Controllability and Observability of Matrix Differential Algebraic Equations

Transcription

1 NERNAONAL JOURNAL OF CRCUS, SYSEMS AND SGNAL PROCESSNG Corollabiliy ad Obsrvabiliy of Marix Diffrial Algbrai Equaios Ya Wu Absra Corollabiliy ad obsrvabiliy of a lass of marix Diffrial Algbrai Equaio (DAEs) ar sudid i his ar h sruur of a losd-form soluio for h sysm is sough via wo o-sidd sub-sysms h soluio is h usd o driv ssary ad suffii odiios for h orollabiliy ad obsrvabiliy of h im-varyig marix DAE sysms Mor sraighforward odiios o h orollabiliy ad obsrvabiliy of liar im-ivaria marix DAE sysms ha oly dd o h sa maris ar also obaid Kywords Corollabiliy, Diffrial Algbrai Equaios, Gramia, Obsrvabiliy 2 Mahmais Subj Classifiaio: Primary 93B5 ad 93B7, Sodary 93C5 M NRODUCON ANY girig sysms, suh as mhaial sysm, lrial iruis ad hmial raio iis, ar modld by ould diffrial ad algbrai quaios (DAEs) ha ao b rasformd io ordiary diffrial quaios Suh DAEs also rfrrd o as sigular sysms hav b sudid xsivly i h viw of umrial simulaio h mos ommoly sudid liar diffrial algbrai quaios ar h vor DAEs li h followig, whr E( ) xɺ ( ) A( ) x( ) = f ( ) () x R, E( ) is sigular for all i h assoiad im irval, E, A R, ad f R A hararizaio of solvabiliy for () ad a gral aoial form rrsaio for solvabl DAEs i h form of () a b foud i [] h DAE sysm () is alld liar imivaria (L) if E ad A ar osa maris For suh L sysms, h roblms of fdba ol lam [2] ad oimal orol [3] hrough sa fdba hav b sudid h aalysis ad orol rsuls for liar im-ivaria sysms hav also b gralizd o im-varyig [4] ad disr-im [5] sysms Corollabiliy ad obsrvabiliy ar of imora ad fudamal roris of orol sysms [6], h orollabiliy ad obsrvabiliy Gramias ar dvisd i h frquy domai for orollr rduio dsig h orollabiliy ad obsrvabiliy roris ar rquird for miimal ralizaio, suh as h mulidimsioal hybrid sysms irodud i [7] Sruural roris of gralizd liar sysms ar sudid i [8] usig h Prro-Siljs igral o obai h iu-ouu ma Nssary ad suffii odiios ar drivd for oml orollabiliy ad obsrvabiliy Diffr orol mhods ar aliabl o orol sysms ha m h orollabiliy ad obsrvabiliy riria for sa fdba ad ouu fdba dsigs, suh as h hrmosyho sysm i [9] h orollabiliy, obsrvabiliy ad ralizabiliy of firsordr marix Lyauov sysms ar firs irodud i [] his ar, w sudy h diffrial algbrai marix Lyauov sysms ovr is solvabiliy ad orol rsivs h rs of h ar is orgaizd as follows A losd form soluio for h roosd marix DAE sysm is rsd i sio 2 his soluio is usd o driv ssary ad suffii riria for orollabiliy ad obsrvabiliy of h marix DAE sysm for boh liar im-varyig ad liar im-ivaria sysms, all rsd i sio 3 Colusio ad rmars o fuur wor ar foud i sio 4 CLOSED FORM SOLUON OF HE MAR DAE SYSEMS h lass of marix diffrial algbrai quaio sysm osidrd i his ar wih iu ad ouu sruurs is dfid as ad whr E ɺ = A EB DU, ( ) = (2) Y = F (22) d ɺ =, h sa offii maris,, E A B R, d h iu sruur marix m m D R, h orol iu q U R, ad h ouu sruur marix F R is also osidrd ha h sa offii maris as wll as h iu/ouu sruur maris ar im-dd f ha is h as, w assum h marix fuios ar oiuously diffriabl ovr = [, ] Morovr, E is sigular for all ssu 3, Volum 5, 2 287

