8th WSEAS International Conference on SIMULATION, MODELLING and OPTIMIZATION (SMO '08) Santander, Cantabria, Spain, September 23-25, 2008

Transcription

1 8h WSEAS Inernaonal Conference on SIMULAION, MODELLING and OPIMIZAION (SMO 08) Sanander, Canabra, San, Seeber 3-5, 008 Obervably of 4h Order Dynacal Sye Jerzy Sefan Reonde Faculy of Auoac Conrol, Elecronc and Couer Scence Slean Unvery of echnology ul. Aadeca 6, Glwce POLAND Abrac: - he aer devoed o analye he obervably of 4 h order ye.. Followng he a of forulaon he condon of nal obervably of enoned fourh order ye ecral heory for lnear unbounded oeraor nvolved. In he arcle he heore on neceary and uffcen condon on nal obervably are forulaed and roved.. Key-Word: - Obervably, Drbued Sye, Secral heory, Lnear Oeraor Proble Saeen Before aeng o chooe he conrol algorh for he echancal ye, neceary o deerne wheher he ye obervable. Deernng a ae of an nfne denonal dynacal ye fro eaureen daa a fundaenal ub-roble n he roble of ablaon or oaon of a feedbac conrol ye. he noon of a nal obervably uable o deerne he nuber a well a he collocaon of he enor. In he aer Reonde [4],[5],[7],[9] we acled lar conrollably and ozaon roble. In he aer Reonde [],[3],[6],[8] we nroduced he nuercal ehod no h feld. In he aer Reonde [] we analyzed he nd order ye wh elfadjon ae oeraor. Le u conder dynacal ye decrbed by he followng lnear abrac dfferenal equaon (): 4 δβ d x() ( β ) γ d x() 4 0 A A d β β β d () δα ( α ) γ α0 αa α A x() Bu(), > 0 wh nal condon: x(0) x0 D( A), ( ) x (0) x,,,3 () where x() ( a Hlber ace ), conan coeffcen: β 0, 0,.., δ, α 0, 0,.., δ β he oeraor A exonen α γ γ are ( α) ( β), R ( α) ( β) conraned: 0 < γ, γ <. Addonally we wll aue ha α coeffcen u no be ulaneouly equal o zero and he egenvalue of he ye () are dnc by ar. he nu oeraor B defned a follow: P P Bu() b u (), B L( R, ) where: B [ b, b,..., bp], u( ) [ u( ), u( ),..., up( ) ], (3) b,,,..., P I aued ha A : D( A) a lnear, generally unbounded, elfadjon and ove-defne oeraor wh doan D(A) dene n and coac reolven R(λ,A) for all λ n he reolven e ρ(a). (auon ). he ae of he ye () oberved by averagng enor whoe ouu are gven by (4): y ( ) c, x( ),,,..., (4) where c,,,..., are fxed eleen n rereenng aal weghng funcon of enor. Ouu of uch enor are aal average of a hycal quany over oe effecve enng regon. I nroduced he ouu vecor y (): y () y() y()... y () Cx () (5) where C: R a bounded lnear oeraor defned by : Cx() c, x() c, x()... c, x() (6) he hycal nerreaon of he equaon () encoae a broad cla of real ye n h for and deend on a arcular for of he A oeraor and of he coeffcen α and β. I well nown ha he oeraor A ha he followng ecral roere []: - Oeraor A ha only urely dcree on ecru conng enrely of dnc real ove egenvalue λ each wh fne ullcy ( < ) : 0 < λ < λ <... < λ < λ <..., l λ (7) ISSN: ISBN:

2 8h WSEAS Inernaonal Conference on SIMULAION, MODELLING and OPIMIZAION (SMO 08) Sanander, Canabra, San, Seeber 3-5, For every x he followng unque exanon hold rue: x < x, φ > φ (8) j - Oeraor A ha he followng ecral reoluon: x D ( A) Ax j λ < x, φ > φ (9) (0) -he fraconal ower of oeraor A defned a follow: β x D ( A ), β (0,) A β x j λ β j < x, φ > j φ j () () -Oeraor A β, 0<β< alo elfadjon and ovedefne wh doan D(A β ) dene n. he ranforaon of he Sae Equaon A fr erfor he ranforaon of gven fourh order lnear abrac dfferenal equaon () o equvalen for of he fr order lnear abrac dfferenal equaon. I can be erfored by nroducng new defned by he forula a follow: ( ) d x () ξ (), 0,,,3 (3) d By h ubuon we can receve he followng fr order lnear abrac dfferenal equaon (4) fro he equaon (3): dς () A ς ( ) (4) d where he ae vecor gven by he equaly (5): () () (3) (4) ς() ξ () ξ () ξ () ξ () (5) and he ae arx oeraor A ha for of he followng arx (6): A δ δβ α ( α) ( β) γ γ α0 αa αa 0 β0 βa βa 0 (6) he ouu equaon can be rewren a (7): y () C x () (7) where C: R defned by (8): () () ( ) C x ( ) c, ξ () c, ξ ()... c, ξ () (8) Ung he ecral reoluon of he ae oeraor A and roere (7)-() he nfne denonal dynacal ye, gven by he abrac dfferenal equaon (4), can be rereened by he nfne ere of he e of fr order fne-denonal ordnary dfferenal equaon wh conan coeffcen n he for (9) a follow: ς () Aς (),,,3,... (9) where he ae vecor gven by he equaly (0): ς () ς() ςj () ς () (0) where he ub-vecor ς () are gven by equaly (): j () () (3) (4) ςj () ξj () ξj () ξj () ξj () (),,3,... j,,..., he ae varable are defned larly le n he equaon (4): d x () ( ) j ξ j () () d where xj () denoe he (j) h coeffcen of he Fourer ere of ecral rereenaon for he eleen x() n he ae ace. he coeffcen are exlcly gven by he nner roduc beween eleen n he ae ace and he arorae egenfuncon φ j of he oeraor A: xj ( ) < x( ), φj >,,3,... j,,..., (3) he ae arx A he followng bloc arce of he for (4): A daga... A,,3,... (4) Exng n he forula (4) ub-arce A are gven by he equale (5): A,,3,... (5) α 0 β 0 Addonally, n he ere of he equaon (5) here ex conan real coeffcen α and β, whch are defned by he equale (6) and (7): δα ( α ) γ α α0 αλ αλ,,3,... (6) δβ 0 ( β ) γ,, 3,... (7) β β β λ β λ ISSN: ISBN:

3 8h WSEAS Inernaonal Conference on SIMULAION, MODELLING and OPIMIZAION (SMO 08) Sanander, Canabra, San, Seeber 3-5, he Jordan Decooon of he Sae Marx. Fr of all, le u calculae he egenvalue of he ae arce A. I can be ealy calculaed, ha h arx e e e e he ranon arx ( A ) ha raher ohcaed for of bloc arx (3): ( A) [ 3 4],,3,... (3) where he ub-arce have for of he followng bloc arx (3): 0,,,3,4,,,3,... (3) 0 e where 0 n he forula (3) denoe he four eleen vercal zero vecor and he bloc are vercal vecor defned by (33): 3 4 α β α β α β α β α β α β α β α β α β α β α β α β (33) 4 Obervably Defnon [0] Dynac ye () wh ouu (4) ad o be nally 0,, > 0 f an nal obervable n he e nerval [ ] ae [ 0 3] he obervaon y () over he e nerval [ ] x x x x can be unquely deerned fro 0,. 5 Man heore he dynacal ye () nally obervable n an arbrary e nerval f and only f he followng ha four dnc egenvalue each wh he ae ullcy : nfne ere of equale (34) fulflled: σ ( A) {,,, },,3,... (8) ran [ C],,,3,... (34) where, are gven by (9): where: β β 4α β β 4α c c c, (9) C c c c,,,3,... (35) Ung he egenvalue, can be found ha he Jordan canoncal for of he ae arce A ha for of he dagonal arx (30): c c c J ( A) dag[ ] (30) cj c, φj,,,3,..., j,,...,,,,.., 5. Proof By lneary, eay o ee ha ye (4),(7) nally obervable n he e nerval [ 0, ] f and only f he null ouu y () for [ 0, ] le ς (0) 0. o rove uffcency le u aue ha y ( ) 0,,,...,, [ 0, ]. Followng he a o calculae he ouu a fr le u calculae he ae () x( ) ξ ( ) fro he rereenaon of he ye (4) by an nfne ere of fne denonal ae equaon. Fro (9) and he Jordan decooon of he ae arx (8)-(33) we have (36): A J ς () e ς (0) A e A,,,3,... ( ) ( ) (36) he xj () coeffcen of he Fourer ere of ecral rereenaon for he eleen x() n he ae ace gven by he fr eleen of he ς j () vecor (). Fro (36) and (0),() we can calculae (37): x () j e e e e () ξ j e e e e () ξ (37) j e e e e (3) ξ j (4) e e e e ξj he ole ae x() can be reconruced bang on he roery (8) a (38): ISSN: ISBN:

4 8h WSEAS Inernaonal Conference on SIMULAION, MODELLING and OPIMIZAION (SMO 08) Sanander, Canabra, San, Seeber 3-5, 008 x () () ξ j e e e e e e e e () ξj e e e e (3) ξ j e e e e (4) ξj (38) h Fnally he ouu fro (7) can be exreed by (39) 0 y ( ) () cjξj e e e e e e e e () cjξj e e e e (3) cjξ j e e e e (4) cjξj (39) where c c, φ,,,..,. Mullyng (39) j j z by e for any colex nuber z uch ha (40): Re z > u Re ±,Re ± (40),,3,... { [ ] [ ] } and negrang reec o e fro zero o nfny, we can oban fro (39): 0 0 () ( z ) ( z) ( z ) ( z) cjξj e e e e ( z ) ( z ) ( z ) ( z) () e e e e cjξj ( z ) ( z) ( z ) ( z) e e e e (3) cjξj ( z ) ( z ) ( ) z ( z) e e e e (4) cjξj,,..., (4) Now le u erfor he negrang n he forula (4) recevng (4): 0 () cjξj z z z z () cjξj z z z z (3) cjξj z z z z (4) cjξj z z z z,,..., (4) We ee ha by analyc connuaon relaon (4) hold rue for all z uch ha z σ ( A ). Le C n be a cloed curve n he colex lane urroundng only he egenvalue n and leavng all he oher, n oude. Inegrang (4) over Cn, n,,3,... lead o (43): n n () () cnjξnj cjξj C j j n zn z n n n () () cnjξnj cjξj C n z n z n n n (3) (3) cnjξnj cjξj z C n n z n n (4) (4) cnjξnj cjξj C n zn z n n () cjξj z z z () cjξj z z z 0 C (3) n c jξj z z z (4) cjξj z z z,,...,, n,,3,... (43) he negrand n he deny (43) are a holoorhc funcon of a colex varable z nde he crcle C n exce he on n. ang h fac no accoun, he nequaly,,,3,... and ung andard ISSN: ISBN:

5 8h WSEAS Inernaonal Conference on SIMULAION, MODELLING and OPIMIZAION (SMO 08) Sanander, Canabra, San, Seeber 3-5, 008 redue heore can be een ha he la negral n he equaon (43) vanhe and fro he reanng negral we can acheve he deny (44): () () (3) cjξj cjξj cjξj (4) cjξj ( ) 0,,,..., 0,,,3,... (44) Perforng he negrang of he deny (4) hree e over he curve Cn, Cn, C nurroundng reecvely he egenvalue n, n, n and leavng all he oher oude we can receve he followng nfne ere of he e of he equaon (45): () () (3) (4) cjξj cjξj cjξj cjξj 0 ( j j j j j j () j j () () (3) (4) cjξj 0 cjξj cjξj 0 cjξj ) () (3) (4) ξ ξ ξ jξj c c c c 0 () (3) (4) c ξ cjξj cjξj cjξj 0 ( ) ( ) 0,,...,,,, 3,... (45) he dealed calculaon goe larly le for he egenvalue n and wll be oed. he nfne ere of he e of he equaon (45) afer roer regroung he unnown ( ξ ) j (0) can be rewren n he arx for (46): cr(, ) cr(, ) c R(, ) cr(, ) cr(, ) c R(, ) (0) ς c R(, ) cr(, ) c R(, ) 0,,,3,... (46) where he vecor of he unnown ς (0) are gven by R, 4 4 arx exreed by (47): (0) and ( ) R(, ),,,3,... (47) he nfne ere of he arx equaon (46) ung he arx Cronecer roduc can be rewren n ore coac for (48): C R(, ) ς (0) 0,,,3,... (48) where C gven by (35). he dynacal ye (4),(7) nally obervable n he e nerval [ 0, ] f and only f ς (0) 0,,,3,.... By he well nown Kronecer-Caelle heore he nfne ere of he equaon (48) have null oluon f and only f (49): ran C R (, ) 4,,,3,... (49) Now le u ae no accoun a arbrary nor of he arx C R(, ) of ze 4 4. By he roere of he Cronecer roduc we can wre for all he oble nor of he arx C R(, ) of ze 4 4 (50): 4 de ( C R (, )) de C de (, 4 4 R (50) Fro he for of he R(, ) arx (47) we can calculae drecly : 3 de R(, ) 6 0, (5) Conderng (5) fro (50) we can ae ha (5): de ( C R (, ) ) 0 de C 0, 4 4 (5) Suarng fro (5) follow ha he ere of he equale (49) are fulflled f and only f (53): ran C (53) [ ],,,3,... Q.E.D. 6 Concluon he a choce of nuercal ool, ogeher wh he bloc arx heory, allowed o concely ulae he ough obervablz of he drbued ye. I hould be oned ISSN: ISBN:

6 8h WSEAS Inernaonal Conference on SIMULAION, MODELLING and OPIMIZAION (SMO 08) Sanander, Canabra, San, Seeber 3-5, 008 ou, ha he reul reened n he arcle ale for very general for of he dznacal ye. he conraned obervably analy can be een a one of he drecon of furher reearch. Reference: [] R. Bellan, Inroducon o Marx Analy, Mcgraw-Hll Boo Coany, New Yor, 960. [] K. Io, N. Kunau, Segrou Model of Srucurally Daed oheno Bea Wh Boundary Inu, In. J of Conrol, Vol. 54, 99, [3] B. Jacob, J.R. Parngon, On conrollably of dagonal ye wh one-denonal nu ace, Sye & Conrol Leer, 55, 006, [4] B. Jacob, H. Zwar, Exac conrollably of dagonal ye wh a fne-denonal ouu oeraor, Sye & Conrol Leer. 43(), 00, [5] Klaa J. Conrollably of Dynacal Sye, Kluwer Acadec Publher, Boon, 990. [6] J. Klaa, Schauder fxed on heore n nonlnear conrollably roble, Conrol and Cybernec, Vol. 9, 000, [7] J. Klaa, Conraned exac conrollably of elnear ye, Sye & Conrol Leer, Volue 47, Iue, 00, [8] S. Labbe, E. rela, Unfor conrollably of edcree aroxaon of arabolc conrol ye, Sye & Conrol Leer, 55 (7), 006, [9] N. I. Mahudov, S. Zorlu, Conrollably of elnear ochac ye, Inernaonal Journal of Conrol, Vol. 78, Nuber 3/0, 005, [0] L. Mller, Non-rucural conrollably of lnear elac ye wh rucural dang, Journal of Funconal Analy, 36 (), 006, [] J. Reonde, Conrollably of Dynacal Sye Wh Conran, Sye & Conrol Leer, Vol. 54/4, 005, [] J. Reonde, Nuercal aroach o he non-lnear dofanc equaon wh alcaon o he conrollably of nfne denonal dynacal ye, Inernaonal Journal of Conrol, Vol. 78, Nuber 3/0, 005, [3] J. Reonde, Nuercal Analy of Conrollably of Dffuve-Convecve Sye wh Led Manulang Varable, Inernaonal Councaon n Hea and Ma ranfer, Vol. 34/8, 007, [4] J. Reonde, Aroxae Conrollably of he n-h Order Infne Denonal Sye wh Conrol Delayed by he Conrol Devce, Inernaonal Journal of Sye Scence, Vol. 39, No. 8, , 008 [5] J. Reonde, Aroxae Conrollably of he n-h Order Infne Denonal Sye, Inernaonal Journal of Aled Maheac and Couer Scence, Vol.8, No., 008. [6] J. Reonde, a Dvdng Mehod In Dohanne Equaon Parallel Solvng, Polh Journal of Envronenal Sude, Vol. 6, No. 4A, 007, [7] J. Reonde, An Algebrac Aroach o he Lnear-Quadrac Oaon of Second Order Dynacal Sye, Archve of Conrol Scence, Volue 7(LIII), No., 007, [8] J. Reonde, he Alcaon of he Decooon of he Real Nuber o he Lnear Cobnaon of hree Naural Nuber Square o he Invegaon of he Conrollably of an Infne Denonal Sye n he hree Denonal Sace Doan, heorecal and Aled Inforac, Vol. 5, nr., 003, [9] J. Reonde, Conrollably of Elac Bea wh One Free End, h IEEE Inernaonal Conference on Mehod and Model n Auoaon and Roboc, Mędzyzdroje, Poland, Augu 8-3, 006, [0] R. rggan, Exenon of ran condon for conrollably and obervably o Banach ace wh unbounded oeraor, SIAM Journal on Conrol and Ozaon, Vol. 4, 976, 3, [] A. Veru, On null conrollably of lnear ye n Banach ace, Sye & Conrol Leer, 54 (4), 005, ISSN: ISBN: