Robust Kernel Clustering Algorithm for Nonlinear System Identificationdownloads.hindawi.com/journals/mpe/2017/2427309.pdf · 2019-07-30 · ResearchArticle Robust Kernel Clustering

Research ArticleRobust Kernel Clustering Algorithm forNonlinear System Identification

Mohamed Bouzbida12 Lassad Hassine12 and Abdelkader Chaari12

1National Higher Engineering School of Tunis (ENSIT) University of Tunis 5 Av Taha Husein BP 56 1008 Tunis Tunisia2Laboratoire drsquoIngenierie des Systemes Industriels et des Energies Renouvelables (LISIER) University of TunisENSIT Tunis Tunisia

Correspondence should be addressed to Mohamed Bouzbida bouzbida_mohamedhotmailfr

Received 12 December 2016 Revised 16 March 2017 Accepted 30 March 2017 Published 14 May 2017

Academic Editor Francisco Gordillo

Copyright copy 2017 Mohamed Bouzbida et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In engineering field it is necessary to know the model of the real nonlinear systems to ensure its control and supervision in thiscontext fuzzy modeling and especially the Takagi-Sugeno fuzzy model has drawn the attention of several researchers in recentdecades owing to their potential to approximate nonlinear behavior To identify the parameters of Takagi-Sugeno fuzzy modelseveral clustering algorithms are developed such as the Fuzzy 119862-Means (FCM) algorithm Possibilistic 119862-Means (PCM) algorithmand Possibilistic Fuzzy119862-Means (PFCM) algorithmThis paper presents a new clustering algorithm for Takagi-Sugeno fuzzymodelidentification Our proposed algorithm called Robust Kernel Possibilistic Fuzzy 119862-Means (RKPFCM) algorithm is an extension ofthe PFCM algorithm based on kernel method where the Euclidean distance used the robust hyper tangent kernel function Theproposed algorithm can solve the nonlinear separable problems found by FCM PCM and PFCMalgorithmsThen an optimizationmethod using the Particle Swarm Optimization (PSO) method combined with the RKPFCM algorithm is presented to overcomethe convergence to a local minimum of the objective function Finally validation results of examples are given to demonstrate theeffectiveness practicality and robustness of our proposed algorithm in stochastic environment

1 Introduction

Modeling and identification are significant steps in the designof the control system Typical applications of these modelsare the simulation the prediction or the control systemdesign Generally the modeling process consists of obtaininga parametric model with the same dynamic behavior ofthe real process However when the process is nonlinearand complex it is very difficult to define the differentmathematical or physical laws which describe its behavior[1 2] In this context the modeling of nonlinear systems bythe conventional methods is very difficult and occasionallyineffective [3 4] So other nonconventional techniques basedon fuzzy logic are used more often in modeling this kindof process due to excellent ability of describing its behavior[4ndash6]

Among the best fuzzy modeling approaches developedin literature we mention the Takagi-Sugeno fuzzy model

In effect this model is described by if-then rules Eachrule includes a fuzzy set antecedent and mathematical func-tions as consequent representing the process behavior ineach region [2 3 7] The identification problem consistsof estimating the model parameters In this context toidentify the parameters of Takagi-Sugeno fuzzy model manytechniques were developed such as the Adaptive schemesheuristic approaches nearest neighbor clustering and sup-port vector learning mechanisms Besides fuzzy clusteringalgorithms are widely used in fuzzy modeling Fuzzy 119862-Means (FCM) Gustafson-Kessel (G-K) Gath-Geva (G-G)Possibilistic 119888-Means (PCM) Fuzzy 119862-Regression Model(FCRM) Enhanced Fuzzy 119862-Regression Model (EFCRM)and Possibilistic Fuzzy 119888-Means (PFCM) are popular clus-tering algorithms used in structure identification part andLS WRLS and orthogonal least square (OLS) techniquewere applied in consequent parameter estimation Amongthe clustering algorithms Fuzzy 119888-Means (FCM) developed

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 2427309 11 pageshttpsdoiorg10115520172427309

2 Mathematical Problems in Engineering

by Bezdek is well-known but this algorithm is sensitiveto noise or outliers and susceptible to local minima [89] However noise in the data sets can make the situa-tion worse by creating many inauthentic minima Theseare able to distort the global minimum solution found byFCM algorithm This flaw has stimulated the researchers toovercome this inconvenience To fight against the effects ofoutlying data various approaches are considered such as thepossibilistic clustering (PCM) proposed by Krishnapuramand Keller [10] and fuzzy noise clustering approach ofDave [11] The possibilistic approach executed a possibilisticpartition in which a membership relation calculates theabsolute degree of typicality of a point in a cluster Althoughthe PCM algorithm is robust against the noise points andallows identifying these outliers it is very responsive toinitializations and occasionally generates coincident clustersTo solve this deficiency of identical clusters a PossibilisticFuzzy 119888-Means (PFCM) algorithm was suggested by Wuand Zhou in 2006 Nonetheless these algorithms are notefficient for unequal dimension clusters and cannot separateclusters that are nonlinearly separable in input space andtheir limits between two clusters are linear In our work toovercome this shortcoming kernelmethods [12] are regardedas the way of dealing with this problem We propose a newclustering algorithm called Robust Kernel Possibilistic Fuzzy119862-Means (RKPFCM) which adopts a kernel induced metricin the data space to replace the original Euclidean normmetric By changing the inner product with an appropriatehyper tangent kernel function one can implicitly affect anonlinearmapping to a high dimensional feature spacewherethe data is more clearly separable However our proposedalgorithm RKPFCM has an iterative nature that makes itsensitive to initialization and sensitive to converge to a localminimum To overcome these problems several solutionshave been proposed in the literature Among them is com-bining the clustering algorithm with a heuristic optimizationtechnique In this context many researches have proposedthe evolutionary computation technique based on ParticleSwarm Optimization (PSO) They have been successfullyapplied to solve various optimization problems [13] Thuswe introduce PSO into the RKPFCM algorithm to achieveglobal optimization The efficacy of our algorithm comparedto many algorithms is tested on noisy nonlinear systemsdefined by recurrent equations and an application to BoxJenkins system This paper will be presented as followsThe second part of work is reserved for introducing theTakagi-Sugeno fuzzy model The third part will be devotedto identifying the premise parameters of this model where weused the proposed RKPFCM algorithm and PSO algorithm isintroduced In the fourth part we will focus on identificationof consequent parameters The simulations results and themodel validity of RKPFCMandRKPFCM-PSO are presentedin part five Finally we conclude this paper

2 Takagi-Sugeno Fuzzy Model

Takagi-Sugeno fuzzy model (T-S) is one of the best tech-niques used for modeling a nonlinear system representedby the recurrent equation 119910(119896) = 119892NL(119909119896) T-S model is

constructed by a rule-based type ldquoif thenrdquo in which theconsequent uses numeric variables rather than linguistic vari-ables (case ofMamdani)The consequent can be expressed bya constant a polynomial or differential equation dependingon the antecedent variables The T-S fuzzy model allowsapproximating the nonlinear system into several locally linearsubsystems [4 17]

In general a Takagi-Sugeno fuzzy model is based on if then rules of the form

119877119894 if 1199091119896 is 119860 1198941 119909119899119896 is 119860 119894119899 then 119910119894 = 119886119894119879119909119896 + 119887119894 (1)

The ldquoifrdquo rule function defines the premise part and ldquothenrdquorule function constitutes the consequent part of the T-S fuzzymodel 119877119894 represents the 119894th rule119886119894119879 = [1198861198941 1198861198942 119886119894119899] is the parameters vector such

as 119886119894 isin 119877119899119887119894 is a scalar119909119896 = [1199091119896 1199092119896 119909119899119896]119879 is observations vector119860 1198941 119860 1198942 119860 119894119899 represents the fuzzy subsetswhere 119894 isin [1 119862]

Here the fuzzy sets are represented by the followingmembership function [7]

119860 119894119895 (119909119895119896) = exp(minus(119909119895119896 minus V119894119895)21205901198941198952 ) isin [0 1](119894 = 1 2 119862 119895 = 1 2 119899)

(2)

where V119894 = [V1198941 V1198942 V119894119895]119879 isin 119877119899 are centers and 120590119894 = [12059011989411205901198942 120590119894119895]119879 isin 119877119899 is the width of the membership functionThe estimated output model is defined by the following

equation [4]

