Parameter estimation methods for fault detection and isolation

Transcription

1 rmeer esimio meods for ful deecio d isolio eres Escoe*, *UC, Uiversi oliècic de Clu Auomic Corol Deprme. Cmpus de errss. errss, Brcelo, Spi eres@eupm.upc.es Louise rvé-mssuès**, **LAAS-CNRS 7 Aveue du Coloel-Roce 3077 oulouse, Frce louise@ls.fr LEA-SICA Europe Associe Loror. Iroducio Ful deecio vi prmeer esimio relies i e priciple possile fuls i e moiored ssem c e ssocied wi specific prmeers d ses of e memicl model of e ssem give i e form of ipu-oupu relio: fu,e,,x were represes e oupu vecor of e ssem, u e ipu vecor, x e se vriles wic re prill mesurle, e o mesurle prmeers wic re liel o cge o e occurrece of ful, d e e modelig errors d/or oise erms ffecig e process. e geerl procedure o deec fuls follows e seps elow: Eslisme of e memicl model of e ssem s orml evior, fu, A is sge, llowle olerces for e ssem s prmeer vlues re lso defied. 2 Deermiio of e reliosip ewee e model prmeers i d e psicl ssem prmeers p j. 3 Esimio of e model prmeers i from mesuremes of, u suile esimio procedure, ˆ g, L,, u, L, u 4 Clculio of e psicl ssem prmeers, vi e iverse reliosip: pˆ f ˆ

2 5 Decisio o weer ful s occurred, sed eier o e cges p j or o e cges i d olerces limis. If e decisio is mde sed o e i e ffeced p i s c e esil deermied from sep 2. is m e cieved wi e id of ful clogue i wic e reliosip ewee process fuls d cges i e coefficies p j s ee eslised. Decisio c e mde eier simpl cecig gis e predeermied resold levels, or usig more sopisiced meods from e field of sisicl decisio eor. e sis of is clss of meods is e comiio of eoreicl modelig d prmeer esimio of coiuous ime models. e procedure is illusred i figure. u G, x e rmeer Esimio Clculio of process coefficies eoreicl modelig pf - p Cges p, p, Ful decisio FAUL Figure.Ful deecio sed o prmeer esimio eoreicl modelig I is pper, e focus is pu o e sud of clssicl prmeer esimio meods. e meods explied re e pplied o e 3-s ecmr [Lue, COSY Becmr rolem]. 2

3 3 2. rmeer esimio for ful deecio I is secio, wo pproces for solvig Recursive Les Squres RLS lgorim re preseed. o ppl is meod, oer impor issues lie implemeio d rousess, o explied ere, mus e e io ccou. Muc more iformio ou ese eciques c e oied i [oulieos & Svris 994]. 2.. Recursive Les Squres lgorims Give e ssem represeed e followig ipu-oupu model: were d re e order model srucure d is e del. e Recursive Les Squres lgorim cosis of: were: ε is e iovio error, is e covrice mrix d is e iovio gi. e lgorim eeds iiil vlues: 0 d 0. ese c e eier provided from owledge of e ssem crcerisics or clculed from iiil d se usig e o recursive les squre meod. e resul miimises e followig expressio: were N is e umer of smples. Uder e followig mild codiios, e LS esime is cosise, i.e. eds o s N eds o ifii: E{ } is o sigulr ε ε [ ] [ ] u u K K K K V N ε 2 N ˆ u u L L

4 E{ ε} 0 e firs codiio gurees e exciio level of e oupu sigls d e secod gurees e sisicl idepedece of e oupu sigls d e error. from e prcicl poi of view, e covergece speed is geerll slow wic mes e sdrd RLS esimio meod ideque for rel-ime ful deecio pplicio. ere re severl pproces for modifig e RLS lgorim o me i suile s rel-ime ful deecio meod: Use of forgeig fcor Use of virul Klm filer Use of slidig widow d 2.2. Forgeig fcor I is cse, e pproc is o cge e loss fucio o e miimied. Le e modified loss fucio e: N s 2 V N s λ ε s is mes e mesures re older 0 /-λ smples re icluded i e crierio wi weig pproximel equl o 36% of of e mos rece mesureme. e 0 mes e memor ime cos of e crierio d reflecs e rio ewee e ime cos of vriios i e dmics d ose of e dmics iself. picl coice of λ re i e rge ewee 0.98 o [Ljug 986]. e RLS meod wi forgeig fcor is: λ λ ε ε Experieces for differe vlues of λ sow decrese i e vlue of e forgeig fcor s wo effecs: e prmeer esimes coverge o eir rue vlue quicer, us decresig e ful lrm del ime. 4

5 2 Bu e expese of icresed sesiivi o oise. If λ is muc less e esimes m eve oscilles roud eir rue vlue. o solve is prolem, ere re differe pproces [oulieos & Svris, 994]. Ol wo of em re preseed ere: e ime-vri forgeig fcor d Klm filers. ime-vrig forgeig fcor Oe lgorim o impleme vrile forgeig fcor is e proposed [Forescue e l. 98]. e recursio cosiss i: I is lgorim, e vlue of e cos σ 0 is e expeced mesureme oise vrice wic mus e cose sed o e owledge of e ssem. e miimum vlue for λ is lso o e cose e user. e iuiive ide eid e ime-vrig forgeig fcor is e forgeig fcor is decresed owrds is miiml llowed vlue s e error icreses. I cosequece, e d correspodig o ig error is forgoe fser. Klm filers redicio Error Gi Forgeig Covrice λ oe :if λ < ë λ e model c e descried s se spce equio: were e se vecor x is give : ˆ ε ˆ ε o model e ime-vrig ses, e se equio c e descried s: em λ ë is mes e prmeer vecor is modeled s o-correled rdom drif wic ssumes slow vriio. e covrice mrix R c e used o descrie ow fs e mi x e x [ K ] K mi { v v s } R s x x v ; E δ, 2 / σ 0 5

6 6 differe compoes of re expeced o vr. e recursive lgorim oied s resul o ppl e Klm filer o e model is: I is lgorim R s similr role s e forgeig fcor λ. ese desig vriles sould e cose rde-off ewee fs deecio wic requires λ smll or R lrge d reliili wic requires λ close o or R smll. 3 Becmr process ful deecio e sudied ssem is preseed i figure 2 [Lue, COSY Becmr rolem]. Figure 2. rocess cofigurio d oios is ssem c e modeled usig e followig lier equios: were: e worig poi s ee cose s proposed i e ecmr. R ε ε S S Q posiio e worig poi, vlve V e vriio of is 2.76*0 roud e worig poi, pump flow e vriio of is 0.m roud e worig poi, e vriio of is 0.5 m roud e worig poi, e vriio of is S Q Q

7 e meods used for ful deecio ve ee: A RLS wi cos forgeig fcor of 0.995, sowed i e figures i lue color; A RLS lgorim wi vrile forgeig fcor, sowed i e figures i gree color; A Klm filer modified RLS lgorim, sowed i e figures i red color; ese meods ve ee pplied o deec fuls i e scerios I, II d III proposed i e ecmr [Lue, COSY Becmr rolem]. 4.. Scerio I Scerio I correspods o vlve V loced closed from ime 000. e figure 3 sows e esime vlues of e prmeers Figure 3. Esimed prmeer vlues usig scerio I 7

8 We e ful is occurs, e recursive esimio lgorim coverges owrds e ew vlues. e re of covergece depeds o e weigig fcor. I is exmple e Klm filer s re of covergece sigificl slower e wo oer oes. Noe i is scerio, e ful vlve V is loced d closed cges e model srucure. As we mii e sme model srucure fer e ful, e esimed vlues of e prmeers ve o relio wi e reli. Eve oug e cge of esimed vlues is idicio of e ful Scerio II Scerio II correspods o vlve V loced opeed. Figure 4 sows e esimed vlues of e prmeers Figure 4. Esimed prmeer vlues usig scerio II 8

9 As for scerio I, we e ful occurs, e recursive esimio lgorim coverges owrds e ew vlues. I is exmple e Klm filer s lso re of covergece muc slower e wo oer oes. I is possile o icrese e R vlue u i is cse e respose vrice icreses. e mjor differece wi scerio I is i is cse e ew vlue of 2 is usle. Acull, o scerios ve covergece prolems due o e o exciio of ipu/oupu sigls Scerio III Scerio III correspods o le i, e ful occurs ime 800 secods. Figure 5 sows e esimed vlues of e prmeers Figure 6. Esime prmeer vlues usig e scerio III 9

10 We e ful occurs ime 800, e esimio respods slowl d covergig o e ew vlues es erl 5 miues. e re of covergece of e esimio could e icresed decresig e weigig fcor or equivle, u e vrice of e esimio respose would e lrger. I e ew sed se e vlues of e prmeers, d 2, wic correspod o e firs, ve differe vlues eir iiil vlues weres e prmeers correspodig o e secod coverge c o eir iiil vlues. is loclies e ful o e firs. 5. Coclusios is pper preses ree clssicl RLS sed prmeer esimio meods ve ee pplied o e 3 s ecmr. I sould e meioed e 3 s ecmr is o idel o es ideificio meods ecuse wo ou of ree of e proposed fuls cge e model srucure. is is ideed difficul prolem, eve wi oer pproces oserver-sed or pri spce pproces. A dvge of ese meods is o pproc simuleousl e ful deecio d ful isolio prolems. is is of course codiioed e owledge of e iverse relio llowig oe o go from model prmeers o psicl prmeers. A prolem mus e repored ou esimio meods is eir ig sesiivi o e prmeriio, i.e. i our cse, e vlues of l d R, wic m compleel cge e lgorim evior, urig usle for isce. Noe oer meods exis s soluio o e prolem of e covrice mrix eig usle or sigulr resul of pour exciio sigls [o e l. 984]. Our experimes sow e covergece re of e lgorims is criicl issue. I sould e oulied our experimes re imed illusrig e esimio lgorims evior u e do o rell ddress e deecio prolem. Ideed, e prolem of deecig cges i sigl e esimed oes is prolem iself. I is ofe solved simpl cecig gis predeermied resolds [qmii, qmxi] u ere is lrge lierure o more sopisiced decisio procedures o deec cges/fuls i sigls [Hägglud 984][Bsseville & Niiforov 993]. Also e ful deecio rousess issue is ddressed, for isce [Wlerg 990][Kwo d Goodwi 990]. Biliogrp Bsseville M., Niiforov I., 993 Deecio of Arup cges : eor d pplicio, Eglewood Cliffs, NJ: reice Hll. Forescue.R., Kerseum L.S. d Ydsie B.E., 98. Implemeio of sefuig regulors wi vrile forgeig fcors. Auomic, 7, 6,

11 Hägglud., 984. Adpive corol of ssems sujec o lrge prmeer cges. roceedigs, IFAC 9 rieil World Cogress, Budpes, Hugr, Jug L., 987. Ssem Ideificio. eor for e user. reice Hll Iformio d Ssem Scieces Series, Uied Ses. Kwo O.K. d Goodwi G.C., 990. A ful deecio meod for uceri ssems wi umodeled dmics, lieriio errors d ois ipus. roceedigs, IFAC rieil World Cogress, lli, Esoi, Lue J., COSY Becmr rolem, Loror 3 Ssem, Becmr for e recofigurio prolem versio.2, i rojecs of p:// o J., Fliower V.M. d Irvig E., 984. Regulio mulivrile dpive des fours. Colloque CNRS Commde Adpive. Aspecs rcique e eoriques, S. Mri d Heres. oulieos A.D., Svris G.S Rel ime Ful moiorig of Idusril rocesses. Kluwer Acdemic ulisers. Wlerg B Rous frequec domi ful deecio/digosis. roceedigs IFAC rieil World Cogress, lli, Esoi, LIENHYEREXE