eh ..... a5. 1111111 . '- H2. 145x. 7»: y c I . Its? . 7......iaanzy! - I \ ‘0‘; Iii. Stunt . J2.5.5 5.35.: I. . _ 90 I I .. trIyI flflaarbqh. y . 3.9 I .. ?.bx!.¢us.$uai. ‘ . . 19 II I I. 951.573 11:55 WI‘c‘I! :- . ..~.IIII.I\ I.t)..sss7aaylxrr..n£ 7’1... .51.. . T .x .. {triageIII-Itanrxvt..:~.blaf:. .upuwwae.‘ : .. .z . .Ir...‘ 9.. IINIOIVIIVIVIIRAPIUZ 25.5%.... gIIINIhIaI: 51.2. .0 IPIOLi—I : . I 9.... .. LI ilk“ )IHBI) ibflflag. in.IVOILH)&IIIH.II‘IIWI’I.2§ 7 _ .. I... a 7. In“. . :xl; . i: II 'I a I .I. huh. .. ..:.........§:.. 733131.»; o..au..l...L32aldrni...‘ ............. I Z, .. .IIIIQIVIIE IIII:¢:J.I§:I9IZHM1..:va‘...‘II:I Ilifislq.i Iv L... .. . . .. . . .. BihwaAézII331 ........... a .1333 {$53! .(HII..II$IIII. 3.55.! If I) I)! (x I .1 .‘1 . : $5.1..le :Ila’latzIII 5.154....5‘255 an aaaaaa a 3.. zzzzzzzzzzz Ia fl: )5 Era-«00,1.SIIIIII52. I II :Cllsl. I 1%.?ax..::>I’IIIII.iIOa-.II ItlI..II..I,1II.fI €250" III..?II lit..v52fi-1 I l: . gtl l! IIItI..I .11....‘II II I n .. I IIIII IIlIII:.iaI$L!pII.III.£.I IIIIIIIQIIIIIC qIIIrlCIIIEIIaIKI IzaaiIIIL. I)\II.ufi:‘IVAIIIn1.fyIIII-E§.I§.~III IZIIIIIGII. Initbg. I. e LFIIIII 3'3. va-Mr‘tiiiIIsr-rk ILJNIII I. I’ll-Inhil If? Idgdhwzi.’ III-Ii: 1'31, I Ill>n.:!;pl. ILI‘IIII Ifiii3~iIz6IDIIIII viii.) I. A I 3:39.32! VIII vahfluarvtlrlflzz d‘ Inklikmi 6H... AIII IIIIIIIJNIIIIIIKsfilirs IL.‘ 5:... 00;: VIII..- .IIZIVII PI)I.I‘?.‘§3§.§ I . .y .PItII’ilIIIII’E‘ c :IlI..I'IOIIIIV.1 “.7 :IY‘IIII...‘ I )WJWIN. “thwart?!” Its...t:0I:.ruN~.aVI7ar§lI.FI 9,! ”I III :gIIEE..«£-‘I\Iflufl . 4 u : III? I... 3.11.!) I If... £29... I. .. I . s 3 ii. I . III-1| 9.. I} II: I: ‘21:}: II III got.- u 2: If I; :93). ......:n;..uu:£u....vihhz Ilkufifwalh 5...},..afichfiiib.r..i.nmmu.!.h.1$intcfii,banisheuixathnf31 .22232Lu..hfbhdfl.3225.10..»l:....s.z§1!.n2hn2..2.....u: ctvl...!m‘vwsnlit§fim.bp nu 2 r .. z a I .1. . 31...... .IIII ' . . 3 . hffigsudama.54ufiflnfiugfiuukrfipfifa1.... ”2.139%..%I§%.fidw.hti»b.~.mfi..mmmwh¥z§:.nnununtfifluvltatylnhhvfsai.w€..s...l..f{..:s§z{2.5:}.2 L3. III’!".’ I. I ~ . v . "f”?\")‘.{|‘.h-n. l’.":.§“\ . I. Igilnli $7: Izllsi‘ Mill!!! Oltz‘ II A. 0 It‘s-IVE... .1513! {II}?! 0.1.}... gluflIIVPIIIII.‘ .III I: a I. a? 31W1712~Ira~7 . 3. .a h. K. E... hazaugxhvl.’ IIIII: Ltnl... z: #1:: .35 . .«t III ENIIIIII 5.))xiiICiII... 5.4., . . 3...: I IVIIIII.IHII.WI=. I. a .85.. I“. III 9 I : :IIIII. I: MIIIIIIvJSIIoixfit. 1.25.3 9| 1 . 5 a... IIfl..f : 3.91! I! QIIIQIIIIMVII 9E1; Oi. iiItIIIlI..I-:‘SII:#8 airtig {II I12. 2.: Ii; It. . . I 21‘ 53.. Eli . .iafimiahhh......,..r..w.;mn.__,.xwfifivffifinflvfihfl flaw... .gnfiflmz,fibhhfi;bfzhe,.9bh..il¢mfifztfirminwfi..stiha..x.n.....¢ tt¢,.3.,...§tnu.fin fi..s.t2.§u$lia III-WW Ia:fi)..fii . .1 YII I lkdleIIIIriv Ila! I 11.7."! :13)...‘ IF!) .. . l i IIII I. . 1) 3 c?‘ I‘lzilglzIyII ., . iiidilzp :0: .1!u5~l££!.u..i I: III II 1.!th II ‘54 L§S§24 IL-IIl‘IIZQIIIII‘IIIn QEIIZIIIIII 1.5.5 :IIIIISIIIVIDIIIIII inn n... .. 1. III I!’?Yli.v:..pivlala $571!..IIQP I II’WI; 9 .......... It? 9)...- 1 h IIIII ..)ia|?.ll.} sg‘zzi;l..cil.$€. I... 971.5}. I.!.$3\IIIIIZ . A fluviIIvesa .33....II:‘:.I§§3§I!ILIIO§II£-KIII IEPILIIIIiEIIIIIId‘Hfl. hitt'sgII—IBWII’ éulNRIIIIIIIy. .I. :52. III}: I I. lifiuk, . . Ilvghsa‘: .‘IIII ‘sazsluti Itiihbfibzitdfiai II ,. Iiiah.5.‘§§?fldw. I M5 II... MM!- Sr IIIIIIIIIIIII‘IIIIIIDE \I’IIII $1§u§u¥l§7un|§p . .. >IvIIVIXI..I€IIIIIZh..¢l§.II?Irv It? .tEIOI 09735.. .II... I237}! INII: .3515 7. I I} ”Rig; 2? I711 .II I‘liltfiegl 5b.. 9311:): If I: .I . . LI $2.15.!!!» .IZ.:r..!.IzyO Iii; I 3.7: tI:#::I I )tIIIIv?‘ 333-8)]. I’llfibglfr‘lh I 1" It! I. II!) Kill. :5 I DIG-DI I law-.132...“493.9“.din.finrfiwflflyfi’wvafiIggrgfifldflnfiu.’finmflvgaduui.i%n.§atn kg“: fiWIfldeeravfififlufivtdflhflhflufiasfifiiflflflbzi.isvtzfitu I Iflkluuyput..:«rh; TNIII? Iaihnflt‘mwuflrezfiifipn , I 3.13.3.2... I In §IIIllm Isvzbalild‘v I) In. II. [In U..I.5)I:IIII.IE.IOIY,|I?I£.IIII.!I$I...I.I:IS..§IJII ..I I... :IIIFIOII‘I. rIgIIIII I: .0! It“! I IIDQ‘OII 22:. III. I III] aflirvsv.hviaflxizalaal .fltgltv 31:31.; a IIIIJWtIiI’I}..Hv~Il-.>III.I "I..- III: 321‘,in .{Iil’tbfin:IzfiIlfiIIIPI-I:i~l.at)z:afl vista? I I)... 2'. IQIIIAIIIII‘III ’IIIVINMI} {(5.11} : IVII.I919:I!IO5III3IIIIII .illlglris: III-3.. Is... 1115....) I I IgitthzyaII :ILIIJSIII 1:2?3IDIIII 3.0.3:]? it I . I. It)... ’4" V gI II!!!) II!!II..\I I. IIIIIIISIIIII 1.3.1.11,{III-[IsiuhznrinI Il:_»vl.$?aa)l9 i‘!’t§25i:il)fizf .- I I glilifiu‘i. IIIIIIIMW‘IXI. 3}:ng I‘lligtg I II‘IIIIMHI’I‘IC: $05.45) flung}: .IIISI‘I sfivnidnvaxthfiiiv I... 9", «XIII-I'M“. :IIII; filIIIIt. Izlflaai11'. ala:?.a\bst§”hi.sixhn .IIIIFO . r i I IIIDI III . §.X{IIK"IIIII.JLHHS.‘.IISII‘¥ >II\ I: a) tIfixll.§I2 $I5IIDII_.I:IID.IFSISIIOIFIVI 9:505:3-9 . RIP! I . I: a h 5 VII£§.I§I I I. I afgIIIIOIIID a; ‘IIIII r Ii»:- biz). . IItatL loiikluiravri ....... Eilauafl Lvtllv. .[ 33.1%}; III): 1!. III-i I a lhzt III: I? Iglhuz‘gggt‘I I I1; 3.131352%.59ittl1. inset}... IIIIIIOIIIIrkuklin stunt-I IIIIIISIIIIIIIIC~,.IOIZ QIJIIEI-u‘ t Vaalvol3~5lx§ IIIII $)1§?‘I£3Ih§ )I‘IIII I,..>|I33\Hhr.I—§III! III vrvv.zn.ll.!.h.:rahh.h;.:vusaciIIi.2.5} v.5 a!.dtrvli:§kl.c Iiigfix , Izziisaifialvaflg W gIIIIII I:nIIIsIIiIIInDDIlIIIlI\¥III-IP$I..QI.I3II\{§IEIIII‘N: {2.73: ztitazI J. 30...,»st III); Erich.“ III... I) It) 5556‘ [:39 n I t“..b§$..§1l)II;I-I . £251.:- SI...‘ )8 II... III... :2. 14‘ IIIBII iiiiii DIIIigz‘sIflIII III: \ .IIIIIIIItl§~N:I.IIr§A. :I, III. :1 SverzI ax 4:32.).va II IIIIIII? it) 1.. it Eligibvyzthflr INIIEIIIIgdnfiIIIIIIIIObIII ii? it“sulglbht.‘ILCIIIQIIOIII.‘I‘I'IM“II 1. :dr; .1... .112. IIIZE.‘ IIEIIIIIII I’telzrbwu-I Kr? II ’dII 48;). III jIlIIIIOIIIIIlurvh III, IIIIIIIOIJIsI :II-jIIIIIQWWDflyfltt-t I‘IIIIE‘ICIIKIIIIII..:I$-I~K. z.- IIxhrér.IIIr.II3IIaco~: Iiaaanltzlsvta .IIIIQ).:..I$ 1:92: Finn‘s. its flirting}; .. [0‘25“ {105.34. :«zsIIIIII {.12.}. II: 15:04 15$ #1! :35: ...... QI1IIII5IIIIII I.) III. III-Dali?! 1 IIIIIIIII‘IFIIIIIII‘ I I I I; ‘zgizéniilg I§I\Ii§|~l§ all If}: «why-14:1 {Juflm‘nflpflflfi ”$3.... It»? Influx.hfl§sfliflijIII 1%IIIIHI‘IEIIINIEIHHW éIIHIIIIHIEIHflVQ‘IliiIOOIbI IIIIAHfinaijHNuI linuhII L. oghndt‘h 2.2.3.5... . V I A‘tttz?’ “a ’01. I Ii I; .5! Eigib 4 . 7300 III-x. 7"! It; in}! ii I if “is rI\I.lIs\I(.LDIIIv..th.I..I§ .5...“ :42.- Ivalv‘. wicks: Ialulavrataysazl : :IIIII. III? 1.01:... g It... III:$II2’5I\H§I|II§I I IIIIIII’IIII:I... IZILVIOE: bit?- vrtrtraiavt. thsnI:..-..Sa.§IIIII: III IIISIIBIIIIIIVIII: IelIIIIIx!‘ , II {SI III I: 61.5.; :ISIIIII £15.23}. .IIIIIIIJIIIIIISI II.) , III I5Q3$213§$~K3 33:). I y 1’33? 911:... z‘iiggaliie I it! I \ gig. 3| .5er III.) $13!:33 1.3222... .KIOIII hut l.- lszviigllih iii: 91. III?! LIFO) is fuil :Dh .IIzIIaIIIPII 111111 I a" I33w§511I I‘ll-I :I I: 1;: sauna. . III-i: alaliaaiIvIID anilvfitylzolbx‘r E. .E rtxxsfbltvilluzz; 7130a I..:ra)\<:w‘(. ‘1 iglgzllglslnalg:zg it III-49.43!»- a Igciggll-J’lz fézgtinIgbb gt LHII 11:21IIIIOLIIIIOII1 3)}?5? .IIrIII: iii-III? allxilvallrlvil. ..0$&.II..IIIII§II¢VI 34 0| Irraiiaa‘vu Iiitx ’III’QIIIIIIYII): IIIIIIIII 2.3m..I..:I!'I£.UVI‘IIIIGI%IIII‘ll. I ...:.....I>I.3Ir234IgnaIIIatIflhtdnuIIIIhIgHI gfi.‘§!¥o Iv. :Ik’ilgltiaal: §Ig§IEII§IIIIIQIII§I . II). ’12:..52‘00 3': rial-i. It I;hflll. II u aaa Q1 I Ii III: Ii )9 I I III . an!!! {I DIVIIASIOIIII ISIfiIQJIEIIJ EEIO V 3 .II‘I’tggg V: :3. 3“. 31‘}: .4) I. ii: {II}... liIIIIIOIlI I I I '5’:th {l3 E‘I‘i; .. %.!)\§§)I§‘I5§OIII§I .OIi I II I .3 433......via-VHS?!Lin-.ryqumHIHnrdutu‘mni liliyfiilngl, . Egg». 3.0!. on ...i...i..(.}5fi258..£3£h§2frvvgunfihiié. I WA} . .. 33: I: It: in... r , . git: III-II. 1. Egg, III. ii. E5 I « \Wgfiruan IRIIIIIIIIIIII..anan.afif)hv:hr.iu. >~II§IIIIIIRILIIIEII tinnitinsfiatgza Sig.§lii§w Izaoyschistmrixiglsiz I “unruflnrikraxlyii..:...1§§$.i~.stiia It.thitinfiio.Khtrflulflifitiihbsiiahrififlaahséjvol‘qlén; glanslnl.l.l§§'it$%§2¥£a .PlxleiniIItuils(ia§ .. = . .. I‘IIII i.5ibzlgil.$lfi)5w.u¢§ IIEIIZII Iggg rfizl’!‘ IIIOI IQIIIt-Iti...-IIIIEI.I.. ‘IIIIIII JIIIIQIFIYIJLSFIIHNVHF..2.2.2:.I: . . .iII..._..II!I.§ 23...? IiAufihaVaflbhgiI I: IItsoilutzitbe is! HEIDIIIIIIIIIIIII’RII! III-“3; I‘Lzelh.’ .1“..than n u x I in}; I I I Ivg1$35571739§ fligdagtz nigh-IIIII‘IKIIIItivhnhrthfi‘glixzhofivrI‘viflboIEKrII It}: I .AI.“’X ‘1 ii: is i); . ‘i'gggf t ‘. l4 - I . .‘IIIKIXIIDIK‘irI‘I’i fast-2",! .rtuu..g§ I. tit. 1X,IIIXIII.IO:I’II§‘IHI.III’I..Q itffi..tlatst trips... ...I .15..) rgisihlliti] Iaxgufib I. iii; Ed I...) I’lin: .IIQIIIRIIItzl’I‘!tI-OI OI‘IzO’OlI fi..7¢‘a.§lni§lglvppi filggz‘gflflznflfihqfig I {It} Jigttig)§lh!§§5f£t :g g g 3 ii; g‘ffg Er: 1E5..III)III)7VIIII‘I3$I5§§IHI)I Infatlav §:::z§r\f‘fip Ibis iIIIAIEIAII:A£II III)» igéigtglimlégtag . 1| I . 3:52;} EgfiivIIztgihilI‘hnf In it. 5| Eggbl-g . I . s tlfap Irv.-. igtihkz‘stfiIIlI-II It‘ltznt Iiasfi . 3591;! . . . r . . ilnigcnirl III... (I . .. KS 2 1 : . gagiinsx‘iét ?§$3I%§ ILIIIIIzIsszE. {Dali-II 3.43.5. . gs} . Iait. it... 2.130... JIHIIIP>IIII£IIIHMWDIZ§fiWIaRII 50:111.?vii).f:£§ra»2lfi)tflifillf.)§a£bfi05III“.at? . If: it. (IN-’IHIVII..LIIIUII($\\5OIVU.|I§I .1. 1.25.5!!! It» . .53415531925z all?! I). fl. H.005) §§§§ .II gig-{III . glaring: ...II¥IIIP.|I||£I3II! .fIDlIIa If} ”nu?! .llCflanfitaahIhihuiufigilIfi vylhflabgh.axn itiizbfiaaaoutitsz‘:a§£§y§: IalsaI.fa&~lr$-Ih.hbwa§ SIKIIIMIAEQ...M#IIIIH:IIR§flog!“Ll..6: .1 , .5: V I. an {a}! a» alvtfiolxraa I §£§.§§§ Is a. ,. . XI! {Itinixa .i I? 2.5?! .aIIIIIIIIIIItIEuriiiI litirjil’l‘nirI 3386.50!!! I‘E 1.197: (I it I; §;§\ 52.2:(1‘1! I/I 1.453).?!»3 ) I = .0. I” It)" mil-Elva“; , 3.35.5156}; It II’VII‘IVOBIISIIIEDI’IIE ll .. a) ‘ I 3.513 :3 S $.2gii? i £1 9: ”YR “flat, 2...... : 324%.; 252...)! it. kn. . 21.2. it. .9 at. ,. 1...; 9? ins; \- ‘ I f in; if I .. :IIL‘IIMIIPIIIII. IIIIr.S5§Ii§I—._.}IISI§J ts WIIiII I . . .545... . . §I§§1I| {IIEII gufsaei. .75.!15)!!>.~.11.1. .ariI‘IIIrJW’I).‘.-.I!D§a I35; . a EII)E.|I.A."I\5 I. . IPIILEIILLII s; «I .J V I . 4 - I I‘ in .. .1. ‘ _ I .. III.).5.XT.II$.Z.II$'E)D\ 41% I 2.)}... 3.x 4! I! . IIICN. . , , , €21t...t.z.r (gajf‘szft k»\t§.l.-i..z¥pbnlzfil§\tif 2.2. I Eifltuls‘nsztth’IIur . . . {Eggkfklie‘ . . .. {31:22:39zif3f! .5 {9... g ., 3.2.1.3.. $3sz§> lists)...“ 5%. teffgssstii-3t [CHIEE . r a. 4 git-.211. , it} E: El; 3&.:$¥$2§£§¢ITE;$§¢<¥§ u§£§£f§E15.§ . 5 align-81.0.1.5) II... :EIIIII; Igl‘gfigft ll 11., .3 ; .. §§.%§.h£§§t £23.33? . fifiyaiigéeii :ttgiéiillililffitygxz sit»! . z...&t2;..€..izfiz.:. .2. . 3...? Egtgggitzs £5... 3.5. 3i. 5.2.51.3 wag. . it...» I Z. IIiiigzgggé §g§§¢IE flIfni‘I. II-IWIJITILK I‘M-.51 \ l. I . >..)I;II..I>\..!..:II$II§ I I: 1 ..l.u;..(qa\JI.JI..)..w.xx.)....\ II). (‘2‘\.r.\rv ..\:..~\i..>l..;‘xh: I: I. 7.: .. t. ,. ..... v)? I 1:39....) I s. ’IIgIlgl 33E»... .17?I¥I:ul. :O? {purity-”31.5.5: ‘ 1‘ Pl F53: lei 1 "d - . tr of v). f l h‘ 41$ .L.‘ 9’ f . I a " . 2;" {7; fix. @ ”vii : I; 'l _ 3'5 5&1.) SF 3‘ " “a ‘g‘: :3! 3:. i w ‘3‘ k, s, ‘33 Eta 56g :8: i i ‘K _,__...... _ fl _ _______ “mm—“J This is to certify that the dissertation entitled STABILITY AND CONTROL OF NONLINEAR SINGULARLY PERTURBED STOCHASTIC SYSTEMS presented by Mohamed Gamal El-Ansary has been accepted towards fulfillment of the requirements for Ph.D. degreein Systems Science W Major 1' Date 1/6/1983 MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES \— RETURNING MATERIALS: ace in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped beiow. STABILITY AND CONTROL OF NONLINEAR SINGULARLY PERTURBED STOCHASTIC SYSTEMS BY Mohamed Gamal El—Ansary A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1983 ABSTRACT STABILITY AND CONTROL OF NONLINEAR SINGULARLY PERTURBED STOCHASTIC SYSTEMS BY Mohamed Gamal El—Ansary A class of nonlinear singularly perturbed systems driven by wide—band noise is considered. The probabilistic behavior of the slow variables is studied when the fast variables are sufficiently fast (represented by p 4 O) and the wide—band noise is sufficiently wide (represented by e 4 O). The possible interaction between the asymptotic phenomena associated with singular perturbations and the asymptotic phenomena associated with fast stochastic fluctuations, is also considered. The slow state which is, in general, not a Markov process, is shown to converge to a diffusion Markov process in the sense of weak convergence as e and H tend to zero and the ratio 5 tends to a nominal value yéE[Yl,m), where Y1 > O is arbitrary but fixed. This limiting process is the solution of a reduced- order diffusion model which is derived explicitly and the interaction between the two asymptotic phenomena described above, has turned out to be important, as it is revealed from the dependence of the reduced order model, in general, CIO) on Y which equals to lim 6,1440 Mohamed Gamal El—Ansary The advantages of having a reduced—order Markov model in hand, to approximate the slow states, are diSplayed by utilizing some of the available work on stability and stabilization of Markov process. Stability properties of the non—Markov slow states are studied through those of the reduced—order Markov states. Design of stabilizing feedback control strategies for the original system is based on well-established stabilization techniques of the reduced—order Markov model. This Dissertation is Dedicated to Hala El-Ansary, my wife for her love and understanding throughout the long years of being a student It is only through her patience that this work was completed. to Tarek, Noha and Sherief, my children for their love and support. ii ACKNOWLEDGMENTS It is a great pleasure to express my sincere appreciation to my major advisor, Dr. Hassan K. Khalil who introduced me to the fields of singular perturbation and stochastic systems. His patience, valuable guidance, expert advice, stimulating discussions, and useful insights made this work possible. I would also like to thank the committee members, Dr. Robert Schlueter, Dr. Robert O. Barr, Dr. David H.Y. Yen and Dr. Habib Salehi. These professors offered inspiration as teachers and continued encouragement throughout my graduate study at Michigan State University. My gratitude extends to Professor John Kreer, the Chairman of the Electrical Engineering and Systems Science Department for his encouragement and support. Lastly, I would like to thank my typist Tammy Hatfield for her excellent work in typing this manuscript. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS I. II. III. IV. LITERATURE SURVEY, BACKGROUND AND INTRODUCTION 1.1 Singular Perturbation Techniques and their Application to Control Systems 1.2 Asymptotic Analysis of Systems Driven by Wide —Band Noise 1.3 Stochastic singularly Perturbed Systems . 