A Robust Fuzzy Control Approach to Stabilization of Nonlinear Time-delay Systems with Saturating Inputs

Transcription

1 5 eaoa Joua of uzzy ye o o ach 8 Robu uzzy Coo ppoach o abzao of oea e-deay ye wh auag pu Che-heg g bac h pape dea wh he abzao of ucea oea e-deay ye ubec o pu auao oea e-deay ye f epeeed by akag-ugeo - fuzzy ode a e of fuzzy pcao whch ae ued o chaaceze oca dyac wh acuao auao aed o a deay depede aby aay a doa of aaco whch he adbe a ae ae eued o coege aypocay o he og deeed he fuzzy coo aw he deeoped o aze he eao of h doa he deed codo ae fouaed e of ea a equae o ha he yhe of fuzzy cooe ad he eao of aby doa ca be caed ou effcey oeoe a obu abzao fo ye wh paaee uceae whch ae e-ayg ad o-bouded dcued ueca eape of uck-ae coo ae poded o deoae he effecee of he deg Keywod: - fuzzy ode obu fuzzy coo doa of aaco oduco he cooe hae bee wdey ued he duy becaue of he pe ucue ad epee co o deg e he hee-e fucoay coo coe eae o boh ae ad eady-ae epoe of pacca egeeg pocee oe ha 9% of dua cooe ae of ype appeaed a couou-e o a dcee-e eo pe of he eee appcao cooe ay uffe codeabe o of pefoace due o egao wdup whe ued a ye wh acuao auao ] he acuao auao ey coo Coepodg uho:che-heg g wh he epae of ecca geeg aoa ooa Uey 64 Wuhua Rd uwe Yu awa 63 -a: g@uwfueduw aucp eceed 8 Oc 7; eed 5 eb 8; acceped ach 8 becaue he acuao cao dee ued eegy o phyca pa he effec of h auao ca age fo degadao of ye pefoace o coed-oop aby abzao of ye wh pu auao ha daw uch eeach aeo he pa yea eeay a ow ga coo aegy peeed o deceae he oupu of cooe owee he ow-ga cooe uuay ha ow ee of pefoace he aay ad yhe of coo ye wh acuao auao oeay ca be cafed o wo a appoache he f oe decy o ake he effec of oea auao degg cooe -4] he ecod ehod deeoped o he aupo ha a cooe ha bee peouy deged o afy oe pecfcao he a a-wdup oop cuded o gae he fuece of auao o he ye aby ad pefoace 5-7] Whe degg cooe fo he ye wh pu auao goba aby of a coce eea eu ae ge 8 9] owee he aby he ude ad oy fo ope-oop abe ye 5] ead oca aby of coed-oop auaed ye egaed ha a ego of aaco whch he adbe a ae w coege aypocay o he og he peece of auaed pu deeed ug deged coo aegy ] he doa of aaco efed by appyg he coeoa cce ad opo cea uga ad abouech 6] oduced a ege a-wdup copeao a ea ye wh pacua auag coo aw fo eao of ego of oca aby oeoe hey eeded h oo o yheze a oea ae feedback cooe o dea wh oupu ackg pobe 7] whee a a-wdup ga deed o aze he ego of aaco 5] ohe appoach fo eag he doa of aaco ge by u e a 3] whee a auay feedback a uzed o ea he coea he eao a wok baed o u ehod ae addeed ] oaby he pa he afoeeoed eaue ea o abzao of oea ye wh coaed pu he eg ude deoed o h ubec ae ey ed ] baed o pu-oupu eazao echque a appoach e- 8

2 C- g: obu fuzzy coo appoach o abzao of oea e-deay ye wh auag pu 5 gag oea ea ode coo wh a-wdup oop deeoped fo ucea oea ye wh pu auao oeoe u 3] epoyed dg ode coo ehod wh a a-wdup copeao o abze a ca of oea cacade ye e deay ofe ecoueed ay pacca ye uch a checa pocee og ad og ao e peuac o hydauc e deay uuay a ouce of aby ad ca deceae ye pefoace heefoe degg coo ye wh e-deay ey chaegg o dea wh he abzao of ea e-deay ye wh pu auao a a-wdup coo appoach baed o a odfed eco codo popoed 4] he age aby ego ca be eaed a he a-wdup cooe he obece of h pape o deg a fuzzy coo aw fo abzao of oea e-deay ye wh auag acuao he oea ye f epeed a akag-ugeo - fuzzy ode 5] e a popey choe yapuo-kaok fucoa caddae uffce codo fo eug aypoc aby of coed-oop ye a ego ae deed addo he fuzzy cooe deeoped o aze he ego of aaco he obaed eu he afoed o a coe pobe ha ca be effcey oed by ug he ea a equay echque edg he deg pcpe we code he pobe of abzao of ucea oea ye wh e deay ad pu auao whee he uceae ae aued o be e-ayg ad o-bouded ay copue uao o uck-ae coo ae peeed o deoae he effecee of popoed ehod oao: deoe he dey a R ad a ad fo he aa ad a egeaue of a epecey b b ] R deoe he aach pace of couou eco fuco appg he ea ] o R wh opoogy of ufo coegece φ up φ c pee o aue of a fuco φ b c by b { φ b ; φ > } b deoe a e defed eay Code a oea e-deay ye decbed by f ϕ ] whee R he ae eco R he coaed coo pu fo a ge deoe he eco of o he ea ] aaed o ] e θ θ fo θ ] a e-ayg deay ad afe ad a ea poe coa peeg he uppe boud of e-ayg deay fuhe aued ha ad a kow coa he coaed pu epeed a u whee u he deged cooe ad a adad auao fuco a : u ] u u u R R deoe wh u g u { u oaby he oao ghy abued fo epeeg boh a caa fuco ad a eco fuco o whou o of geeay h udy aue ha he auao ee equa o oe o a o-uy auao he ee of auao ca be abobed o he pu ˆ U uˆ U u whee U dag u a ad u a he auao apude of he h pu he aby aay fo e-deay ye geeay cafed o wo caegoe aey deay-depede cea ad deay-depede cea he deay-depede cea aow a age e deay ad appy o he ye whch o a po kowedge of deay e aaabe 6] coa he deay-depede cea cude foao o he deay ad ca hade he ye whoe aby deped o he ze of he e deay 7 8] h egao deoed o he deay-depede abzao o decbe oea ye he - fuzzy ode 5] egaded a a effece oo becaue ca appoae a cope o defed ye o a copac e o abay accuacy aed o h appoach he - fuzzy ode fo oea e-deay ye epeed a a ue : f z adad z he C ϕ ] whee ad ae ye ace of copabe deo he fuzzy e z he pee aabe of fuzzy pcao he ube of he fuzzy ue aued ha he pee aabe do o deped o he pu he cee of gay defuzzfcao yed he oupu

3 5 eaoa Joua of uzzy ye o o ach 8 of he fuzzy ye w z ] w z whee w z z ad z deoe he gade of ebehp fuco coepodg o z e z w z w z 3 he he oea fuzzy ye ca be peeed by z ] 4 oaby z ad z fo oeoe he fuzzy cooe accodg o he paae dbued copeao C ge by Coo ue : f z adad z he u o equaey u 5 he goba aby codo of oea e-deay ye whou acuao auao ae deed 6] whch baed o he deay-depede aay he peece of pu auao he codo cao be apped he degg wok ead o code he oca abzao o he oea e-deay ye 4 ge he aupo θ > he doa of aaco defed a he ego coag a a cod- c o θ b uch ha he coepodg ae aecoy w coege aypocay o he og ce deeao of he eac doa of aaco paccay pobe 4] a pobe of ee o eae a e of he adbe a codo Ω { b : θ δ} uch ha he aypoc c aby eued heefoe he obece of h udy o deg he fuzzy cooe 5 o aze he doa of aaco oaby a ude h fed hae bee epoed 4 9] whee he aayzed ye ae ea e-aa oupu-deayed o a - fuzzy ode epecey he popoed ehod ca be ewed a a eeo of he wok efoe poceedg he deao oe ue eaed o he deeope of he popoed ehod eed o be addeed e he h ow of a a R be f efe he yec poyhedo { R : f } 3 aby aay ad deg of fuzzy ye ca be eay efed ha h pe pacua deoe he ego whee he coo u doe o auae ha f he he coed-oop dyac of 4 becoe ] 6 h cae he aby ca be eued by a popey choe coo aw 6] e be he e of dagoa ace whoe dagoa eee ae ehe o o eape uppoe he Coequey hee ae eee eoe each eee of a e obouy ea ]: e R ad R f he co { } whee co deoe he coe hu ha ca be epeed a 7 whee Chooe ad a ad epecey y ea he fuzzy coo aw 5 ca be we a

4 C- g: obu fuzzy coo appoach o abzao of oea e-deay ye wh auag pu 53 8 oaby he codo ao pe e a poe defe a R ad a poe caa a epod defed a } : { R oeoe de f ad oy f he foowg codo ae afed 9]: f f whee f he h ow of ea 7]: o ay R y ad ay poe defe a R he foowg equay hod: y y y y y ea 3 ]: e ad be ea ace of appopae deo wh he fo ay poe defe a ad ay caa > afyg > he foowg equae hod: a fo ay caa > ; b ubug 8 o 4 yed he coed-oop ye dyac ]} ] { ] > 9 whee deoe efe he ]} ] { > o oaoa coeece he foowg deao a fuco of w be ucccy epeed by doppg ague he peeed appoach deed a foow heoe Code he e-deay fuzzy ye decbed 4 uppoe ha he poe defe ace auay ace ad a caa > e ad he foowg codo hod: / ; / ad he fo a a ae Ω θ wh a a δ 3 he coed-oop ye guaaeed o be aypocay abe by he fuzzy coo aw 8 oof: Chooe he yapuo-kaok fucoa a / d 4 he / / / 5 ubug o 5 ge /

5 54 eaoa Joua of uzzy ye o o ach 8 / 6 y ea he foowg eao hod 7 he 8 f he codo ad ae afed he > h pe κ c fo a poe caa whee / a a κ ece foow ha κ δ a peeed 3 o ay ] θ θ Ω he aecoy cofed aed o Kaok heoe ] oe ay cocude ha fo ay a codo beogg o Ω he ye dyac aypocay abe h copee he poof oaby he codo h pode a ba fo he aby aay ad he coe epeo of u Whe he pu uauaed u ou o be a doa of aaco ge a popey deged coo h kd of eao coeae pacuay fo he auag uao o oecoe he defcecy a e coeae agoh fo eao of he ego of aaco oduced 3 9] whee a auay a uzed o deee f he deed epod coace poed ha he eaed ego of aaco ca be eaged ode o appy opzao echque fo yhezg fuzzy cooe he aby coa heoe ae afoed o a ea a equay pobe y chu copee he codo ad ae equae o he foowg fo: W 9 W whee W W W ad * deoe he apoed eee he yec poo o he -pocedue he codo ha he epod beog o h ca be depced a Z γ whee γ Z he h ow of Z ad Z Reca ha obece of h udy o deg a fuzzy cooe o aze he doa of aaco o h ed a opzao ceo oduced he coe of heoe ceδ pee a eaued ze of he doa of aby ae aue of a ad a w pode a age aue ofδ o equaey a age doa of aaco ece he opzao pobe aed a foow 4]: { } c c ubec o he coa 9- ad W whee c ad c ae ug faco

6 C- g: obu fuzzy coo appoach o abzao of oea e-deay ye wh auag pu 55 oaed by he eu of heoe he deg pcpe eeded o he - fuzzy ye wh uceae Code he - fuzzy ode ] he ad ad : f z z R ϕ whee ad ae coepodg uceae ha epee paaee aao he dyac of he fuzzy ye epeed a ] z 3 addo he uceae ae epeed he fo of ] ] b Ξ whee ad b ae kow ea ace of appopae deo ad Ξ deoe ukow e-ayg a fuco afyg Ξ Ξ Wh he fuzzy coo aw 8 he coed-oop ye ca be we a ]} ] ] 4 whee heoe o he ucea fuzzy ye decbed 3 uppoe ha poe defe ace auay ace ad poe caa ad e uch ha he foowg codo hod: ; / 5 / b b b b 6 whee ad h he fo a a ae Ω θ wh a a δ 7 he aypoc aby of he coed-oop ye guaaeed by he fuzzy coo aw 8 oof he yapuo-kaok fucoa caddae choe a d 8 he / 9 Ug ea ad ea 3 oe ca efy he foowg eao: b b ] ubug 4 o 9 ad appyg he aboe eao ge } ] ] b b b b 3 f he codo 5 ad 6 ae afed

7 56 eaoa Joua of uzzy ye o o ach 8 he heefoe bouded ad deceag ha κ c > aed o he Kaok heoe ] he aypoc aby guaaeed fo ay a codo beogg o Ω o chu copee he ea a equae 5 ad 6 ca be afoed o he foowg fo: U Y b U Y b W Ψ Y b W W Ψ W Ψ 3 W 3 whee U U Y Ψ W W W W ccodgy he au doa of aaco obaed by a feabe ouo of he opzao pobe { c c } ubec o he coa 3 3 ad 33 W whee c ad c ae ug faco 4 ueca eape h eco copue uao o backg up coo of a uck-ae 7] apped o deoae he effecee of he popoed agoh he e-deay uck-ae ode decbed by a q a 3 q a q a q q q a q a q q a q a q a q a q a dq a q ] q dq q a q a q a q a q a dq a whee a 7 q π / 7 44 π / he coo pu aued o be cofed wh uy auao ee o he ge aue of he uceae efed ha ad o equaey 95/ / 5/ ad 95/ / 5/ he - fuzzy ode fo h ye 7]: Rue : θ a q / a q / abou

8 C- g: obu fuzzy coo appoach o abzao of oea e-deay ye wh auag pu 57 Rue : θ a q / a q / abou π o π he ye ace ae ge beow: q a q a q a q q a q a dq dq a 59 5δ δ δ δ q q a q a q a q a q a dq a whee d / π ad δ he ebehp fuco ae chaacezed by ep 3 θ 5π ep 3 θ 5π he fuzzy coo aw ge by u he oa codo he feabe ouo of obaed a foow: ] ] ] ] δ 535 he a ae e a 45π 75π ad 55 ece ] Ω oeoe he popoed cooe copaed wh he fuzzy cooe deeoped by Cao ehod 6] whch he feedback ga ae ] ad ] gue -4 how he coed-oop ye pefoace; he od e pee he pefoace of he popoed appoach ad he dahed e efe o he ye epoe by ug Cao ehod he uao eu eea ha boh cooe ca abze he uck-ae backg up coo owee he cooe baed o Cao ehod doe o cude he effec of he pu auao o oupu eache he auao ee a dcaed g 4 h codo udeabe becaue he pu auao gh degade he ye pefoace ee caue he coed-oop ye uabe coa he popoed cooe ede a afacoy pefoace ad cofe coo aue wh he uy he aby doa a eaed eued by he popoed cooe he peubed codo he cuded he uao oe he opzao pobe 33 ad yed he foowg eu:

9 58 eaoa Joua of uzzy ye o o ach ] ] ] ] δ he uao eu egadg he a ae 75 ] π π ae poed g 5-8 whee he od e deoe he ye epoe of he popoed appoach ad he dahed e dcae he ye epoe baed o Cao ehod Obouy he coo aue wh he age of uy ad he aypoc aby guaaeed he effecee ad coece of popoed ehod deoaed e gue ae epoe of e gue 3 ae epoe of e gue ae epoe of Coo pu e gue 4 e epoe of he coo pu

10 C- g: obu fuzzy coo appoach o abzao of oea e-deay ye wh auag pu 59 gue 5 ae epoe of ude he peubed codo gue 6 ae epoe of ude he peubed codo gue 7 ae epoe of 3 ude he peubed codo gue 8 e epoe of he coo pu ude he peubed codo 5 Cocuo h egao pee a fuzzy coo chee fo abzao of oea e-deay ye ubec o pu auao he cooed ye aayzed ad deged baed o he - fuzzy ode coe epeo fo auag pu fouaed by oducg a auay feedback a aed o he popoed appoach uffce codo ha eue he coed-oop ye o be aypoca abe ae deed hough yapuo-kaok fucoa aay oeoe he pobe of eag he age ego of aaco oe he deged coo aw oed by ug ea a equay echque ay he effecee of popoed ehod deoaed by copue uao o a uck-ae coo Refeece ] ode ad C a aabe-ucue o pee egao wdup a dua ecoc o 48 o pp ] C e abouech ad C uga aby ego fo ea ye wh auag coo a cce ad opo cea oc of 36 h Cofeece o eco ad Coo pp a ego C U 997 3] u Z ad Che aay ad deg ehod fo ea ye ubec o acuao auao ad dubace uoaca o 38 o pp ] u ad Z Coo ye wh cuao auao: ay ad eg oo : khäue 5] J oe a a J ad abouech wdup deg wh guaaeed ego of ab-

11 6 eaoa Joua of uzzy ye o o ach 8 y: a appoach a uoac Coo o 5 o pp 6-5 6] C uga ad abouech ege a-wdup fo ye wh pu agude auao J Robu oea Coo o 8 pp ] abouech C e ad C uga Oupu ackg pobe fo ye wh pu auao a oea egag aco J Robu oea Coo o pp ] Kapoo R ee ad oud a-wdup deg fo ea ye wh pu auao uoaca o 34 o 5 pp ] ude Kohae ad oa uaabe a-wdup cooe yhe ug ea a equae uoaca o 37 pp ] Y Y Cao Z ad Wad awdup appoach o eagg doa of aaco fo ea ye ubec o acuao auao a uoac Coo o 47 o pp 4-45 ] Y Y Cao Z ad Wad -wdup deg of oupu ackg ye ubec o acuao auao ad coa dubace uoaca o 4 pp -8 4 ] Q u ad Ragaah -wdup chee fo ucea oea ye oc Coo heoy pp o 47 o 3 pp ] J u Robu coo of a ca of oea cacade ye: a oe dg ode appoach oc Coo heoy pp o 49 o pp ] abouech J oe a a J ad aca eay-depede a-wdup aegy fo ea ye wh auag pu ad deayed oupu J Robu oea Coo o 4 pp ] K aaka keda ad O Wag ufed appoach o coog chao a a -baed fuzzy coo ye deg a Ccu ad ye o 45 o pp ] Y Y Cao ad ak ay ad yhe of oea e-deay ye a fuzzy coo appoach a uzzy ye o 8 o pp - 7] Che ad u eay-depede obu coo fo - fuzzy ye wh e deay a uzzy ye o 3 o 4 pp ] J Yoeyaa Robu aby ad abzao fo ucea akag-ugeo fuzzy e-deay ye uzzy e ad ye o 58 pp ] Y Y Cao ad Z Robu aby aay ad fuzzy-chedug coo fo oea ye ubec o acuao auao a uzzy ye o o pp ] u Z ad Che ay ad deg fo dcee e ea ye ubec o acuao auao y Co e o 45 pp 97- ] Yue Robu abzao of ucea ye wh ukow pu deay uoaca o 4 pp ] J K ae heoy of ucoa ffeea quao pge ew Yok 977