DESIGN OF OBSERVERS FOR A CLASS OF NONLINEAR SYSTEMS IN ASSOCIATIVE OBSERVER FORM

Size: px

Start display at page:

Download "DESIGN OF OBSERVERS FOR A CLASS OF NONLINEAR SYSTEMS IN ASSOCIATIVE OBSERVER FORM"

Carmel Strickland
5 years ago
Views:

1 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES DESIGN OF OBSERVERS FOR A CLASS OF NONLINEAR SYSTEMS IN ASSOCIATIVE OBSERVER FORM Ü Koa T Mllar R Pearso Absrac Coos or he esece o a observer orm or olear screeme amc moels are kow o be resrcve movag varos eesos eg geerale observer orms o elarge he class o ssems or whch observers wh lear error amcs ca be esge Ths paper roces a alerave approach base o replacg he sal ao operao + wh a more geeral bar operao ha s assocave coos a cacellave These reqremes lea o a smple represeao or he operao erms o a coos srcl moooc co Ths orm s calle a assocave observer orm a s emosrae ha he kow resls or observer esg ee easl o hs class o olear ssems a el lear error amcs A cosrcve algorhm s escrbe ha eermes wheher he orgal olear ssem ca be rasorme o he assocave observer orm The propose approach s compare wh he geerale observer approach volvg boh sae a op rasormaos a s show ha boh approaches el ecal resls O he oher ha or approach smples he compaos o he op rasormao whch are oe wo epee seps a o o reqre he solo o a ssem o ereal eqaos as he geerale observer approach oes Kewors: Nolear corol ssem Dscree-me ssem Geerale observer orm Assocave bar operao INTRODUCTION Coser he olear ssem G h where s he sae are respecvel he p a op o he ssem I hs paper we coser he esg o observers or scree-me olear ssems o he orm b meas o he so-calle assocave observer orm whch s a geeralao o he saar observer orm Roghl speakg a ssem observer orm s a lear observable ssem ha s ercoece wh a op a pepee olear: AT&P oral PLUS 7 8

2 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES Observers or hs k o ssems ma be cosrce b blg a classcal lear Leberger observer or he lear par a ag he measremeepee o-lear o hs observer The problem o rasormg he screeme olear ssem o he observer orm has bee se or ssems wh oe op a who ps [9] a [] A eeso o ssems wh ps a a olear op co h s gve b Igebleek [7] Uorael he coos or he esece o a observer orm are eremel resrcve Thereore ere k o geeralaos have bee cosere o elarge he class o ssems or whch oe ca esg a observer wh lear error amcs: eher he class o rasormaos allowe was elarge or geerale observer orms were roce [] For eample beses sae rasormaos also op rasormaos [6] ssem mmerso o a hgher mesoal ssem [8] or op-epee me scale rasormaos [5] were cosere Fall he paper [] aresses he problem o rasormg he olear ssem o olear observer caocal orm he eee saespace b agmeg he orgal ssem wh some alar saes a eg vral ops As a geeralao o he observer orms he so-calle geerale op eco was roce ha beses he ops a he ps epes also o a e mber o her me ervaves or shs he scere me case Ths paper roces a alerave geeralao o he amlar observer orm The geeralao presee here I [9] he orer o he sae varables s perme s base o replacg ao observer orm wh a more geeral bar operao reqre ol o be assocave coos a cacellave These reqremes he lea o a sel smple represeao or he operao erms o coos srcl moooc co Ths orm s calle a assocave observer caocal orm a or ask s observer esg or sch ssems Or movao s o eplore he ee o whch kow resls or observer esg o or o o ee o hs class o olear ssems ASSOCIATIVE BINARY OPERATORS The bar operaors cosere here ma be vewe as a mappg rom some oma D I I o I where I s a erval o real mbers ha ma be e or e b ms be ope o a leas oe se Frher s assocave sases or all a I Eqvalel hs bar operao ma be wre as F recg o he assocav eqao []: F F F F or all I Frher s coos he map F : I I I s coos a cacellave eher o he ollowg coos mples : or I has bee show [] ha he bar operaor s coos assocave a cacellave o I a ol 4 where s srcl moooc a coos o I The mos commo AT&P oral PLUS 7 9

3 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES eamples are ao correspog o a mlplcao correspog o l ; he oher eamples are he parallel combao ee as arsg rom he parallel combao o ressaces elecrcal eworks a ee b he co a he proecve ao operao ee as [] whch correspos o he co / For coveece he class o all assocave coos a cacellave bar operaors wll be eoe I ollows rom 4 ha a bar operaor s also commave: a as a coseqece he combao: s vara er arbrar permaos o he erms Aoher eremel sel coseqece o he represeao 4 s ha he bar operao s verble wh a verse operao gve eplcl b: 5 I ollows recl rom 4 a 5 ha Whe eoes ao or mlplcao he verse operaos o sbraco a vso are well-kow As less obvos eamples oeha he verses o he parallel combao a he proecve ao operao are gve b: OBSERVER DESIGN FOR SYSTEMS IN ASSOCIATIVE OBSERVER FORM The assocave observer orm s ee b replacg all aos he observer caocal orm wh arbrar bar operao rom : ˆ ˆ 6 Relae o ssem 6 he ollowg amcal ssem ˆ ˆ k ˆ k k ˆ 7 calle observer a ee or he error erm e ˆ Proposo Observer 7 garaees or ssem 6 assocave observer orm a lear error amcs e e k e e e k e 8 e ke Proo B 4 we ca rewre eqao 6 as ollows ˆ AT&P oral PLUS 7

4 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES where s a srcl moooos co ee b he bar operaor Aalogosl b 4 a he observao ha elg we ca rewre eqao 7 as ollows ˆ ˆ ˆ ˆ k ˆ ˆ k ˆ k ˆ 9 The sraghorwar compao els ow 8 or error amcs Sppose sases he geerale homogee coo [] p 45: k k or all k I Le a oe ha k k k K I ollows rom hs observao ha sases Cach s power eqao: k k k k or whch he ol coos solos are kow o be [] p : sg where s a real cosa I ollows rom hs resl a he moooc reqreme o ha hs co ms be o he orm: sg 4 or some oero real a some Noe ha er hese coos ollows ha: / sg 5 a ha / ~ k k sg k k 6 The avaage o hese observaos s ha sases hese coos we ca raw wo coclsos Frs he cl erm appearg he observer eqao 7 above or becomes: ~ k ˆ k ˆ ~ k ˆ 7 so he observer eqaos or ow have he orm: ~ ˆ ˆ k ˆ a ~ ˆ k ˆ The oher avaage o hs coo s ha mples ha I orms a algebrac rg where represes orar scalar mlplcao The ke les he ac ha he geerale homogee coo s sce o mpl ha s srbve over or he proo see Appe We coecre ha hs coo s also ecessar e ha sases he geerale homogee coos I orms a rg However he proo s le or re research AT&P oral PLUS 7

5 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES I ssem oes am a assocave observer orm a observer or ssem ma he be obae b rs cosrcg a observer 7 or he ssem he assocave observer orm 6 he ew cooraes T a he leg ˆ T ˆ be he esmae o We hs see ha observer esg or ssem s relavel eas whe ca be rasorme o assocave observer caocal orm Ths rases he qeso er wha coos ca be p o assocave observer orm SYSTEM TRANSFORMATION INTO ASSOCIATIVE OBSERVER CANONI- CAL FORM The crcal po he cosrco o a olear observer o he orm 7 wh lear error amcs 8 s he rasormabl o he scree-me olear ssem o assocave observer orm 6 I [] p-op erece eqaos wh assocave amcs 8 were se wh respec o realabl/realao a was show ha he assocave moels o he orm 8 o have a classcal sae space realao he assocave observer orm 6 Wha s especall mpora wh regar o he opc o hs paper s ha oce he assocave srcre o he p-op moel correspog o s recoge he sae space moel cosrco a assocave observer orm 6 s rec allowg a smple raslao rom pop moel 8 o sae space moel 6 So or approach s o he p-op eqao correspog o he sae eqaos or whch we wa o cosrc he observer a check hs eqao ca be p o assocave /o orm 8 Ceral s o alwas eas o recoge he assocave moel srcre p-op eqao 9 b smple speco sce epes o esece o cera cos o ee avace I [] a algorhm was gve o check a hgher orer /o erece eqao ca be wre he assocave orm 8 Ths algorhm perms compao o he reqre cos sep b sep wheever he es The algorhm s cosrcve p o egrag some oe-orms whch s ver commo olear seg To make hs paper sel-sce we recall hs algorhm below Algorhm Calclae or Check: I o sop; oherwse I hols or all he accorg o ormla he oal ereal o co AT&P oral PLUS 7

6 AT&P oral PLUS 7 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES ca be wre as 4 Coseqel or all 5 However he problem o g he co ha eermes he assocave operaor s sll a ope qeso Thogh a complee solo remas a sbec or re research we sgges he ollowg approach Problem Gve /o eqao 9 whch s kow o am he assocave /o orm 8 a he cos he co 8 Solo From 7 oe ca easl compe he ollowg epressos ' ' 8 or Tha s rom 7 a ' ' a ow b ' ' elg 8 From eqaos 8 s oe possble o he co We wll emosrae he compaos Seco 6 o hree eamples Uorael corar o wha was clame [] coos are o sce o rasorm he p-op eqao 6 o he orm 8 The ollowg ssem proves a coereample or whch he ecessar coo s sase b he eqao ca o be wre he orm 8 Accorg o he -orms are a b he cos a he egrag acors are respecvel as a Compe

7 4 AT&P oral PLUS 7 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES 9 The rec compao shows ha or However we r o we oba he ollowg relaos The seco ormla leas o ' ' a sce coas a varable here oes o es as a sglevarable co The same happes wh he hr ormla ssem Coseqel he co: : oes o es or hs eample Reall hols or he e o co ca be wre as a compose co b o e as he orm 8 a he egrag acors ca be epresse as compose cos as well Or e ask s o he ecessar a sce coos see Theorem below o rasorm he p-op eqao o he orm 8 We sar b provg a lemma ha s arl sraghorwar eeso o Hbers resl [7] Theorem 6 Noe however ha he assocave amcs orm s ere rom he srcre o he op eqao cosere [6] Ths lemma wll be sel provg he Theorem below Lemma I he p-op eqao ca be rasorme o he assocave amcs orm 8 here ess a co S sch ha or S 4 Proo B 8 5 Takg he ereal els b ' ' ' 6 Coseqel or

8 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES ' ' 7 Sce he rgh ha se o 7 s a oal ereal also he le ha se has o be a oal ereal a s eeror ereal eqals ero ' ' 8 From 8 or l ' S Theorem The coos 4 are ecessar a sce o rasorm he p-op eqao 6 o he assocave amcs orm 8 Proo Scec The proo alls arall o hree seps O he rs sep we wll show ha er he coos o Theorem here ess beg a oal ereal o a cera co S sch ha 4 hols O he seco sep we wll prove ha 4 els a all o he las sep we wll show ha Frs oe ha case 4 els Takg he eeror ereal o we oba l 4 Obvosl cao be ake eqal o l sce here s o reaso o assme ha all egrag acors are eqal However we ma search he orm l A 4 The we have 4 For 4 o hol we have o prove ha er 4 44 or all Sce he mber o cooraes he /o space s a he mber o -orms k k s he o o orm he bass a ever -orm cao be wre as he lear combao o he -orms However as we wll show he seqel hols he - orms ca be epresse as he lear combaos o B 4 a A k k k A k k k k A where k k k k k k 45 k Ak B sbsg rom 45 o 4 we ge k k k a sbsg he las resl o 4 we ge or all AT&P oral PLUS 7 5

9 6 AT&P oral PLUS 7 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES k k k k whch els 44 To cocle he rs sep oe ha rom 4 l A A ollows mmeael ha A or are oal ereals Thereore s also oal ereal a ca be wre as S Sce b s a oal ereal s eeror ervave S S eqals ero a b Cara s Lemma } { spa S Thereore a co S ca be epres-se as a compose co T S The laer allows s o ee a co sch ha S e ' 46 The ' l S 47 Accorg o 4 a 46 or ' l 48 rom where b rec compao we ge ' 49 Thereore here es cos sch ha ' 5 Mlplcao o b ' gves ' 5 elg 5 Fall we have o show ha or 5 De o 5 a 5 ' ' ' 54 elg 55 Necess Deoe The where ' 56 a l 57 Drec compao ow gves l 58 From 56 ' ' ' ' ' ' l l where

10 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES ' ' ' ' ' ' 59 Sbsg 59 o 4 a akg o acco ha '' ' '' 4 ollows mmeael Remark I he case coo s boh ecessar a sce I ollows rom he ac ha er a he elg 4 4 COMPARISON OF TWO METHODS I hs seco we compare he meho se o rasorm he p-op erece eqao 6 o he assocave observer caocal orm escrbe he prese paper wh he meho gve [6] Noe ha Hbers cosers he eqaos 6 who ps a searches or he op rasormao p : so ha he co 6 sases p The correspog sep or meho s o rasorm eqao 6 o he orm 8 Sce 6 oes o coa he ps we reqre 6 Comparso o 6 a 6 emosraes he aalog bewee wo mehos a gves p 6 To eerme he co p Hbers ees he -orms 64 The correspog sep or meho s ormla o ee he -orms 65 Obvosl he -orms or meho a he -orms se he Hbers meho are relae he ollowg wa 66 I orer o rasorm 6 o he orm p all he -orms ~ p p' or have o be oal ereals hereore or he ollowg has o hol: p' p' The laer els he ssem o ereal eqaos o eerme he co p : l p' 67 The -orms are geeral o eac b accorg o [6] he mlplcao o wh he co p' gves s he eac -orms ~ p' 68 Usg 67 oe ca he co p' a he va egrag p b boh seps are o eas aks geeral Sce p' have o be oal e-reals or he same has o hol or AT&P oral PLUS 7 7

11 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES p' p' see 66 Accorg o ormlae 66 a 67 oe ma also wre l p' 69 Usg or meho he correspog sep s o check or he coo 7 whch sase els 7 The laer meas ha b mlplg he -orms b he egrag acor gves s also he eac ereals Ne we look or he relaoshp bewee he coeces a p ' 68 Accorg o 6 oe ca wre eqao 6 also he orm p p p Takg he paral ervave wh respec o els accorg o 64 p' p' Comparg he obae resl wh 7 we ge p' 7 p' O corse oe ca also he ere egrag acors Takg o acco ha p oe ca p rom he ssem o eqaos 8 5 EXAMPLES Eample Coser he op eqao To calclae he -orms we ake he paral ervaves a o oba The coeces a are a we all oba ha a To compe b 8 ' ' elg ' Oe ca as 7 Ths choce wll el : Y a sce AT&P oral PLUS 7 8

12 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES Y Y We ca also calclae co p sg Hbers meho Calclag -orms accorg o 64 we ge 74 To he co S l p' 75 ecessar or calclag p we se ormla 67 a calclae he eeror ereals o a The S 76 Coseqel S ess b 74 a 76 S S wh S 77 Sce s a eac -orm coo 67 gves S elg S 78 The co S ca be o b solvg he ssem o ereal eqaos 77 a 78 Noe ha he geeral case hs ca be a ver complcae ask Takg o acco also 75 we ge ' p 79 To eerme he co p he rgh ha se o 79 has o be epresse erms o o Ths sep aga e-remel complcae leas o he resl p' or p' 8 The egrao gves p 8 he same resl as 7 obae b or meho Eample Coser he /o eqao b c a 8 Compe b c a b a c a 8 rom whch b c a a Thereore b 8 AT&P oral PLUS 7 9

13 4 AT&P oral PLUS 7 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES ' ' The laer els ' a l Applg he Hbers meho oe compes 84 a a c b So S a b akg o acco /o eqao 8 a relaoshp 84 he laer els l q a c b S From here s geeral o possble o he co q Eample Coser he /o eqao a a 85 Compe a a a a 86 rom whch a Applg ow ormla 8 we oba ' ' From above we ge ' a all 87 Applg he Hbers meho oe compes 88 a S a a S a a a 89 Comparg ormlas a 86 we ge a a a S elg l l ' l g a a p S where he las erm s a co epeg ol o The ' ag a p Sce he las epresso has o be epresse va 85 we ake

14 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES g a ge p' elg p / Ths solo agrees p o he sg wh he solo or 87 6 CONCLUSIONS Ths paper has roce a ew class o olear observers ha ehb lear error amcs The bass or hs observer class s he replaceme o he sal ao operaor + wh a more geeral operaor he caocal observer orm or olear scree-me amc moels ha has bee cosere prevosl b a mber o ahors The reslg srcre calle he assocave observer orm s sgcal more leble ha he caocal observer orm greal elargg he class o olear moels ha ca be represee The operaor o whch hs eeso s base sll ehbs a mber o characerscs o he sal ao operaor: parclar s reqre o be assocave coos a cacellave mplg ha ma be epresse erms o a coos srcl moooc co Takg reces o + a reces he assocave observer orm roce here o he well-kow caocal observer orm Allowg o be more geeral b reqrg o sas he geerale homogee coo scsse Sec mples ha he operaos a scalar mlplcao sll orm a rg as he case where s he sal ao operaor a leas o a assocave observer ha correspos o he sal lear observer b wh slghl moe gas wh + replace b a wh replace b he verse o he operaor whch s also smpl epresse erms o he co Relag he geerale homogee coo leas o he mos geeral case where he assocave observer orm s slghl more comple b sll reasoabl sraghorwar I all cases he error amcs rema lear as show Seco I ao o eg he assocave observer class s escrbe we have also presee a realao procere or pg a gve olear screeme amc moel o assocave observer orm We emosrae hs meho or hree smple eamples a also compare wh he meho o Hbers whch leas o he same resl or wo o hese hree eamples b als o el a solo or he hr I ao he compaos volve or procere are sbsaall smpler 7 APPENDIX Proposo Uer he geerale homogee coo I orms a algebrac rg Proo A rg see [4] p s ee as a se X wh wo bar operaos a ha sas he ollowg aoms: A: s commave: or all X ; A: s assocave: or all X ; A: a ero eleme ess Z X sch ha Z or all X ; A4: ave verses: or all X here ess X sch ha Z ; AT&P oral PLUS 7 4

15 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES M: he operao s assocave: or all X ; D: boh operaos sas le srbv: ; D: boh operaos sas rgh srbv: I or paper he operao s ee o be assocave a s show Seco ha s also commave To show he esece o a ero eleme oe ha Z mples Z Z Z 9 Frher sce ehbs he verse operao ollows ha he ave verse or a eleme X s smpl Z Ths or operao _ alwas sases coos A hrogh A4 I he case o eres here he operao s smpl scalar mlplcao whch s boh assocave a commave so ha coo M s sase a coos D a D are eqvale Ths he colleco X ees a rg a ol srbv coo D s sase I erms o he co hs coo s: Ne sppose sases he geerale homogee coo: 9 a oe ha he rgh-ha se o Eq 9 he reces o φ 9 Seg v a applg he verse co o boh ses o Eq 9 els he resl ha v v 94 Combg Eqs 9 a 94 he gves 95 esablshg ha srbv coo D hols Ths sases he geerale homogee coo 9 ollows ha X ees a rg Ackowlegemes Ths work was parall sppore b he Esoa Scece Foao Gra r 69 REFERENCES [] J ACZÉL: A Shor Corse o Fcoal Eqaos Reel Dorrech 987 [] J ACZÉL J DHOMBRES: Fcoal Eqaos Several Varables Cambrge Uvers Press New York 989 [] ST CHUNG J W GRIZZLE: Sample-aa observer error learao Aomaca 6: [4] LL DORNHOFF FE HOHN: Apple Moer Algebra Macmlla New York 978 [5] M GUAY: Observer learao b op eomorphsm a opepee me-scale rasormaos Pergamo Press Oor AT&P oral PLUS 7 4

16 MODELLING SIMULATION AND IDENTIFICATION OF PROCESSES [6] HJC HUIJBERTS: O esece o eee observers or olear sree-me ssems I: H Nmeer a T I Fosse eors New Drecos Nolear Observer Desg volme 44 o Lecre Noes Cor a I Sc Sprger-Verlag 999 [7] R INGENBLEEK: Trasormao o olear scree-me ssem o observer caocal orm I Prepr o h IFAC Worl Cogress 4 Se Asrala Ise o Cberecs a TUT Akaeema ee 68 Tall Esoa Tal: Fa: e-mal: koa@ccocee ael@parsekee Roal Pearso ProSaos Corporao 5 Marke S Se 5 Harrsbrg PA 7 USA Fa: e-mal: roalpearso@prosaoscom [8] P JOUAN: Immerso o olear ssems o lear ssems molo op eco SIAM J Corol Op 4: [9] W LEE K NAM: Observer esg or aoomos scree-me olear ssems Ss Corol Le [] T LILGE: O observer esg or olear scree-me ssems Er J Corol [] D NOH NH JO JH SEO: Nolear observer esg b amc observer error learao IEEE Tras o Aomac Corol [] RK PEARSON Ü KOTTA S NÕMM: Ssems wh assocave amcs Kbereka [] EI VERRIEST: Lear ssems over proecve els I Proc 5h IFAC Smposm o Nolear Corol Ssems NOLCOS Sa-Peersbrg 7 Ülle Koa Tael Mllar AT&P oral PLUS 7 4

Continuous Time Markov Chains

Continuous Time Markov Chains Couous me Markov chas have seay sae probably soluos f a oly f hey are ergoc, us lke scree me Markov chas. Fg he seay sae probably vecor for a couous me Markov cha s o more ffcul ha s he scree me case,