119910 (119896) = 119862sum119894=1

120573119894 (119896) 119910119894 (119896) (3)

As

120573119894 (119896) = prod119899119895=1119860 119894119895 (119909119895119896)sum119862119894=1prod119899119895=1119860 119894119895 (119909119895119896) 119896 = 1 2 119873 (4)

so

119910 (119896) = 119888sum119894=1

120573119894 (119896) [119886119879119894 119909119896 + 119887119894] (5)

3 Identification Algorithm forPremise Parameters

To identify the premise parameters of a Takagi-Sugeno fuzzymodel described by equation (1) we used the PossibilisticFuzzy 119862-Means (PFCM) algorithm and our proposed algo-rithms (RKPFCM and RKPFCM-PSO)

Mathematical Problems in Engineering 3

31 Possibilistic Fuzzy 119862-Means (PFCM) Algorithm The Pos-sibilistic Fuzzy 119862-Means (PFCM) algorithm which usesEuclidean distance finds the partition of the collection 119883 =1199091 119909119873 sub R119901 of119873measures specified by 119896-dimensionalvectors 119909119894 = [1199091119896 1199092119896 119909119894119896]119879 into 119862 fuzzy subsets byminimizing the following objective function [18]

119869PFCM (119880 119879 119881) = 119862sum119894=1

119873sum119896=1

(119886120583119898119894119896 + 119887119905119894119896)1198632119894119896+ 119862sum119894=1

120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896) (6)

where 1 le 119862 le 119873 the number of clusters120583119894119896 the membership of 119909119896 in cluster 119894 satisfying0 le 120583119894119896 le 1

119862sum119894=1

120583119894119896 = 1 1 le 119896 le 119873 (7)

119905119894119896 the typicality of 119909119896 in classes 1198941198632119894119896 = 1003817100381710038171003817119909119896 minus V11989410038171003817100381710038172 (8)

119881 the set of cluster centers (V119894 isin 119877119901)120578119894 the suitable positive numbers described by

120578119894 = 119870sum119873119896=1 120583119898119894119896FCM1198632119894119896sum119873119896=1 (120583119898119894119896FCM) 119870 gt 0 (9)

Typically 119870 is chosen to be 1120583119894119896FCM are the terminal membership values of FCM119898 is a weighting degree this parameter has a signifi-cant impact on the form of clusters in data space

To minimize equation (6) we take its partial derivativeof variables 120583119894119896 119905119894119896 and V119894 equal to zero and obtain thefollowing equations

120583119894119896 = [[119862sum119895=1

(119863119894119896119863119895119896)2(119898minus1)]]

minus1 (10)

119905119894119896 = exp(minus1198871198631198941198962120578119894 ) forall119894 119896 (11)

V119894 = sum119873119896=1 (119886120583119898119894119896 + 119887119905119894119896) 119909119896sum119873119896=1 (119886120583119898119894119896 + 119887119905119894119896) forall119894 (12)

PFCM Algorithm Steps Given a set of observations 119883 =[1199091 119909119873]119879

Initialization (119897 = 0)Set the number of clusters 119862 1 le 119862 le 119873Set the level of weighting119898 2 le 119898 lt 4Set the parameters 119886 119887Set the stopping criterion 120576 120576 gt 0Execute a FCM clustering algorithm to find initialfuzzy partition matrix 119880 and cluster centers 119881Initialize the typicality matrix 119879 randomlyCompute 120578119894 by (9)

Repeat for 119897 = 1 2 Step 1 Compute the cluster centers by (12)

Step 2 Compute the membership matrix 119880 = [120583119894119896] by (10)Step 3 Compute the typicality matrix 119879 = [119905119894119896] by (11)

Until 119880119897 minus 119880119897minus1 lt 120576 then stop Otherwise set 119897 = 119897 + 1and return to Step 1

32 Proposed Robust Kernel Possibilistic Fuzzy 119862-Means(RKPFCM) Algorithm The PFCM can deal with noisy databetter than FCM and PCM nevertheless these conventionalclustering algorithms becomemore effective when applied onlinearly separable data or with a reasonable quantity of errorsIn reality the linearly separable data are rareTherefore FCMPCM and PFCM share the same negative point in that theyare unable to get good separation of data that are nonlin-early separable in input space To correct the imperfectionsfound in PFCMparticularly the nonlinear separable problemkernel [7] methods are regarded as the way of dealing withthis problem In this context we proposed a new extensionof Possibilistic Fuzzy 119862-Means algorithm based on kernelmethod (RKPFCM) The present work proposes a way ofincreasing the accuracy of the PFCM algorithm by exploitinghyper tangent kernel function to calculate the distance usedin its objective function

The kernel function is defined as a generalization of thedistance metric that measures the distance between two datapoints mapped into a future space in which the data are moreclearly separable [12 19ndash21]

Define a nonlinear map as Φ 119909 rarr Φ(119909) isin 119865 where119909 isin 119883 and 119865 is the transformed feature space with higheror even infinite dimension119883 denotes the data space mappedinto 119865 [20ndash22]

The RKPFCM algorithm minimizes the following objec-tive function119869RKPFCM (119880 119879 119881)

= 119862sum119894=1

119873sum119896=1

(119886120583119898119894119896 + 119887119905119894119896) 1003817100381710038171003817120601 (119909119896) minus 120601 (V119894)10038171003817100381710038172+ 119862sum119894=1

120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896) (13)


Then 119909119896 minus V119894 is mapped into space 119865 [22]1003817100381710038171003817119909119896 minus V1198941003817100381710038171003817 997888rarr 1003817100381710038171003817120601 (119909119896) minus 120601 (V119894)1003817100381710038171003817 (14)

where1003817100381710038171003817120601 (119909119896) minus 120601 (V119894)10038171003817100381710038172 = (120601 (119909119896) minus 120601 (V119894))sdot (120601 (119909119896) minus 120601 (V119894))= 120601 (119909119896) sdot 120601 (119909119896) + 120601 (V119894) sdot 120601 (V119894)minus 2120601 (119909119896) sdot 120601 (V119894)= 119870 (119909119896 119909119896) + 119870 (V119894 V119894)minus 2119870 (119896119896 V119894)

(15)

119870(119909119896 V119894) = Φ(119909119896)119879Φ(V119894) is an inner product kernel functionIf we adopt the hyper tangent kernel function that is

119870(119909119896 V119894) = 1 minus tanh(minus1003817100381710038171003817119909119896 minus V119894100381710038171003817100381721205902 ) where 120590 gt 0 (16)

then119870(119909119896 119909119896) = 119870(V119894 V119894) = 1 Thus (15) can be written as1003817100381710038171003817120601 (119909119896) minus 120601 (V119894)10038171003817100381710038172 = 2 (1 minus 119870 (119909119896 V119894))= 2 tanh(minus1003817100381710038171003817119909119896 minus V119894100381710038171003817100381721205902 ) (17)

Considering (17) the objective function (13) is transformedas follows119869RKPFCM (119880 119879 119881)

= 2 119862sum119894=1

119873sum119896=1

(119886120583119898119894119896 + 119887119905119894119896) tanh(minus1003817100381710038171003817119909119896 minus V119894100381710038171003817100381721205902 )+ 119862sum119894=1

120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896) (18)

The derivation of the objective function (18) according to 120583119894119896119905119894 and V119894 defines the relationship update of cluster centers andmembership coefficients

(i) Derivative of 119869RKPFCM(119880 119879 119881) with respect to V119894120597119869RKPFCM (119880 119879 119881)120597V119894 = 2 119873sum

119896=1

(119886120583119898119894119896 + 119887119905119894119896)sdot (1 minus tanh2(minus1003817100381710038171003817119909119896 minus V119894100381710038171003817100381721205902 ))(119909119896 minus V1198941205902 )

= 2 119873sum119896=1

(119886120583119898119894119896 + 119887119905119894119896)(1 minus tanh(minus1003817100381710038171003817119909119896 minus V119894100381710038171003817100381721205902 ))sdot (1 + tanh(minus1003817100381710038171003817119909119896 minus V119894100381710038171003817100381721205902 ))(119909119896 minus V1198941205902 )

(19)

So

120597119869RKPFCM (119880 119879 119881)120597V119894 = 2 119873sum119896=1

(119886120583119898119894119896 + 119887119905119894119896)119870 (119909119896 V119894)sdot (1 + tanh(minus1003817100381710038171003817119909119896 minus V119894100381710038171003817100381721205902 ))(119909119896 minus V1198941205902 )

