ControlEngineeringPractice14(2006)699717Aneuro-fuzzymultiple-modelobserverapproachtorobustfaultdiagnosisbasedontheDAMADICSbenchmarkproblemFaiselJ.Uppala,,RonJ.Pattona,MarcinWitczakbaControl&IntelligentSystemsEngineering,UniversityofHul,Hull,HU67RX,UKbInstituteofControlandComputationEngineering,UniversityofZielonaGora,ul.Podgorna50,65-246ZielonaGora,PolandReceived29October2004;accepted22April2005Availableonline3August2005AbstractThispaperpresentsanewframeworkforfaultdetectionandisolation(FDI)basedonneuro-fuzzymultiplemodellingtogetherwithrobust optimal de-couplingof observers. ThisnewparadigmiscalledtheNeuro-FuzzyandDe-couplingFault DiagnosisScheme (NFDFDS). Multiple operating points are taken care of through the NF modelling framework. The structure also providesresidualsthat arede-coupledtounknowninputs, makinguseof theearlierresearchonunknowninput de-coupling. TheNFparadigm exploits the combined abilities of neural networks and fuzzy logic and is an efcient modelling tool for non-linear dynamicsystems because of its approximation and reasoning capabilities. The paper also provides a comparative study of NFDFDS with theExtendedUnknownInputObserver(EUIO)forFDI,usingtheDAMADICSbenchmarkexample.r 2005ElsevierLtd.Allrightsreserved.Keywords:Fault detectionandisolation; Discretemultiple-model observers; Neuro-fuzzynetworks; DiscreteunknowninputobserversforFDI;Discretenon-linearobservers;RobustFDI1. IntroductionItiswell knownthatthesolutionstotherobustnessproblemoffaultdetectionandisolation(FDI)fornon-lineardynamicsystems dependuponreliablediscrimi-nationbetweentheeffectofuncertainmodelbehaviourandfaults (Chen&Patton, 1999). Most model-basedFDImethods(Chow&Willsky, 1984; Frank&Ding,1997; Gertler Ja nos, 1998; Isermann &Balle , 1997;Patton,Frank,&Clark,1989;Patton,Frank,&Clark,2000)relyonalinearstate-spacemodel ofthesystem.However, for non-linear systems, the standard approachistolinearisetheprocessmodel aroundtheoperatingpoint andmake use of model-basedmethods derivedfromlinearsystemstheory.However,linearisationdoesnot provide a good model for the processes withstrongly non-linear behaviour. The robust observerapproaches, particularlytheunknowninput observer(UIO) areundoubtedlythemost commonlyusedFDImethods. These linear systems methods can, tosomeextent compensate for model uncertainty thus increasingthe reliability of FDI. The model-reality mismatchisrepresentedbytheso-calledunknowninput. MostoftheworkontheUIOhasbeendirectedtowardslinearsystems. There are some observer-based approaches(Chen&Patton, 1999)forcertainclassesofnon-linearsystems but the systems that can be represented by thesenon-linearobserversarelimitedtoa fewstandard typesofnon-linearity(Ashton,Shields,&Daley,1999;Chen&Patton,1999;Frank&Ding, 1997;Frank&Seliger,1991;Hou&Pugh,1997;Thau1973).Furthermore, the non-linear observer approachcanbeusedonlywhenthenon-linearsystemsdynamicsareknownwithsufcientcondence; thisisrarelythecaseforreal systemapplications. ThusthemainmotivationARTICLEINPRESSwww.elsevier.com/locate/conengprac0967-0661/$ - seefrontmatter r 2005ElsevierLtd.Allrightsreserved.doi:10.1016/j.conengprac.2005.04.015Correspondingauthor.E-mailaddresses:f.uppal@hull.ac.uk(F.J.Uppal),r.j.patton@hull.ac.uk(R.J.Patton),m.witczak@issi.uz.zgora.pl(M.Witczak).forthisresearchistoexploretheaddedvalueofusingcomputational intelligence(CI)approaches(Jang, Sun,& Mizutani, 1997; Brown & Harris, 1994; Patton,Lopez-Toribio, &Uppal, 1999; Witczak &Korbicz,2002) asanextensiontothemodel-basedcounterpart,for cases of non-linearity in which the non-lineardynamicsareinsufcientlyknown. Inthiscontext, theneuro-fuzzy (NF) methods are known to overcome someof theproblems facedbythemodel-basedtechniques.NFmodels combine the approximationcapability ofneural-networkswiththereasoningoffuzzylogic. Thecombinationgivesrisetoapowerful fromofmultiple-model approximationthatisespeciallyattractivewhennon-linear systems areconsideredandfor whichonlyglobal modelling can be valid. The FDI problembecomesaspecialcaseofthismodellingapplication.TheproposedFDIscheme(Fig.1)comprisesabankof (M 1)NNFbasedde-couplingobserverswhereMisthenumberoffaultscenariosconsideredandNisthe number of operationpoints. It generates aset ofresiduals(r0k . . . rMk) at thekthsamplinginstant, intheform of a structured residual set (Chen & Patton, 1999),whilstthediagnosticlogicunitperformsananalysisofthe residuals to determine the nature and location of thefaults. Inthis scheme, eachresidual is designedtobesensitive to a subset of faults (or the residuals aredesignedtohavedifferentresponsetodifferentfaults).Ideally, eachresidual is sensitive toall but one fault,called a Generalized Residual Set (based on thegeneralizedobserverschemeofFrank(1987).Eachof theM 1Fault DiagnosisObservers isanon-linear systemcomprising Nlinear (integratedbyfuzzy fusion) sub-observers each one corresponding to adifferent operatingpoint of theprocess. Theinputstoeach observer are the process inputs and outputs, ukandyk, respectively, and the number of sub-observersdependsonthenumberof operatingpointsneededtoachieve required approximation. Their outputs arecombined by fuzzy fusion to generate the outputestimates.Computingthedifferencebetweentheactualandestimatedoutputsgeneratesasetofresiduals. Theset of fuzzyobservers together withthe NFmultiple-modelanddiagnosticlogicformthenewschemecalledtheNFDFDS.The NFDFDS is based on the identication of aparticular type of NFmodel (a simpler formof TS(Takagi Sugeno) (Takagi & Sugeno, 1985). A method togeneratelocallyoptimal de-coupledobserversautoma-tically fromthe NF model is proposed. The stepsinvolvedinthisschemeareshowninFig.2.Inthis proposedscheme, the computational limita-tions and stability constraints associated with theconventionalfuzzyobserver(Wang,Tanaka,&Grifn,1995;Tanaka,Takayuki,&Wang,1996)arealleviated.Thedesignof conventional fuzzyobserverinvolvesaniterative procedure, selecting eigenvalues arbitrarily,pole-placement andsolvinginequalities usingLMI tondapositivedeniteLyapunovmatrixcommontoallsub-models, which ensures stability of the global system.Ifsuchamatrixdoesnotexist,anothersetofarbitraryeigenvalues are selected and the whole procedure isrepeatedtill asolutionisfound. Thiskindof