GLOBAL EXPONENTIAL OBSERVERS FOR TWO CLASSES OF NONLINEAR SYSTEMS

Transcription

1 GLOBL EXONENTIL OBEE FO TO CLE OF NONLINE YTEM Iasso aralls * ad Cosas ravars ** * arm o Evromal Egrg Tal vrs o Cr 7 Caa Gr mal aral@vg..gr ** arm o Cmal Egrg vrs o aras 65 aras Gr mal ravars@mg.aras.gr sra Ts ar dvlos s odos or s o gloal oal osrvrs or wo lasss o olar ssms lass o ssms w a gloall asmoall sal oma s ad lass o ssms a volv o a o s. I rs lass drvd oos-m osrvr also lads o osro o a ros gloal samld-daa oal osrvr dr addoal odos. Two llsrav amls o alaos o gral rsls ar rsd o s a ssm w mooo olars ad or s mosa ssm. words osrvr dsg olar ssms oal osrvrs.. Irodo O o ggs allgs o mamaal orol or as rolm o osrg sa osrvrs or olar ssms. Ts rolm as arad a lo o ao lrar as dads; as aroad w a var o mods ad rom a var o os o vw s or sa [578678] ad rrs r. I s wor w os o olar orward oml ssms o orm. m wr s a o-m s s a smoo vor ld ad o s gv. wr s a smoo mag. T am s o osr gloal oal osrvrs. valal mods or gloal oal osrvrs ld g-ga osrvrs or gloall Ls ssms [8] as wll as rl-rro osrvrs rmarl or olar ssms w mooo olars [7]. I rasormao-asd osrvrs orgall dvlod loal orm [] ad ssql gloal orm [] ssm s mad o a lar ssm ad dsg o osrvr s rormd rasormd oordas wr oal ovrg s mosd. I s wor w rs s odos or s o oal osrvrs or wo mora lasss o olar ssms w ar o ovrd aov mods Nolar ssms w a asmoall sal oma s Nolar ssms volvg o o ss For o lasss o ssms roosd osro o gloal oal osrvr sars w a adda osrvr w s ssql modd addg a orro rm ordr o sas arora Laov

2 qals. I sold masd a l ormla or osrvrs ar rovdd a as ad ror orol raor a drl al rsls o ar. I o wr w sd rs lass o ssms adda osrvr s a loal osrvr ovr a ra oma s wras orro rm ors rajor o r oma s m. T drvd oos-m osrvr a also lad o osro o a ros gloal samld-daa oal osrvr dr addoal odos. T samld-daa oal osrvr s ros w rs o rraos o samlg sdl ad w rs o masrm rrors s also [9] or samld-daa osrvrs. o sds sod lass o ssms w ror o volvg o a o ror ss o. Hr adda osrvr dos o gara a osrvr rajors l w o s ad s s aomlsd addg a arora orro rm. T dsg o orro rm s rormd ar rasormg ssm rog a arora smoo jv ma a mas o s oo v og oal ovrg s ord orgal oordas. T rsls o o ar mora as or ma lasss o ssms sa volvs a o s or aml ologal ssms sall volv o rs qadra. Howvr r s aor raso a movas rsls o o. I a ag o oordas X a od wr s a smoo jv mag sasg or all or ra Hrw mar ad ra mag mag a sd or dsg o a osrvr or.. dr addoal oss s []. T rsls o o sow a w do o av o assm a.. s oo sad w a rqr a s a o s ad al Torm.. Fall o w rs wo llsrav amls o alao o gral rsls. T rs aml s a ssm w mooo olars ad w al rsls o o o drv a gloal oal osrvr rs dr oos-m masrms ad ssql dr samld masrms. T sod aml s a oraor ollowg mosa modl w osv sa varals volvg o o rs qadra o. lg rsls o o lads o a gloal oal osrvr w osv sa smas. Noao. Trogo s ar w ado ollowg oao [. B C ; Ω w do lass o oos os o w a vals m Ω. B C ; Ω wr s a gr w do lass o os o w oos drvavs o m ordr w a vals Ω. B w do ror o s. For a vor w do s rasos ad s Elda orm. dos m m rasos o mar ad dos dd orm o mar.. m { } s. o wll alld osv d ad > or all. o wll alld radall odd ss { M } M. For a o C ; grad o a. m ar r m or odd or all dod s row vor

3 . sms w a Gloall smoall al Coma Cosdr orward oml ssm... Or ma oss s so garas a r ss a oma s w s rosl gloall asmoall sal adjv ros mas orm o all masral ad loall ssall odd s. H Tr s a radall odd o ssarl osv d o C ; a osv d o C ; ad a osa > s a ollowg qal olds or all w. Idd oss H garas a or vr al odo ad or vr masral ad loall ssall odd solo o. rs oma s { } ar a ras rod.. r ss T C ; s a or all T. Morovr o a oma s { } s osvl vara. Ts a s garad ollowg lmma w s rovd a d. Lmma. Cosdr ssm. dr oss H. T r ss T C ; s a or vr ad or vr masral ad loall ssall odd solo o. w al odo ad orrsodg o sass ma or all ad T. or all Or sod oss garas a w ar a oso o osr a arora loal oal osrvr or ssm... H Tr s a smmr ad osv d mar osas > > ad a smoo mag w or all w s a ollowg qal olds or all w ad. Idd oss H ojo w oss H garas a or vr { } ad or vr masral ad loall ssall odd solo o ssm.. w wll sas a sma o orm M σ. or all or arora osas M σ > rovdd a al smao rror s sl small. Ts s w ssm. s rmd as a loal oal osrvr. T radr sold o a oss H olds aomaall or olar ssms o orm or vr > > ad or vr o-m s... m wr... ar smoo mags. I ordr o al o osr a olar oal osrvr or ssm.. w d a addoal al oss..

4 H Tr s osas a < s a ollowg qal olds or all w a < < ad.5 Hoss H moss osras or volo o rajors o loal osrvr.. Idd qal.5 moss a od o drvav o Laov o C ; alog rajors o loal osrvr. or s rgos o sa sa. ar ow rad o sa ad rov ma rsl o rs so. Torm. Cosdr ssm.. dr oss H-. loall Ls mag ˆ ˆ or all w.6 ϕ ˆ or all w >.7 wr ϕ s dd ϕ ma.8 ad [] s a arrar loall Ls o a sass s or all s ad s or all s a. T r ss M C ; ad σ > s a or vr masral ad loall ssall odd ad or vr solo o.. w ˆ.9 al odo orrsodg o sass σ M or all. mar. a Torm. sows a dr oss H- a orro rm s dd ordr o al o osr a ϕ gloal oal osrvr or ssm... T orro rm oms av rgo > a ad s ma as s o gara vald o dral qal ˆ or all w. T rvos dral qal ojo w Lmma. garas a solo rs a arora oma s m ad s arora oma s loal oal osrvr wors. Iqal. garas a oss H olds rovdd a r s osas a < s a ollowg qal olds or all w a < ad.

5 5 roo o Torm. Frs o a or all w ollowg qal olds ˆ. Idd do.7 mls ˆ ϕ. B dsgsg ass ad > sg do.8 ad og a w old a. olds. N w sals ollowg qal ˆ or all w ad. No a qals.. ad dos ml a qal. olds or as a. Tror w os o as a <. o.7 gvs ˆ ϕ. Iqals.. ad a a ϕ mls a. olds. Morovr qals.. sow a. olds ϕ. I rmas o osdr as < ad > ϕ. I s as do.8 mls > ϕ. Iqal.5 gvs ϕ.5 sg.5. ad a a w oa ϕ Comg.. ad aov qal w old a. olds. L arrar masral ad loall ssall odd ad arrar ad osdr solo o.. w.9 al odo orrsodg o. Lmma. ojo w. ad. mls r ss ; C T s a or vr ad or vr masral ad loall ssall odd solo o.. w.9 w al odo ad orrsodg o sass ma ma or all or all T ad or all T

6 sg. ad asoll oos o w old a σ or all.6 wr T T ma. σ ad > ar osas a sas or all { } M σ ma T T ma ma ma.7 o.7 ojo w.6 ad a a ma ma all mls a. olds. T roo s oml. or advaag o osrvr dsg rovdd Torm. s a a osrvr a mlmd w samld masrms. T ollowg rsl garas dsg o a gloal samld-daa oal osrvr. Torm. Cosdr ssm.. dr oss H- ad sos a ollowg addoal oss olds H ad r ss a vor L oma or mag s dd o. s a L. Morovr r m s L C ; σ r γ > s a or vr masral ad loall ssall odd w ad or vr w M ad or vr loall odd s solo w o.. w ˆ w w [ τ τ w τ τ τ r τ τ w τ w w τ.8 sass σ γ s M s or all.9 s mar.5 a T qas o masrm rrors. I s lar a samld-daa osrvr.8 sass a -o-o sal ror w rs o masrm rror. T w qas o rraos o samlg sdl. Mor sall dsrd qal.9 s garad or vr samlg sdl w damr lss or qal a r >.. or vr s o samlg ms { } τ w τ ad τ r τ.8 s a rd ssm w varal samlg aro s []. s s also [9]. T ovrall ssm.. 6

7 Hoss H olds aomaall or olar ssms o orm. wr... ar smoo mags. I sold od a samld-daa osrvr.8 s smlar o samld-daa osrvrs osrd [9]. Howvr rsls rsd [9] ao sd ordr o rov Torm.. T raso s a qal. [9] dos o old or all ms as rqrd [9]. aalog o qal. [9] olds ar a al ras rod. T ras rod s dd so a sa o orgal ssm ad osrvr sa r a arora oma s. roo o Torm. s roo o Torm. w rs o a or all ollowg qal olds w w ˆ w. Morovr al as roo o Torm. w sow a. olds. os ad oss H ml s o a osa G s a ˆ w ˆ G w or all w w. G or all w ad. Lmma. ojo w. ad. mls r ss T C ; s a or vr w or vr masral ad loall ssall odd ad or vr loall odd s w solo w o.. w.8 al odo w w ad orrsodg o s w sass ma ma or all T ad or all T or all I ollows rom aov smas a ollowg qal olds or all { ma ma } ma. sg.. ad asoll oos o w old a G w s s σ s σ s s or all. wr { τ τ ma T T } m or all σ ad > ar osas a sas. Fall o a or vr ollowg sma olds wr τ { τ τ } w s s rg s s s s τ s ma. No a rom qal τ r ad.5 w oa σ w σ s s rg σ τ s σ s s rg σr s s s τ s σs s s τ s σs s s.5.6 7

8 Iqals. ad.6 ml ollowg qals or all s s s G σ w s s.7 s s s σ s σs s s w s s σ s s rg σr s σs s s σ.8 s Fall w assm a r > s sld so a s s G r σr <.9 Iqals.7.8 ad.9 gv ollowg sma or all s s γ s s s σ γ σ s s σ. G s wr γ G G. { τ τ ma T T } T T r r σ r m ollows a ma. Comg. ad. w old a.9 olds w { } γ M G σ r σ ma T T ma ma ma T roo s oml.. Gloal Eoal Osrvrs or sms o O s Cosdr orward oml ssm X F X X. wr s a o s s a o-m s gv m F s a smoo vor ld ad o s H X. wr H s a smoo mag. assm owldg o a smoo jv mag w ad d or all wr s Jaoa o mag s a ssm.. dr ag o oordas X s rssd.. wr ad ar smoo mags sasg F or all. H or all. wr s jaoa o mag. 8

9 T ollowg oss mls s o a adda gloal oal osrvr or ssm... Tr s a smmr ad osv d mar a osa > ad a smoo mag H w F or all H w H s a ollowg qal olds H X F X X X or all X.5 Idd oss garas a or vr X ad or vr masral ad loall ssall odd solo o ssm.. w wll sas a sma o orm X M X.6 σ or all or arora osas M σ >. Howvr ssm.6 s o ssarl a osrvr s w ao gara a or all. I ordr o sa rolm a dr wa s ov o s ag o oordas or osrvr.6.7 wr s a smoo mag. Now rolm a sad as ollows log ssm.. s orward oml ssm.. w.7 s o ssarl orward oml ssm.. s orward oml rsls [] gara s o a radall odd o ssarl osv d o C ;[ a oos o ad a osa s a or all w.8 T rolm a w osdr s so s rolm o s/dsg o a osrvr w sa w garas gloal oal ovrg orgal X oordas asd o owldg o o ad adda osrvr.6. Or ma rsl garas a dr som addoal assmos s/dsg rolm o gloal oal osrvr s solval. Torm. os a r s osas { q j } j j a > ε a smmr ad osv smd mar... w q C ; ad a oos o H [ s a ollowg qal olds or all w a ad < ε.9 { a } C. os a r s C s m or a < or all C H. 9

10 loall Ls mag H G G λ or all H. wr H λ s dd > λ or all H. ad ] [ s a arrar loall Ls o a sass s or all a s ad s or all a s. T r ss > M s a or vr masral ad loall ssall odd ad or vr X solo X o.. w G. al odo X X orrsodg o sass X M X ε or all. roo Iqal.5 ojo w dos.. ad do mls a ollowg qal olds or all. vala qa ˆ or all w a wr ˆ λ. g λ a ad ] [ s a arrar loall Ls o a sass s or all a s w oa. B dsgsg ass > ad w ma old a ollowg qal olds or all w a.5 lam a ollowg ssm s orward oml λ.6

11 T lam s rovd a d. ssm.6 s orward oml ad sg ag o oordas X w old a solo X o.. ad.6 sarg rom arrar al odo X ad orrsodg o arrar masral ad loall ssall odd ss or all ad sass X or all. vala qa ˆ wr ˆ λ. Cosdr ass a. I s as λ ad ˆ. Tror qal. mls a ˆ. a >. I s as ˆ λ ad w g ˆ λ λ.7 No a λ. I.7 mls ˆ. I < ad λ.7 mls ˆ. Ts ol as a rmas o osdrd s as < ad > λ. I s as w av λ ad >. Iqal.9 gvs ε λ T aov qal ojo w. ad.7 mls ˆ ε.8 ε w old a aov qal olds or all. sg ag o oordas X ad dral qal.8 w old a solo X o.. ad.6 sarg rom arrar al odo X ad orrsodg o arrar masral ad loall ssall odd sass or almos all ε wr X X. T s o a osa > M sasg. s a dr osq o aov dral qal. T roo s oml.

12 . Eamls I rs so w wll al rsls o rvos os o wo s amls. T rs aml s a ssm w mooo olars ad w wll al rsls o Torms. ad.. T sod aml s a mosa modl w osv sa varals ad w al osro o o. Eaml. Cosdr olar laar ssm ] [. sm. s a ssm o orm. w mooo olars. T olars ar o gloall Ls; owvr a oos gloal oal osrvr a dsgd sg modolog roosd [7]. Hr w wll dsg a oos gloal oal osrvr sg Torm. ad w wll sow a w a also dsg a ros gloal oal samld-daa osrvr sg Torm.. wll sow a oss H- old or ssm.. Frs o a oss H olds w. Idd o a qals ad gv s 5 5 wr. T aov qal sows a qal. olds w ad. N w sow a oss H olds. L q w q > > ad L L wr L L q ar osas o sld. g q L L ql L sg qals q q w oa q L L ql L

