State and Parameter Estimation for a Partially Observable System Subject to Random Failure POM-2001 ORLANDO
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1 Stte nd Pmete Etmton fo Pty Obevbe Sytem Subect to Rndom Fue POM-2 ORLADO Opeton Pnnng Schedung nd Conto Abtct: We conde pty obevbe ytem ubect to ndom fue The ytem montoed t equdtnt pont of tme nd the nfomton obtned tochtcy eted to the ytem' tte whch unobevbe except the fue tte By combnng the fue nfomton nd the nfomton obtned though condton montong nd ppyng the chnge of meue method we deve ecuve fte nd deveop pocedue fo pmete etmton bed on the EM gothm The pocedue teted ung e dt et Dmng Ln Vm M nd Andew KS Jdne Deptment of Mechnc nd Indut Engneeng Unvety of Toonto Intoducton We conde pty obevbe ytem ubect to ndom fue The tte of the ytem decbed by homogeneou Mov chn wth fnte tte pce The ytem' fue obevbe but the wong tte not The nfomton bout the ytem decbed by n obevton poce obtned t equy pced npecton tme Condton montong technque hve been we deveoped nd e fequenty ued n pctce The mot commony ued method ncude pectometc ny of engne o nd vbton montong Howeve thee e t vey few mthemtc mode cpbe of utzng the nfomton obtned on-ne fo n effectve mntennce decon-mng Seve type of mntennce mode wth pt nfomton hve ppeed n the tetue focung mny on the decon pect eg ee Sond (97 Jenen nd Hu (993 Stde (994 Aven (996 nd M et (998 The condton-bed mntennce (CBM mode ppcbe to e tuton ncude the tte-pce mode fo funce eoon pedcton by Chte et (997 nd the popoton hzd mode by M nd Jdne (992 The ppe by M nd Jdne ( povded theoetc b fo the deveopment of oftwe ced EAKT (EAKT Mnu 2 by the CBM botoy etbhed t the Unvety of Toonto n 994 nd uppoted by eve mo ndut ognzton MMO nd SERC The EAKT oftwe now we deveoped nd the CBM botoy h hd ch expeence n nyzng CBM dt nd povdng conutncy to the mntennce engnee ung the oftwe In th ppe we tudy the etmton pobem n pty obevbe ytem decbed by hdden Mov mode (HMM wth obevbe fue tte HMM h ttcted condebe ttenton n the pt two decde Howeve thee h been tte ttempt to ppy HMM to mntennce/epcement pobem To the utho' nowedge the wo by Fenández- Guchend et (993 the ony pubhed ppe tht tuded both etmton nd conto n Poceedng of the Twefth Annu Confeence of the Poducton nd Opeton Mngement Socety POM-2 Mch 3-Ap 2 2 Ondo F
2 2 mpe epcement mode decbed by two tte HMM wth unobevbe fue tte Wth the pd deveopment of ft compute we deveoped nfomton ytem n mny compne nd the mpementton of ophtcted condton-montong technque HMM w ey ttct condebe ttenton n the mntennce/epcement e n the ne futue Thee e two mn etmton pobem n n HMM: tte nd pmete etmton The etmte bed on poceng pt nd peent obevton ced fte In th ppe we ppy the chnge of meue method (Eott et 995 to obtn the tte nd pmete etmte of the (pty hdden Mov poce decbng the tte of the ytem The ppe ognzed foow We fomute the mode of the deteotng ytem n next ecton Then new meue contucted unde whch the obevton e d Unde the new meue we deve gene fte nd obtn pec ce fte fo tte tnton nd occupton tme Bed on thee fte pmete e-etmton ung the EM gothm dcued Fny numec exmpe uttng the pocedue gven The Mode We conde deteotng ytem ubect to ndom fue The ytem (pty