In engineering field, it is necessary to know the model of the real nonlinear systems to ensure its control and supervision; in this context, fuzzy modeling and especially the TakagiSugeno fuzzy model has drawn the attention of several researchers in recent decades owing to their potential to approximate nonlinear behavior. To identify the parameters of TakagiSugeno fuzzy model several clustering algorithms are developed such as the Fuzzy
Modeling and identification are significant steps in the design of the control system. Typical applications of these models are the simulation, the prediction, or the control system design. Generally, the modeling process consists of obtaining a parametric model with the same dynamic behavior of the real process. However, when the process is nonlinear and complex, it is very difficult to define the different mathematical or physical laws which describe its behavior [
Among the best fuzzy modeling approaches developed in literature we mention the TakagiSugeno fuzzy model. In effect, this model is described by ifthen rules. Each rule includes a fuzzy set antecedent and mathematical functions as consequent representing the process behavior in each region [
TakagiSugeno fuzzy model (TS) is one of the best techniques used for modeling a nonlinear system represented by the recurrent equation
In general, a TakagiSugeno fuzzy model is based on if
Here, the fuzzy sets are represented by the following membership function [
The estimated output model is defined by the following equation [
To identify the premise parameters of a TakagiSugeno fuzzy model described by equation (
The Possibilistic Fuzzy
Typically,
Set the number of clusters
Set the level of weighting
Set the parameters
Set the stopping criterion
Execute a FCM clustering algorithm to find initial fuzzy partition matrix
Initialize the typicality matrix
Compute
Compute the cluster centers by (
Compute the membership matrix
Compute the typicality matrix
The PFCM can deal with noisy data better than FCM and PCM; nevertheless, these conventional clustering algorithms become more effective when applied on linearly separable data or with a reasonable quantity of errors. In reality, the linearly separable data are rare. Therefore, FCM, PCM, and PFCM share the same negative point in that they are unable to get good separation of data that are nonlinearly separable in input space. To correct the imperfections found in PFCM particularly the nonlinear separable problem, kernel [
The kernel function is defined as a generalization of the distance metric that measures the distance between two data points mapped into a future space in which the data are more clearly separable [
Define a nonlinear map as
The RKPFCM algorithm minimizes the following objective function:
If we adopt the hyper tangent kernel function, that is,
Set the number of clusters
Set the level of weighting
Set the parameters
Set the stopping criterion
Execute a FCM clustering algorithm to find initial fuzzy partition matrix
Initialize the typicality matrix
Compute
Compute the cluster centers by (
Compute the membership matrix
Compute the typicality matrix
The Particle Swarm Optimization is a heuristic search method proposed by Kennedy and Eberhart (1995). This technique uses random population solution particles to find an optimal solution to problems. Each particle moves in the search space with a dynamically adjusted position and velocity for the best solution. The particle is characterized by data structure that contains the coordinates of the current position in the search space, the best solution point visited so far, and the subset of other agents that are seen as neighbors. These adjustments are based on the historical behaviors of itself and other agents in the swarm. The change of speed (acceleration) and the position of each particle in the optimization landscape (search space) are iteratively [
where
The fitness function defines our optimization problem described by the following expression:
Select the number of clusters
Set the 1st particle generation clusters centers.
Initialize the fitness function and speed of each particle.
Compute
Compute the fuzzy partition matrix
Compute the typicality matrix
Calculate the new value of fitness for each particle using (
Compare the fitness of each particle with
Compare the fitness value of
Update position and speed of each particle by (
So this algorithm is converged when
The defuzzification method, used in the TakagiSugeno fuzzy model, is linear with the consequent parameters
In this example we have used 12 data which are composed of 10 models and two noises; this data set (
In this example the parameters settings are
Figure
Prototypes results of clusters centers.
Algorithms  Centers 
Error ( 

FCM [ 

0.414 
PCM [ 

1.416 
PFCM [ 

0.3796 
KPFCM [ 

0.0005 






Clustering by RKPFCMPSO algorithm.
According to Table
Figure
After applying the identification algorithm, it is necessary to validate the TakagiSugeno fuzzy model. Several validation tests of the model are used. Among them, we cited the Mean Square Error (MSE) test, Root Mean Square Error (RMSE), and the Variance Accounting For (VAF) test.
Consider a nonlinear system described by the following difference equation [
200 samples were generated by simulation and were used, where the selected input variables are chosen
The complete data set has been used to train the model. The noise influence is analyzed with different SNR levels (SNR = 10 dB and SNR = 5 dB).
In this part, we have applied various algorithms and our proposed clustering RKPFCM and RKPFCMPSO which approximate the nonlinear model (
The used parameters are
The shape of the excitation signal used for identification is illustrated in Figure
Inputoutput sequences.
The simulation result given by the RKPFCMPSO algorithm is illustrated in Figure
Identification results for the RKPFCMPSO algorithm with SNR = 10 DB.
Table
Comparison results without noise.
Algorithms  Number of rules  MSE  VAF (%) 

Cluster fuzzy [ 
6  1.9 × 10^{−3}  — 
Kalman cluster fuzzy [ 
2  2.2 × 10^{−5}  — 
FCM  2  2.1422 × 10^{−5}  99.9581 
GK  2  2.1142 × 10^{−5}  99.9592 
KFCM  2  2.1011 × 10^{−5}  99.9601 
PFCM  2  2.0152 × 10^{−5}  99.9611 








Tables
Comparison results with SNR = 10 dB.
Algorithms  Number of rules  MSE (10^{−4})  VAF (%) 

FCM  2  9.8397  98.0345 
GK  2  9.8831  98.0303 
KFCM  2  9.9492  98.0289 
PFCM  2  9.8313  98.0362 








Comparison results with SNR = 5 dB.
Algorithms  Number of rules  MSE (10^{−3})  VAF (%) 

FCM  2  1.7594  96.0655 
GK  2  1.7552  96.0668 
KFCM  2  1.7572  96.0659 
PFCM  2  1.7534  96.0783 








The local linear models identified are given as follows:
Consider a highly complex modified nonlinear system described by the following difference equation [
The following input signal is expressed as
The noise influence is analyzed with different SNR levels (SNR = 20 dB, SNR = 10 dB, SNR =5 dB, and SNR = 1 dB).
The obtained identification results by RKPFCMPSO are, respectively, shown in Figures
Inputoutput sequences.
RKPFCMPSO performances for test data with SNR = 10 dB.
The evaluation performance index (RMSEtrn and RMSEtest) stands for training and testing data, respectively.
Tables
20 dB Noise.
Algorithms  RMSEtrn  RMSEtest  Number of rules 

FCM [ 
0.1393  0.3004  4 
GK [ 
0.1326  0.1484  4 
FMI [ 
0.1457  0.2378  4 
FCRM [ 
0.1291  0.1495  4 
MFCRMNC [ 
0.1183  0.1395  4 








10 dB Noise.
Algorithms  RMSEtrn  RMSEtest  Number of rules 

FCM [ 
0.3453  0.7150  4 
GK [ 
0.3330  0.3489  4 
FMI [ 
0.3956  0.4416  4 
FCRM [ 
0.3312  0.3694  4 
MFCRMNC [ 
0.3233  0.3465  4 








5 dB Noise.
Algorithms  RMSEtrn  RMSEtest  Number of rules 

FCM [ 
0.6170  1.0364  4 
GK [ 
0.5855  0.6505  4 
FMI [ 
0.6274  0.7021  4 
FCRM [ 
0.5805  0.6485  4 
MFCRMNC [ 
0.5658  0.5998  4 








1 dB Noise.
Algorithms  RMSEtrn  RMSEtest  Number of rules 

FCM [ 
0.9330  1.3646  4 
GK [ 
0.8932  1.0226  4 
FMI [ 
0.8953  1.0320  4 
FCRM [ 
0.8872  1.0095  4 
MFCRMNC [ 
0.8699  0.9947  4 








We consider the Box Jenkins gas furnace data set which is used as a standard test for identification techniques. The data set is composed of 296 pairs of inputoutput measurements. The input “
all the 296 data pairs are used as training data and
Identification results for the RKPFCMPSO Algorithm (SNR = 10 dB).
Based on the comparison presented in Table
Comparison results for Box Jenkins system.
Algorithms  Number of inputs  Number of rules  MSE 

Tong (1980) [ 
2  19  0.469 
Pedrycz (1984) [ 
2  81  0.320 
Xu (1987) [ 
2  25  0.328 
Sugeno (1991) [ 
2  2  0.359 
Yoshinari (1993) [ 
2  6  0.299 
Joo (1997) [ 
2  6  0.166 
Zhang (2006) [ 
2  2  0.160 
Glowaty (2008) [ 
2  2  0.391 
Andri (2011) [ 
2  10  0.167 
Tanmoy Dam (2015) [ 

Without noise  2  2  0.152 
SNR = 30 dB  2  2  0.153 
SNR = 10 dB  2  2  0.277 


Without noise 



SNR = 30 dB 



SNR = 10 dB 



SNR = 5 dB 



SNR = 2 dB 





Without noise 



SNR = 30 dB 



SNR = 10 dB 



SNR = 5 dB 



SNR = 2 dB 



When we use our algorithm RKPFCMPSO the local linear models identified are given as follows:
In literature, various clustering algorithms have been proposed for nonlinear systems identification. In this work, we developed a new clustering algorithm called RKPFCMPSO for the nonlinear systems identification. Our algorithm is an improvement of the Possibilistic Fuzzy
In the future, we will integrate other optimization methods such as the gravitational search algorithm to optimize our hybrid method and we will apply this algorithm for identification of some complex nonlinear real systems as the robotic or the mechatronic systems.
The authors declare that they have no conflicts of interest.