RATIONAL APPROXIMATION AND IDENTIFICATION OF DISTRIBUTED PARAMETER SYSTEMS
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1 oeed of he 7h Wold Coe he Ieaoal edeao of Auoa Cool RAIOAL AROXIAIO AD IDEIICAIO O DISRIBUED ARAEER SYSES V Gubaev O Zhukov Sae Reeah Iue of ASU SAU 4 Aad Gluhkov Kev 368 S Ukae e-al: vf@aekevua Aba: he ufed fe deoal odel uue whh aue o bae o develo he ehod aloh of e aoal aoxao defao ooed fo dbued aaee e wh dee u ouu he odeed uaed ealzao ovee o fe-deoal o-aoal odel of e fo ulea e oeao Aoxao eeeed b ee exao o deede ba fuo whh ae fudaeal oluo of oda dffeeal equao he u of oda ealzao have ueeded eao of eave defao aloh ad equeal odel eouo b eaae a o of oe o eveal ode Kewod: dbued aaee e odel aoxao eave defao oda ealzao uea Gee fuo IRODUCIO hee ae a aoahe ehod deal wh aaee defao oble fo dbued aaee e I oe of hee aoahe he odel uue eleed a a bouda value oble fo aal dffeeal equao (DE A he ae e hee ae a ublao whee odel baed o fe-deoal aoxao o fe elee dezao of DE ae odeed A ahe oeheve eae of he defao oble fo dbued aaee e ha bee evewed fo exale (Bak Kuh 989 he aoal aoxao defao of he fe deoal e ae odeed h ae I ooed o al he uveal aoah fo odel uue eleo whh eealze he u of u ouu elao fo wde la of fe fe-deoal e lud vaou DE e b ea of Gee fuo ufed adzed fuo Suh odel uue ake oble uaed aoal aoxao oveee o he oal e wh ulea e oeao (Glove 988 akla 99 all ohooal ba fuo ae eloed fo oal aoxao of able e hee ae a ublao devoed o h oble (ee e Wahlbe 994; akla 99 Heubee 995 Va De Hof eal 995 Whe aoxae odel eoued u defao o he bae of u- ouu daa oued b exeeal eo o oe hee ae addoal obale oe odel olex well-oede (akla 99 I ode o oe wh h oble he ef eave defao aloh whh e o fd aoal aoxao aee wh uea avalable daa develoed he dea of he offeed aoah o aoxae e b ea of ee exao wh ee o deede ba fuo whh ae fudaeal oluo of fe-deoal oda dffeeal equao he defao edued o aoxao of exeeall obaed ouu b ea of uh fe ee wh ukow o ol exao oeffe bu alo eevalue whh ae aaee of ba fuo he oble uh eae a be ouvel olved f aoal aoxao wll be ake he fo of ouda ealzao Al of uh ealzao allow u eleed daa al odal aal o eoe odel b eaae a fo whh we ue he e ubodel beaue he oa oe o eveal e ode ouda fo ovde alo ohe efeee whh wee ulzed ooed aloh HE ROBLE SAEE o odel defao ool of he olex eal e equed a f o hooe he aoae odel uue I ve had o do h whou a o foao eeal kowlede abou he la If he obe dbued ae aaee o /8/$ 8 IAC /876-5-KR-956
2 haae ae e-deede he we deal wh he ae-e e whh a be haaezed b ala fuo he le ae b veo fuo ha deed o boh he aal vaable z defed oe aea lud he bouda e o eax ( eeal ae A a ule he oee uh e ae alo deeed b he bouda al odo uhe lea e wll be odeed ol ha alo a o foao I h ae he ufed ad fo of odel uue uveal eouh well ahed wh ex ehod of odel defao he dea of uh adzao wa offeed b Bukovk he wde la of e wh dbued lued aaee wa olleed uued ad fo (Bukovk 979 he a haae of e ad fo he Gee fuo o he ule eoe fuo he Gee fuo alled alo a he fluee fuo o he oue fuo Alo wh he ala o veo vaable w ha haaeze he ae-e e ae le odue he adzed fuo f ha eeed a eealzed fuo b ea of whh a exeal volue bouda o al fluee (u o he e a be we dow he ufed fo If he Gee fuo H ζ τ he adzed fuo f ae kow he e ae deeed b equao w H ζ τ f ( ζ τ dζ dτ ( whee he loue of he e he uue ( lude he a quaza odel A la ae he e l a aaee he ( eeeed a w ( z H ζ f ( ζ dζ ( Se wh lued aaee ae he eal ae of ( w ( H ( τ f ( τ d τ (3 he equao (3 well-kow Cauh foula w ( Ф( w Ф( τ u( τ dτ (4 whee Ф ( τ he ao ax of lea dffeeal equao e whh eeae he Gee fuo he eealzed adzed fuo f ( τ exeed a ( τ u( τ w δ ( τ f whee δ eealzed Da fuo I veo ae w ol( w w K w k f z ol( f f K f So w L L H ζ τ f ( ζ dζ dτ k K k l k l l (5 I wde-ead ae whe e hf-vaa he Gee fuo ha exeo H ζ τ H ζ τ o aal u ofe eouh o kow o he feld w bu loal o eal haae whh a be eaued o eaed Reee he a he ouu vaable ol(( K o loal w z wheea eal eauee we have ( ae ( z w( z dz ψ whee ( z fuo defed o he e z / he e ode wh Se ( z ψ weh ψ ( z whe a be oe ube of o f eealzed fuo he bouda o ohe ula ao defed b uue Howeve he e deedee f oeed a a ule wh exeal lued aaee he fo f R u f ( z whee u defe he vaable of he exeal ao o he e f ( z debe dbued fluee o exale e wh eleoae oee he ue wd ae exeal lued u aal haae ae defed b ofuao of ol wd A a eul fo uh e wh he fe ube of u ouu (wdeead ae ae he odel uue a be we a ( H u τ τ dτ ; ; R (6 whee H ( τ ψ ( z dz H ζ τ f ( ζ dζ Hee ψ ( z ae odeed a eealzed fuo ha ovde all eauee lud owe adz fo I hould be oed ha H ( τ ae eeaed b he ae Gee fuo Coequel he e haae of elee H ( τ hould be he ae o olee eah ohe hu he weh fuo ψ (z f ( ζ wll deee he obevabl o oollabl oee of eleva eealzed deee of feedo 3 RAIOAL AROXIAIO Lea e-vaa e ude defe auo uh ha odel (6 a be we he ovoluo fo wh o-aoal (fe-deo ae o aoal (fe-deoal e afe ax fuo G ( wll be odeed Suh e fo wde eouh la of ax-valued afe fuo ae quae of lea oeao (ae A B C D a bewee dffee 6453
3 fe-deoal (o fe-deoal lea veo ae o ha G( D C( I A B A f h la lude he e wh oeao of ulea e ha due bouded Hakel oeao (Glove e al 988 whh ula value σ ( σ af σ < I kow ha he Hakel oeao Г ulea f G( oeod ax-valued afe fuo C( I A B a be eeeed he fo G ( Re( ξ ( ξ G( [ G ( ] (7 ude auo ha olex ube ae loaed he lef half-lae (able e ula value ae dffee ( C Г whee C o Г σ a ulea o of Г I le h eul alo ead o he ae of ulle ula value b u he ha of Shd a (Adaa e al 97 Howeve ol le ula value ae wll be odeed hee he ee (7 ufol ovee Re > he ulea o of aoaed Hakel oeao he ule eoe ax H ( of e-vaa e he ae of ulea oeao Г o of he elee H ( ha a be exeed b deooo H ( ξ ( Re( ξ e (8 he fuo H ( ae ouou alo evewhee o he eax af H u l η { R H l η l C η C } Γ / fo all Мo Aodl wh (Glove e al 988 ee (7 ovee H H H ; oveee of ule eoe ax H ( fo exao (8 o L H Hakel L ovded b he laal loue heoe (ee e akla 99 heefoe he afe ax wh elee ( G ( Re( ξ ( ξ ( o oeod ule eoe ax ha o of elee ( ξ H Re( ξ e ( a be eleed a uaed odel of e Hee oveee G ( G ( H H fo uaaed whee oe of afoeeoed o I fa foula ( debe he aoa fedeoal daal e wh afe ax-valued elee we he fo of aal fao deooo Eah e ( aalooul ( oeod o he eeal deee of feedo o ode of he (9 e Exeo ( ( ae equvale o A H C e B G ( C ( I A B ( whee ae A B C defe he fedeoal ae-ae e e dx A x B u d (3 C x he deo of veo u ae equal o R eevel he oal ouu ealzao (Glove e al 988 balaed ealzao ae uuall ulzed aoxao heo he uaed e ( C A B ude uh ealzao ode wh ( bu fd he equao whh lk elee of ae C A B wh ( he eevalue ξ oeffe oval ak Coeo bewee odel (3 ( beoe le eouh f we ake he oda fo of ealzao ead of balaed o oal ouu ealzao I h ae he ax A a be we a A da{ S S S } whee daoal blok α β S ξ α ± β β α he ae C B ae alo exeed he blok fo C [ C C C ] B [ B B B ] whee L b b L b R C B L b b L b R he aoe oeao So ( a be we a ( α H ( o β β e (4 whee ( b b b b oeove uable o ele he obevable oal oda ealzao f oe of olu of eah ax C aued ; o oollable oal oda ealzao el b b ; R eah blok B I he f ae equao oe he oeffe of deooo (4 wh elee of ae C B ae b R R ( b ( ( R ( ( ( ; R ; ( ( R ( (5 eod ae exeed a ( ; ; ( ( ; ; ; ( 6454
4 b b ( ( ( ( ( ( ( ( ( ( ( ( (6 If he e ha oe u a ouu he eaoable o ue he obevable oal oda ealzao whe he e ha oe ouu a u eaoable o ue oollable oal oda ealzao I hee ae exao oeffe elee of ae of he ae ae odel have he uque oul I eeal ae fo e wh a u ouu we have he ovedeeed e of oul equao o oluo (5 (6 wee alulaed b LS ha led o he avea oedue fo defe e of oeffe Bede he eal eevalue deooo (4 ae he eal ae ha eeved a β b e ( o β Coequel he odel deo o ube of deee of feedo equal o whee oeod o he olex eevalue o he eal oe eevel heefoe (4 aoxae uue fo elee of ule eoe ax ha we a deooo H ( wh ee of deede ba fuo whh ae fudaeal oluo of he equvale lea e of dffeeal equao he Lalae afoao of (4 [ ]( LH lead o aoal afe fuo ha eeeed a aal fao ula deooo wh ole α ± β he lef half-lae ; 4 IERAIVE IDEIICAIO Ohooal ba fuo have eloed he effeve ool fo he uoe of e aoxao defao I eablhed fa ha eve able e ha a uque ee exao e of uh ba a fe-leh ee of uh exao a eve a a aoxae odel Howeve eal udeable ha he aua of he aoxao wll be eeall deede o he hoe of ba fuo If he da of he ba eea e he da e o be odeled ae loed we wll have he fa oveee So ooed fo aoxao eave defao o al deede bu o eeal ohooal ba whh loe o eefuo of he ode e he ule eoe fuo (4 ae exeed a deooo of deede fuo ha ae fudaeal oluo of (3 Hee ae kow o ol ( exao oeffe bu eevalue ( aaee α β alo If eevalue ae kow he oal aoal aoxao would be aaloou o he ohooal ba ae So offeed o eae a f α β ( afe ha fd he oal value of ( he develoed ehod e defao a be aled fo ealz of h aoah he eauee obaed fo exee ae avalable daa If equed o eablh he aoal aoxao of kow bouda-valued oble fo DE wh fe ube of exeal u ouu he eleva daa a be obaed fo ouaoal ulao o eal vual la defao oble ea he ae he u a lude dffee al bu hould be foave exe all fa e ode o eablh he foaabl odo le ode a f he ex fluee ( ( u ( u u o ω ω (7 aled o eve eaae u Iu (7 allow o fo vae dffee fluee Releva eoe a he -h ouu ( ( ( ( d d β ( d ω d ( 3 ( 4 oω o β e α (8 whee ( ( ( d [ u ( u ( ] 3 4 ( ( ( [ u ( u ( ] 3 4 ( ( ( ( d [ u ( u ( ] 3 4 ( ( ( [ u ( u ( ] 3 4 ( ( ( 3 ( d u ( ( 3 4 ( ( u ( ( 3 4 ( ( ( 4 ( d u ( ( 3 4 (9 ( ( ( u ( ( 3 4 α α α ( β ω α ( β ω β ω 3 β ω 4 α ( β ω α ( β ω he ouu (8 o of boh eefuo whh defe ae fuo ha laf he ead-ae oe whh ae aued b (7 Bu efeed o ele fo (8 ehe foed oo o ae oe ue he eaael o ele he foed vbao fo able e eouh o ake he daa fo >> whee ae e he oal 6455
5 aoxao of (>> b uaed u a be aed ou b fe-feque defao ehod Howeve defao o he bae of foed oo hee ex he oble of odel deo eleo Aael he eave defao o he bae of ae daa oe efeable fo eouo of aoxae odel I ode o ele he ae behavo le efo he uleea eao of alo ov eval π A a eul we eeve a ew al ω ha equal o ( α d β d o β e ( π ω dθ π θ ω dθ θ ( θ θ ω π ( ( θ dθ ( Coeffe d d ( ae exeed leal va d d bu ubeoe wa Afe e he al fo exee alula he oval ak of aaee α β d d evalua fo ( hould be olved Code ow he ouao of fo (9 h ak a be oel olved f deea ( ( [ ] u u 3 4 [ ] ( ( u u 3 4 equal o zeo Whe aaee α β ae ukow he eeal ae dfful o eae he alude ( d d whh exe all ode ude odeao I que oble ha he ae u a ve exao axu fo oe ode whle ohe ode wll be o he level of dubae So ueed o ue al ha ae le fo aal whh ae able o exe he ode wde eevalue ae he eaula ule wh alude u ha aahed o he -h u o he eval [ ] afe hee odo o eave defao hee a be ued he fee oo a ha a be we a ( u e α ( e α α ( β ( ( α ( { [ β ( θ ] o[ β ( ] θ} e o β ( ( o he ae behavo a equal o ( ( α ( u { ( e [ β ( θ ] α β ( ( α ( θ ( e o β ( θ oθ [ ] } Aod o ( ( he alude of ex ode deeae veel oooall o α β eae del oooall o u Bede eaoable o eae he wdh of ule fo all value of α he fa eevalue ha a be defed ou he lae ( α β eal doa ha lude he ode wh foave al o he bakoud of daa eo he o boadb he u ake a deal ule u u δ ( ha ve he ouu ( ( u ( [ o ( ( β ( α ( ] e β (3 Code ow ea oble eave hee of aoxae odel eouo I ueed u aalal ouu ( ( o (3 oeod exeeal daa o he ae [ ] o [ ] o eou odel eavel b dee a eah eao aaee of oe o eveal ode Oe hould a wh ode ha ve he al o oe ao eouh lae a he ed of eval ha eal Obvoul ode wh he alle value of α wll ve a obuo hee Whe we oe ea o o he ube of eeall foave ode wll ow o aou of laeα h wll be ued eave defao he dea o eee he odel a aeao of ubodel Eah ubodel wll be eoued eaael Daa o he ubeval eal allow o deee all ukow aaee of he f ubodel Afe ha we evaluae he ouu of h ubodel u ( ( o (3 uba fo ha wa obaed exee hu we ( ( fd a ew al whh adble fo eao of eod ubodel aaee h al wll eal have he ow foave eval [ ] o [ ] we ele ubeval fo eod ubodel efo he ae ao a fo f ubodel Suba al ( ( fo we defe ew oeod ubeval ollow hd ubodel a be defed Suh eao ae eeaed ul he al ( ( ( K beoe duhable o q he oe bakoud Aeao of all ubodel whh wee defed wll ve he ouda ealzao of fedeoal aoxao ha oe wh he ex uea Le u o aloh of ubodel aaee eao I offeed he hee o of wo ae A he f ae all ubodel α β hould be deeed he eod ae ha addeed o exao oeffe evaluao a be ealzed b ad LS ehod Se he eod ae he well-kow ak we debe ol aloh of ubodel eevalue eao Dffee aoahe wee odeed oe of whh o he follow q q ( ( ( ( o K whee q 6456
6 ( ( we fo ew al d ( ( ( ( q ( γ γ d (4 q q ( d ( d ( ( q q ( α α β q q α β d α β d ( ( ( ( γ γ ( τ d (5 whee τ q 3 q q ( q 4 ( α β ( α β ( q ( τ ( ( τ dτ q α ( q ( τ dτ α β γ ae vaed Le ake he equee of ( ( γ fo 3 aled daa { } q { ( ( β } α fo 4 wh he ubeal ae q of def ubodel Cooe he Hakel ae ( ( ( K q ξ q q ( ( Y 3 L ( ξ q q q q ( ( ( ( η η L ξ η q q q (6 ξ η 4 ( Le he SVD of Y be ve b q ( ( ( ( Y U V (7 q q q q ( ( whee U q V ae ohooal ae q Σ ( q a daoal ax wh he ula value o-ea ode o he daoal Le aalze behavo of ula value whe α β γ ae vaed Reul ehe oe ula value (fo ae wh γ o wo (fo α β ed o zeo Coeod lluao ae eeeed o f f whee ula value behavo fo ae of eal eevalue how oeod o dffeeal f - o eal afoao Suh aloh beoe oe effeve f a o oble o fd he ouh eao fo eevalue o o o ou afflao eval fo he Soee h a be doe del fo ae behavo o ubeval 3 4 Reak If we va aaeeα β γ hee ae eaaed o ol ula value of ubodel o be defed bu alo he ula value of he ode wh weak eoe o dubae I a ae oble o eae he eevalue oe eel b aalz of o ula value afoao Reak Soe ouu fo dffee a oa o-foave al of defe ode due o he bad oollabl o bad obevabl So he olee odel oled fo all def ode wh ee o R u-ouu 5 COCLUSIO Seleve daa hoe fo ubodel defao ea ea la o ohooalzao oedue So e of oohooal ee exao due o ef al beoe oble o ealze eave odel eouo b eaae a he a d of he develoed eave defao ehod ha eao ed o aoal aoxae odel wh aaee ba fuo whh ve al devao bewee da of eah eal e eealzed deee of feedo he odel oeove e eao ae eaed whe all foave ouu al of ode beoe exhaued we oba he odel deo ha full oe wh he eo avalable daa REERACES Adaa V DZ Aov G Ke (97 Aal oee of Shd a fo a Hakel Oeao he eealzed Shu-aka oble ah USSR-Sb Bak H K Kuh (989 Eao ehque fo Dbued aaee Se Bkäue Boo Bukovk AG (979 Dbued aaee Se Chaae 4 ( ua See oow Glove K R Cua R ao (988 Realzao Aoxao of Lea Ife- Deoal Se wh Eo Boud SIA Co OzVol akla (99 O Idefao of Sable Se Oal Aoxao Auoaa akla (99 Aoxao of Sable Se b Laee le Auoaa Vol Heubee S Va de Hof OH Boa (995 A eealzed Ohooal Ba fo Lea Da Se IEEE a Auo Cool AC Wahlbe Bo (994 Laee Kauz odel e h IAC S o Se Idefao SYSID 94 Vol3 - Deak Coehae 6457
X-Ray Notes, Part III
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