Multi-Sensor Degradation Data Analysis

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1 A publcaon of CHEMICAL ENGINEERING TRANSACTIONS VOL Gues Edors: Enrco Zo Pero Barald Copyrgh 23 AIDIC Servz S.r.l. ISBN ; ISSN The Ialan Assocaon of Chemcal Engneerng Onlne a: DOI:.333/CET3336 Mul-Sensor Degradaon Daa Analyss Dngguo Hua a Khalfa N. Al-Khalfa b Abdelmagd S. Hamouda b E. A. Elsayed *a a Deparmen of Indusral and Sysems Engneerng Rugers Unversy 96 Frelnghuysen Road CoRE Buldng 2nd floor Pscaaway NJ 8854 USA b Deparmen of Mechancal and Indusral Engneerng Qaar Unversy 35 New Engneerng Buldng (female campus) Doha Qaar elsayed@rc.rugers.edu Mulple sensors are commonly used for degradaon monorng of crcal componens. The degradaon ndcaors derved from hese sensors are sgnfcanly dfferen n erms of scale and nerpreaon. Snce hese ndcaors reflec he same degradaon pah bu have dfferen scales and nose he fuson of he daa correspondng o hese ndcaors s mporan for he condon monorng and accurae relably esmaon of he componens. In hs paper we develop a unfed degradaon pah ha consders sascs-based degradaon and physcs-based degradaon models. We also esmae he relably of such componens. Smulaon resuls show ha he unfed degradaon model approxmaes he acual degradaon pah effecvely. Addonally more accurae esmaes of relably are obaned when compared wh ndvdual degradaon pahs.. Inroducon For a crucal componen of a sysem s real me degradaon condon should be monored n order o mae manenance decsons n a mely manner. As more advanced sensors are developed becomes more economcal and feasble o ulze mulple sensors on crucal componens n order o oban more accurae relably esmaes and perform manenance and replacemen n a mely fashon. The maor reason for mul-sensor degradaon monorng s ha degradaon nformaon provded by a sngle sensor may be unrelable and probably ncomplee. Mul-sensor degradaon monorng also generaes new ssues and challenges one of whch s he fuson of dfferen sources of nformaon. The challenge les n hree aspecs. Frs colleced from dfferen ypes of sensors worng based on dverse physcs phenomena he scales of dfferen degradaon ndcaors may vary dramacally. Thus he nerpreaons of hese degradaon ndcaors may dffer oally from each oher. Second some sensors may be more accurae han ohers snce hey are more sensve n ceran degradaon sages. Thrd hese ndcaors are usually correlaed wh each oher snce hey reflec he same underlyng degradaon process and hence s necessary o nerpre hese ndcaors ogeher. Therefore s of grea sgnfcance ha a unfed degradaon pah can be obaned from mulple degradaon ndcaors. Addonally s mporan o derve relably nformaon mercs.e. relably and MTTF from he mulple sources of degradaon nformaon o asss manenance decson mang. Degradaon pahs are usually modelled by sochasc processes and physcs-based models. Brownan moon and gamma processes are wdely used sochasc processes n degradaon modellng. McPherson (2) proposes hree mos ofen occurrng physcs-based degradaon models. However so far he modellng of mul-dmensonal correlaed degradaon processes has seldom been consdered. Moreover he degradaon modellng combnng physcs-based and sochasc models has no been nvesgaed. Even hough sgnfcan research has been conduced on mul-sensor daa fuson for faul dagnoss and prognoss sparse wor has addressed daa fuson of mul-sensor degradaon nformaon or unfed degradaon pah. We e al. (2) wegh dfferen sensor sgnals by mnmzng he uncerany of unfed measuremens bu he daa from dfferen sensors are consdered as ndependen nsead of correlaed wh each oher. Please ce hs arcle as: Hua D. Al-Khalfa K. Hamouda A.M. Elsayed E.A. 23 Mul-sensor degradaon daa analyss Chemcal Engneerng Transacons DOI:.333/CET3336 3

