H OBSERVER DESIGN FOR A CLASS OF TIME-DELAY NONLINEAR SYSTEMS WITH EXTERNAL DISTURBANCE

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1 H OSEE DESGN FO CSS OF ME-DEY NONNE SYSEMS H EXEN DSUNCE H, a G a 3 School o ormaio Scic & Eiri, Norhar Uivriy, Shya, iaoi, 89, Chia School o uomaio a Elcroic ormaio, Sichua Uivriy o Scic & Eiri, Zio, Sichua, 643, Chia pihc@yahooc 3 riicial llic o Ky aboraory o Sichua rovic, Sichua Uivriy o Scic & Eiri, Zio, Sichua, 643, Chia q77@6com civ uu ; accp 6 Fbruary SC hi papr i cocr wih h problm o H obrvr i or a cla o im-lay oliar ym wih ral iurbac h purpo o hi problm i o i a H obrvr uch ha, or h ar oliarii a wll a all amiibl ral iurbac i a quaio, h obrvaio proc rmai lobally poially abl, iply o h im lay h oliarii ar aum o aiy h lobal ipchiz coiio, a h ral iurbac ar allow o b im varyi, urucur a orm bou civ liar mari iqualiy mhooloy i vlop o olv h propo problm riv h coiio or h ic o h ir robu oliar obrvr, a h characriz h aalyical prio o h obrvr Numrical ampl mora h valiiy a applicabiliy o h pr approach i hi papr Kywor: H Obrvr, im-lay oliar ym wih ral iurbac, iar mari iqualiy M NODUCON Much rarch ha b o o obrvr i or oliar ym ovr h la hr ca hau 973; Zak a alco 99; ajamai 998 a obrvr or oliar ym wa pr by hau 973 a by Kou, Ellio a ar 975 h mho hav o, i ral, iclu a ymaic chiqu or h corucio o a obrvr Howvr, h obrvr aiy a uici coiio or h aympoic abiliy o h rror ym ac, hr i o a raihorwar mho or lci h obrvr ai o aiy h uici coiio Mo o h mho ar ba upo a mil coiio, uch a wh J o ppl Mah a Mch 86: 67-78,

2 68 H a G a h o-liarii i boh h yamic a h maur oupu quaio o h ym, aiy h local or lobal ipchiz coiio h abiliy o h rror ym i achiv ui liar mari iqualiy M chiqu ajamai 998 ui h hau 975 obrvr by coiri h iac o uobrvabiliy Eii 984 a h mari coiio umbr o h ivcor mari o h rror ym Zha Zha al 9 whr hr ar urhr rrc cocr h abiliy a abilizaio o a liar ym wih boh a a ipu lay For icr-im ym wih irval-lik im-varyi lay, a improv lay-p abiliy cririo i obai i rm o a liar mari iqualiy achai hi papr al wih h problm o H obrvr i or a cla o im-lay oliar ym wih ral iurbac h robu oliar obrvr ar ai chul, ha i h ruli obrvaio proc rmai lobally poially abl or h ar oliarii a wll a all amiibl ral iurbac i a quaio, iply o h im lay impl albraic paramriz approach i ploi, which abl u o characriz boh h ic coiio a h o pc robu oliar obrvr or a cla o oliar a lay ym wih ucrai ral iurbac i how ha a ir oluio i rla o a liar mari iqualiy which ar o iicul o olv h rmair o h pr papr i arra a ollow h robu H oliar obrvr i problm i ormula i cio or a cla o im-lay oliar ym wih ral iurbac h mai rul a wll a ail rivaio ar iv i cio 3, iclui h ic coiio a h plici prio o h ir robu oliar obrvr mora h valiiy a applicabiliy o h propo hory i cio 4 by o ampl cio 5, om coclui rmark h papr wih poii ou poibl io a uur rarch ircio OEM FOMUON Coir a cla o im-lay oliar ym wih ral iurbac crib by, [,], y C u D, whr i h a vcor o ym, u p i h corol ipu vcor, m i h ral iurbac, y i h oupu vcor, i h im-lay coa o p k h ym, maric,,,,, D, C m ar kow coa maric Noliar vcor ucio aiy ollowi lobal ipchiz coiio, rpcivly: a,,,,, J o ppl Mah a Mch 86: 67-78,

3 H Obrvr Di or a Cla o im-dlay Noliar Sym 69 Ur h ipchiz coiio or oliar ucio vcor, h ym ubaially ca b aalyz liarly a hi papr, w i a ic cririo or h H obrvr i or hi cla o imlay oliar ym wih ral iurbac ur h coiio h ull-orr oliar a obrvr ur coiraio i o h orm ˆ ˆ yˆ Cˆ ˆ ˆ ˆ u Kyˆ y, 3 whr ˆ i h a vcor o h obrvr, p rpo ym, u i h corol ipu, h coa mari K ai o b chul m yˆ i h oupu vcor o h m i a obrvr applicaio, oc h obrvr ai K i pr-chul, o ca upa h rla ym maric o h obrvr by ui h maur valu o h prurbaio h rwal irval may b rmi by iri rquirm, a h pc prormac ca alway b ur rarl o h amiibl prurbaio Di h rror a ˆ, h i ollow rom h ym cripio a h obrvr quaio 3 ha KC ˆ ˆ D 4 hi papr, h oal o a robu H oliar obrvr i problm or h im-lay oliar ym wih ral iurbac i, iv a prcrib lvl o iurbac auaio, o i h obrvr ai K, uch ha, or h oliarii a a all amiibl ral iurbac, h rror-a ym 3 wih rmai lobally poially abl, iply o h im lay, ha i lim, S 5 ur zro-iiial coiio or all ozro [,, whr S i a poiiv ymmric mari a [, i h pac o quar irabl vcor ucio ovr [, hi ca, h rror ym 4 i ai o b aympoically abl wih H prormac h ollowi iiio a lmma will b uul i rivi a ir robu oliar obrvr ai or h im-lay oliar ym wih ral iurbac J o ppl Mah a Mch 86: 67-78,

4 7 H a G a Diiio Soorvol 99 h H orm i i a S whr i a rar ucio mari rom o For a iv lvl, ca b ra i h quival orm i iquaio 5 o i H S 6 h, h rlaio i iquaio 5 ca b rpr by H 7 mma a al 99 G,, E a F b ral maric o appropria imio wih F bi a mari ucio h, w hav a or ay a F F, F E E F E E, b or ay uch ha F F a E E, G F EG F E G E E G mma Nolu 985 w choo S,, h, S S mma 3 Kohkoui al 9 Schur-complm h maric i, i,, 3 b ucio o uch ha a 3 3 h M 3 J o ppl Mah a Mch 86: 67-78,

5 H Obrvr Di or a Cla o im-dlay Noliar Sym J o ppl Mah a Mch 86: 67-78, 7 i quival o h ollowi coiio 3 a a, 3 b a 3 h cio, hrouh a plo o H vru im, h rlaio i iquaio 7 i vrii 3 MN ESU ND OOF h horm a blow will orc h ir robu lobal poial abiliy or hi cla o im-lay oliar ym wih ral iurbac corai upo a liar mari iqualiy h M problm or achivi robu H oliar obrvr i i pr i h ollowi: horm For iv a S S, i hr ar i,,,, a M uch ha S D D 8 whr M C MC

6 H a G a J o ppl Mah a Mch 86: 67-78, 7 h h rror ym 4 i aympoically abl wih H prormac Furhrmor, h obrvr ai K i iv by M K 9 roo Fir, o how h H prormac or h rror ym 4, o ca coir h ollowi yapuov-kraovkii ucioal 4 3 whr 3 4 h im rivaiv o alo h rajcory o h rror ym 4 i ˆ ˆ ˆ ˆ D D K C KC w u h lmma, w hav ˆ ˆ ˆ ˆ ˆ ˆ a

7 H Obrvr Di or a Cla o im-dlay Noliar Sym J o ppl Mah a Mch 86: 67-78, 73 ˆ ˆ ˆ ˆ ˆ ˆ a DD D D 3 From iquaio, a 3, w hav DD K C KC w u h lmma, w hav ha h im rivaiv o i wri a h im rivaiv o 3 i wri a 3 h im rivaiv o i wri a 4 h, h im rivaiv o 4 i wri a 4 3 S S 4 whr

8 H a G a J o ppl Mah a Mch 86: 67-78, 74 S DD K C KC Sco, l K M h iquaio 8 i quivally cha io h liar mari iqualiy o iquaio which i iv by S D D 5 whr K C KC w u h lmma 3 Schur-complm, h mari iqualiy i iquaio 5 i quival o 6 h, w hav S S 7

9 H Obrvr Di or a Cla o im-dlay Noliar Sym 75 w ira boh i o iquaio 7 rom o S Sic a, h, w hav, w hav h rlaio i iquaio 5 h h rror ym 4 i aympoically abl wih ai K M H prormac, a h obrvr mark : h am problm i ivia Kohkoui al 9, howvr, which hav oly coir coiio which h liar par coai lay, a or oliar cio coai h lay i o icu h pciic corol chm i o propo, a h cop ivia i rmly rric Obviouly, i i a pcial ca o hi papr 4 SYSEM SMUON hi cio, w mora h hory vlop i hi papr by ui o umrical ampl Eampl Coir h im-lay oliar ym wih ral iurbac wih 3 i 4 i i u, i, [,], y 8, o maiu Fiur pr a iurbac ial,, a im pric o o mor ha, ri h a pac by ma o mari D Fi Diurbac ial,, ri h a pac or a im prio o J o ppl Mah a Mch 86: 67-78,

10 76 H a G a a ra poi, compui h ial rom o : H orm, o ca achiv h ollowi rlaio or h S 5 No ha h la rlaio i coi wh compar o 4456 h, accori o h horm, h robu oliar obrvr ar ai chul: 646 K 594 Ur h obrvr o h ym 8 i i h rror ym o h ym 8 i aympoically abl, h yamic rpo how a how i h iur Fi Dyamic rpo o rror ym J o ppl Mah a Mch 86: 67-78,

11 H Obrvr Di or a Cla o im-dlay Noliar Sym 77 mark : h am problm i ivia Zha al, howvr, which i coir or a cla o im-lay oliar ym wihou ral iurbac h cop ivia i rmly rric Numrical imulaio i hi papr i h provi o how h civ a aibiliy o h propo mho Obviouly, i i a pcial ca o hi papr 5 CONCUDNG hi papr ha ui h problm o robu H oliar obrvr i or a cla o im-lay oliar ym wih ral iurbac ai-chul obrvr rucur ha b aop liar mari iqualiy approach ha b vlop o olv h abov problm Spciically, h coiio or h ic o h pc robu oliar obrvr hav b riv i rm o a liar mari iqualiy lo, h aalyical prio o h ir obrvr ha b characriz O ampl ha how h civ o h pr i approach ha b mora ha h ir robu oliar obrvr o im-lay ym wih ral iurbac, wh hy i, ar uually a lar, a h rmaii rom ca b u o m ohr pc prormac rquirm, uch a h corai o h H orm o h rar ucio rom poibl oi ipu o obrvaio rror oupu h mai rul ca alo b o icr-im ym a ampl-aa ym h will b h ubjc o urhr iviaio 6 CKNOEDGMENS h auhor woul lik o hak h rr or hi/hr hlp aiio, h auhor woul lik o pr hi icr apprciaio o roor Xiao-Ho or om valuabl uio owar achivi h work o hi papr hi work wa uppor by h Op Fouaio o riicial llic o Ky aboraory o Sichua rovic YJ7 EFEENCES Eii 984 w corollabl a ucorollabl Sym & Corol r 45, pp Kohkoui J a urham KJ 9 Dicoiuou obrvr or o-liar im-lay ym raioal Joural o Sym Scic 44, pp Kou S, Ellio D, a ar J 975 Epoial obrvr or o-liar yamic ym ormaio a Corol 93, pp 4 6 Nolu E 985 Sabilizaio o a cla o iribuioal covoluio quaio raioal Joural o Corol 44, pp ajamai 998 Obrvr or ipchiz o-liar ym EEE raacio uomaic Corol 43, pp J o ppl Mah a Mch 86: 67-78,

12 78 H a G a achai K M approach o abily a abilizaio i lay corollr i or icr-im ym raioal Joural o ppli Mahmaic a Mchaic 69, pp -8 Soorvol 99 h oo H Corol roblm: Sa-pac pproach ric Hall, hau FE 973 Obrvi h a o o-liar yamical ym raioal Joural o Corol 73, pp a YY, Xi H, a Souza CE 99 obu corol o a cla o ucrai oliar ym Sym & Corol r 9, pp Zak SH a alco 99 Sa obrvaio o o-liar corol ym via h mho o yapuov i Drmiiic Corol o Ucrai Sym, Ziobr S oo: r rriu, pp Zha XM, i MM, u M, a Sh JH 9 Furhr rul o abiliy a abilizaio o liar ym wih a a ipu lay raioal Joural o Sym Scic 4, pp - Zha Y, Ya HJ, a iu Obrvr i or ipchiz oliar ym wih im-lay Joural o Norhar Uivriy Naural Scic 3, pp 5-54 i chi J o ppl Mah a Mch 86: 67-78,