iterativeprocedureisnotrequiredintheproposedapproach.Asthe global state estimation is not necessary for FDIsimple sub-models are used that can easily andefcientlybe obtained. The removal of these stabilityconstraints simplies and speeds up the approach to thesolutionof identicationandFDIproblems. Sincetheglobal stateestimationproblemis avoided, thedesignprocedureoftheobserverissimplerandhencetheon-line computational burden is smaller. This speeds up theresidual processing and consequently on-line FDI tasks.Moreover, the reasoning and approximation capabilitiesoftheNFnetworkscanbeexploited.AnoutlineoftheNFDFDSapproachwaspresentedat IFACSymposiumSAFEPROCESS 2003 (Uppal,Patton,&Witczak,2003)andthispaperfocusesontheapplicationstudytotheelectro-pneumaticowcontrolvalve while giving an overview and summary ofNFDFDS. AcomparisonwithanotherFDIapproachusingthesameapplicationisalsopresented.Inparticular, acomparativestudyisperformedwiththefaultdetection schemethat isbasedon the so-calledextendedunknowninput observer (Witczak, Obucho-wicz, &Korbicz, 2002) and the state-space modelsdesigned with genetic programming (GP) (Korbicz,Kos cielny, Kwalaczuk, &Cholewa, 2004). The paperisorganisedasfollows.ARTICLEINPRESSNomenclaturek sampleinstante() expectationoperatoruk c Rrinputvectorxk c Rnstatevectoryk c Rmprocessoutputvectorymk c Rmmodeloutputvectorek c Rnstateestimationerrordk c Rqunknowninputvector,whereqpmEk c Rnq,Hkunknowninputdistributionmatrix,fk c RgfaultvectorF1k,F2kfault distribution matrices: matrices thatrepresenttheeffectoffaultsonthesystemw1k,w2kprocess(w1)andmeasurementnoise(w2)Qk,Rkcovariancematricesofw1andw2nd numberofdatasamplesF.J.Uppaletal./ControlEngineeringPractice14(2006)699717 700Section 2 summarises the formulation of the proposedFDIscheme. Theissuesregardingthemodel structure,determination of disturbance and fault distributionmatrices, global stabilityandobserverconvergencearediscussed. Section 3 presents the application studyconsidering NFDFDS for the DAMADICS valveactuatorbenchmarkproblem.Thebenchmarkproblemitself is describedindetail by Bartys, Patton, Syfert,Heras,andQuevedo(2005)intheleadingpaperofthisissue. The FDI task, determination andperformanceevaluation of the NF model and residual generation andisolation are summarised. Section 4 presents a compara-tive study regarding the application of the proposed NFobserverandanotherobserver-basedtechniqueappliedfor the DAMADICS benchmark problem. Section 5providesconcludingremarks.2. FormulationofNFDFDS(summary)2.1. ThemodelstructureTheNFmodel usedhereisaspecial formoftheTSfuzzy model (also referred to as multiple-model scheme)with linear models as fuzzy rule consequents. The modelcanberepresentedasavelayerneuralnetworkwhereeachlayercorrespondstoafunctionofthefuzzylogicinferencesystem(layer-1:input;layer-2:inputmember-shipfunctions; layer-3: rules; layer-4: output member-ship functions; layer-5: defuzzication). Thisrepresentationcanbeusedfortrainingortuningmodelparameters online. Theequivalent fuzzylogicdescrip-tionof the networkconsists of a set of rules inthefollowingform:Rule l : Ifua1 is m1;i . . . uana is mna;jthen gm = glm, (1)where ua= [ua1 . . . uana] c u is the antecedent input vector,na is the number of antecedents, m1;i . . . mna;jare theinput membership functions for the lth rule,ARTICLEINPRESS1O1MNyk0kr1krkrM+1ky0kyMky1ydiagnosisperformsfuzzyrulemost dominantregarding theinformationand theresidual setbased on theLogicDiagnosticSet of Residuals validity of each sub-observer parameters & fuzzy measure oflocal observerprovidesNF-ModelFault Diagnosis Observerseach row (0M) corresponds to a specific fault conditioneach column (1N) corresponds to an operating pointFuzzy Fusioneach block performs a weighted sumof estimates of i-thro w of observers,0>>=>>>>;. (11)Since ^ yk =PNl=1al^ ylk, where0palp1, it canbeseenthatLimkoXNl=1alylmk ^ ylk

oXNl=1alel; whereXNl=1alel= eoo,(12)LimkoXNl=1alylmk ^ ylk

oe; eoo. (13)Remark. . If the NF model approximates the non-linearsystem well (with local dynamic linear models) and if alllocalobserversconvergetothelocalNFmodels,itcanbe expected that the global observer output willconvergetothesystemoutput.ARTICLEINPRESSO11kE 2kENkEO2OMFig.3. Observeroutputspace.F.J.Uppaletal./ControlEngineeringPractice14(2006)699717 7033. Applicationstudy:FDIofthenon-linearvalveThis section presents the results of applying theNFDFDStothenon-linearsugarplantvalveactuatorproblem. This study is based on the DAMADICSvalvebenchmarkmodel (details of process andapplicationcan be found in the leading paper of this issue i.e. Bartysetal.(2005),Syfertetal.(2003)).3.1. TheFDItaskTheprocesstobemodelledhasfourinputsandtwooutputs.Theinputsoftheprocessarethecontrolvalue(u1 = cv), inlet pressure (u2 = P1), outlet pressure(u3 = P2) and temperature (u4 = T), while the outputsare the stemdisplacement of electro-pneumatic servomotor(y1 = Xsd)andliquidowthroughthevalve(F).Notethatforcomparisonpurpose(e.g.rise/falltimeofdifferent variables) y2 is taken as 1F instead of F, whichcan always be converted back if required. Table 1 showsthelistofthefaultsconsideredforFDItask.Thesearecategorised into three groups namely: control valvefaults, positioner faults andgeneral/external faults. Anon-linear SIMULINKmodel (testedagainst the realdata)isusedtogeneratethefaultdata.Thesefaults aredescribedindetail inDAMADICSbenchmark(Syfertetal., 2003).Theidenticationdataare collectedusingthe non-linear SIMULINKmodel(testedagainsttherealdata)oftheactuator.3.2. NFmodelstructureA 4-input 2-output NF model is constructed togenerate residuals (r1; . . . ; rM). The inputs of the NFmodel are u1, u2, u3and u4, while the outputs are y1andy2. For theantecedents DoubleGaussianmembershipfunctions are used with u1being the antecedent variable.The consequents are linear models with variables [y1, y2,u1, u2, u3, u4]. The antecedent variable was selected as u1as thenon-linear valvewas foundtoshownon-linearbehaviour only with respect to u1. In the proposedapproach, linear models integratedtogether by fuzzyfusionfordifferentvaluesofu1approximatethevalvebehaviour.The functionf1inEq. (14) was foundtobe morelinear (The function can be approximated well by alinearfunction.)andthereforeonesub-modelwasusedfor y1 over the whole antecedent input space whereas thefunction f2in Eq. (15) was found to be more non-linearandcouldnotbeapproximatedwell byjustonelinearrelation. (Notethatthetermfunction usedhereistorefertothenatureofdependenceoftheoutputsy1andy2ondifferentvariables.)Therefore,vesuitablelinearsub-models(i.e.N = 5)integratedbyfuzzy-fusionwereusedfory2.Notethatindicesi,j innui;jandnyi;jmeanthat nuandnyvalues areusedfor the ithconsideredvariableuorywhenthejthoutputisestimated.The initial structure, the number of rules (sub-models)andconsequent variables (without past values) of theNFmodelwerefoundexperimentally.Theconsequents(local models) are dynamic in nature as a result of usinglocal feedbackandpastvaluesofconsequentvariablesas shown by Eqs. (14) and (15). This structure wasdetermined based on a compromise between the numberof past values of consequent variables and errorreduction.