2 Relably esmaon for sochasc degradaon models s found n many references. A recen sudy by Moln e al. (2) presens he dsrbuon for frs passage me of Brownan moon wh me dependen drf and dffuson coeffcens. However he research on he relably esmae for mul-dmensonal correlaed degradaon processes s rare. The frs passage me dsrbuon of even 2-dmensonal Brownan moon wh drf s que complex (Domné and Peper 993). I s more dffcul o oban explc relably funcons for more dmensons of more advanced degradaon processes. Therefore we nvesgae he use of mul-sensor nformaon o oban a unfed degradaon pah usng hree man seps. Frs degradaon ndcaors from mulple sensors are modelled as a - dmensonal correlaed Brownan moon. Addonally physcs-based degradaon model s consdered n combnaon wh sochasc process based models. Second a unfed degradaon pah s obaned by averagng dfferen dmensons of degradaon pahs consderng weghs ha are deermned ulzng lead-lag correlaon beween dfferen pahs. Thrd relably nformaon s esmaed based on mul-sensor degradaon nformaon by usng movng bloc boosrappng mehod whch s verfed o be more accurae han ha derved from sngle degradaon pahs. 2. Mul-sensor Degradaon Modelng 2. Mul-dmensonal correlaed Brownan moon based degradaon modellng For crcal componen degradaon ndcaors can be exraced from mulple sensors. These degradaon ndcaors wh dfferen scales and nerpreaons are usually correlaed wh each oher snce hey reflec he same underlyng degradaon process. Brownan moon wh drf has been wdely used o model sngle degradaon pah. For mulple degradaon ndcaors Mul-dmensonal Correlaed Brownan moon wh drf (MCB) s used o model he degradaon process under he assumpon of lnearly ncreasng degradaon. MCB can be specfed as (Brbec Unversy of London 29) W () = HM [ + Z ()] + w () where W() s he - dmensonal MCB M = [ μ μ2... μ ]' s he vecor of drf coeffcens H s he Cholesy decomposon of correlaon marx Ρ among dfferen dmensons of W () Z() s a vecor of Brownan moons.e. Z() = [ σb() σ2b2()... σb()]' where σ s he dffuson coeffcen of he h Brownan moon and w = [ w w2... w ]' s he vecor of nal values. W () has dmensons each can be descrbed as a Brownan moon wh drf. W() = λ + θ B() + w (2) where W () s he h dmenson of W () and B () s a sandard Brownan moon. = λ and θ are s drf and dffuson coeffcens respecvely = 2 2 From Eq. () and (2) follows ha λ = h μ and θ = h σ and W () = h μ + h σ B () + w = = (3) 2.2 Physcs-based degradaon model wh randomness In hs paper physcs-based model s also consdered o model degradaon processes. The power law model s used snce s he mos wdely observed degradaon model n pracce (McPherson 2). Combnng randomness he power law degradaon model s specfed as Eq. (4) Y() = β + δ β B() + Y (4) m m where Y () s he degradaon process β and m s power law coeffcens Y s s nal value of degradaon δ β m s he dffuson coeffcen a me and B() s a sandard Brownan moon. 3. Unfed degradaon pah based on correlaon nformaon In order o deermne he degradaon condon of a crcal componen s necessary o derve a unfed degradaon pah from mul-sensor degradaon nformaon. The correlaons among dfferen degradaon 32

