Real time estimation of modal parameters of non-stationary systems using adaptive wavelet filtering and Recursive Least Square algorithm.

Transcription

1 Rel tme estmto o modl prmeters o o-sttory systems usg dptve wvelet lterg d Recursve Lest Squre lgorthm. Adrzej Klepk AGH Uversty o Scece d echology, Krkow, Pold deusz Uhl AGH Uversty o Scece d echology, Krkow, Pold ABSRAC: he pper presets method o modl prmeter estmto bsed o RLS (Recursve Lest Squre) lgorthm, d wvelet lterg. he wvelet lterg gves possblty to decouplg requecy compoets o sgl respose o structure. hs operto c lso reduce the order o the sgl model estmted by RLS lgorthm. A ddtol dvtge o ths method s the possblty o dptg the wvelet lter prmeters to the chgg prmeters o the system. Reduced model order sgctly reduces the tme o estmto o modl prmeters, whch ebles the rel tme mplemetto o the method. Due to recursvely updted covrce mtrx o model prmeters, the codece tervls o modl prmeters c be lso estmted. All routes hve bee mplemeted d tested MALAB. he hrdwre relzto o the lgorthm hve bee cheved employg FPGA (Feld Progrmmble Gte Arry) techology. he method hve bee tested o smulted dt delvered by AIRBUS tem d o the test bed wth vrble stess. IRODUCIO My prctcl egeerg systems chge dymc prmeters durg ther operto. Possble reso o prmeters chge c be dmge, chges o opertol codtos or occur some physcl pheome cusg chges dymcs o the system. Alyss o osttory systems re more dcult th sttory cse d requre dedcted method o detcto. osttory ler mechcl systems re those systems whose prmeters s stess, turl requecy, mss d dmpg rtos or sttstcl propertes o put sgl (such s me vlue d vrce) re tme depedet. Chges o ths prmeters c be cused by my ctors, rom chges o evrometl codtos lke temperture, humdty, chges o structure geometry, vrble opertol codtos to dmge. I prctce, my egeerg structures lke trc excted brdges, rottg mchery workg wth vryg speed, rcrts, robots, cres d my others should be treted s o sttory systems. Alyss o o sttory systems requre pplcto o dedcted method whch c ot be bsed o vergg whch s most populr wy o mprovg qulty o prmeters detcto results. I some cses, whe chges o prmeters re slow, the system c be treted s sttory gve tme tervl. Frsts ttempts o o sttory systems prmeters detcto cossts dptto o clsscl detcto method o Ler me Ivrt (LI) systems. M problem o ths pproch ws to dpt clsscl lgorthm to operte o smll umber o smples, where qussttorty codto ws ssumed. Wth the spce o the lst severl doze yers my method or o sttory system detcto were creted. he method bsed o deret lgorthms lke or exmple BLS (Bootstrpped otl Lest Squres), RLS (Re-

2 IOMAC' 4 th Itertol Opertol Modl Alyss Coerece cursve Lest Squre), ARMA (me-depedet Autoregressve Movg Averge), Recursve Subspce d my others. I the lterture my o pplcto o ths method c be oud. Oe o exmple o pheomeo cusg osttory behvor o rcrts (but ot oly) c be lutter pheomeo. Ustble vbrtos o rple c be reso o ctstrophc lure o the rcrt (Kehoe 995). A crtcl stblty pheomeo s kow s ero-elstc lutter. I the lterture (Breer M.. 997, Reed 98, Cooper 006) my cses o lutter pheome re creully studed. he mportce o ccurte deto eroelstc eects such s lutter o lght vehcles re evdet d ct c be trced erlest dys o med vto tsel. For prevetg rom lutter pheomeo, the rple s submtted to lght lutter testg procedure, wth cremetlly cresg lttude d rspeed. Flght lutter testg procedure c be ormulted s procedure o stblty test. Oe possble soluto s to d stble poles (whch re resposble o lutter pheome) o the rcrt structure wth employg modl model prmeters. Importt chlleges o the -lght modl lyses re the lmted choces or mesured exctto puts, d the presece o umesured turl exctto put (e.g. turbulece). A better explotto o lght test dt c be cheved by usg output-oly system detcto methods, whch explots recorded vbrto dt uder turl exctto codtos, wthout rtcl cotrol surce exctto d other types o exctto puts (Kehoe 995, Reed 98). here re my deret modl prmeters detcto methods tht could be used or lght lutter testg (Breer 003, Cooper 006, Breer M.. 997, Klepk A. Uhl 008, Verbove P. 004). hs pper presets recursve modl prmeters detcto method o osttory systems. he clsscl RLS lgorthm s supported wvelet trsorm whch llows decouple compoets o the sgl respose d reduce the order o the model. hs operto gretly ccelerted the process o modl prmeter estmto, whch s crtcl prt o the lgorthm. Such pproch llow rel tme mplemetto o the lgorthm. he method ws ppled to detcto o modl prmeters o rple model d to trck requecy chges o the system wth vrble stess. RECURSIVE MEHOD OF MODAL PARAMEERS ESIMAIO he proposed lgorthm cossts o three m prts. I the rst step the sgl s decomposed by wvelet trsorm. I secod step model prmeters re estmted. I thrd step modl prmeters d ther stdrd devto re determed. Orgzto o the lgorthm show Fgure. Fgure : Orgzto o the lgorthm. All prts o the lgorthm re descrbed detl the ext subsectos.. Wvelet rsorm he wvelet lyss s method o sgl decomposto. As result o the wvelet lyss, cotrdcto to the Fourer trsorm, elemetry sgls so clled wvelets re obted. Wvelet uctos re cotuous, oscllted wth vrous durto tmes d spectrums. From the mthemtcl pot o vew, cotuous wvelet trsorm (CW) o sgl x(t) c be deed s (Uhl. 005) (K. A. Uhl. 006): x, b xt W g * g t b dt ()

3 3 Usg propertes o wvelet trsorm, c be proved mthemtclly tht ths kd o tme requecy lyss decouplg requecy compoets o sgl. Utlzg bove metoed eture s sgl processg method ech compoet o the sgl c be lyzed seprtely. Addtolly order o the model usg to detcto process s reduced to model or oly gve requecy compoet o the sgl. hs pproch decrese computtol eort whch c be sgct or hgher model order systems especlly durg dg roots o the chrcterstc polyoml. Besdes, t s ot lwys demd to trck ll turl requeces d dmpg rtos o the system. Usg wvelet ltrto oly crtcl modes (rom the pot o vew o process) c be trck. It lso reduces demd or computtol cpblty whch crese pplcblty o method d mkes rel tme mplemetto process eser. Schemtclly process o requecy compoet decouplg s show Fgure. More detled ormto bout requecy compoets decouplg c be oud t (Uhl. 005). Fgure : Wvelet bsed sgl compoets decouplg.. Recursve Lest Squre lgorthm Method o estmto model coecets bsed o RLS lgorthm. Schemtclly the lgorthm s preseted Fgure 3. Fgure 3 : Orgzto o Recursve Lest Squre method. he dcte successve umber o smple rom A/D coverter or memory buer. Determto o modl prmeters rom the receved sgl model s tme cosumg or o hgher orders models d requre to d the roots o the chrcterstc polyomls. By usg wvelet lterg, oly oe requecy compoet o the sgl s lyzed. hs determes the order o estmtg model. It s worth tkg to cosderto tht by usg wvelet trsorm the problem o selecto the model order s solved. hs problem (though very mportt) wll ot be dscussed ths rtcle.

4 4 IOMAC' 4 th Itertol Opertol Modl Alyss Coerece s s 4 rct 4 l, 4 rct 4 l 4 l s s s E P 0 0 P E P 0 0 m m m.3 Modl prmeters d ther stdrd devto hks to low (equl to ) model order, estmto o modl prmeters s reduced to the pplcto o smple mthemtcl ormuls. For exmple, the turl requecy o the system d the correspodg dmpg coecet c be determed rom the depedece: () where:, model coecets, - turl requecy, - dmpg rto, s smplg tme. Whe lytcl depedeces betwee modl d model prmeters re kow t s possble to clculte covrce mtrx o modl prmeters (Aderse P. 007, Aderse 997). (3) where: E - s the expected vlue, ],,...,, [ s - s vector o estmted modl prmeters, 0 - s vector o true modl prmeters, P - s covrce mtrx o modl prmeters. Usg ylor seres expso method, the covrce mtrx o modl prmeters c be expressed s: (4) Where s the ollowg cob mtrx: (5) d - s the vector o model prmeters. I geerl cse the cob mtrx s mpossble to determe, tht s why the cosdered cse the umercl deretto wth the Cetrl Derece method ws ppled. Whe the stdrd devto o modl prmeters s kow, the codece bouds c be esy determed usg sgm method. I ths work three sgm ws ssumed. It mes tht bout 99,7% o the vlues le wth three stdrd devto rge rom expected vlue. 3 ADAPIVE WAVELE FILERIG M dculty o the detcto process wth usg wvelet trsorm lgorthm s selecto o prmeters resposble or decouplg o sgl requecy compoets. he ssue coected wth proper wvelet ucto prmeters selecto mly cocer the Heseberg relto. Accordg to ths rule the sgl cot be lyzg wth the sme good resoluto tme d requecy dom smulteously. Selecto o wvelet ucto requre some compromse