obevbe t tme 2 L nd the obevton poce (Y tochtcy eted to the tte poce ( Suppoe tochtc pocee e defned on pobbty pce ( Ω F P Denote the tte of the ytem t tme by 2 L The tte of the ytem t tme gven o t dtbuton nown The tte poce { 2 L} decbed by homogeneou Mov chn wth fnte tte pce S { e e2 L e } whee e unt -dmenon vecto whoe th eement e epeent the th pobe tte of the ytem The t tte e epeent the fue tte of the ytem nd th tte obevbe The ft tte { e L e } e not obevbe We ume tht the fue tte e n bobng tte e the ytem cnnot be ecoveed to ny wong tte when t f Let A ( be the tnton pobbty mtx of the Mov chn 2 L nd A be the mtx contng of the ft coumn of A whee P{ e e} 2 L ( Obvouy fo 2 L nd e A ( A Let Y be the meuement ten t tme gven tht the ytem opetng When the ytem f ony the fue nfomton vbe Suppoe Y h M pobe vue Y SY { f f 2 L f M } whee f unt M-dmenon vecto wth n the th poton Suppoe futhe tht P { Y f L Y L Y } P{ Y f } Wte C ( c M ( whee c P{ Y f e } 2 L 2 L M (2 e Poceedng of the Twefth Annu Confeence of the Poducton nd Opeton Mngement Socety POM-2 Mch 3-Ap 2 2 Ondo F
3 3 Bed on the bove umpton the nfomton vbe cont of two pt: the meuement nfomton (Y nd the fue nfomton Wte g ( f 2L M nd et g M be n (M -dmenon vecto wth n the t poton nd eewhee To combne the obeved meuement nfomton nd the fue nfomton we defne new ndom vbe S g g L g } uch tht { 2 M P{ g e } P{ Y f e } c < M < M o M P{ g e} M Thu epeent the ove nfomton vbe t tme When g ( M the ytem n ome wong tte nd the meuement Y f When g M the ytem n the fue tte nd no meuement vbe Let F nd G be the compete ftton geneted by ( L ( 2 L nd ( L 2 L epectvey Then the ytem dynmc decbed by the foowng HMM: A V 2 L (3 C W 2 L (4 whee S S A ( C ( c ( M nd the eement tfy (5 M c c 2 L (6 < M o M c (7 M V nd W e mutuy ndependent mtnge ncement tfyng E[ V F ] E[ W F G ] V E[ V V ] dg( A Adg( A (8 W E[ W W ] dg( C Cdg( C (9 whee dg( z the dgon mtx wth vecto z on t dgon Chnge of Meue We begn wth defnng new pobbty meue P uch tht unde th meue the obevton equence d nd ( Mov chn wth the tnton mtx A Then we contuct meue P unde whch the ytem' dynmc decbed by (3 nd (4 It tun out to be much ee to deve the fte unde the new meue nd then tnfom the eut bc Poceedng of the Twefth Annu Confeence of the Poducton nd Opeton Mngement Socety POM-2 Mch 3-Ap 2 2 Ondo F
4 4 to the e tuton unde meue P We ue pme to denote tnpoe nd b b to denote nne poduct Futhe we ntoduce the foowng notton: e g c C M g λ ( M c nd λ fo 2 L Fom Komogoov' dp extenon theoem thee ext new pobbty meue P uch tht The eut dp G howng tht the tte poce nd the obevton poce foow (3 nd (4 e ummzed n the foowng emm Lemm Unde P E [ G ] A Lemm 2 Unde P E ] C [ Fte Fo ny G-dpted c poce φ defne ( φ E[ φ ] The etmte of φ denoted by φˆ gven by ˆ E[ φ ] φ E[ φ ] E[ ] We c E[ φ ] the unnomzed etmte of φ nd E[ ] the nomzng fcto Cey φ E[ φ ] E[ φ ] whee vecto wth ( eement Theefoe we ony need to obtn ( φ nd ( to ccute φˆ Suppoe H c poce of the fom H H α β δ ( whee α β e G meube δ ( nown vecto functon of α M c β nd δ ( e vecto of dmenon Wte d ( Mc e Theoem The ecuve fomu fo the fte of H ( H d ( { ( H e E[( α β e δ ( e By ppyng Theoem wth ome pecfc α β nd δ ( we obtn pec ce fte fo tte tnton nd occupton tme Fte fo the tte Let q ( e E[ e ] q ( q ( e q ( e2 L q ( e ( Then the ecuve fomu fo ccutng q the unnomzed tte fte ]} Poceedng of the Twefth Annu Confeence of the Poducton nd Opeton Mngement Socety POM-2 Mch 3-Ap 2 2 Ondo F