2 NERNAONAL JOURNAL OF CRCUS, SYSEMS AND SGNAL PROCESSNG his sio, our mai objiv is o obai a gral form of soluio for (2) h losd-form soluio lays a imora rol i drivig h odiios o h orollabiliy of (2), ad obsrvabiliy of (2) ad (22) i sio 3 h soluio of (2) is osrud via h soluio of wo o-sidd subsysms of (2) o s h sag, w bgi wih som basi dfiiios assoiad wih h soluios of liar firs-ordr marix diffrial quaios Dfiiio 2 A marix fuio Z ( ) R is a fudamal soluio of a liar firs-ordr marix diffrial quaio L( ɺ,, A, A, A ) = DU, (23) 2 r whr A R, i=, 2,, r, D R ow as h iu i m m sruur marix, ad U R is h iu marix, if Z saisfis (23) wih D=, d( Z ( )) for all, ad vry soluio o (23) a b wri as h= Zϒ, whr ϒ is a arbirary by osa marix ad is a ariular soluio o (23) Obviously, h marix diffrial algbrai quaio (2) is a sial as of (23) h o of sa rasiio marix assoiad wih a sysm of ordiary diffrial quaios was irodud i [] W xd h sa rasiio marix o (23) via h fudamal soluio of (23) Firs, rwri (23) as a iiial valu roblm, ɺ 2 r, ( ) L(,, A, A, A ) = =, (24) Aordig o Dfiiio 2, h uiqu soluio o (24) is = Z( ) Z ( ) =Φ (, ) (25) h marix fuio Φ (, ) a b gralizd as Φ (, s) = Z ( ) Z ( s), whih is ow as h sa rasiio marix assoiad wih (24) Dfiiio 22 h sa rasiio marix assoiad wih (24) is dfid as Φ (, s) = Z ( ) Z ( s) (26) whr Z( ) is a fudamal soluio of (24) is asy o h ha Φ saisfis h marix diffrial quaio (24) i, ad i saisfis h followig hr roris: Lmma 2 h sa rasiio marix saisfis h followig roris (i) Φ (, ) =, Φ(, ) Φ (, ) =Φ (, ),, 2, 3 (ii) (iii) Φ (, ) =Φ (, ),, 2 W will xlor wo sial liar marix diffrial quaios of (23), whih lad o h soluio of (2) h firs o is h sadard liar marix diffrial quaio as follows, ɺ = A (27) whr A, R, ad (27) is a liar im-ivaria (L) marix diffrial sysm if A is a osa marix; ohrwis, i is ow as liar im-varyig (LV) sysm f (27) is L, h fudamal soluio of (27) is Z ( ) = h marix xoial owr sris, A is formally dfid by h ovrg 2 A 2 = A A A 2!! hr ar may umrial mhods availabl for omuig A A i uorial rviw a b foud i [2] h sa rasiio marix assoiad wih (27) is his is baus sa sa ( ) = ( s) A Φ (, s) = Z( ) Z ( s) = A ad sa ommu wih ah ohr ad h soluio o (27) is mor omliad if i is LV is wri as a iiial valu roblm, ɺ = A( ), ( ) =, (28) h gral soluio of (28) is of h form ( ) (, ) ( ) =Φ (29) whr Φ is h sa rasiio marix assoiad wih (28) h im-varyig as, h sa rasiio marix Φ has a aalogous form similar o h L as, i A( τ ) Φ (, ) = (2) A ssu 3, Volum 5, 2 288

3 NERNAONAL JOURNAL OF CRCUS, SYSEMS AND SGNAL PROCESSNG if A( ) ad A( τ) ommu Ohrwis, h marix xoial (2) is gralizd o h so-alld Pao-Bar formula as a xsio o h owr sris xasio for Φ, τ τ τ τ τ2 τ2 τ Φ (, ) = A( ) d A( ) A( ) d d τ τ A( ) A( 2 ) A( ) d d τ τ τ τ τ si of h la of losd-form xrssios for Φ from im-varyig sysms, h sa rasiio marix is a usful ool for sudyig h roris of soluios of (2), whih lads o h xloraio of orollabiliy ad obsrvabiliy roris of h marix DAE sysm h sod lass of marix diffrial quaio off (23) o b osidrd hr is h so-alld marix diffrial algbrai quaio as follows E ɺ = A DU ( ), ( ) =, (2) Rall ha E is a sigular marix Wh sysm (2) is imivaria, i E, A ad D ar osa maris, h soluio of (2) a b drmid by h assoiad marix il, se A, as a rsul of Lala rasform Dfiiio 23 A marix il se A is rgular if d( se A) for som s C h marix DAE (2) is solvabl if ad oly if h il se A is rgular, whih is aalogous o h rsul from [3] ordr o obai a xlii form of soluio for (2), o ds o rasform (2) io a aoial form o b mor sifi, suos h ra of marix E saisfis r( E) = r<, ad d( se A) is a ozro olyomial of dgr m, m r, h hr xis o-sigular maris, P, Q C, suh ha (2) is rasformd io h followig aoial form by rmulilyig (2) by P ad a oordia hag wih Q, ɺɶ = A ɶ D U N ɺɶ 2= ɶ 2 D2 U ( ) (22) m whr ɶ ( m) C, ɶ ɶ 2 C, = Q, ad N is a ɶ 2 ( m) ( m) marix of iloy idx κ, whih i mas N for i< κ ad N κ = A algorihm for osruig h similariy rasformaio maris P ad Q is dvlod i [4] h as of m= r, marix N is idially zro, ad h sod quaio i (22) boms a algbrai marix quaio Mawhil, if m< r, marix N is i h Jorda aoial form wih zros o h mai diagoal ay v, h ODE subsysm i ɶ is oally dould from h DAE subsysm i ɶ 2, ad hy a b solvd saraly h soluios ar giv as follows, ad A( ) A( τ ) = ɶ ( ) ( ) D U( τ), (23) whr κ i ( i) ɶ 2 ( ) = N D2U ( ), (24) i= ( i U ) ( ) dos h i h drivaiv of h iu marix fuio U ( ) h DAE sysm (2) has smooh soluios if h iiial odiio ɶ 2 ( ) saisfis (24) Furhrmor, if h iloy idx κ is grar ha o, h h iu marix fuio U ( ) is oiuously diffriabl u o h ordr of κ For liar im-varyig DAEs, w rwri (2) as E( ) ɺ = A( ) G( ), ( ) =, (25) whr G( ) = D( ) U ( ) h ssary ad suffii odiios for h soluio of (25) a b ddud from [] Firs, for ay ˆ, w obai aylor sris of h marix fuios, E, A, G ad Du o hir similariy i aylor sris xasios, w us h followig xrssio H ( ) = H ( ) ( ) H ( ) whr H=, H= E, A, G,ad h aylor! sris xasios ar subsiud i (25), h for ah j>, bu lss ha h ordr of smoohss of E, A, G ad, ad, h DAE (25) is rasformd io a sysm of algbrai quaios, whr j j j j ψζ = ϕ g (26) E E A 2 E ψ j= E A E A E E j Aj 2 2 E j2 Aj 3 je ssu 3, Volum 5, 2 289

4 NERNAONAL JOURNAL OF CRCUS, SYSEMS AND SGNAL PROCESSNG 2 ζ j=, ϕ j Dfiiio 24 h marix j A A =, ad g A j j G G = G j ψ j is alld smoohly -full if hr xiss a smooh o-sigular marix fuio P( ) o suh ha P( ) ψ j ( ) = R( ) Wih ( ) = ( ), h algbrai sysm a b usd o driv a quaio for ɺ ( ) xliily wh ψ j is -full, i ɺ ( ) = Aɶ ( ) ( ) Gɶ ( ), whih a b solvd aordigly horm 2 h sysm (25) wih E, A ral aalyi is solvabl if ad oly if hr is a igr l [, ] suh ha (i) rψ ( l ) is osa o ; (ii) ψ l is -full o ; ad (iii) j R ( ψl ) R ( ϕl ) = R o, whr R ( ) is h olum sa of h marix h roof of horm 2 is similar o h o i [], as suh omid hr wha follows, w assum solvabiliy for ah of h wo o-sidd marix diffrial quaios basd o h rvious disussios Our goal is o osru a losd-form soluio for (2) L b h fudamal marix soluio of righsid diffrial algbrai sysm E ɺ = A ad b h fudamal soluio of h lf-sid sysm ɺ = B W hav h followig rsul horm 22 h omlmary soluio of h homogous quaio has h uiqu form Eɺ = A EB (27) = C, whr rrss h oraio of omlx ojuga rasos ad C is a by osa marix Proof: Noi ha ɺ B si B is a ral marix = H, i is sraighforward o show ha = C is a gral soluio o show vry soluio of h homogous soluio is of h form = C L Z( ) b a soluio ad Z( ) = Y sr Zɺ = Y ɺ Yɺ i (27), o has EZɺ = EY ɺ EY B= AY EY B Si Z ( ) is a soluio, i rquirs EY ɺ = AY Also, baus is a fudamal soluio of h righ-sid sysm, w hav Y = C, whr C is a arbirary osa marix ( by ) his lads o Z( ) = C, h h uiquss Our x s is o osru a soluio for h ohomogous sysm (2) horm 23 Evry soluio of (2) is of h form of ( ) = C, whr is a ariular soluio of (2) Proof: is sraighforward o show ha ( ) = C is a soluio of (2) is also asy o s ha hrfor, o horm 22 saisfis h homogous quaio (27) a oly b wri as C aordig Similar o solvig o-homogous ODEs, h ariular soluio a b foud via variaio of aramrs Assum = C, whr C dds o h, ɺ = ɺ C C ɺ Cɺ afr subsiuig i (2), o has Eɺ C E C ɺ E Cɺ = A C E C B DU H, E C ɺ = DU Cosidr EP = DU, l D rrs h s of all m by maris suh ha for ay U, h olums of DU is i h olum sa of E, i D R ( E) Si U is a orol iu marix, ad wih h assumio of solvabiliy of (2), D is o-my h s D is idd guarad o b o-my if R( E) R ( D) Howvr, his odiio is o ssary hrfor, P= E DU, whr E is h sudo-ivrs of E Now, w obai a xlii quaio for C ɺ as follows, Cɺ = E DU ssu 3, Volum 5, 2 29

5 NERNAONAL JOURNAL OF CRCUS, SYSEMS AND SGNAL PROCESSNG Afr igraig h quaio ovr, ad alyig C( ) i (2),, w obai h xrssio for h ariular soluio of ( ) = ( ) E DU ( ) (28) h fial soluio of (2) wih a iiial odiio ovr is summarizd i h followig horm horm 24 h soluio of (2) wih iiial odiio ( ) = is giv by τ ( ) =Φ (, ) Φ (, ) Φ (, ) E DUΦ (, τ) (29) whr Φ ad Φ ar sa rasiio maris assoiad wih h righ-sid ad lf-sid subsysms, rsivly Proof: Aordig o horm 23, h gral soluio of h marix DAE (2) is giv by, ( ) = C, whr is giv by (28) is obvious ( ) = hrfor, afr usig h iiial odiio, h soluio of h VP (2) is ( ) = ( ) ( ) ( ) ( ) ( ) E DU ( ) H h rsul (29) Si h sa rasiio maris saisfy Lmma 2, h soluio a also b wri i a alraiv form, ( ) =Φ (, ) Φ (, ) τ E DU τ Φ (, ) Φ (, ) Φ (, ) Φ (, ) (22) whih will b usd wh w xlor h orollabiliy roris of (2) CONROLLABLY AND OBSERVABLY OF MAR DAE SYSEMS his sio, w xlor h orollabiliy ad obsrvabiliy roris of h marix DAE sysm (2) ad (22), whil (22) is osidrd wh h obsrvabiliy is ord h as of im-varyig sysms, h orollabiliy ad obsrvabiliy ar xamid ovr h im irval Wh worig wih orol sysms, h firs s is o drmi whhr a rsribd orol objiv a b ahivd by maiulaig h orol iu A dir aroah is o osru a orol iu ha will driv h sysm sa rajory o h dsird sa Mor ovi riria a also b obaid o s h orollabiliy of h sysm as show i his sio Dfiiio 3 h marix DAE sysm (2) ad (22) is said o b omlly orollabl if for ay, ay arbirary iiial sa ( ) =, ad ay arbirary fial sa, hr xiss a fii im irval = [, ] ad a orol U ( ),, suh ha ( ) = h orollabiliy rory is imora for a orol sysm f h sysm is o orollabl, a orol U ( ) may o xis o ahiv h orol objiv hr ar ohr ys of orollabiliy Coml sa orollabiliy oly rquirs (2); o h ohr had, oml ouu orollabiliy rquirs aaim of arbirary ouu, whr (22) has o b osidrd h orol iu U ( ) a b iwis oiuous ovr h orollabiliy of a orol sysm usually boils dow o hig whhr a s of fuios assoiad wih h sa rasiio marix ad h iu sruur marix ar liarly idd ovr A wll-ow ad mor ovi s for liar idd of a s of fuios is by way of h Gramia marix [] Dfiiio 32 h Gramia marix assoiad wih a s of fuios { f ( ) } i ovr is dfid as i= G= g ij, whr = g f ( τ) f ( τ) ij i j As suh, a s of fuios{ f ( ) } i i= ar liarly idd ovr if h Gramia marix G is o-sigular or osiiv dfii si G is a o-gaiv Hrmiia marix horm 3 h marix DAE sysm (2) is omlly sa orollabl o if ad oly if h orollabiliy Gramia marix τ ℵ (, ) = Φ (, ) E DD E Φ (, τ ) (3) is osiiv dfii Proof: Suos ℵ (, ) is osiiv dfii L ( ) = ad ( ) = b wo arbirary iiial ad fial sa ovr h irval = [, ] Nd o show ha hr is a orol iu U ( ) ovr ha will driv o i W hoos ssu 3, Volum 5, 2 29

6 NERNAONAL JOURNAL OF CRCUS, SYSEMS AND SGNAL PROCESSNG U ( ) =D E Φ (, ) ℵ (, )[ Φ (, ) Φ Φ (, )] (, ) (32) suffis o show ha h righ-sid of (22) yilds if = ad subsiu (32) for U i (22), usig Lmma 2 o simlify h xrssios ivolvig h sa-rasiio marix, Φ (, ) Φ (, ) Φ (, ) τ E DU τ τ Φ (, ) ( ) Φ (, ) Φ (, ) =Φ (, ) Φ (, ) Φ (, ) τ Φ (, ) E DD E Φ (, τ) ℵ (, ) Φ (, ) Φ (, ) τ Φ (, ) E DD E Φ (, τ) ℵ (, ) Φ (, ) Φ (, ) Φ (, ) = hrfor, h marix DAE sysm (2) is omlly sa orollabl o Covrsly, assum h sysm (2) is omlly sa orollabl o, w d o show ha h orollabiliy Gramia ℵ (, ) is osiiv dfii is obvious ha h marix ℵ (, ) is symmri ad o-gaiv, all w d o show is ha h marix is ivribl Assum h marix ℵ (, ) is o ivribl, or quivally, is ull sa is o-my As suh, hr xiss a o-zro vor suh ha H, ℵ = Φ Φ θ (, ) θ θ (, τ ) E DD E (, τθ ) = θφ (, τ) E D = 2 θ R θ Φ (, ) E ( ) D( ), (33) is ow ha h sysm (2) is omlly sa orollabl hrfor, hr xiss a orol U suh ha h iiial sa ( ) = θθ is driv o h fial sa ( ) = H, from (22), θθ =Φ (, ) Φ (, ) τ E DU τ Φ (, ) Φ (, ) Φ (, ) Φ (, ) whih a b simlifid as θθ = Φ (, τ) E DUΦ (, τ) Pr-mulily boh sids by θ, o has 2 θ θ = θ Φ (, τ) E DUΦ (, τ) = du o (33), whih imlis θ=, h h oradiio From h roof of horm 3, h orollabiliy of marix DAE sysm from a arbirary iiial sa o a arbirary fial sa is quival o h orollabiliy from a arbirary iiial sa o h origi is usually umbrsom o sablish orollabiliy of a orol sysm hrough h orollabiliy Gramia Our x s is o dvlo a alraiv ririo for sig orollabiliy wihou igraio W would assum diffriabiliy of h marix offii fuios i (2) Evually, his w ririo will b alid o s h orollabiliy of liar im-ivaria marix DAEs Dfiiio 33 Cosidr h marix DAE sysm (2), wih h assumio ha all offii maris i (2) ar diffriabl, a squ of marix fuios P ( ) ovr ar dfid rursivly as follows, (i) P ( ) = E D ( ) D ( ) (34) (ii) Pɺ ( ) P ( ) E = A( ) P ( ) P ( ) B( ) =,2, (35) h followig lmmas will b usd o rov a drivaiv formula ivolvig h marix fuios P ( ) ad sarasiio marix fuios Φ ad Φ Lmma 3 L ( ) b a ivribl marix fuio ovr h, Proof: Si rs o o g ɺ ɺ (36) ( ) = ( ) ( ) ( ) ( ) ( ) =, diffria boh sids wih ɺ ɺ ( ) ( ) ( ) ( ) = ssu 3, Volum 5, 2 292

7 NERNAONAL JOURNAL OF CRCUS, SYSEMS AND SGNAL PROCESSNG Lmma 32 h followig holds for all, s Φ (, s) P ( s) Φ (, s) (, s) P ( s) (, s) =Φ Φ, =, 2, (37) Proof: Us mahmaial iduio Wih =, Φ (, s) P ( s) Φ (, s) = ( Φ ) PΦ Φ P ( s) Φ Rall ha P ( Φ Φ ) ɺ (38) Φ (, s) = ( ) ( s), whr is h fudamal marix of h righ-sid subsysm E ɺ = A Aordig o Lmma 3, Φ (, s) =Φ (, s) E A O h ohr had, Φ (, s) = ( s) ( ), whr is h fudamal marix of h lf-sid subsysm Aordig o Lmma 3, i is asy o s ɺ ( s) ( s) ɺ ( s) ( s) = ɺ = B H, Φ (, s) = B Φ (, s) Afr subsiuig hs xrssios i (38), o has Φ (, s) P ( s) Φ (, s) =Φ [ E AP P P B] Φ s P s =Φ (, ) ( ) Φ (, s) from (35) Now, assum Φ (, s) P ( s) Φ (, s) (, s) P ( s) (, s) =Φ Φ s Coiu o diffria boh sids, Φ (, s) P ( s) Φ (, s) = [ Φ (, s) P ( s) Φ(, s)] = ( Φ ) PΦ Φ Pɺ ( s) Φ Φ P ( Φ) =Φ [ E AP Pɺ P B] Φ =Φ (, s) P ( s) Φ (, s) ɺ h maris P ( ) irodud i Dfiiio 33 ar usful for fidig alraiv ririo for sa orollabiliy of (2) alog wih addiioal smoohss odiio o h sysm maris all du o Lmma 32 ad Dfiiio 33 horm 32 Suos h marix fuios i h liar imvaryig marix DAE sysm (2) saisfy h smoohss odiio, i, l l b a osiiv igr, l A( ), B( ) C ad D( ), E ( ) C l h, sysm (2) is omlly sa orollabl if hr xiss α suh ha h orollabiliy marix ( ) = [ P ( ) P ( ) P l ( )] (39) α α α α is full row ra, i r( ( )) = α Proof: Assum h DAE sysm (2) is o omlly sa orollabl Aordig o horm 3, h assoiad orollabiliy Gramia ℵ (, ) mus b sigular H, hr xiss a o-zro osa vor υ R, suh ha υ ℵ (, ) υ=, whr ℵ (, ) is giv by (3), whih imlis ha υ Φ (, ) E ( ) D( ) = for all H, υ Φ (, ) E DD Φ (, ) =, or from (34), υ Φ (, ) P ( ) Φ (, ) =, (3) a -drivaivs of boh sids of (3) wih rs o, aordig o Lmma 32, w hav υ Φ (, ) P ( ) Φ (, ) =,, =,2,, l (3) L β b a arbirary oi i, i < <, du o Lmma 2, (3) ad (3) a b ombid as β u Φ (, ) P ( ) Φ (, ) =,, =,, 2,, l (32) whr u = υ Φ (, β ) u is a o-zro vor baus υ ad Φ (, β ) is ivribl Now, subsiu = β i (32) o g u P ( ) Φ (, ) = or, si β β β Φ (, ) is o-sigular, u P ( β ) =, =,,, l his is wri as β β l β β u [ P ( ) P ( ) P ( )] = u ( ) = β du o (35) wih = ssu 3, Volum 5, 2 293

8 NERNAONAL JOURNAL OF CRCUS, SYSEMS AND SGNAL PROCESSNG whih imlis ha ( β ) ao b full row ra, i r( ( )) <, whih oradis h ra odiio of h β orollabiliy marix horm 32 a b usd o driv orollabiliy ririo for liar im-ivaria (L) marix DAEs (2), i whih all offii maris ar osa maris o his d, a xlii formula is drivd for h P maris from h rursiv rlaio (34) ad (35) as follows, i i P = ( ) ( E A) E DD B, =,,2 i= i (33) wih h udrsadig ha A =, h idiy marix Afr P big subsiud i h orollabiliy marix (39), wih h hl of lmary olum blo oraios o, w obai a orollabiliy s for h L marix DAE (2) i h followig horm horm 33 h liar im-ivaria marix DAE sysm E ɺ = A EB DU, ( ) = is omlly sa orollabl ovr if ihr of h followig orollabiliy maris is full ra, i r( ) = or r( ) =, whr r l= DD E AE DD ( E A) E DD r= DD E DD B E DD B (34) h obsrvabiliy of a orol sysm is of osidrd as a dual roblm of orollabiliy for liar ODE sysms Howvr, his may o b ru for DAEs [6] Obsrvabiliy is imora for a orol sysm baus, if h sysm is obsrvabl, h ouus of h sysm a omlly drmi h sas of h sysm O h ohr had, if a sysm is o obsrvabl, i mas som of h urr sas ao b drmid by h masurm of h ouus hrough ssors As suh, a orollr osrud basd o hs ouus do o fulfill h orol sifiaios rlad o hos uobsrvabl sas A formal dfiiio for obsrvabiliy is giv blow Dfiiio 34 h marix DAE sysms (2) ad (22) is said o b omlly obsrvabl if for ay ad ay iiial sa ( ) =, hr xiss a fii im > suh ha, l = [, ], h orol U ( ) ad ouu Y ( ) for suffi o drmi h iiial sa l Wihou loss of graliy, i a b assumd ha h orol U ( ) is idially zro hroughou h im irval [5] W hav h followig rsul o h obsrvabiliy of marix DAE (2) ad (22), wih zro orol iu, via h obsrvabiliy Gramia marix horm 34 h marix DAE sysm (2) ad (22) is omlly obsrvabl o if ad oly if h obsrvabiliy Gramia marix τ ℵ o (, ) = Φ (, ) F ( τ ) F ( τ ) Φ ( τ, ) (35) is osiiv dfii Proof: Assum h obsrvabiliy Gramia marix is osiiv dfii, osidr (2) ad (22) wih U ( ) = ovr Aordig o (22), h, h ouu Φ (, ) =Φ (, ), o has ( ) =Φ (, ) Φ (, ) ovr Y ( ) = F( ) Φ (, ) Φ (, ) Si F( ) Φ (, ) = Y ( ) Φ (, ) Pr-mulily boh sids of h abov quaio by Φ (, ) F ( ) ad igra ovr, w hav =ℵo Φ Φ (, ) ( τ, ) F ( τ) Y ( τ) (, τ) hrfor, h marix DAE sysm (2) ad (22) is omlly obsrvabl Covrsly, assum h sysm is omlly obsrvabl, w d o show ha h obsrvabiliy Gramia marix ℵ o (, ) is osiiv dfii Obviously, ℵ o (, ) is symmri Assum, howvr, ℵ o (, ) is sigular h, hr xiss a o-zro vor v C suh ha (35), ℵ 2 τ τ v o (, ) v = From vℵ o (, ) v = F ( ) Φ (, ) v = H, F( τ) Φ ( τ, ) v ovr f w hoos h iiial odiio is vv =, whih is a o-zro marix h ouu Y ( ) = F( ) Φ (, ) v vφ (, ) i H, h iiial sa = vv ao b uiquly drmid from h abov quaio his oradis h odiio ha h sysm is omlly obsrvabl ssu 3, Volum 5, 2 294

9 NERNAONAL JOURNAL OF CRCUS, SYSEMS AND SGNAL PROCESSNG Similar o h ram of orollabiliy, w will driv algbrai odiios o h obsrvabiliy of (2) ad (22) Dfiiio 35 A squ of marix fuios Q ( ) ovr assoiad wih h marix DAE sysm (2) ad (22) ar dfid rursivly as follows, wih h assumio ha all offii maris i (2) ad (22) ar diffriabl (i) Q ( ) = F ( ) F ( ) (36) (ii) Qɺ ( ) Q ( ) Q ( ) E = A( ) B( ) Q ( ) =,2, (37) Lmma 33 For all, s, h followig holds, Φ (, s) Q ( ) Φ (, s) (, s) Q ( ) (, s) =Φ Φ, =,2, (38) Noi ha h rursiv rlaio (36), (37) alog wih h drivaiv rlaio (38) ar diffr from hos for orollabiliy, i (34), (35), ad (37) h roof of Lmma 33 is similar o ha of Lmma 32 wih h xio ha Lmma 3 is o dd i h roof Lmma 33 a b usd o rov h followig obsrvabiliy horm for imvaryig marix DAE sysms (2) ad (22) horm 35 Suos h marix fuios i h liar imvaryig marix DAE sysm (2) ad (22) saisfy h smoohss odiio, i, F( ) C A( ), B( ), E ( ) C l ad l h, sysm (2) ad (22) is omlly obsrvabl if hr xiss β suh ha h obsrvabiliy marix Q ( ) β Q ( β ) o ( β ) = Ql ( β ) is full olum ra, i r( ( )) = o β (39) For liar im-ivaria DAE sysm (2) ad (22), a biomial formula for Q is obaid from (36) ad (37), Q m m m m m = B F F( E A) = Fially, w obai h obsrvabiliy riria for h L DAE sysm (2) ad (22) horm 36 h liar im-ivaria marix DAE sysm wih /O sruurs is omlly obsrvabl ovr if ihr of h followig obsrvabiliy maris is full ra, i r( ) = or r( ) =, whr lo ro F F F FE A lo= F F( E A) ad V CONCLUSON F F BF FE ro= B F FE Corollabiliy ad obsrvabiliy riria for h marix diffrial algbrai sysms (2) ad (22) ar drivd for liar im-varyig ad liar im-ivaria ass, rsivly Du o h uiqu sruur of h sysm, losd form soluios ar obaid o osru h Gramias ad rursiv rlaios for h orollabiliy ad obsrvabiliy maris h roblm boms mor omliad if h dsrior marix dos o aar o h righ sid of (2) as a losd form soluio may o b availabl for h w sysm his is a ogoig rsarh ad our rlimiary rsuls idia ha rai sysms a b rasformd io h form of (2) ACKNOWLEDGMEN h auhor would li o ha h rviwrs for hir hlful omms REFERENCES [] S L Cambll, A Gral Form for Solvabl Liar im Varyig Sigular Sysms of Diffrial Equaios, SAM J Mah Aal, 8, -5, 987 [2] E B Casla, V G da Silva, O h Soluio of a Sylvsr Equaio Aarig i Dsrior Sysms Corol hory, Sysms Corol L, 54, 9-7, 25 [3] E Johr, Variaioal Calulus for Dsrior Problms, EEE ras Auoma Cor, 33, , 988 [4] S L Cambll, Sigular Sysms of Diffrial Equaios, 6, Rsarh Nos i Mahmais, Pima Boos Ld, Lodo, 982 [5] L Lai, Sigular Corol Sysms, 8, Lur Nos i Corol ad formaio Sis, Srigr-Vrlag, brli, Hidlbrg, 989 [6] R Sadghia, P Karimagha, ad A Khayaia, Corollr Rduio of Disr Liar Closd Loo Sysms i a Crai Frquy Domai, J Ciruis, Sysms, ad Sigal Prossig,, 45-49, 27 [7] V Prlia, Miimal Ralizaio Algorihm for Mulidimsioal Hybrid Sysms, WSEAS ras Sysms, 8, 22-33, 29 [8] V Prlia, Sruural Proris of Liar Gralizd Sysms, WSEAS ras Sysms ad Corol, 3, 7-7, 28 [9] Y Wu, O h Modlig ad Corol of Could Muli-Loo hrmosyhos, i Pro Amria Cof Alid Mah, Puro Morlos, Mxio, 2, 5- [] K N Mury ad L V Faus, Som Fudamal Rsuls o Corollabiliy, Obsrvabiliy ad Ralizabiliy of Firs-Ordr Marix Lyauov Sysms, Mah Si Rs J, 6, 47-6, 22 [] Kailah, Liar Sysms, Pri Hall, Eglwood Cliffs, NJ, 98 ssu 3, Volum 5, 2 295

10 NERNAONAL JOURNAL OF CRCUS, SYSEMS AND SGNAL PROCESSNG [2] C Molr ad C Va Loa, Ni Dubious Ways o Comu h Exoial of a Marix, wy-fiv Yars lar, SAM Rviw, 45, - 46, 23 [3] E L Yi ad R F Siov, Solvabiliy, Corollabiliy, ad Obsrvabiliy of Coiuous Dsrior Sysms, EEE ras Auoma Cor, 26, 72-77, 98 [4] S L Cambll ad L R Pzold, Caoial Forms ad Solvabl Sigular Sysms of Diffrial Equaios, SAM J Alg Dis Mhods, 4, 57-52, 983 [5] S Bar ad R G Camro, roduio o Mahmaial Corol hory, Oxford Al Mah ad Comu Si Sris, Clardo Prss, Oxford, 985 [6] CJ Wag, Corollabiliy ad Obsrvabiliy of a Liar im-varyig Sigular Sysms, EEE ras Auoma Cor, 44, 9-95, 999 Ya Wu rivd h BS dgr i Alid Mahmais ad Comur Si from Bijig Uivrsiy of hology, Bijig, Chia, h MS dgr i Alid Mahmais ad PhD i Alid Mahmais ad Elrial Egirig from Uivrsiy of Aro, Aro, OH, i 992, 996, ad 2, rsivly 2, h joid h Darm of Mahmaial Sis, Gorgia Souhr Uivrsiy, Sasboro, GA, whr h urrly is a Assoia Profssor H also holds a adju rofssorshi wih h Darm of Mhaial Egirig, Uivrsiy of Maioba, Wiig, Caada His urr rsarh irss ilud adaiv orol, dralizd orol, samlig hory, digial filr dsig, ad sh/imag rossig ssu 3, Volum 5, 2 296