Nonlinearity-Compensation Extended Kalman Filter for Handling Unexpected Measurement Uncertainty in Process Tomography

Transcription

1 oieariy-compesaio Exeded Kama Fier for Hadig Uexpeced Measureme Uceraiy i Process omography Jeog-Hoo Kim*, Umer Zeesha Ijaz*, Bog-Seo Kim*, Mi-Cha Kim**, Si Kim*** ad Kyug-You Kim* * Deparme of Eecrica & Eecroic Egieerig ** Deparme of Chemica Egieerig *** Deparme of ucear ad Eergy Egieerig Cheju aioa Uiversiy, Jeju , Korea (e: ; E-mai: yugy@cheju.ac.r) Absrac: he objecive of his paper is o esimae he coceraio disribuio i fow fied iside he pipeie based o eecrica impedace omography. Specia emphasis is give o he deveopme of dyamic imagig echique for wo-phase fied udergoig a rapid rasie chage. oieariy-compesaio exeded Kama fier is empoyed o cope wih uexpeced measureme uceraiy. he oieariy-compesaio exeded Kama fier compesaes for he ifuece of measureme uceraiy ad soves he isabiiy of exeded Kama fier. Exesive compuer simuaios are carried ou o show ha oieariy-compesaio exeded Kama fier has ehaced esimaio performace especiay i he uexpeced measureme evirome. Keywords : oieariy-compesaio Exeded Kama Fier, Exeded Kama Fier, Eecrica Impedace omography, Dyamic Image Recosrucio, Process omography, Roo Mea Square Error.. IRODUCIO Process omography (P) ivoves usig omographic imagig mehods o maipuae measureme daa from sesors i order o obai precise quaiaive iformaio o he iaccessibe regios. he regio may be for exampe, a furace, a mixig chamber or a pipeie, ad he omography imagig ca be based o eecromageic or acousic soudig or radioscopic imagig. I essece, he goa is o esimae compuaioay he muidimesioa disribuio of some physica parameer based o idirec observaios from he boudary of he objec []. ypica feaures of idusria processes are a high oise eve ad rapid chages i he objec. hus, he imagig modaiy has o be sufficiey fas ad robus for proper dyamica chage of he arge. We cosider he probem of imagig he coceraio disribuio of a give subsace i a fuid movig i a pipeie based o saic or ow frequecy measuremes o he surface of he pipe. Se of coac eecrodes are aached o he surface of he pipe, ad are eecroicay isuaed from he pipe. Eecric curres are ijeced hrough hese eecrodes ad he correspodig voages eeded o maiai he curres are recorded. Hece he imagig modaiy used i his case is Eecrica Impedace omography (EI). As compared o he radiioa EI, i he prese case, he objec is very rapidy chagig durig he daa acquisiio; hece a reasoabe spaioempora resouio is desirabe. Raher ha cosiderig he iverse probem as a radiioa omography recosrucio probem, we view he probem as sae esimaio probem. he coceraio disribuio is cosidered as a sochasic process, or a sae of he sysem, ha saisfies a sochasic differeia equaio. his equaio is referred o as sae evouio equaio. We mode he coceraio disribuio by he covecio-diffusio equaio, which aows a approximaio of he veociy fied. We cosider here approximaig a fas fow wih a amiar fow ad compue he veociy fied by sovig he avier-soes equaios umericay. Coveioay, he sae esimaio is performed by usig Kama fier, fixed-ag Kama smooher or exeded Kama fier (EKF) agorihm. I our case we have used oiear-compesaio exeded Kama fier (CEKF). he wor fow is expaied i Fig.. EI Imagig Modaiy is used o measure voages o eecrodes. Probem domai is discreized ad soved usig Fiie Eeme Mehod. his sep is caed Forward Sover. omographic image recosrucio is doe hrough dyamic fier. his sep is caed Iverse Sover. Fig.. Worfow of he ypica recosrucio process i Process omography (I his case, a sraigh pipe is cosidered). he purpose of he prese wor is o appy CEKF o dyamic P for performace ehaceme of he dyamic image recosrucio i he presece of uexpeced measureme uceraiy. his uexpeced measureme

2 uceraiy ca be ay exera shor-ivig perurbaio i he measureme daa. Usuay such perurbaios cause he coveioa EKF o diverge ad esimaio performace is deerioraed drasicay. he isabiiy of EKF i such cases is a major boe ec for such perurbed sysems. I order o ace his probem, CEKF is empoyed i Iverse Sover which has aready proved is migh compared o EKF i opimizaio probems reaed o oher was of ife, especiay arge Moio Aaysis. See []. he res of he paper is orgaized as foows. I secio, we have expaied he discree sae-space dyamic mode cosiderig covecio-diffusio mode. For he breviy of discussio, we have ep our discussio shor. Furher deais o P ca be foud i [6-]. Secio 3 deas wih EI appied o P. Oy he Observaio mode is discussed. Secio 4 deas wih is ad ous of CEKF. Secio 5 deas wih he simuaio ad compariso of resus.. DISCREE SAE-SPACE DYAMIC MODE I he case of movig fuids io he sraigh pipe he coceraio disribuio c = c( x, ca be modeed by he sochasic covecio-diffusio equaio as foows c = κ c v c µ () κ = is he diffusio coefficie, = v(x) µ = is he where κ (x) v is he veociy of he fow ad µ ( x, modeig errors. Icompressibiiy is defied as v = () Which represes ha desiy of fuid is same hroughou he fied ad i does o chage wih ime. Boudary codiio is defied as c = a x ( Ω Ω ) \ (3) iwa which meas ha here is o diffusio hrough he pipe was ad he ipu boudary, so ha he ouward ui orma is orhogoa o he veociy of he fow i he wa. Iiia codiios are c ( x,) = c ( x) (4) c( x, = c ( a x Ω i i (5) (4) represes he iiia vaue a = ad (5) represes he Diriche codiio which ca be ae io accou by usig he Perov-Gaeri mehod. () ca be soved i discree form usig he Perov-Gaeri mehod ad he bacward (impici Euer mehod as c (6) = Fc s w where F is he sae rasiio marix, s is he ipu vecor ad w is he R disurbace vecor. Here, we assume a iear mode saisfyig R ( x, = λc( x, (7) he reaso for his assumpio of coceraio c ( x, is o esimae i by eecrica impedace omography. Sice here is a direc iear reaioship bewee coduciviy ad coceraio, hece by usig EI, we ca recosruc coduciviy ad he ca map coceraio agais i. his is he mai reaso why EI is used as imagig modaiy. We ca obai he discree sae-space mode as foows = F s w (8) where F, s ad w are fucios o reae bewee he resisiviy ( x, ad c x ieary. (, ow, e us cosider he case i which he ime sep is oo arge i compariso o he veociy of fuid, for ha case, he bacward Euer mehod is iaccurae whie sovig he covecio-diffusio equaio umericay by he evouio mode. Assume he ime sep / is sma eough o obai a feasibe umerica souio for he sochasic covecio-diffusio equaio. Here, he sae equaio correspodig o he ime sep / is used as foows. [8] (9) = F s w is he ime sep used i he evouio mode. Where We ca obai he ex sep as = F s w = F( F s w ) s w = F ( Fs s ) ( Fw w ) () Simiary, 3 = F ( F s Fs s ) 3 3 ( F w Fw w ) () 3 Furhermore, = F where ( F F ) s ( F F ) w ( s w ) = () F Γ Γ is ( F F ).

3 Hece, we ca obai he sae evouio equaio as Where ( s w ) = F Γ (3) F is he evouio marix, s w is he disurbace is he ipu vecor ad vecor. 3. EECRICA IMPEDACE OMOGRAPHY Whe eecrica curres I ( =,,..., ) are ijeced io he objec Ω hrough he eecrodes e ( =,,..., ) aached o he boudary Ω wih he iera srucure, he coduciviy disribuio ( x, is ow for Ω, he correspodig eecrica poeia u ( x, o he Ω ca be deermied uiquey from he paria differeia equaio, which ca be derived from he Maxwe equaios as ( u) = i Ω (4) wih he foowig boudary codiios based o he compee eecrode mode: u ( ) u z = U o e, =,,..., (5) u ( ) ds = I, =,,..., (6) e u = o Ω \ e (7) = where, z is he effecive coac impedace bewee h ( ) ( ) eecrode ad eecroye, U = U ( ) is he poeia o ( ) ( ) he h eecrode a ime, I = I ( ) is he ijeced curre o he h eecrode a ime, e is h eecrode, is ouward ui orma, ad is he oa umber of eecrodes. Furhermore, he foowig wo cosrais for he ijeced curres ad measured voages are eeded o esure he exisece ad uiqueess of he souio: = = ( ) I = (8) ( ) U = (9) he compuaio of he poeia u ( x, o he Ω ad () he voages U o he eecrodes for he give coduciviy disribuio ( x, ad boudary codiios is caed he forward probem. I geera, he forward probem cao be soved aayicay, hus we have o resor o he umerica mehod. here are differe umerica mehods such as he fiie differece mehod (FDM), boudary eeme mehod (BEM), ad fiie eeme mehod (FEM). I his paper, we used he FEM o obai umerica souio. I FEM, he objec area is discreized io sufficiey sma eemes havig a ode a each corer ad i is assumed ha he coduciviy disribuio is cosa wihi each eeme. he poeia U a each ode ad he eecrodes a ime, defied by he vecor U = R( ) I () where, R( ) ad I are he fucios of he coduciviy disribuio io he objec ad he ijeced curres hrough he eecrodes a ime, respecivey. For more deais o he forward souio ad he FEM approach, see [8,] Here, e U R, defied as [ U U U ] U... () be he measureme voages o he surface ad iera h eecrodes iduced by he curre paer. he he observaio equaio ca be described as he foowig oiear mappig wih measureme oise U = V ( ) v () where he measureme oise v is aso assumed o be whie Gaussia oise wih covariace. For deais o FEM forward sover for EI, cosu chaper 3 i []. 4. IVERSE SOVER BASED O OIEARIY-COMPESAIO EXEDED KAMA FIER 4. oiear-compesaio Exeded Kama Fier agorihm From (3 ) ad (), we ca obai he dyamic equaios as foowigs ( s w ) = F Γ (3) U = V ( ) v (4) I EKF he sae esimaio is opimized as miimizig he cos fucioa as foows J ( ) = E{ ε } ε = {( z h ( )) R ( z h ( )) ( ) P ( )} (5)

4 where E { } is he expecaio, is he aes prediced sae ad is E { }, so ha he R measureme oise covariace. P R is he ime-updaed error covariace marix, which is defied by P E{( )( ) } (6) iearizig (8) abou he curre prediced sae we obai U = V ( ) H ( ) H O. s. (7) v where H. O. s represe he higher-order erms which wi be cosidered as addiioa oise, H is he Jacobia marix defied by H V ρ ρ = ρ where ρ is he resisiviy i.e,. ow we defie he pseudo-measureme as y U V (8) ( ) H (9) Ad hece, we ca deveop he iearized observaio equaio as foowig y = H v (3) I compariso o he cos fucioa defied for image recosrucio for EKF, he cos fucioa for CEKF is compued as foows J ( ˆ ) = {( z H xˆ ) R ( z H xˆ ) ( ˆ ˆ ) P ( ˆ ˆ )} (3) By miimizig he cos fucioa ad sovig for he updaes of he associaed covariace marices, we obai he CKEF agorihm which cosiss of he foowig wo seps simiar o EKF. (i) Measureme Updae Sep (Fierig) [ H P H R ] αk ( y H ) K (3) = P H i = = ( I K H ) P (33) C β (34) (ii) ime Updae Sep (Predicio) P = F P F Γ Q Γ (35) = F s (36) where F F s is he ipu vecor. α is used o adjus he Kama gai = is he evouio marix ad K i equaio (33), he rage of α is ~ ad β is deermied by α : = α : α β (37) < α Here, he coefficie α adjuss a opimizaio vaue of he Kama gai accordig o he uceraiy of a measureme vaue. Whe α =, he resus obaied from sae esimaio probem are equa o he resu of coveioa EKF. his meas ha CEKF is worig ie coveioa EKF. Whe α =, he sae is o updaed. So, a prediced sae is used isead of a fiered sae. his meas ha whe he sysem is esimaed by he ucerai measureme oise he prediced sae is o updaed. Sice β is a parameer adjusig he error covariace marix of equaio (34) depedig o α so he more α is far from he more β is decreased. Aso, he process error covariace measureme error R is deermied by Q ad he Q = E{ w w } (38) R = E{ v v } (39) Where w is he Whie Gaussia oise for he process a ime sep ad v is he Whie Gaussia oise for measureme daa a ime sep. 4. empora Reguarizaio Because he dyamic recosrucio is depede o ime, so for recosrucio we jus eed empora reguarizaio, o spaia reguarizaio. empora reguarizaio is cosidered i hree compoes as foows Q β µ µ = I (4) Q β ηi = (4) η Q β η η I = (4) (4) is he sochasic aure of he diffusio, we assume he oise µ is ucorreaed ad havig cosa variace i a

5 pars of he phaom. (4) represes a ucerai osciaory compoe i he pipe ie ad (4) meas he ipu sream is assumed o be very sowy varyig i he ime scae. Hece, process error covariace is represeed as foows Q = YQ Y DQ D HQ H (43) µ η η where he marices Y,D ad H are he fiie eeme marices mappig he radom vecors µ, η ad η, respecivey. [8,] Here, β, µ β ad η β is obaied empiricay. η 5. SIMUAIO RESUS We have carried ou he compuer simuaios o syheic daa o evauae he recosrucio performace of CEKF. he compuer simuaio was carried ou o a sraigh pipe icudig varyig measureme oise. Paraboic veociy fied are aso cosidered. Fig.. Sraigh pipe-ype FEM mesh(mesh for iverse probem) ad eecrodes. he FEM meshes used for he iverse sovers are show i Fig.. We have used he sraigh pipe-ype mode wih a mesh size of 394 eemes ad 5 odes. We have used a fie mesh ear he boudaries i order o mae a good sesiiviy aaysis cosiderig he compicaios ivoved i measureme. Eecrodes are ocaed o each side of pipe as a se of 8, he oa umbers beig 6. vaue of coduciviy disribuio is se o / 4 Ω cm. he ijecio paer uses he opposie mehod. he ime o measure voages of a paer is se o 5 ms. ex, simuaios were carried ou o aayze effecs o he image recosrucio by he ucerai measureme oise o he foowig daa. Iiia assumed coduciviy is =. 43 Ω cm, iiia assumed covariace for he iiia sae vecor is C = (. I, he average veociy i x -direcio is ) assumed cosa for give ime. he covariaces wih respec 3 o he empora reguarizaio are β = 5 ad 6 β η =, = η β. he measureme oise ν is se o.% of he differece bewee he maximum ad he miimum vaue of he voage wihou he oise. he uexpeced measureme uceraiy oise cosised of Whie-Gaussia oise ha occurs for 5 ime seps from 3 h sep oward. For he sae of compariso of performace of he recosrucio agorihm, roo mea square error (RMSE) is defied as foowig RMSE V ( ( )) = ( U rue V ( )) ( U rue V ( )) ( U ) ( U ) rue µ rue (45) We have cosidered wo cases: 3% ad % of uexpeced measureme uceraiy oise of he differece bewee he maximum ad he miimum vaue of he voage wihou he oise. 5. Simuaio Resus : Aaysis of uexpeced measureme uceraiy Fig. 3. Compued veociy fied iside sraigh pipe. he veociy fied is assumed o saisfy he codiios of paraboic fow as show i Fig 3. Here, he equaio wih respec o he fow across x -direcio is deveoped as foows v ( x, = v x x, mea y y R (44) Where v x, is he spaia average veociy i x -direcio. mea y is he idex of y as disace from ceer of he pipe ad R is he ier radius of he pipe. I is aso assumed ha he iiia average veociy i x -direcio, v, is 45 cms x mea. he iiia seig for parameers used i he simuaio is as foowig. he coac impedace z used i he simuaio is. Ω. he covecio coefficie χ is 5. umber of frames for curre ijecio is 5. he miimum vaue of coduciviy disribuio is se o / Ω cm ad he maximum (a) RMSE for V ( ) (b) α updae i CEKF (c) RMSE for V ( ) (d) α updae i CEKF Fig. 4. (a) ad (c) represe RMSE for V ( ) wih he ucerai measureme oise 3% ad% respecivey. (b) ad (d) represe he variaio i α cases for he wo cases

6 7. ACKOWEDGMES he wor was suppored by gra o. R (4) from he Basic Research Program of he Korea Sciece & Egieerig Foudaio. (a)rue (b)ekf (c)cekf (d)ekf (e)cekf Fig. 5. Image recosruced accordig o each ucerai measreme oise (bewee he ierva of 8 h ad 5 h ime seps). (a) rue Image Frame. (b) ad (c) Image recosruced wih 3% ucerai measureme oise. (d) ad (e) Image recosruced wih % ucerai measureme oise. I Fig. 4 ad Fig. 5, we ca see ha whe uexpeced measureme uceraiy occurs, here is a ie fucuaio i RMSE wih CEKF as compared o EKF sice EKF agorihm jus seecs he Kama gai ha opimizes ieary ad ca o opimize agais he oieariy pheomeo. O he corary, CEKF modifies he Kama gai by α, ad esimaio quaiy is beer ha EKF i case of oieariy. 6. COCUSIOS A dyamic impedace imagig echique is appied o he visuaizaio of wo-phase fow fied udergoig rapid rasie. I h i s paper, oiear-compesaio exeded Kama fier is empoyed o cope wih he uexpeced measureme uceraiy. We have poied ou he ehacemes i he esimaio for he cases whe ucerai oise exiss i he sysem. I hose cases, oiear-compesaio exeded Kama fier is far more effecive ha coveioa exeded Kama fier i erms of spaia resouio of recosruced image. For he verificaio of our hypohesis, we have simuaed a bubby fow ad a sug fow ad recosruced he pipe-ype images wih syhesized daa ad have compared he resu based o roo mea square error. he recosruced images idicae a good possibiiy of dyamic process omography sysem wih iegraed oiear-compesaio exeded Kama fier o he visuaizaio of rapid rasie wo-phase sysem udergoig sudde perurbaio. 8. REFERECES [] Arhur Geb, Appied Opima Esimaio, MI Press, Cambridge,974. [],,,,,,,,,,! E(Eecrica omograph " # $ % &!, ( ) *, -. /, Ju.. 7 [3] = >?A@, 3 4 C 5 6! 8 9 : ; < ", B ) D 8 8 E F ) ( G ) H I, Dec.. [4] J KA, Dyamic Eecrica C Impedace omography wih Prior Iformaio, B ) D 8 8 E F ) ( G ) H I, Dec.. [5] R.A.Wiiams, M.S.Bec, Process omography : Pricipes, echiques ad appicaios, Buerworh-Heiema d, 995. [6] A. Seppäe, M. Vauhoe, J.P. Kaipio, E. Somersao, "Iferece of veociy fieds based o omographic measuremes i process idusry", 4h Ieraioa Coferece o Iverse Probems i Egieerig, Rio de Jaeiro, Brazi,. [7] A. Seppäe, M. Vauhoe, P.J. Vauhoe, E. Somersao ad J.P. Kaipio, "Fuid dyamica modes ad sae esimaio i process omography: Effec due o iaccuracies i fow fieds", Deparme of appied physics repor series ISS , Uiversiy of Kuopio, Mar.. [8] A. Seppäe, M. Vauhoe, P.J. Vauhoe, E. Somersao ad J.P. Kaipio, "Sae esimaio wih fuid dyamica evouio modes i process omography - EI appicaio", Deparme of appied physics repor series ISS , Uiversiy of Kuopio, Ju.. [9] A. Seppäe, M. Vauhoe, E. Somersao ad J.P. Kaipio, "Sae space modes i process omography - approximaio of sae oise covariace", Deparme of appied physics repor series ISS , Uiversiy of Kuopio, Oc.. [] A. Seppäe, M. Vauhoe, E. Somersao ad J.P. Kaipio, Sae esimaio wih fuid dyamica evouio modes i process omography A appicaio wih impedace omography Iverse Probems, vo. 7, pp ,. [] oieariy-compesaio Exeded Kama Fier ad Is Appicaio o arge Moio Aaysis, Oi Eecric Idusry Co. d., o. 59, Vo. 63, Ju 997. [] M. Vauhoe, Eecrica impedace omography ad prior iformaio, Ph.D hesis, Kuopio Uiversiy, Kuopio, 997.