure shows that a model can never be totally valid (the validities are different to 1). While, really, there is one model which is totally valid (see figure 3 and the expressions of the three transfer functions H1(z), H2(z) and H3(z)).

Fig. 5: Evolution of the validities calculated by the residues method.

5.1.2 New approach of validities computation

The proposed approach for validities calculation uses the classes’centres obtained in the stage of determination of a models’base. The coordinates of the three obtained centres c1, c2, c3 are the following ones :

c1(0.6327; 0.6321); c2(0.0005; 0.0008);

c3 (0.5478; 0.5469)

The evolution of various validities of models in this case are given by the figure 6. This figure shows that a model can be totally valid (validity equal to the unity). For example: between 0 and 200, the model M3 is totally valid, then, the corresponding validity is equal to 1.

Fig. 6 : Evolutions of the validities ( new approach).

The validation result obtained by exploitation of the input’sequence giving by the relation ( 13 ), shows that the multimodel output yfn(k) obtained by utilisation of the new validities expressions follows the real output of the uncertain process with a relatively negligible error. Indeed, the prediction errors er1 and er2 of the two multimodel outputs respectively yfc(k) and yfn(k) with regard to the real output show well the precision brought by the new approach of validities computation (fig. 7).

Fig. 7 : evolution of the relative prediction errors between the real output and the multimodel outputs

yfc (k) and yfn(k).

5.2 Example 2

We consider now a second-order system described by the relation (12) but in this case, the variation laws parameters [24] are given by figure 8.

Fig. 8: The variation laws of the considered process parameters

The structural and parametric estimation of the base’models to be elaborated from three data sets relative to the various classes obtained after learning of Kohonen's network gives the three following second-class transfer functions :

The effective multimodel output is obtained by fusion of the three models outputs of the base following the relation (13), by using the validities of models calculated by both methods described previously namely; the method of residues and the new proposed method.

The results of validation, obtained by exploitation of another sequence of input given by:

show that the multimodel output yfn(k) follows the real output of the uncertain process with a negligible error. While, the output obtained from the classic expressions of validities yfn(k) follows the real output but with a relatively important error. Indeed, the prediction errors of the two outputs yfc(k) and yfn(k) with regard to the real output show well the precision broughtby the new approach of validities calculation (fig. 9).

Fig. 9: evolutions of the relative prediction errors between the real and the multimodel outputs

yfc(k) and yfn(k).

6 Conclusion

A new technique of validities computation is presented in this paper. This technique consists on minimising a quadratic criterion. The criterion exploits the classes’centers obtained in the stage of determination of the models’base of uncertain processes. The multimodel output is then obtained by fusion of the models’base outputs weighted by the obtained degrees of validities. The application of this new technique of validities’computation was made on two simulation examples. These applications showed the efficiency of the proposed technique in term of precision with regard of the residue’method.

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