Fault Diagnosis Using Feature Vectors and Fuzzy Fault Pattern Rulebase

Transcription

1 Fault Dagoss Usg Feature Vectors ad Fuzzy Fault Patter Rulebase Prepared by: FL Lews Updated: Wedesday, ovember 03, 004 Feature Vectors The requred puts for the dagostc models are termed the feature vectors The feature vectors cota formato about the curret fault status of the system Feature vectors may cota may sorts of formato about the system Ths cludes both system parameters relatg to fault codtos (bulk modulus, leakage coeffcet, temperatures, pressures) as well as vbrato ad other sgal aalyss data (FFT, eergy, kurtoss) Feature vector compoets are selected usg physcal models ad legacy data such as that avalable the US SH-60 HIDS study Physcal models show that compoets such as bulk modulus ad leakage coeffcet should be cluded, ad HIDS shows the mportace of vbrato sgature eergy, kurtoss, etc Dfferet feature vectors wll be eeded to dagose dfferet subsystems The feature vector ϕ (t) s a suffcet statstc for the full state ad hstory X(t) of the system that determes the fault/falure codto Z(t) That s the fault lkelhood fucto PDF ca be decomposed as f ( z / x) = g( x) h( ϕ( x),, so that kowledge of ϕ (t) s suffcet to dagose the fault state Z(t) It s ot ecessary to kow the full system state ad hstory X(t) The feature vectors are extracted usg system detfcato ad dgtal sgal processg techques ϕ (t) s motored cotuously ad s a tme-varyg vector At each tme, the fault status wll be determed by comparg ϕ (t) to a lbrary of stored fault patters ϕ (t) cotas the symptoms of the fault status, ad the fault patter lbrary (FPL) cotas dagostc formato terms of kow fault patters May techques have bee used to create the FPL, cludg physcal modelg [Skorm], statstcal techques, fault test data (eg HIDS) rule-based expert systems, reasog systems, fuzzy logc, eural erks, etc We wll use adaptve fuzzy logc systems ths SBIR work We wll show that AFL systems ca clude the best features of rule-based systems ad eural erks, whle they also allow the cluso of physcal modelg formato, expert systems, ad statstcal legacy falure data a very straghtforward maer A AFL s a model-based fuzzy reasog system some or all of whose parameters ca be tued to lear o- ad/or off-le Fuzzy Logc Dagostc Systems A rather stadard applcato of FL systems to CBM was gve [Flppet] The essetal FL rulebase from that paper s show Fgure ote that t corporates some formato o cpet falures, whch meas that the FL system ca cota progostc formato, though we are prmarly usg t for dagoss ths secto A FL system [Lews ] s descrbed mathematcally as f ( x) z = = = = = µ ( x ), () µ ( x )

2 where we are usg product ferecg ad cetrod defuzzfcato The cotrol represetatve values are z ad the -D membershp fuctos are µ () The x are the compoets of the -vector x Tradtoal membershp fuctos ca be tragular, gaussa, etc Fgs ad 3 llustrate these for the -D case Geerally the FL system s multvarable wth puts a -vector where > Data Fuso The data to be corporated to the FPL comes from may sources Ma fuctos of the FPL are to () fuse the data, ad () provde a dagoss of system codto based o the feature vector The FPL must clude rules from expert small medum large Sdebad compoet I oe cp oe or oe oe or oe cp oe Sdebad compoet I small medum large Fg FL rulebase to dagose broke bars motor drves usg sdebad compoets of vbrato sgature FFT [Flppett 000] umber of broke bars = oe, oe, Icp = cpet fault Fg FL system wth tragular MFs Fg 3 FL system wth Gaussa MFs techcas who have experece o the system, ad rules derved from physcal models, tme fuctos of vbrato aalyss data (eg HIDS data), observed fault/falure legacy data, ad statstcal formato terms of umerous test ad actual cases Samples of some data appear the tables ad fgures Geerally, the data compoets are gog to be vectors For stace, Table the feature vector wll have compoets, whle Table t wll have 7 compoets Legacy statstcal data wll geerally have may compoets observed for each fault mode Fg 4 shows a represetatve sgle data compoet IF (BM s egatve medum) ad (LC s egatve small) THE (fault s ar cotamato) IF (BM s postve) ad (LC s ormal) THE (fault s water cotamato) IF (BM s ormal) ad (LC s postve medum) THE (fault s excessve leakage) Table Sample of dagoss rules for a hydraulc pump obtaed from expert techcas [Skorm ] BM= bulk modulus, LC= leakage coeffcet Fault/Falure descrpto Perturbatos of the physcal model parameters J B C K L M p B p Hydraulc system leakage +0% Cotrol surface loss -0% -0% Excessve hydr cyl frcto +0% +0% Rotor mechacal damage -0% -0% Motor magetsm loss -0% -0% Ar hydraulc system -0% -0% Table Sample of dagoss rules obtaed from physcal models of a actuator system [Skorm]

3 AFL systems are model-based reasog systems allow oe to fuse all of ths data to a sgle ufed dagostc lbrary that classfes the observed feature vector ϕ (t) to oe of may kow fault/falure codtos The AFL system ca cota rules geerated from all the sources metoed It s obvous how to costruct FL rules from expert systems data By assgg degrees to the percet chages Table, oe ca easly assg FL rules for data from physcal models Statstcal Legacy Fault Data To corporate statstcal legacy data as appears Fg 4 to the FL system, oe must frst determe whch data compoets to clude the feature vector ϕ (t) Ths ca be doe usg Pearso s Correlato Compoets that are sgfcatly correlated wth the fault of terest should be cluded the feature vector Bootstrappg techques ca be used to determe cofdece levels for the correlato The MATLAB Statstcs Toolbox wll be used ths SBIR Statstcal data s corporated to a falure Drve tra gear tooth wear Vbrato magtude Fg 4 Sample of Legacy statstcal fault data FL rule base as follows [Wag ] Gve statstcal sample values of vectors x R of measurable data ad ther assocated faults z, t s kow that a cosstet estmator for the ot PDF s gve by T ( x x ) ( x x ) ( z z ) P( = exp exp ( + ) / + (π ) σ = σ σ Substtutg ths to the formula for the codtoal expected value zp( dz E[ z / x] =, () p( dz oe obtas the expected value of the fault state z gve the actual observed data x as E[ z / x] = = = T ( x x ) ( x z exp σ T ( x x ) ( x x exp σ x ) ) (3) Remarkably, ths s a specal case of the FL system equato () That s, the statstcal fault data ca drectly be corporated to a FL system usg gaussa membershp fuctos wth rules defed by the legacy data pars Ths developmet holds for o-gaussa as well as Gaussa statstcal data The smoothg parameter σ s selected by smulato to gve a good compromse betwee accuracy of fault dagoss ad the ablty to geeralze to ew data sets Stadard FL clusterg techques [Wag] ca be used to derve a reduced umber of FL rules from statstcal data, as depcted Fg 5 3

4 Cofdece Itervals ca easly be establshed for the fuzzy FPL, we beleve A good measure of cofdece s the covarace matr extesvely used Kalma Flterg [Lews] The varace of the statstcal data ca be foud by usg a formula lke (3) that s of secod order about the mea Ths formula s drect to derve usg the plug correspodg to () for the varace Ths exteds the oto of fuzzy systems, ad we have ot see ths dea the lterature I ths SBIR work we wll pursue the dea of determg cofdece tervals for FL systems Fault codtos oe three Vbrato magtude Fg 5 Clusterg of statstcal fault data Maufacturg varablty ad part usage varablty are easly corporated to the fuzzy FPL I fact, varablty effects determe how wde the membershp fuctos should be Ths s related to selectg thresholds Bayesa ad eyma-pearso decso-makg for target detecto I ths research we wll exame selectg the MFs for reduced probablty of false alarms Fgure 93- from Chestut [] shows that the FPL membershp fuctos wll have to be updated over tme due to usage ad msso stress eural erk Patter Classfcato Due to ther learg, classfcato, ad geeralzato propertes, eural erks () are extremely useful for dagoss It s show [Lews] that fuzzy systems are a geeralzed case of structured The AFL system used ths work for dagoss cludes all the best features of, as follows A mult-put, sgleoutput, -layer eural et s descrbed by the equato L f ( x) = z σ v x + v0 + z0, (4) = = wth v, v0 the frst-layer weghts ad thresholds, z, z0 the secod-layer weghts ad thresholds, the umber of puts, ad L the umber of hdde-layer euros The actvato fuctos σ () may be selected as hard lmters, sgmods, tah, etc Gve statstcal sample values of vectors x R of measurable data ad ther assocated faults z, t s well kow [Lews] how to tue the weghts ad thresholds to partto the output space to regos, each o cetered at the fault z Ths s performed tally to tra the The, operato, whe a feature vector ϕ (t) s preseted to the, t s classfed to oe of the fault regos Trag ca be by the backpropagato algorthm, the Leveberg-Marquardt algorthm, etc The MATLAB Toolbox has all of these ad wll be used ths SBIR For a that has p outputs, L hdde layer euros, ad hard lmter actvato fuctos, proper trag parttos the space R usg L hyperplaes to p decso 4 Drve tra gear tooth wear low med severe

5 regos Clusterg techques appled to the data ca be used f the are multple patters correspodg to the same fault mode [Wag] Adaptve Fuzzy Systems The key pot here s that the FL system () s a geeralzed case of the (4) where the thresholds are vectors ot scalars [Lews] I fact, the humps Fgs ad 3 are the actvato fuctos (Each hump correspods to a fault mode ote that geerally the FL system has equal to more tha dmesos) Therefore, all the stadard tug algorthms ca be used to update the fuzzy system represetatve values ad membershp fuctos, whch meas that as the FPL lears, the humps Fgs ad 3 wll move aroud to the best locatos for falure dagoss Ths s the motvato for the termology adaptve fuzzy systems I ths SBIR work we wll explore may optos for learg the FPL We wll add some extra membershp fuctos the FPL after corporatg the expert, physcal model, ad statstcal fault data These extra MFS wll be tued to mmze the overall mea-square fault classfcato error to reduce false alarms probabltes ad maxmze detecto probabltes These are extra rules added to the FPL that wll be leared usg tellget systems techques We wll also vestgate o-le tug of the rules geerated by the expert, physcal model, ad statstcal fault data Ths amouts to fe-tug the expert opos ad legacy data formato to get better fault dagoss performace It could compesate, for stace, for accurately kow parameters physcal system modelg 5