(20)

Equating (20) to zero leads to

120597119869RKPFCM (119880 119879 119881)120597V119894 = 0 (21)

Then

V119894

= sum119873119896=1 (119886120583119898119894119896 + 119887119905119894119896)119870 (119909119896 V119894) (1 + tanh (minus 1003817100381710038171003817119909119896 minus V11989410038171003817100381710038172 1205902)) 119909119896sum119873119896=1 (119886120583119898119894119896 + 119887119905119894119896)119870 (119909119896 V119894) (1 + tanh (minus 1003817100381710038171003817119909119896 minus V11989410038171003817100381710038172 1205902)) (22)

(ii) Derivative of 119869RKPFCM(119880 119879 119881) with respect to 120583119894119896 In thispart we used the Lagrange multiplier

119869RKPFCM (119880 119875 119881)= 2 119862sum119894=1

119873sum119896=1


120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896)minus 119873sum119896=1

(120582119896 sdot ( 119862sum119894=1

120583119894119896 minus 1))

(23)

120597119869RKPFCM (119880 119879 119881)120597120582 = minus( 119862sum119894=1

120583119894119896 minus 1) = 0 (24)

120597119869RKPFCM (119880 119879 119881)120597120583119894119896= 2119898119886 (120583119894119896)119898minus1 tanh(minus1003817100381710038171003817119909119896 minus V119894100381710038171003817100381721205902 ) minus 120582 = 0 (25)

From expression (25) we can write 120583119894119896 in this form

120583119894119896 = ( 1205822119898119886 tanh (minus 1003817100381710038171003817119909119896 minus V11989410038171003817100381710038172 1205902))1(119898minus1) (26)


Substituting expression (26) in expression (24)

120597119869RKPFCM (119880 119879 119881)120597120582 = 119862sum119895=1

120583119895119896= 119862sum119895=1

( 1205822119898119886 tanh (minus 10038171003817100381710038171003817119909119896 minus V119895100381710038171003817100381710038172 1205902))1(119898minus1)

= 1(27)

It is also

( 1205821198962119898119886)1(119898minus1)= 1sum119862119895=1 (1 tanh (minus 10038171003817100381710038171003817119909119896 minus V119895100381710038171003817100381710038172 1205902))1(119898minus1)

(28)

The two expressions (26) and (28) give the following expres-sion

120583119894119896 = 1sum119862119895=1 (tanh (minus 1003817100381710038171003817119909119896 minus V11989410038171003817100381710038172 1205902) tanh (minus 10038171003817100381710038171003817119909119896 minus V119895100381710038171003817100381710038172 1205902))1(119898minus1) (29)

Therefore the updating relationship is

120583119894119896 = [[119862sum119895=1

( (1 minus 119870 (119909119896 V119894))(1 minus 119870 (119909119896 V119895)))1(119898minus1)]]

minus1 (30)

(iii) Derivative of 119869RKPFCM(119880 119879 119881) with respect to 119905119894119896120597119869RKPFCM (119880 119879 119881)120597119905119894119896= 2119887 tanh(minus1003817100381710038171003817119909119896 minus V119894100381710038171003817100381721205902 ) + 120578119894 log 119905119894119896 = 0 (31)

119905119894119896 = exp(minus2119887 tanh (minus 1003817100381710038171003817119909119896 minus V11989410038171003817100381710038172 1205902)120578119894 ) forall119894 119896 (32)

Therefore the updating typicality matrix is

119905119894119896 = exp(minus2119887 (1 minus 119870 (119909119896 V119894))120578119894 ) forall119894 119896 (33)

Similarly (9) is rewritten by

120578119894 = 119870sum119873119896=1 120583119898119894119896FCM2 (1 minus 119870 (119909119896 V119894))sum119873119896=1 (120583119898119894119896FCM) (34)

RKPFCM Algorithm Steps Given a set of observations 119883 =[1199091 119909119873]119879Initialization (119897 = 0)

Set the number of clusters 119862 1 le 119862 le 119873Set the level of weighting 119898 2 le 119898 lt 4Set the parameters 119886 119887 and 120590Set the stopping criterion 120576 120576 gt 0

Execute a FCM clustering algorithm to find initialfuzzy partition matrix 119880 and cluster centers 119881Initialize the typicality matrix 119879 randomly

Compute 120578119894 by (34)Repeat for 119897 = 1 2

Step 1 Compute the cluster centers by (22)



33 Robust Kernel Possibilistic Fuzzy 119862-MeansAlgorithm Based on PSO (RKPFCM-PSO)

331 PSO The Particle Swarm Optimization is a heuristicsearch method proposed by Kennedy and Eberhart (1995)This technique uses random population solution particles tofind an optimal solution to problems Each particle movesin the search space with a dynamically adjusted position andvelocity for the best solution The particle is characterized bydata structure that contains the coordinates of the currentposition in the search space the best solution point visitedso far and the subset of other agents that are seen asneighbors These adjustments are based on the historicalbehaviors of itself and other agents in the swarmThe changeof speed (acceleration) and the position of each particle inthe optimization landscape (search space) are iteratively [612 23]

V119896(119897+1)

= 119870 [119908V119896(119897) + 1205881 (119901119896(119897) minus 119909119896(119897)) + 1205882 (119901119892(119897) minus 119909119896(119897))] 119909119896(119897+1) = 119909119896(119897) + V119896(119897+1)

(35)


where

119896 = 1 NP size of particles119863 size of the landscape (search space)V119896 = (V1198961 V119896119889 V119896119863) the speed of particle119909119896 = (1199091198961 1199091198961 119909119896119889 119909119896119863) the position ofparticle119901119896 = (1199011198961 1199011198961 119901119896119889 119901119896119863) the best previousposition of particle119892 index represents the best particle among all parti-cles in the group119870 the constriction factor described by the followingrelationship

119870 = 210038161003816100381610038161003816100381610038162 minus 120593 minus radic1205932 minus 41205931003816100381610038161003816100381610038161003816 where 120593 = 1198881 + 1198882 (36)

where 1198881 and 1198882 are two positive constants satisfying thefollowing relationship

120593 = 1198881 + 1198882 gt 4 (37)

1205881 and 1205882 random variables defined as follows

1205881 = 1199031 times 11988811205882 = 1199032 times 1198882 (38)

where 1199031 and 1199032 are two random variables between 0 and 1119908 the weight of inertia according to this equation119908 = 119908max minus 119908max minus 119908min

itermaxtimes iter (39)

where 119908max and 119908min are the initial and final weightitermax is the maximum iterations and iter is thecurrent iteration number

332 Fitness Function The fitness function defines ouroptimization problem described by the following expression

Fitness = 119866119869RKPFCM (119880 119879 119881) (40)

where

119866 is a positive constant119869RKPFCM(119880 119879 119881) represents the objective function ofthe RKPFCM algorithm

RKPFCM-PSO Algorithm Given a set of observations 119883 =[1199091 1199092 119909119873]119879 the RKPFCM-PSO algorithm is describedby the following steps

Initialization (119897 = 0)Select the number of clusters 119862 fuzzy degree 119898the parameters a and b the population size NP theconstants 1198881 and 1198882 the randomvariables 1199031 and 1199032 theweight of inertia119908max and119908min the size of the searchspace119863 the constant 119866 and the stopping criterion 120576Set the 1st particle generation clusters centersInitialize the fitness function and speed of eachparticleCompute 120578119894 by (34)

Repeat 119897 = 119897 + 1

Step 1 Compute the fuzzy partition matrix119880 = [120583119894119896] by (30)Step 2 Compute the typicality matrix 119879 = [119905119894119896] by (33)Step 3 Calculate the new value of fitness for each particleusing (40)

Step 4 Compare the fitness of each particle with 119901best if thevalue is better than 119901best and then set the 119901best valueStep 5 Compare the fitness value of 119892best with the followingif the value is better than 119892best 119892best then is set equal to thisvalue

Step 6 Update position and speed of each particle by (35)

So this algorithm is converged when V119896(119897+1) minus V119896(119897) lt 120576

that is to say stop iteration and find the best solution in thelast generation If not go back to Step 1

4 Identification for Consequent Parameters

The defuzzificationmethod used in the Takagi-Sugeno fuzzymodel is linear with the consequent parameters 120579119894 = [119886119879119894 119887119894]which can be obtained as a solution of aweighted least squaresproblem according to the following equation

