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2004 IEEE International Conference on Systems, Man a nd Cybernetics*A Fault Diagnosis Model Through G-K Fuzzy Clustering

Ning Lv Xiaoyang Yu J u n f e n g WuDepartment of Automation Department of Measurement-control Department of AutomationHarbin University of Science and and Instrument, Harbin University of Harbin University of Science andTechnology, Harbin, C hina Sci. and Tech, Harbin, China. Technology, Harbin, Chinaning-lv@sina.com yuxiaoyangok@163.com yjswjfl230@163.com

Abstract - A new method fo r ast building the knondedge-based fault diagno sis model by means of fuz zy clustering isproposed. The scheme is contrived by Gustafon-Kessel(GK) algorithm, which is of many goodproperties. In thispaper, it is jirst investigated how to integrate theproperlies of fault diagnosis systems into the GKclustering algorithm in the product space of input andoutput voriables. Then the way to convert the fizz y clustersto the fault diagnosis model is suggested. Hence, aneffic ient olgorithm to acquire the knowledge-based faultdiagnosis model fi om observations is worked out. As aresult, the obtained fault diagnosis m odel can ident% foultpatterns of different shape and orientotion in one data s et.Moreover, by introducing the concept of thejiury degreeo faultiness (DaF), the proposed approach seems to bemuch moreflex ible and with more power@ ability to dealwith data contominafed by noise compared with thetraditional fau lt diognasis method. Finolly, A n experimentof thefault diagnosis of a satellite power supply subsystemdemonstrates the effectiveness of the proposed fmltdiagnosis model.Keywords: Fault diagnosis, knowledge-based system,fuzzy lustering, degree of faultiness (DoF)1 Introduction

Due to the increasing complexity and riskiness ofmodem control systems, fault diagnosis has become animportant issue in modem automatic control theory, andduring the last two and a half decades, an immense amountof research has been done in this field resulting in a greatvariety of different methods with increasing acceptance inpractice [1,2]. The core of the fault-diagnostic methodologyis the so-called model-based approach, where either theanalytical or knowledge-based models, or combinations ofbath, are used in combination with analytical or heuristicreasoning [l]. However, in the case of fault diagnosis incomplex systems, one is faced with the problem that no, orno sufficiently accurate mathematical models are available.The use of knowledge-based models in the kamework ofdiagnosis expert systems is then the only feasible way.Nevertheless, the major difficulty in how ledge-mode l-hasedtechniques is the knowledg e acquisition, which is known asan extremely difficult task. To copy with this difficulty, some

soft computing techniques, such as fuzzy logic, neuralnetworks, and rough set theory are applied (3-51.The fuzzy expert system combines the expertise andexperience of the diagnosis experts with the diagn osis systemusing fuzzy $then rules so that it can tackle the uncertaintiesin fault diagnosis. Some people use the general fuzzycluster ing approach to bu ild the fuzzy relational models fromobservations [6,7], but few further researches that integratefuzzy clustering into fault diagnosis models have been doneso far despite of some work for rough data mod els done byHuang [8]. For the fault diagnosis models, the input variablesare usually continuous measures, while the output variablealways takes discrete values, e.g. some integers like 1,2, , .,that represent the corresponding fault types, e.g. fault I ,fault 2 , . Based on these properties, in this pap er, a novelapproach to knowledge-based fault diagnosis models issuggested through Gustafson-K essel (GK) fuzzy clustering[9] in the product space of input and output variables. Theresulting fault diagnosis model not only can identify faultpattems of different shape and orientation in one data set, butalso has a strong ability to deal with a v ariety of noises. Anexperiment of the fault diagnosis of a satellite power supplysubsystem demonstrates the effectiveness of the proposedfault diagnosis mode.The rest of the paper is organized as follows: Section2 presents the main idea of integration fuzzy clustering intofault diagnosis models after a brief review of Gustafson-Kessel (GK) fuzzy clustering algorithm. Section 3 desc ribesthe methodology of the proposed approach to building thefuzzy fault diagnosis model. Experimental results of thefault diagnosis of a satellite power supply subsystem areshown in Section 4, and final conclusions are discussed inSection 5 .2 Adaptive Fuzzy Clustering inCombination with Fault Diagnosis

Model2.1 Gustafson-Kessel (GK) algorithmFuzzy clustering is an important tool to identify thestructure in data. The Gustafson-Kessel (GK) algorithm [7],which is the fuzzy generalization of the Adaptive DistanceDynamic Clusters algorithm [7], searches for ellipsoidal

* 07803-8566-7104/$20.00 0 2004 IEEE.5114

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clusters. It can be used for linear or planar clusters becausethis type of cluster can be viewed as a special case ofellipsoids for which one or more radii are zero.In the GK algorithm, the distance from a point x k toa cluster prototype (center) vi is a squared inner-productdistance norm,

2DzhWj = l l X k - v i / l ~ ~( X k - v i ) T h f i ( X k - v ; ) (1)

where M i =det(Fi)'/" F;' is a positive-definite symmetricmatrix related to the covariance matrix Fi of the: ithprototype, and n is the dimension of the input-output productspace. A partition of data set {XI, ... , x N } into c fuzzyclusters is performed by minimizing the objective function

where U=[uik]s a fuzzy partition matrix, andZ u i k = l , l < k < N (3 )i= l

ui k ~[0 ,1 ] m e[ l , m ] is a weighting exponent whichdetermines the fuzziness of the resulting clusters, (for acrisp model m=l, fuzzy model m>l, but mo stly m=2). Thestationary points of the objective function (2) can be hundby adjoining the constraint (3) by means of Lagrangemultioliers:

(4)and by setting the gradients of 7 with respect to U,V and hto zero. If D& >O , V i , k and m>l, then (U,V) mayminimize (2) only if

(5)an d

An advantage of the GK algorithm over FCM is thatGK can detect clusters of different shape and orientation inon e data set. This is due to the fact that the eigenstruc tureof the cluster covariance matrix provid es information aboutthe shape and orientation of the cluster. The ratio of thelengths of the cluster's hyperellipsoid axes is give n by theratio of the square roots of the eigenvalues of F, . Thedirection s of the axes are given by the e igenvectors of F ; sshown in Figure 1. Linear subspaces of the data space arerepresented by flat hyperellipsoids, which can be seen ashyperplanes. The eigenvector Corresponding to the smallesteige nva lue determines the normal to the hyperplane , and can

be used to compute optimal local linear models h m hecovariance matrix.

F

Fig. 1 Equation ( ~ - v ) ~ F - ' ( x - v ) = Iefines ahyperellipsoid. The length of the jth axis of thishyperellipsoid is given by and its direction isspanned by@ j, where l, andq+j are the jth eigenvalueand the correspo nding eigenvector of F, respectively.

In sum, the GK algorithm may be formulated asfollows: given a set of data { x k I k = 1,2,...,N) , choose thenumber of clusters I W N , the weighting exponent m>land the termination tolerance E>O. Initialize the fuzzypartition matrix randomly, such that (3) holds. Repeat for/=1,2, ...Step 1: comp ute the cluster centers

Step 2: comp ute the cluster covarian ce matricesC ( U " : - " ) y X t - v f ) ) ( x i -v i 4 TN

, l < i < c, = *= I NE ( U p " ) "

D W g = ( x k - V : ' ) ) ~ M ~ ( X ~"!'I),

k=IStep 3: compute the distancesM i =det(Fi)""F,-I,

l < i

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Assume that x E R" , s the input data vector, y E R , sthe category data, i.e., the output data, which are someintegers. Denote Z, =[X, y, I T , k indicates the kth datapoint, and defme the type of fuzzy cluster Ci as thecomponent of the correspond ing cluster center

T

TV, = [Vi, .vi2 I . . . , Vi."+lI .Proposition I The fault diagnosis model of highquality means that the classification accuracy of everycluster is high. This is exhibited intuitively in clusteringresults that almost all the category values of the data pointsof cluster Ci are equal, and they nearly equal to the typecomponent of the cluster centerVi =[vi l ,v iz ,..., v ~ , , ~ , ] ~ ,.e., = y e . This leads to thefuzzy covariance matrix Fiof cluster Ciwith the followingform r. ... * 01

F . = (7)

where the last row and column of F, correspond to the typeIt shows that: (a) the covariances of fault categoryvariable y and the other data point components are nearlyzeros, i.e., CO~(X,J,)=0; (b) The variance of fault categoryvariab ley of cl usterC i is nearly zero, i.e.,

D ( ~ ~ ~ , ~ ) = c o v ( y , y ) = F ~ ( n + l , n + l ) o O8 )

of c,.

Membership unclion plots.:mi0 v1,n+1 vz.n*1 . v c , n+1Fig. 2 Membership function of C j n the fype vLml f faultdiagnosis model of high quality (i=l,2,.., )Thus, the Gaussian membership b c t i o n of clusters C i

(1 5 i 5 c) is some narrow-pulses with the ir centers equal tothe type components

Proposition 2 The fault diagnosis model of low qualitymeans that the classification accuracies of the most clustersare low. This is exhibited in clustering results that most faultcategory values of the data points of cluster Ci are of muchdifference, and they diverge from the type componentv ~ , ~ + ~f the cluster center Vi = [ v i l , iz ,...,

of clus ter centers (Fig.2).

TI greatly.

This implies that (a) the covariances of fault categoryvariable y and the other data point components of he samecluster Ciare usually some nonzero values, i.e. cov(x,y)>o,(b)The variance of fault category variable y of a clusterCi isa large value, i.e.,D ( ~ , , + ~ ) = c o v ( y , y ) =i ( n + l , n + l ) > O (9)

MembershiD undion dots

v1. +1 v2, +1 .. vc,n+1Fig. 3 Membership function of C , n the fype of faultdiagnosis model of low q uality in v2,,,+,

Therefore, the Gaussian membership function of clusterscorresponding to fault diagnosis model of low quality hassome flat curves whose centers diverge from their truecategory values (Fig.3).AAer clustering with GK algorithm in the productspace, we in fact obtain a set of fuzzy clusters Ci ,i=1,2 ._.,.In our approach, we set a tolerance vector, rolSig220,ER', where c is the number of the clus ters, for the varianceD(v.+J of the c cluster centers in the product-space. Onlythe clustering results, in which all the variances of thecluster centers satisfyare accepted for building the fault diagnosis model; otherwise,we should increase the number of clusters c and perform thefuzzy clustering algorithmonce more.

So, this fuzzy clustering in product space of input andoutput variables under the supervision of the resulting faultdiagnosis model quality is here referred to as thesupervised fuzzy clustering.

D(~ , ,~~)

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improved increasingly. When the number of clusteis ischosen equal to the number of fault pattem groups thatactually exist in the data, it can be expected that theclustering algorithm will identify them correctly, and theaccuracy of the fault diagnosis model becomes the bes t.3 Convert Fuzzy Clusters to FaultDiagnosis M odel

Numb er section and subsection headings consecutivelyin Arabic numbers and type them in bold. Use point size 14for section beadings and 12 for subsection beadings. Avoidusing oo many capital letters. Keep section and subsectionheadings always flushed left. If any further subdivision of asubsection is needed the titles should be 12 point. Thefuzzy cluster C , can be characterized by its center vector[v,], , ,] and variance vector [U : , . ,a i in the inputspace. If we assign the Gaussian type membership funi:tionto each cluster component, A , ( x y ) = expi = l , _ _ . ,, j = l , .., , kl,.., , X=[xi ... xJ, where thesemembership functions can be obtained by projecting clusterC , to its every dimension], then we will have the fiuzzyfault diagnosis rules in which each antecedent propositionis expressed as a logical combination of propositions withunivariate ficzzy sets defined for the individual comporientsofX , and usually in the following conjunctive form:R,: If xI is A,,(xl) and . and x. is A,(x,),

In this case, the degree offaultiness of a data X k ithrespect to the fault cluster C , , DoF8(X k ) , an be definedas the product o f the individual membership degrees in theprojecting space

then fau/f-ppe(C,)=c,.

DoF,(X,= 4 (XI (10)]=I

While in the Cartesian product-space, DoFX Xk issimply defined as the membership degree of themultidimensional fuzzy set C,

j= lwhere D;wj is the distance between xk and the center ofcluster Ci in GK algorithm (Ref. to Eq.(2)).Now, we can identify the type of a fault sample usingDoF by predefining a constant threshold vector TH. For anydata Dufu(9, if DoF,(Dufo(i)) -THO, j=1,2,..., , thenDufu(i) C, ; therwise Datu(i) e C , .The fml type o:fthe

data sample Do~o(J]s synthesized according to theDoF, (Duta(i)) =1,2,. ..,c as follows.

Cuse I: There exist one or more DoF, Dafo(i)), uchthat DoF,(Dufu(i)) >THG),j=I,Z ,..., .In this case, if the consequent values (i.e., , andu&+~f the cluster centers) of all these cluster rules arethe same or approximately equal to each other, then takethe cluster with the maximumDoF,(Dafa(i)) value as thecluster to which Dafu(i) belongs; otherwise, Do/u(i) istaken as the data point which can not b e identified.

Case2: There exist no DoF, (Dofo(i)),uch thatDoF,(Dafu(i)) T H ( J ) , =1,2 ,...,

This means that no faults have been detected, and thesystem may operate in normal state.4 Experiments Study

We consider the fault diagnosis of a satellite powersupply subsystem. Table 1 gives a measure data setgenerated by the fault simulator stand of the satellite powersupply subsystem. 15 samples are given, and the last columnis the fa ult type: 1-faults in power attenuation, 2 - aults inlinear shunt current controller, 3 -no faults.

Table 1 Data set for fault diagnosis of a satellite powers u u ~ l vubsvstem.No ICM Icnb Vdl Isc Fault

1 0.18 0.23. 30.5 0.32 12 0.17 0.21 33.6 0.32 13 2.97 2.81 31.6 0.32 24 2.15 1.86 30.8 0.32 35 0.16 0.19 34.3 0.32 16 0.18 0.22 29.7 0.29 3I 2.66 2.97 33.9 0.32 28 2.69 2.84 34.6 0.28 39 2.93 2.78 30.8 0.30 3I O 2.85 , 2.68 33.6 0.32 211 0.22 0.19 32.5 0.32 112 2.76 2.83 34.2 0.32 213 0.22 0.18 32.1 0.32 114 2.85 0.22 28.6 0.32 315 2.78 2.67 34.7 0.32 2

Applying ou r approach on the data set, feature Icnaand lcnb are selected to build the fault diagnosis model,and seven fuzzy fault pattem clusters are produced asshown in Table 2.So, according to Table2, the fault d iagnosis rules for thesatellite power supply subsystem are obtained as follows:IfX EC l or XE G, Thenfaulf-Qpe=l;IfXEC5, Then mm_tJpe=2;I fXECzorXEC~orXECsorXEC7,henfault-1p~3;where, X=[Icna, Icnb]. When setting foN=O.Ol andto/Sig2=0.01, the identifymg accuracy of the above fuzzyfault diagnosis model on the training data (Table 1) reaches100%. Another 15 data samples are generated by the fault

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simulator stand as a test data set. The valid identifjmgaccuracy of the obtained fault diagnosis model is 93.3% . Theexperimental results show that our method is effective.[3] Plamen P. Angelov, An ev olutio nary appro ach to fuzzyrule-based model synthesis using indices for rules.Fuzzy Sets and Systems, Val: 137, Issue: 3, pp. 325-338,2003141 S. Jakubek and T. Strasser, Neural networks applied toable 2 Fuzzv clusters for fault diarmosis model

[0.1800[0.2200[2.1885[2.8040[2.8190

5 Conclusions

IVariance VectorSig2: (Icna Icnb)14.4439e-09 7.1 102e -

[6.5718e-16 3.3270e-[6.2500e-40 6.2500e-[2.4810e-05 2.6913e-[1.1202e-04 1.482%[2.0859e-04 8.1480e-11.2969e-08 3.9074e-

_ .- automatic fault detection. The 2002 45th MidwestSymposium on Circuits and Systems, 2002.[5] E. L. Bona ldi, L. E. B. da Silva, G. Lambert-Torres, L.E. L. Oliveira, and F. 0. Assunco, Using rough setstechniques as a fault diagnosis classifier for inductionmotors. IEEE 2002 28th Annu al Con ference of theIndustrial Electronics Society, IECON 02. vo1.4,pp.3383- 3388,2002[6] P. A m , . M. Perronne, G. L. Gissinger, and P. M.

fault detection. Control Engineering Practice. Val: 9,Issue: 5, pp. 555-562,2001[7] Jinjie Huang, Sbiyong Li, Chuntao Man. A T-S Typeof Rough Fuzzv Controller Based on Process Inout-

FaultBP5 MWSCAS-2002. vol.l,639-64 2,2002

3132_. , Frank, Identification of fuzzy relational models for

In the field of fault diagnosis systems, there is a rapiddevelopment 60m the well-established but limited efficiencytraditional methods of signal-based fault diagnosis, towardsmodel-based approaches, using analytical an d or know ledge-based models. In this paper, a novel knowledge-model-basedapproach through Gustafson-Kessel (GK) fuzzy clusteringalgorithm is developed, which integrates the properties ofboth fault diagnosis systems and GK algorithm. Theproposed fault diagnosis model consists of a set of fuzzyrules, and as a result, it can detect fault patterns of differentshape and orientation in a data set. Moreover, by introducingthe concept of the fuzzy degree offmlrirress (DoF), theapproach seems to be much flex ible and with more powerfulability to handle the noise data compared with the traditionalfault diagnosis models. The proposed approach is really asofter technique to build the fault diagnosis model in asense.Acknowledgement

The authors thank to the su ppolts from National 973Science Research Plans of China (No. 2002cb312200-01-1)and the Provincial Natural Science Foundation ofHeilongjiang, China (F0202).References[l ] Paul M Frank, B irgit, Koppen-Seliger. New

Developments Using AI in Fault Diagnosis.Engineering App licatio ns of Artificial Intelligence, V a l10, Issue: 1, pp. 3-14, 1997[2] A. Siddique, G. S. Yadava, and B. Singh, Applicationsof artificial intelligence techniques for inductionmachine stator fault diagnostics: review. 4th IEEEIntemational Symposium on Diagnostics for ElectricMachines, Power Electronics and Drives, 2003.SDEMPED, pp. 29- 34,200 3.

.,output Data. Proceedings of the 42n d IEEE C onferenceon Decision and Control. Maui, Hawaii USA, pp.4729-4734.2003

[SI Jinjie Huang, Shiyong Li. A G A-based Approach toRough Data Model. The 5th World Congress onIntelligent Control and Automation (WCICA2004),Hangzhoy China, pp. 1880-1884,2004191 D. E. Gustafso n, and W. C. Kes sel, Fuzzy clusteringwith a fuzzy cova riance matrix. In: Proc. IEEE CD C,San Diego, CA, USA, pp. 76 1-766 ,1979

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