5 5 q d q ( e ( Fte fo the numbe of ump Let J e e be the numbe of ump fom tte e to tte e up to tme fo 2 L nd 2 L The ecuve fomu fo the fte of J Fte fo the occupton tme ( J ( ( J e d ( d ( Let O e be the numbe of occon up to tme tht the ytem h been n wong tte e fo 2 L The ecuve fomu fo the fte of O ( ( ( e O O d d ( q e (2 Fo the ppcton of the EM gothm n the next ecton we woud need nothe veon of the occupton tme defned by O e whch count the numbe of occon fom tme up to tme tht the ytem h been n the wong tte e fo 2 L The ecuve fomu fo the fte of O ( O d ( ( O e d ( q e (3 Fte fo the tte to obevton tnton Let T e g be the numbe of tme up to tme tht the obevton poce n tte f gven the ytem n the wong tte e fo 2 L nd 2 L M The ecuve fomu fo the fte of T ( T d ( ( T e ce q Pmete e-etmton The hdden Mov mode decbed n th ppe detemned by the et of pmete e (4 Poceedng of the Twefth Annu Confeence of the Poducton nd Opeton Mngement Socety POM-2 Mch 3-Ap 2 2 Ondo F
6 6 θ { 2 L 2 L ; c 2 L 2 L M} tfyng contnt M 2 L 2 L (5 c c 2 L 2 L M (6 We now wnt to ue the EM gothm to deve new pmete et ˆ θ {ˆ ( 2 L 2 L ; cˆ ( 2 L 2 L M} tfyng contnt (5 nd (6 wth nd c epced by updte to â we defne dpˆ θ nd et dp θ F ˆ ( e e â nd ĉ epectvey To The bove defnton of the new pobbty meue P utfed by the θˆ foowng emm Lemm 3 Aumng e ( we hve E ˆ[ ] ˆ F ( ( θ whee E the expectton unde pobbty meue P θˆ θˆ Theoem 2 Bed on the EM gothm the new etmte fo gven obevton up to tme ( e g e gven by o M ˆ J ( J ˆ ( ˆ O ( O In the next theoem we obtn the etmte of c Theoem 3 Bed on the EM gothm the new etmte fo c gven obevton up to tme ( e g e gven by o M ˆ T ( T cˆ ( ˆ O ( O umec exmpe To utte the etmton pocedue deveoped bove we ued e dt et obtned fom the Cdn Rve Co (CRC n Cnd It contned nfomton on the wong ttu nd o ny eut fo whee moto The whoe CRC dt et conted of 75 htoe We eected one of them fo ou ny O npecton ntev (meued n wong ge n the w dt et ved qute bt To educe the vton n npecton ntev to cetn degee we deeted npecton wth hot ntev o to me the umpton of n equ ntev between npecton ppoxmtey vd Afte the modfcton the eected htoy hd 97 Poceedng of the Twefth Annu Confeence of the Poducton nd Opeton Mngement Socety POM-2 Mch 3-Ap 2 2 Ondo F
7 7 ecod of meuement nfomton (eut fom the ny of o mpe nd the 98th obevton w fue Ech meuement conted of 2 dt vue (eve of met ptce n ppm edment eve etc We found two of them Fe (on nd Sed (edment gnfcnt by ung EAKT oftwe (EAKT mnu 2 To dcetze the meuement we defned new meuement vbe Y wth M 4 tte: Y f f Fe men(fe nd Sed men(sed ; Y f 2 f Fe men(fe nd Sed > men(sed ; Y f 3 f Fe > men (Fe nd Sed men(sed ; Y f 4 f Fe > men(fe nd Sed > men(sed Hee men( denote the men vue of meuement vbe n the dt et Ou hdden Mov mode w ppcbe to the modfed dt et nd we pped etmton pocedue deveoped bove to the dt We umed 3 tte fo whee moto the ft two e wong tte nd the thd one the fue tte The ft tte epeent good wong