13 I w rr assm a ad wr > s a osa o sld w oa T slo q L ql L L 8 q 6q L q q 6q L. q q ojo w aov qal mls a. olds w oss H olds w arrar as wll o a.. Tror w av sowd a q sasg q > > ad >. Morovr oss H olds Fall w sow a oss H olds. Mor sall w wll sow a r s osas a < s a. olds or all [ ] w a < ad w ad. Iqal. s qval o ollowg qal L L. 8 sg qals ad L L ma L L w old a. olds rovdd a ollowg mor dmadg qal olds or all w a < ad 5 L L ma.. ms old or all w a < ad < w old a. olds aomaall rovdd ollowg qal olds ma L L a os. ml a qal.5 olds rovdd a > a > ad q > ar sld so a 6 q > ad a 5 a 5 q 6 m q.6 6 q q 6q a 5 For aml all qals old or a q mag. Tror w ar a oso o d

14 L ˆ L or all [ ] w.7 ϕ L ˆ or all [ ] w >.8 ϕ L wr ϕ [ ] s dd ϕ ma g L L.9 ad g [] s a arrar loall Ls o a sass g s or all s ad g s or all s. Torm. garas s o r > s a ssm ˆ w ˆ w w w τ τ τ τ r τ w τ [ τ τ. s a ros gloal samld-daa oal osrvr. Eaml. Cosdr mosa modl [5] X X X X. w o X ad s [ [ wr > ar osas ad [ ma ] s a loall Ls odd o w ad > or all >. sall ssm sas X ad rrs omass orao ad ssra orao rsvl o osv qas ad s a osvl vara o s a oas sall magl rajors o ssm. T rm X rrss grow ra o mroorgasms ad s a masral qa oraors w a gasos rod l aaro dgsrs wr ogas rodo ra s rooroal o mroal grow ra [6]. T ollowg adda osrvr. sass oss w sas rqrm I >. Howvr s lar a adda osrvr. dos o or all al odos ad all ms. Hr w wll al rsls o o sg smoo jv mag dd

15 5. wll osr a gloal oal osrvr or ssm. dr assmo a r ss * > s a or all ] [ *... w wll assm a s o-drasg o rval ] [ *. Idd smoo jv mag dd. allows s o drm vor lds so a all qaos.-.7 old w ad [ [. radall odd o osv d o ;[ C ma dd.5 ] ma w g or all ma ma.6 ] [ ma s a loall Ls odd o w ad > or all > r ss a osa > γ s a γ or all >. Tror.6 mls a ollowg dral qal olds or all ma ma γ sg qals γ γ ojo w aov dral qal w old a.8 olds w ma ma ma γ ad arrar osa. Fall w vala qa or all ad > sg qals ad aov qal w g or all ad > ma.7.8

16 ga sg a a r ss a osa γ > s a γ or all > ad qal γ γ w oa rom.7 ad.8 or all ad > ma γ m.9 γ ma wr > sass * l ad s o sld. Iqal.9 mls a < ad.9 olds w arrar ε or all ad > w. I ollows rom.9 a.9 olds rovdd a r ss osa ε s a ollowg qal olds or all w < < ε. sg a a s o-drasg o rval [ ] w old a. olds or all w < < rovdd a ollowg qal olds * ε. o o mls a r ss sl larg > s a ε w drl mls qal.. Tror w old a.9 olds w arrar a. T gloal oal osrvr wll gv qaos λ. wr λ s dd. ad [] s a arrar loall Ls o a sass s or all s a ad s or all s a. 5. Coldg mars Ts wor dvlod s odos or s o gloal oal osrvrs or wo lasss o olar ssms. T rs s lass o ssms w a gloall asmoall sal oma s. T sod s lass o ssms a volv o a o ror ss o. I o ass osro sars w a adda osrvr w s ssql modd addg a orro rm ordr o sas arora Laov qals. I rs lass o ssms adda osrvr s a loal osrvr ovr a ra oma s wras orro rm ors rajor o r oma s m. I sod lass o ssms adda osrvr dos o gara a osrvr rajors l w o s s s aomlsd rog a arora orro rm. T dsg o orro rm s rormd ar rasormg ssm rog a arora smoo jv ma a mas o s oo. T drvd oos-m osrvr a lad o osro o a ros gloal samld-daa oal osrvr. T das dvlod o adl sod lass o ssms old d oal s o o rasormao-asd osrvrs rlag rqrm o a domorsm o oo allowg mag o vrs ma o a o ss o. 6

17 rs [] ra M. Nolar Osrvrs Crl Crro sg ad osss alss omaa [] dr. ad L. ral O Es o a aas-ravars/lrgr Osrvr IM Joral o Corol ad Omao [] gl. ad E.. oag Forward Comlss odd Osrval ad r Laov Cararaos sms ad Corol Lrs [] sol. ad L. ral Gloal Coml Osrval ad O-o-a al ml Es o a Gloall Covrg Osrvr Mamas o Corol gals ad sms [5] Bosos. ad J. Tsas Codos o Es o wg Osrvrs or Nolar Tm- arg sms arxv.v. [6] oa oma Corol o Borosss l-ite Lodo 8. [7] Fa X. ad M. ra Osrvr sg or sms w Mlvaral Mooo Iqals sms ad Corol Lrs [8] Gar J.. ad I. a rms Osrvao Tor ad laos Camrdg vrs rss. [9] aralls I. ad C. ravars From Coos-Tm sg o amld-aa sg o Osrvrs IEEE Trasaos o oma Corol [] aralls I. ad.-. Jag al ad alao o Nolar sms. Lodo rgr-rlag. [] aas N. ad C. ravars Nolar Osrvr sg sg Laov s lar Torm sms ad Corol Lrs [] rrr M. ad. oa Evalao o Corol rags or aro gso rosss Iraoal Joral o dav Corol ad gal rossg [] osoa. Commad osro d osrvars or ls ssèms o léars ras à doés éalloés résa ss vrsé ars-d L-CN ELEC 9. [] osoa. T. md-l ad F. Lama-Lagarrg Osrvrs or lasss o olar word ssms IEEE Iraoal Ml-Cor o sms gals ad vs jra Tsa 9 [5] m H. ad. alma T Tor o Cmosa. ams o Mroal Como Camrdg ds Mamaal Bolog Camrdg vrs rss Camrdg 995. [6] Tsas J. Osrvr sg or Nolar sms sms ad Corol Lrs [7] Tsas J. Frr sls o Osrvr sg rolm sms ad Corol Lrs d roo o Lmma. T a a oma s { } 7 s osvl vara or vr masral ad loall ssall odd s a dr osq o dral qal.. N w osdr solo orrsodg o arrar. o. w arrar al odo.. > ad Tr ss ] s a solo s dd o ad ao dd <. ma { [ } [ ma ma. wll sow a. os a. Ts mls a > or all. Iqal. mls a or almos all w [ ma mls or all. C ; s radall odd ollows rom [ ma qal a solo s odd o. Ts sadard or o ordar dral qaos mls a. L > ma δ dd m{ } ma [ ma [ ma δ. Idd o a osv o δ > s a dr osq o o o ad omass o s { }. ral qal. ojo w ad > mls a δ or almos all [. Cosql w oa < or all w s a orado. δ

18 { [ } ma w d. Co o mls a > ad. Morovr osv vara o oma s { } mls a solo ss or all ad sass or all. sg a argm smlar o o sd or as w ar a oso o sals a or all [ ad wr δ { } δ m. Fall d T or all ad { } T δ m. T aov aalss garas a or all wr δ or all T ad a ma or all. Co o o T s a dr osq o o o ad a a lvl ss o ar oma ss. T roo s oml. roo o lam a ssm.6 s orward oml Frs o a or vr al odo ad vr omo o solo o ssm.6 s dd or all. Ts ollows rom a a.. s orward oml. B vr o do λ ad a a s a radall odd o w gara s o a oos o H [ s a or all w a. Iqal. ojo w.5 sows a r ss a oos o ˆ H [ s a ˆ or all. T dral qal. sows a solo o ssm.6 sass ollowg qal or almos all or w solo ss β. wr ad β ˆ. T dral qal. sows a s ds β or all or w solo ss. Morovr qal s ds β sows a rmas odd o odd rvals o m. sg a a s a radall odd o ad a sadard orado argm w old a solo o ssm.6 s dd or all. T roo s oml. 8