^ y1k = f1(u1k1 u1knu1;1; u2k1 u2knu2;1 ; u3k1 u3knu3;1; u4k1 u4knu4;1 ; y1k1 y1kny1;1,y2k1 y2kny2;1), (14)^ y2k = f2(u1k1 u1knu1;2; u2k1 u2knu2;2 ; u3k1 u3knu3;3; u4k1 u4knu4;2 ; y1k1 y1kny1;2,y2k1 y2kny2;2). (15)The antecedent membership functions (MFs) em-ployed are of double Gaussian form, which is acombination of two Gaussian membership functionsthe rst one deningthe shape of the leftmost curvewhilethesecondonedenestheshapeoftherightmostcurve.ThiscanbedescribedasmDGMF(o) =exp (om1)22s1n o h i; \opm1;1; \m1ooom2;exp (om2)22s2n o h i; \oXm2;8>>>>>:(16)wheremDGMF(o)isthemembershipgradeforaninputo, m1andm2are the centres ands1ands2are thestandarddeviationsofthetwoGaussianfunctions.The antecedent MFs for the liquidowmodel aregiven in Fig. 4 as an example. These were obtained usingthe structure identicationtechniques as describedinBabuska(1998).ARTICLEINPRESSTable1FaultlistfortheFDItaskFaultscenariosControlvalvefaultsF1:ValvecloggingF2:ValveplugorvalveseatsedimentationF3:ValveplugorvalveseaterosionF6:Internalleakage(valvetightness)F7:MediumevaporationorcriticalowPositionerfaultsF13:RoddisplacementsensorfaultGeneralfaults/externalfaultsF17:UnexpectedpressurechangeacrossthevalveF18:FullyorpartlyopenedbypassvalvesF19:FlowratesensorfaultF.J.Uppaletal./ControlEngineeringPractice14(2006)699717 704The identied local linear models (consequents) are inthefollowingform:ylmk =ylm1kylm2k2435 =P1Pl22435ylm1k1ylm1k2ylm2k1 . . .ylm2k2u1k1u1k2u2k1 ::: u2k2u3k1u4k11T,(17)whereylmkistheoutputvectorwithylm1kandylm2kbeingthemodel output estimatesforstemdisplacement andliquidow,P1andPl2are the consequent parametervectorsfory1andy2, [ylm1k1ylm1k2ylm2k1ylm2k2u1k1u1k2 . . . u2k1u2k2u3k1u4k1 1] is the set of conse-quent variables (and their past values) at kth instant andl denotesthesub-model.NotethatP1\l = 1; . . . ; Nis sameas y1is approxi-matedbyalinearfunctionwhereasy2isapproximatedby a non-linear function which is a convex combinationof N-linear functions with parametersPl2wherel = 1; . . . ; N.3.3. PerformanceoftheidentiedmodelModel performance was evaluated with simulated andreal data from the plant for normal operation. DifferentdatasetsusedforthisaresummarisedinTable2. Theperformance of the model was measured (with fault freedata) usingthemean-squared-errorMSEperformancefunction MSE = 1=(nd 1)Pnd1k=0 (yk ^ yk)T(yk ^ yk),where^ yk = ymk.Figs. 5 and 6 showthe model performance withData-1.TheMSEforthetwooutputswasfoundtobe2.56 104and 7.20 104respectively for the normal-isedsetofdata.Thedifferencebetweentheprocessandmodel output values (error) canbe attributedtoun-modelleddynamics andnoise, it canbeseenthat theerrorsforbothstemdisplacementandowaresmallinmostoftheoperatingspace.Figs. 7and8showmodel performancewithData-2.The MSEfor both outputs was higher (5.53 104,9.09 103) as comparedwiththe previous result. ItARTICLEINPRESS1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0Antecedent variable (control value)00.10.20.30.4membership grade0.50.60.70.80.91Antecedent membership functionsMF1MF2MF3MF4MF6Fig.4. Antecedentmembershipfunctions.Table2DatasetsusedforperformanceevaluationData-1 SimulatedtrainingdatawithmultiData-2 SimulatedtestingdatawithsinusoidalinputData-3 SimulatedtestingdatawithinputshiftedinstantaneouslyfromzerotodifferentvaluesData-4 One-hourrealdatafromplantcollectedon11thOctober2001(fordetailsseeSyfert,Bartys,QuevedoandPatton,2003)2000 1800 1600 1400 1200 1000 800 600 400 200 00Data samples (sample time = 1s)0.10.20.30.4measurement amplitude0.50.60.70.80.91real value: solid line (-)estimated value: dotted line (--)Stem displacement, MSE = 2.56exp-4Fig.5. NFmodelperformanceforstemdisplacement(Data-1).2000 1800 1600 1400 1200 1000 800 600 400 200 0Data samples (sample time = 1s)00.10.20.30.40.50.60.70.80.91measurement amplitudeLiquid flow, MES = 7.20exp-4real value: solid line (-)estimeted value: dotted line (--)Fig.6. NFmodelperformanceforliquidow(Data-1).F.J.Uppaletal./ControlEngineeringPractice14(2006)699717 705was also observed that the error is large when the ow isdecreasedfromitsmaximumvalue. Thisisduetothebacklasharisingfromthevalve-stictionwhenthevalveisfullyopenedandcanbeignoredasacontrolvalveisoperated to regulate the ow and not usually operated inafullyopenedstate.Figs.9 and 10show modelperformancewith Data-3.ItwasobservedthattheNFmodelhasalargeerroratthesesuddenchangesbuttheerrorconvergestozeroinashort time. Inthis case, theMSEset was foundas[7.49 104, 2.11 103]. The higher MSE is due to theun-modelled dynamics that can be ignored as thesesudden changes rarely occur in real system applications.Figs. 11 and12 showthe performance of the NFmodel using Data-4. The MSE set was found as[8.0 104,1.29 102].ThehighervaluesoferrorforData-4(realdatafromplant) canbeattributedprimarilytothehighapprox-imationerror of thesimulationmodel (70.1forbothstemposition and liquid ow), which was used togenerate the training data for the NF model. Moreover,ARTICLEINPRESS1500 1000 500Data samples (sample time = 1s)000.10.20.30.40.50.60.70.80.91measurement amplitudeStem displacement, MSE = 5.53exp-4real value: solid line (-)estimated value: dotted line (--)Fig.7. NFmodelperformanceforstemdisplacement(Data-2).1500 1000 500Data samples (sample time = 1s)000.10.20.30.40.50.60.70.80.91measurement amplitudeLiquid Flow, MSE = 9.09exp-3real value: solid line (-) estimated