condton nd the econd tte ndcte tht the whee moto opetng n poo condton It eonbe to ume tht new whee moto wong n good condton e e We ccuted the fte of the tte nd othe quntte of nteet nd then etmted the tnton pobbty mtce A nd C The etmton pocedue w coded n Mtb The nt etmte of the mode pmete e et 2 23; c The Mtb pogm w un 3 ung the modfed dt et Afte 7 teton the dffeence between the peent etmte nd 5 the pevou etmte fo A nd C wee e thn The ccuton too bout 2 mnute of CPU tme The fn etmte of pmete A nd C wee: ˆ ˆ A C Fom the computed eut we obeve tht the whee moto h vey hgh pobbty (9469 to emn n the good wong tte t the next npecton epoch f t n good condton t peent If the whee moto n the poo wong tte t w hve bout 5% chnce to go bc to the good tte t the next tge (due to outne mntennce The pobbty of fue t the next tge when the moto n poo condton 242 whee t coe to zeo when t n good condton Summy nd concuon In th ppe we hve tuded the etmton pobem fo fue-pone ytem ubect to condton montong t dcete pont of tme To mode the ytem we hve condeed n HMM wth obevbe fue tte We hve combned the fue nfomton nd the nfomton obtned though condton montong nd pped the chnge of meue method to obtn gene ecuve fte nd the fte fo tte tnton nd occupton tme A pocedue bed on the EM gothm utbe fo off-ne pmete etmton h been popoed The pocedue h been pped to e dt et (fte ome modfcton nd enbe eut hve been obtned Poceedng of the Twefth Annu Confeence of the Poducton nd Opeton Mngement Socety POM-2 Mch 3-Ap 2 2 Ondo F
8 8 Thee e cetn mtton to ou mode Ft we hve umed tht the npecton ntev e of n equ ength whch my not wy be tue n e dt Second the nge of meuement h been condeed to be fnte et whch my eque ome dcetzton of the obevton pce befoe the mode cn be pped to e tuton We w ex ome of thee umpton n ou futue wo Refeence Aven T Condton Bed Repcement Poce---A Countng Poce Appoch'' Rebty Engneeng nd Sytem Sfety Vo 5 (996 pp Chte A H W Wng nd J M Shp A Stte Spce Condton Montong Mode fo Funce Eoon Pedcton nd Repcement Euopen Joun of Opeton Reech Vo (997 pp -4 Eott R J L Aggoun nd J B Mooe Hdden Mov Mode: Etmton nd Conto Spnge-Veg 995 EAKT Mnu Condton-Bed Mntennce Conotum Lbotoy Deptment of Mechnc nd Indut Engneeng Unvety of Toonto 5 Kng' Coege Rod Toonto O Cnd M5S 3G8 Em: cbm@meutoontoc Web te: wwwmeutoontoc/cbm 2 Fenández-Guchend E A Apotth nd S I Mcu Any of n Adptve Conto Scheme fo Pty Obeved Contoed Mov Chn IEEE Tncton on Automtc Conto Vo 38 o 6 (993 pp Jenen U nd G H Hu Optm Stoppng by Men of Pont Poce Obevton wth Appcton n Rebty Mthemtc of Opeton Reech Vo 8 (993 pp M V nd A K S Jdne Computton of Optm Poce n Repcement Mode IMA Joun of Mthemtc Apped n Bune & Induty Vo 3 (99 pp M V nd A K S Jdne Optm Repcement n the Popoton Hzd Mode IFOR Vo 3 o (992 pp M V Jng nd A K S Jdne A Condton-Bed Mntennce Mode IMA Joun of Mthemtc Apped n Bune & Induty Vo 9 (998 pp 2-2 Sond E J The Optm Conto of Pty Obevbe Mov Pocee PhD detton Stnfod Unvety Cfon 97 Stde W Mxm Weng-Out of Deteotng Sytem Euopen Joun of Opeton Reech Vo 73 (994 pp Poceedng of the Twefth Annu Confeence of the Poducton nd Opeton Mngement Socety POM-2 Mch 3-Ap 2 2 Ondo F
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