120579119894 = [119883119890119879120595119894119883119890]minus1119883119890119879120595119894119884 (41)

where 119883119890 = [119883 1] represents an extension of regressionmatrix119883 = [1198831119879 1198832119879 119883119873119879]119879119884 = [1199101 1199102 119910119873]119879 is the output vector120595119894 is a diagonal matrix of dimension (119873 times 119873)containing the coefficients 120583119894119896 of fuzzy memberships

The RKPFCM and RKPFCM-PSO clustering algorithm areused to find width of the membership functions by thefollowing equation [24]

120590119894119895 = radic 2 lowast [sum119873119896=1 120583119894119896 (119909119895119896 minus V119894119895)2]sum119873119896=1 120583119894119896 (42)

Mathematical Problems in Engineering 7x2

minus2

0

2

4

6

8

10

12

20 4 6minus4 minus2minus6x1

Figure 1 Clustering by RKPFCM-PSO algorithm

5 Simulation Results and Validation Model

51 Identical Data with Noise In this example we have used12 data which are composed of 10 models and two noises thisdata set (11988312) is presented in [25] The FCM PCM PFCMKPFCM and our algorithms RKPFCM and RKPFCM-PSOwere used in clustering the data set in tow groups (119862 = 2)

In this example the parameters settings are 119886 = 1 119887 = 2119898 = 2 119866 = 10 NP = 20 119863 = 2 119908max = 09 119908min = 041198881 = 205 1198882 = 205 120590 = 20 itermax = 1000 and 120576 = 10minus9Figure 1 shows the clustering results for our proposed

method RKPFCM-PSO The Ideal (true) centroids are

119881ideal = [minus334 0334 0] (43)

Table 1 shows the results of center clusters using the sixalgorithms The error between the results prototypes andideal center clusters is calculated by the next expression

119864lowast = 1003817100381710038171003817119881ideal minus 119881lowast10038171003817100381710038172 (44)

where lowast is the FCM PCM PFCM KPFCM RKPFCM andRKPFCM-PSO

According to Table 1 our proposed algorithm RKPFCM-PSO gives the best prototypes of centers

Figure 1 shows the effectiveness of our approach as well

52 Identification of T-S Fuzzy Model After applying theidentification algorithm it is necessary to validate the Takagi-Sugeno fuzzy model Several validation tests of the model areused Among them we cited the Mean Square Error (MSE)test Root Mean Square Error (RMSE) and the VarianceAccounting For (VAF) test

MSE = 1119873 119873sum119896=1

(119910 (119896) minus 119910est (119896))2 RMSE = radic 1119873 119873sum

119896=1

(119910 (119896) minus 119910est (119896))2

Table 1 Prototypes results of clusters centers

Algorithms Centers 119881lowast Error (119864lowast)FCM [14] (minus298 054298 054) 0414

PCM [14] (minus215 002215 002) 1416

PFCM [14] (minus284 036284 036) 03796

KPFCM [14] (minus333 003333 003) 00005

RKPFCM (minus3337 00023337 0002) 179 times 10minus5RKPFCM-PSO (minus33385 0000133385 00001) 423 times 10minus6

VAF = 100[1 minus var (119910 minus 119910est)var (119910) ]

(45)

where ldquo119910rdquo is the real output and ldquo119910estrdquo is the estimated output

521 Example 1 Consider a nonlinear system described bythe following difference equation [15]119910 (119896)= 1199102 (119896 minus 3) + 1199102 (119896 minus 2) + 1199102 (119896 minus 1) + tanh (119906 (119896))1 + 1199102 (119896 minus 1) + 1199102 (119896 minus 2) + 1199102 (119896 minus 3)+ 119890 (119896)

(46)

where 119910(119896) and 119906(119896) are the output and the input of thesystem respectively119890(119896) is a noise119906 (119896) = 06 sin (3120587119896119879119904) + 02 sin (4120587119896119879119904)+ 12 sin (120587119896119879119904) (47)

119879119904 = 001200 samples were generated by simulation and were used

where the selected input variables are chosen 119910(119896 minus 1) 119910(119896 minus2) 119910(119896 minus 3) 119906(119896)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 andour proposed clustering RKPFCM and RKPFCM-PSOwhichapproximate the nonlinear model (46)

The used parameters are 119886 = 1 119887 = 3119898 = 2 119866 = 10 NP= 30 119863 = 4 119908max = 09 119908min = 04 1198881 = 205 1198882 = 205 120590 =20 itermax = 1000 and 120576 = 10minus9

The shape of the excitation signal used for identificationis illustrated in Figure 2


Input signal (u)

minus2minus1

012

50 100 150 2000Sample

040608

11214

50 100 150 2000Sample

Output signal (y)

Figure 2 Input-output sequences

Error signal

y

05

1

15

50 100 150 2000Sample

minus002

0

002

004

50 100 150 2000Sample

yest

Real output (y) and estimated output (yest)

Figure 3 Identification results for the RKPFCM-PSO algorithmwith SNR = 10DB

The simulation result given by the RKPFCM-PSO algo-rithm is illustrated in Figure 3

Table 2 shows the various modeling performance resultswithout noise obtained by different algorithms this com-parison results demonstrate that the best MSE and bestVAF are obtained by the proposed methods (RKPFCM andRKPFCM-PSO)

Tables 3 and 4 present the various modeling performanceresults with noise influence (SNR = 5 dB and 10 dB) obtainedby the different algorithmsHowever our proposed algorithmRKPFCM-PSO retained the best performance with a higherlevel of noise

The local linear models identified are given as follows

1199101 (119896) = 03167119910 (119896 minus 1) + 00263119910 (119896 minus 2)minus 00482119910 (119896 minus 3) + 01183 (119896) + 06916

Table 2 Comparison results without noise

Algorithms Number of rules MSE VAF ()Cluster fuzzy [15] 6 19 times 10minus3 mdashKalman cluster fuzzy [15] 2 22 times 10minus5 mdashFCM 2 21422 times 10minus5 999581GK 2 21142 times 10minus5 999592KFCM 2 21011 times 10minus5 999601PFCM 2 20152 times 10minus5 999611RKPFCM 2 18498 times 10minus5 999723RKPFCM-PSO 2 18412 times 10minus5 999742

Table 3 Comparison results with SNR = 10 dB

Algorithms Number of rules MSE (10minus4) VAF ()FCM 2 98397 980345GK 2 98831 980303KFCM 2 99492 980289PFCM 2 98313 980362RKPFCM 2 85158 981628RKPFCM-PSO 2 85149 981637



1199102 (119896) = 04045119910 (119896 minus 1) + 00665119910 (119896 minus 2)minus 00304119910 (119896 minus 2) + 01519119906 (119896)+ 05417(48)

522 Example 2 Consider a highly complex modified non-linear system described by the following difference equation[16]

119910 (119896) = 119910 (119896 minus 1) (119910 (119896 minus 2) + 2) (119910 (119896 minus 1) + 25)10 + 1199102 (119896 minus 1) + 1199102 (119896 minus 2)+ 119906 (119896) + 119890 (119896) (49)

where 119910(119896) is the model output and 119906(119896) is the model inputwhich is bounded between [minus1 +1] The 119890(119896) is a noise

The following input signal is expressed as119906 (119896)=

sin(2119896120587250 ) 119896 le 50008 lowast sin(2119896120587250 ) + 02 lowast sin(211989612058725 ) otherwise

(50)


Input signal (u)

Output signal (y)

minus15minus1

minus050

051

15

minus20246

500 1000 15000Sample

500 1000 15000Sample


Error signal

minus202468

minus1minus05

005

1

200 400 600 800 10000Sample

200 400 600 800 10000Sample

y

yest


Figure 5 RKPFCM-PSO performances for test data with SNR =10 dB

1500 samples were generated by simulation in which 1000samples were used to train the model Fuzzy model param-eters have been identified once testing of model was doneby the remaining 500 samples and 119910(119896 minus 1) 119910(119896 minus 2) 119906(119896)119906(119896 minus 1) are chosen as input variables In this example theparameters settings are 119886 = 1 119887 = 2 119898 = 2 119866 = 10 NP =30 119863 = 4 119908max = 09 119908min = 04 1198881 = 2 1198882 = 21 120590 = 20itermax = 1000 and 120576 = 10minus9

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 RKPFCM-PSO arerespectively shown in Figures 4 and 5