value: dotted line (--)Fig.8. NFmodelperformanceforliquidow(Data-2).2500 2000 1500 1000Data samples (sample time = 1s)500 000.10.20.30.40.50.60.70.80.91measurement amplitudeStem displacement, MSE = 7.49exp-4real value: solid line (-)estimated value: dotted line (--)Fig.9. NFmodelperformanceforstemdisplacement(Data-3).2500 2000 1500 1000Data samples (sample time = 1s)500 000.10.20.30.40.50.60.70.80.91measurement amplitudeLiquid flow, MSE = 2.11exp-3real value: solid line (-)estimated value: dotted line (--)Fig.10. NFmodelperformanceforliquidow(Data-3).3500 3000 2500 2000 1500 1000 500 00Data samples (sample time = 1s)0.10.20.30.40.50.60.70.8MagnitudeStem position, MSE = 8.0exp-4real value: solid line (-)estimated value: dotted line (--)Fig.11. NFmodelperformanceforstemdisplacement(Data-4).F.J.Uppaletal./ControlEngineeringPractice14(2006)699717 706therewas slight differenceintheoperatingconditionsthat were used for modelling as compared to theones observed in Data-4. Model performances withthese tests indicate that the model has reasonableapproximation.3.4. StatespacerepresentationofthelocalmodelsThe local models were rst converted to an auto-regressive exogenous (ARX) MIMOformand thenconverted to the state space formfor computationalconvenience. It was veried that the local modelsobtainedwerestable.Anexemplarysub-modelisgiveninAppendixA.3.5. FaultanddisturbancedistributionmatricesThe fault distributionmatrices for each sub-modelwere obtainedfromthestatespace descriptionof themodels as describedinSection2. Basedonthe statespacerepresentationof thelocal models andthefaultanddisturbancedistributionmatrices,thedesignofde-coupling observers is straightforward as described inChen and Patton (1999), Uppal et al. (2003). Anexampleof asub-model withfault distributionmatrixisgiveninAppendixA.3.6. ResidualgenerationandfaultisolationTheNFDFDS,basedon the NFmodel,localmodelsinthestatespaceformandthede-couplingobservers,was applied to the control valve (based on theDAMADICS benchmark problem (Bartys et al.,2005)) inorder togenerateaset of tenresiduals (onefor the fault-free case). These residuals were analysed toisolateninesimulatedfaults(Table1)inthesystem.This fault diagnosis scheme consists of collectinginformationfromtheresiduals(r0; . . . ; rM) intermsoffault effect at different operating points (M = 9 in thisstudyas9faultswereconsidered).AnexampleinFig.13showsr0inthepresenceofF1(Table1) withoutde-coupling (thedisturbancedistribu-tion matrix E0, being a null matrix of appropriatedimensions). Mediumfaults (fault strength 0.5) wereanalysed(seeFig.15)inthiscase.Itwasobservedthatall the faults have a signicant effect on the residual as aresultofnotusingde-coupling.The vertical lines inthe gure distinguishbetweendifferent operating points according to the mostdominantfuzzyruleinthatregion.Themostdominantrule inafuzzysystemis denedas the one withthemaximumringstrength.In this study, a constant threshold value of 0.1, basedon visual analysis of model residuals, was used althoughsomemoresophisticatedmethods canalsobeappliedfor enhancing the isolation. Note that the residual ismainly affected inoperating pointsfourand ve(OP-4,OP-5). The effect in OP-2 when the valve is being closedcanbeattributedtoun-modelleddynamics. Moreover,itseemsthattheresidual tendsto0andthereforethiseffect was ignored. It is interesting to note that the faulthas different effect at different operating points, asexpected for a non-linear system. Therefore thisinformationcanbecollectedfromthisresidualasasetof operating points at which it exceeds the threshold i.e.[4,5]andthisisenteredinTable3forF1andR0.Asanotherexample,Fig.14,showsresponseofr0inthe presence of medium fault F6. It can be seen that theresidualexceedsthe thresholdinoperating pointsthree,fourandve(OP-3,OP-4andOP-5).Thiscorrespondsto the Table 3 entry for F6 and R0 (note that capital R isusedheretodenotethesymptomsassociatedwiththeresidualsratherthanresidualitself).The fault isolation tables (Tables 35) were completedasdescribedinaboveexamples. Eachcolumninthesetables constitutes a fault symptom. These symptomsARTICLEINPRESS3500 3000 2500 2000 1500 1000 500 00Data samples (sample time = 1s)Magnitudereal value: solid line (-)estimated value: dotted line (--)0.511.5Liquid flow, MSE = 1.29exp-2Fig.12. NFmodelperformanceforliquidow(Data-4).2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 420000.20.40.60.811.21.4Data samples (sample time =1s)residual amplitudeOP-1 OP-2OP-3OP-4OP-5OP-4OP-3OP-2OP-1Threshold Linesdistinguishing the operatingpoints (OP)Fig.13. Responseofresidualr0inpresenceofmediumfaultF1.F.J.Uppaletal./ControlEngineeringPractice14(2006)699717 707werecollectedformedium,intermediate(betweensmall& medium)and smallsizefaults,for faultstrengths 0.5,0.4and0.25, respectively. Thesestrengthsarestandar-dised to a range [0,1] where the limiting values 0 and 1correspond to some pre-dened states or physical values(fsmin, fsmax) as shown in Fig. 15. For more details on thephysicalinterpretationoffaultstrengths,pleaserefertoBartysetal.(2005),Syfertetal.(2003).Itcanbeseenthateachfaulthasauniquesignatureand therefore it can be isolated. Indeed, as can be seen inTable3, residualsR0toR9provideuniquesymptomsfor eachof thefaults,and hence effectivefault isolationispossible.Inotherwords,thesymptomsareperceivedhere as a reaction of the residuals R0 to R9 to theparticularfault, comparee.g. thesecondandthesixthcolumnofTable3.Ontheotherhand, it canbeseenthat thediagonalentriesofTable3areclosetozero.Thisshowsthatthedecoupling quality is very good and the proposedapproaches permits robust fault detectionandconse-quentlyfaultisolation.It can also be seen fromthe Tables 35 that thesymptoms aresimilar for various fault strengths. Thiswasexpectedasaresultofusingde-couplingwithfaultdistribution directions. The symbol k in the tablesdenotes that residual is non-zeroonlyfor valvebeingclosed. The bold entries show that the residual isdesignedtobezero(ideally), emptyentries meanthatthe residual is close to zero and the symbol shows thediagonal entries(residual closetozero). Itcanbeseenthat eachfault hasauniquesignatureandthereforeitcanbeisolated.3.7. SummaryofresultsusingNFDFDSapproachFault diagnosis innon-linear systems is challengingprimarily because the faults may have different effects atdifferent operating points. The previous section pre-sented the application of NFDFDS to the DAMADICSactuator valve benchmark problem, for nine realisticactuatorfaults,listedinTable1.Fault symptoms were initially obtained using the datawithmediumfaults(seeTable3).Theresultsshowthatall nine faults were isolated as they have differentsymptoms. Aset of different inputoutput data withintermediateandsmallfaults(Tables4and5)wasalsoapplied to verify the FDI scheme and it was seen that thesymptoms obtained with thesedata were very similar tothatofobtainedinTable3,whichmadefaultisolationpossible.As aresult of de-coupling, it canbe expectedthatfaultswithintherangeoffaultstrengthsusedinabovethreecases canbeisolated. For verysmall faults, theisolationbecomes difcult as the noise level becomescomparable to the fault effect. However, for larger faultsisolationcanbe expectedtobe relativelyeasier. It isinteresting to note that although each residual isdesignedideallytobeinsensitivetoaparticular fault,in practice the residual is insensitive to a subset of faultsasseenintheaboveresults.ARTICLEINPRESS1.28 1.3 1.32 1.34 1.36 1.38 1.4 1.42 1.44x 104-0.100.10.20.30.40.50.60.70.8OP-1OP-2OP-3OP-4OP-5 OP-4OP-3OP-2OP-1residual apmlitude Data samples (sample time =1s)Threshold Fig. 14. Response of residual r0in presence of F6 (faultstrength = 0.5).Table3SymptomsformediumfaultsF1F2F3F6F7F13F17F18F19R04,5 2,3,4 3 3,4,5 41,2,3,4,5 3,4,5 1,2,3,4,5 3,4,5 1,2,3,4R12k 2 1,2,3,4,5 4 1,2,3,5 1,2R25 5 1,2,5 5 1,2,3,4,5 5 1R34,5 2 1,2,3,4,5 1,2,3,4,5 1,2R44,5 2 1,2,3,4,5 1,2,3,4,5 1,2R55 5 2,5(small) 2,3,4,5 5R64,5 2 1,2,3,4,5 1,2,3,4,5 1,2R71,2,3,4,5 4 4 1R84,5 2,5 1,2,3,4,5 1,2,3,4,5 1,2R94,5 2,4 4 1,2,3,4 4 1,2,3,4,5 4 2,4(small)F.J.Uppaletal./ControlEngineeringPractice14(2006)699717 7084. Acomparativestudywithanotherobserver-basedtechniqueThe main purpose of this section is to performacomparative study regarding the application of theproposed NF observer and another observer-basedtechnique(Witczak&Korbicz, 2004; Witczak, Patton,&Korbicz, 2003)appliedfortheDAMADICSbench-mark problem. The latter technique is based on thestate-space model developedwithGP(Witczaket al.,2002) and the so-called extended unknown inputobserver (EUIO) (Witczak&Korbicz, 2004; Witczaket al., 2003). In order to make this paper self-contained,ashortoutlineoftheGPandtheEUIOisgiveninthesequel.4.1. Designofstate-spacemodelswithGPGP belongs to the class of the stochastic optimisationtechniques that are widely known as evolutionaryalgorithms (EAs) (Korbicz et al., 2004). EAs areinspired by some biological processes, which allowpopulationsoforganismstoadapttotheirsurroundingenvironment. SuchalgorithmshavebeeninuencedbyDarwinstheoryofnatural selection, orthesurvival ofthe ttest. In the GP algorithm, an individual isrepresented by a tree. The nodes of such a tree areformedwith the so-called terminal andfunctionsets(Koza, 1992). The process of the evolution starts from arandomly generated population of trees. These trees can,of course, by createdinmany different ways (Koza,1992) and hence they may constitute various initialsolutionsto theproblembeingconsidered.ThepurposeARTICLEINPRESSTable4Symptomsforintermediate(betweensmall&medium)faultsF1F2F3F6F7F13F17F18F19R04,5 2,3,4 3 3,4,5 1,2,3,4,5 3,4,5 1,2,3,4,5 3,4,5 1,2,3,4R124k 2 1,2,3,4,5 4 1,2,3,5 1,2R25 5 1,2,5 5 1,2,3,4,5 5 1R34,5 2 1,2,3,4,5 1,2,3,4,5 1,2R44,5 2 1,2,3,4,5 1,2,3,4,5 1,2R55 5 2,5(small) 2,3,4,5 5R64,5 2 1,2,3,4,5 1,2,3,4,5 1,2R71,2,3,4,5 4 4 1R84,5 2,5 1,2,3,4,5 1,2,3,4,5 1,2R94,5 2,4 4 1,2,3,4 4 1,2,3,4,5 4 2,4(small)Table5SymptomsforsmallfaultsF1F2F3F6F7F13F17F18F19R05 2,3 3 3,4,5 1,2,3,4,5 3,4,5 1,2,3,4,5 3,4,5 1,2,3R12k 2 1,2,3,4,5 4 1,2,3 1,2R25 5 1,2,5 5 1,2,3,4,5 5 1R34,5 2 1,2,3,4,5 1,2,3,4,5 1,2R44,5 2 1,2,3,4,5 1,2,3,4,5 1,2R55 5 2,5(small) 2,3,4,5R65 2 1,2,3 1,2,3,4,5 1,2R71,2,3,4 4 4 1R85 2,5 1,2,3,4,5 1,2,3,4,5 1,2R9 2,4 4 1,2,3,4,5 4 1,2,3,4,5 4 20 0.25 0.50.75 1.0fsmin fsmaxNormalised valuesmall medium bigFault strengthPhysical valueFig.15. Faultstrengthinterpretation.F.J.Uppaletal./ControlEngineeringPractice14(2006)699717 709of the GP algorithm is to evolve a population of trees insuchawaysoastogetanoptimal (inthesenseofthecriterionchosen) solution. Inorder torealise suchachallenging task, the GP algorithm repeats the followingsteps: application of genetic operators (usually crossoverand mutation), tness calculation (according to thepredenedcriterion), andselection. Adetaileddescrip-tionoftheGPalgorithmcanbefoundinmanyjournalpapers and monographs, e.g. Korbicz et al. (2004), Koza(1992), Gray, Murray-Smith, Li, Sharman, andWien-brenner(1998),Witczaketal.(2002).SinceanoutlineoftheGPalgorithmisgiven, letusconsider thenon-linear discretetime systemdescribedbythefollowingequations:xk1 = A(xk)xk h(uk) w1k,yk1 = Cxk1 w2k1, (18)ThediagonalmatrixA()andthevectorfunctionh()consistofnon-linearfunctions. Anapproximatemodelof(18)canbe^ xk1 = A( ^ xk) ^ xk h(uk),^ yk1 = C ^ xk1, (19)where ^ xkstandsforthestateestimate. InWitczakandKorbicz(2004) andWitczaket al. (2002), it isproventhat themodel (19) isgloballyasymptoticallystableifmax[ai;i[o1, whereai;iisadiagonal entryofA(). Thisimplies the following structure of these diagonal entries:ai;i( ^ xk) = tanh(si;i( ^ xk)); i = 1; :::; n, (20)wheretanh( ) stands for thehyperbolictangent func-tion, and si;I() denotes a non-linear function that can beobtained with the GP algorithm. In the case of applyingthe GP algorithm to design the model (19), each entry ofA()andh()shouldberepresentedbyapopulationoftrees. Thedetails regardinganimplementationof thistechniquecanbefoundinWitczakandKorbicz(2004),Witczaketal.(2002).4.2. DesignofanextendedunknowninputobserverLet us consider the following class of non-linearsystems:xk1 = g(xk) h(uk F1;kfk) Ekdk,yk1 = Ck1xk1 F2;k1fk1, (21)where g() stands for anon-linear function, fkis thefault,dkisanunknowninput,F1;k,F2;k,Ekareknowndistribution matrices of faults and unknown input,respectively. Thestructureof anobserverdedicatedtothestateestimationandresidual generationof (21) isgivenby^ xk1=k = g(xk) h(uk),^ xk1 =^ xk1=k Hk1ek1=k K1;k1ek, (22)where ek1=k = yk1 Ck1 ^ xk1=k, and the residual isek = yk ^ yk.ThetaskofdesigninganobserverreducestotheproblemsofobtainingthematrixK1;k,insuchaway so as to minimise the difference between^ xkand xk,as well as the matrix Hkin such a way so as to decouplethe effect of anunknowninput onthe residual. Thedetails regarding an implementation of this observer canbefound inWitczakandKorbicz(2004),Witczaketal.(2002).Themaindrawbackrelatedwiththeapplicationoftheobserver(22)tothesystem(21)isthatit requiresthe knowledge regarding the covariance matrices QkandRkof process noise w1kandmeasurement noise w2k,respectively. The problemis that the system(21) isdeterministic whichmeans that QkandRkshouldbezero matrices. Unfortunately, such a setting of thesematrices mayleadvarious computational problems aswell as tothedivergenceof theobserver. Inorder toovercomethesedifculties, theLyapunov approachcanbeutilisedtoestablishtheconvergenceconditionoftheobserver (Witczak &Korbicz, 2004; Witczak et al.,2002).Asaresult,itcanbeshownthatthematricesQkandRkshouldbesetasfollows:Qk = b1eTkekI d1I; Rk = b2eTkekI d2I, (23)where I stands for the identity matrix, b1, b2aresufciently large positive constants and d1, d2aresufcientlysmallpositiveconstants.Theaboveparametershavetobeselectedempiricallyastheyareproblem-dependent.Ontheotherhand,thevaluesarenotcriticalandthealgorithmisnotsensitiveforsmallchangesofthem.4.3. DevelopmentandtestingofthefaultdiagnosisschemeThe rst step to develop the fault detection scheme forthe DAMADICS benchmark is to obtain the state-spacemodel with the GP approach. For the sake ofcomparisonwithnon-linearstate-spacemodeldesignedwith GP, the linear state-space model was obtained withtheuseofMATLABSystemIdenticationToolbox.Inparticular, the N4SID identication procedure wasemployed for that purpose. In both the linear andnon-linear cases the order of the model was testedbetween2and8. Unfortunately, therelationbetweenthe input (control value) and liquid owcannot bemodelled by a linear state-space model. Indeed, themodellingerror was approximately35%, thus makingthelinearmodelnotacceptable.Ontheotherhand,therelationbetweentheinput(controlvalue)andthestemdisplacementcan accurately be modelled with the linearstate-spacemodel. Bearingthisinmind, theidentica-tionprocesswasdecomposedintotwophases,i.e.:+derivation of a relation between rod displacementandtheinputwithalinearstate-spacemodel;ARTICLEINPRESSF.J.Uppaletal./ControlEngineeringPractice14(2006)699717 710+derivation of a relation between juice ow and the inputwithanon-linearstate-spacemodeldesignedbyGP.Experimental results showed that the best-suitedlinear model is that of thesecondorder. After the50runs of the GPalgorithmperformedfor eachmodelorder, it was foundthat the order of the model thatprovides thebest approximationqualityis equal to2.The mean-squared output error for this model was7.9 103. Acomparisonbetweentheresponseof themodelandthesystemisgiveninFig.16.The maindifferences betweenthe behaviour of themodel and the system can be observed for the non-linearmodel(juiceow)duringthesaturationofthevalve.Inorder toovercomethis modellinginaccuracy, anextended unknown input observer, described earlier,was employed. An estimation of an unknown inputdistribution matrix Ekconstitutes a very important partof theobserver designprocedure. It is knownthat toobtain this matrix it is necessary to estimate anunknowninput.In a general case, the problemof unknown inputestimation can be perceived as a non-linear optimisationtaskformulatedasfollows:^d+k = argmin|{z}^dc+qeTk1ek1. (24)ItcanbeshownthatsolvingEq.(24)isequivalenttosolvingthefollowingsetoflinearequations:Ck1 ^dk = ek1=k. (25)Itshouldbeclearlyunderlinedthatanexactsolutionof (25) canbe obtainedif the rankof the matrixCkequals n. Unfortunately, this condition is usuallyimpossibletoattain. Eq. (25) canbesolvedefcientlyusing an orthogonal factorization of the matrix (Lawson& Hanson, 1974). Note that the condition rank(Ck) = mcan be relaxed and a singular value decomposition(SVD)ofmatrixisthenusedinsteadofitsorthogonalfactorization. Moredetailsabout thesetechniquescanbe foundin(Lawson&Hanson, 1974). Knowinganestimateofdk, k = 1; . . . ; nd, thetaskofobtainingthematrices Ekand Hkcan be solved according to theapproachdescribedinChenandPatton(1999).Know-ing the matrix Hkit is possible to determine theconstants related with Eq. (23). Aparameter choicefortheDAMADICSbenchmarkunderconsiderationisb1 = 10, b2 = 103and d1 = d2 = 0:01. The values ofthese parameters were obtained empirically by manycomputersimulationsandtheywereselectedinsuchawaysoastoensuretheconvergenceoftheobserveraswell as to provide its fast convergence. For suchobserver parameters, the mean-square error was reducedfrom7.9 103to 2.2 103. This increase in themodeling quality can also be observed by comparing theresultsshowninFigs.16and17.Since a robust residual generator is available, theremainingproblemboilsdowntodeterminingathresh-oldfor the purpose of fault detection. Totackle thisproblemaconstantthresholdcanbeusedbutfromtheauthors experience it is known that better results can beobtained with the technique of a simple adaptiveARTICLEINPRESSFlowDiscrete time Discrete timeRod displacementFig.16. System(dotted)andmodel(solid)outputs(ow[left],roddisplacement[right]).F.J.Uppaletal./ControlEngineeringPractice14(2006)699717 711polynomial threshold described in Witczak and Korbicz(2004).Table 6 shows the results of fault detection for a set ofabrupt faults (Bartys et al., 2005). It canbeseenthatthere are alsosome problems with fewsmall and/ormediumfaults. However, itshouldbepointedoutthatall faults, which are considered as large can be detected.Undoubtedly, a further improvement in fault detect-abilitycanbe achievedbyintroducingmore sophisti-cateddecisionmethods.4.4. ComparisonSincethesimulationstudyhasbeenperformedusingboth the techniques presented in this paper, i.e.NFDFDSandEUIOwithGP, acomparisoncanbeperformed.Inordertodothis,appropriatecriteria thatexhibit the drawbacks and advantages of the approachesare established. Each of these criteria is accompanied bya discussion regarding the performance of the ap-proaches being considered. The criteria being consideredareasfollows:4.4.1. DifcultyindesigningthemodelsNFDFDS: The approach requires designing a NFmodel resulting in a number of linear state-space modelsthat canrelativelyeasilybeobtained, i.e. thedevelop-ment of the software that has to be used for that purposeis not complex and some existing routines can beadopted.EUIOwithGP: Theapproachrequiresanon-linearstate-spacemodel obtainedwiththeGPapproach, i.e.the software requiredis verycomplexandspecializedandhenceitsappropriatedevelopmentisveryhard.Superiorapproach:NFDFDS.4.4.2. ComputationaldesigncomplexityNFDFDS: The design complexity is adequatelyillustratedbyFig. 