Properties of a Generalized Impulse Response Gramian with Application to Model Reduction

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1 56 Ieriol Jorl of Corol, Yoseo Aomio, Choo d d Jeho Sysems, Choi vol, o 4, pp 56-5, December 4 Properies of eerlized Implse Respose rmi wih Applicio o Model Redcio Yoseo Choo d Jeho Choi Absrc: I his pper we ivesie he properies of eerlized implse respose rmi he recrsive relioship sisfied by he fmily of rmis is esblished I is show h he eerlized implse respose rmi cois iformio o he chrcerisic polyomil of lier ime-ivri coios sysem he resls re pplied o model redcio problem Keywords: eerlized implse respose rmi, Lypov eqio, Mrov prmeer, model redcio, ime-mome INRODUCION I ideificio or model redcio problems, impor s is he compio of he chrcerisic polyomil of he oriil or redced-order sysem Severl lierre hve show h he chrcerisic polyomil of sysem c be exrced from he iformio eered by he implse respose d For coios sysems, he rm mrix [,] d he implse respose rmi [3] re ood exmples of iformio from which he chrcerisic polyomil of sysem c be obied For discree sysems, he Hel mrix [4] d he implse respose rmi [5] possess he sme properies Recely ew implse respose rmi ws irodced i [6] h c lso be ilized o compe he chrcerisic polyomil of discree sysem I ddiio o compi he chrcerisic polyomil, hose implse respose d re sefl for he order redcio of lier ime-ivri sysems [6-] I his pper we ivesie he properies of eerlized implse respose rmi [], which icldes he rm mrix of [,] d he implse respose rmi i [3] s specil cses he recrsive relioship sisfied by he fmily of Mscrip received Je, 3; revised Mrch 7, 4; cceped Ocober 3, 4 Recommeded by Edioril Bord member Yo Il Lee der he direcio of Edior Ch Choo Ch his pper ws ccomplished wih he help of reserch fd provided by he Kore Cocil for Uiversiy Edcio, sppor for 3 Domesic Fcly Exche Yoseo Choo is wih he School of Elecroic, Elecricl d Comper Eieeri, Hoi Uiversiy, S 34, Si- Ri, Jochiwo-Ep, Yeoi-, Chm 339-7, Kore (e-mil: yschoo@wowhoicr) Jeho Choi is wih he School of Elecricl d Elecroic Eieeri, Chb Niol Uiversiy, S 48, esi- Do, Hedeo-, Cheoj, Chb , Kore (e-mil: choi@powerchbcr) rmis is esblished I is lso show h he eerlized implse respose rmi cois iformio o he chrcerisic polyomil of lier ime-ivri coios sysem he resls re pplied o model redcio problem his pper is orized s follows I Secio, some prelimiries re preseed he properies of eerlized implse respose rmi re sdied i Secio 3 A pplicio o model redcio problem is cosidered i Secio 4 d he pper is coclded i Secio 5 PRELIMINARIES Coicl relizios Cosider sble h-order lier ime-ivri sysem described by he rsfer fcio bs + b s + + b s+ b s = s + s + + s+ or by he miiml se-spce relizio () x () = Ax() + br(), () y () = cx (), (3) x ( ) R he rsfer fcio H( s ) c be expded io he followi wo forms s = s s s, (4) m m m3 s = + + +, (5) s 3 s s i s d m i s respecively deoe he imemomes d Mrov prmeers of he sysem, d re comped from he coefficies of H( s ) or from

2 Properies of eerlized Implse Respose rmi wih Applicio o Model Redcio 57 he sysem mrices s follows: i i + i + i j j j= i i i i j j j= = ( b + ), (6) m = b m, (7) wih =, i = for i <, b i = for i i (6), d = b = for i > i (7), or i i i i = ca b, (8) i i m = ca b (9) Le C be he sdrd corollbiliy mrix for he relizio ( A, bc, ) ive i () d (3) As is well ow, he sysem provided i () d (3) c be rsformed o he followi corollbiliy coicl form by he similriy rsform [] x () = x() + br ˆ (), () y () = cˆ x (), () ˆ A C AC = =, () ˆ b= C b=, (3) [ ] [ ] cˆ = cc = m m m (4) For ech Ν, le ˆ b ˆ = A b d c ˆ ˆ = ca he we hve clss of coicl relizios { ( A ˆ, b, c ), Ν } of H( s ) I is esily see h, for, b es he form sch h ll elemes re zero excep for he (+)h eleme which is eql o oe, d b = [ ] Coversely, from he Cyley-Hmilo heorem, we hve i i i + = i ca b ca b ca b ca b (5) he, si (8), (9) d (5), i c be show h c s e he followi form: (i) For c = m m m : [ ] (ii) For ( ) : [ + ] : c = [ ] c = m m (iii) For + eerlized implse respose rmi For he sysem described i () d (3), he implse respose is ive by A h () = ce b (6) For i, recrsively defie [-3] h ( i+ ) () = h i ( α ) dα, (7) d hi+ () = hi(), (8) d wih h () () Defiiio: For ech Ν, he h-order eerlized implse respose rmi, is defied by, = h() h() h+ () h() h () + h() h+ () h+ () h+ () h+ () d h() h+ () h+ () h+ () h+ () (9) he (+)h-order eerlized implse respose rmi, + is defied similrly he rmi of he form ive i (9) ws firs irodced i [] i relio o he model redcio problem However, deiled lysis ws o performed o he properies of he rmi Noe h ( ), correspods o he rm mrix of [,] d, is he implse respose rmi defied i [7] I priclr, ( ), + d, + respecively deoe he chrcerisic rm mrix [,] d he chrcerisic implse respose rmi [3] of he sysem 3 MAIN RESULS I his secio some impor properies of he eerlized implse respose rmi re sdied Lemm []: For ech Ν d he relizio (, b, c ), he h-order eerlized implse respose rmi, is iqe posiive defiie solio o he Lypov eqio

3 58 Yoseo Choo d Jeho Choi ˆ A + = c c (),, I some pplicios sch s model redcio, i is ecessry o compe he eerlized implse respose rmi, for differe vles of, which reqires he repeed solvi of () However, i is sfficie o solve () oly oce for some s idiced i he followi lemm Lemm : For ech, we hve + =, Ν (),, Proof: Premliplyi A ˆ d posmliplyi  o boh sides of (), we hve c c () (, ) + (, ) = = c+ c+ Sice ech, is iqe solio o (), he resl follows I [-3], i ws idiced h he rm mrix d he implse respose rmi coi iformio o he chrcerisic polyomil of lier ime-ivri coios sysems I c be demosred h he eerlized implse respose rmi possesses he sme propery heorem : For ech Ν, priio he (+)horder eerlized implse respose rmi, + s,,, + =, (3),, +, he he coefficies h() h+ () h () h () = + + d (4) h+ () h+ () i s i () or () re ive by [ ] = = (5),, Proof: Sice ech h ( ) sisfies he chrcerisic eqio, we hve h+ () = h() h+ () h+ () = [ h() h+ () h+ () ] (6) he, h () h () + = h+ () d (7) h+ () =, d we hve (5) A differe forml c be derived from he recrsive relioship ive i () For he ese of preseio, le ij deoe he (i,j)h eleme of he (+)h-order eerlized implse respose rmi, +, d le, + heorem : he by, + l,,, l [, l] = (8) l +, l ll i s i () d () re comped, + [, + ] =, (9) [, ], + =, +, +, [, ]( ), (3) [ ] =, [ ], = 3,,, + 3, +, + = Proof: From (), we obi eqios s follows: + [, ] + =, (3),, +, +,, +, + = (3) Sice, + [, + ] is posiive defiie,, [, ] + exiss d (3) follows from (3) Sbsii (3) io (3), we obi he qdric eqio of he form d d + =, (33)

4 Properies of eerlized Implse Respose rmi wih Applicio o Model Redcio 59 =,, + [, ], d =, +, + [, ], [, ] =,, + [, ], +, + d =, +, + [, + ] [, ] he,, h + () h () + = h() d (39) h+ () = ˆ, d he proof is compleed he (9) follows from he fc h is posiive sice he ive sysem is sble For he sysem mrix  ive i (),  is ive by, (34) = which deermies he chrcerisic polyomil of he reciprocl sysem [] heorem 3: For ech Ν, priio, + s he, +,,, =, (35),, h + () h () = + h() d (36) h+ () ˆ = =,, (37) Proof: From (6), we hve h() = h+ () h+ () (38) = () () () ˆ [ h h h ] APPLICAION O MODEL REDUCION I his secio he resls of Secio 3 re pplied o he model redcio problem cosidered i [7,9,] For h-order sble sysem described by () or () d (), he objecive is o fid redced model h preserves implse eeries s well s some imemomes d Mrov prmeers of he oriil sysem he rmi echiqe of [7] yields he horder redced model h preserves he firs elemes of he implse respose rmi d he firs Mrov prmeers of oriil sysem I [9], i ws reveled h he redced model reii he firs elemes of he rm mrix d he firs ime-momes c be derived by pplyi he sme echiqe o he reciprocl sysem of he oriil sysem I [], more eerl redced models were obied bsed o he mehods of [7] d [9] Assmi wo differe forms of he redced model, d pplyi he echiqes of [7] d [9] seprely o (7) or (8), mliple redced models were derived so h some ime-momes d/or Mrov prmeers s well s diol elemes of implse respose rmi d/or rm mrix re preserved i he redced models I will be show h he redced models of [] c be obied more efficiely by pplyi he resls of he previos secio From lemm,, d +, re respecively iqe solios o he followi wo Lypov eqios for he relizio (, b, c ) ˆ A + = c c, (4),, A ˆ + A ˆ = c c = A ˆ c c A ˆ (4) +, +, + + ˆ A Premliplyi d posmliplyi boh sides of (4), we hve +, +,  o + = c c (4) he Lypov eqios derived i [7] d [9] re specil cses of (4) d (4) respecively wih = d =

5 5 Yoseo Choo d Jeho Choi Now le (,, b,, c, ) be he relizio i coicl form for he h-order redced model d deoe he pricipl ledi le, sbmrix of, If + = c c, (43),,,,,, he he redced model clerly preserves he firs elemes of he oriil h-order implse respose rmi, Similrly if ˆ ˆ A + A = c c, (44) +,,, +,,, he he firs elemes of +, re reied i he redced model If b, d c, re chose ppropriely, he some ime-momes d/or Mrov prmeers of he oriil sysem lso c be preserved i he redced model Coseqely wo h-order redced models, deoed by H, d H, respecively, c be obied for ech Le, = (45) For H,, priio he oriil (+)h-order eerlized implse respose rmi, + s d compe =,,, +,, + i s i (45) s i heorem, ie, [ ],, (46) = (47) Le c, be he -dimesiol row vecor h cosiss of he firs elemes of c d le ˆ b, = A, b,, b, deoes he - dimesiol colm vecor h cosiss of he firs elemes of b ˆ he he firs redced model is compleed I is esily see h H, H, preserves he firs ( ) ime-momes d he firs ( + ) Mrov prmeers of he oriil sysem Alerively, i c be show h he Lypov (43) holds Hece he firs elemes of he oriil h-order implse respose rmi, re lso reied i he redced model For H,, we compe, ised of A ˆ, si heorem 3 Le A ˆ = =, A, d priio, + s he, + (48),, = (49),, [ ] = (5),, d he secod redced model H, is compleed by compi b, d,, A, = d choosi he vecors c s i he firs model I is i esily see h H, preserves he firs ( ) imemomes d he firs ( + ) Mrov prmeers of he oriil sysem However he firs elemes of +, re miied i he redced model sice he Lypov eqio (44) holds i his cse Applyi he bove mehod for differe vles of, we c obi fmily of redced models Noe h he h-order redced models derived by he mehods of [7] d [9] respecively correspod o H, d H O he oher hd, redced models H,,,,, H ( ), re he sme s hose obied by he echiqe of [] Exmple: Cosider forh-order sysem wih he rsfer fcio [] s + s+ s = 4 3 s + 3s + s + 3s+ (5) he firs wo ime-momes d he firs for Mrov prmeers of (5) re respecively comped by

6 Properies of eerlized Implse Respose rmi wih Applicio o Model Redcio 5 =, =, m =, m = 86, m3 = 66, m4 = 7893 he he se-spce relizio (, bc ˆ, ˆ) of (5) i corollbiliy coicl form is ive by 8645 ˆ 983 A = c ˆ = [ ], b ˆ = [ ], Solvi he Lypov () for =, we obi he forh-order eerlized implse respose rmi by,4, = (5) From he recrsive relioship ive i (), we hve,4, =, = (53) (54) Now we derive for secod-order redced models H,, H,, For H,, we hve from (46), (47) d (5) = = (55) Hece H, is ive by Similrly s, 54 = 7, b, =, [ ] [ ] c, = m m = 86 H, is obied from (46), (47) d (53) For ˆ 9 A, = 354, ˆ b, = Ab, =, c, = [ m ] = [ ] H,, we obi from (49), (5) d (53) = = he ˆ A, = 84 = 688 b, d c, re ive s i H, Usi he sme procedre, H, is obied from (49), (5) d (54) s ˆ 365 A, =, 948 ˆ 365 b, = A b, = 948, c, = [ ] = [ ] Noe h for secod-order redced models cqired bove re he sme s hose derived i [] 5 CONCLUSIONS I his pper we sdied he properies of eerlized implse respose rmi he recrsive relioship sisfied by he fmily of rmis ws esblished I ws show h he chrcerisic polyomil of he lier ime-ivri coios sysem c be deermied from he eerlized implse respose rmi Usefless of he eerlized implse respose rmi for model redcio ws lso discssed REFERENCES [] V K Ji d R D p, Ideificio of lier sysems hroh rmi echiqe, Ieriol Jorl of Corol, vol, pp 4-43, 97 [] V Sreerm d P Aholis, O he properies of rm mrix, IEEE rs o Circis d Sysems I, vol 4, o 3, pp 34-37, Mrch 994 [3] V Sreerm d F K Yp, Chrcerisic implse-respose rmi, Elecroics Leers, vol 7, pp 85-87, 99 [4] V Sreerm d A Y Zomy, Noe o he Hel mrix, Elecroics Leers, vol 7, pp , 99

7 5 Yoseo Choo d Jeho Choi [5] V Sreerm, D Beriso, d Y H Le, Implse-respose rmi for discree sysems, Elecroics Leers, vol 7, pp , 99 [6] S Azo, P Brehoe, P Vilbe, d L C Clvez, A ew discree implse respose rmi d is pplicio o model redcio, IEEE rs o Aomic Corol, vol o 3, AC-45, pp , Mrch [7] P Aholis d V Sreerm, Ideificio d model redcio from implse respose d, Ieriol Jorl of Sysems Sciece, vol, pp 54-55, 99 [8] V Sreerm d P Aholis, Model redcio of lier discree sysems vi weihed implse respose rmi, Ieriol Jorl of Corol, vol 53, pp 9-44, 99 [9] V Sreerm d P Aholis, O he compio of he rm mrix i ime domi d is pplicio, IEEE rs o Aomic Corol, vol AC-38, o, pp 56-5, Ocober 993 [] W Krjewsi, A Lepschy, d U Viro, Model redcio by mchi Mrov prmeers, ime momes, d implse-respose eeries, IEEE rs o Aomic Corol, vol AC-4, o 5, pp , My 995 [] C Levi d V Sreerm, Model redcio vi prmeer mchi si rmi echiqe, IEE Proc D, vol 4, pp 86-96, 995 [] Kilh, Lier Sysems, Preice Hll Ic, New Jersey, 98 Yoseo Choo ws bor i Dejeo, Kore, o Oc, 957 He received he BS deree i Elecricl Eieeri from Seol Niol Uiversiy i 98, d MS d PhD derees from he Uiversiy of exs Asi, USA, i 99 d 994, respecively From Dec, 979 o A, 989, he wored for ADD s Resercher He held he Posdocorl posiio ERI from Sep, 994, o Feb, 995 Sice Mrch, 995, he hs bee wih he School of Elecroic, Elecricl d Comper Eieeri, Hoi Uiversiy, Kore, he is crrely Associe Professor His reserch ieress re i he res of sochsic corol, dpive corol d model redcio Jeho Choi received he BS, MS, d PhD derees i Elecricl Eieeri from Seol Niol Uiversiy, Seol, Kore, i 979, 98, d 989, respecively From 98 o 983, he ws Fll-ime Lecrer i he Deprme of Elecroic Eieeri, Jyo echicl Collee, Dejeo Sice he he hs bee wih he School of Elecricl d Elecroics Eieeri, Chb Niol Uiversiy, he is crrely Professor From 993 o 994 d from 998 o 999, he ws Visii Professor Uiversiy of oroo, oroo, Cd I, he ws Visii Professor Albor Uiversiy, Demr His reserch ieress re i he res of DSP-bsed UPS sysem desi, cive power filer d recive power compesio, power qliy isses, eery sore sysems, d he pplicios of sysem heory He is member of KIEE, KIPE, IEEE, JIEE, d EPE He is he Pblicio Edior for he Jorl of Power Elecroics (JPE)