The evaluation performance index (RMSE-trn andRMSE-test) stands for training and testing data respectively

Tables 5ndash8 show the comparative performance ofRKPFCM and RKPFCM-PSO with different existingalgorithms such as FCM G-K Fuzzy Model Identification

Table 5 20 dB Noise

Algorithms RMSE-trn RMSE-test Number of rulesFCM [16] 01393 03004 4GK [16] 01326 01484 4FMI [16] 01457 02378 4FCRM [16] 01291 01495 4MFCRM-NC [16] 01183 01395 4RKPFCM 01167 01377 4RKPFCM-PSO 01165 01370 4

Table 6 10 dB Noise


Table 7 5 dB Noise


Table 8 1 dB Noise


(FMI) FCRM and MFCRM-NC It is clearly seen fromthe results that our algorithm RKPFCM-PSO gives the bestperformance in noisy environments The local linear modelsidentified by RKPFCM-PSO are given as follows

1199101 (119896) = 08562119910 (119896 minus 1) + 00548119910 (119896 minus 2)+ 20967119906 (119896) minus 16609119906 (119896 minus 1) + 008141199102 (119896) = 08058119910 (119896 minus 1) minus 00924119910 (119896 minus 2)+ 13488119906 (119896) minus 03718119906 (119896 minus 1) + 08032


Error signal

4550556065

50 100 150 200 250 3000Sample

minus2minus1

012

50 100 150 200 250 3000Sample

y

yest


Figure 6 Identification results for the RKPFCM-PSO Algorithm(SNR = 10 dB)

1199103 (119896) = 05673119910 (119896 minus 1) + 01896119910 (119896 minus 2)+ 12330119906 (119896) minus 08613119906 (119896 minus 1) + 005201199104 (119896) = 08881119910 (119896 minus 1) minus 00571119910 (119896 minus 2)+ 15400119906 (119896) minus 09195119906 (119896 minus 1) + 04536(51)

523 Example 3 We consider the Box Jenkins gas furnacedata set which is used as a standard test for identificationtechniques The data set is composed of 296 pairs of input-output measurements The input ldquo119906rdquo is the gas flow rate intoa furnace and the output ldquo119910rdquo is the CO2 concentration in theoutlet gases In order to take all the above-mentioned issuesinto account we simulated the following experimental case[16]

all the 296 data pairs are used as training data and119910(119896 minus 1) 119906(119896 minus 4) are selected as input variables tovarious algorithms while we use two rules (119862 = 2)In this example the used parameters are 119886 = 1 119887 = 2119898 = 2 119866 = 10 NP = 30 119863 = 3 119908max = 09 119908min =04 1198881 = 195 1198882 = 21 120590 = 10 itermax = 1000 and 120576 =10minus9

The simulation result given by the RKPFCM-PSO algorithmis illustrated in Figure 6

Based on the comparison presented in Table 9 it is clearthat the proposed algorithmRKPFCM-PSO ismore robust tonoise than the other algorithms found in literature

When we use our algorithm RKPFCM-PSO the locallinear models identified are given as follows

1199101 (119896) = 06777119910 (119896 minus 1) minus 09593119906 (119896 minus 4) + 1730071199102 (119896) = 05560119910 (119896 minus 1) minus 13297119906 (119896 minus 4) + 235880 (52)

Table 9 Comparison results for Box Jenkins system

Algorithms Number of inputs Number of rules MSETong (1980) [16] 2 19 0469Pedrycz (1984) [16] 2 81 0320Xu (1987) [16] 2 25 0328Sugeno (1991) [16] 2 2 0359Yoshinari (1993)[16] 2 6 0299

Joo (1997) [16] 2 6 0166Zhang (2006) [16] 2 2 0160Glowaty (2008)[16] 2 2 0391

Andri (2011) [16] 2 10 0167Tanmoy Dam(2015) [16]Without noise 2 2 0152SNR = 30 dB 2 2 0153SNR = 10 dB 2 2 0277RKPFCMWithout noise 2 2 01458SNR = 30 dB 2 2 01512SNR = 10 dB 2 2 02342SNR = 5 dB 2 2 02922SNR = 2 dB 2 2 03014RKPFCM-PSOWithout noise 2 2 01455SNR = 30 dB 2 2 01503SNR = 10 dB 2 2 02338SNR = 5 dB 2 2 02902SNR = 2 dB 2 2 03001

6 Conclusion

In literature various clustering algorithms have been pro-posed for nonlinear systems identification In this work wedeveloped a new clustering algorithm called RKPFCM-PSOfor the nonlinear systems identification Our algorithm is animprovement of the Possibilistic Fuzzy 119862-Means Clustering(PFCM) where we used a hyper tangent kernel function tocalculate the distance of data point from the cluster centersand a heuristic search algorithm PSO to reach the globalminimum of the objective function The proposed algo-rithm provides better results of fuzzy modeling of unknownnonlinear systems The robustness and the quality of thisproposed method are demonstrated by simulation results ofnoisy nonlinear systems described by recurrent equationsand application to a Box Jenkins gas furnace system Thusthe proposed methods show favorable results in noisy envi-ronments compared with the techniques mentioned in theliterature

In the future we will integrate other optimization meth-ods such as the gravitational search algorithm to optimizeour hybrid method and we will apply this algorithm for


identification of some complex nonlinear real systems as therobotic or the mechatronic systems

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] E Kim M Park S Kim and M Park ldquoA transformed input-domain approach to fuzzy modelingrdquo IEEE Transactions onFuzzy Systems vol 6 no 4 pp 596ndash604 1998

[2] M Sugeno and G T Kang ldquoFuzzy modelling and control ofmultilayer incineratorrdquo Fuzzy Sets and Systems vol 18 no 3 pp329ndash345 1986

[3] J C Bezdek Pattern Recognition with Fuzzy Objective FunctionAlgorithms Plenum Press New York NY USA 1981

[4] M Sugeno and T Yasukawa ldquoA fuzzy-logic-based approach toqualitative modelingrdquo IEEE Transactions on Fuzzy Systems vol1 no 1 pp 7ndash31 1993

[5] J-Q Chen Y-G Xi and Z-J Zhang ldquoA clustering algorithmfor fuzzy model identificationrdquo Fuzzy Sets and Systems vol 98no 3 pp 319ndash329 1998

[6] T Ahmed H Lassad and C Abdelkader ldquoNonlinear systemidentification using clustering algorithm and particle swarmoptimizationrdquo Scientific Research and Essays vol 7 no 13 pp1415ndash1431 2012

[7] J Abonyi R Babuska and F Szeifert ldquoModified Gath-Gevafuzzy clustering for identification of Takagi-Sugeno fuzzy mod-elsrdquo IEEE Transactions on Systems Man and Cybernetics PartB (Cybernetics) vol 32 no 5 pp 612ndash621 2002

[8] H Frigui and Krishnapuram ldquoA robust competitive clusteringalgorithm with applications in computer visionrdquo IEEE Transac-tions on Pattern Analysis andMachine Intelligence Computationvol 21 no 5 pp 450ndash465 1999

[9] K C Ying S W Lin Z J Lee and I L Lee ldquoA novel functionapproximation based on robust fuzzy regression algorithmmodel and particle swarm optimizationrdquo Applied Soft Comput-ing vol 38 no 2 pp 1820ndash1826 2011

[10] R Krishnapuram and J Keller ldquoA possibilistic approach toclusteringrdquo IEEE Transactions on Fuzzy Systems vol 1 no 2pp 98ndash110 1993

[11] R Dave ldquoCharacterization and detection of noise in clusteringrdquoPattern Recognition Letters vol 12 no 11 pp 657ndash664 2011

[12] J Zhang and L Shen ldquoAn improved fuzzy c-means clusteringalgorithm based on shadowed sets and PSOrdquo ComputationalIntelligence and Neuroscience vol 2014 Article ID 368628 10pages 2014

[13] T Niknam and B Amiri ldquoAn efficient hybrid approach basedon PSO ACO and k-means for cluster analysisrdquo Applied SoftComputing vol 10 pp 183ndash197 2010

[14] M Tushir and S Srivastava ldquoA new Kernelized hybrid c-meanclusteringmodelwith optimized parametersrdquoApplied SoftCom-puting Journal vol 10 no 2 pp 381ndash389 2010

[15] S Bidyadhar and J Debashisha ldquoA difierential evolution basedneural network approach to nonlinear identificationrdquo AppliedSoft Computing vol 11 no 1 pp 861ndash871 2011