2. Althoughthe number of designsteps is relatively large, each of the steps does notinvolvealargecomputationalburden.EUIO with GP: The design complexity is very hard asit involves the GP algorithmthat is extremely timeconsuming (see (Koza, 1992) for a comprehensiveexplanation), i.e. the computation of an adequate modelARTICLEINPRESSFlowDiscrete time Discrete timeRod displacementFig.17. System(dotted)andobserver(solid)outputs(ow[left],roddisplacement[right]).Table6Resultsoffaultdetection(Ddetectable,N)Fault F1F2F7F13F17F18F19Small D D D D DMedium D D D D DF.J.Uppaletal./ControlEngineeringPractice14(2006)699717 712may even take a fewdays on a modern computers.Moreover, theresultingmodel requires further simpli-cation with the help of a symbolic computationpackage like MAPLE or MATHEMATICA. The designof EUIO is easy as it is designed for the nominal systemoperatingmodecontrarytoNFDFDS.Superiorapproach:NFDFDS.4.4.3. Free parameters that has to established by the userNFDFDS:Themainparametersthatmustbesetbytheusercanbecategorisedasfollows.+Structureidentication: Initial numberof sub-mod-els, sub-model orders, coefcient of fuzziness andinputs.+Transformation into simplied TS NF model: Outputerrorandcomplexitytolerance.+Designoflocal optimal decouplingobservers: Para-metersregardingtheinstrumentalmatricesQandR,initialcovariancematrixP,initialstate.+Online collection of symptoms using NFDFDS:Residualthreshold.Aprecisenumberoffreeparametersthatmustbesetbytheuserdependsontheapplication.EUIOwithGP: Thefreeparameterscanbedividedintotwodistinctgroups,i.e.:+TheparametersregardingtheGPalgorithm: Cross-over and mutation probabilities, population size, typeof initial model creation, avariance for parameterestimation.(subtotalnumber:5).+The parameters regarding the EUIO: Parametersregarding the instrumental matrices Q and R (two foreach of the matrices), initial covariance matrix P,initialstateestimate(subtotalnumber:6).Thus, thetotal numberoffreeparametersthatmustbesetbytheuseris11.Superiorapproach:NFDFDS.4.4.4. DatasetsrequiredNFDFDS: Theapproachrequiresdatasetsforbothnormal andfaultybehavior of the system. Moreover,the data sets contain the subsets that correspond todifferent operatingmodes of the systems. Please notethat the data for faulty operating mode may not beavailableinpractice, andthisisthemaindrawbackoftheapproach.EUIO with GP: A data set for normal operating modeof thesystemis required. Suchadataset is availablewithoutanyproblems.Superiorapproach:EUIOwithGP.4.4.5. Requirements regardingcomputational power foron-lineapplicationsNFDFDS: The maincontributiontothe computa-tional complexityisduetotherelativelylargedescrip-tion of the system (8). Indeed, the size of the matrices (8)is large and it requires a large amount of computermemory to be implemented. On the other hand, some ofthe matrices (8) are sparse and hence effective techniquesforsparsematricescanbeusedwhileimplementingtheapproach in practice.Undoubtedly, such techniques areprotable since the on-line computation of localobserversrequiresmatrixinversion.EUIO with GP: An on-line computation regarding theapproachis verysimple andthe maincomputationalburdenis duetothematrixinversionrequiredbytheobserver.Superiorapproach:EUIOwithGP.4.4.6. RobustnesstomodeluncertaintyNFDFDS: Robustness tomodel uncertaintyis rea-lizedbyemployingthe concept of anunknowninputresulting ina number of unknowninput distributionmatricesdesignedfordifferent operatingmodesof thesystem.EUIO with GP: Similarly, robustness to modeluncertaintyisrealizedbyemployingtheconcept of anunknown input resulting in one disturbance distributionmatrix. ContrarytoNFDFDS, the maindrawbackisthat theunknowninput maychangeits behavior andhenceasingledisturbancedistributionmatrixisnotasgoodasasetofsuchamatricesdesignedfordifferentoperatingmodes.Superiorapproach:NFDFDS.4.4.7. ReliabilityoffaultdiagnosisinthecontextoftheDAMADICSbenchmarkThe comparison with respect to fault detection isbased on a selected set of abrupt faults that are given inTable 6. This table presents the results of fault detectionobtainedwith EUIO. Whilst Tables 3 and5 presentresultsofFDIobtainedwithNFDFDS.For the convenience of the reader, we provide Table 7thatsummarizestheresultsoffaultdetection.NFDFDS: AscanbeseeninTable7Theapproachprovides reliable fault detection for a set of abrupt faultsbeingconsidered. Moreover, as canbeseeninTables35,allthefaultscanbeisolatedaswell.EUIO with GP: Fault detection can reliably beperformed while perfect fault isolation is impossible.This is mainly because the approach provides oneresidualonlythatisnotsufcientforfaultisolation.Superiorapproach:NFDFDS.The criteria listed above provide a means forestimatingthepractical applicabilityof NFDFDSandEUIO. As aresult of comparingtheapproaches withrespect to these criteria it can be said that the NFDFDSARTICLEINPRESSF.J.Uppaletal./ControlEngineeringPractice14(2006)699717 713has certainadvantages over EUIOdesignedwithGP.Indeed, the EUIOis superior over NFDFDS withrespect to 4th and 5th criterion. The 4th criterionconcerns the availability of the data regarding the fault-operatingmodethatisrequiredbyNFDFDSwhilethedesignof EUIOdoesnot requiresuchadata. Ontheother hand, thesedata are required to perform the faultisolation and this is the main reason (cf. criterion 7) whyEUIOisnotabletoprovidefaultisolation.Thismeansthat NFDFDS can be designed only using data fornormaloperatingmodesbut,inthiscasefaultisolationis impossible. The EUIO is also superior with respect torequirements regarding computational power for on-lineapplications (cf. criterion 5). However, this superiority isof less practical importance since modern computers