3 pahs are mporan nformaon and are used o oban wegh for each pah; hence he wegh-averaged unfed degradaon pah approach s ulzed n hs paper. The weghs are calculaed by employng and modfyng he leadershp score proposed by Wu e al. (2) consderng he PageRan proposed by Brn and Page (998). The leaders n a group of me seres (degradaon pahs) behave ahead of oher me seres and are consdered as good represenaves of he whole group. In hs paper he normalzed leadershp score s perhaps an effecve approach o wegh he dfferen degradaon pahs n order o oban a unfed degradaon pah. Suppose here are sngle degradaon pahs he leadershp score a me s obaned usng he followng four seps. Sep. A sldng wndow of lengh g for he h degradaon pah D () s defned as a subsequence of he degradaon pah whch s sg = ( D g+... D ). The lagged correlaon beween wo sldng wndows s g and sg of wo degradaon pahs D and D a lag l denoed as ρ g () l can be calculaed by Eq. (5) see (Wu e al. 2) l ( D l s g l)( D s ) τ = g+ τ+ τ l g l l g () l g l lg l ρ = σ σ ρg ( l) l < (5) c c c where s ab and σ ab are he mean value and sandard devaon of sldng wndow s ab. I should be noed ha lag l should no exceed ( g /2) and has o sasfy lag span l q g /2. Sep 2. The lagged correlaon values beween any wo degradaon pahs over he enre lag span are aggregaed by calculang he expeced correlaon value condonng on l. Tha s (Wu e al. 2) E = max{ E ( ρ l ) E ( ρ l< )} (6) where q E ρ l = max ρ ( l) Pr( l l ) ( ) l = { } and ( ρ ) l q { ρ = } E l < = max ( l) Pr( l l < ). Noe ha he unformly dsrbued condonal probables Pr( l l ) = / ( q+ ) and Pr( l l < ) = / q. Sep 3. Leadershp marx L s deermned based on he values of E ( ρ l ) and E ( ρ l < ) he hrd creron s added by auhors o enhance he robusness of hs mehod. E ρ l < > E ρ l he h degradaon pah leads he h one and L = ; () If ( ) ( ) (2) If E ( ρ l ) E ( ρ l ) (3) If E ( ρ l ) E ( ρ l ) L < < he h degradaon pah s led by he h one and L = ; < = he h degradaon pah leads or s led by he h one wh probably ½.e. Pr( = ) = Pr( L = ) = / 2. Snce he calculaon of E and E s no fully symmerc consderng he case of l = s possble ha L = L. Therefore he auhors propose wo crera o revse L as follows. (a) If L = L = hen L s se as f E < E and L s se as f E E ; (b) If L = L = hen L s se as f E < E and L s se as f E E. The reason for creron (a) s ha E E means he evdence of leadng s more powerful han ha of leadng. The reason for creron (b) s ha E E means he evdence of beng led by s more powerful han ha of beng led by. The marx of aggregaed lagged correlaon marx E s revsed by seng E max { = E E }. Sep 4. Accordng o Wu e al. (2) The leadershp score weghs have smlar relaonshp as hose of PageRan proposed by Brn and Page (998) and s obaned as: L Score = ( d) + d Score ε + E ( L ) (7) where Score s he leadershp score of degradaon pah a me d s a dampng facor whch can be se beween and and s se as.85 as suggesed (Brn and Page 998) ε s a small number say 33

4 o avod zero valued denomnaor. Eq. (7) forms a sysem of lnear equaons whch can be solved mmedaely. The leadershp scores are hen normalzed o oban he weghs for any dmensons of degradaon ha s = Score Score. The unfed degradaon pah s hen obaned as = η = D () = η W() (8) However due o he dfference n scales of dfferen degradaon pahs he unfed degradaon pah should be modfed consderng he scales. The modfcaon s carred ou by Eq. (9) ( ϕ ) ( ) η ϕ = = (9) D () = () D () () where ϕ () s he slope for he h degradaon pah a me. Slope s used snce afer degradaon value D () beng dvded by slope ϕ () he scale relaed nformaon s elmnaed from D (). 4. Boosrappng based approach for relably esmaon Due o he dffculy n dervng explc form of relably funcon for mul-dmensonal degradaon process Movng Blocs Boosrap (Efron and Tbshran 993) s used o oban he dsrbuon of he frs passage me and relably esmae. 4. Boosrappng based relably esmaon Suppose degradaon sgnals are colleced from sensors. The frs passage me of he degradaon process s defned as he me when any one of he degradaon pahs reaches s crcal hreshold h. Suppose a me n= / Δ daa are colleced from each sensor. Thus an n degradaon marx s obaned. Le b= % n be he lengh of a movng bloc he h bloc conans he o + b rows of degradaon marx. Mulple blocs are resampled from he degradaon marx and added o he end of he marx sequenally so ha frs passage me s obaned for he overall degradaon process. Noe ha he daa n each bloc should be modfed before beng added o he end of he degradaon marx. Frs he nal values n each resampled bloc should be elmnaed by subracng he row us pror o he bloc from each row of he bloc. Second he row a he end of he presen marx should be added o each row of he bloc o follow. The resample process s carred ou for V mes o oban V realzaons of frs passage me he dsrbuon of frs passage me and he relably esmae. 4.2 Explc form of relably funcon for Brownan moon When a degradaon pah s modelled as a Brownan moon wh drf hen he relably funcon can be specfed explcly snce he dsrbuon of he frs passage me of Brownan moon wh drf coeffcen λ and dffuson coeffcen θ s Inverse Gaussan and he relably can be obaned by Eq. () h w λ 2λ h w + λ R () =Φ exp ( h w 2 ) Φ θ θ θ () where Φ () s he CDF of normal dsrbuon h s he crcal hreshold and w s he nal value of degradaon. For - dmensonal correlaed Brownan moons each dmenson can be descrbed as a Brownan moon wh drf as specfed n Eq. (2). Thus he relably funcon for each ndvdual degradaon pah can be calculaed by Eq. (). The correspondng parameers can be esmaed usng MLE and he daa colleced by me. The esmaes of parameers λ θ and w can be obaned for each dmenson respecvely. For a sngle dmenson of he mul-dmensonal correlaed Brownan moon process he MLEs for he hree parameers wh n daa are ˆ n ( ) 2 n 2 = ΔW nδ ˆ θ ( ˆ = ΔW λδ) ( nδ) and wˆ = W. λ = 4.3 Explc form of relably funcon for physcs-based degradaon model If a degradaon pah s modelled by power law degradaon model combnng randomness as specfed by Eq. (4) he relably can be esmaed by Eq. () (Moln e al. 2) = m m mh ( Y) mβ 2( h Y) mh ( Y) mβ R () =Φ exp Φ m 2 m δm β δ δm β () 34

5 where he degradaon pah parameers m β δ and Y have he same meanng as n Eq. (4); and h s he crcal hreshold. The MLE parameers esmaes wh n daa are obaned by maxmzng he lelhood funcon n Eq. (2) and Y ˆ () = Y. m ( ΔY () ( mβ ) 2 ) Δ nδ L( ΔY( ) Δ ; β m δ) = exp (2) =Δ m 2 m 2 πδ ( mβ 2 ( ) Δ δ mβ ) Δ 5. Numercal Smulaon 5. Unfed degradaon pah for mul-dmensonal correlaed Brownan moon A numercal smulaon of 4-dmensonal Brownan moon process s conduced wh parameers shown n Table and he correlaon marx Ρ. Table : Parameer sengs for 4-dmensonal Brownan moon process Dmenson Dmenson 2 Dmenson 3 Dmenson 4 μ σ Ρ = Addonally he nal values for he four dmensons of degradaon process w are all se as. In he case of - dmensonal correlaed Brownan moon he slope of each degradaon pah s us he drf coeffcen ˆ λ esmaed a me he unfed degradaon pah s modfed as ˆ ( ) (3) D () = λ ( η W () ˆ λ ) = = Dm Dm 2 Dm 3 Dm 4 3 Dm Dm 2 Dm 3 Dm 4 Degradaon 2 Degradaon 2 Unfed pah Tme Tme Fgure. Unfed degradaon pah and hreshold Fgure 2. Movng bloc boosrappng a me 5 For he - dmensonal correlaed