[16] T Dam and A Kanti Deb ldquoBlock sparse representations inmodified fuzzy c-regression model clustering algorithm forts fuzzy model identificationrdquo in Proceedings of the IEEESymposium Series on Computational Intelligence pp 1687ndash1694IEEE Cape Town South Africa December 2015

[17] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications formodeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[18] X H Wu and Zhou ldquoA possibilistic c-means clustering algo-rithm based on kernelmethodsrdquo inComputer Design andAppli-cations (ICCDA rsquo10) International Conference pp 2062ndash2066IEEE Guilin China June 2006

[19] S R Kannan S Ramathilagam R Deviand and A SathyaldquoRobust kernel FCM in segmentation of breastmedical imagesrdquoExpert Systems with Applications vol 38 pp 4382ndash4389 2011

[20] P Kaur I M S Lamba and A Gosai ldquoNovel kernelized type-2 fuzzy c-means clustering algorithm in segmentation of noisymedical imagesrdquo in Proceedings of the Recent Advances in Intel-ligent Computational Systems (RAICS rsquo11) IEEE TrivandrumIndia November 2011

[21] B Mohamed T Ahmed H Lassad and C Abdelkader ldquoAnew extension of fuzzy c-means algorithm using non euclideandistance and kernel methodsrdquo in Proceedings of the ControlDecision and Information Technologies (CoDIT rsquo13) Interna-tional Conference IEEE Hammamet Tunisia May 2013

[22] X Zhao and J Zhou ldquoImproved kernel possibilistic fuzzyclustering algorithm based on invasive weed optimizationrdquoJournal of Shanghai Jiaotong University vol 20 no 2 pp 164ndash170 2015

[23] R C Eberhart and Y Shi ldquoComparing inertia weights and con-striction factors in particle swarm optimizationrdquo in Proceedingsof the Congress on Evolutionary Computation pp 84ndash88 LaJolla CA USA July 2000

[24] R Babuka and H Verbruggen ldquoNeuro-fuzzy methods fornonlinear system identificationrdquoAnnual Reviews inControl vol27 pp 73ndash85 2003

[25] N R Pal K Pal J M Keller and J C Bezdek ldquoA possibilisticfuzzy c-means clustering algorithmrdquo IEEE Transactions onFuzzy Systems vol 13 no 4 pp 517ndash530 2005

Submit your manuscripts athttpswwwhindawicom

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


by Bezdek is well-known but this algorithm is sensitiveto noise or outliers and susceptible to local minima [89] However noise in the data sets can make the situa-tion worse by creating many inauthentic minima Theseare able to distort the global minimum solution found byFCM algorithm This flaw has stimulated the researchers toovercome this inconvenience To fight against the effects ofoutlying data various approaches are considered such as thepossibilistic clustering (PCM) proposed by Krishnapuramand Keller [10] and fuzzy noise clustering approach ofDave [11] The possibilistic approach executed a possibilisticpartition in which a membership relation calculates theabsolute degree of typicality of a point in a cluster Althoughthe PCM algorithm is robust against the noise points andallows identifying these outliers it is very responsive toinitializations and occasionally generates coincident clustersTo solve this deficiency of identical clusters a PossibilisticFuzzy 119888-Means (PFCM) algorithm was suggested by Wuand Zhou in 2006 Nonetheless these algorithms are notefficient for unequal dimension clusters and cannot separateclusters that are nonlinearly separable in input space andtheir limits between two clusters are linear In our work toovercome this shortcoming kernelmethods [12] are regardedas the way of dealing with this problem We propose a newclustering algorithm called Robust Kernel Possibilistic Fuzzy119862-Means (RKPFCM) which adopts a kernel induced metricin the data space to replace the original Euclidean normmetric By changing the inner product with an appropriatehyper tangent kernel function one can implicitly affect anonlinearmapping to a high dimensional feature spacewherethe data is more clearly separable However our proposedalgorithm RKPFCM has an iterative nature that makes itsensitive to initialization and sensitive to converge to a localminimum To overcome these problems several solutionshave been proposed in the literature Among them is com-bining the clustering algorithm with a heuristic optimizationtechnique In this context many researches have proposedthe evolutionary computation technique based on ParticleSwarm Optimization (PSO) They have been successfullyapplied to solve various optimization problems [13] Thuswe introduce PSO into the RKPFCM algorithm to achieveglobal optimization The efficacy of our algorithm comparedto many algorithms is tested on noisy nonlinear systemsdefined by recurrent equations and an application to BoxJenkins system This paper will be presented as followsThe second part of work is reserved for introducing theTakagi-Sugeno fuzzy model The third part will be devotedto identifying the premise parameters of this model where weused the proposed RKPFCM algorithm and PSO algorithm isintroduced In the fourth part we will focus on identificationof consequent parameters The simulations results and themodel validity of RKPFCMandRKPFCM-PSO are presentedin part five Finally we conclude this paper

2 Takagi-Sugeno Fuzzy Model

Takagi-Sugeno fuzzy model (T-S) is one of the best tech-niques used for modeling a nonlinear system representedby the recurrent equation 119910(119896) = 119892NL(119909119896) T-S model is

constructed by a rule-based type ldquoif thenrdquo in which theconsequent uses numeric variables rather than linguistic vari-ables (case ofMamdani)The consequent can be expressed bya constant a polynomial or differential equation dependingon the antecedent variables The T-S fuzzy model allowsapproximating the nonlinear system into several locally linearsubsystems [4 17]

In general a Takagi-Sugeno fuzzy model is based on if then rules of the form

119877119894 if 1199091119896 is 119860 1198941 119909119899119896 is 119860 119894119899 then 119910119894 = 119886119894119879119909119896 + 119887119894 (1)

The ldquoifrdquo rule function defines the premise part and ldquothenrdquorule function constitutes the consequent part of the T-S fuzzymodel 119877119894 represents the 119894th rule119886119894119879 = [1198861198941 1198861198942 119886119894119899] is the parameters vector such

as 119886119894 isin 119877119899119887119894 is a scalar119909119896 = [1199091119896 1199092119896 119909119899119896]119879 is observations vector119860 1198941 119860 1198942 119860 119894119899 represents the fuzzy subsetswhere 119894 isin [1 119862]

Here the fuzzy sets are represented by the followingmembership function [7]

119860 119894119895 (119909119895119896) = exp(minus(119909119895119896 minus V119894119895)21205901198941198952 ) isin [0 1](119894 = 1 2 119862 119895 = 1 2 119899)

(2)

where V119894 = [V1198941 V1198942 V119894119895]119879 isin 119877119899 are centers and 120590119894 = [12059011989411205901198942 120590119894119895]119879 isin 119877119899 is the width of the membership functionThe estimated output model is defined by the following

equation [4]

119910 (119896) = 119862sum119894=1

120573119894 (119896) 119910119894 (119896) (3)

As

120573119894 (119896) = prod119899119895=1119860 119894119895 (119909119895119896)sum119862119894=1prod119899119895=1119860 119894119895 (119909119895119896) 119896 = 1 2 119873 (4)

so

119910 (119896) = 119888sum119894=1

120573119894 (119896) [119886119879119894 119909119896 + 119887119894] (5)

3 Identification Algorithm forPremise Parameters

To identify the premise parameters of a Takagi-Sugeno fuzzymodel described by equation (1) we used the PossibilisticFuzzy 119862-Means (PFCM) algorithm and our proposed algo-rithms (RKPFCM and RKPFCM-PSO)



119869PFCM (119880 119879 119881) = 119862sum119894=1

119873sum119896=1

(119886120583119898119894119896 + 119887119905119894119896)1198632119894119896+ 119862sum119894=1

120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896) (6)


119862sum119894=1

120583119894119896 = 1 1 le 119896 le 119873 (7)



120578119894 = 119870sum119873119896=1 120583119898119894119896FCM1198632119894119896sum119873119896=1 (120583119898119894119896FCM) 119870 gt 0 (9)



120583119894119896 = [[119862sum119895=1

(119863119894119896119863119895119896)2(119898minus1)]]

minus1 (10)

119905119894119896 = exp(minus1198871198631198941198962120578119894 ) forall119894 119896 (11)

V119894 = sum119873119896=1 (119886120583119898119894119896 + 119887119905119894119896) 119909119896sum119873119896=1 (119886120583119898119894119896 + 119887119905119894119896) forall119894 (12)










= 119862sum119894=1

119873sum119896=1

(119886120583119898119894119896 + 119887119905119894119896) 1003817100381710038171003817120601 (119909119896) minus 120601 (V119894)10038171003817100381710038172+ 119862sum119894=1

120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896) (13)




(15)





= 2 119862sum119894=1

119873sum119896=1


120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896) (18)



119896=1


= 2 119873sum119896=1


(19)

So

120597119869RKPFCM (119880 119879 119881)120597V119894 = 2 119873sum119896=1


(20)


120597119869RKPFCM (119880 119879 119881)120597V119894 = 0 (21)

Then

V119894



119869RKPFCM (119880 119875 119881)= 2 119862sum119894=1

119873sum119896=1


120578119894 119873sum119896=1


(120582119896 sdot ( 119862sum119894=1

120583119894119896 minus 1))

(23)

120597119869RKPFCM (119880 119879 119881)120597120582 = minus( 119862sum119894=1

120583119894119896 minus 1) = 0 (24)






120597119869RKPFCM (119880 119879 119881)120597120582 = 119862sum119895=1

120583119895119896= 119862sum119895=1


= 1(27)

It is also


(28)




120583119894119896 = [[119862sum119895=1


minus1 (30)
















V119896(119897+1)

= 119870 [119908V119896(119897) + 1205881 (119901119896(119897) minus 119909119896(119897)) + 1205882 (119901119892(119897) minus 119909119896(119897))] 119909119896(119897+1) = 119909119896(119897) + V119896(119897+1)

(35)


where




120593 = 1198881 + 1198882 gt 4 (37)


1205881 = 1199031 times 11988811205882 = 1199032 times 1198882 (38)





Fitness = 119866119869RKPFCM (119880 119879 119881) (40)

where




Repeat 119897 = 119897 + 1








120579119894 = [119883119890119879120595119894119883119890]minus1119883119890119879120595119894119884 (41)





minus2

0

2

4

6

8

10

12







119881ideal = [minus334 0334 0] (43)







MSE = 1119873 119873sum119896=1


119896=1

(119910 (119896) minus 119910est (119896))2



PCM [14] (minus215 002215 002) 1416

PFCM [14] (minus284 036284 036) 03796

KPFCM [14] (minus333 003333 003) 00005



(45)



(46)









Input signal (u)

minus2minus1

012

50 100 150 2000Sample

040608

11214

50 100 150 2000Sample

Output signal (y)


Error signal

y

05

1

15

50 100 150 2000Sample

minus002

0

002

004

50 100 150 2000Sample

yest




















(50)


Input signal (u)

Output signal (y)

minus15minus1

minus050

051

15

minus20246

500 1000 15000Sample

500 1000 15000Sample


Error signal

minus202468

minus1minus05

005

1

200 400 600 800 10000Sample

200 400 600 800 10000Sample

y

yest








Table 5 20 dB Noise


Table 6 10 dB Noise


Table 7 5 dB Noise


Table 8 1 dB Noise





Error signal

4550556065

50 100 150 200 250 3000Sample

minus2minus1

012

50 100 150 200 250 3000Sample

y

yest














6 Conclusion







References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119869PFCM (119880 119879 119881) = 119862sum119894=1

119873sum119896=1

(119886120583119898119894119896 + 119887119905119894119896)1198632119894119896+ 119862sum119894=1

120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896) (6)


119862sum119894=1

120583119894119896 = 1 1 le 119896 le 119873 (7)



120578119894 = 119870sum119873119896=1 120583119898119894119896FCM1198632119894119896sum119873119896=1 (120583119898119894119896FCM) 119870 gt 0 (9)



120583119894119896 = [[119862sum119895=1

(119863119894119896119863119895119896)2(119898minus1)]]

minus1 (10)

119905119894119896 = exp(minus1198871198631198941198962120578119894 ) forall119894 119896 (11)

V119894 = sum119873119896=1 (119886120583119898119894119896 + 119887119905119894119896) 119909119896sum119873119896=1 (119886120583119898119894119896 + 119887119905119894119896) forall119894 (12)










= 119862sum119894=1

119873sum119896=1

(119886120583119898119894119896 + 119887119905119894119896) 1003817100381710038171003817120601 (119909119896) minus 120601 (V119894)10038171003817100381710038172+ 119862sum119894=1

120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896) (13)




(15)





= 2 119862sum119894=1

119873sum119896=1


120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896) (18)



119896=1


= 2 119873sum119896=1


(19)

So

120597119869RKPFCM (119880 119879 119881)120597V119894 = 2 119873sum119896=1


(20)


120597119869RKPFCM (119880 119879 119881)120597V119894 = 0 (21)

Then

V119894



119869RKPFCM (119880 119875 119881)= 2 119862sum119894=1

119873sum119896=1


120578119894 119873sum119896=1


(120582119896 sdot ( 119862sum119894=1

120583119894119896 minus 1))

(23)

120597119869RKPFCM (119880 119879 119881)120597120582 = minus( 119862sum119894=1

120583119894119896 minus 1) = 0 (24)






120597119869RKPFCM (119880 119879 119881)120597120582 = 119862sum119895=1

120583119895119896= 119862sum119895=1


= 1(27)

It is also


(28)




120583119894119896 = [[119862sum119895=1


minus1 (30)
















V119896(119897+1)

= 119870 [119908V119896(119897) + 1205881 (119901119896(119897) minus 119909119896(119897)) + 1205882 (119901119892(119897) minus 119909119896(119897))] 119909119896(119897+1) = 119909119896(119897) + V119896(119897+1)

(35)


where




120593 = 1198881 + 1198882 gt 4 (37)


1205881 = 1199031 times 11988811205882 = 1199032 times 1198882 (38)





Fitness = 119866119869RKPFCM (119880 119879 119881) (40)

where




Repeat 119897 = 119897 + 1








120579119894 = [119883119890119879120595119894119883119890]minus1119883119890119879120595119894119884 (41)





minus2

0

2

4

6

8

10

12







119881ideal = [minus334 0334 0] (43)







MSE = 1119873 119873sum119896=1


119896=1

(119910 (119896) minus 119910est (119896))2



PCM [14] (minus215 002215 002) 1416

PFCM [14] (minus284 036284 036) 03796

KPFCM [14] (minus333 003333 003) 00005



(45)



(46)









Input signal (u)

minus2minus1

012

50 100 150 2000Sample

040608

11214

50 100 150 2000Sample

Output signal (y)


Error signal

y

05

1

15

50 100 150 2000Sample

minus002

0

002

004

50 100 150 2000Sample

yest




















(50)


Input signal (u)

Output signal (y)

minus15minus1

minus050

051

15

minus20246

500 1000 15000Sample

500 1000 15000Sample


Error signal

minus202468

minus1minus05

005

1

200 400 600 800 10000Sample

200 400 600 800 10000Sample

y

yest








Table 5 20 dB Noise


Table 6 10 dB Noise


Table 7 5 dB Noise


Table 8 1 dB Noise





Error signal

4550556065

50 100 150 200 250 3000Sample

minus2minus1

012

50 100 150 200 250 3000Sample

y

yest














6 Conclusion







References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







(15)





= 2 119862sum119894=1

119873sum119896=1


120578119894 119873sum119896=1

(119905119894119896 log (119905119894119896) minus 119905119894119896) (18)



119896=1


= 2 119873sum119896=1


(19)

So

120597119869RKPFCM (119880 119879 119881)120597V119894 = 2 119873sum119896=1


(20)


120597119869RKPFCM (119880 119879 119881)120597V119894 = 0 (21)

Then

V119894



119869RKPFCM (119880 119875 119881)= 2 119862sum119894=1

119873sum119896=1


120578119894 119873sum119896=1


(120582119896 sdot ( 119862sum119894=1

120583119894119896 minus 1))

(23)

120597119869RKPFCM (119880 119879 119881)120597120582 = minus( 119862sum119894=1

120583119894119896 minus 1) = 0 (24)






120597119869RKPFCM (119880 119879 119881)120597120582 = 119862sum119895=1

120583119895119896= 119862sum119895=1


= 1(27)

It is also


(28)




120583119894119896 = [[119862sum119895=1


minus1 (30)
















V119896(119897+1)

= 119870 [119908V119896(119897) + 1205881 (119901119896(119897) minus 119909119896(119897)) + 1205882 (119901119892(119897) minus 119909119896(119897))] 119909119896(119897+1) = 119909119896(119897) + V119896(119897+1)

