New Allied Fuzzy C-Means algorithm for Takagi- Sugeno Fuzzy model Identification BOUZBIDA Mohamed, TROUDI Ahmed HASSINE Lassad, CHAARI Abdelkader Higher School of Sciences and Techniques of Higher School of Sciences and Techniques of Tunis (ESSTT) Tunis (ESSTT) Research unit (C3S) Research unit (C3S) Tunisia Tunisia bouzbida_mohamed@hotmail.fr, troudi.ahmed@yahoo.fr houcinelassad@yahoo.fr , nabil.chaari@yahoo.fr Abstract— Takagi--Sugeno (TS) fuzzy model have received particular attention in the area of nonlinear identification due to their potentialities to approximate any nonlinear behavior [1]. In literature, several fuzzy clustering algorithms have been proposed to identify the parameters involved in the Takagi- Sugeno fuzzy model, as the Fuzzy C-Means algorithm (FCM) and the Allied Fuzzy C-Means algorithm (AFCM) . This paper presents the New Allied Fuzzy C-Means algorithm (NAFCM ) extension of the AFCM algorithm . Then an optimization method using the Particle Swarm Optimization method (PSO) combined with the NAFCM algorithm is presented in this paper (NAFCM- PSO algorithm).The simulation’s results on a nonlinear system shows that the New Allied Fuzzy C-Means algorithm combined with the PSO algorithm gives results more effective and robust than the Allied Fuzzy C-Means algorithm. Keywords— nonlinear system, TS fuzzy model, Fuzzy identification, fuzzy clustering, non-Euclidean distance, Particle Swarm Optimization I. INTRODUCTION Fuzzy model identification is an effective tool for the approximation of uncertain nonlinear systems on the basis of measured data [2]. Among the different fuzzy modeling techniques, Takagi-Sugeno (TS) fuzzy model drawn the attention of several research, this is to their effectiveness in the nonlinear system modeling [1]. In this context, the fuzzy clustering technique [2] [3] constitute one of the best approaches used for the representation of such process. Indeed, this technique is to approximate the nonlinear system overall by Takagi-Sugeno local linear models, in this case, each model represents by a fuzzy rule [4]. The number of rules (clusters) is fixed by an expert according to the type of application considered and the performances required by this last. Several clustering algorithms exist in literature allowing the identification of the parameters intervening in the TS fuzzy model. We can quote as an example the Fuzzy C-Mean algorithm (FCM) [4]. However, FCM is sensitive to noises. To resist the noises some fuzzy clustering algorithms have been proposed. A novel fuzzy clustering model, called allied fuzzy c-means (AFCM) clustering [5], has been proposed by Wu and Zhou to deal with noisy data. AFCM can produce memberships and possibilities simultaneously and it overcomes the noise sensitivity shortcoming of FCM. However, both FCM and AFCM are all based on Euclidean distance in their objective functions. In real world, the Euclidean distance is not complex enough to deal with more sophisticated problems. Wu and Yang have proposed a non-Euclidean distance to replace the Euclidean distance [6] in FCM. Inspired by Wu and Yang’s algorithm, we introduce the new distance into AFCM to replace the Euclidean distance in it and propose a new fuzzy clustering algorithm called New Allied Fuzzy C-Means algorithm (NAFCM) but the application is limited because of their convergence to local optima and their sensitivity to initialization (random choice of number of clusters). To remedy this problem, a combination of the New Allied Fuzzy C-Means algorithm (NAFCM) and the PSO algorithm, is used. The effectiveness of this algorithm (NAFCM-PSO) compared to the AFCM algorithm is tested on a noisy nonlinear system. This paper is organized as follows: The second part of this work is devoted to formulating a Takagi-Sugeno fuzzy model and identifying the premise parameters of this model using the new Allied Fuzzy C-Means algorithm (NAFCM), and identifying the consequent parameters by RWLS. The third part is dedicated to presenting the particle swarm optimization and the NAFCM-PSO algorithm. The forth part is dedicated to presenting the results of simulation and model validity of AFCM, NAFCM, NAFCM-PSO algorithms. Finally, we conclude this paper with a conclusion. II. TAKAGI-SUGENO FUZZY MODEL Takagi-Sugeno fuzzy model (TS) is one of the best approaches for modeling and identifying a nonlinear system, defined by the recurrent equation () ( ) NL k y k g x = . TS model is constructed by a rule-based type If ... Then in which the consequent uses numeric variables rather than linguistic variables (case of Mamdani). In general, a Takagi-Sugeno fuzzy model is based on rules of the form: U.S. Government work not protected by U.S. copyright