Keywords: fuzzy clustering, stabilization diagrams, system identification, modal analysis, genetic algorithms.

In the structural dynamics area, system identification represents one of the most critical steps. Therefore, a great research effort has been made in the last decades to improve the accuracy and reliability of this process. Observing and measuring the responses of the physical structure to natural or forced excitations lead to the extraction of modal parameters of the system such as resonance frequencies, damping coefficients and mode shapes. To this aim, a widespread tool is the so-called stabilization diagram where the system's poles are represented for several model orders. In real applications, the assessment of this diagram is often extremely demanding due to the high number of poles amongst which only a few represent the true or physical ones. A number of attempts has already been made to provide automated procedures to assess stabilization diagrams and/or identify spurious modes. In Lanslots and Scionti [2], Scionti, Lanslots, Goethals et al. [3], such automated, deterministic procedures are presented. In this paper the non-deterministic fuzzy clustering technique is introduced to automatically assess stabilization diagrams. It can be used to identify clusters that represent physical system poles.

Fuzzy clustering has already been successfully applied in various fields including economics, finance, marketing, speech recognition, pattern recognition, data mining, image processing. It is a technique that differs from hard clustering in the sense that it tries to split a data set into a number of overlapping subsets. The notion fuzzyness can thus be found in the idea that data points are allowed to be part of multiple clusters. The basis of fuzzy clustering lies in the Fuzzy C-Means algorithm (FCM), and was in its present form introduced by Bezdek [1] in 1981. Since then, a number of enhancements has been proposed based on this basic FCM algorithm, such as cluster merging and volume prototypes.

Standard FCM has two main drawbacks which often lead to sub optimal results. The first one is that it can only deal with hyper spherical partitions. The second disadvantage is that, due to its Euclidian distance metric, FCM naturally evolves to equally important clusters, meaning that each cluster has almost the same density. Solutions are presented to overcome these drawbacks, by using different distance metrics.

A number of criteria is available to measure the validity of the clustering results. This validity mainly depends on the initial choice of the number of clusters and on cluster center initialization. When a good initialization procedure is applied, given that a correct number of initial cluster was provided, then the resulting initial cluster centers will be a good estimate of the true clusters. Therefore, the validation problem reduces to the choice of a good initialization scheme. A number of such procedures are discussed, including random and hybrid (using basic FCM in combination with advanced algorithms) initialization.

Genetic Algorithms (GAs) are based on the most powerful mechanism of nature: evolution. Evolution implements the principle `Survival of the Fittest'. Species that do not adapt will die out, and with adaption the best will come up. Genetic Algorithms can be used as a stand-alone clustering technique in which it learns cluster centers, or to initialize FCM.

To explore the use of FCM-like algorithms in a practical case, a data set was used from a real world in-flight flutter test, with white noise excitation. The Gustafson-Kessel FCM, combined with GA initialization was the most successful approach.

As a conclusion, it was proven feasible to automatically extract the correct modal parameters using fuzzy clustering in a real world noisy test case. The proposed fuzzy clustering techniques proved to be very promising for system identification and modal analysis in applications where experts are not available and/or lack of time prevents the use of their expertise.

1

J.C. Bezdek. "Pattern Recognition with Fuzzy Objective Function", Plenum Press, New York, 1981.

2

J.P. Lanslots and M. Scionti. "Automatic assessment of stabilization diagrams." Tech. Rep., LMS International, TST Research & Technology Development, Leuven, Belgium, 2002.

3

M. Scionti, J.P. Lanslots, I. Goethals, A. Vecchio, H. Van der Auweraer, B. Peeters, and B. De Moor. "Tools to improve detection of structural changes from in-flight flutter data." To appear in Proceedings of the VIII International Conference on Recent Advances in Structural Dynamics, Southampton, U.K., to appear: july 2003.

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