5 5 betwee qulty o ormto rom requecy d tme dom. Addtol coveece o usg costt wvelet ucto or detcto process s tht t hs lmted bdwdth. I stuto where chges o sgl requecy compoet re grter the hl o ssumed bdwdth prmeter (to ssume tht cetrl requecy o wvelet lter s tued to requecy o gve compoet) ltrto process c be correct d obted results c be oly lter respose. hs problem s preseted gure. Detled ormto bout o dptve wvelet lterg method or system prmeter detcto c be oud t (Klepk A. Uhl 008, Uhl. 008). Soluto o costt lter bdwdth problem c be mke bdwdth prmeter codtol o detcto process. hs requres the determto o wvelet uctos or deret momets tme. Applyg tht, ormul must be rewrtte s: W x, b x t g where s gve tme tervl. Determto o wvelet uctos whch llows ltrto o the gve requecy compoet requres deto o scle prmeter ssocted wth the requecy by the ormul: s (7) where s s smplg tme d s requecy correspodg to scle prmeters. From the bove relto shows tht the scle prmeter chges wll lso chge the requecy o the sgl lterg. hs llows to chge the bd wvelet lter durg detcto process. I the preseted lgorthm, the scle prmeter s determed bsed o the requecy estmted rom recursve model. g * t b dt (6) Fgure 4: Scheme o dptve procedure orgzto. Adptto process s relzed by comprg wvelet requecy d requecy obted rom RLS lgorthm. I the requeces hve the sme vlue or ther derece cot gve rge, detcto process s cotued wthout y chges, but ths two requeces hve deret vlue or ther derece hs vlue greter th ssumed, the requecy o wvelet ucto s chged to vlue correspodg to requecy vlue estmted rom RLS lgorthm. Schemtclly, process dptto d dgrm o method wth dptve wvelet lterg s preseted Fgure 4, where e d w re curret estmted requecy d wvelet requecy respectvely. 4 HARDWARE IMPLEMEAIO OF MODAL PARAMEERS RACKIG ALGORIHM he tme-dom method, bsed o clsscl recursve RLS lgorthm, s ppled s the proposed soluto d s ecet eough d reltvely esy to mplemet. he de o the proposed method s bsed o pplcto o wvelets lterg or decomposto o mesured system respose to compoets relted to prtculr vbrto modes. hese compoets c be extrcted prllel wy or ll modes. he procedure c be ppled to y umber o mesured sgls d cossts o steps lsted the dgrm (Fgure. 5b).

6 6 IOMAC' 4 th Itertol Opertol Modl Alyss Coerece he hrdwre rchtecture s show dgrmmtclly Fgure 5. I proposed soluto hrdwre s lmted to two log puts d A/D sgl coverso wth 00 Hz smplg rte d two log outputs wth vlue o turl requecy d dmpg rto or chose orml mode. ) b) Fgure 5 : Hrdwre mplemetto: ) Scheme o mplemeted hrdwre, b) Dgrm o ppled procedure he USB dgtl output s mplemeted to preset results or ll vestgted vbrto modes. Used FPGA Alter chp helps to motor mxmum 60 modes smulteously o-le wth 5 ms perod. 5 CASE SUDIES 5. A ero elstc model he eroelstc equtos dee the tme evoluto o the vector q (t) o the structure geerlzed dsplcemets by the secod-order deretl equto (Klepk A. Uhl 008). he let-hd sde o ths equto cocers the structurl eorts whle the rght-hd sde s the sum o vrous exterl orces. he vector o the mesuremets Z (t) depeds lerly o q (t) d ts rst d secod dervtves ccordg to equto : M q ( t) ( A G) q ( t) K q( t) = F ( t) F ( t) F ( t) Z( t) = t t q q q t q (t) - Vector o the geerlzed dsplcemets, M, A, K - Structurl mss, dmpg d stess, G - Gyroscopc terms due to the eges, F_(t) - Ustedy erodymc orces, F_c(t) - Cotrol surce orces ucto o the delectos t, F_t(t) - urbulece orces ucto o the wd speeds W t. Bsed o preseted model the smulto ws perorm. he scero o smulto ssumed tht speed o ple would be uormly cresed orm 330 kts to 360 kts. he results o modl prmeters detcto o the model or mode resposble or lutter pheome show Fgures 6 d 7. ) b) c t (8) Fgure 6 : Comprso o detcto results: ) dmpg rto, b) requecy.

