### Fuzzy Model Identification: Selected Approaches

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D Vol. Jagannathan S. Neural Netw. Johansen T. Kambhampati C. Mason J. Kang H. Kwon C. Lee H. Knospe C. Kwanghee N. Leland R. Li H-X and Deng H. Liao H. Lightbody G.

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Rovatti R. Slotine J. Sugeno M. Wang L. Design and Stability Analysis. Ying H. Man Cybern. Part C: Applic. Zeng X. Finally, the matrix that induces the new distance in the GK process is defined by. From the matrix M i , the updated distance measure is defined as. The procedure can be repeated, ending when an established criteria for stop, s , is set, and the error.

After the fuzzy clustering process is finished, the fuzzy inference of Takagi and Sugeno TS An example of the fuzzy identification method is shown in Figure 4 , supposing two clusters, C 1 and C 2 , obtained by the clustering process. Source: by the authors. The rules established by the Takagi-Sugeno inference method are for this example.

The model performance is evaluated by the value in percentage of variance VAF, Variance Accounted For which is given by the expression. The closer the value 3. The decision of the best number of clusters to perform the GK algorithm, is based on the best overall VAF. Once the decision of the number of clusters is made, and a dynamic formula is obtained, next step is to test the efficiency of the approach, through the cross-validation, so call k -fold method 6. The main objective of this method is to verify the model behavior on multiple data groups, to be used for training and testing.

In this work a 4-fold is applied for the validation of the data. The procedure is given as follows:.

The data group is divided into four parts, group A, B, C, and D, composing by 9, inputs each, from the total of 38, data inputs;. In the first step, the data groups B, C, and D are together for training, and the group A for validation. The VAF rate is calculated corresponding to this simulation.

## Analysis and Synthesis of Fuzzy Control Systems: A Model-Based Approach

This number is calculated VAF1;. A new arranging of the groups A,B,C, and D is obtained, making the circulate arrange of the letters, so that, in the second step, the group D is the validation data and the groups A, B, and C are the training data. The calculated validation rate is VAF2;. Summarizing the process, the entire algorithm to obtain the fuzzy identification of the model, as has been detailed in this section, follows the steps shown in Figure 6.

It is shown in Figure 7 the three-dimensional graphic of the data input,output, time , and the centers of the 20 clusters. One of the clusters and its level of membership degree is shown in Figure 8. Each layer is a surface correspond to a degree of membership. For instance, the outside darkest surface is the level 0. The minor ellipsoid in black corresponds to the membership degree 1. The scale on the right side specifies the corresponding membership degree of the ellipsoid surface.

The 20 clusters projection on the plane input,output is shown in Figure 9.

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The different gray colors indicate the membership degree of the points to the cluster. Once made the system identification through the algorithm, developed in C, a cross validation technique is applied. The results of the 4-fold validation technique is shown in Table 1. The fuzzy identification result compared with the real data is shown in Figure In order to identify simulated data of a hydrogen fuel cell dynamics, a fuzzy technique is applied. It is important to notice that in this type of system identification, black box type, there is no previous information about the model.

Furthermore, the behavior that the data show is strongly nonlinear. Fuzzy technique has the proper pliability for these type of models, which allows to reach faithful formulas, and with capacity to describe the real phenomenon. The identification process that has been applied in this work is divided into three main parts: the fuzzy clustering, a Takagi and Sugeno inference method, and a cross validation of the type k -fold. It is important to highlight, as a mathematical contribution of this work, the three-dimensional graphic, which is an interpretation of the geometry of the data clustering.

The engineering relevance of the study rests in the use of fuzzy identification in data from a hydrogen fuel cell. The two novel aspects of this proposal are, to the best of our knowledge, uncommon in the literature. Fuzzy methodology has shown good performance for identification tasks, validated by the k -fold technique, specially in cases when there is little information about the process, as is the case of this study. The improvement on the codification, for better performance in accuracy and computer speed, is a research currently is progress.

Furthermore, a future survey is among the purposes, involving a comparative study with other techniques and its combinations applied to same and different data. Introduction to System Identification. Study and development of an energy system based in a hydrogen fuel cell for the power injection in electrical net. Abonyi, R. Modified Gath-Geva fuzzy clustering for identification of Takagi-Sugeno fuzzy models.

IEEE Trans. Neuro-Fuzzy Methods for Modeling and Identification.

## Fuzzy Model Identification: Selected Approaches - Google книги

Springer Verlag, Heidelberg, Plenum Press, New York, Costa, F. Comparative analysis of clustering methods for gene expression time course data. Genetics and Molecular Biology, 27 4 , Dovzan, V. Cybern, 3 3 , Fuzzy clustering with fuzzy covariance matrix.

A unifying view of Wiener and Volterra theory and polynomial kernel regression. Hyde, P. Fully online clustering of evolving data streams into arbitrarily shaped clusters. Elsevier Information Sciences, , Algorithms for Clustering Data. Prentice Hall, Englewood Cliffs, System Identification Theory for the User. Maciel, F. Risk management using evolving possibilistic fuzzy modeling. Some Methods for classification and Analysis of Multivariate Observations.

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Pereira, V. Mendes, J. Figueiredo, J. Numerical methods for fuzzy clustering. Information Sciences 2 , Lecture of 13 March Berichte der Verhandlungen der naturforschenden Gesellschaft in Basel, 4 , Parametric system identification using neural networks. Applied Soft Computing - Elsevier, 47 , Nonlinear Problems in Random Theory. This is an open-access article distributed under the terms of the Creative Commons Attribution License.