(35)


where




120593 = 1198881 + 1198882 gt 4 (37)


1205881 = 1199031 times 11988811205882 = 1199032 times 1198882 (38)





Fitness = 119866119869RKPFCM (119880 119879 119881) (40)

where




Repeat 119897 = 119897 + 1








120579119894 = [119883119890119879120595119894119883119890]minus1119883119890119879120595119894119884 (41)





minus2

0

2

4

6

8

10

12







119881ideal = [minus334 0334 0] (43)







MSE = 1119873 119873sum119896=1


119896=1

(119910 (119896) minus 119910est (119896))2



PCM [14] (minus215 002215 002) 1416

PFCM [14] (minus284 036284 036) 03796

KPFCM [14] (minus333 003333 003) 00005



(45)



(46)









Input signal (u)

minus2minus1

012

50 100 150 2000Sample

040608

11214

50 100 150 2000Sample

Output signal (y)


Error signal

y

05

1

15

50 100 150 2000Sample

minus002

0

002

004

50 100 150 2000Sample

yest




















(50)


Input signal (u)

Output signal (y)

minus15minus1

minus050

051

15

minus20246

500 1000 15000Sample

500 1000 15000Sample


Error signal

minus202468

minus1minus05

005

1

200 400 600 800 10000Sample

200 400 600 800 10000Sample

y

yest








Table 5 20 dB Noise


Table 6 10 dB Noise


Table 7 5 dB Noise


Table 8 1 dB Noise





Error signal

4550556065

50 100 150 200 250 3000Sample

minus2minus1

012

50 100 150 200 250 3000Sample

y

yest














6 Conclusion







References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






120597119869RKPFCM (119880 119879 119881)120597120582 = 119862sum119895=1

120583119895119896= 119862sum119895=1


= 1(27)

It is also


(28)




120583119894119896 = [[119862sum119895=1


minus1 (30)
















V119896(119897+1)

= 119870 [119908V119896(119897) + 1205881 (119901119896(119897) minus 119909119896(119897)) + 1205882 (119901119892(119897) minus 119909119896(119897))] 119909119896(119897+1) = 119909119896(119897) + V119896(119897+1)

(35)


where




120593 = 1198881 + 1198882 gt 4 (37)


1205881 = 1199031 times 11988811205882 = 1199032 times 1198882 (38)





Fitness = 119866119869RKPFCM (119880 119879 119881) (40)

where




Repeat 119897 = 119897 + 1








120579119894 = [119883119890119879120595119894119883119890]minus1119883119890119879120595119894119884 (41)





minus2

0

2

4

6

8

10

12







119881ideal = [minus334 0334 0] (43)







MSE = 1119873 119873sum119896=1


119896=1

(119910 (119896) minus 119910est (119896))2



PCM [14] (minus215 002215 002) 1416

PFCM [14] (minus284 036284 036) 03796

KPFCM [14] (minus333 003333 003) 00005



(45)



(46)









Input signal (u)

minus2minus1

012

50 100 150 2000Sample

040608

11214

50 100 150 2000Sample

Output signal (y)


Error signal

y

05

1

15

50 100 150 2000Sample

minus002

0

002

004

50 100 150 2000Sample

yest




















(50)


Input signal (u)

Output signal (y)

minus15minus1

minus050

051

15

minus20246

500 1000 15000Sample

500 1000 15000Sample


Error signal

minus202468

minus1minus05

005

1

200 400 600 800 10000Sample

200 400 600 800 10000Sample

y

yest








Table 5 20 dB Noise


Table 6 10 dB Noise


Table 7 5 dB Noise


Table 8 1 dB Noise





Error signal

4550556065

50 100 150 200 250 3000Sample

minus2minus1

012

50 100 150 200 250 3000Sample

y

yest














6 Conclusion







References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





where




120593 = 1198881 + 1198882 gt 4 (37)


1205881 = 1199031 times 11988811205882 = 1199032 times 1198882 (38)





Fitness = 119866119869RKPFCM (119880 119879 119881) (40)

where




Repeat 119897 = 119897 + 1








120579119894 = [119883119890119879120595119894119883119890]minus1119883119890119879120595119894119884 (41)





minus2

0

2

4

6

8

10

12







119881ideal = [minus334 0334 0] (43)







MSE = 1119873 119873sum119896=1


119896=1

(119910 (119896) minus 119910est (119896))2



PCM [14] (minus215 002215 002) 1416

PFCM [14] (minus284 036284 036) 03796

KPFCM [14] (minus333 003333 003) 00005



(45)



(46)









Input signal (u)

minus2minus1

012

50 100 150 2000Sample

040608

11214

50 100 150 2000Sample

Output signal (y)


Error signal

y

05

1

15

50 100 150 2000Sample

minus002

0

002

004

50 100 150 2000Sample

yest




















(50)


Input signal (u)

Output signal (y)

minus15minus1

minus050

051

15

minus20246

500 1000 15000Sample

500 1000 15000Sample


Error signal

minus202468

minus1minus05

005

1

200 400 600 800 10000Sample

200 400 600 800 10000Sample

y

yest








Table 5 20 dB Noise


Table 6 10 dB Noise


Table 7 5 dB Noise


Table 8 1 dB Noise





Error signal

4550556065

50 100 150 200 250 3000Sample

minus2minus1

012

50 100 150 200 250 3000Sample

y

yest














6 Conclusion







References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





minus2

0

2

4

6

8

10

12







119881ideal = [minus334 0334 0] (43)







MSE = 1119873 119873sum119896=1


119896=1

(119910 (119896) minus 119910est (119896))2



PCM [14] (minus215 002215 002) 1416

PFCM [14] (minus284 036284 036) 03796

KPFCM [14] (minus333 003333 003) 00005



(45)



(46)









Input signal (u)

minus2minus1

012

50 100 150 2000Sample

040608

11214

50 100 150 2000Sample

Output signal (y)


Error signal

y

05

1

15

50 100 150 2000Sample

minus002

0

002

004

50 100 150 2000Sample

yest




















(50)


Input signal (u)

Output signal (y)

minus15minus1

minus050

051

15

minus20246

500 1000 15000Sample

500 1000 15000Sample


Error signal

minus202468

minus1minus05

005

1

200 400 600 800 10000Sample

200 400 600 800 10000Sample

y

yest








Table 5 20 dB Noise


Table 6 10 dB Noise


Table 7 5 dB Noise


Table 8 1 dB Noise





Error signal

4550556065

50 100 150 200 250 3000Sample

minus2minus1

012

50 100 150 200 250 3000Sample

y

yest














6 Conclusion







References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Input signal (u)

minus2minus1

012

50 100 150 2000Sample

040608

11214

50 100 150 2000Sample

Output signal (y)


Error signal

y

05

1

15

50 100 150 2000Sample

minus002

0

002

004

50 100 150 2000Sample

yest




















(50)


Input signal (u)

Output signal (y)

minus15minus1

minus050

051

15

minus20246

500 1000 15000Sample

500 1000 15000Sample


Error signal

minus202468

minus1minus05

005

1

200 400 600 800 10000Sample

200 400 600 800 10000Sample

y

yest








Table 5 20 dB Noise


Table 6 10 dB Noise


Table 7 5 dB Noise


Table 8 1 dB Noise





Error signal

4550556065

50 100 150 200 250 3000Sample

minus2minus1

012

50 100 150 200 250 3000Sample

y

yest














6 Conclusion







References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Input signal (u)

Output signal (y)

minus15minus1

minus050

051

15

minus20246

500 1000 15000Sample

500 1000 15000Sample


Error signal

minus202468

minus1minus05

005

1

200 400 600 800 10000Sample

200 400 600 800 10000Sample

y

yest








Table 5 20 dB Noise


Table 6 10 dB Noise


Table 7 5 dB Noise


Table 8 1 dB Noise





Error signal

4550556065

50 100 150 200 250 3000Sample

minus2minus1

012

50 100 150 200 250 3000Sample

y

yest














6 Conclusion







References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Error signal

4550556065

50 100 150 200 250 3000Sample

minus2minus1

012

50 100 150 200 250 3000Sample

y

yest














6 Conclusion







References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Documents

Robust Kernel Clustering Algorithm for Nonlinear System Identificationdownloads.hindawi.com/journals/mpe/2017/2427309.pdf · 2019-07-30 · ResearchArticle Robust Kernel Clustering