1.4 Objective of the Thesis REDUCED —ORDER MODEL AND CONVERGENCE RESULT 2.1 Introduction 2.2 Problem Formulation and Assumptions 2.3 The Convergence Theorem . . . Appendix A Appendix B Appendix C STABILITY 3.1 Introduction 3.2 Stochastic Asymptotic Stability 3.3 Mean Square Boundedness . 3.4 Examples . . . . STABILIZING CONTROL 4.1 Introduction 4.2 State Feedback Stabilizing Control 4 3 Output Feedback Stabilizing Control 4 4 Example DISCUSSION AND CONCLUSION 5.1 Discussion 5.2 Conclusion and Future Research BIBLIOGRAPHY iv 12 19 22 22 23 27 44 76 87 87 102 110 115 115 118 122 135 138 138 142 146 CHAPTER I LITERATURE SURVEY. BACKGROUND AND INTRODUCTION 1.1. Singular Perturbation Techniques and their Application to Control Systems. It is a common practice of control engineers to simplify mathematical models which represent physical systems under investigation. The singular perturbation approach outlined in this section provides tools for simplifications in control systems analysis and design. Accordingly a typical simplification is to neglect some small time constants, masses, momentscflfinertia, some parasitic capacitances and inductances, and a number of unimportant parameters. The presence of such parameters increases the dynamic order of the model and introduces fast modes which make the model stiff, that is, difficult to handle on a digital computer. Consider a dynamic system which is modeled by the following initial value problem: X r,- ll xO (1.1) g(x(t).y(t).u(t)) y(to) = y0 (1.2) f(x(t),y(t),u(t)) X(tO) "C <0 ; II where u is a small positive parameter representing para- sitic elements, x and y are n- and m-dimensional vectors, respectively, and u is an r-dimensional deterministic input vector. For n = O, the order n4-m of (1.1) and (1.2) reduces to n, that is (1.2) becomes o = g(§(t) .§(t) .E(t)) (1.3) Suppose that (1.3) has an isolated root along which 3% is nonsingular, Wt) = h(§(t) .E(t)> (1.4) Substituting (1.4) into (1.1) we obtain the reduced system x; = f(>_<(t) ,‘fi’(t)) 320:0) = x0 (1.5) Reducing the order (mi-n) of (1.1) and (1.2) to n of (1.5) is not the only advantage of (1.5). Another advantage can be realized when we notice that in (1.2) we actually have y = g/u, that is, if u is very small and g # 0, then y is increasing very rapidly. This explains, in a sense, what we mean by the stiffness of (1.1) and (1.2) which is eliminated from (1.5). To see the effect of this simplification procedure on the variable y, which has been excluded from the simplified model (1.5), we notice that 37 which is given by (1.4) starts at tO from y(t the original variable y which starts at t O) = h(x(to), u(tO)), 1n contrast to 0 from a pre— scribed value yO, where there may be a large discrepancy between yO and y(to). Thus the best that one can hope for is that y(t) is a good approximation to y(t) every- where except near t = t0 and that x(t) is a good approximation to x(t) everywhere. To study the behavior of y near t = t the time scale is stretched by of introducing the transformations T = (1.6) In terms of T, (1.1) and (1.2) becomes dx _ _ a;- - uf(x.y.u) x(o) — x0 (1.7) gl=g(xyu) y(0) = (l 8) dT I I yo 0 Setting H = O in (1.7) and (1.8) we get that x(T) = x0. Then (1.8) can be written in a more convenient form in terms of q = y-—y as 3—2 = g 33(t). (1.10) y(t) ———> Y(t) (1.11) We notice that, actually y(t) is approximated by y(t)+-n(T) for all t€E[tO,tf] but n(T) 4 O as u 4 o (i.e. T 4 w). The essential conditions are stability type conditions which are imposed on the boundary-layer system (1.9). The two time scale phenomena accompanied the solution of the initial value problem is at the heart of the singular perturbation approach to stability and control problems. In a typical control problem one starts by defining separate reduced and boundary—layer problems. Assuming the existence of solutions for these problems, an approximate solution is postulated by combining the separate solutions. The validity of the approximations as H 4 O is established via asymptotic analysis (cf. [1—31). In general if the singularly perturbed system, which is represented by (1.1) and (1.2) is asymptotically stable, the fast states represented by the vector y are important only during a short initial period. After that period they are negligible and the behavior of the system can be described by its slow states represented by x. In many applications the fast states y are basically parasetics, that is, for example the equation (1.2) can represent the model of an actuator in a control system which can be neglected. Neglecting the fast modes is equivalent to assuming that they are infinitely fast, that is letting H 4 O in (1.2). 1.2. Asymptotic Analysis of Systems Driven by Wide-Band Noise: In this section we study and review some of the work that has been done concerning dynamic systems with external influences which are approximately white noise (wide-band noise). In this thesis, our main concern will be the asymptotic analysis of a class of systems having the above property. Let us first introduce the basic topics and definitions that will be used and then we will review the work done which is related to our work. Itb's Stochastic Differential Equation: It is of the form dx= ffi¢fldt+G(LxhhMt) togth (122) x is a vector (the system state) in Rn, the vector-— valued function f(t,x) is usually called the drift coefficient, G(t,x) is an n xm. matrix—valued function and w is a Wiener process, (Brownian motion), usually taken to be Gaussian, in Euclidean m-space. Equation (1.12) was originally studied in [4,5] and later, under less restrictive conditions, in many text bodks [cf. 6-8]. Equation (1.12) is interpreted as a stochastic integral equation t t x(t) =x(tO) +f f[s,x(s) ]ds +j G[s,x(s) ]dw(s) (1.13) t t O O It is assumed that f and G are measurable in (t,x) for t.€[tO,T], xéERn; and satisfy (1) a growth condition lf(t,x)l4—IG(t,x)l g_K(1+-1x1), t€E[tO,T], n (1.14) XGER and (ii) a uniform Lipschitz condition ‘f(t,X) _f(tIY) ! + lG(tIX) —G(tIY)l g K‘X -Y‘I (1.15) tE [tO,T], x,y€Rn In (1.13) x(t ) is any (finite-valued) random vector 0 independent of the increments dw. Under these conditions (1.13) determines a unique stochastic Markov process x which is also called a diffusion process. For $62C2(Rn), the differential operator associated with the process x is defined by: £m(x) = f’ O and n2 > 0 there exists a o > 0 such that if 1x0) < a then: 6t (i) P[ix(t)l g nze , t 2.0] 2.1-nl for some 9 > 0. (ii) P[lim [x(t)] = o) = 1. t-OOO Definition of Asymptotic Stability in the Mean Square: The equilibrium solution is said to be asymptotically stable in mean square if there exists constants d > 0, K1 2_O and t) lngl+K25at Vt 2 o and then K2 > 0 such that E]x( the process x(t) is said to be exponentially bounded in mean square with exponent d. This form of the definition is stated in [18] and we are going to use it, as it is, later. Review of Related Work: The mathematical theory of stochastic differential equations is concerned almost exclusively with the study of Its equations and the associated Markov processes. This theory has found many useful applications and has become a powerful tool in the study of diffusion processes (cf. [7],[8]). However, many of its aspects are somewhat drastic idealizations of physical processes in the sense that the noise affecting the physical system is approximated by white noise which is not a physical process but an abstraction. This was the motivation for later work which led to modeling dynamic systems with external noise which are approximately white noise, by systems of ordinary differential equations with wide-band noise as input so that Makov process techniques can be used. Several powerful methods for doing this have been developed. The problem has been initiated by [19] and then developed more (cf. [ll-15]). In [19], the Langevin scalar equations: dxn(t) = m(xn(t))dt+o(xn(t))dyn(t) (1.17) has been considered as a mathematical representation of a physical model, where yn(t) 4 y(t) in the mean square sense as n 4 m, and y(t) is a scalar Brownian motion process. It has been shown that the solutions xn(t) of (1.17) converge to the diffusion process x(t) as n 4 m, in the mean square sense, where x(t) satisfies the Ito equation: lO dx = [m(x) +3120 (mg—f((x) ]dt+o(x)dy (1.18) This says that the Langevin equation cannot replaced by an Itb differential equation without realizing the necessity for the correction term, %o(x)g%(x) in (1.18). More work has been developed along that line. All the authors in [ll—15] have treated the problem of weak convergence of x€(-) to a diffusion where x€(-) is defined as the solutions of ordinary differential equations with wide—band random hand sides. More specifically the system that they have all considered is of the form: 6 d l e e e e 3:375:21 = €F O, y€(t) = y(t/ez) where y(t) has been taken to be, in general, a stationary process (other hypotheses has been introduced in those references). The process y€(t) is, in a sense (will be made precise in the next chapter), a wide—band noise, and the system (1.20) is a wideéband noise system. The parameter c > 0 measures departure from the white noise. Another interpretation for e is that it differentiates between the time scale of fluctuations of the coefficients and the solution. In [11] and [12] y(t) is considered to be a Markov process, ergodic, bounded and satisfies other assumptions so that under certain smoothness conditions on F and G, 11 (x€(t),y€(t)) are, an (nt-m), jointly Markov process. It has been proved, using partial differential equations and perturbation techniques, that x€(t) converges weakly to a diffusion process x(t), as e 4 O on [0,T] where T < m, but arbitrary. At this point, it seems to be interesting to make analogy between this asymptotic analysis that has been carried out in [11] and [12] and those of the deterministic singular perturbation. We notice that the solution x€(t) of (1.19) is not exactly a Markov process, but it can be considered as components of a higher-dimensional Markov 6 process, as was the case when (x€(t),y (t)) was treated as a jointly Markov process. Then approximating x€(t) by the Markov process x(t) explains, in a sense, that an order—reduction procedure has been taken place which is in analogy to the order-reduction that occurs in deterministic singular perturbation. In [13—15] the same system, which is roughly represented by (1.19) has been studied but with different assumptions on the process y(t). Semigroup techniques due to [10] and Martingale approach have been employed in [13] and [14,15] respectively, to prove that x€(t) converges weakly to the diffusion x(t) whose differential operator [cf. 13] takes the form: Af = EG’(x.y(s))£X?:(t) = a(x(t))+b(x(t))v€(t) (1.27) where v€(t) is exponentially correlated noise with correlation time c. It has been suggested that for suffi— ciently small e and u, x(t), the solution of (1.27) can be approximated by a diffusion process, defined by an Ito equation. Moreover, this diffusion process cannot be obtained as the asymptotic limit which results either by letting e 4 0 first then u 4 O or by letting. u 4 0 then 6 4 0, since two different limits are expected. In deriving the reduced-order model corresponding to (1.27), an intuitive reasoning has been employed. It has been assumed that over a time interval At which is very small with respect to the relaxation time of x(t) while very large with respect to u and e, the process x(t) will behave like a continuous Markov process. With that 17 assumption the following well known definitions, from the theory of Markov process [cf. 6,7], A(x) = lim B(X(t+AEZL_-X(t) / x(t) = x), (1.28) At4O 2 B(x) = lim n{[X(t+At) ”(tn / x(t) = x), (1.29) At At4O has been used to calculate the drift coefficient A(x) of the approximating diffusion and its diffusion coefficient B(x). In the calculations of the conditional moments given by (1.28) and (1.29), there has been no demand for finding an exact solution of (1.27), it has been enough to solve (1.27) on a small interval At satisfying 1.30 Trel >> At >> max(u.€) ( ) where, the relaxation time Trel of x(t) is defined by: da —1 Trel m1n(—a§(x)) Equation (1.27) has been integrated over the interval [t,t+—At] and after applying the basic assumption (1.30), the result of integration has been simplified to: >Mt+AU =xHfl+ahdtHAt (1.31) nt+At X t+At-T)/|.l b J j e‘ (x(T))v€(T)dT t t + 14H The integral on the right—hand side of (1.31) considers the correlation of b(X(T)) with v€(7). Since this 18 integral is not a stochastic integral, it has been considered as a Riemann integral. Then successive approximation has been used [cf. 32], with initial solution x0 = x(t), and Taylor series expansions around x have been employed. 0' With the aid of (1.30), only the terms of order At has been retained anui a second order approximation has been obtained. It has been claimed that higher order approx- imations have the same accuracy 0(At) that the second order one has. Finally it has been shown that the results of calculations of (1.28) and (1.29) are: A(x) = a(x) +fi4375 g—§(x)b(x)sm). (1.32) 2 B(X) = b (X)S(O): (1.33) Where S(w/€) is the spectrum of v6, so that the suggested reduced order-model corresponding to (1.27) has been represented by the following Itb equation: d§(t) = A(§(t))dt +./B('§£(t)) dw(t) (1.34) There has been no rigorous proof, in that paper, to validate that the process x(t), defined by (1.27), converges to the diffusion process x(t), defined by (1.34), as €,u 4 O in any stochastic sense. The remarkable feature about the suggested reduced— order model (1.35) is its dependence on the ratio 3, as it is apparent from (1.32). hinting to the interaction between the two asymptotic phenomena. 19 1.4. Objectives of the Thesis: Our main Objective, which has been motivated by [30], is to generalize the reduced-order model, that has been suggested by [30], to a wider class of singularly perturbed systems and to provide a rigorous proof of convergence of the slow states to the diffusion process defined by the reduced-order model and then, to explore the possible application of the reduced—order model in stability and control problems. In this thesis we consider the nonlinear singularly perturbed system: >'<(t) = al(x(t)) +A12(X(t))y(t) +131 (x(t))v€(t) (1.35) uSHt) = a21(x(t)) +A2y(t) + B2(x(t))v€(t) (1.36) where v€(t) is a wideéband zero-mean stationary vector process with correlation matrix 6 e I l T Eiv (t)v (t+T) ]= ENE) More assumptions will be imposed on the process v€ in the next chapter. This class of singularly perturbed systems is similar to the deterministic one studied in [2] from the vieWpoint of allowing nonlinearity in the slow variable x while assuming linear dependence on the fast variable y. We allow the input noise to be state dependent by letting the input matrices B1 and B2 be functions in x: we do not, however, allow them to be function in y since that will destroy the linearity in y which is very desirable feature as it is apparent from [2]. 20 The accomplishments reported in this thesis are summarized as follows: (a) The asymptotic behavior of the slow variables, defined by (1.35) and (1.36), has been studied when the fast variables are sufficiently fast (represented by u 4 O) and the wide—band noise is sufficiently wide (represented by e 4 O). A reduced—order model to represent the behavior of the slow variables has been derived. It has been shown that the slow variables converge weakly to the solution of this reduced—order model as e 4 O and H 4 0. However, our proof cover the two cases: 4 O as c 4 0, (i) (T) IT: (ii) a and u of the same order, i.e., there exists positive constants K1 and K2 such that o < K1 3 E g K2 < w. The third case, namely: (iii) $40 as 1140 Follows essentially as a special case of [33] after applying results of [12] or [13]. This case is briefly discussed in chapter 2. The proof adapts a martingale method developed by [14] for proving weak convergence of a sequence of non—markovian processes to a diffusion process. (b) (C) 21 The use of the reduced-order model in stability of the full-order system, given by (1.35) and (1.36), has been examined. A result has been obtained which provide stochastic asymptotic stability of the origin of the full system if the origin of the reduced—order model is so, provided that the parameters 6 and u are sufficiently small. The main advantage of using the reduced—order model is that it is a Markov model, and the theory of stochastic stability [cf 16,20] applied to stochastic differential equations of Itb type is rich. Applying the reduced—order model in control problems has been considered. A stablizing output feedback control has been designed, using a nonlinear Observer, for the reduced—order model. We have been motivated by the work of [18], in which a stabilizing feedback control for a system represented by an Itb equation has been designed using an observer. The designed control law has been implemented to the full—order system, with an observer, and conditions, under which the closed loop system is stable, have been spelled out via the use of our stability result. CHAPTER II REDUCED-ORDER MODEL AND CONVERGENCE RESULT 2.1. Introduction. This chapter is concerned with sutyding the asymptotic behavior of the slow variables of a singularly perturbed system driven by wideéband noise, when the fast dynamics are too fast, represented by u 4 O, and the wide—band noise is too wide, represented by 6 4 O. A reduced- order diffusion model that approximates the behavior of the slow variables is derived together with a rigorous proof of convergence. Our proof covers the two cases u/e 4 O as e 4 O and 6 and u being of the same order of magnitude, i.e, Kl g_% g_K2 for some positive constants K1 and K2. It is also shown that the case S 4 O as u 4 O, which is not covered by our proof, can be deduced from results already available in the literature. This chapter is arranged in the following way. In the second section we introduce the singularly perturbed model and list all the assumptions that are needed for the convergence proof. In section 3, the basic theorem is stated and proved. To make the proof more readable some lengthy details which are not very essential to follow the 22 23 logic of the proof, have been given in separate appendices at the end of the chapter. 2.2. Problem Formulation and Assumptions: Consider the singularly perturbed system i(t) = al(x(t))-+A12(X(t))y(t)4—Bl(x(t))v€(t), (2.1) x(0) = x0 49(t) = a21(X(t))-+A2y(t)4-B2(X(t))v€(t). (2.2) n where x€ER , yéiRm and are bounded random vectors. Xo’yo The stochastic process veéERr is defined as v (t) = -%: v(t/e) (2.3) v'EI where v(t) satisfies (Al) v(t) is a stationary, zero mean, right continuous, uniformly bounded process on [0,“). The o—algebras induced by v(t) are assumed to have a mixing property with an exponential mixing rate [9], i.e., sup |P(A2/Al) -P(A2)] g e_afr A.,t 1 for some a > 0, where A.l €o[v(s), s g t] and A. Eo[v(s), s 2_t-tT]. The process v€(t) is said to 2 24 be wide-band noise since its power spectral density matrix S€(w) = S(w/e) will have a frequency band of wO/e when S(w), the spectral matrix of v, has a frequency band wo. Indeed, the process v€(t) converges to Gaussian white noise by the central limit theorem [11]. The coefficients of (2.1) and (2.2) are assumed to satisfy (A2) The coeff1Cients al, a21, A12, B1 and B2 are continuous in x and have continuous partial derivatives up to the second order which are bounded uniformly in x. Moreover, a21 and B2 are bounded uniformly in x. (A3) The constant matrix A2 is Hurwitz, i.e. Rex(A2)<(O. (A4) The vector al(x) and the matrices A12(x) and Bl(x) are required to satisfy 1al(x)1+1A12(X)1+1Bl(x)1gK(1 +1x1) Vx ERn (2.4) and the vector aO(x) and the matrix B (x) which are O defined by: a = a -—A A_la (2 5) O 1 l2 2 21 ' and —l B =13 —A A B (2.6) 0 1 12 2 2 25 are required to satisfy 1a0(x) -aO(z)1+1BO(x) —BO(z)1gK1x—z1 ]ix,zERn for some positive constants K. We notice that from (A2) and (A4) growth conditions similar to (2.4) will be satisfied for a0 and B0' (A3) is needed to guarantee the asymptotic stability of the boundary layer phenomena associated with y. Under the assumptions (A2)-(A4), the usual existence and uniqueness theory for ordinary differential equations gives us a solution for (2.1) and (2.2) on [0,T] for sufficiently small M and for each sample path of v(-). This follows by minor modification of the technique used in proving the basic result of [34]. Our objective is to study the asymptotic behavior of x(-) as e 4 O and u 4 O. The main result of this chapter shows that x(o) converges weakly to a diffusion process §(-). The infinitismal generator associated with §( ), whose form will follow from the prOof of the result, is given by n LYf(X) = Z3 b.(x) —— (x) =1 1 (2.7) 26 U 3“, n aO(x) + hl( I — BO(X)S(O)BO(X) as 5, I [aij(x)1, S(w) is the spectral matrix of v, _ ’ I I -l l h1i — tr[DiBOW +DiA12A 22] , _ I I I —l h2i — tr[EiBOW -tEiA12A 22], h = tr[—F’B W'B'(A’)-l —F’B .V-.’(A’)'l 3i i 0 2 2 i 0" 2 I —1 + FiA12A 2P], ___ ., l . I____I ., Di [inii : vaiz: :Vx 1r] ' nxr B = [V 2] I l l] nxr E1 = [Vxfl 1 : VXni23_——_lvxnir] 7 nxr B = [h ] . 2 1] mxr F1 = [V gil : Vx§i2:_—_-:ngim] ' nxm A _ [5 l . 12 13 nxm l(I) denotes transposition. x).—A12(x)A'§h2(x)-+h3(x), (2 .9) .10) .11) .12) .13) .14) .15) 27 O a AZYT ’ Z = I e BZR (T)dT, for some v€E[y ,w), O l Y1.> O (2.16) an A2k I I AZIX and p = 1‘ e (1322 + 2132)e d1. (2.17) O We require that the coefficients A(x) and b(x), defined above, satisfy the following conditions (A5) 1A(x)1 gC(1+lx12), xeRn (A6) gC(1+1x12), XERn where C is some positive constant. These two conditions in addition to (A2) guarantee that the martingale problem corresponding to (2.7) is well-posed [8]. 2.3. The Convergence Theorem: Theorem: Under the assumptions (Al)—(A6), _x(') converges weakly to '§(-) as e 4 o, u 4 o and E 4 y where yéE[Yl,°), Y1 > O is arbitrary, but fixed. Proof: We utilize a technique for proving weak convergence of a sequence of non—Markovian processes to a diffusion process which was introduced in [10] and further developed in [13-15]. The version used here is _ ... art—v." “ll-1'11... ..I- «uteri. ... ..l.. _l . I 28 due to [14]. The main step in the proof is finding a sequence of test functions fe'“(t) for a given function f(x) such that certain conditions are satisfied. We use the so called perturbed test function which was used for similar purposes in [12-15]. Before we get to the technical details of the proof we need to introduce some definitions and terminology. Truncated Processes: For every positive integer N, let SN = [xéiRn, HXH g_N] and define the truncated process xe’;(t) to be the solution of °€.H_ 6.11 6.11 (2,11 6.11 x N—qN(X N)[a‘1(X N)+A12(X N)y N (2.18) 6.11 € €.u _ -€.u_ 6.11 6.11 6.11 6 MY N— [a21(x N)+A2y +lex N)v ]. (2.19) 6. y §(O) = y0 __ _ n_ where qN(x) — l for xéESN, qN(x) — O for XGER SN+l and qN(x)€E[O,l] and has third derivatives that are bounded uniformly in x and N. For each N, {x€';(-)] is bounded uniformly in u and e. As it will be seen the actual technical proof involves only the truncated processes {x€'§(°)]. See [14,15] for similar treatment. Terminology: Let (Q,P,J) be the probability space in which v(°) is defined and let JE': be the o-algebra 29 induced by [x€’;(s), ye';(s), v€(s), O g_s g_t] and EE’; the corresponding conditional expectation. Let £9 be the class of measurable (w,t) real valued functions such that if f(°) 0, such that p—lim [f€';(t) —f(x€';(t))1 = o, (2.20) e,u40 e/u-W p-lim [A€';f€’;(t) -L§f(x€'§(t))] = o, (2.21) 6.1140 e/H4Y P[sup1f€';(t) —t(x€'§(t))1 2 n) 4 o as KT (2.22) 6.11 4 O. 6/14 4 Y sup 1A€';f€’§(t)1 gMe'fIfm), (2.23) th and sup P[Mt’,LIl.(f) 2 K) 4 o as K 4 m (2.24) c,u then [x(-)] converges weakly to '§(.) as e 4 O and u 4 O and e/b 4 v. For notational convenience we write x(t), y(t), Ae'“, LY, fi(t) and Et instead of x€'§(t), ye';(t), Ae':, Lg, f:’§(t) and E:’§ respectively but we are always working with the truncated process [x€’;(-)]. Moreover, we omit the qN terms for further simplification. Now we proceed with the proof of the A theorem. Let féEGg be given, then Ae'“i(x(t)) = if (x(t))[a (x(t))+A (X(t)) (t) a 1 12 Y (2 ) - .25 + Bl(X(t))V€(t)] 31 we observe that y(t), the solution of (2.2) is given by: AZt/Ll 1 t A2 (t — “/11 Y(t) = e Yol'fi o e 321(X(T))dT t A (t-T)/u + 3:1” e 2 B2(X('r))V(T/€)d'r c 0 Now, since a21 and B2 are bounded uniformly in x and v(t/e) is uniformly bounded on [0,”), we have: -OI t/H t —d (t-T)/u 2 2 2 lY(t)l§K1e ‘YO‘+F I0 e d7 K3 t -az(t-'r)/u + -—-:-_f e dT dye 0 _ 'E :Kl+.—_2_ V/e Bf Then, by the compact support of 5;, the last two terms on the right hand side of (2.25) are of order lA/E and cannot be part of the operator LY, so they are averaged out by defining fl(x,t) as: f (x t) =1 E (x)E [A (x) (A(t+s x) +A—la ()) 1 ' o ax t 12 Y ' 2 21 X e (2.26) + Bl(x)v (t+s)] where A Azs/u Azs/u —1 y(t+s,x) = e y(t) + (e -I)A 2a21(x) l t+s A2(t+s-T)/u € (2‘27) + [If e B (X)V (T)dT t 2 32 Subtracting the term -A_§a21(x) in (2.26), in a /\ sense, centers y at its steady state mean. Setting X = x(t) in (2.26) and defining fl(t) = f1(x(t),t) we claim that lfl(t)l S K:1_\/E'*'K2\/J (2.28) where K1 and K2 are positive constants independent of T and w. Proof of the claim: From (2.26) and (2.27) we have: w éfi AZS/H f (X(t).t) = f (X(t))Et[A12(X(t))e (y(t) OOX + §A12(x(t)) i0 e 132(x(t))v€(t+1)d1 + B1(x(t))v€(t+s)]ds then by using the bound on [y(t)] we have: °° at “125/11 — E2 [fl(x(t),t)ng1 [a—X(x(t))l[lA12(x(t))le (Kl+-_) O V/€ 1A12(x(t))1 s -02(S->\)/l-l e + ___—11 10 e 1B2(x(t))1IEtv (t+11d11ds = I +1 33 Then, from (A2), (A4), the boundedness of the truncated process x(t) and the compact support of Bf we have Bx after integrating with respect to 5: ~ ~ 11 ~ +3” GI -— ~— 1Il1 g Klu + K6? 3 K111 K2./ E V11 g K2¢u where we used that E is bounded. The mixing property implies lEtV€\ = K I” e-CIX/EZ dk = KlyE 1f1(x(t),t)1 g K1VE+ KZV/LI, [which proves (2.28) . We next show that fl(t) ED(A€’H). We have [13] 34 6.11 _. A f1(t) p 11m [Etf (x(t+ 6) ,t+ 6) —fl(X(t) .t)]/6 640 1 p—lim [Etfl(x(t+ 6) ,t+ 6) -fl(x(t) ,t+ 6) 1/6 640 + p—lim [E f (x(t).t+ 6) -f (x(tLtH/é 640 t 1 1 (2.29) if the limits exist and are in £0. We first show that the second limit exists and is in £0. From (2.26), fl(x(t),t) can be written in the form: 0 fl(x(t),t) = 10 Etgl(x(t),t+s)ds where gl(x(t),t+-s) is equal to the integrand in the right—hand side of (2.26). So the second term of (2.29) is 12 = p-Zéim [Etfl(X(t) .t‘l’ 5) -fl(X(t) .t)]/5 4O m = p_§i$ [Et 10 Et+6gl(x(t),t4-64-s)ds co - I Etgl(x(t),t-ts)ds]/o O Setting u = 6-ts, we get 35 ca 12 = p-lim [I6 Etgl(x(t),t+u)du-—fO E (X(t).t4-S)ds]/6 640 tgl 6 . 1 ._ 13-151,“ 510 Etgl(x(t) ,t+s)ds - -g1(x(t),t+s) l = _%§(X(t))[A12(x(t))y(t)-+A (x(t))A-2a21(X(t)) 12 + Bl(x(t))v€(t)]. Therefore, the second limit exists. By the compact support of g; and the right continuity of v€(t) it is obvious that E1g1(x(t),t-ts) —gl(x(t),t)1 4 O as s 4 0+ and that sup E1gl(x(t),t)1 < m and this implies t that the second limit in (2.29) belongs to i . For the O first limit we have: p-lim [Etfl(x(t+ (5) ,t+ 6) —fl(x(t) .t+ 5) ]/6 640 1 f6 Bfl = p—lim — E [— (x(t+u),t+ 6) (a (x(t+u)) 5‘0 6 O “t 5x 1 + A12(x(t+u))y(t+u)+Bl(x(t+u))v€(t+u))]du Bf = 75%(x(t),t)(al(x(t))-+A12(x(t))y(t)i-Bl(x(t))v€(t)) which shows that the limit exists. See [13] for a similar treatment. Now by an argument similar to the one that has been used to show that the second limit is in £ we 0’ can show that the first limit also is in £0. We conclude 36 that f1(t) €D(A€’H). Then, from (2.29) and the above limits we have, (with x(t) = x) €.H _ _§£ -l A fl(t) — 5X (X)[A12(X)Y(t)-+A12(X)A 2a21(X) a + Bl(x)v€(t)] + 3% (x.t) [a1(X) + A12(x)y(t) +Bl(x)v€(t)] (2.30) Adding (2.25) to (2.30) we get 6'“ f f (t)) — 5f () afl A ( (X) + 1 — '3? x aO(X) + a—X (X.t) [al(X) (2.31) + A12 (X)y (t) + B1 (x)v€ (t) 1 The last two terms of (2.31) cannot be part of the operator LY, so we average them out by defining £2, for every XERn and tE[O,T] as: w afl A f2(x,t) =10 [Et E (x,t+s)(A12(x)y(t+s,x) + A12 (x)A-%a21(x) + B1 (x)ve (t + s)) at (e/u) + 5; (X)aO(X) —L f(x)]ds (2.32) (6/11) - . The form of L , as defined by (2.5—2.17) With e/u replacing Y, results as a by—product of showing that 37 1f2(x(t),t)1 is 0(u4-e), i.e.. by identifying the parts of the first three terms on the right—hand side of (2.32) which are not 0(6) or 0(u). Using that fX and fXX have compact support and the mixing property (2.4), and (A2)-(A4), it is shown in Appendix B that: 1f2(t)1 g K364-K4H (2.33) where, f2(t) = f2(x(t),t) and K and K are positive 3 4 constants independent of T and w. Following the same steps, we used to show that fl 0 such that: hfefl”fm)—LYfm)1gch§—y1 (24m Now, we verify condition (2.21) of lemma (1) as follows: 1A€IHf€'H(t) —LYf(x(t))1 g_1A€'“f€'“(t) -L(€/“)f(x(t))1 + 1L(€/“)f(x(t)) —LYf(x(t))] (2.41) 40 But, from (2.35), (2.36) and by applying (2.37)—(2.39) we get: Of 1Ae'ufe'u(t)-L(€/u)f(X(t))131-63%- (x(t),t)ao(x(t))1 0f + 16_X2 (x(t) ,t)a1(X(t))l Of + 13,—X2 (X(t).t)(A12(X(t))Y(t) +B1(X(t))v€(t))1 g ED/E +E2\/1T+ E3e+R4u (2.42) where all Ki > O and independent of T and w. Then, from (2.40) and (2.42) and by taking expectation, we get: €.H c,u v —— ,— ._ _ _ _ ElA f (t) —L f(x(t))1 g Kl\/€+K2\/11+K3€+K4LJ + CIE - Y1 4 O as e,u 4 O and E 4 y, and since this expected value is finite for all t, condition (2.21) of lemma (1) is verified. From (2.36), (2.28) and (2.33) it is obvious that: (sup 1f€'“(t) —f(x 0 it follows from [12] or [13] that if the coefficients of (2.1) and (2.2) and the process v(t) satisfy the appropriate assumptions then the corresponding solutions of those equations converge weakly as e 4 O to the solutions of the singularly pertubed Ito model: (See Appendix C for derivations). dx = [31(x) +A12(x)y]dt+Bl(x)\/de (2.43) udy = (3210;) +A2y]dt+B2(x)\/mdw (2.44) where 31(x) = al(x) +31 (x), (2.45) _21(x) = a21(x) +h2(x), (2.46) 42 and h" . = tr[Ei’B w’], (2.48) 21 1 and Di' Ei and W are as defined in Section 2.2. The system (2.43) and (2.44) is a special case of the system studied in [33] (See (2.1.1) of [33]). Under the assumptions of [33] the process X(-) converges weakly °(-) to the diffusion process X with differential operator L given by (See Appendix C for details) n n 2 _ — 0f 1 8 f(X) Lf(x) — ‘_ b.1(x) 8—357”) +9: . 