caneasilytackleon-linecomputationregardingNFDFDS.Furtherincrease with respect tothe efciency of on-linecomputation can be achieved by applying techniques forsparse matrices as well as in implementing a mechanismfor parallel computation regarding local observers (notethattheyworkindependently).5. ConcludingremarksInthispaperanovel multiple-model fault detectionandisolation(FDI)schemeispresentedfornon-lineardynamic systems called neuro-fuzzy and de-couplingfault detection scheme, which is a hybrid techniqueincorporating both neuro-fuzzy and model-basedmethods. Anapplicationof FDI for anelectro-pneu-matic valve actuator in a sugar factory is presented. Thekeyissues of ndingasuitablestructurefor detectingand isolating nine realistic actuator faults are alsodescribed.The technique presentedinSection4basedonthestate-spacemodeldevelopedwithgeneticprogrammingand extendedunknown inputobserver, wassuccessfullyusedforfaultdetectionandparticularlyindeterminingthe faults that are detectable. Unfortunately, thisapproach does not allow reliable fault isolationtherefore a realistic comparison was not possible.Indeed, in this case, the NFDFDS is much betterchoice. For fault isolation NFDFDS frameworkproposes the use of data from different operating pointsasshowninTables35.In real applications the disturbances, noise andmodelling errors must all be taken into account.NFDFDSemploys locallyoptimal observers designedaccording to minimum state estimation variance.Throughthisapproachtheapproximationandreason-ing capabilities of neuro-fuzzy models are combinedwith the de-coupling capabilities of optimal observers toperform reliable FDI. Theneuro-fuzzymodelpresentedhereisaspecialformoftheTSfuzzymodel,whichcanbe used to represent the system by a fuzzy fusion of locallinear sub-models. The necessary condition for theapplicationof this FDI schemeis that thenon-linearsystem can be represented by this special form of the TSmodel, whichis truefor manypractical control loopprocesses. It is shown that the powerful neuro-fuzzymodelling techniques can be applied for identication ofnon-linear dynamic systems for FDI. If all the localmodels are stable and the sub-observers converge to thelocal models it can be expected that the overall model isstable and the global observer will converge, asdiscussedinSection2. This is veriedby the resultsobtainedinSection4.As mentionedbeforecomputational limitations andstability constraints are alleviated in the proposedscheme, whichmakestheidenticationandFDIeasierandfaster. This achievement canbe attributedtothefactthattheglobalstateestimationisnotnecessaryforFDIandhencethereisnoneedforunnecessaryglobalfeedbacks.Generallyspeaking, the neural networkandneuro-fuzzybasedFDIapproachesrequirelargetrainingdatasets, as the data for different fault strengths andscenariosareusuallyneeded. Oneoftheadvantagesoftheproposedapproachisthat it de-couples certainsub-observers from some fault directions and therefore thesefaults with any magnitude are de-coupled provided theydo not change their distribution direction (with differentmagnitudes). NFDFDS is expected to require muchARTICLEINPRESSTable7PerformancecomparisonregardingfaultdetectionFault F1F2F7F13F17F18F19EUIOwithGPSmall D D D D DMedium D D D D DNFDFDSSmall DI DI DI DI DI DI DIMedium DI DI DI DI DI DI DIDdetectable,Iisolationpossible.F.J.Uppaletal./ControlEngineeringPractice14(2006)699717 714smallerdataset(asthedataforonlyonefaultstrengthvalue are sufcient). Moreover, the structure of the FDIscheme is transparent as compared to the neuralnetworkblackbox methodandit canalsodeal withnon-linear systems incomparisontothe model-basedmethods.Themaindrawbackofusinglocal optimal observersisrelatedtotherestrictiveassumptionsconcerningthenoise distribution. The noise sequences are generallyassumed to be zero-mean white noise processes withGaussian distribution. In practice, both deterministicand stochastic errors are expected. In this approach it isimpossibletoisolatetwofaultswiththesamedistribu-tion direction. In such cases other fault information e.g.the frequency domain must be used to achieve isol-ability.The proposed NFDFDS can be used online forcollectingthesymptomsandperformingfaultisolationbycomparisonwithfault signatures. If afault occurs,andithappensthataftersometimeenoughsymptomshave not beencollected, the proposedalgorithmwillsuggest the possibility of a number of faults. This mightbe the case that the systemhas not operated in theregionsinwhichfaultshaveanyeffect.AcknowledgementsTheauthorsacknowledgefundingsupportundertheECRTNcontract (HPRTN-CT-2000-00110) DAMA-DICS. Thanks are expressedtothe management andstaff of the Lublin sugar factory, Cukrownia Lublin SA,Polandfor their collaborationandprovisionof man-power and access to their sugar plant. Faisel UppalacknowledgesfundingsupportthroughanORS(Over-seas Research Scholarship) award, together with theHull Universitys OpenScholarship. The authors alsoexpress many thanks to Assitant Professor SuzanneLesecq of Laboratoire dAutomatique de Grenoble,UMRCNRS-INPG-UJF5528andDrLetitiaMireaoftheUniversityofIasifortheirimportantsuggestions.ARTICLEINPRESSAppendixAAnexampleofasub-modelandfaultdistributionmatrix:x1k1x2k1x3k1x4k126666643777775=0:3523 0:4676 0:1744 0:79980:1016 0:0893 0:2229 0:19070:0031 0:1950 0:7409 0:03320:0191 0:4340 0:1960 0:351726666643777775x1kx2kx3kx4k266666437777750:1680 0:7961 0:0395 0:0513 0:03470:2571 0:0007 0:6321 0:3573 0:21640:0201 0:0462 0:1663 0:2397 0:01500:4542 0:2491 0:4774 0:3039 0:328426666643777775u1ku2ku3ku4k126666666643777777775,y1ky2k" # =0:5741 0:8545 0:2340 1:36180:0394 0:1513 0:5093 0:1438 x1kx2kx3kx4k266664377775,E1=0:0161 0:0135 0:0262 0:0076 0:07270:0050 0:0882 0:0175 0:0056 0:01250:0006 0:0537 0:0071 0:0689 0:04680:0037 0:0565 0:0168 0:0339 0:078426666643777775.F.J.Uppaletal./ControlEngineeringPractice14(2006)699717 715It was veriedthat the local models obtainedwerestable and as an example Fig. (A.1) shows the polepositionsoftherstsub-model.ReferencesAshton, S.A., Shields, D.N., Daley, S. 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Understanding neural networks:Computerexplorations.MassachusettsInstituteofTechnology.ARTICLEINPRESSF.J.Uppaletal./ControlEngineeringPractice14(2006)699717 717