CONTROLLABILITY OF A CLASS OF SINGULAR SYSTEMS

Transcription

1 44 Asa Joural o Corol Vol 8 No 4 pp December 6 -re Paper- CONTROLLAILITY OF A CLASS OF SINGULAR SYSTEMS Guagmg Xe ad Log Wag ASTRACT I hs paper several dere coceps o corollably are vesgaed or a class o lear sgular sysems whch sysem parameers are pecewse cosa Necessary ad suce geomerc crera or C-corollably ad R-corollably o such sysems are esablshed respecvely These codos ca be easly rasormed o algebrac orm y applyg he prcple o dualy C-observably s dscussed as well Furhermore he rsc relaoshp bewee hese resuls ad exsg resuls are also dscussed The a ovel ecessary ad suce crero or C-corollably o lear me-vara sgular sysems s derved as a byproduc KeyWor: Sgular sysems corollably pecewse cosa reachably I INTRODUCTION I rece years aalyss ad syhess problems o sgular sysems have bee well suded [-8] Ths class o sysem has bee exesvely suded pas years as hs class s more suable ha he sadard represeao modellg praccal sysems such as ecoomcal sysems large scale sysems ewors power sysems see or sace [] ad reereces here There are several deos o corollably or dere purposes For a lear me-vara sgular sysem he sysem s called compleely corollable (C-corollable) [] ca be drve o ay ermal sae rom ay admssble al sae; he sysem s called R-corollable [] ca be drve o ay ermal sae he reachable se rom ay admssble al sae; he sysem s called mpulse corollable (I-corollable)[] or every al codo here exss a smooh (mpulse-ree) corol u() ad a smooh sae rajecory x() soluo; ad he sysem Mauscrp receved November 9 4; revsed Aprl 7 5; acceped Sepember 8 5 The auhors are wh he Ceer or Sysems ad Corol LTCS ad Deparme o Mechacs ad Egeerg Scece Peg Uversy ejg 87 Cha (e-mal: Ths wor was suppored by Naoal Naural Scece Foudao o Cha (No 37 No 644 ad No 674) ad Naoal Key asc Research ad Developme Program (C3) s called srogly corollable (S-corollable)[3] s boh R-corollable ad I-corollable [] vesgaed C-corollably o sgular sysems wh sgle me-delay corol ad he ecessary ad suce codos were esablshed The [] exeded he resuls [] o a mulple me-delayed case ad ecessary ad suce crera or R-corollably ad I-corollably were derved as well [3] ad [4] suded he ssues o corollably ad observably or aalycally solvable lear mevaryg sgular sysems bu he model cosdered [3] ad [4] was assumed o be he sadard caocal orm Despe hese mpora resuls abou corollably aalyss o me-vara or me-varyg sgular sysems very ew papers cosder pecewse cosa sgular sysems I hs paper he auhors am o derve ecessary ad suce crera or corollably o pecewse cosa sgular sysems For a pecewse cosa sgular sysem a dsc eaure s ha he rajecory o he sysem s dscouous ad jumps a he dscouous po C-corollably ad R-corollably o such sysems s vesgaed ad ecessary ad suce geomerc codos are esablshed The he algebrac crera are obaed as well y prcple o dualy he C-observably s dscussed as well Furhermore he rsc relaoshp bewee hese resuls ad he exsg resuls or lear me-vara sgular sysems ad pecewse lear sysems s also addressed A ovel ecessary ad suce crero or C-corollably o lear me-vara s-

2 G Xe ad L Wag: Corollably o a Class o Sgular Sysems 45 gular sysems s derved as a byproduc Ths paper s orgazed as ollows Seco II ormulaes he problem ad preses some prelmary resuls C-corollably ad R-corollably are vesgaed Seco III The relaoshp bewee hese resuls ad he exsg oes are dscussed Seco IV Seco V preses wo llusrag examples Fally Seco VI cocludes he whole paper by II PRELIMINARIES Cosder he pecewse cosa sgular sysem gve Ex () = Ax () + u () y () = Cx () [ ) = where x() R s he sae vecor u() R p s he pu vecor y() R q + s he oupu; x( ):= lm + x( + h) h x( ):= lm + x( h) x( ) = x() mples ha he soluo o sysem () s le couous E A C are he h ow p ad q cosa real marces or = E may be sgular ad de(se A ) / ad he dscouous pos < < < where < ad < = < The sysem s sad o be regular each quadruple (E A C ) s regular as a me-vara sgular sysem or = I hs paper oly regular sysems are cosdered Sce each quadruple (E A C ) s regular here exs osgular marces P ad Q such ha I G QEP = QAP = N I Q = CP = [ C C ] ( ) p p where N R s lpoe R ( ) p q R C q ( ) R C R ad < < The marx P ad s verse are decomposed as () () U P = [ P P ] P = (3) U P P ( ) where R R R ad U ( ) R Moreover deoe h = ad U H = P exp( G h ) U = I he res o he paper deoe U he se o ucos wh pecewse dereable mes As usual s assumed ha all he corol pu u() U Gve a pu uco u() U u () () deoes he h dervave o u() = Le = A be he marces produc A A ad A = be he marces produc A A Now cosder he geeral soluo o sysem () Lemma For ay ( ] gve he al sae x ad a pu u U he geeral soluo o sysem () s gve as ollows: (a) = G ( s) x () = P exp[ G( )] U x ( ) + P e (b) = 3 j ( j) u() s P ( N ) u () G( ) x () = P e U H j x ( ) m Gm( ms) + H j Pm e m u m m= m j ( j) Pm ( Nm) m u ( m) G( s) j ( j) ( ) u G ( s) P e u + P e u P N + j ( j) () (4) P ( N ) u ( ) (5) Proo See Appedx A Remar y Lemma oe ows ha he soluo o sysem () s dscouous a = A he dscouous po he sae jumps rom x( ) o x( + ) Oe par o x( + ) s hered rom x( ) ad he oher par s correspodg o he corol pu u( ) Now some mahemacal prelmares wll be gve as he basc ools he ollowg dscusso Gve marces A R ad R p deoe Im() he rage o e Im() = {y y = x x R p } ad deoe A he mmal vara subspace [] o A o Im() e A = A Im() =

3 46 Asa Joural o Corol Vol 8 No 4 December 6 The ollowg lemma s a geeralzao o Theorem 78 [9] whch s he sarg po or dervg he corollably crera p Lemma Gve marces G R R N p R R P R ad P R where + = or ay < < + oe has G ( s) j ( j) x x = P e u P ( N) u ( ) u U G = [ P P] (6) N Proo See Appedx Lemma 3 Gve marces A Q R ad R p oe ca see QA I Q = Q AQ (7) Proo See Appedx C I he ollowg corollably ad observably o sysem () a me sa wll be dscussed I = he he sysem s reduced o a lear me-vara sgular sysem or whch may corollably deos ad crera have bee esablshed [47] Thus he remag par o he paper he auhors cocerae o he case whe = 3 III CONTROLLAILITY Frs he reachably o sysem () wll be dscussed For sysem () a sae x s called reachable rom al sae x R a me sa ( < ) here exss a pu u() U such ha he sysem s drve rom x( ) = x o x( ) = x Le R [ ]( x) be he se o reachable saes rom x The reachable se o he sysem s R [ ] = R ( ) x x [ ] Theorem For sysem () he reachable se rom sae x [ ] s gve by R ( x ) = I( x ) + P Q ( A + E ) P Q [ ] + H j Pm Qm ( Am + Em) PmQm m m= where H x I x ( ) = j (8) Proo See Appedx D Deo C-corollably Sysem () s sad o be compleely corollable (C-corollably) [ ] ( < ) or ay sae x x R here exss a pu u() U such ha he sysem s drve rom x( ) = x o x( ) = x Corollary Sysem () s C-corollable [ ] ad oly H j PmQm ( Am + Em) PmQm m + PQ ( A+ E) PQ = R (9) m= Proo Sysem s C-corollable ad oly or ay x x R Eq (5) has a soluo u() U y Theorem hs s equvale o x x H P Q A E P Q I ( ) j m m ( m + m) m m m m= + PQ ( A+ E) PQ or ay x x R Thus hs s equvale o (9) Remar Corollary s he geomerc orm y smple rasormao oe ca ge a algebrac orm codo I ac or = deoe ( ) U = PQ ( A + E) PQ ( A + E) PQ + V = H ju () The deoe Q = [ V V V U ] () [ ] I s easy o very ha ImQ ( [ ]) = R [ ] () Thus oe ges he ollowg algebrac orm crero Corollary Sysem () s C-corollable [ ] ad oly ra( Q[ ]) = Deo R-corollably Sysem () s sad o be corollable he se o reachable saes (R-corollable) [ ]( < ) or ay al sae x R ad ay ermal sae x R [ ] here exss a pu u() U such ha he sysem s drve rom x( ) = x o x( ) = x

4 G Xe ad L Wag: Corollably o a Class o Sgular Sysems 47 Corollary 3 Sysem () s R-corollable [ ] ad oly Im H H P Q A E P Q j j m m ( m) m m m m= PQ (3) Corollary 5 Sysem () s C-observable [ ] ad oly m m= T T T T T T T j m m m m m m m H Q P ( A + E ) Q P C T T T T T T ( ) + Q P A + E Q P C = R (7) Proo y Theorem s easy o see ha = Im H j + PQ A+ E PQ R[ ] ( ) + H P Q ( A + E ) P Q m= j m m m m m m m (4) The he sysem s R-corollable ad oly or ay x R ad ay x R [ ] Eq (5) has a soluo u() U Ths s equvale o x x H P Q A E P Q I ( ) j m m ( m + m) m m m m= PQ (5) Ths s also equva- or ay x R ad ay R le o R x [ ] [ ] ( ) H jpmqm A EmPmQm m m= PQ (6) Obvously hs s equvale o (3) Smlar o Corollary by he oaos () ad () Corollary 3 ca be rasormed o he algebrac orm as ollows IV RELATIONSHIP ETWEEN THESE RESULTS AND EXISTING RESULTS I Seco III ecessary ad suce codos or corollably o pecewse cosa sgular sysems are derved Sce pecewse cosa sgular sysems are exesos o lear me-vara sgular sysems ad pecewse cosa sysems hese resuls geeralze he exse resuls o corollably o lear me-vara sgular sysems ad pecewse cosa sysems 4 Exeso rom lear me-vara sgular sysems For sysem () (E A ) = (E A ) = he he sysem s reduced o a lear me-vara sgular sysem We ll show ha crera (9) ad (3) are reduced o he radoal oes I ac hs case oe ca assume ha (P Q ) = (P Q) (G N ) = (G N) = U where G R P = [ P P] P = ad U Q = They wll be dscussed respecvely: (a) C-corollably Crero (9) s smpled as m= H j PQ ( A + E) PQ PQ = R (8) Corollary 4 Sysem () s C-corollable [ ] ad ra H j j Q [ ] = ra( Q [ ]) oly ( = ) Remar 3 I he above aalyss reerece s made o reachably ad corollably oly I should be oced ha he observably couerpars ca be addressed dualscally Deo 3 Observably The sysem () s sad o be C-observable o [ ]( < ) ay al sae x R ca be uquely deermed by he correspodg sysem pu u() ad he sysem oupu y() or [ ] Frs s easy o see ha PQ ( A + E) PQ = P G + P N Nex or ay R oe ges ( + ) exp( ) P G P P G P N exp( ) exp( ) = P G P P G + P G P P N = P exp( G) G P G Thus oe ca see ha he le par o he Eq (8) s jus P G + P N Sce P s osgular (8) s equvale o G N = R (9)

5 48 Asa Joural o Corol Vol 8 No 4 December 6 I s obvous ha (9) s jus he radoal crero or C-corollably o lear me-vara sgular sysems Thus he exsg resul s a specal case o hs paper s resul Moreover oe ges a ew crero or C-corollably o lear cosa sgular sysems as ollows Corollary 6 A lear me-vara sgular sysem (E A ) s C-corollable ad oly oe o he ollowg codo hol: ( A+ E) PQ = R () ( ( ) ) ra ( A+ E) PQ ( A+ E) PQ = () ( ) ra [( A + E) PQ I s ] = s () (b) R-corollably Crero (3) s smpled as I m H j H j PQ ( A+ E) PQ m= The (3) s equvale o PQ (3) Pexp G hj I m( P ) P G + P N (4) Moreover sce ImP ( ) I mp ( ) = (4) s equvale o Pexp G hj I m( P ) P G (5) Sce P P ad exp( G hj ) are all ull ra (5) s equvale o R G Obvously hs s also equvale o G = R Ths s jus he radoal crero or R-corollably o lear me-vara sgular sysems 4 Exeso rom pecewse lear sysems For sysem () E E are osgular he he sysem s reduced o a pecewse lear sysem The auhors wll show ha he crera (9) ad (3) are also reduced o he radoal oes I ac hs case oe ca assume ha (P Q ) = (I E ) = he (9) s rewre as m= j j j m m m exp( A h ) E A E m + E A E = R (6) I parcular E = I = he (6) s jus m= exp( Aj hj) Am m + A = R (7) I s easy o see ha (7) s jus he radoal crero or corollably o pecewse lear sysems [] As o crero (3) oe has Im exp( Aj hj) exp( Aj hj) Am m K m= + A (8) Obvously (8) s also equvale o (7) Thus s show ha C-corollably ad R-corollably are boh reduced o he geeral corollably o pecewse lear sysems V ILLUSTRATING EXAMPLES I hs seco wo umercal examples are gve o llusrae how o ulze hs crera Example Cosder a 6-dmesoal pecewse cosa sgular sysem wh E = A = = (9) E = A = = (3) where = = ad = = Now he crera s used o sudy he corollably o he sysem Example y smple calculao we ge P = P = Q = Q = I 6 = ad = 4 Moreover s easy 6 o very ha H A+ E + A + E = R y Corollary he sysem s C-corollable I ac he corol pu s ae as

6 G Xe ad L Wag: Corollably o a Class o Sgular Sysems 49 c ( ) ; c( ) + c3 = ; u () = c4 ( ) ; c5( ) + c6 = The oe has c c3 c4 c5 c 6 (3) x() = H H x() +Φ where Φ = e ee e e e I s easy o very e ha he marx Φ s osgular Ths shows ha he sysem s C-corollable deed Example Cosder a 6-dmesoal pecewse cosa sgular sysem wh E = A = = (3) E = A = = (33) where = = ad = = y smple calculao oe ges P = P = Q = Q = I 6 = ad = Moreover s easy o very ha 6 H A + E + A + E R bu I mh ( H) H A+ E + A + E y Corollares ad 3 he sysem s R-corollable bu o C-corollable I ac c he corol pu s ae as c ( 5] ; c ( 5 ] ; u () = c3 ( ) ; c4 = (34) The oe has x() = H H x() + Φ[c c c 3 c 4 ] T where Φ= Meawhle by smple calculao oe ges R [] = I m I s easy o very ha Im(Φ) = R [] Ths shows ha he sysem s R-corollable deed As o C-corollably s obvous o see ha he 6 h varable o he sae remas zero all he me I ca o be aeced by ay pu I s easy o see ha he sysem s o C-corollable VI CONCLUSIONS Ths paper has deal wh he corollably o pecewse cosa sgular sysems Necessary ad suce geomerc crera or C-corollably ad R-corollably have bee esablshed respecvely These crera are easly rasormed o algebrac orms Furhermore he relaoshp bewee hese resuls ad he exsg resuls leraure have also bee dscussed A ovel ecessary ad suce crero or C-corollably o lear me-vara sgular sysems has bee derved as a byproduc APPENDIX A Proo o Lemma For = le z () = P x() z () [ ) decompose z () as z () = where z () () z R ad () z R he ge

7 43 Asa Joural o Corol Vol 8 No 4 December 6 z () = Gz () + u() Nz () = z () + u() (35) z( ) = P x( ) [ ) The soluo o (35) s ( ) G G( s) e z ( ) + e ( ) u s z () = j ( j) ( N) u ( ) (36) or ( ) The oe ges G( ) G ( s) j ( j) G( ) G( s) ( ) ( ) j ( j) P ( N) u ( ) x () = P z () + P z () = P e z ( ) + P e u( s) P ( N ) u ( ) = Pe U x + P e us Sce z ( ) = U x( ) or ( ) oe has G( ) G( s) j ( j) P ( N) u ( ) x() = P e U x( ) + P e u( s) Sce x( ) = x( ) he above equao also hol or = Thus oe ows ha (a) hol For = 3 use mahemacal duco () For = ( ) G( ) G( s) j ( j) P ( N) u ( ) x () = P e U x ( ) + P e us ( ) = P exp[ G( )] U P exp( G h) U x( ) + P exp[ G ( s)] u( s) j ( j) ( ) u ( ) j ( j) P N u P N + P exp[ G ( s)] u( s) ( ) ( ) Sce x( ) = x( ) he above equao also hol or = The s ow ha (4) hol or = () Suppose ha (5) hol or 3 he auhors wll prove ha (5) hol or For ( ) G( ) G ( s) j ( j) x() = P e U x( ) + P e u P ( N ) u ( ) G( ) Gh = P e U P e U x( ) G( s) + P e u j ( j) P ( N) u ( ) G ( s) j ( j) + P e u P ( N ) u ( ) = G( ) Gh j j = P e U Pj e U j x( ) m Gh j j Gm( ms) + Pj e U j Pm e m u() s m= m m j ( j) Pm ( Nm) m u ( m) exp[ ( )] j ( j) P N u + P G s u ( ) ( ) exp [ ( )] + P G s u j ( j) P ( N ) u ( ) Sce x( ) = x( ) he above equao also hol or = Thus oe ows ha (5) hol or APPENDIX Proo o Lemma y Theorem 78 [9] oe has { exp [ ( )] ( ) U } x x = G s u s u = G Meawhle oe has x x N u u N j ( j) = ( ) ( ) U = (37) (38)

8 G Xe ad L Wag: Corollably o a Class o Sgular Sysems 43 Thus s easy o see ha G ( s) x x = P e u() s j ( j) P ( N) u ( ) u U G ( s) Px + Px x = e us () j ( j) x = ( N) v ( ) u v U Moreover oe P G + P N (39) E G ( E s) { x x = e u() s u U } G ( s) { () U } E = x x = e u s u = G (4) where E = ( + )/ The or ay x P G + P N rewre x as x = P x + P x where x G ad x N The here mus exs x () U [ E ] so ha G ( s) x x = P e u() s P N u u P G Sce j ( j) ( ) ( ) U + P N (4) G P G + P N = [ P P] N by (39) ad (4) oe ha (6) hol Proo o Lemma 3 Sce APPENDIX C I = = ( ) I ( ) = QA Q = ( QA) m( Q) = ( QA) QIm( ) = Q AQ m = Q AQ Oe oly ee o very ha A I = A I ac s easy o see ha A I A+ I A The oe sees A = A I + I A I Hece A I = A j ( j) = x ( N) u ( ) E Moreover deoe z = exp[ G ( s)] u ( s ) G Sce xz G here mus exs u () U [ E ) so ha x z = exp[ G( s)] u( s) The oe ca ae u() [ E ) v () = u () [ E ] so ha j ( j) P exp[ G ( s )] v ( s ) P ( N ) v ( ) = P( x z + z ) + P x = x Ths mples ha E x x x = P exp[ G( s)] u( s) j ( j) P ( N) u ( ) u U I ollows ha APPENDIX D Proo o Theorem Frs cosder R [ ]() y Lemma le x( ) = he x ( ) ( m Gm m s ) H j P m e = m u ( s ) m m= m j ( j) Pm ( Nm) m u ( m) + P exp [ G ( s)] u( s) j ( j) P ( N ) u ( ) (4) I ollows ha m Gm( ms) R [ ] = x x = H j Pm e m u m= m m j ( j) Pm ( Nm) m u ( m) + P exp[ G ( s)] u( s) j ( j) P ( N) u ( ) u U

9 43 Asa Joural o Corol Vol 8 No 4 December 6 m Gm( ms) = H j x x = Pm e mu() s m m= m j ( j) Pm ( Nm) m u ( m) u U y Lemma oe ges + x x = P exp[ G( s)] u( s) j ( j) P ( N) u ( ) u U R [ ] = H P G + P N m m j m m= N m m For m = by Lemma 3 G m m N m m G = Q ( A + E ) P I Q m m m m m m = Q ( A + E ) P Q m m m m m m I s easy o very ha (8) hol or zero sae For ozero sae x he proo s smlar ad hus omed REFERENCES Da L Sgular Corol Sysems Volume 8 o Lecure Noes Corol ad Iormao Sceces Sprger-Verlag New-Yor (989) Yp EL ad RF Scovec Solvably Corollably ad Observably o Couous Descrpor Sysems IEEE Tras Auoma Cor Vol 6 pp 7-77 (98) 3 Verghese G Levy ad Th Kalah A Geeralzed Sae-Space or Sgular Sysems IEEE Tras Auoma Cor Vol 6 pp 8-83 (98) 4 Cobb D Descrpor Varable Sysems ad Opmal Sae Regulao IEEE Tras Auoma Cor Vol 8 No 5 pp 6-6 (983) 5 Cobb D Corollably Observably ad Dualy Sgular Sysems IEEE Tras Auoma Cor Vol 9 No pp 76-8 (984) 6 Zhou Z MA Shayma ad T Tar Sgular Sysems: A New Approach he Tme Doma IEEE Tras Auoma Cor Vol 3 No pp 4-5 (987) 7 Chu KE A Corollably Codesed Form ad a Sae Feedbac Pole Assgme Algorhm or Descrpor Sysems IEEE Tras Auoma Cor Vol 33 No 4 pp (988) 8 Tag W ad G L The Crero or Corollably ad Observably o Sgular Sysems Aca Auomaca Sca Vol No pp (995) 9 L C J Wag ad C Soh Necessary ad Suce Codos or he Corollably o Lear Ierval Descrpor Sysems Auomaca Vol 34 No 3 pp (998) Ishhara JY ad MH Terra Impulse Corollably ad Observerbly o Recagular Descrpor Sysems IEEE Tras Auoma Cor Vol 46 No 6 pp () We J ad W Sog Corollably o Sgular Sysems wh Corol Delay Auomaca Vol 37 pp () Guagmg Xe ad Log Wag Corollably o Lear Descrpor Sysems IEEE Tras Crcus Sys I Vol 5 No 3 pp (3) 3 Wag C-J Corollably ad Observably o Lear Tme-Varyg Sgular Sysems IEEE Tras Auoma Cor Vol 44 No pp 9-95 (999) 4 Wag C-J ad H-E Lao Impulse Observably ad Impulse Corollably o Lear Tme-Varyg Sgular Sysems Auomaca Vol 37 pp () 5 L C JL Wag G-H Yag ad C Soh Robus C-Corollably ad/or C-Observably or Ucera Descrpor Sysems wh Ierval Perurbaos All Marces IEEE Tras Auoma Cor Vol 44 pp (999) 6 Ho DWC ad Z Gao ezou Idey o Reduced-Order Observer-ased Corollers or Sgular Sysems Auomaca Vol 37 No pp () 7 Gao Z ad DWC Ho Proporoal Mulple-Iegral Observer Desg or Descrpor Sysems wh Measureme Oupu Dsurbaces IEE Proc Cor Theory Appl Vol 5 No 3 pp (4) 8 Raou J ad EK ouas Observer-ased Coroller Desg or Lear Sgular Sysems wh Marova Swchg Proc 43rd IEEE Co Decs Cor Nessau ahamas pp (4) 9 Huag L Lear Algebra Sysem ad Corol Theory Scece Press P R Cha (984) Woham WM Lear Mulvarable Corol: A Geomerc Approach Sprger-Verlag New Yor 3rd Ed (985) Ezze J ad AH Haddad Corollably ad Observably o Hybrd Sysems I J Cor Vol 49 No 6 pp (989)