21 aij(x) 8x.8x. (2.49) 1—1 1 1,3—1 1 j where B(x) = 51(x) -A12(x)A‘§321(x)+h‘3(x), (2.50) A(X) = [aij(x)l = BO(X)S(O)B6(X). (2.51) ii (x) = tr[—F.’B 8(0)13’(A’)‘l —F’A 15(A’)’1] (2 52) 3 1 l 2 2 i 12 2 ° and P satisfies PAz-tAZP = -B28(O)B2 (2.53) It is interesting to notice that the reduced order model corresponding to the operator L in (2.49) can be obtained Y from L in our reduced-order model (2.7) by letting 43 Y 4 O (or E 4 0). So, generally speaking, we can say that the operator LY of (2.7) gives the right form of the reduced—order model for all values of v, i.e. Y E [0,”) . Remark 2.2: There are special cases where the infinitismal generator associated with E, given by (2.7), is independent of the parameter Y. Such special cases and their significance will be discussed in Chapter 5. Here we would like to point out that in such cases the convergence theorem reduces to the statement: "under the assumptions (Al)—(A6), X(') converges weakly to §(-) as e 4 O and u 4 0, provided that E 2 Y1 > O. APPENDIX A To verify inequality (2.37), let us consider f x,t) l( as defined by (2.26) and (2.27). So, we have: fl(x,t) = 1: g; bd.A12(x)eAZS/pds(y(t)-+A—§a21(x)) + g§(X)A12(X)& I: :+S eA21ds+-0(e>+-o (B—l) so that, if we define f2(x,t) by (2.32), (2.33) will be satisfied. Let I = the left hand—side of (8—1), and let /\ .. g(x,t+—s) = A12(X)y(t+-s,X)-+A12(X)A :a21(x) 6 (3-2) -+ Bl(x)v (t-+s) w afl I = $0 Et[-a—x_ (x,t+s) -g(X,t+S)]dS ° (B-3) no I = ID Et[g (x,t-+s)oVXfl(X.t-+S)]d5 48 49 _5_ I . where VX — (ax .3;—,‘°-,5§—). From (A-3), the inner l 2 n f x product g' -V can be expressed as: l g’(X,t-+s) ~vxf (x,t-rs) l E) E) 52f(x) A = [—Lig.(x,t+s) -—-—— §. (X)c1 (y (t+s,x) i,j=l k,L=l 1 BXiBXj gt Ik k m Bg- A 5f 31 + \EEI GkV§V(X)) ‘Hgi(X:t+S) ag(x)§§:(x)azk(yk(t+sox) + g a Q (X)) -—ng (X t+s) 9—12-02) 5. (x)cx V21 kV V i I Bj ji Lk m BQ ' E --—v (X)] v=l qu axi + 2% i [g.(xt+s) 62f”) e (x) foo E vemdx i,j=l k=1 1 axiéxj 3k t+s t+s k + g (x.t+s> 83% ) figIll-fix) fa Et+ vfiumx] X3 ax. t+s S 1 03—4) Let us define _ /\ _ y(t+s,x) = y(t+s,x) +A éa21(x) (B—S) Then, it can be seen easily that the first term of (B-4) takes the form: 50 I -l'— - pg (x,t4-s)fXX(X)A12(X)A 2y(t+s,x) (B—6) where = (ax 2f( 3x).) . The second term of 3 nxn <2: 5 1— (34)=-w - OQgW(xt+S) §»(M(N'yW+snd) j=l L=l ij X 3; 2 2 —l- . th where (A 2y(t4—s,x))£ is the 2 component of the m-vector A_%y(t-+s,x). Then, summing over i and using (2.15) we get: The second term of (8—4) n Bf I -l— = Z —ua— g (x,t+s)F.(x)A 2y(t+s,x) (B—7) j=1 Xj 3 where Fj(x) is defined by (2.15). The third term of (B-4) = -ug'(x,t4-s)(a21(x))é(Aé)_2A A12(X)VX f(X ) (B-8) The fourth term of (B—4) can be reduced to I E: g (x,t-+S)fXX(X)BO(x) f Et+sv (X)dx (B—9) t+s ae.k For the last term of (B-4), we need to look at T§%T (x) 1 Since BO(X) = Bl(x).—A12(x)A‘§BZ(x) = (ejk(x))nxm it is seen from (2.13)—(2.15) that m (X) = $- (X)- Z) Z} §j £(X)G n (x ), and therefore 3k 1: 1 q=l iq qk 51 66 aw. 5% ‘a‘l’km TEE”) ‘ 2 5x! “L n k(X) X i £,q=l 1 q q m an — Z s (X)0 k(X) £,q=l z I 8x. Denoting the last term of (B-4) by I4 we have: n r of aw,k w 6 I4 = Z Z [gi(x,t+s) 5X—-(X) 45x (X) I Vk(>‘)d)‘ i,j=l k=l j i t+s g M agje ._ _ gi(x,t+s) 5327“!) 5X. (x) op nqk(x) Po --1 J 1 q j Et+sv£(1)dk t+s m 5f an k °° e - 23— g (x,t+s) ax.(x) up J—ax. (x) J“ Et+svk(>.)dx] p,q—l j q i t+s = I4l+I42+I43 (8-10) 513 i of I J.” 6 I = (X) g (x,t+s)V 1). (x) E v (de 41 j=l k=l ij x 3k t+s t+s k but from (2.13) and by summing over k we get g 5:? , f e I ——(x) g (x,t+s)D.(x) E v (de (B—ll) 41 j=1 ij j t+s t+s where Dj(x) is defined by (2.13) n m 5f 55. -1 I42 = ‘. ;_ § 53:7(X)9i(x't+s) ax. (X) (A 2B2(X) 1,3—1 p—l 1 co 6 E v (Mdk) ft+s t+s p 52 Q where (A_%B2(x) f v€(x)d1)£ is the Lth component of t+s m ' —1 e . the m-vector A 2B2(x) f Et+sv (1)d1. Then by summing t+s over i and then over p, and using (2.15) we get: n on of I -l e I = -Z ——-(x)g (x,t+s)F.(X)A B (X) I E V (Mdl 42 j=1 ij j 2 2 t+s t+s (B—12) 2% g 5f I I = - —(x)§. (x)c1 (g (x,t+s)E . a . 43 3:1 p,q=l X3 JP pq q f Et+sv€(x)dk) t+s where Eq is given by (2.14). n 5f —1 I43 = .331 SEER) (A12(X)A 2w(x,t+s))j (B—l3) where w(x,t4—s) is a vector whose ith component is given by Q g'(x.t+s)E. f v€()\)d). (B—14) l t+s Then, from (B—3), (B—4) and (B-6)—(B—13), we have: H II IO Et[g’(x,t+ 8) ° fol(x,t+s) ]ds f0 Et[-ug’(x,t+-s)fxx(x)A12(x)A‘%§(t+-5.x) of I —l— H 53—;(X)g (x,t+S)Fj(X)A 211(1‘—+s,x) j I w: j 53 - ug' (x.t + s) (a21(x> );((A2’> ‘2A1’2(x)vxf (x) m I e + g (x.t+—s)fxx(x)BO(x) (t+s Et+sv (1)dx E: af I” e + —(x)g’(x,t+S)D.(X) E V ()x)d)\ j=l axj 3 t+s t+s 1‘ 5f I -l m e - 1? 5—7(x)g (x,t+s)Fj(X)A 2B2(X) I Et+sv (de ]_]_ j t+s n af -1 _ jg: 5;;(x)(A12(x)A 2w(x,t+s))jds (B-lS) From (B—2) and (8-5) we have: g(x,t+s) = A (X)§7(t+s,x) +Bl(X)v€(t+s) 12 Substituting 9 into (B—15) and then after simple manipulation we have: Q I = f [—utr(£X -l O 2 (x)A12(x)A Etmt + s.x)§’ (t + s.x> >A’ x 12 - utr (fXX(X)A (x)A‘§Et(§(t + s.x>v’€ (t + s))B‘l’ 12 n of I — -I ‘1 I _ utr(j§1 5;;(X)Fj(x)A12Et(y(t4-s,x)y (t-ks,x))(A 2) ) n af 6 -—, —1 , —Lmr(2 . anHxnsme(v(t+my W+snd)m ) ) j=l oXj j l t 2 54 .Bl (x)) £(x)Dj’ (x)A12 (x) (t Et37(t + stxw' 6( axj +s tr( 1)d1 .Wv of 8. X3 E v€(t+ s)v’€(>.)d>.)‘ tr( t 12W (X)Df (x)B (x) 3 l ft+s 5f I an S;§(X)Fj(X)A12(X) [t+s Ety(t4-s,x)v tr( JEN” B§(A‘§>’> n on Bf I 8 I6 - tr( 2 ——(X)F.(X)B (X) E v (t+s)v d 3:1 axj 3 1 ft+s t (x) 1 ---7 Et(A12(X)A_:w(x,t+ s) )j — €( I (x)Ety(t+ s,x) + B1(X)EtV t+ s)] >’(A’>‘2 (a21 x 2 fx(XHdS I (B—l6) 55 °° I Now let us consider f Ety(t+s,x)y (t+s,x)ds, where O 1 we set {f(t) = y(t) +A-2a21(x) on I E 37(t+ s,x)y'(t+ s,x)ds O t °° A s/LJ_ —/ A’s/Ll = f EtIe 2 y(t)y (t>e 2 O s A (s—fl/u __ A'S/u + if e 2 B (X)V€(t+T)y’(t)e 2 GT H o 2 A s/u s A'(s—1)/u + -l- e 2 y(t)f V'€(t+X)B'(X)e 2 d1 (1 o 2 s s A (s -T)/).l c + %J‘ J" e 2 B2(x)v€(t+T)v'V(t+1) u 0 A'(S-X)/u , Bé(x)e 2 dldk]ds (B-l7) From the estimate on y(t) and that a21(x) is bounded, we get °° A s/u_ __ A’s/u , K ‘2 °° —2o s/u (f e 2 y(t)y’(t)e 2 ds)gKKl+—_2_) J e 2 dsglf-ggK' 0 \/€ 0 (B-18) and for the second term in (B-17) we have, (notice that B is bounded) 2 56 I 1 w s .A2(S-T)/H e -—I .A2s/u ‘— I I e B (x)E v (t-tT)dTy (t)e ds] *1 o o 2 t m s -d (Zs-T)/u w -d s/u —2d s/u g,3% I I e 2 des = E I [e 2 -—e 2 ]ds “ o o o S. K?“ 3 K (B—l9) e K H We used that (v (t+T)‘ g —: and that E 3 K. \/€ From (B—17) and (B—l6), the significant term of I1 is: 0° 5 s A (S—T)/).l u -l 2 L = — f trf (X)A A f I e 1 :2 0 xx 12 2 o o e 16: I AZWS-M/Ll I 82(X)Et(v (t+ T)V (t+ 1))B2(X)e A12(x)d'rd)\ds (B—ZO) Subtracting and adding to (B—20), a term equal to the one that appears on its right hand side but with E(v€(t+T)v’€(t+i)) replacing E (v€(t+T)v’€(t+i)), t Ll can be written as: L — 1 mfg [St (E €(t+ ) '€(t+)\) E €(t+ ) 'et 1—11‘fooortv TV ”V TV(+1)) , A§\ds (B—21) =L +L ll 12 57 From the mixing property (2.4), it follows [c.f. 36] that IEtv€(T)v€(1)—Ev€(T)V€(1)) g E éO‘WtVE (is-22> for every 0 g t g T g 1. Then from (B—22),(A4) the bounded truncation state x(t) = x and the compact support of fXX(x), we have: s s X —a (S-x)/u —G (s—T)/u K —a(1 e) 2 2 )Lnlififofofoe / e e dede ° 8 1’? -0 (S-M/Ll —0 (S-T)/l~l + l: f I J éG(T/€) e 2 2 e dXdes Lie 0 (B—23) changing the order of integration in (B—23). Then, for example, the first term in the right hand side becomes: K I” K °° 411/6 —02(s—1)/u -a2(s—'r)/u E? I f e e e dsdex O O )\ 0‘2 d w - ——4—— A d = ZoFH~ I fl G(Li e) e zT/Hdek 2““ o o a d1 2 d , ... —— —(—+—)1 =1-‘e—“J (ee-e“ 6 )dxsKe Similar estimate can be obtained for the second term of (B—23). Then, we have: (we used that u/e g Kl) [L11] 3 Ke (B—24) 58 Now we consider w s 1 A (S-T)/H _ _J-_ -l 2 I X-T L12 - we (0 I0 Io terX(x)A12(x)A 2e B2(X)R ( e ) I A2'(s—>\)/H I B2(x)e A12(x)dede l m S T _l A2(S-T)/H x_T - JE Io I0 (0 terX(x)A12(X)A 2e B2(X)R( ) , Age-m , B2(x)e A12(x)d1des = T1+T2 where, the correlation matrix R(T) = E(v(t4-T)v’(t)) satisfies, (also follows from (2.4) and [36]): T (Rm) 3 K50 (B—26) setting 1,—T = w in T1 and changing the order of integrabon, we get: w s s A (s+w—X)/u _ l -l 2 I w Tl — ”HE (0 IO fw terX(x)A12(x)A 2e B2(x)R (E) , Age-w , B2(x)e A12(x)d1dwds Integrating by parts just once, gives 59 l m S -l A2w/Ll I w T = E f0 f0 trfxx(x)A12(x)A 2e B2(X)R (E) I—lI 2) A12 I 32 (x) (A (X) dwds w s _ A S/H — % f0 Io terX(x)A12(x)A %e 2 B2(X)R’(§) I A2(S-W)/LlL I -l I B2(x)e (A2) A12(x)dwds A2(s+w—x)/u 1 an S S I W + J? f0 f0 jw terX(x)A12(x)e B2(x)R ‘5’ A2 (S-X)/L1(A' -1 Bé(x)e 2) I A12(x)d1dwds (B—27) Similarly, setting T —x = w in T2, changing the order of integration and then replacing the dummy variable T by 1 we get: 1 an .S as 1 A2(S-)\)/|J. 0 w xx , Ag (8+w—1)/H I B2(x)e A12(x)dxdwds and integration by parts once implies: _ l °° “S —l W I AZIW/Ll I --1 T2 — E [0 Jo terX(x)A12(x)A 2B2(X)R(E)B2(x)e (A2) Ai2(x)dwds m as .A2(s—w)/h .Azs/u 60 co 5 s A (S-X)/Ll + if f J trf (X)A12(X)e 2 B2(x)R( w 0 0 w xx E ) A2(s+w-1)/u B2’(x)e (A, -l 2) A12(x)d1dwds (B-28) Employing the facts that for any matrix A, trA = trA' and for any matrices X and Y , tr(XY) = tr(X’Y’) nxm mxn = tr(YX). We observe the following: (i) The first two terms in (B—27) are equal to the first two terms in (B-28) respectively. (ii) The third term of (B—27) = —T2 and the third term of (B—28) = -Tl. From these observations and (B-25), we conclude that: m A w/u _1 S -1 2 Iw I L12 _ E f0 f0 terX(x)A12(x)A 2e B2(X)R (E)B2(x) (A’)‘1A (x)dwds 2 12 1 no 8 —l 2228/)L I W I — E JO (0 terX(x)A12(x)A 2e B2(x)R (E)B2(x) Ads-WV” , —1 , e (A) A (x)dwds (B-29) 2 12 The first term in B-29 61 co _ -1 I I -l _ Io terX(x)A12(x)A 2 ZIB2(X)(A2) A12(x)ds co 0 -l A2W(E) I I + I f terX(x)A12(x)A 2e B2(X)R (W)B2(X) O s/e (Aé) -1A12(x)dwds (8—31) It can be seen that the second term in (B-30) g_Ku. And the second term in (B-29) is bounded by Klei-Kzu where we used (B-26), (A4), boundendess of x(t), compact support of fXX(x) and that % g_K. So, from (B—l6), (B-l8), (B—l9), (B-21), (B-24), (B—29), (B—30) and (B—31) we conclude that: °° 1 I = IO [—utr(fXX(x)A12(x)A_2Et(§(t+-s,x)yv(t-ts,x))A£2(x))ds °° —1 I I —1 = f0 tr(fXX(X)A12(X)A 2 2 32m) (A2) A12(X))ds+el (B-32) where, Z; is defined by (2.16) and (ell g_Kl€4-K2HI (B—33) where K ,K are some positive constants independent of 1 2 T and m 62 Now, let us consider I of (B—l6). From (B-l7) 3 we see that the important term in I3 is similar to (B-20), namely, a n s s A (5 T)/u 1 8f I 2 L = —— I tr( Z3 -——(x)F.(x)A (x) I I e 2 p 0 3:1 xj j 12 I A (s-X)/u _ B2(x)Et(v€(t + T)V’€(t + 1))B2’(x)e 2 (Ag) l)d'rd>.ds (B—34) Repeating similar steps to the ones that has been made to get from (B-ZO) to (B-21), we can express (B—34) as the sum of two terms, i.e., L = L + L (B-35) and from (8-22), the boundedness of Fj(x) and the same reasoning as before, we get (note: L21 involves the expression appearing in (B—22)) (L21) 3 K6 and similar to (B-25), we get: 1 w s 1 n af A2(s—T)/u L — ---— I Z .-—-(x)trF (X)A (x)e 22 “6 IO 0 J10 j=1 a 3 12 A (8—1)/u B2( )R’<) T>Bz’(x>e 2 (Aé)-lde1ds 63 n \ A (s-T)/u - — I IS IT 23 Of (x)trFf(x)A (x)e 2 O O O j=1 axj j 12 A'(s-x)/u (x)R’(AJLI)Bé(x)e 2 (Aé)_lde1ds 1_ w s T n. 5f I A2(s-T)/h — —— I I I Z} ———(x)trF.(x)A (x)e us 0 O O j=1 ij j 12 A'(s—1)/u 132mmT —X)Bé(x)e 2 (Aé)-lde1ds (B—37) After some manipulations it can be shown that °° n °° A 1 °° A (—)w 5f I 2 2 LJ. L = — Z -—(x)trF.(x)A (x) e e 22 IO j=l ij j 12 IO (IO g(§)w . B2(X)R’ (w )de’ (x) )f: R(w w)B2(X X)e aw) A51 e d1(A2) ds+e2 I” 2: 5f “a AZX z; z; = - -—-—(x)trF.’(x)A (x) I e (_, B'(x)+B(x) 1’) O j=l ij j 12 O 2 A’1 e 2 dx(Aé)—ld54-e2 — — In 23 iflitrF’ (X)A (X)P(A_l)’ds+-e (B—38) _ O j=1 5x j 12 2 2 where P is defined by (2.17) and ]eZI g Ktl (B-39) 64 Similar to (B-18) and (B—l9), it follows that the contribution of the first three terms of (B-l7) to I 3 of (B-16) is 0(u). Combining this, with the results of (B—35)—(B—39) we get the following estimate: °° n at I =J“ —utr(>3 —A (x) 3 O j=1 ij j 12 Et(§(t+ s,x)§’(t + s,x) (A-g) ')ds °° n at 1 _ _ _ I I — I _ _ trI I 5X.(X)Fj(x)A12(x)P(A 2) ds+e3 (B 40) 0 3—1 3 where 1e3l 3 K16 + K2H (B—41) for some positive constants K1 and K2 independent of T and w, Now let us consider an _ [e co AZS/Ll_ IC‘ I E y(t+s,x)v (t+s)ds =I e y(t)E v “(t+s)ds o t o t °° s A2(s —T)/u + L1: I I“ e B2(X)Etv€(t+T)v’€(t+s)des (B-42) Using the mixing prOperty and that y(t) is O(%:) we get: on A2s/u_ —(0 +%)s IIO e y(t)EtV€(t-ts)dsI g E IO e 2/u 65 Let T* equal to the second term of (B-42) then we have: °° s A (s-fl/u T* = 31' I I e 2 Bz(x) (Etv€(t+ T)v’€(t+s) O °° s A (s-T)/u - Ev€(t4-T)v€(t+-s))des+-l I I e 2 H o o B2(x)E(v€(t-+TWTJ’€(t+-s))des = Tl + T2 (B-44) Using (B—22) and that Bz(x) is bounded we have: I< m ‘s ‘4:](2(S_T)/Ll -GS/€ ]TlI g E? IO IO e e des 3 K (B—45) If we consider 12 in (8-16) with (B—42)—(B-45) we get: Q 12 = -u f0 tr(fXX(x)A12(x)A_%Et(y(t4-s,x)v’€(t+-s))Bi(x))ds ” _1 1 s A2(s-T)/u = —u f0 tr(fXX(x)A12(x)A 2 ’E IO e B2(x) E(v€(t+T)v’€(t+s))B]:(x)des)+e3 (B-46) Where Ie3I g K1€ + K2u. First term of (B—46) co m (E = _tr IO IO fXX(x)A12(X)A-:e 2 “ B2(x)R’(w)Bi(x)dwds 66 w m —l 2(E)w / , + tr IO I/e fXx(X)A12(x)A 2e B2(x)R (W)Bl(x)dwds m —l-« , = - IO terX(x)A12(x)A 224B1(X)dS4-e4 (3-47) where le4‘ 3 K16 + KZL‘L (B-48) Hence, it follows from (B—46)—(B—48) that: I = —f trf X(x)A12(x)A-%2Bi(x)ds+e (B—49) x 5' g Kle + KZH (B—50) Following steps similar to what has been done in (B—42)— (B—50) we conclude that: m n \ _I 14 = - (i [O tr(EiL aoxf‘(X)Fjl(x)Bl(x)Et(v€(t+s)y (t+s,x)) 3— 3 ’)ds °° n M 1 —1 = _j tr[j§l g(xmj’mml’m) 1’ (A 2) ’]ds+e6 (B—51) where 1e61 g_Kle + K2u. (B—52) 67 Then, we consider the integral: co 00 OD AZS/H co Et§(t+-s,x)v’€(1)d1ds = I e y(t) I Etv’€(x)d1ds O t+s O t+s a m A (s-T)/u l 2 6 I6 + 5 I0 I I e B2(x)Etv (t+-T)v (t+—x)dexds (B—53) = CP1 + a2 K m w 4125/"1 -ox/€ lei.£ E I S e e dkds g K16 + KZH (B—54 where we have used the mixing property and that y(t) is o(%—_). V/€ 0° °° s A (S—T)/LJ 1 p 2 a w = _ I J e B ( )(E v (t-tT) (t-+1) 2 u 0 s O 2 t - Ev€(t-+T)v’€(t+-1))dwdids °° °° as A (S-T)/Ll - + g I I I e 2 B2(X)E(vb(t-+T)v’€(t-+1))de1ds = Lp21 + Cp22 (B—55) From (B-22) we get: 8 —c (s—T)/u @211 g— II: I: I e 2 sax/e delds g K164-K2U 0 (3-56) 68 Changing order of integration, we get A2(S-T)/H gp = .1. I” I} IX e B (X)R’()k -T)deTdk 22 He 0 O T 2 e 0 l I = I (A‘ Z -A-1W )ds+e (B—57) 2 2 7 0 where ‘e7I g_Kl€4—K2H (B—58) and w is defined by I R(¢)d¢. Then I in (B—16), in o 7 View of (B—53)—(B-58) can be written as: an n of 1 co —' [C :7 =f tr( 25X (X)D.(X)A12(X) f Ety(t+s,x)v (x)dx o j=l j 3 t+s on n n _ a I -1"" bf I — f0 [tr(jElg-S(X)Dj(X)A12(X)A 22. -51 agnj —l I A 2B2(X)W )]ds + e8 (B—59) where le8] 3 K16 + Kzu (B—60) where K1’ K2 are some positive constants independent of T and m. We have used that fX(X) has compact support, (A2) and (A4). Similarly I5 can be handeled as I7 and we get: 69 on I = IO tr(fXX(X)BO(X) I E v€(X)y”(t+-s,x)dxA’ (X))ds t+s t 12 a: _1 » , —]_ I I I0 tr[fXX(X)A12(X)A 2ZBO(X) -fXX(X)Al (X)A 2W BO(X) ] ds 2 + e9 (B-6l) where Iegl g-Kle + Kzu (B—62) and K1’ K2 are some positive constants independent of T and w. Also, in the same way: I = 1xL-Z r—OUF(MA (m I Ia§u+saov”(de 9 f0 3:1 dxj 12 t+s t I -l B2(X)(A2) ]ds — a 3? 5f( >F’( >A <>A‘IZB’( >(A’ )‘ld _ _ IO tr j=1 5§_ x x 12 x 2 2 x 2 s a E; 8f I -].WI I I -1 + IO tr ._ BX (X)F (X)A12(X)A 2 B2(X)(A2) ds-I—elo 3—1 J (B-63) where IelOI S.Kl€4'K2H (B-64) Let us consider the integral 70 V a: co J = I I E v€(t+-s)v' t O t+s €(x)dlds Q Q 6 I I Etv (t+-s)v'€(t+-X)dde O s I I [Etv€(t+-s)v’€(t4-X)-Ev€(t+-s)v’€(t+-x)]dde O s Q @ 6 I6 _ + I I Ev (t4—s)v (t4—X)dkds — Jl+J2 (B—65) O s From (B—22), we have lJlI g’Ke (B—66) 1 Q on w (9 r‘: I I J2 = E I I R (E)dwds = I W ds (B—67) O O 0 Then, from (B—65)—(B—67), (8-16), compact support of fxx’ (A2) and (A4), we get: I = "‘ tr(f (X)B (x) Ev€(>\)v’€(t+s)d)\B'(x))ds 6 J0 xx 0 {+5 t l = IO tr(fXX(X)B1(X)W’Bé(X)) ds+ell (8-68) where Ielfl 3 K16: + Kzu (B—69) and K1, K2 are some positive constants independent of T and a. Also, we have from (B—65)-(B-67), (B-l6), 71 compact support of fx’ (A2) and (A4) we get: an n 0° ‘ Bf ’ 6 I6 I = tr 2 5—(x) D.(X)B (x) E V (t+s)v (Md): 8 I0 j=1 Xj J l £+s t a n af 2 IO tr .§; ‘§T(X) D5(X)B1(X)W'ds-+e12 (B-70) 3—1 3 where Ielzl g_Kl€4-K2H (B-71) and K]. I K2 T, w. From (B—65)—(B—67), (B-16) and the same assumptions are some positive constants independent of as before, we get: co n on \f I E I6 I = -I tr 23 to (x) F.(X)B (x) I E v (t+S)v mm 10 O j=l oxj j l t+s t I —l I B2(x) (A 2) ds = _I tr 2 g(x) F’(X)B (X)W’BI(X)(A’)— ds+e ._ a l 2 2 l3 O 3—1 J (B—72) where Iel3I g Kle4-K2p (B—73) and K1' K2 are some positive constants independent of T and w. From (8—14) and (B-l6) 72 °° af —1 I = - Z) ———(X)E (A (X)A w(x,t+-s)). I” 21% 5f 1 = - ———(X)(A (X)A— h (x)).ds-+e (B—74) O j=1 axj 12 2 2 3 14 where \el41 g Kl€+K2Ll (3-75) and w(x,t4—s), h2i(x) are defined by (B—14) and (2.11) respectively. Finally, it can be shown that ”12‘ 3 K16 + K2“l for some positive constants K1' K2 independent of T and w. Now we add (B—32), (B—40), (B-49), (B—Sl), (B—59), (B-6l), (B-63), (B—68), (B—70), (B—72) and (B-74), and let e denote the smnn of all ei that appear in the above equations and then from (8—3) and (B—16) we have: 2 w ofl I = 1:1 Ii = IO Et[7§;(x,t%-s)-g(x,t+—s)]ds _ at: -l" I I -1 I _ IO [terX(X)A12(x)A 221B2(x)(A2) A12(X) — tr 2n: £(X)F’(X)AI (X)P(A’)-:L ._ 8X. j 12 2 3—1 J — t f (X)A (X)A—IZBWX) -tr E i( )F'() r XX 12 2 1 j=1 ij j X Bl(X) 22’ (Ax—é), 73 + tr 2 Pimm’mm (X)A—12 ._ 8X. j 12 2 3—1 3 n a 1 - tr é 5——.(X)D3(X)A12(x)A—ZBZ(X)W’ 3—1 J -l I —1 + terX(x)A12(X)A 22:BO(X)-terX(x)A12(x)A 2 BZ(X)W'Bé(x) n of I -l-’ I I -l - tr :13 53?} )Fj(X)A12(x)A 22413200042) 3-1 3 n 5f I -]_W I I -l + tr 3E: 5;;(X)Fj(X)A12(X)A 2 B2(x)(A2) n \f + terX(X)Bl(X)W Bé(x)+-tr Z) 5; (x)D'(x)B1(x)W' J=l j n of I I I I -l — t .2 SCOOP (X)Bl(X)W B 2(X) (A ) 3=1 3 n 6f —1 _ jg: S§f(x)(A12(X)A 2h2(x))j+e (B-77) where e is the sum of all e which appears in the k right hand side of each Ii for i = 1,2,°--,12, and it satisfies le\ 3 Kl€+K2H (B-78) where K1' K2 are some positive constants independent of T and w. From the definition of B x), the first term, the 0( third term and the seventh term of (B—77) will cancel. 74 Moreover, we notice that 1 X)A_2 trfxx(x)[Bl(x¥WIBé(x)-A12( B2(X)WIB6(X)] = trf X)BO(X)W’B6(X) = % trf (X)BO(X)S(O)B6(X) XX ( XX % terX(x)A(x) (B—79) where A(X) is defined by (2.9) From the definition of P in (2.17), P satisfies the Lyapunov equation I _ _ / I _ PA2+A2P _ (2132+ 1322) (B 80) From (8—80) and (B-77) we have: tr[—Fj' (X)A12(X)A_§ZB2’ (x) (A-%) ’ —Fj'(X)A12(X)P(A-:) ’ I I -l I - Fj (X)B1(X) Z (A 2) 1 = tr[-F5(x)BO(x)ZL’(A‘§)’-+F5(x)A12(x)A‘§p] (B—81) Then, from (B—77) and (B-79)-(B—81) we have: I - I” [4% -a—f—(x) (h (X) -(A (X)A-1h (X)) +h (X)) _ O j=l 5X. lj 12 2 2 j 3j 2 1 n a f + 2 . Z: aij(x) 3§73§7(x)]ds+-e (B—82) 1,3-1 1 j 75 where hij' h2j and h3j are given by (2.10), (2.11) and (2.12) respectively. Then from (2.7)—(2.9) and (B-82) we have: I = I (Le/“f(x)._§§(x)ao(x))ds+-e (B-83) 0 Therefore, by defining (x = x(t)) ” af1 af f2(x,t) = IO [Et[-5-)-(—(X,t+s)g(x,t+s)]+3;(x)a0(x) - LEI/Hf (X) ]ds It follows directly from (B—83) and (B-78) that If2(x,t)] 3 K16 + Kzu where K1 and K2 are some positive constants independent of T and w as required. APPENDIX C We first derive equation (2.43) and (2.44): X Let X = (3’), then (2.1) and (2.2) can be written in the form: al(x)-+A12(x)y Bl(x)ve X: +—_ (C-l) V/€ h <>+A > E()€ H a21 X 2y 2 x v where u > O is small, arbitrary but fixed and 52(x)==&B2(x). Equation (C-l) is of the form: >°<=i_ F(X.V) +G(X.V) (c—2) V,€ which has been considered in [ll-l3], but let us apply, for example, the result of [13] on the system (C-2), where B1(X)V e F(X.v) = . v (t) = v(t/e) (C-3) B2(x)v€ and a (x)-+A (x)y G(X,v) _ 1 12 (C—4) 77 The convergence result of [13], says that under certain conditions X(-) converges to a diffusion process XO(-) whose differential operator is given by: Af(X) = EG’(x,v(s))fs(X) + I: EF’(X,V(S)) (F’(X,v(s + T))fX(X))X)dT (c—s) = Il + 12 where fx (Si’ 3;), 12 = E I: (v’(s>B{(x>v’Eé> + v’(s)Bl’(x)vX(v’(«r+s)E2 y + v'(s)g£(x)7y(V/(T'tS)Bi(X)fX(X)) + v'(s)g2’(x)Vy(v'(T+s)g2'(x)fy(x)))d7 (c—e) Consider the first term of, 12 78 IO Ev'(s)Bi(X)VX(V’(T4'S)Bi(X)fX(X))dT co = tr IO EBl(x)v(s)(VX(f;(X)Bl(x)v(T-+s))'dT Q = tr IO EBl(x)v(s)(fXX(X)Bl(X)v(T4-s) i¥ivxwij +§ +A2y> ’fy and therefore, by adding the expressions in (C-7)-(C-11), we get: that the operator A, corresponding to X0, Which is defined by (C-S) is 80 Af (X) (51 (x) +A (x)y)fX(x) + (321m) +A2y)fy(X) 12 + 12.- tr[B1(X)S(O)B]:(X)fxx(X) + 2Bl(x)S(O)Ez'(x)ny(X) + 152 (x)s<0)'§§(x)fyyS(o>BZ’(x)va) (c—18) __ — I l I £3 — (al(x)) -v +—§ tr(Bl(X)S(O)Bl(x) .VXX) (c-19) a a I 5 5 I V = —-‘—I...I_ I = —'I°°'I‘— I Where x (5x1 axn) vy (Byl Bym) a2 vxx = Vx(vx) = (SiTEET) ' etc. 1 j nxm For any smooth function f(x), XEERn, which has compact support, the function ¢él)(x,y) has been introduced in [33] and it has been required that Vél) must satisfy: 2 ¢(l)(x y)4—£ f(x) = o (c-20) l f ' 2 which can be written, via (C—l7) and (C-18), as ’ .v ((1) l ’ ‘(l) (Azy) ykf (X,y)+2 tr(B2(X)S(O)B2(x)vyywf (x.y)) I — + (A12(X)y) fX(X) — O (C-Zl) By means of the linearity in y and the fact that the constant matrix A is nonsingular, it can be seen that 2 ¢%(x,y) has to be linear in y. So we suggest the following form for If 83 “(1) I Vf (XIy) = g (X)y (c-22) where g(x) is a vector function of x, to be determined later. Substituting from (C—22) into (C-Zl) we get: (Azy)'g(X)-+(A12(X)y)'fX(X) = O I y'mggm +A12(X)fx(x)) = o (c-23) (C-23) has to be true Vy ERm, this implies that I —1 g(X) = -(A2) A x)fX(X) I 12‘ Then, from (C—22) we have: wél)(X.y) = -f;(X)A12(X)A_%y (C—24) which is defined up to an additive constant. We proceed as in [33], by defining Y(t:x) to be the diffusion process in Rm generated by £1 given by (C—l7). This process is actually a Gaussian Brownian motion process which possess an invariant measure given by: 1 I -l l -§(y Q y) m 1 e dy VA c R (c-25) 2 A ((2w)m detQ) Q is the variance matrix of Y(t;x) and it is dependent on 84 the parameter xEERn. Q is in fact the solution 3 of the Lyapunov equation (2.53). We notice that I F(dy;x) = 1 (C-26) Rm As in [33] the diffusion operator L (see (2.2.5), (2.2.6) in [33]). is defined by: Lf(x) 1) I F(dy;x)[aél(x) °Vy¢é (XIY) Rm + tr(Bl(X)S(O)Bé(X)V ¢é1)(x.y)) XY -VX¢él)(X.y)i-§i(X) ~fX(X) -fXX(X))] (C—27) Substituting from (C—24) into (C-27) and making use of (C-26), then, the first term of (C—27) is 1 R — -I I ‘1 I = — I m P(dy:x)a21(X) (fX(X)A12(X)A 2) R = _ f}:(X)A12(X)A_%5-21(X) . (0-28) It can be shown after simple manipulations that the second term of (C—27) is: 85 = _ I B(dy;x)tr(Bl(X)S(O)B£(X) Rm 1 . v (f (X)A12(X)A-2y)) xy x 1 I 2) A12(X) II l H- H H} X §‘ w (X)S(O)B£(X)(A- n 5f -1 - .2 tr —.-(x)B(x)S(O)B2’(X) (A 2) ’Fi’m fXX(x)A12(x>A'zy m R 5f I I - _ é Si—'-(x)y A12(X)Fi(X)A 2y 1-1 1 I = tr(-Q(A_%) [A12(X)fXX (X)A12(X) n 8f —1 -l I I - El 5X.(X)Q(A 2)Q(A 2) Fi(x)A12(x)) 1—1 1 = -1 tr((Q(A‘l)’-+A"1Q)A’ (x)f (X)A (X)) 2 2 2 12 xx 12 n A -l _ tr .2 gflxmm 2) ’Fi’O, Ial(x)I-+IA12(X)[-+]Bl(x)l g_MIxI (5) The coefficients aO(x) and B (x) which are defined 0 by (2.5) and (2.6) are required to satisfy: Ia (x)-aO(z)Ii-IBO(X)-—BO(z)I§;le-zI Vx,z€ERn O and for some K > 0. Now we may consider the diffusion process x(t), generated by LY and given by (2.7)-(2.17), to be the solution of the Ito equation: d§(t) = b(§(t)) +o(§(t))dw(t), Em) = x (3.3) 90 where b(X) is defined by (2.8) and O(X) = BO(X)\/S(O) (3.4) and we require: (6) The coefficients b(x) and O(X) satisfy ‘b(x)‘+-‘o(x)‘ g_M]x‘ VxéiRn, for some M > O and (b(x) -b(z)l+ lo(x) —o(z)| gxix-z\ VX,ZERn for some K > 0. We will consider functions V(X), XGERn with the following properties: (a) V(x) is real—valued, positive definite, V(x)==O Ll X ll 0 V(x)-+w as (x1-aw and has continuous partial derivatives up to the third order. (b) For any vector or matrix valued function g(x,t)==O(x) for t O is arbitrary but fixed. Moreover L is defined by: n. a l n 82 L") = ._ 131‘") 57%”? . 2. an”) W“) 1-1 1 1.3—1 1 3 (3.8) where A — ( sog’( '3 (X) — Bo X) () o X) — [aij(X)]. (3.9) The vector b(x) is defined by (2.8)—(2.17). Now we state the theorem: Theorem 1: Suppose that there exists a Lyapunov function V(x) on Rn satisfying (a),(b) and, for some X > O. 92 LV(X) g_—AV(x) VxeéRn (3.10) Suppose that all the assumptions (1)-(6) are satisfied. Then, there exist €O'HO and Y0 such that for all . . e €,u satisfying 0 < e g 60’ O < H g “0 and lJ-Y':;YO: x(t), the solution of (3.1) and (3.2), is uniformly stochastically asymptotically stable as t-4w, i.e. for any n1 > O and n2 > 0 there is a 6 > 0 such that if 1x0] < 6 then: St (I) P(lx(t)l g n25 , t 2_O} 2_1 -n1 for some 6 > 0. (II) P(lim lx(t)l = o) = 1. tam Moreover, if V(x) satisfies, in addition, 2 nl n czlx( g_V(x) g cllx‘ VXEER for some positive constants c1 and c2 and some positive integers nl and n2 then (I) and (II) will be satisfied in the large, i.e. independent of the initial condition x0. Remark: The condition (3.10), under the assumptions stated on the coefficients of L, guarantees that the limiting diffusion process x(t) is uniformly stochastically asymptotically stable as t-4”, see [7] or [20]. Proof of Theorem 1: In the proof we adOpt the same terminology and definitions, concerning the operator AC’H, which has been used in Chapter II. In fact we are going to 93 repeat the averaging method but with the truncated function VN(x) instead of f(x). We follow the basic idea of the proof of [12 Ch. 5]. Operating, now on VN(x) by A€.H’ (using from now on x for x(t), for 0 g t g_T where T is arbitrary), we-get €oH A V (X) = \ (X) [al(x) +A12(x)y(t) +B (3.11) Averaging out the last two terms by defining N /\ _ V (x,t) = —(x) j [A12(x) (E y(t+s,x) +A Nyl t (3.12) + B x)E v€(t+-s)]ds l( t where A S/u _ _ 2 (y(t) +A gamma) -A Eamon (D A y(t+—s,x) = 2(t+s _ r)/H p (3.13) + & f e B (x)v'(T)dT EV m A -1 ‘VN,1(X't)‘ g [a(x) f0 [A12(x) (Ety(t+s .x) +A 22:121(x)) + Bl(x)E v€(t+s)]ds‘, t but from (1)—(4) and (3.13) the integral is bounded by (KlV—€+K2\/U) 1x1. So it follows from condition (b) that: 94 IV (x,t)! g_(KlvFEA-K2VKE)V(X) for xéES (3.14) N,1 N where K1 and K2 are independent of N and T. Operating 6:H on VN,1(X't) by A we get Ae'“v ( t) — -a—v§< )[A (H O, to be determined later, we have: A av 6&1 A _ €/Ll N,1 (A -+x)VN(x,t) — L VN(x)-FXVN(x,t)-+ Eh< (x,t)aO(X) BVN 2 C + ___—BX: (x,t)(al(x)+A12(X)y(t)+B1(X)Vv(t)) (3.22) 96 A Our goal at this point is to choose 1 appropriately such em A . . . e/M that (A -+x)VN(x,t) g 0. From the definition of L , (1)—(4), (3.5), (3.6) and the smooth dependence of Le/H on e/u(E[Yl,w), there exists a constant c > 0 such that for xESN (Le/“VN(x) -LvN(x)\ chE—ylwx) (3.23) From (3.14) and (3.18) we have for x(ESN: VN(X,t) _>_ (1—Kl\/€-K2\/LL-K3€—K4H)V(x) (3.24) Similarly we have for x(ESN: V (x,t) g (1+Kl\/—€+K2V’I+K €+K H)V(x) (3.25) N 3 4 Now we want to find upper bounds for the last two terms in (3.22) similar to the upper bounds in (3.14) and (3.18), and this follows from the definitions of VN l(x,t), VN 2(x,t), from the assumptions (1)-(4) and from (3.5)—(3.7). So we have: BVN 1 __ __ 1 5X (x,t)aO(x)l g (K5\/€+K6¢ (1)V(X) vxesN (3.26) 6V 5V N,2 N 2 a ( ax (x,t)al(x) ( +1 5X (x,t) (A12(X)y(t) +Bl(X)v (t))) (_T /_ 1. g (K7e4-K8u4-K9V(t4-K10V u)v(x) vxéESN (3.27) 97 All the Ki in (3.26) and (3.27) are independent of N and T. Now it follows from (3.22),(3.23),(3.26),(3.27), (3.25) and (3.10) that for XESN we have: (A€’“+/>:)V( t) [-)\+C1E— 1+3? "72+? _+K€+K NX' g (1 Y l\/ 2\/Ll 3 4‘1 A __ ,__ + 1(1+Kl\/ €+K2\/H+K3€+K4H)]V(X) (3.28) There exist 80 > 0, MO > 0 and Y0 > 0 sufficiently small such that the following conditions are satisfied: (1) l -KlV/eo -K2V’HO-—K3€O-K4uo = C1 > 0 (ii) For all e and u satisfying 0 < e g 60, A 0 < u g “0 and (E'-Y( 3 YO’ A can be taken small enough such that: _)\+c1§ ‘Y‘ +El\/_e+Ez\/U+E3€+E4H A _ _ +X(1+Kl\/€+K2\/H+K3€+K4Ll) go. (3.29) Then by this choice of 60, H0 and Y0’ (3.28) reduces to: 6M A (A -+x)VN(X,t) g 0 for x ESN (3.30) For 0 g t g T and for each N, the function 98 is a zero mean martingale even if t is replaced by a stopping time T with B(T) < a, [c.f. 10]. Let us now redefine MN as in (3.31), but with VN A replaced by eXtVN. We have: But from the definition of A, we have: Ae’HV (x(s),s)e)‘S N A E) /\ (eM8+ )VN(X(s+ 6).s+ a) -e)‘SV (x(s),s)) . N = p—lim Et 690 6 A A = p-lim % [ex<9+é) —eXS]EtVN(x(s-+6),s-+6) 640 A As . 1 -I- e p—lim 5 [EtV (x(s+ 6), s+ 6)-V (x(s) ,s)] N N 640 a Q Q _ __ s s €,u —— ds e VN(X(s),s)+e A VN(x(s),s) A A = eAS(A€'P+ ))vN(x(s) ,5) Then, we replace t by thTN = min(t,TN) in (3.32) where TN = inf{t :x(t) ¢ SN). It is obvious that B(tIWTN) < m. Then (3.32) becomes: 99 A A(tnTN) MN(t(WTN) = VN(x(t flTN),t PTN)e -VN(X(0),O) tn'T /\ A _ I exs(A€’H4-X)VN(X(S),s)ds 0 from which we have: A x(TNflt) e VN(x(t flTN),t flTN) = VN(x(0),O) tnTN /\ (3.33) + f esx(A€’H-+§)V (x(s) s)ds-+ (t 0T ) O N ' MN N Since x(s) ESN for 0 g s g tflTN, (3.30) and (3.32) give A A(TNflt) e VN(X(t DTN),t flTN) g_VN(X(0),0)-+MN(t flTN) (3.34) Let C2 = 1-+KlV/€O4-K2V/uO-+K3€ -+K4HO then from (3.24), condition (i), and (3.25) we have V(x) for x 68 (3.35) c 2 N lV(x) g VN(X,t) g c So (3.34) and (3.35) imply: A A(TNflt) 0 g cle V(x(t DTN)) g c2V(X(0))+-MN(t DTN) (3.36) The right hand side of (3.36) is a nonnegative integrable martingale. Using Kolomogrov's inequality for nonnegative martingales and that E(MN(t()TN)) = 0, (3.36) gives 100 A A(TNflt) le V(x(tr33N)) > n, o g t g,T) P(c g P(c2V(x(0))-+MN(t(WTN) > n, 0 g t g'T} g c2E(V(X(O)))/n (3.37) Letting N 4 w implies that TN 4 w, since by the linear growth assumptions on the coefficients there is no finite escape time. Then from (3.37) we have: At molex V(x(t)) > n. 0 get 3 T) g c2E(V(xO))/n (3.38) Letting T 4 m we get: A P{cleXtV(X(t)) > n. t 2,0) g c2E(V(XO))/n (3.39) By the smoothness assumptions on V(x) we have: _ n1 Cl‘X‘ g_V(x) g_c21x‘ for (x) g_rO (3.40) for some rO > 0, 5i > 0, 32 > 0 and some positive integers nl and n2. Then A A _ I'll __gAt gAt (cllx(t)) g C n. tZO7DfV(x(t)) g—n. tgo? C1 Hence by (3.39) and (3.40) we have: /\ -).t/nl 77 l/nl p(|x(t)( g_e ( __) . t 2 o) 2 1-—c2E(v(xO))/n c c l 1 c E n 2 2 2 21- n B(IXOI ) (3.41) For any ml and n2, choose n so small that (3.41) gives ‘ c E n Pa’lx(t)lgé'et n2. t>o)>1— 2213(12):]2 (3.42) 9 n2 where 6 = —L Choosing 6 so small that c 5'6 /n<(n , n1 2 2 1 we get that for all xO with 1x0) < 6 P (x(t)! eetn t O) > 1 -n (3 43) ' S 2' 2- J — 1 ' and this proves (I). Now since (lim (x(t)) = o} = (11m V(X(t)) taw tam A 2 {sup extc V(x(t)) $.01 tZO l (V (X) of the above statement) where C (3.39), (3.40) imply that for P(lim (x(t)) = o} 2 1.___75"_ t'fico . . _ 0) being radially unbounded is necessary for the validity is any positive constant, IXOI < 6 we have 102 Then as C 4 w we get P(lim lX(t)l = O) = l (3.44) tfi“ which proves II. Now if V(x) satisfies (3.40) VxéiRn and if (x01 < m, (3.43) and (3.44) follow in the large. 3.3. Mean Square Boundedness: The stability result presented in section 3.2 was concerned with establishing the asymptotic stability of the origin in a stochastic sense where the origin x = 0 is an equilibrium point of the system for any driving input noise. That is, if the initial condition X0 = 0 then x(t) = 0 for all t 2 0. A key assumption there was the requirement that al(x), A12(x) and Bl(x) vanish at x = 0. While requiring al(x) to vanish at X = 0 is a typical and acceptable assumption because it can be always achieved by shifting the origin to the equilibrium point of the unforced system, requiring A12(x) and B1(X) to vanish at X = 0 is not always valid. In many cases driving inputs do not vanish at x = 0 and one cannot discuss asymptotic stability of x = 0 because x(t) does not necessarily tend to x = 0 as t 4 m. For deterministic systems the appropriate concept of stability is bounded—input bounded- output stability, i.e., to establish that for any bounded input the trajectories of the system remain in a bounded set. 103 A stochastic version of this concept is the mean—square boundedness where one shows that x(t) has a bounded mean square. The objective of this section is to study the stability of x(t) when A12(x) and Bl(x) do not necessarily vanish at x = 0. We now state the following alternative assumption. (4’) The coefficients al(x), A12(x) and B1(X) are required to satisfy, for every x€ERn and for some positive constants M1 and M2. (al(x)]+1A12(x)‘+1Bl(x)(g Ml‘.x\ +M2. The Lyapunov function V(x) will be taken to be a quadratic form, namely V(x) = X’Qx, Q > 0. So it is obvious that V(x) in this form satisfies conditions (a) and (b) including (3.5)—(3.7), which are stated in section 1. Now we are ready to state theorem 2. Theorem 2: Suppose that there exists a positive definite n.xn matrix Q such that V(x) = X’QX satisfies LV(X) g K—xwx) VXERn (3.45) for some K 2 0 and A > 0. Moreover assume that all the assumptions (1)—(6), with (4’) replacing (4), are satisfied. €O'H0 and Y0 . . e satisfying 0 < o 3-60' 0 < u g “0 and ‘fl"Y‘ g YO’ the Then there exist such that for all €,u process x(t), defined by (3.1) and (3.2), is bounded in 104 the mean square, i.e., there exists a positive constant K > 0 such that E{Ix(t)12} g_K, provided that B(1X012)< 9. Remark 2: The quadratic Lyapunov function V(x) satisfies the following conditions: E(V(x(t))). B((LV(x(t))() and B((——7(x(t))o..(x(t))(2) are bounded in t in any bounded time interval, and that V(x) 2_c(xl2 VxéiRn and for some c > 0. Similar conditions to these and to (3.45) have been required [c.f. 18,31] to guarantee that, the solution to an Ito equation is exponentially bounded in mean square with some positive exponent, i.e., Ef]x(t))2 g K14-K2eat for some K1 2 0, K2 > 0 and a > 0, where x(t) is the solution of an Ito equation. This is actually the case for x(t), the solution of our reduced-order model. The av)O’..] above conditions are valid because V,‘LV‘ and ‘(BX l] i are dominated by polynomials and c is in fact equal to 1min(Q) which is positive since Q is positive definite. Proof of theorem 2: Using the fact that V(x) satisfies 2 cllx12 g V(x) g c2‘xl VxéERn (3.46) for some positive constants c1 and c2, and then preceeding in the same way as in the proof of theorem 1 of 105 section 3.2, we get the following inequalities which are similar to the ones given by (3.14), (3.18), (3.23), (3.26) and (3.27) respectively. The new inequalities take the forms: h%ulbmt)‘g_UfiyFE+KT/E)Hx]2+‘xh, (347) IVN’2(x,t)1 g (K364-K4H)(‘X124-‘Xl4—1), (3.48) For some c > 0 (Le/HVN(X) — LVN(X)] g_c]E-—y1(1x124-lxl), (3.49) avN,1 —— r— 2 and finally av av . 1 §;2(x,t)al(x)14-1—7§§3(x.t)(A12(x)y(t)+—Bl(x)v°(t))) 3 (K76 +K8Ll +K9¢3+ KlO\/Ti_) (1x12+ 1x1+ 1) (3.51) All the positive constants Ki in (3.47)—(3.51) are independent of N and T. Defining V (x,t) as in N (3.21), we have 5V 8V €,u __ e/u N,1 N,2 A VN(X.t) — L VN(X) + 5X (x,t)aO(X) + 5X (x,t) (al(X) + A12(x)y(t)+-Bl(x)v€(t)) (3.52) Then from (3.49), (3.45), (3.50) and (3.51), (3.52) implies that 106 Ae'qu(x.t) g K-xc11x12+c(3 -y( ((x(2+ (xl) + (K7€+K8H+K9\/€+KlO\/H)( (1x12 +(x(+1) g K+ (El‘E_Y‘+Cl(€'H) -AC1)1X12+52‘E-Y‘+C2(€,(J) (3.54) There exist €0 > 0, “0 > 0 and Y0 > 0, sufficiently small, such that for 0 < e g 60’ 0 < u g.uo and e ‘fl"YI < Y0 we have — € — __ _. cl(J-Y]-+cl(€,u)-xcl — K2 for some K2 > 0. Then it follows from (3.54) that €,u — — 2 A VN(x,t) gKl—K21x1 (3.55) for some positive constants K: and Ké. Also the above 0 and Y can be made small enough that 0' 0 0 by the aid of (3.47), (3.48) and (3.46) we have choice of e dlle2-—d2 g_VN(X,t) g 61(X124-82 VxéESN (3.56) where di and Bi are some positive constants. Now let us introduce a set QO as follows: _ 1 K .2 _ n . _ 1 Q0 - (X‘ER .(X( < KO} where KO-— (:7-) K2 We define the starting time T as follows: 0 107 T = o if x(0) i QO = inf (t :(x(t)( = K) if x(0) e Q t20 O and let be defined as the first time that x(t) T1 enter QO for t 2 TO, i..e., T1 = inf(t :tQZTO,x(t) 0, 109 independent of T. Therefore letting T a a we get 2 — - Elx(t)l gK Vtzo and this completes our proof. Remark 3: Suppose that K== 0 in (3.45), i.e., LV(x) g -XV(X), and that the constant M2 in (4') vanishes, i.e. the coefficients al(x), A12(x) and Bl(x) vanish at the origin and satisfy (al(x)]+‘A12(x))+lBl(x)\ ng(x] 'v’xERn. Then, if we proceed in a way similar to the steps of the proof of theorem (1) we conclude the following inequalities which are similar to (3.30) and (3.35) respectively 6 u A , (A ’ -+1)VN(x,t) g 0 for Xt:SN (3.62) C1V(X) g VN(x,t) g c2V(x) for x(ESN (3.63) CI and c2 are positive constants independent of T and N. But since V(x) = x’Qx, we have; — 2 — 2 / cllxl g_VN(x,t) g €21X1 for x<:SN (3.64) Then similar to (3.34) and by the aid of (3.62) we have: Q n (TN t) e VN(X(tflTN)ItflTN) ng(x(0),0) +MN(tflTN) (3.65) then from (3.64) we have: 110 A MT 0t) _ o 56 N Ix(tflT)l2gc]x(2+ (tat) (366) i 1 N 2 0 MN N ° Taking unconditional expectation we get: A( ()) 1 T t —— N 2 — 2 — clE(e (X(t(1TN)( )g_c2E(XO‘ g_C By the monotonic convergence theorem we have: A( ) A A T flt '- 2 t 2 C limE(e N (x(tflTNH ) =E(e)‘ (x(t)! ) g;— N#” c 1 2 Qt then E(lx(t)l ) 3K6 3'03th since K is independent of T, it follows that 2 Qt E((x(t)l ) 3K6 ‘v’tZO. 3.4. Examples:_ Example 1. Consider the system: x = —5x+xy+xv€ (3.67) My = —y-+e v (3.68) We would like to study the stability of x(t) when 6 and u are sufficiently small, and let us take Y = .1, as a nominal value of the ratio e/u. v€(t) = A: v(t/e) \/e and v(t) is a zero mean, stationary, uniformly bounded process for t.€[0,®) and satisfies a mixing condition with decaying exponential, so that, if R(T) is the correlation function, then we have 111 ]R(T)I g’eT (taking the exponent to be 1 here). According to the results of Chapter 2 the state x(t) of (3.67) and (3.68) can be approximated by a diffusion .process X(t) whose diffusion Operator is given by: (Let us assume that S(O) = l, where S(w) is the spectrum of v) 2 3—X2 3—2x2 —2x2 —x2 -2x e X -+2x e -+xe -+xe ] L(-) = (—5x+%[x+xe (3.70) _ 2 (o)4—%X2(1+-ex2) 533 2 co + 2x352X j 5'1(T)R(T)dT]a§ 2(-) X 0 d To establish the stability properties of x(t) we need to study the stability of the diffusion X(t) and this can be done, if we can find a Lyapunov function V(x) which satisfies LV(X) g_—XV(X), V3 0. Let us choose V(x) = x2, then from (3.70) we have: 2 2 2 2 2 LV(X) = ~10X24—[x24—x2ex --2X4eX 4-2x452X 4—x2e2X 4—Xzex ” - 1(7) 4—2x2 2 —x2 2 +4] e' R(T)dToxe +x (1+e ) (3.71) 0 2 2 2 m (1 1 2 g_-10X 4—4.36X +-.75x I 5 ° )TdT-t4X 0 2 ” g—X = —V(X) vxEJR 2—2X2 We have used that max(x e ) 3 .18. X Then it follows that, [c.f. 7,20], the solution X of the reduced—order model given by L of (3.70), is stochastically asymptotically stable. Then, it follows by 112 theorem 1 of this chapter, since all the assumptions of the theorem are satisfied, that the process x(t) is stochas- tically asymptotically stable for e and H sufficiently small and for E sufficiently close to Y = .1. Remark: It is interesting to notice that, if we allow 3, for example, to take values in [.05,w) (say), then we see from (3.71), that the term: 6 e 2 _ 2 w -(—)T _ 2 w —(-+l) 4-2x 4X482X I e H R(T)dT g4x4e2X f e H = £%?E___ O O (-+ 1) u 2 4X4é2X i "1.05 Since 3 Z_.05 and the above conclusion is valid for sufficiently small 6 and u and for any 2 in [.05,”). Although theorem 1 is valid only for the case when E is close to a norminal value Y in [Yl,m) for some Y1 > 0, the proof can be modified to show that if LV g_—1v is satisfied uniformly in y then the statement of the theorem holds for all 3 2 Y1 > 0. Example 2: Consider the system: X = —2X+xy+v€ (3.71) . 6 (1y = —y+v (3.72) This system is different from the one which has been considered in example 1 in that, the right-hand side of (3.71) does not vanish at x = 0 which means that x = 0 113 is not an equilibrium point and the best that we can hope to establish is to show that x(t) is bounded in the mean square, for all t 2.0 and for e and u sufficiently small and for E sufficiently close to a nominal value Y = .5 (say). Let us assume that v6 satisfies the same assumptions as in example 1. The process x(t) of (3.71) and (3.72) can be approximated by a diffusion process X 1). whose differential operator is given by: (We take S(0) = L . — ( 2 +1[ +1]+2“) d(- +l(1+ )2d2(-) ( ) - (- X 5 X 4.)a§ ) 5 X 5;: (3.73) where m 5T 2:] R(¢)5' d: (3.74) 0 Then if we choose the Lyapunov function V(x) = X2, (3.73) implies: LV(X) = -4x2+-x24-x+-2x2:+-(1+—X)2 -3x2+ (x2+ (3+2Z)x+1) 2 3+2ZZ_(3+2Z)2 -3X + (X-+ 2 ) 4 +—l 2 Using the fact that (a-I-b)2 g 2(a 4-b2) for any real numbers a and b and that L») 12):] g(l.5)Td =2 0 we get LV(X) g 12 -x2 = 12 —V(X) ‘7'XEIR 114 This implies [c.f. 18] that X(t) is exponentially bounded in mean square with exponent l, i.e. E(]X(t)]2) g_K14-Kzet V'tZJD, for some K1 2_0 and K2 > 0. Then, since all the assumptions of theorem 2, of this chapter, are satisfied, it follows that the process x(t) is bounded in the mean square sense, V't 2,0, for sufficiently small 6 and u and for sufficiently close to 0.5. E u CHAPTER IV STABILIZING CONTROL 4.1. Introduction: It is a well known fact that an important aspect of feedback design, is the stability of the control system. Whatever has to be achieved with the control system, its stability must be assured. Actually, sometimes, the main goal of a feedback design is to stabilize a system if it is initially unstable. Let us recall that the two types of feedback designs are the state feedback, in which it is assumed that the complete state of the system can be accurately measured at all times and is available for feedback, and the output feedbackq which is the much more realistic case where there is an observed variable whose dimension is, in general, less than that of the states and it serves as input to the controller. The observed variable is usually corrputed by an observation noise. The states of the system, which cannot be measured accurately in this case, can be reconstructed from the observed variables and the feedback control, in this case, is a function of the reconstructed states. For example, in the case of linear systems, where both the state equation and all the output variables are corrupted by 115 116 additive white noise (the state equation is an Ito equation) one can use a Kalman filter [c.f. 37] for a state recon— struction and then a state feedback control can be designed to achieve certain prespecified objectives. Stabilizing nonlinear stochastic systems via the use of an asymptotically stable stochastic observer have been considered recently [18]. The work which is done in that paper is a generalization of the Kalman filter structure. Until recently, singular perturbation techniques have primarily focused on state feedback design of linear systems. Advantages of these techniques, such as order reduction and separation of time scales, are expected to have a more dramatic effect on feedback design of nonlinear systems. Stabilizing deterministic nonlinear singularly perturbed systems have been considered, for example, in [2] and [38]. In this chapter we consider the stochastic stabilization problem for nonlinear singularly perturbed systems driven by wide-band noise. We consider the following system: 6 $4 = a1 (x) +A12(X)y+Bl(X)vll+Gl(X)u (4.1) 6 My = a21(x) +A2y + B2(X)v1l +G2(X)u (4.2) 2 = cl(x)+c2y+B3(X)v22 (4.3) where u is a control vector in Rp, z is the output E: 2. vector in Rq (g g n), vlléff:and v22 ER; are independent and have the same properties, as v6 defined in Chapter 2, 117 where 61 and 62 are different in general, so that, if e the observation noise v22 has spectrum which is wider 6 than that of the system noise v11 then we expect 62 to l'GZ'Cl and B3 are, in general, functions in x and are required to be much smaller than 61' The matrices G satisfy certain smoothness conditions which are specified later. The outline of this chapter is, roughly, as follows: 1. We begin with the open—loop full—order system (4.l)—(4.3) and we aticipate an open-loop reduced order model (0LROM) in the form of an Ito equation. 2. We design a stabilizing feedback control for the above (0LROM) model which will result in a stochastically asymptotically stable closed—loop reduced order model (CLROM). Work similar to that of [18] has been done, in that regard. 3. We apply the feedback control which we obtained in step 2 to the full—order open—loop system (4.1),(4.2), and we obtain a full—order closed loop system (FOCLS) which will be of the form (2.1) and (2.2). 4. We apply results of Chapter 2 to identify the reduced- order closed—loop model (ROCLM) corresponding to (FOCLS) which has been obtained in step 3. 5. We require that the (CLROM) be the same as the (ROCLM) and this results in some conditions which will be referred to as the consistency conditions under which the OLROM will be identified completely. 118 6. We apply the results of Chapter 3 to obtain conditions, under which the(FOCLS)is stoachastically asymptotically stable. Remark: In the second section we will study the case when all the states of (4.1) and (4.2) are available for perfect measurement and a stabilizing feedback controller has been designed according to the above outline. In the third section we repeat the same procedure but in this case we assume that the states are not available for perfect measurement and an output feedback controller via an observer is employed. In section 4 we illustrate the procedure by an example. 4.2. State Feedback Stabilizing Control: Let us write 6 again the full—Order system (4.1) and (4.2) with v11 written simply as v6 . e X — al(x) +A12(x)y+Bl(x)v +Gl(x)u (4.4) My = a21(x)-+A2y-+B2(X)v€-+Gz(x)u We assume that the slow state variables x(t) are available for measurement. Since the results of Chapter 2 indicate that x(t) tends in the limit to a diffusion process, it is reasonable to anticipate that the open—loop reduced—order model corresponding to (4.4) and (4.5) takes the Ito form: 119 d; = b(§)dt+E(X—)udt+g(x_)dw (4.6) where the exact forms of the vector b and the matrices G and 0 will be determined later. Let us assume that there exists a sufficiently smooth function g(x) such that, when the feedback control u = g(x) is applied to the system (4.6), the resulting closed-loop reduced—order model dx = (36%;) +E(§)g(§))dt+6(§)dw (4.7) is asymptotically stable in some stochastic sense (see Chapter 3). Then, we apply the control law u = g(x) to the Open-loop full order system (4.4) and (4.5) to obtain the closed-loop full order system . N e x — a1(x) +A12(x)y+B1 (X)V (4.8) (15; = 3210:) +A2y+ B2(X)v€ (4.9) where a1 = al-rGlg (4.10) and a2l = a21+GZg (4.11) Equations (4.8) and (4.9) are in the form of (2.1) and (2.2) respectively. Then, the reduced—order closed—loop model corresponding to (4.8) and (4.9) can be obtained by applying the results of Chapter 2, assuming that the A coefficients and the process vc satisfy the required 120 assumptions which will be spelled out later. So, the reduced-order closed—loop model corresponding to (4.8) and (4 .9) is: 10 (§))g(§)dt +\/A(§) dw (4.12) a; = b(§)dt+ (ol(§) —A (§))z>.‘2 2 12 where b(x) and A(x) are defined similar to (2.8) and (2.9). Now we impose a consistency condition which is stated as follows: The closed-loop reduced-order model, which is obtained by applying the control u = g(;) to the open-loop reduced—order model, is the same as the reduced— order closed—100p model corresponding to the closed-loop full—order system (obtained by applying the same control u = g(x) to the open-loop full-order system). This, condition says that the coefficients of (4.6) must be the same as those of (4.12) for any g(x) and this implies: b(x) = b(x) 'V’XERn (4.13) ~ —1 .. n G(X) = Gl(X)-A12(X)A 2G2(x) I7x€ER 3 GO(X) (4.14) g(X) =\/A(X) (4.15) Hence the open—loop reduced-order Ito model that approximates the slow states of the non-Markov open—loop full-order system (4.4) and (4.5) is given by: a)? = b(§)dt+oo(§)udt +./A(§)dw (4.16) 121 With the (0LROM) (4.16) in hand we can proceed now to design the feedback control u = g(x) to stabilize x of (4 .16). This control task is much simpler than the original task of stabilizing x of (4.4), (4.5) since now we are dealing with the ito equation (4.16) for which stability and stabilization techniques exist in the literature [c.f. 7, 16, 18, 20]. Suppose now that we succeeded in finding a sufficiently smooth function g(x) with g(0) = 0 such that the application of the feedback control u = g(X) to (4.16) results in a stochastically asymptotically stable (CLROM) with a diffusion operator L’ given by: O(X)g(X)) . fX(X) + NIH Lf(x) = (b(x) +G tr (A(X)fX (X)) (4.17) X The use of the feedback control u = g(x) with the full system (4.4) and (4.5) is justified by the following theorem whose proof is a straight forward application of Theorem 1 of Chapter 3. Theorem 1: Suppose that there exists a Lyapunov function V(x) for x E Rn which satisfies all the assumptions of Theorem 1 Chapter 3 and that LV(X) g -AV(X) for some A > 0. Moreover, suppose that assumptions similar to (1)-(6) stated in Chapter 3 are satisfied where, al,a21 and a0 IV ~ ~ — ~ —l~ are replaced by al,a21 and a0 — al-—A12A 2a21, respec- tively. Then the solution, x(t), of (4.4) and (4.5), with the control u = g(x), is uniformly stochastically 122 asymptotically stable as t 4 w, for sufficiently small 6 and H and for E sufficiently close to a normal value Y€5[Yl:”) for some Yl > 0. 4.3. Output Feedback Stabilizing Control: Let us consider the full—order open loop system (4.1) and (4.2) with 2, given by (4.3), representing the 6 observed variables, where vll(t) = 5%: vl(t/6l), 62 1 €1 V”:1 v2 (t) = 4:: V2(t/€2), IT€E[Y1'm) for ‘Y1 > O arbitrary V/€2 but fixed and vl(t) and V2(t) satisfy all the assumptions given in Chapter 2. Moreover, let Rl(T) = B(v1(t)vi(t+-T)), to 03 I R2(T) — E(V2(t)v2(t+T)), wl —]" Rl(T)dT, w2 —f R2(T)dT 0 0 and Sl(w) and 82(w) denote spectrum matrices of V1 and v2, respectively. The main assumption in this section is that the states of the system are not available for perfect measurement. Then, we follow essentially the basic steps of section 4.2 to stabilize the initially unstable system (4.1) and (4.2). So we proceed in doing that as follows: We anticipate that the open-loop reduced—order model of (4.1)—(4.3) takes the form: dX = fl(x)dt+Fl(X)udt+01(x)dwl (4.18) of = £2 (E) dt + F2 (§)udt + 02 (E) dwl + 03 (E) dw2 (4 .19) We consider a controller of the form u = g(X) where X is the output of the observer: 123 61;: = fl (X)dt+ Fl(§<)g(§c)dt+ K[dz - f2(;()dt —F2(§<)g(§<)dt] (4.20) for some constant gain matrix K. The vectors f1(X), f2(x) and the matrices F1,F2,01,02 and 03 are to be determined by applying a similar consistency condition to the one stated in section 4.2. The closed-loop reduced- order augumented model which follows from (4.18)-(4.19) with u = g(X) is: a; = fl(§)dt+Fl(§)g(§<)dt+ol(§)dwl (4.21) 6);: = fl(§<)dt+Fl(§<)g(§<)dt+K(f2(§) —f2(;<))dt (4.22) K (F2 (35) — F2(X) )g (3}) dt + K02(;)dw1 + K03 (x) dw2 To determine the exact form of fl’fZ’Fl’FZ'Ol’OZ and 03 we propose an observer for the full system (4.l)-(4.3), to reconstruct the states x, in the form: § = f (§)-+F (X)u+—K(z-—f l 1 2(32’) —FZ(§)u) (4.23) The gain matrix K is the same as the one appearing in (4.22). Now applying the same control law u = g(X) to (4.1), (4.2) and (4.23), as a function of the reconstructed states 2, then the closed—loop full-order augumented system takes the form: X = al(x) +G (X)g(§)+A (x)y+B 1 12 x2- I — £16.“) +Fl(§<’)g(x) +K(C1(X) —f2(’>2) 426666?) E: 2 + Kc2y+KB3(x)v2 124 6 - _ 'v l uy — a21(x) + 02(x)g(x) +A2y + B2(x)vl which can be simplified to X = al(X)-+A12(X)y-tBl(x)v (4,24) . ~ ~ 6 0y = a21(X)-+A2y-+B2(X)v (4.25) x v61 e l where X== , v = 2 V62 2 '51 (X) = £162“) + Fl(§)g(’>‘£) + x(cl(x) -f2(’>‘€) —Fz(’§)g(’>‘<’)) ‘ (4.26) 32100 = (a21(x) +62(x)g(x)) (4.27) 1‘12”” A12(XJ = (4.28) K02 Bl(x) 0 B1(X) = (4.29) O KB3(X) and B2(X) = (B2(X) O) (4.30) 125 System (4.24) and (4.25) are basically in the form of (2.1) and (2.2) and under additional assumptions, which will be stated later, it can be shown by the convergence result of Chapter 2 that x(t) of (4.24) and (4.25) x converges weakly to a diffusion X = as 61 4 O, A X 6 6 4 O, u 4 0 and I} 4 Y. If we trace the steps of the 2 convergence proof in the case of only two parameters 6 and H we will find that, in the case of three parameters, 6 the ratio is given by 1% and it does not depend on 62, 6 and so we will require 1% 4 Y_<) —F l 3 0 2 The design problem, is to choose a function g(x) which is smooth enough and a constant matrix K such that both the state x and the error e = X-—x will be stochastically asymptotically stable. Let us write the It0 equations X (4.62) and (4.69) in the form (using X = A ) X (33?: §(§)dt+3(3€)dw (4.70) 3300 = b(E) +60(§)g(§<) \b(§<) +GO(§<)g(§<) +K(co(§) —co(§<)) +K(FO(§) -FO(§<))g(§<) (4.71) 132 and 01(x) O 6(X) = (4.72) K02(X) K03(x) Suppose we succeed in finding u = g(x) and the gain matrix K to stabilize (4.70), then the next step is to apply the same control law to the open-loop full order system (4.l)—(4.3) where u = g(x) and x is the reconstructed states and satisfies the equation of the following observer: x==bk)+GOmMflX)+KmlMJ—COM))4beOgW) ~ €2 + Kc2y+KB3(x)v2 . (4.73) where K in (4.73) is the same gain matrix obtained above. Then we would like to spell out all the conditions, under which the stability of (4.72) would imply that of (4.1), (4.73) and (4.2), when u = g(§) is applied to (4.1) and (4.2). This will be done with the aid of the results of Chapter 3. We state here the assumptions that will imply asymptotic stability in probability according to Theorem 1 of Chapter 3. This will require that we consider the case when c2 5 O (in the case when c2 4 O assumptions can be made to show boundedness in the mean square sense according to Theorem 2 of Chapter 3.) So considering c2 E O we require the following assumptions: (A) B (O) = 0, 83(0) = 0, A (0) = 0, c (0) = 0, a (0) = 0, and 12 133 (B) b(x) and G(X) are required to satisfy: [lo/(X) —g(Y)]+ [g(X) —;(Y)] 3 C]X—Y] vx,Y6R2n and (”66612.4 (“56612 g C(l+ (x12) 33.6122“ (C) The coefficients ;i(X), Ai2(X), Bi(x), 521(X) and A4 B2(X) are continuous and have continuous partial derivatives up to the second order which are uniformly bounded in X€R2n in addition to a21(X) and 320:). ~ ~ ~ - 2 (D) 1a(X)]+]A (X)[+[B(X)nglx[ V'XERn l 12 l and for some K > 0. (E) (1; (X) -5 (Y)(+ [E (X) -8 (Y)] g K]X -Yl VX YER2n 0 0 0 0 ' for some K > O. (F) vl(t) and v2(t) satisfy the same type of conditions as stated in (A1) of Chapter 2. (G) The constant matrix A2 is Hurwitz, i.e., 1%2A(A2) 0 134 EV(X) g -XV(X) )(XfERzn (4.74) Then there exist 6%, no, and Y0 such that for all i €1 x(t) 0 < 61 g 60' O < u :40 and ]F-Y] 3 YO. x(t) — 35(t) . the solution of the closed—loop system resulting from (4.1), (4.2) and (4.73) after applying the feedback control u = g(x) is stochastically asymptotically stable as t 4 w. nggf: From the assumptions and (4.76), the solution of the closed-loop reduced-order model represented by (4.72) is stochastically asymptotically stable then following exactly the steps of the proof of Theorem 1 of Chapter 3 after the necessary modification concerning the following estimates, which are similar to the estimates given by (3.14), (3.18), (3.23), (3.26) and (3.27) respectively. 2n (SN here is subset of R ) )V1,N(X’t)] g (KP/77+ K2\/_E; + K3\/_H)V (X) . (v2,N(X.t)l 3 (K 6 +K €2+K u)V(X). 4 1 5 6 €1/Ll €1- IL VN(X) -—LVN(X)] g c]-D--YIV(X), avN 1 ... _ _ _ I 636 (x,t)aO(X)[ 3 (K7\/ el+K8VI €2+K9\./).l ) V(X), and 0V 0V ~ ~ 6 6 N,2 ~ N,2 1 2 I ]—-;3—x---(X.t)<’:1l (X) l + )778‘55-(X’t) (A12(X)y+Bl(X) (v1 .v2 ) )l 3 (K10\/ 61+Kll\/ 62 +K12V/u +Kl3€l +Kl4€2 + Kl5p)V(X) 135 all the above inequalities are true for X O and G2 > 0. Let Sl(w) and 82(w) denote 136 the corresponding power spectrum functions respectively. The open-loop reduced-order model is given by (we take 2 _ _ 1— dx — udt + —xdw (4.78) 2 1 dz = xdt—i—xdw2 (4.79) We want to find a control u = g(g) where ; satisfies the equation of the observer d§< = udt+K[dz —§—< W1 where w = w2 By taking V(X,e) = x2-+e2, we have: £V(§,e) = + l§24-l§2+-K2§2 137 where i. is the diffusion operator corresponding to the Ito equation (4.81). Choosing K = 1, F = 2 we get £v<§£,e) = 433122 —4§e+2e2) g -.23(§2+e2) = -.23v<§,e) V(x,e)€CR2 §' Then it follows that, the solution of (4.81) is e stochastically asymptotically stable. Then applying the control law u = -2x to the open-loop full-order system (4.75)-(4.77), where § satisfies the equation of the observer ~ x = —3x+u+ (2 —§) (4.85) will stabilize the system according to theorem 2 of this chapter, since all the assumptions of the theorem are obviously satisfied. This conclusion holds for sufficiently small 61,62 and M. We notice that in this particular example we do not have to require that the ratio el/u be sufficiently close to a nominal value Y since the reduced—order model is independent of Y. CHAPTER 5 DISCUSSION AND CONCLUSION 5.1. Discussion: In this section we discuss the reduced—order model defined by the operator LY of (2.7) and explore various special cases of practical significance. Inspecting the drift coefficient b(x) and the diffusion coefficient A(X), given in chapter 2, shows that the wide-band nature of v€(t) affects only the drift coefficient. In other words, if one had tried to obtain a reduced-order model by following the intuitively appealing, but wrong, procedure of simply replacing the wide—band noise by its limit white noise and then applying the order reduction procedure of singularly perturbed deterministic systems [1], he would have obtained a reduced-order model with drift coefficient aO(x) and diffusion coefficient A(X). The differences between the two drift coefficients are the terms hl' —A12A-:h2 and h3. These terms depend, respectively, on the partial derivatives of B1,B2 and A12 with respect to x. The appearance of the partial derivatives of B1 and B should be expected in view of the asymptotic 2 analysis of nonlinear systems driven by wide-band noise [ll—l3]. The appearance of the partial derivatives of 138 139 A12 is less obvious. However, if we take into consideration that as e and u tend to zero the process y(t) itself tends to white noise, we can see that A12 plays the role of an input matrix multiplying wide-band process, similar to the roles played by B1 and B2. It is interesting to notice that if the matrices A12,B1 and B2 are constant (independent of x), the terms hl'h2 and h3 will vanish. In this special case applying the intuitive procedure of formally setting H = O and formally replacing the wide- band noise by its limit white noise, would lead to the correct reduced—order model. One disturbing fact about the reduced—order model (2.7) is that the drift coefficient b(x) depends on Y = lim €,p40 through the matrices Z1 and P. This is the consequence ‘CIm of the interaction between the asymptotic phenomena associated with singular perturbations on one hand, and the asymptotic phenomena associated with rapid stochastic fluctuations on the other hand. The dependence of LY on Y has important impact on the engineering practice of neglecting parasitic elements when writing down differential equations representing electrical networks, mechanical systems, etc. It is apparent now that if one would be interested in solving those equations when driven by wide-band noise and using the usual white noise approximations, the parasitic elements should not be neglected from the outset. Rather, they should be included in the system description and their relationship with the wide-band noise be studied in order to obtain the right reduced-order 140 diffusion model. Fortunately, there are interesting classes of systems for which the engineering practice will work out without causing trouble. These are systems for which the operator LY will be independent of Y. Using the explicit form of the operator LY given by (2.7)-(2.17), we can easily identify classes of systems for which this is true. Essentially, we need to look for special cases when 2 = O or when the partial derivatives multiplying Z and P vanish. For example, when B2 = O, the matrix 2 = 0. That is intuitively clear since B2 = 0 means that y(t) would be a smooth process whose elimination from (2.1) can be done using the usual singular perturbation routine. Indeed, we do not need B2 = O for y(t) to be a smooth process in the limit. We only need that B2 takes the special form B2(x) = pagé(x) or B2(x) = €G§2(x) for some constant a > Checking the proof of the theorem, NIH it can be seen that the terms containing B2, 2 or P drop out. In addition, we have already seen that for the class of systems in which AlZ'Bl and B2 are constant matrices; the terms hl'h2 and h3 vanish and the drift coefficient b(x) reduces to aO(X) which is independent of Y. In Chapter 1 we have outlined Blankenship's ppproach [28] and implied that it is valid when E a O as e a O. This can be verified for our problem by applying Blankenship's procedure to our system. The algebraic equation 141 al(x)+Ay+B (x)v€=O (5.1) 2 2 2 has the unique solution -1 € 2[a21( y = -A X) +B (X)V ]. (5.2) 2 Using (5.2), an outer solution for x is defined by 2(t) = a (X(t))+—B (X(t))v€(t). (5.3) O O As 6 4 O, x(t) tends to a diffusion process g(t) whose infinitismal generator has drift and diffusion coefficients defined by b==a +h —A A- and X = BOS(O)B6 (5.5) Where $1 = tr[DinW], (5.6) $2 = tr[E£BOW], (5.7) $3 = —tr[F£BOWBéA_:] (5.8) It can be easily verified that this is exactly our reduced— order model when Y 4 w ( 142 5.2. Conclusions and Future Research: In this thesis, a class of nonlinear singularly per- turbed systems driven by wide-band noise has been considered. It has been shown that the probabilistic behavior of the slow variables can be predicted from a reduced-order diffusion model which has been derived explicitly. The use of the reduced-order model in studying stability of the full-order system, has been examined. Then the possible application of the reduced-order model in control problems has been con— sidered. Stabilizing state feedback and output feedback controls have been designed, where for the latter a non- linear stochastic observer for the reduced-order model has been used. The importance of these results is that of getting an explicit form of a reduced-order model, where the use of this reduced—order model may lead to considerable simpli— fication in solving problems. It is obvious,ikn:example, that any simulation involving the full—order systems will, computationally, be much more difficult than working with the reduced-order model, because of the higher dimension and the ill-conditioning caused by the small parameters 6 and u. The reduced-order model, being a Markov model, has an important significance in its own, and this stems from the fact that the mathematical theory of stochastic differential equations is concerned mainly with the study of Ito equations and the associated Markov processes. This Markov model and its dependence, in general, on the ratio 143 E contradicts the engineering practice of neglecting parasitic elements when writing differential equations representing physical systems as we pointed out in the above discussion. What this reduced-order model tells us is that before neglecting any parasitic elements one has to study their relationship with the wideeband noise. An important step towards the effective use of the reduced-order model has been explored in chapters 3 and 4, where stability properties of the non-Markov full-order system has been established from stability properties of the reduced-order model, and stabilization of the full- system via the use of the Markov reduced—order model has been, also, estabilished. Several additional work and topics are worth of future study, among those are: l. Studying nonlinear systems which are more general than the one that has been considered here and represented by (2.1) and (2.2), in the sense that the system may be non— linear in y and also nonlinear in the driving noise. In this case different stability conditions have to be imposed to guarantee the stability of the boundary layer system. The time varying case may also be considered. We studied here the case when the input noise is bounded and satisfies a certain mixing condition, so one may consider the case when the noise is unbounded in addition to some different conditions other than the mixing one, for example, the case 144 when v(t) is the output of a linear system driven by white noise. 2. Studying systems of the form (2.1) and (2.2) but with \/TI multiplying the coefficient B2(x) in (2.2). This case does not have the trouble caused by the fast variable y(t) which, in our case, tends to white noise as u and e 4 O, [c.f. 27]. One may also be able to say more about the asymptotic behavior of y(t) as u and e 4 O. 3. Studying the possibility of obtaining a near Optimal control by optimizing an appropriate cost function for the corresponding reduced-order model. One may also consider an approach to the output feedback problem different from the one that has been studied in chapter 4 of this thesis. The suggested approach is as follows: 1) Design a stabilizing control law for the open—loop reduced—order model based on state feedback, assuming the states Q, of that system, can be measured. 2) Construct an observer which generates a vector x such that for any u the error x(t)-x(t)-+O as t-+0° in some stochastic sense. 3) Apply the previously determined control law to x(t) then a stability result may be established for the augmented system including the states x . . If this scheme works for the open—loop reduced- X \ order model then this control law may be applied to the full order model in a way similar to what we have established in chapter 4 or in a way similar to the above procedure. This suggested approach is well established for linear time 145 invariant systems [c.f. 37], and it is done in the spirit of the separation principle. A recent work following the above procedure has been done by [39] for deterministic nonlinear systems. 4. An important task for future research is identifying physical systems that can be treated using the results that has been established in this thesis. A conceivable class of a systems, may be represented by the following blockdiagram e s4-v u x ———>®———9 L13} = Ay+Bu -———‘> >1: = F(X.y) ———-T-> /\ actuator plant G(X) (4 controller f‘ a n C . . where s is some reference input and v 18 a Wide-band stochastic process. 10. 11. 12. BIBLIOGRAPHY P.V. Kokovtovic, R.E. O'Malley, Jr. and P. Sannuti, "Singular Perturbation and Order Reduction in Control Theory-4An Overview," Automatica, Vol. 12, pp. 123—132, 1976. J.H. Chow and P.V. Kokotovic, "A Two-Stage Lyapunov— Bellman Feedback Design of a Class of Nonlinear Systems," IEEE Trans. Automatic Control, Vol. AC-26, pp. 656—663, June 1981. A. Saberi and H. Khalil, "Quadratic-Type Lyapunov Functions for Singularly Perturbed Systems," IEEE DCD Conference, San Diego, California, Dec. 1981. K. Ito, "Stochastic Differential Equations in a Differentiable Manifolds", Nagoya Math. J. 1(1950), 35-47. K. Ito, "On Stochastic Differential Equations", Mem. Amer. Math. Sco. 4, 1951. E.B. Dynkin, "Markov Processes", Academic Press, New York, 1965. - I.I. Grihman and A.V. Skorohod, ”Stochastic Differential Equations" Springer—Verlag Berlin Heidelberg New York, 1972. D.W. Stroock and S.R.S. Varadhan, "Multidimensional Diffusion Processes,“ Springer-Verlag New York Inc., 1979. P. Billingsley, "Convergence of Probability Measures,“ John Wiley, New York, 1968. T.G. Kurtz, "Semigroups of Conditioned Shifts and Approximation of Markov Processes, Ann. Probability, 1975, Vol. 3, pp. 618-642. G.C. Papanicolaou and W. Kohler, "Asymptotic Theory of Mixing Stochastic Ordinary Differential Equations," Comm. Pure Appl. Math., Vol. 27, pp. 641-668, 1974. G. Blankenship and G.C. Papanicolaou, "Stability and Control of Stochastic Systems with Wide—Band Noise Disturbance," SIAM J. Appl. Math., Vol. 34, pp. 437-476, May 1978. 146 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 147 H.J. Kushner, "Jump-Diffusion Approximations for Ordinary Differential Equations with Wide-Band Random Right Hand Side," SIAM J. Control and Optimization, Vol. 17, pp. 729-744, November 1979. H.J. Kushner, "A Martingale Method for the Convergence of a Sequence of Processes to a Jump-Diffusion Process," Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 53, 207—219, 1980. H.J. Kushner and Y. Bar-Ness, "Analysis of Nonlinear Stochastic Systems with Wide-Band Inputs," IEEE Trans. Automatic Control, Vol. AC—25, pp. 1072-1078, 1980. F. Kozin "A survey of Stability of Stochastic Systems," Automatica, J. IFAC, 5, 1969, pp.95-112. E. Lukacs "Stochastic Convergence,” Academic press, 1975. T. Tarn and Y. Rasis, "Observers for Nonlinear Stochastic Systems," IEEE Trans. Automatic Control, AC-21, pp. 441— 448 (1976). E. Wong and M. Zakai, "On the Relation between Ordinary and Stochastic Differential Equations," Internat. J. Engin. Science, 3, pp. 213-229, 1965. H.J. Kushner, "Stochastic Stability and Control," Academic press, New York, 1967. A. Haddad, "Linear Filtering of Singularly Perturbed Systems," IEEE Trans. Automatic Control, AC—21, pp. 515—519, 1976. A Haddad, P. Kokotovic, "Stochastic Control of Linear Singularly Perturbed Systems," IEEE Trans. Automatic Control, AC-22, pp. 815-821, 1977. D. Taneketzis and N. Sandell, "Linear Regulator Design for Stochastic Systems by Multiple Time-Scale Method,” IEEE Trans. Automatic Control, AC-22, pp. 615-621, 1977. H. Khalil, "Control of Linear Singularly Perturbed Systems with Colored Noise Disturbance,” Automatica, Vol. 14, pp. 153—156, 1978. H. Khalil, A. Haddad, and G. Blankenship,” Parameter Scaling and Well—Posedness of Stochastic Singularly Perturbed Control Systems," Proc. Twelfth Asilomar Conference, Pacific Grove, California, Nov. 6—8, 1978. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 148 H. Khalil and Z. Grajic, "Near-Optimum Regulators for Stochastic Linear Singularly Perturbed Systems," The let IEEE Conference on Decision and Control, Orlando, Florida, pp. 1317—1321, December 1982. A. Bensoussan, "Singular Perturbation Results for a Class of Stochastic Control Problems," IEEE Trans. Automatic Control, Vol. AC—26, pp. 1071-1080, October 1981. G. Blankenship, "On the Separation of Time Scales in Stochastic Differential Equations," Proc the 7th IFAC Congress, pp. 937-944, Helsinky, 1978. Hoppensteadt, "Properties of Solutions of Ordinary Differential Equations with Small Parameters," Comm. Pure Appl. Math., 24, pp. 807—840, 1971. V.D. Razvig, "Reduction of Stochastic Differential Equations with Small Parameters and Stochastic Integrals," Int. J. Control, Vol. 28, pp. 707—720, 1978. H.J. Kushner, "Asymptotic Distribution of Solutions of Ordinary Differential Equations with Wide-Band Noise Inputs: Approximate Invariant Measures," Stochastics, 6, PP. 259—277, 1982. A.G. Korn and T.M. Korn, ”Mathematical Handbook for Scientists and Engineers," McGraw Hill Co., New York, 1968. G.C. Papanicolaou, D. Stroock and S.R.S. Varadhan, "Martingale Approach to Some Limit Theorems," Statistical Mechanics and Dynamical Systems, Duke Turbulence Conference, M. Reed, ed., Duke University Mathematics Series, Vol. 3, Durham, NC, 1977. J.J. Levin and N. Levinson, "Singular Perturbations of Nonlinear Systems of Differential Equations and an Associated Boundary layer Equation," J. Ration. Mech. Ana1., Vol. 3, pp. 247—270, 1954. A.P. Sage and J.L. Melsa, "Estimation Theory with Applications to Communications and Control,“ McGraw— Hill Book Co. 1979. P. Hall and G.C. Heyde, "Martingale Limit Theory and its Application," Academic Press, 1980. H. Kwakernaak and R. Sivan, "Linear Optimal Control Systems," John Wiley and Sons, Inc., 1972. 38. 39. 149 J. Chow and P. Kokotovic, "Two—Time Scale Feedback Design of a Class of Nonlinear Systems," IEEE Trans. Automatic Control, AC-23, pp. 438—443, 1978. M. Vidyasagar, "On the Stabilization of Nonlinear Systems Using State Detection," IEEE Trans. on Automatic Control, Vol. AC—25, No 3, June 1980. LBR 54 MICIHIIGIINHTIIMITTHWVIWWI IN 3 1293 0307123