7 7 ) b) Fgure 7 : Ideted tme hstory o modl prmeters o sttory ero - elstc system wth estmted codece bouds: ) dmpg rto, b) turl requecy.. As c be otced some peks ppered estmted codece bouds. hs occurrece s cused by wvelet dptto process. I the ext step ter wvelet ucto chgg, the RLS lgorthm s clled wth tl prmeters coected wth old wvelet ucto. As result o t covrce mtrx vlues crese temporry. hs problem ws solved hrdwre verso where prllel computto c be perorm. 5. System wth vrble stess he hrdwre mplemetto o lgorthm ws tested o the DOF system wth vrble stess. Expermetl setup o expermet s presets Fgure 8. Fgure 8 : Expermet rrgemet. A electromgetc shker d sgl geertor were used to excte the structure. Whte ose sgl ws used s exctto sgl. Durg expermet, pressure metl bellows ws chged. As result o system stess chges turl requecy o the system ws shted.. me hstory o system respose, result o turl requecy detcto re preseted gure 9. ) b) Fgure 9 : System respose (), Comprso o deted turl requecy o the system (b).

8 8 IOMAC' 4 th Itertol Opertol Modl Alyss Coerece Addtolly, wvelet dptto process ws preseted Fgure 9b (red dshed le). 6 COCLUSIOS All perormed test showed tht dptve wvelet ormul combed wth RLS lgorthm gves stsctory results. turl requecy o the systems re deted much better the dmpg rtos. Good results o turl requecy detcto gves possblty o use ths kd o lgorthm other pplctos. A exmple c be vbrtg strg sesors where requecy o oscllto o sesor resotor c gve ormto bout chges o opertol codto. Aother exmple c be process o rottol speed trckg structures lke wd turbe, ger boxes or mg mches. A ormulted lgorthm llows computg modl prmeters o structures rel tme. Hrdwre mplemetto o the lgorthm s proposed wth the Hrdwre-Sotwre Co-desg pproch,.e. prt relzed by hrdwre d the remg prt by sotwre rug o os II sot-processor coted the FPGA. Usg dptve ormul o wvelet lter, process o wvelet ucto selecto ws smpled. he method eble to trck modl prmeters o the o sttory system eve ther chges re sgct. REFERECES Aderse P., Brcker R., Gourst M., Mevel L., 007. Automted Modl Prmeter Estmto For Opertol Modl Alyss o Lrge Systems, Proceedgs o the d Itertol Opertol Modl Alyss Coerece (IOMAC),. Aderse, P., 997, Idetcto o Cvl Egeerg Structures usg Vector ARMA Models, Alborg Uversty. Breer M.., R.C. Ld d D.F. Vorcek, 997, Overvew o recet lght lutter testg reserch t ASA Dryde, ASA M/479. Breer, M.., 003, o-sttory dymcs dt lyss wth wvelet-svd lterg, Mechcl Systems d Sgl Processg, 003: Cooper, A.A. Abbs d.e., 006, Curret Sttus d Chlleges or Flght Flutter estg, Proceedgs o ISMA 006 Coerece,: Kehoe, M.W., 995, A hstorcl Prevew o lght lutter testg, AGARD Coerece Proceedgs 566, Advced Aeroservoelstc estg d Dt Alyss,: 5. Klepk A. Uhl, Vcher P., 008, O-le lght lutter mrg detecto lgorthm d cse study, Structurl helth motorg: proceedgs o the ourth Europe workshop, 008: Reed, I.E. Grrck d W.H. "Hstorcl developmet o rcrt lutter." ourl o Arcrt, 98: Uhl, Klepk A., 005, Applcto o wvelet trsorm or detcto o modl prmeters o osttory systems, ourl o theoretcl d ppled mechcs, 005: Uhl., Klepk A., 006, he use o wvelet trsorm or -lght modl lyss XII Itertol Coerece o ose d Vbrto Egeerg,. Uhl., Petko M., Krpe Gl, Klepk A., 008, Rel tme estmto o modl prmeters d ther qulty ssessmet, Shock d Vbrto,: Verbove P., B. Cuberghe, P. Gullume, S. Vldut, E. Prloo., 004, Modl prmeter estmto d motorg or o-le lght lutter lyss, Mechcl Systems d Sgl Processg,: