Novel Quadratic Tracker and Observer for the Equivalent Model of the Sampled Data Linear Singular System *

Transcription

1 Applie Maheaical Sciece, Vol. 6, 0, o. 68, Novel Quaaic acke a Obeve o he Equivale Moel o he Saple Daa Liea Sigula Sye * Jao Sheg-Hog ai, Chia-Hig Che a Mig-Je Li Cool Sye Laboaoy, Depae o Elecical Egieeig, Naioal Cheg Kug Uiveiy, aia 70, aiwa, Republic o Chia hai@ail.cku.eu.w Shu-Mei Guo Depae o Copue Sciece a Ioaio Egieeig, Naioal Cheg-Kug Uiveiy, aia 700, aiwa, Republic o Chia guo@ail.cku.eu.w Leag-Sa Shieh Depae o Elecical Egieeig, Uiveiy o Houo, Uiveiy Pak, Houo, X , U.S.A. lhieh@uh.eu Abac hi pape oe a ovel quaaic acke a obeve o he equivale oel o he aple-aa liea igula ye. Fi, oe exiig echique ae uilize o cove he liea igula ye io a equivale egula oel coaiig a iec aiio e o ipu o oupu. he we popoe a ovel quaaic ubopial acke o hi equivale egula oel by iceizig he liea aalogue quaaic peoace iex io a equivale icee oe iecly. Fuhe, we evelop he ovel aalog a igial obeve o he equivale egula oel, epecively, a he ae o hi equivale egula oel i ieauable. A illuaive exaple eoae he eecivee o he popoe eig. * hi wok wa uppoe by NSC ue coac NSC99--E MY3.

2 338 Jao Sheg-Hog ai e al Keywo: igula ye; acke; obeve; igial eeig Ioucio he igula ye aually aie i ecibig lage-cale ye, a hee ae eveal exaple occu i powe a iecoece ye. I geeal, a iecoecio o ae vaiable ubye i coveiely ecibe a a igula ye, eve hough a oveall ae pace epeeaio ay o eve exi. Ove he pa ecae, ay eeach eul egaig he igula ye have ucceully olve lo o coplicae poble uch a abiliy [4], ipulive oe [], coollabiliy, obevabiliy [5], a he uicie a eeial coiio o ipule coollabiliy a obevabiliy o he ie-vayig igula ye [3]. Howeve, he ai pupoe o he above pape ae eihe o abilize he igula ye o o pove coollabiliy a obevabiliy o he igula ye. Hee, he key oe o hi pape i abou he ackig iue. A i, a igula ye wih a egula pecil ha bee aoe io a aa oe by uig a block oal aix couce via he exee aix ig ucio. Subequely, he igula ye i he ecopoe io a low ubye wih he iie oe a a a ubye wih he iiie oe, whee he a ubye ay iclue he ipulive oe a oyaic iiie oe. Whe he igula ye oe o coai he ipulive oe, he i ca be aoe io a euce-oe egula ye a a oyaic a ubye. I i, he a ae oly epe o he ipu. Coaily, i ipulive oe i coaie i he igula ye, he a peliiay eeback gai [] i ue o eliiae i. A eioe above, he igula ye ca be covee io a equivale egula oel coaiig a iec aiio e o ipu o oupu. I hi pape, we ai a evelopig he opial liea quaaic aalogue acke a obeve o hi equivale egula oel. Fi, a ovel quaaic ubopial acke ha bee popoe o he equivale egula oel, by iecly iceizig he liea quaaic peoace iex peciie i he coiuou-ie oai io a equivale icee-ie peoace iex. While, a he ae o he coiuou-ie egula oel a he coepoig aple-aa egula oel wih a iec aiio e ae o available o he eauee, we evelop ubopial aalog a igial obeve o boh oel, epecively. Fially, a ae-achig igial eeig eho [3] i uilize o evelop he obeve-bae igial coolle o he equivale egula oel wih a iec aiio e o ipu o oupu. he igially eeige coolle i able o cloely ach he ae bewee he igially coolle aple-aa egula oel a he aalogouly coolle coiuou-ie egula oel. A a eul, he popoe obeve-bae liea quaaic igial acke i able o ake he equivale egula oel ack he eie eeece igal vey well. o he auho kowlege, he opial acke

3 Novel quaaic acke a obeve 3383 o he egula oel coaiig he iec aiio e o ipu o oupu ha o bee popoe i lieaue. hi pape i ogaize a ollow. I Secio, he ecopoiio o igula ye i oulae wih oivaio a backgou o ubequely wok. Secio 3 popoe he blocke-iaga a eig ehoology o he iceize quaaic ubopial acke o he aple-aa liea egula oel wih a iec aiio e o ipu o oupu. Secio 4 pee he obeve-bae igial acke accoig o he popoe peicio-bae igial obeve eig, a he ae o he ye eioe i Secio 3 i ieauable. A illuaive exaple a iulaio eul ae povie i Secio 5, a Secio 6 coclue he pape. he elae icuio o he picipal h oo o a aix a aociae aix eco ucio ae collece i Appeix A. Poble oulaio he objecive o hi pape i o pee a ovel quaaic ubopial acke a obeve o he equivale oel o he aple-aa liea igula ye. Owig o well evelope o egula ye i heoie a pacical eho, i i expece o exe he eablihe eul o he equivale egula oel o he igula ye. Bae o hi oivaio, we aop a block oal aix, which i couce via he exee aix ig ucio, o ecopoe he igula ye io a euce-oe egula ubye a a oyaic ubye. he i ca be covee io a equivale egula oel coaiig a iec aiio e o ipu o oupu. A a eul, we ca exe he igial eeig poble o igula ye.. Ioucio o he aix ig ucio Le u coie he igula ye a ollow Ex () = Ax () Bu (), (a) y() = C x(), (b) p whee x ( ) R i he ae veco, u ( ) R i he ipu a y ( ) R i he oupu. he coa aice, he E, A, B a C all have appopiae ieio, a E i a igula aix. he aix ig ucio o a quae aix A C wih Re ( σ ( A)) 0 eie i Robe 6 ha bee how a ollow ig( A) = ig ( A) I, () whee I i a ieiy aix a ig ( A) = ( λi A) λ, π i (3) c whee c i a iple cloe coou i igh-hal plae o λ a ecloe all

4 3384 Jao Sheg-Hog ai e al igh-hal plae eigevalue o A. Noig ha, he aix ig ucio [4], [8], [] i alo eie a ig( A) A( A ) = = A ( A ), (4) whee he aix A eoe he picipal quae oo o A. Peevig he eigeveco o he oigial aix i a ai eaue o he aix ig ucio. hi popey i ueul o uyig he eigeucue o aice a evelopig applicaio. he igula aix E ca be oiie by uig he biliea aoaio E = ( E ρi)( E ρi), (5) whee ρ i he aiu o a cicle wih oigi a i cee. Sice hi ki o cicle oly coai zeo eigevalue o E, o hee will be o eigevalue locae o he cicueece. heeoe, he eigevalue wihi he cicle ae appe io he le-hal plae o he coplex -plae, a he ohe ae appe io he igh-hal plae o he coplex -plae.. he egula pecil a he aa pecil Deiiio [] Le E a A be wo quae coa aice. I e( E A ) 0, o all, he ( E A ) i calle a egula pecil. Deiiio [6] Le ( E A) be a egula pecil. I hee exi cala α a β uch ha α E β A = I, he ( E A) i calle a aa pecil. Noe ha o ay egula pecil, ( E A) ca be eaily aoe io a aa oe by uliplyig ( αe βa) o E a A epecively, whee α a β ae cala uch ha e( α E β A) 0. Hece, he aix coeicie o a aa pecil ( E A) becoe E ( ) αe β A E, (6) A ( ) αe β A A. (7) he oiie ye eai i ae veco x( ) a he aice ( E, A ) have he ollowig popeie. Lea [] ): he coa aice EA = AE, which ea ha E a each ohe. ): he aice E a A have he ae eigepace. A coue he above popeie eable u o ecopoe a igula ye io a euce-oe egula ubye (low ubye) a a oyaic ubye (a ubye).

5 Novel quaaic acke a obeve Decopoiio o igula ye Coie he coiuou-ie igula ye i (). I i well-kow ha he igula ye ca be ecopoe io a low a a a ubye. Fo (6) a (7), he egula pecil ( E A) ca be aoe io a aa oe ( E A). Noe ha ice E i a igula aix, heeoe i ha a lea oe zeo eigevalue, β ca o be equal o zeo. Hece, uliplyig (a) by ( αe βa), we ca ge he ollowig equaio Ex () = Ax () Bu (), (8) whee he ( α β ) E = E A E, he A = ( αe βa ) A a B = ( αe βa) B. Becaue α E β A = I, he pecil ( E A) i a aa oe, which ha he popeie eioe i Lea. I oe o ecopoe ye (8), we cove he ae x( ) io x() by x() = Mx(), (9) whee he coa aix M i a block oal aix o E a i eeie by ea o he exee aix ig ucio [Appeix A]. he M aix o ae pace aoaio i give a ollow Sep : Fi ig( E ) uig he exee aix ig ucio wih a aequae ρ, whee = ( ρ )( ρ ). E E I E I Sep : Fi ig ( E ) = [ I ig( E )] a ig ( E ) = [ I ig( E )]. Sep 3: Couc he aix M = [i(ig ( E )) i(ig ( E ))], whee i( ) epee he collecio o he liealy iepee colu veco o ( ). Subiuig (9) io (8) a uliplyig M o he le, he equaio ca be ewie a M EMx () = M AMx () M Bu () ( ) ( ) = I ( ) α E x M Bu, β i.e. ( Iκ α E ) O E O β B x () = x() u(), (0) O E B ( O Iκ α E ) β whee x () = [ x, x ], a M EM = block iagoal { E, E }. he E i iveible wih ak( E) = eg{e( E A)} κ, he [ B, B ] = M B a

6 3386 Jao Sheg-Hog ai e al E i a ilpoe aix wih ieio ( κ) ( κ ). Noice ha ice e( I α ) κ E =, he aix i iveible. Sipliyig (0) by uliplyig he block iagoal { E, β( I κ αe) } o boh ie, oe ha ( I E I ) O κ O α κ x() β x() O β( I κ αe) E = O I κ E B u (), β( Iκ αe) B Iκ O A O B x() x() u() O E =, () O Iκ B whee he E = β( I κ αe) E, ( A E α I κ ) =, B = E B, β B = β( I αe ) B. κ o ae he igula ye () io a equivale egula oel, hee ae wo cae o be icue. Cae : E i a zeo aix. Sye () ca be wie a Iκ O A O B x() x() u() O O =, () O Iκ B a (b) ca be ewie a y() = C x() = C M x() Cx(), (3) x () whee C = CM a x () = x (). he, he coiuou oel () a (3) ca be covee a ollow x () = A x () B u(), (4) P κ whee C = C R, x () = B u(), (5) y () Cx () [ C C] x C x () = = (), y () = Cx() Du (), (6) P ( κ ) p R, D =CB R. Cae : E i a Joa aix. I i eakable o oe ha ice

7 Novel quaaic acke a obeve 3387 { } ak( E) eg e( E A ) = ak( E ), (7) whee he ak( E ) i calle he geealize oe o he igula ye, a he eg{ e( E A) } i ai o be he oe o he low ubye. Reeig o (), oe ca eeie he ube o he ipule oe eaily by uig he above equaio. Fi o all, we aue he igula ye ha q ipulive oe, a he ak( E ) = q. Wha ollow i o i peliiay eeback gai K ie a o pove ha K ie ca eliiae ipulive oe by he ollowig ep. Le x() = L xˆ (), (8) I κ O whee x ˆ() = x ˆ (), ˆ x () = x (),( U x()) a L = O U. he U i a oal aix o E wih ieio ( κ) ( κ) uch ha U E U i i he Joa block o. Subiuig (8) io () a uliplyig i by L, we obai he ollowig equaio I ˆ ˆ κ O A O B ˆ() ˆ ˆ x = x() u(), (9) O E ˆ O Iκ B whee ˆ E = U E U, Aˆ = A, ˆ B = B, ˆ B = U B. Noice ha E ˆ i i he Joa block o wih block o ize u, u,, u, whee i= ui = colu(ow) ube o E ˆ. akig he Laplace aoaio o he a ubye Ex ˆ ˆ () ˆ () ˆ = x Bu () i (9), oe obai ˆ xˆ () = ( E I ) ( Eˆ xˆ (0) Bˆ U()) κ l i ˆ i = ˆ ˆ ˆ i= 0 E ( E x (0) B U( )), (0) whee xˆ () a U () eoe he Laplace aoaio o xˆ () a u ( ), epecively. he x ˆ (0) eoe he iiial value o xˆ (), a l i calle o be he ilpoecy iex o E ˆ. akig he ivee Laplace aoaio o above equaio, oe ha l l ˆ i ( i) ˆ i ˆ ( i) xˆ () = E xˆ (0) δ () E B u (), () i= 0 i= 0 () i () whee δ () a δ eoe he ela ucio a he i h eivaive o he ela ucio, epecively. Fo he above equaio (), he ipulive oe o he a ae ae iuce o icoie iiial coiio o he a ae o

8 3388 Jao Sheg-Hog ai e al icoiuou cool ipu (o i eivaive). Hee, a peliiay eeback eig eho o eliiae he ipulive oe ha bee eive i []. Deeiaio o he peliiay eeback gai K = [ K, K, K ], whee K i o ieio o ie ie ie ie κ ( κ ) j =,,, ( κ), i uaize a ollow. ). I ui, whee i, a i coepoig Joa block i a ull aix, he Kie = O, K u u u i ie u u u i K = O = O ieu u u i, ). I u i >, whee i, a i coepoig Joa block i o a ull aix, he δ ( bˆ ( )) u u ui ( ˆ δ b( u u u )) i Kie =, u u u i δ ( bˆ ( u u u ) ) i Kie = O, u u u i Kie = O, u u u i bˆ κ ˆ whee ˆ b κ B bˆ ( κ ) 0, i bˆ ij = 0 ( ˆ δ b ), i ˆ ij bij >0., i bˆ ij < 0 Le. ie j bˆ bˆ bˆ b ˆ a, i i, i,, i

9 Novel quaaic acke a obeve 3389 u () = K xˆ () v () ie = O K κ ie x ˆ( ) v (). () Subiuig () io (9) yiel Ex ˆ () ˆ k = Ax k () Bv k (), (3) Ik O Aˆ ˆ BK ie Bˆ whee Ek = O Eˆ, Ak =, Bk =. ˆ O Iκ B Kie ˆ B he igula ye i (3) i obaie by applyig he liea peliiay eeback cool law u ( ) o () o he ye i (9). I (3) ha he q iie oe (whee q = ak( E ) = ak( E ˆ ))a he κ oigial iie oe. All hee iie oe ae guaaee o be coollable. he ex i o ecopoe he igula ye io a euce-oe egula oel wih ( κ q ) coollable iie oe a he oyaic equaio wih ( κ q) iiie oyaic oe. I ca be accoplihe by uig peviouly oulie ep agai. A i, we ao he egula o io a aa oe by uliplyig (3) by ( γek ηak), whee γ a η ae abiay cala uch ha ( γ Ek η Ak) i iveible. heeoe, we obai. ˆ ˆ k k k k k k k k k ( γe ηa ) E x( ) = ( γe ηa ) A x( ) ( γe ηa ) B v( ). (4) Le xˆ( ) = M x (), (5) whee he coa aix M i eeie by uig he exee aix ig ucio. he poceue i he ae a he peviou illuaio o iig M, excep ha i opeae o ( γek ηak) Ek. Subiuig (5) io (4) a uliplyig i by M yiel. M ( γe ) ˆ k ηak EkM x( ) = M ( γe ηa ) AMx ( ) M ( γe ηa ) Bv ( ) k k k k k k = I γm ( γek ηak) EkM x() M ( γek ηak) Bkv() η o ( Iκ q γ Ek ) O E. k O η Bk x () = x () v(). O E k B ( ) k O I κ q γ E k η Fuhe,

10 3390 Jao Sheg-Hog ai e al ( Iκ q γ Ek ) O Ek O. η Bk x () = x () v() O O, B κ q k O I κ q η whee ( ) ( ), ( ) x = x x, M ( γek ηak) EkM =block iagoal{ E, } k E k = block iagoal { Ek, O( q κ )} k k. he E k i iveible wih ak = eg{e(e A )} = ( q κ), a he E k i a ull aix wih ieio ( κ q) ( κ q) a Bk, B k = M ( γek ηak) Bk. Moeove, uliplyig block iagoal { Ek, I κ q } o he above ye η yiel I q O κ. Ek ( Iκ q γ Ek ) O x() η x() O O = κ q O I κ q Ek B k v () I κ q B, (6) k η x () κ q κ q whee x ( ) R, x () R a x () = x (). hu, ye (6) ca be ecopoe io he ollowig o x () = A x () B v(), (7) x () = B v(), (8) whee A = Ek ( Iκ q γ Ek ), B = Ek Bk a B = I κ q Bk. η η a y() = Cx() = CMLMx () = Cx (), (9) whee C = C MLM. Applyig he ae poceue a (4)-(6), he equivale oel o he igula ye ca be ogaize a ollow x () y() = Cx () = [ C C] = Cx() Cx () x () = Cx () ( CB ) v () p ( κ q) whee C = C R, = Cx () Dv(), (30) C p ( qκ ) p R, D=CB R. Ae he aoaio, he igula ye ca be covee io a

11 Novel quaaic acke a obeve 339 equivale egula oel which coai a iec aiio e o ipu o oupu. he ipulive oe ca be eliiae by ea o he eho []. Hee, i hi pape, we oly icue he cae ha igula ye oe o coai ipulive oe, ice he icee-ie oel o a igula ye wih ipulive oe oe o exi. Reak he igula ye ca be covee o a equivale egula oel which coai a iec aiio e o ipu o oupu, i geeal. Howeve, o oe cae, uch a I O x () A O x() B u () O O x() = O I x () B, x() x() y () = [ C 0] [ C 0] x () = Bu() = Cx () 0 u (), whee he iec aiio e o ipu o oupu i vaihe. heeoe, he igula ye wihou he ipulive oe i ju a cla o he egula oel wih a iec aiio e o ipu o oupu. 3 A ovel acke o he egula oel wih a eehough e Coie he aleaive o o (7) a (30), he equivale oel o he igula ye, ecibe a ollow x () = Ax() Buc (), (3a) yc() = Cx() Duc(), (3b) whee 0 <. Fo hi ye, le he aociae co ucio be iiize J = {[ yc() ()] Q[ yc() ()] uc () Ruc() }, (3) 0 p whee A R, B R, C R, Q i a poiive eieie yeic aix, R i a poiive yeic aix, () i a eeece ipu, a ial iex i iie, ie.. <. Le he coiuou coolle be uc() = Kc() x() Ec() (), (33) whee Kc( ) i he coiuou eeback gai, Ec ( ) i he coiuou eeowa gai, a ( ) i eeece ipu veco. he ubopial liea quaaic aalogue acke o he oel (3) i how i Figue.

12 339 Jao Sheg-Hog ai e al D () E c u () c x () = Ax() Bu () c x() C yc () K c Figue. he oigial aalog ye wih a aalog coolle. Le he coepoig igially coolle oel o (3) be ecibe a x () = Ax () Bu (), x (0) = x, (34) y () = Cx () Du (), (35) whee u ( ) R i piecewie-coa, uch ha u ( ) = u ( k ) o k < ( k ), a > 0 i he peio o aplig a hol. Le u () be a icee-ie ae-eeback cool law o he o u ( k) = K x ( k) E * ( k), k < ( k ), (36) whee p K R a E 0 R ae he eeback a eeowa igial gai, epecively, a * ( k ) i a piecewie-coa eeece ipu veco o be eeie i e o k ( ) o ackig pupoe. he igial eeece ipu veco wih ackig pupoe i peciie a * ( k) = ( k ). he viewpoi ha bee pove i [3]. he oveall igially coolle cloe-loop ye becoe * x ( ) = Ax( ) B[ K x( k) E ( k)], x ( 0) = x0, (37) o k < ( k ), whee he coolle i ealize uig a zeo-oe hol evice a how i Figue. D * ( k) E u ( ) k Z. OH.. u () K x () = Ax () Bu () x ( k ) x () C y () Figue. Peicio-bae igial acke.

13 Novel quaaic acke a obeve 3393 he icee-ie oel i ecibe a x(( k ) ) = G x( k ) H u( k ), (38) whee A G = e (39) a H = ( G I) A B, i A exi o i i H = A B, i A oe o exi. (40) i= 0 ( i )! I he ollowig paagaph, he iceize quaaic ubopial acke o a coiuou liea egula oel wih a iec aiio e o ipu o oupu will be popoe i eail. Fi, cove he oigial coiuou quaaic co ucio io he icee o, a olve he opiizaio i he icee oai. Whe he egula oel i ealize by he aa-aplig chee i Figue, u ( ) i a piecewie-coa ipu ucio. he, he co ucio J give by (3) ca be iceize a ollow. Whe = N, we ca ewie J a N ( k ) * * J = {[ C x( ) Duc( k) ( k)] Q [ Cx( ) D uc( k) ( k)] k= 0 k uc ( k) Ruc( k)}. (4) he eie x( ) o k < ( k ) ca be wie a whee A ( k) A ( τ ) () = ( ) ( τ ) k x e x k e Bu τ = ζ ( k) x ( k) η( k) u ( k), (4) ζ ( k) e A ( k) =, A ( τ ) k e B k η( ) = τ. Subiuig (4) io (4), oe ca euce N * J = { x( k ) Qx ( k ) x( k ) Mu( k ) x( k ) M ( k ) k = 0 * * * u ( k) M3 ( k) u( k ) Ru ( k ) ( k ) R ( k )}, whee ( k ) Q = [( Cζ( k)) Q( Cζ( k))], k ( k ) M = [( Cζ( k)) Q( Cη( k) D)], k ( k ) M = [( Cζ ( k)) Q], k

14 3394 Jao Sheg-Hog ai e al ( k ) M 3 = [( Cη ( k) D) Q], k ( k ) { η η } R = R [ C ( k) D] Q[ C ( k) D], R k ( k ) = Q. k Coequely, he above equaio (4) ca be ewie a Q M M x ( k) N * * J = [ x ( k ) ( k ) u ( k )] M R M 3 ( k ). (43) k = 0 M M3 R u ( k) I i ieeig o oe ha he iegal e i (43) iclue co e ivolvig x ( k ) a u ( k ). Alo, aice Q, M, M, M 3, R a R ae oiie i e o Q a R i (4). o eliiae he co e o he co ucio J (43), le u eie _ Q M M Q= R M M3 M R M, 3 x ( k ) R = R, x ( k) = * ( k) a vk ( ) = R M M 3 x( k) u( k) o u( k) =R M M 3 x( k) v( k). (44) he, equaio (43) ca be wie a N _ J = [ x ( k) Qx( k) v ( k) Rv( k)]. (45) k = 0 Suppoe he coiuou-ie cool egula oel i appoxiae by i icee equivale, we obai he augee oel a x( k ) G 0 x( k) H x( k) = vk ( ) R * M M 3 * k ( ) 0 I ( k) 0 ( k), x (( k ) ) Gx ˆ ( k) Hu ˆ ( k), (46) 0 whee ˆ G G = 0 I, ˆ H H = 0. Ogaizig (46), oe ha x (( ) ) ( ) ˆ k = Gx k Hv( k ), (47) whee ˆ ˆ G = GHR M M 3. Le he Hailo ucio epe o he co ucio (45)

15 Novel quaaic acke a obeve 3395 H ( k ) = [ x( k ) Qx( k ) v( k ) Rv ( k )] λ ( k )[ Gx ( ) ˆ k Hv( k)], (48) whee λ ( k ) i a co-ae (Lagage uliplie). Bae o he well-evelope cool heoy, we have he ae equaio x (( ) ) ( ) ˆ k = Gx k Hv( k) (49) a he co-ae equaio _ λ( k ) = G λ(( k ) ) Q x ( k ), (50) wih he aioay coiio _ ˆ 0 = H λ(( k ) ) R v( k) o ( ) ˆ vk = R H λ(( k ) ), (5a) a he bouay coiio _ λ ( N ) = Q x ( N ). (5b) Aue ha λ ( k ) ca be wie a he ollowig o λ ( k ) = P( k ) x ( k ), (5) whee Pk ( ) i a eal yeic aix o appopiae ieio. hu a, he oigial ubopial ackig poble ha bee covee io a egula poble. o eive he ubopial egulao, ubiuig (5) io (50) yiel _ Pk ( ) x( k) = GP (( k ) ) x(( k ) ) Qx( k). (53) Subiuig (5) a (5) io (49) yiel ˆ ˆ x(( k ) ) = [ I HR H P(( k ) )] Gx ( k). (54) Siilaly, ubiuig (54) io (53) give _ ˆ ˆ { Pk ( ) GP (( k ) )[ I HR H P(( k ) )] G Qx } ( k) = 0. (55) he above equaio u hol o all x ( k ). Hece, we u have _ ˆ ˆ Pk ( ) Q GP(( k ) )[ I HR = H P(( k ) )] G. (56) Equaio (56) i calle Riccai equaio. Ree o (5) a (5) a k = N o have o _ λ ( N ) = Q x ( N ) = P( N ) x ( N ) PN ( ) = Q. (57) Fo (57) a (56), we ca obai all he p( k ) o 0 k N. By eeig o (53) a (55), he eie ubopial viual cool ipu give by (5a) becoe vk ( ) = Kk ( ) x ( k), (58) whee ˆ ( ) (( ) )[ ˆ ˆ Kk = R HP k I HR HP(( k ) )] G. Fo (44), he _

16 3396 Jao Sheg-Hog ai e al oigial coolle u ( k ) ca be how a ollow u( k) =R M M 3 x( k) v( k) x( k ) x( k ) =R M M 3 K( k) * * ( k) ( k) x( k ) x( k ) =R M M 3 * [ K K] * ( k) ( k), (59) ( ) whee K R p, K R p, K R. A a eul, he coolle ca be ewie a * u( k) = Kx( k) E ( k), (60) whee K K R = M a E = K R M3. I i well-kow ha he high-gai coolle iuce a high qualiy peoace o ajecoy ackig eig a ae eiaio, a i alo ca uppe ye uceaiie uch a oliea peubaio, paaee vaiaio, oelig eo a exeal iubace. Fo hee eao, he igial coolle wih a high-gai popey i aope i ou appoach. he high-gai popey coolle ca be obaie by chooig a uiciely high aio o Q o R (o be how i Lea ) i (4) o ha he ye oupu ca cloely ack he pe-peciie ajecoy. Lea [0] Le a pai o weighig aice { QR, } be give a iagoal aice Q= qip R a R= I > 0. hee exi he lowe bou o Q, R, i.e. Q = q I a R = I, eeie by p weighig aice { } BB CC q κ =, AA a log a he popey o he high-gai cool ill hol, ha i, P ζ P, o BB CC q BB CC q q q ζ = κ κ = = a AA AA κ > κ κ, whee P a P ae he yeic poiive-eiie oluio o he ollowig Riccai equaio, epecively, AP PA PBR BP CQC = 0, AP PA PBR BP CQC=. 0 4 A ovel peicio-bae igial obeve o he egula oel wih a eehough e Auig ha he ae x() o he coiuou-ie egula oel (3) i

17 Novel quaaic acke a obeve 3397 ieauable, a le he coiuou-ie obeve, a how i Figue 3, be xˆ () = Axˆ() Bu () L [ y () yˆ ()], (6a) c c c c yˆ () = Cxˆ() Du (), (6b) c c () Liea Quaaic acke E c u () c D x () = Ax() Bu () c x() C y () c Aalog Obeve L c xˆ () = Axˆ() Bu () L y () yˆ () ( ) c c c c xˆ( ) C yˆ c () K c D Figue 3. Obeve-bae liea quaaic aalogue acke whee x ˆ( ) R i he eiae ae, yˆ c() i oupu o he obeve, a p L c R i he obeve gai. he, ipliyig (6) yiel xˆ () = Axˆ() Bu () [ () () ( ˆ c Lc Cx Duc Cx() Duc())] = Axˆ() Bu () ( () ˆ c Lc C x C x()). Le he obeve eo be e () = x () x ˆ(), o ha he eo equaio o he obeve ca be eive a e () = ( A LCe c ) (). (6) Accoig o he ual cocep o ubopial acke eig eho, oe obai he ubopial obeve gai L c i (6) a Lc = PoC R o, (63) whee P o i he yeic a poiive eiie oluio o he ollowig Riccai equaio APo PoA PoC Ro CPo Qo = 0. (64) Wha ollow i o i a igial obeve o he above aalogue obeve. A i, eie he icee-ie obeve eo a

18 3398 Jao Sheg-Hog ai e al e ( ) ( ) ˆ k = x k x( k). (65) I i expece ha he icee-ie eo ae ca cloely ach he coiuou-ie eo ae a each aplig ia, i.e. e ( k) e( k), o ha he igial ae xˆ ( k ) ca cloely ach he aalogue ae xˆ( ) a each aplig ia, which alo iplie xˆ ( ) ˆ k x( k ) x( k ). Equaio (6) ca be ewie a e () = Ae () Lu c c, (66) whee u c =Ce( ). Moeove, uig he ae eivaio a how i (34) o (38), oe ca ge he icee-ie eo oel o (66) e( k ) = ( G MN) e( k), (67) whee M = ( G I) A Lc, i A exi o i i M = AL, i A oe o exi, (68) ( i )! c i= 0 N ( I p CM) CG =. (69) Fuhe, eiig L M I CM oe ha M N = L CG. (7) he icee-ie oel coepoig o he aalog oe (3) i give a x( k ) = Gx( k ) H u( k ), (7) y( k ) = C x ( k ) Du( k ). (73) Fo (7) a (73), oe ca obai CGx ( k ) = y( k ) CH u( k ) D u( k ). (74) Subiuig (7) io (67) yiel e ( ) ( )( ( ) ˆ k = G LCG x k x( k)) = ( G L ) ( ) ( ) ˆ CG x k G LCG x( k). (75) Siilaly, ubiuig (74) io (75) oe ha = ( p ), (70) e ( ) ( ) ˆ k = x k x( k ) = [ Gx ( ) ( )] ˆ k Hu k x( k ) = Gx( k) Ly( k ) LCH u( k) ( G L ) ˆ CG x( k) LDu( k ). (76) Fo (75) a (76), he icee-ie obeve ca be how a xˆ ( ) ( ) ˆ k = G L C G x( k ) ( I L C) H u( k ) Ly( k ) LDu( k )

19 Novel quaaic acke a obeve 3399 = G xˆ ( k ) H u ( k ) L ( y ( k ) Du ( k )), (77) yˆ (( ) ) ˆ k = Cx( ( k ) ) Du( ( k ) ), (78) whee L = ( G I ) A L ( I C( G I ) A L ), (79) c p c H G = G L CG, (80) = ( I L C H. (8) ) * ( k) Digial acke E u ( k) Z. OH.. u () Digial Obeve D x () = Ax () Bu () C y () Z u ( k ) L y ( k) Du ( k) xˆ ( k ) = G xˆ ( k ) H u ( k ) L ( y ( k ) Du ( k )) K xˆ ( k ) D Figue 4. Peicio-bae igial obeve a acke he obeve-bae igial acke chee o he aple-aa egula oel, coaiig a iec aiio e o ipu o oupu, i how i Figue 4. 5 A illuaive exaple Coie a coiuou-ie liea igula ye ecibe i [] wih E =, A = I6, B =, C, = whee he eeece ipu i ( ) = [5co(3 ) 5i(3 ) 5], he iiial coiio i x(0) = ( MLM )[ x (0) x (0) ] =[ ], x (0) i

20 3400 Jao Sheg-Hog ai e al [ 0.5 ] a x (0) i [0 0], epecively. Sice a he iiial ie = 0, beoe he eig poce, u c (0) = 0, i iplie x (0) = 0. Howeve, a he ia o iiial ie = 0, we will eig a o-zeo cool ipu u c (0) o ha i will yiel he eie ackig pupoe a = 0. Nevehele, u c (0), he eige o-zeo ipu, ca o be ue o eeie he iiial ae x, o, he iiial coiio houl be give cauiouly. Sice 0E A = A = I6, a accoig o he eiiio o he aa o, E, A i i aa o. I we ake α = 0 a β =, he heeoe { } E = E, A = A a B = B. Becaue E i igula. i.e. E iclue oe zeo eigevalue, uig he biliea ao o i he iilaiy aoaio aix M o E i eceay. Aue ρ = 0.5, uilizig he algoih ecibe i Secio, oe ha E = ( E ρi6)( E ρi6) =, ig( E) =, ig ( E) =, ig ( E) =,

21 Novel quaaic acke a obeve M = i(ig ( E)) i(ig ( E)) = Fo (0), oe ha E O I 3 O M E M = =, M A M = O E O I, M B = B B = Bae o (9) a he ac ha E i i he Joa o, i yiel ˆ L= I6, E = 0 0, A A E I = = =, ˆ B ˆ = B = E B = , B B B 0 0 = = = By ea o he aociaio bewee (7) a (), a ice ak ( E ) = q =, he igula ye ha oe ipule oe. he peliiay gai K ie i eige o eliiae he ipulive oe o he igula ye. Becaue u =, Kie = O, u =, a i coepoig Joa block i o a ull aix, he ie loop coolle gai ca be give a δ ( b( u u)) δ ( b3) Kie = K = = u ie δ ( b( u u)) = δ ( b3), Kie = K u u ie = O 3. hu, we ge he peliiay eeback gai 0 0 K ie = 0 0 a he cool law

22 340 Jao Sheg-Hog ai e al u () = O 3, Kie v () x ˆ() v = () Subiuig he above cool law o he igula ye yiel he cloe-loop o i (3), whee I3 O Ek = ˆ =, O E Aˆ ˆ B K ie A k = =, O I ˆ B K ie Bˆ Bk = =. Bˆ ao he egula o { k, k} E A io a aa o oce agai wih γ = a η =, a ue he exee aix ig ucio o i a iilaiy aoaio aix M o E k, whee M = Le xˆ( ) = M x () a copue (6) o have

23 Novel quaaic acke a obeve E k O M ( γek ηak) EkM = =, O O ( I4 γ Ek ) O η [ I6 M ( Ek Ak ) γ γ η EkM ] = η O I η =, B k M ( γek ηak) Bk = =. B k heeoe, he euce-oe egula ubye a he oyaic ubye ca be how a x () = ( Ek γ I4) x () Ek Bkv() η = A x () B v() k k = x () v(), a x () = ηb kv () = v () Fo (), we ca obviouly obeve ha he ieeiaio o he a oe

24 3404 Jao Sheg-Hog ai e al wih he ipule oe i iiie. hi iplie he coepoig icee-ie oel oe o exi. heeoe, o have he eie acke o he equivale oel o he aple-aa igula ye, i i eaoable o aue he ipulive oel ha bee eliiae by he above appoach. Subequely, he coepoig oupu ca be ewie a x () yc() = Cx() = CMLMx () = x (). 0.5 Fo (30) i Secio, yc() = Cx () Dv(), whee C = a D = 0 0. Fially, he igula ye ca be ecibe a x ( ) = A x ( ) B v( ) a 4 x () R, k k yc() = Cx () Dv(), whee () c v () R. he iiial coiio o x (0) a y (0) = Cx (0) Dv(0) ae y R, a he cool ipu [ 0.5 ] a [0.5 0], epecively. Accoig o (79)-(8), he paaee o icee-ie obeve ae c L G = = , H = , Hee, le Q 8 = 0 I, R I =. he coepoig igial cool gai aice ( K, E ) accoig o (60) ae K = a E = he, he eul by ea o he popoe eho i a how i Figue 5. Nevehele, i we uilize he eho ha eglec he iec aiio e, he he eul will be ivege.

25 Novel quaaic acke a obeve y 4 y (), () ie(ec) (a) y 0 8 y (), () ie(ec) (b) Figue 5. he copaio bewee oupu y ( ) a he eeece ( ) by ea o he popoe eho: (a) y () v. (), (b) y () v. (). 6 Cocluio he ovel liea quaaic obeve a acke o he equivale oel o he aple-aa liea igula ye ae peee i hi pape. he piciple iea i o ao he igula ye io a equivale egula oel wih a iec aiio e o ipu o oupu. he, boh he egula oel wih a iec aiio e a i aociae coiuou-ie peoace iex ucio ae covee io he equivale icee-ie ype, epecively. Sice he peoace iex ucio i couple wih ae, ipu a eeece, a viual cool ipu i uilize o cove he peoace iex ucio io a

26 3406 Jao Sheg-Hog ai e al ecouple peoace iex ucio. A a coequece, he well-evelope icee-ie opial cool heoy i iecly applie o he ecouple ye oel o he eig o opial igial acke o he aple-aa egula oel coaiig a iec aiio e o ipu o oupu. Whe he ae o he coiuou-ie egula oel wih a iec aiio e a i coepoig aple-aa egula oel ae o available, hi pape oe ubopial aalog a igial obeve o he, epecively. Fo he iulaio eul, we ca ee ha he popoe obeve-bae liea quaaic igial acke o he equivale oel o he igula ye ca ack he eie eeece igal ucceully. APPENDIX A: HE PRINCIPAL h ROO OF A MARIX AND HE ASSOCIAED MARIX SECOR FUNCION Deiiio A. [] Le a aix A C, eigepecu σ ( A) = { λ i, i =,,, }, eigevalue λ i 0 a ag( λ i ) π. he picipal h oo o A i eie a ( A) A A C, whee i a poiive iege a (a) =, (b) ag( σ ( A)) ( π /, π / ). he picipal h oo o a aix i uique. Deiiio A. [9] Le a aix A C, σ ( A) = { λi, i =,,, }, λ i 0 a ag( λ i ) π ( k / ) / o k [0,]. I aiio, le M be a oal aix o A, i.e. A = MJ A M, whee J A i a aix coaiig Joa block o A. he, he aix eco ucio o A, eoe by eco ( A ) o S (A), i eie a S( λ ) S( λ ) 0 eco ( A) = S( A) = M M, (A) 0 0 S( λ) whee S ( λ i ) i he cala -eco ucio o λ i, which i eie a j kπ / λi / i S ( λi ) = e = λ wih k [0,] whee λ i lie i he ieio o he io eco i C boue by he eco agle π ( k / ) / a π ( k / ) /, a λi i he picipal h oo o λ i.whe =, he cala -eco ucio o λ i becoe he cala ig ucio o λ i, eoe jk by Sig( λ i ), i.e. eco ( λi) = S( λi) = e i/ π = λ λi ig( λi) wih k [0,]. he aix eco ucio S (A) eie i Deiiio A. ca be expee

27 Novel quaaic acke a obeve 3407 a ( A ) whee A i he picipal h oo o ucio, eoe by ig( A ), becoe S ( A) = A, (A) A. Alo, he aociae aix ig ( ) ig( A) = S ( A) = A A. (A3) he paiioe aix eco ucio o A ca be ecibe a ollow: Deiiio A.3 [9] Le a aix A C, σ ( A) = { λi, i =,,, }, λ i 0 a ag( λ i ) π ( p / ) / o p [ 0, ]. Alo, le M be a oal aix ( ) o A. he, he q h aix -eco ucio o A, eoe by S q ( A), i eie a ( q) S ( ) 0 0 λ ( q) ( q) 0 ( ) 0 ( ) S λ S A = M M, (A4) ( q) 0 0 S ( λ ) ( q) whee he q h cala -eco ucio o λ i, eoe by S ( λ i ), i, π( q ) < ag( λ ) ( ) o [0, ], ( q) i < π q q S ( λi) = 0, ohewie. he q h aix -eco ucio o A ca be obaie by he ollowig equaio ( q) j qπ / i S ( A) = [ S ( A) e ] o q [ 0, ]. (A5) i= Whe = a q =, he q h aix -eco ucio o A becoe () he aix ig iu ucio o A, eoe by ig ( A) o S ( A) a () ig ( A) = S ( A) = I ig( A). (A6a) [ ] he aix ig plu ucio o A i ig ( A) = I ig ( A). (A6b) A geealize a a able algoih wih h-oe covegece ae o copuig he picipal h oo o a give aix A i lie below : Whe he oe o he eie covegece ae =, { [ ] [ ] } Gl ( ) = Gl ( ) I ( ) Gl ( ) I ( ) Gl ( ), (A7a) R( l ) = R( l) G ( 0) = A, li G ( l) = I ; l [ I ( ) G( l) ] [ I ( ) G( l) ], (A7b)

28 3408 Jao Sheg-Hog ai e al R ( 0) =, li R ( l) = A. I l Whe he oe o he eie covegece ae = 3, G( l ) = G( l) 3I ( ) G l G ( l) 34 3 I G() l G () l G ( 0) = A, li G ( l) = I ; 5 l 5 6 R( l ) = R( l) 3I G( l) G ( l, (A8a) 34 3 I Gl () G() l, (A8b) R ( 0) =, li R ( l) = A. Alo, a geealize a a able algoih wih h-oe covegece ae o copuig he aix eco ucio o a give aix A i a ollow : Whe he oe o he eie covegece ae =, Q( l ) = Q( l) [ I ( ) Q ( l) ][ I ( ) Q ( l) ], (A9) Q ( 0) = A, liq( l) = S ( A). l I l Whe he oe o he eie covegece ae = Ql ( ) = Ql () 3 I Q() l Q () l 34 3 I Q () l Q () l, (A0) Q ( 0) = A, liq( l) = S ( A). l ) Reeece [] S. L. Capbell, Sigula ye o ieeial equaio II, Pia, New Yok, 98. [] F. R. Gaache, he heoy o aice II, Chelea, New Yok, 974. [3] S. M. Guo, L. S. Shieh, G. Che a C. F. Li, Eecive chaoic obi acke a peicio bae igial eeig appoach, IEEE aacio o Cicui a Sye-I, Fuaeal heoy a Applicaio, 47(000),

29 Novel quaaic acke a obeve 3409 [4] H. J. Lee, J. B. Pak a Y. H. Joo, A eicie obeve-bae aple-aa cool: igial eeig appoach, IEEE aacio Cicui a Sye-I, Fuaeal heoy a Applicaio, 50(003), [5] B. G. Mezio, M. A. Chiooulou, B. L. Syo a F. L. Lewi, Diec coollabiliy a obevabiliy ie oai coiio o igula ye, IEEE aacio Auoaic Cool, 33(988), [6] R. Nikoukhah, A. S. Willky a B. C. Levy, Bouay-value ecipo ye: well-poee, eachabiliy a obevabiliy, Ieaioal Joual o Cool, 46(987), [7] J. D. Robe, Liea oel eucio a oluio o he algebaic Riccai equaio by ue o he ig ucio, Ieaioal Joual Cool, 3(980), [8] L. S. Shieh, Y.. ay a R. E. Yae, Soe popeie o aix-ig ucio eive o coiue acio, IEE Poceeig Pa D, 30(983), -8. [9] L. S. Shieh, Y.. ay, C.. Wag, Maix eco ucio a hei applicaio o ye heoy. IEE Poceeig D-Cool heoy a Applicaio, 3(984), 7-8. [0] J. S. H. ai, Y. Y. Du, W. Z. Zhuag, S. M. Guo, C. W. Che a L. S. Shieh, Opial ai-wiup igial eeig o uli-ipu uli-oupu cool ye ue ipu coai, IE Cool heoy Applicaio, 5(0), [] J. S. H. ai, L. S. Shieh a R. E. Yae, Fa a able algoih o copuig he picipal h oo o a coplex aix a he aix-eco ucio, Ieaioal Joual o Copue a Maheaic wih Applicaio, 5(988), [] J. S. H. ai, C.. Wag a L. S. Shieh, Moel coveio a igial eeig o igula ye, Joual o Fakli Iiue, 330(993), [3] C. J. Wag a H. E. Liao, Ipule obevabiliy a ipule coollabiliy o liea ie-vayig igula ye, Auoaica, 37(00), [4] H. J. Wag, A. K. Xue, Y. F. Guo a R. Q. Lu, Ipu-oupu appoach o obu abiliy a abilizaio o uceai igula ye wih ie-vayig icee a iibue elay, Joual o Zhejiag Uiveiy-Sciece A, 9(008), Receive: Febuay, 0