Brownan moon specfed by Table he unfed degradaon pah s shown n lne wh crcle marers n Fgure. As can be seen from he fgure he wegh-averaged hreshold for he unfed degradaon pah flucuaes around an average hreshold lne hrough me and converges due o more avalable daa. The frs passage me of he unfed degradaon pah s approxmaely n he mddle of he frs passage mes of he four sngle degradaon pahs. Ths shows ha he unfed degradaon pah can represen he underlyng degradaon process que well. 5.2 Movng bloc boosrappng mehod based relably esmaon Boosrappng based relably nformaon esmaon s carred ou a mes 3 5 and 7 respecvely wh g = and q =.6( g / 2). Fgure 2 shows he boosrappng degradaon pahs a me 5. The esmae for MTTF (Mean Tme o Falure) becomes more accurae when more daa are avalable. The rue frs passage me s 7.75 and MTTF s esmaed as and 7.69 a mes 3 5 and 7 respecvely. 35

6 The boosrappng based relably values a me 3 and 7 are shown n bold lnes n Fgure 3. From he fgure can be observed ha he dsrbuon of he frs passage me.e. he falure me converges o he rue frs passage me. The relably values based on sngle degradaon pahs are calculaed usng Eq. () a dfferen me pons as shown n Fgure 3 n dashed lnes. By comparson can be seen ha any relably funcon obaned from sngle degradaon pahs s no as accurae as he relably obaned from - dmensonal correlaed degradaon pahs especally a he early sage of degradaon. Relably Tme Relably Tme Fgure 3. Relably values a me pons 3 (lef fgure) and 7 (rgh fgure) 6. Conclusons Mul-sensor degradaon daa analyss s suded n hs paper. Mul-dmensonal correlaed Brownan moon and power law model (a physcs-based degradaon model) have been used o model mulple degradaon pahs. A unfed degradaon pah s obaned by weghng mulple degradaon pahs where weghs are calculaed by usng lead-lag correlaon nformaon among dfferen pahs. The numercal smulaon resul shows ha he unfed degradaon pah can represen he underlyng degradaon que well and hence may provde more useful nformaon for manenance. For mul-dmensonal correlaed Brownan moon based degradaon daa relably and MTTF are esmaed ulzng movng bloc boosrappng mehod. I s shown ha he esmae converges wh me o he rue values and he relably esmae s more accurae han usng daa for any of he sngle degradaon pahs. Acnowledgemen Ths paper was made possble by he suppor of NPRP gran from Qaar Naonal Research Fund (QNRF). The saemens made heren are solely he responsbly of he auhors. References Brbec Unversy of London 29 Brownan Moon and Sochasc Calculus < accessed Brn S. Page L. 998 The Anaomy of a Large-Scale Hyperexual Web Search Engne Sevenh Inernaonal World-Wde Web Conference (WWW 998) 998 Brsbane Ausrala. Domné M. Peper V. 993 Frs Passage Tme Dsrbuon of a Two-Dmensonal Wener Process wh Drf Probably n he Engneerng and Informaonal Scences Efron B. Tbshran R.J. 993 An Inroducon o he Boosrap London UK Chapman & Hall 99. Mcpherson J.W. 2 Relably Physcs and Engneerng: Tme-o-Falure Modelng Sprnger 2. Moln A. Talner P. Kaul G.G. Porporao A. 2 Frs Passage Tme Sascs of Brownan Moon wh Purely Tme Dependen Drf and Dffuson Physca A: Sascal Mechancs and s Applcaons We M. Chen M. Zhou D. Wang W. 2 Remanng Useful Lfe Predcon usng a Sochasc Flerng Model wh Mul-Sensor Informaon Fuson Prognoscs and Sysem Healh Managemen Conference (PHM-Shenzhen) 2 May Wu D. Ke Y. Yu J. Yu P. Chen L. 2 Deecng Leaders from Correlaed Tme Seres. In: Kagawa H. Ishawa Y. L Q. Waanabe C. (eds.) Daabase Sysems for Advanced Applcaons. Sprnger Berln Hedelberg. 36