System Realization Using 2-D Output Measurements

Transcription

1 Systm Ralzaton Usng -D Output asumnts J.E. ou assahustts Insttut of Thnology Lnoln Laboatoy Abstat A two-dmnsonal (-D) systm alzaton va output masumnts s dvlopd. Th thnqu s basd on mult-nput mult-output (IO) systms wh two sts of mats (A,B,C) that haatz th systm dynams of a tagt n a -D spa a obtand fom th ow olumn dtons of th output masumnts. A modal domposton appoah s usd to oupl th two sts of mats obtan th stats pang wthout ffot. Efftvnss of th thnqu s onfmd, sults a psntd fo stat-ang ada masumnts tan on a anonal tagt. I. ITRODUCTIO SYSTE alzaton s a poblm of onsdabl ntst n many aas of ontol sgnal possng. In nt yas, systm dntfaton basd on -D nputoutput data has vd a lot of attnton fo applatons n atv ontol of flxbl stutus mobl ommunatons suh as llula odlss tlphony. In applatons suh as syntht aptu of ada magng, tagt dntfaton of ballst mssls, dtmnaton of potn stutu fom nula magnt sonan (R) that us only -D output data sts, systm dntfaton s aly nvstgatd. Th two-dmnsonal natu of th poblm th absn of th nput data ma systm dntfaton a hallngng tas. owv, Kung t al. [] dvlopd a mthod that an b gadd as a -D systm alzaton. Rfn [] dsbs an ffnt mthod fo omputng th stat-spa mats fom a -D output data st by xplotng th stutu nhnt n th masumnts. owv, n ths appoah th -D systm alzaton s obtand fom two spaat sts of -D systms. Th thnqu bas down f th on o both opn-loop mats psnt multpl Ths sah ffot was suppotd und th ausps of th Lnoln Laboatoy w Thnology Intatvs ogam. Th w Thnology Intatvs ogam s suppotd pnpally by th Dpatmnt of th A Fo und Contat F968--C-. Opnons, ntptatons, onlusons, ommndatons a thos of th autho a not nssaly ndosd by th Untd Stats Govnmnt. J.E. ou s wth Lnoln Laboatoy, assahustts Insttut of Thnology (IT), 44 Wood Stt, Lxngton, A USA; phon: ; fax: ;-mal: jpou@ll.mt.du gnvalus; moov, th ouplng of th two -D systm alzatons s not fully addssd. In ths pap, th matx-nhand matx pnl (E) thnqu psntd by ua [] s followd th stutu nhnt n th data matx poposd n Rfn [] s xplotd to dvlop a novl -D systm alzaton mthod. Th thnqu s basd on th opnloop mats of two spaat mult-nput mult-output (IO) systms that a oupld va a modal domposton ad on on of th opn-loop mats to obtan th stat pa at no ost. Th sultng thnqu povds nhand -D stat stmats. Und tan umstans th -D systm alzaton algothm ylds asymptotally unbasd mnmum vaan stmats. It s tstd on stat-ang ada masumnts tan on a anonal tagt. Th m of th pap s oganzd as follows: Ston II dsbs th -D output data fomulaton, Ston III psnts a mathmatal famwo fo th IO systms, Ston IV dsbs th ouplng of th two -D systm alzatons that gvs s to th -D systm alzaton thnqu, Ston V psnts sults fom stat-ang output masumnts olltd on a anonal tagt. Conlusons a psntd n Ston VI. II. -D OUTUT DATA FORULATIO It s assumd h that th -D data sampls y( mn, ) ouptd wth wht Gaussan nos w( mn, )hav th followng fom: m n ymn (, ) = as p + wmn (, ); = m =,, ; n =,, wh dnots th numb of satts mbddd n th data; a fs to th th omplx ampltud assoatd wth th th sattng nt wth pol pa bs, pg s dfnd by jφ a = a, () a wh φ dnots th th phas. Th pols s obtand fom th gnvalus of th opn-loop mats p ()

2 ad on th ows olumns of th output data, sptvly. Th loaton of th th satt n th -D angl( s), angl( p)g spa s gvn by b. Ths pap psnts a thnqu that povds dt stat pang basd on th ouplng btwn p. On s, pg th pol pas b a dtmnd, th stats an b pad. Th omplx ampltud may b xtatd usng a last-squas ft of a sum of pols, ah wghtd by a n th fom of Eq. (). Ston III psnts th two IO systms that gv s to th -D systm alzaton algothm. Basd on th -D data fomulaton dfnd by Eq. (), th matx notaton of masumnts tan at loos may b wttn y(,) y(, ) Y = L y(, ) y(, ) y(, ) y(, ) y(, ) y(, ) (3) Thus on an fom anl mats wth vy ow olumn of th data matx, whh s dfnd by Eq. (3). Fo xampl, th anl mats of th mth ow nth olumn a gvn by ym (, ) ym (, L) ow m = ol n = L L ym (, ) yml (, + ) ym (, L+ ) ym (, ) s y(, n) y( J, n) y(, n) y( J +, n) III. IO SYSTES y( J +, n) y(, n) sptvly. aamts L J that appa n Eqs. (4) (5), sptvly, dnot th olaton wndows n olumn ow dtons. Thy a hustally hosn to L= J =, wh th bats b [ ] ow m [ ] O Q. O Q O Q (4), (5) dnot th smallst ntg lss than o qual to th nstd quantty. Th pmay ntst n ths ston s modlng -D data by usng two sts of omplx mats. Th fst st s dvd fom a IO systm ad on a ow-nhand ow data matx wth ; m =,, as nts. Th sond st of mats s obtand fom a IO ol systm dvd fom a olumn-nhand data matx ol wth ; n =,, as matx lmnts. xt, th ow ow = A + BW + BW n two systms a oupld to obtan th -D systm alzaton algothm. Th ow-nhand data matx an b obtand by stang th anl mats dsbd by Eq. (4) nto a olumn vto suh that ow ow ow =. (6) ow On an nhand data matx s dfnd, an autogssv movng avag (ARA) modl may b dvd. Th fst IO systm poposd may b sn as a latonshp btwn an mpuls matx W may b haatzd by th ARA usv quaton L L l l l= l= l Bl B l l wh A,, f to th matx offnts ow, (7) of th ARA modl. A sngl-nput sngl-output (SISO) systm of an ARA haatzng Eq. (7) was dvlopd n Rfn [], psntd by a st of stat vaabls [,3]. Thfo, th ARA modl that s dfnd by Eq. (7) allows th followng stat-spa psntaton: X = AX + + BW (8) ow = C X + DW, (9) wh X nput wth Da funtons as nts on ts man dagonal zo lswh; A C, B C L (-, C C L+) D C (-L+) L a onstant mats. Followng th stps dsbd n Rfns [,3], t s asy to s that ow D () L L L C s th stat, W C dnots th = C A B; =,, ow Eq. () ndats that an nhand anl matx. () ow fomd fom a squn of mpuls matx sponss suh that

3 = ow ow ow J ow ow ow ow 3 J + ow ow ow J+ J+ ow an b fatod. Th domposton of podut of two mats s gvn by =ΩΓ wh Γ= ow C C A Ω = C A J J B AB A B nto a () (3) (4). (5) In lna systm ontol thoy, Ω Γ a nown as th obsvablty ontollablty mats, sptvly. By omputng th sngula valu domposton of th nhand anl matx ow ts low an tunaton [,3] fo a SISO systm, th ow followng an duton of s obtand: ~ ow = U Σ V. (6) In Eq. (6), U sn sn sn sn dnots th sgnal omponnts of th lft-untay matx (U ), Σ s a dagonal matx wth th sgnal sngula valus of sn o w aangd n dasng od as nts on ts man dagonal. Futhmo, V s th sgnal omponnt of th ght-untay matx (V ) dnots onjugat tanspos. Thfo, th obsvablty ontollablty mats a gvn by / Ω = U sn Σ sn (7) / Γ = Σsn Vsn. (8) sptvly. Basd on th sults psntd n Rfn [4], th st of omplx mats ( A, B, C ) of th fst IO systm dsbd by Eqs. (8) (9) may b dvd fom Ω o Γ. Fst, th dvaton of ths mats a basd on Ω; thy may b wttn wh ow h A = Ω l Ω l Ω l Ω B ( ) ow = Ω Ω Ω C = Ω( : L+,:) s dfnd by Eq. (6) sn (9) (), () Ω = Ω( L+ :( J + )( L+ ),:)() Ω l = Ω(:( J)( L+ ),:). (3) Ω s an xtndd obsvablty matx that s dfnd by C Ω C A =. (4) C A In Eqs. () though (3), Ω( : l,:) dnots th matx obtand by png th ows to l of Ω. Th stat-spa mats dfnd by Eqs. (9) though () may also b dvd fom th ontollablty matx Γ. It s asy to s that A = Γ Γ l Γ Γ l ol Th sond IO systm s dvd fom by followng stps smla to thos dsbd n Eqs. (7) though (8). Th st of omplx mats A, B, C ( ) an b obtand fom Ω by usng Eqs. (9) though () o Γ by followng Eqs. (5) though (7). Th two IO systms a ady to b oupld. Th oupld -D systms that gv s to th -D systm alzaton algothm a psntd nxt. h (5) B = Γ(:, : L) (6) ow C = Γ( ΓΓ), (7) wh Γ = Γ(:, L+ : LJ) (8) Γ l = Γ(:, : LJ ( )), (9) wh Γ s an xtndd ontollablty matx that s dfnd by Γ = B A B A B. (3) In Eqs. (8) (3), Γ(:, : l) fs to th matx obtand by png th olumns to l of Γ. Th olumnnhand data matx s fomd by stang th ol anl mats dsbd by Eq. (5) nto a ow vto. athmatally, ol ol ol ol =. (3) IV. COULED -D SYSTE REALIZATIO A thnqu to pa th gnvalus s that a p assoatd wth th th satt s psntd h. Ths gnvalu-pang thnqu s basd on th gnvalu domposton of th opn-loop matx A [whh an b omputd fom Eq. (9) o (5)] of th fst IO systm.

4 Th modal matx of A gnvalus of. A Th gnvalu domposton of wll b usd to od th Λ = A, (3) wh Λ s a dagonal matx wth th gnvalus of A dfnd by γ { A } = { s s,, s} (33), A A = A. (35) Th mpotant lmnts of th pdng matx a ts dagonal nts, whh a wttn dagla q= la (,),, A(, ) q. (36) Th gnvalus of th sond IO systm a omputd aodng to γ { A} = { γ, γ,, γ }. (37) To assoat th gnvalus γ wth th ospondng, th odng of th lmnts of γ a dfnd wth spt to th poston stngth of th nts of angl(dag{ angl(dag{ A }) by{ γ { A } A }). Fo xampl, f angl( dag{ A (, ) }) (.., ) s th stongst lmnt of angl(dag{ }), th th nty l q dag A angl( l q ) γ, γ,, γ = lp, p,, p a A s wttn as nts on ts man dagonal. In Eq. (3) dnots A th modal matx of s wttn = [ v, v,, v ], (34) wh v psnts th th gnvto assoatd wth th s gnvalu. atx s fomd dfnd by s { } of γ { } A A A must b th lmnt that xhbts th hghst angl stngth. Th pols assoatd wth Eq. (33) a obtand fom Eq. (37) aodng to q. (38) ot that Eqs. (33) (38) povd th ot gnvalu pa b s, p g assoatd wth th th sattng nt. Thfo to obtan th pop stat pang btwn X X on only nds to od th latt aodng to Eq. (38). Th omplx ampltud assoatd wth th gnvalu pa b s, p g an b gadd as th th lmnt of an ampltud vto that an b xtatd usng a last-squas ft of a sum of gnvalus to th data st; t s tatd lswh (Rfn [4]). Th -D systm alzaton thnqu s summazd n th followng algothm: ) Fom th ow- /olumn-nhand data matx ow ol / by usng Eq. () omput th stat mats ( A ) / ( ) A, ( ) B / / ( B ) ( C ) ( ) C fom Ω by usng Eqs. (9) though () o Γ usng Eqs. (5) though (7). ) Comput th gnvalu domposton of A by usng Eq. (3) to obtan th omplx pols dag { A } s th nts of aodng to Eqs. (33) (36), sptvly. 3) Comput th omplx pols aodng to Eq. (38) od th stats X. V. EXALE FRO STATIC-RAGE RADAR OUTUT EASUREETS Stat-ang output masumnts tan on th.6 m long monoon nty vhl dsbd n Rfn [5] shown n Fgu s onsdd to onfm th fftvnss of th -D systm alzaton algothm. Fo ths xpmnt, fous was on a sgmnt of Rang (m) atv Rl Extndd tun p dbody goov Sond goov Fst goov Bas dg Sond goov Extndd tun dbody goov Fst goov ostp Rlatv Coss-ang (m) Fgu. Canonal tagt phas plot of gnvalus of - D systm alzaton algothm sald to tagt latv physal dmnsons; data usd n ov Gz bwdth ov dg vwng angl. Gz data olltd fom 3 Gz n 4 z nmnts a tagt vwng angl angng fom 5 to 5 dg n.5 dg stp sz. A modl od of 4 (.., = 4) s sltd a po to stmat th mpotant fatus

5 (nostp, goovs, bas dg) of th tagt. Fgu shows th -D phas plot of th pol pas s, p sald wth spt to th latv tagt physal dmnsons. Th dots that dsb th mag of th tagt n th -D physal spa a labld to ndat th loatons of th tagt fatus. Eah goov n th mag s dfnd by two satts (on at ah dg), wh th nn dstan btwn thm dfns th latv damt of th on ston whn th goov s loatd. Futhmo, th algothm psnts th bas dg by a st of fou satts to ndat ts stong ontbuton to th ada tuns th latv lag damt of th on bas. Th two xtndd tuns that appa n ba of th bas an b lmnatd by hoosng a small modl od. owv, th pnalty fo suh a ho s a lss auat psntaton of th modld data. As xptd, n th mag th nostp s dfnd by only on satt. Bas dg (xt m ght) Bas dg ( xt m ght) Ral -.8 Imagnay Fquny (Gz) Ral Imagnay Vwng angl (dg) Fgu. Stat spons fom bas dg (xtm ght satt) of anonal tagt fo al (sold uv) magnay (dashd uv) omponnts as a funton of Fquny vwng angl. Fgus show th stat assoatd to th bas dg (xtm ght satt) as a funton of fquns vwng angls, sptvly. Th plots also show th al (sold uv) magnay (dashd uv) omponnts of th stat. Th stat assoatd wth th nostp s plottd n Fgus (3). It s mpotant to not that baus of th hust ho of th olaton wndow b g lngth L= [ ] J [ ] o -. =, th al magnay omponnts fo th stats psntd n Fgus () (3) a plottd only fo half th fquny angula xtnts. owv, on an us th stat mats to ov th full bwdth vwng angl fo th stats latd to th sattng nts. Th 7th olumn of th data matx s hosn to ompa th modl obtand fom th -D systm alzaton algothm th masud data fo a modl od = 4. Fgus 4 llustat th ompasons btwn th ft (dashd uv) th data (sold uv) fo th I Q hannls, sptvly. In ths fgus th fttd modl dvats fom th masumnts, pmaly du to a low valu of. xt, th pvous xpmnt s patd usng ostp p st Fquny (Gz) Ral Imagnay Ral Imagnay Vwng angl (dg) Fgu 3. Stat spons fom nostp of anonal tagt fo al (sold uv) magnay (dashd uv) omponnts as a funton of fquny vwng angl. a modl od = 4. Fgus 5 show xllnt agmnt btwn th modl th masud data. It s mpotant to not that vn though th hgh modl od has th bnft of png th fttd data vy los to th masumnts, t s ws to hav a tad-off btwn modl od th -D psntaton of th tagt n physal spa.

6 VI. COCLUSIOS A two-dmnsonal systm alzaton thnqu basd on nhand data mats has bn dvd to xtat statspa mats n th ow olumn dtons of a masumnt matx stats assoatd wth th man sattng nts that fom a tagt. Th thnqu s omputatonally ffnt baus an algba mthod s usd to oupl th two sts of mats. Stat-ang ada masumnts tan on a anonal tagt hav bn usd to judg th patablty of th algothm. It has bn dmonstatd that whn nhand data mats a usd, th -D systm alzaton thnqu povds auat loatons of th sattng nts vy good agmnt btwn th fttd modl th matx of masumnts. nnl I Cha Q Channl asumnts odl Vwng angl (dg) asumnts odl Vwng angl (dg) Fgu 4. Compason of fttd modl (dashd uv), usng an od = 4, masumnts (sold uv) fo 7th olumn of data matx al magnay omponnts. [3] J.E. ou, K.. Cuomo, J.T. ayhan, A Stat- Spa Thnqu fo Ultawd-Bwdth Cohnt ossng, IT Lnoln Laboatoy, Lxngton, ass., Thnal Rp. TR 54 ( July 999), ESC- TR [4] J.E. ou, J.T. ayhan, K.. Cuomo, Algothm Dvlopmnt foman Bounds fo Spas- B, Spas-Angl ossng, IT Lnoln Laboatoy, Lxngton, ass., ojt Rp. T-4 (4 Jun ). [5] K.. Cuomo,J.E. ou, J.T. ayhan, Ultawd- B Cohnt ossng, IEEE Tans. Antnnas opag., 47, 94-7 (Jun 999). nnl I Cha Q Channl asumnts odl Vwng angl (dg) asumnts odl Vwng angl (dg) Fgu 5. Compason of fttd modl (dashd uv), usng an od = 4, masumnts (sold uv) fo 7th olumn of data matx al magnay omponnts. REFERECES [] S.Y. Kung, K.S. Aun, D.V.B. Rao, Stat-Spa Sngula Valu Domposton-Basd Appoxmaton thods fo th amon Rtval oblm, J. Opt. So. Am., 73, (Dmb 983). [] Y. ua, Estmatng Two-Dmnsonal Fquns by atx Enhanmnt atx nl, IEEE Tans. Aoust., Sph, Sgnal oss., 4, 67 ptmb 99).