Direct Partial Modal Approach for Solving. Generalized First Order Systems

Transcription

1 Iteratoal Matheatcal Foru Vol. 7 o Drect Partal Modal pproach for Solvg Geeralzed Frst Order Sstes ha. l-saed ad M. Modather M. dou Departet of Matheatcs College of Scece ad uatara Studes Sala B dulazz Uverst l-khar Saud raa stract. I ths Research we propose a explct soluto to the partal egevalue assget prole for o-setrc geeralzed frst order cotrol sstes sgleput case usg orthogoalt relatos etwee egevectors. Our soluto ca e pleeted wth ol a partal owledge of the egevalues ad the correspodg egevectors of geeralzed frst order cotrol sstes. We show that the uer of egevalues ad egevectors that eed to rea uchaged wll ot e affected feedac atrces. We prove the feedac atrx ust e of real for. uercal exaple s gve to llustrate the proposed ethod. Kewords: explct soluto- partal egevalue assget- sgle put. Itroducto he frst order atrx dfferetal equato v& t v t (.) ( ) ( ) eads wth the separato of varales v( t) t xe x a costat vector to the prole of fdg the egevalues ad egevectors of the lear pecl P( ) (.) he scalar s called a egevalue ad the correspodg vector x s called a egevector f the satsf P( ) x (.3) ere we assue that are real o-setrc costat atrces wth s osgular. he sste (.) ca e cotrolled wth applcato of a forcg Correspodg author ha. l-saed: -al address: ha_ath@ahoo.co Peraet address: Departet of Scece ad Matheatcs Facult of Petroleu geerg Suez Caal Uverst gpt

2 .. l-saed ad M. M. M. dou fucto u () t replaced R a costat ad ( t) ( t) v( t) u( t) u a scalar whch case (.) s v& + (.4) hs prole s called geeralzed lear state space sste also called descrptor sste. he sste (.4) s cotrollale.e. ra ( ) C. for Descrptor sstes arse for exaple odelg tercoected sstes ad ecooc processes [89]. B susttutg u fv to (.4) ad the choce of the cotrol vector f R costat leads to closed loop sste ( t) ( f ) v( t) v& (.5) hs prole leads wth the separato of varales of closed pecl P c ( ) ( f ) ; (.6) I a realstc stuato however ol a few egevalues are troulesoe so t aes ore sese to alter ol those troulesoe egevalues whle eepg the rest of the spectru varat. hs leads to ow as the partal egevalue assget prole [3]. Matheatcall gve the lear pecl Ω ( ) { } ad set{ } + P( ) whose spectru closed uder coplex cougato. he the partal egevalue assget prole for the geeralzed lear sste requres us to fd the feedac vector ( f ) { } + Ω. f such that M.. Raada ad ha. l-saed [4] troduced a explct soluto to the partal egevalue assget prole of the sste (.4) the specal case f I where I s dett atrx the sste (.4) has the stadard frst order sste the for v & t v t + u t. (.7) ( ) ( ) ( ) he soluto of the partal egevalue assget prole ca e otaed usg ol a partal owledge of egevalues ad egevectors of the lear

3 Drect partal odal approach pecl P( ) wthout usg a proecto ethods [5 6]. soluto techque of ths tpe wll e called a drect partal odal approach t s drect ecause the soluto s otaed drectl wthout usg a tpes of reforulatos or proecto. It s partal odal ecause ol a part of the egevalues ad egevectors s eeded for soluto [].. Orthogoalt Relatos For Geeralzed No Setrc geprole hs secto troduce orthogoalt relatos for geeralzed egevalue prole Defto (). et ad e ( ) x ( ) x P. (.) atrces. scalar s called a geeralzed egevalue ad a o zero colu vector x s called the rght geeralzed egevector correspodg to f Defto (). et ad e x x (.3) atrces. scalar s called a geeralzed egevalue ad a o zero colu vector satsfg (.4) s called the left geeralzed egevector correspodg to where s the cougate traspose of the vector. Defto (3). scalar C s a egevalue of the lear pecl P( ) wth the correspodg rght egevector x ad the left egevector. Defto (4). he trplet ( x ) Defto (5). he pars ( x) ad ( ) egepars of P. s called the egepar of P. Defto (6). he pecl P s called sgular f for a sgular.e. det ( ) are called respectvel rght ad left the C the atrx P ( ) s. Otherwse the pecl P s called regular. I ths paper we restrct ourselves to regular P.

4 .. l-saed ad M. M. M. dou If s sgular we ca have the fte egevalue. Note that f s o-sgular whch the equvalet prole x x s perfectl well defed ad the fte egevalue correspods to. he geeralzed setrc defte egeprole has ol real egevalues. he geeralzed o-setrc egevalue prole ca have real coplex or fte egevalues. D.R. Sarssa [7] state ad prove a well ow result o the orthogoalt relatos of the egevectors for a atrx. Datta ad et al [] troduced orthogoalt relatos of the egevectors for the setrc defte quadratc pecl. hrough ths paper we cosder the atrx s o-sgular ad postve defte we ca have ol fte egevalues (real or coplex). he followg theore estalshes the orthogoalt relatos of the egevectors for par ( ). heore. et e the egevalues of the lear pecl wth R ad let X ad Y e respectvel the rght ad the left egevector atrces. ssue that { } { } Φ partto ( ) + X X X ad Y ( Y Y ) where X ( x ); X ( ) Y ( ) ( ) Y +. he ad x x + x ad Y X (.5) Y X (.6) If addto ( ) s real setrc the Proof. et Λ ( ) dag sste of the equatos d X X ad X X (.7). We ca rewrte (.) ad (.) the for X XΛ (.8)

5 Drect partal odal approach 3 Y hs ples that ΛY ΛY (.9) X Y he equato (.) ca e wrtte as ΛN NΛ where N XΛ Y X ust e dagoal sce t coutes wth a dagoal Λ. (.) N Y X Y Y Y ( X ) X Y X Y X X Y X he Y X sce Y Y ultplcato X o the rght we ota Λ where Λ dag( ). Fall f ( ) Y X ΛY X s real setrc they X whch proves (.7). he relato (.6) turs out to pla a e role our later developets. Note: the stadard egevalue prole I the aove theore where I dett atrx. x x s the specal of ths theore; put 3. Partal gevalue ssget Prole Suppose R are osetrcal ad s postve defte. et the sste of equatos where X X XΛ (3.) Λ dag C dstct e a C ad ( ) egedecoposto of the pecl P ( ). usg the orthogoalt relato P( ) (3.) Y X (3.3) we ow preset a soluto to the partal egevalue assget prole for the atrx pecl (3.).

6 4.. l-saed ad M. M. M. dou Gve coplex uers { } setrc aout x-axs ad a vector R we are requred to fd f R whch are such that the closed loop pecl. has spectru P c ( ) ( f ) ; (3.4) { } + (3.5) hs s the partal pole assget prole whch we use the vector f R to replace the egevalues { of the pecl ( ) whle leavg the } other egevalues uchaged. et us partto the egevector atrx P { } rght egevector atrx X the left Y ad egevalues atrx Λ as follows: X ( ) X X Y Y Y Λ dag ( ) Λ Λ where X ( x x ); X ( x + x ) Y ( ) ad Y ( + ) wth Λ ( ) ad Λ dag( ). dag heore 3. et he for a choce of we have + f Y (3.6) ( f ) X X Λ (3.7) I words ths theore assures us that a choce of wth f as (3.6) guaratees that the last egepars ( Λ X ) of (3.) are also egepars of the closed loop pecl (3.4). Proof et ( Λ) he X e the egevector-egevalue atrx par of the pecl P( ) X XΛ

7 Drect partal odal approach 5 et X ad Y e respectvel the rght ad the left egevector atrces of the pecl P( ). o prove ths expadg the left had sde of (3.7) susttutg (3.7) we ota f Y ( f ) X X Λ X X Λ Y X ( Y ) X Because X X Λ. Furtherore Y X fro the theore. hus he theore s proved. ( f ) X X Λ Rear. Fro heore 3.we see that there s a paraeters fal of feedac vectors that ca e used to chage soe egevalues the closed loop pecl whle the reag uchaged. 3. Choosg I order to use heore3. to solve the partal pole assget prole we eed to choose whch wll ove { } of the pecl ( ) { } P c ( ) there exst a egevector atrx f that s possle. If there s such a vector Z C ; Z ( z z z ) z ad atrx D dag( ) ( f ) Z ZD. whch are such that Susttutg for f ad rearragg we have P to the (3.8) Z ZD Y Z G Where G Y Z ad c G s a vector that wll deped o the scalg chose for the egevectors Z. o ota Z we ca solve for each of the egevalues z usg the equatos c ( ) z (3.9)

8 6.. l-saed ad M. M. M. dou hs correspods to choosg the vector ( ) egevectors we could solve the c. So havg coputed the square lear sste ( ) G (3.) for ad hece detere the vector f. he aove dscusso leads us to forula the followg algorth for the soluto of the partal egevalue assget prole. lgorth. he sgle put partal pole assget algorth Iputs: R ad a -vector D dag( ) coplex cougato. closed uder ssupto: Nuers ; are all dstct where are the egevalues of P( ) ad s osgular postve defte. Output: the feedac vector P c ( ) ( f ) ; f such that the spectru of the closed-loop pecl s{ ; + } where + are the last egevalues of the pecl P( ). Step. Ota the frst egevalues of the pecl P( ) that eed to e reassged ad the correspodg left egevectors. For Λ dag( ). Y ( ) Step. Solve for Step 3. For z z ; ( ) z G Y Z where Z ( z ) ad Y ( ) z. Step 4. Solve for : G ( ) Step 5. For f Y.

9 Drect partal odal approach 7 3. xplct xpresso For heore 3. Suppose the pecl (3.) has egedecoposto (3.) ad f s chose as (3.6) wth the copoets of chose as. (3.) he the closed loop pecl (3.4) has spectru (3.5) ad ts frst egevectors ca e scaled to satsf (3.9). Proof. I vew of heore 3. we eed ol show that ( ) ( ) [ ]. z f Φ (3.) where ( ) z (3.3) for the choce of f ad dcated. Now ( ) Φ wth f replaced the expressos (3.6) gves ( ) ( ) [ ] z Y Φ he fro (3.3) ( ) [ ] z Y Φ (3.3). Now susttutg for usg (3.) gves ( ) z Φ ( ) ( ) z Φ he

10 8.. l-saed ad M. M. M. dou ( ) ( ) z Φ he -th colu of (3.) ca e wrtte as. x x ece for a choce of ( ) x x + the ( ) ( ) x x (3.4) Susttutg (3.4) to the last expresso for ( ) Φ gves ( ) ( ) z Φ usg (3.3) ( ) Φ ( ) ( ) Φ Cacelg the coo ter we get ( ) Φ ( ) Φ.

11 Drect partal odal approach 9 We ow show that (3.5) for a sets of { } ( ) ad{ } Φ vashes as requred. Defe the oc poloal P whch the are dstct ad thus estalsh that () t ( t ). (3.6) he agrage poloal whch terpolates P at the dstct pots t recovers P tself so we a wrte P ( ) t () t P ( ) ( ) P () t a ( t ) (3.7) where Moreover a P ( ) ( ) a (3.8) ecause P () t s oc. quatg the two fors (3.6) ad (3.7) of P () t at t gves

12 3.. l-saed ad M. M. M. dou a fro whch (3.5) follows (3.8). hs copletes the proof. Fro the expresso (3.) t s clear those suffcet codtos for the exstece of ad cosequetl for a soluto to the partal pole assget prole to exst are that heore et (a) o vashes () the { } are dstct (c) ust e ot orthogoal to. + e the egevalues of the lear pecl wth R. et the egevalues to e chaged to ad the reag egevalues to sta varat. he partal egevalue assget prole s solvale for a artrar set { } f ad ol f for all ad. Proof We frst prove the ecesst. Suppose that. Sce for all ad. hs eas that for a f we have c ( f ) ( f ) whch ples that s a egevalue of par for ever f ad thus caot e reassged. Next we prove the suffcec. We assue that Y. he we eed to prove that there exsts a feedac f whch assgs the egevalues Λ artrarl whle eepg all the other egevalues uchaged. Deote ( ) ad Y ( ) wth ( ) Y + Λ dag

13 Drect partal odal approach 3 Λ dag( ) ad D dag( ). et Y ( ) + egevector atrx of. Sce ( Y ) Y I the the spectru of { } Y Y e left ad Y dag( ) Y Λ Λ or equvaletl the atrx par ( ) +. We assue that there exsts a feedac vector that the closed-loop atrx he dag Multplg ths o the rght dag s Φ such Λ Y Φ has the desred egevalues. ( D Λ ) dag( Λ Λ ) Y ( ) Φ Y gves ( D Λ ) Y dag( Λ Λ ) Y Y ( ) Y Φ Sce Y Y Y the dag Y Y ( D Λ ) Y Y Y ( Φ ) Multplg ths o the left gves ( Y ) Deote f ( Y ) dag( D Λ ) Y ( ΦY ) ( Y ) dag( D Λ ) Y ( ΦY ) ΦY ad c f the the spectru of c { } +. s 4. Real For Of Feedac Vector f. I ths secto we prove that f s real f all the{ setrc aout x-axs. heore 4. et { } ad { } closed uder coplex cougato ad let { } } are coplex uers e two dstct sets of coplex uers e a part of egevalues

14 3.. l-saed ad M. M. M. dou of lear pecl P( ) wth colu vectors { } R. We assue that o zero are the left egevector correspodg to{ }. he ad wheever where s a copoet of as (3.). Proof Frst Sce (4.) We cougate to oth sdes of (4.) (4.) (4.3) Sce the R ad Copare (4.) wth (4.4) we ota (4.4) (4.5) B traspose we ota Secod Fro the relato (3.). (4.6) he Sce ad the. (4.7). (4.8)

15 Drect partal odal approach 33 ad hece.. he theore s proved. Now we descre how to trasfor a coplex cougate set of ad the set of left egevectors Y to the real oes the followg theore. hs wll e requred to ota real feedac vector f fro coplex oes. heore 4. et { } e a set of coplex uers closed uder coplex cougato { } ad let vectors { } e such that ad wheever. he I- here exst a osgular atrx C such that R Y Y Where { } ad Y { } C ad Y R are real atrces. C R (4.9) ad oth R II-here exst real feedac vector f such that f Y (4.) R R Proof I-Defe Where S followg propertes. S M S S ad S M O O O S S M (4.) S the the atrx S has the thus the atrx s osgular ad

16 34.. l-saed ad M. M. M. dou We rewrte { } other for { } where { } a Sce we assue that + ad the { a + a } ad S { a + a } S { a } + + { S 34S + S S } R. We rewrte Y { } loc for Y { 34 + } where { } + +. Sce c + d we assue that + ad where c [ c c c ] ad d [ d d d ] the + { c + d c d }. + S { c + d c d } S { c d } { S 34S + S S } Y R Y. (4.) B usg traspose cougate of (4.) we ota Y Y R II-ow we show that feedac vector f ust e real. Fro the secto 3 we show that spectru of the closed loop pecl ( ) ( f ) ; { } + such that f Y. P c s Sce R Y Y R the f Y Y RY R where oth R ad Y are real atrces the there exst real feedac f Y such R R R that the spectru of the closed loop pecl ( ) ( f ) ; { } P c s +. he theore s proved. Clearl f all the{ } are real the Y s real as well. If addto all the { are real the ad so f s also real. }

17 Drect partal odal approach Nuercal xaple he pecl show ale P( ) has egevalues ordered accordg to ther real parts ale We ow copute the feedac vector gevalues of P( ) f

18 36.. l-saed ad M. M. M. dou [ ] f ale gevalues of P( ) gevalues of ( ) ( f ) ; P c

19 Drect partal odal approach 37 6 Cocluso I ths paper we derved a explct soluto to the partal egevalue prole usg oe of the orthogoalt relatos etwee the egevectors for the lear pecl P( ). We eed ol a partal owledge of the spectru (ad the assocated left egevectors) of the atrx. hese egevalues ad egevectors are requred to e reassged. We proved that the soluto (feedac vector ) for ths prole s the real for. cowledgeets he authors gratefull acowledge Research Cetre Sala B dulazz Uverst Saud raa for supportg ad ecourageet durg ths wor Refereces [] B. N. Datta Recet developets o oodal ad partal odal approaches for cotrol of vrato ppled Nuercal Matheatcs 3 () (999) 4-5. [] B. N. Datta S. lha Y. M. Ra Ortogoalt ad Partal Pole ssget for the Setrc Defte Quadratc Pecl. lg. ppl. 57 (997) [3] B. N. Datta D. R. Sarssa Partal gevalue ssget ear Sstes: xstece Uqueess ad Nuercal Soluto Proc. MNS' Notre Dae ug () [4] M.. Raada ad ha. l-saed Partal egevalue assget prole of lear cotrol sstes usg orthogoalt relatos cta Motastca Slovaca() (6) 6-5

20 38.. l-saed ad M. M. M. dou [5] M.. Raada ad ha. l-saed Proecto lgorth for Partal gevalue ssget Prole Usg Iplctl Restarted rold Method Joural of Vrato ad Cotrol (7) () [6] Y. Saad Proecto ad deflato ethods for partal pole assget lear state Feedac I ras. uto. Cotrol Vol. 33(3) (988) [7] D. R. Sarssa heor ad Coputatos of Partal gevalue ad gestructure ssget Proles Matrx Secod-Order ad Dstruted-Paraeter sstes Ph. D. thess orther llos uverst [8]. Varga Roust pole assget for descrptor sstes : Proceedgs of the MNS Sposu Perpga Frace (). [9]. Varga uercall relale approach to roust pole assget for descrptor sstes Future Geerato Coputer sstes 9 (3). -3. Receved: prl