Journal of Engineering Science and Technology Review 7 (1) (2014) Research Article

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1 Jes Jounal o Engneeng Scence and echnology Revew Reseach Acle JOURNAL OF Engneeng Scence and echnology Revew Sudy on Pedcve Conol o ajecoy ackng o Roboc Manpulao Yang Zhao Dep. o Eleconc and Inomaon echnology Jangmen Polyechnc Jangmen 599 Chna Receved 7 Ocobe 3; Acceped Febuay Absac Model pedcve conol MPC des om ohe conol mehods manly n mplemenaon o he conol acons. In hs pape he ackng eo peomance ndex o ollng opmzaon o pedcve conol s desgned o a class o nonlnea sysems by he elave degee hen employed o desgn conolle o obo manpulao. In ohe wods he obo manpulao s eeence ajecoy s based on aylo expanson and he acual ajecoy s subsued no he peomance cos uncon whch s devaed o mnmum o oban he conolle. hen he peomance analyss o he closed sysem s made. Smulaon esuls demonsae he eecveness o he mehod based on aylo expanson. Keywods: Robo Manpulao Pedcve Conol aylo Expanson Sably Analyss. Inoducon In 978 Gae e al. poposed he model pedcon-elaed heusc conol algohm MPHC. Snce hen pedcve conol has undegone gea developmen successvely geneaed he Model Algohm Conol MAC Dynamc Max Conol DMC Inenal Model Conol IMC Genealzed Pedcve Conol GPC and ohe dozens o algohms. hough paccal ndusal applcaons hese dozens o algohms have been poved o have bee eecs. Inally hese above-menoned pedcve conol mehods wee amed a lnea sysems. Howeve n pacce ndusal poducon pocesses ae oen manesed as nonlnea chaacescs. Alhough mos o he ndusal poducon pocesses can be modeled n he vcny o wokng pons by local lneazaon mehods some song nonlnea conolled objecs o nonlnea objecs wh specal sucues ae dcul o oban sasacoy conol eecs by usng convenonal lnea conol mehods. A pesen hee ae no many eseach esuls o pedcve conol mehods o nonlnea sysems. Beng mplemened hough s ollng opmzaon he pedcve conol sel has a cean degee o obusness. I a sysem only has weak nonlnea chaacescs can be egaded as a model msmach; a sysem exhbs song nonlnea chaacescs usng a convenonal lnea pedcve conol due o he model havng a compaavely gea acual devaon wll no able o each he eec o opmzng he conol so ha he use o a non-lnea model wll be needed o pedcon and opmzaon hus poducng a nonlnea pedcve conol mehod. In vew o he pesen eseach pogess a home and aboad snce he sucue dencaon and paamee esmaon o a nonlnea conolled objec have gea dculy he onlne nonlnea * E-mal addess: zhaoyang9783@gmal.com ISSN: Kavala Insue o echnology. All ghs eseved. ollng opmzaon mplemenaon o an algohm wll expeence anohe dculy even he dynamc chaacescs and pedcve model o a pocess ae obaned n some way. Fo a nonlnea conolled objec because o s own me-vayng couplng and ohe chaacescs hee s no ye a uned pedcve conol mehod now o conol. hs eques a moe eecve nonlnea pedcve conol mehod. In ode o uhe educe he onlne calculaed amoun o pedcve conol and make be appopae o nonlnea conolled objec n ecen yeas many scholas have conduced a good deal o eseach no nonlnea pedcve conol and s mpovemen mehods. hs pape daws on he cuen-oupu all-ode devaves o consuc he uue-oupu aylo-ode pedcon model o he smooh ane nonlnea sysem. hen deves he conolle soluon wh he pedcve oupu ackng eo nom as he mnmum ndcao whch could mpove accuacy o oupu pedcon usng he hgh conol odes.in ecen yeas obo ajecoy ackng conol has obaned a lo o achevemens Howeve due o he a obo havng song couplng hgh nonlneay me-vayng and ohe chaacescs s model paamees vay wh he changes n s poson aude and load and ousde neeence model unceany and ohe acos make he dculy o conolle desgn ncease. In ode o acheve he onlne compensaon o unceany vaous conol saeges have been pu owad n successon. Among hem sel-adapve conol 3 s manly used o a case o conanng paamee unceany and o he unknown paamees o a sysem beng lneazed bu has sc eal-me equemens makng s ealzaon moe complex and s dcul o ensue he sably o a sysem and o acheve a cean conol peomance ndex especally when hee s non-paamee unceany; obus conol 5 can acheve eecve conol ove a manpulao sysem bu hs mehod

2 Yang Zhao/Jounal o Engneeng Scence and echnology Revew needs knowng he uppe bound o an unceany and n paccal applcaon he uppe bound o an unceany s dcul o ge. akng no accoun he deecs o he above mehods many scholas combne 678 an nellgen conol mehod and obus conol wh adapve conol o make he wo coodnae each ohe guaaneeng ha a conol sysem has good dynamc peomance and obus peomance. he nellgen conol mehod epesened by a neual newok 9 and uzzy logc does no need a mahemacal model o a sysem bu can accuaely appoxmae a nonlnea sysem povdng an eecve way o deal wh an uncean obo sysem. Howeve hs algohm has he lage amoun o calculaon and poo ealmeness. hs pape combnes he aylo-based mechancal hand ajecoy and eeence ajecoy and negaes hem no he pedcve conol peomance ndcaos wh ollng opmzaon. hen ake he devave o o ge he conolle whch could pove he sably o he conolle. Meanwhle he algohm s hghly eal-me. Assumpon. he sysem s eeence sgnal and oupu o me ae connuous. Assumpon. he sysem s all saes can be obseved. Assumpon 3. he sysem s zeo dynamc sably y has a elave degee and he sysem s sable n zeo dynamcs. Fo he denon o he elave degee see Denon. Denon. I a nonlnea sysem lke ha o omula n aea D D hen j L L h x = j = L g LL h x x D - g k k L x x n he omula h = L L h L x = h. he sysem has a elave degee. ha s omula3 s esablshed equvalen o: 3. Conolled Objec Model A obo can be egaded as an open-chan gd and mullnk mechansm. Is ajecoy ackng conol poblem s ha gven a jon angle ajecoy veco o be acked and he nal sae o he jon angle s equed o desgn a conolle and povde a jon conol oque o make a jon angle o a obo mee a cean ack condon. he knec equaon o a obo havng wo degees o eedom can be epesened as: Lh g x = Lad x gh = M L - h x = ad g L - h x ad g whch s esablshed o equvalenly has: M θ θ C θ θ θ G θ = τ whee M θ :dene symmec nea max C θ θ :cenugal oce and cools oce veco G θ :gavy veco τ = τ τ :jon conol oque veco θ = θ θ :obo jon angle poson θ = θ θ :obo jon angle velocy θ = θ θ : obo jon angle acceleaon. 3. Desgn o Pedcve Conolle Fo he ollowng MIMO he nonlnea sysem s: x = x g x u y = h x = m whee h x x g x d g x d g x a a L a a d gx = L x n he omula a x. x s an appopae uncon. Denon. Suppose he consecuve pedcon o he uue conol amoun o he conol sysem u τ n τ mees: d d k u τ τ dτ u τ = τ k > dτ hen s he conol sep o he pedcve conol sysem. A s dene he pedcon oupu and eeence ajecoy a momen τ especvely as y τ and y τ and dene he pedcon eo as: e τ = y τ y τ n x R : sae vaables m u R :sysem conol npu m y = y y y m R :sysem oupu. Consde a class o nonlnea sysems lke hose o omula and make he ollowng assumpons. Se he peomance ndcao o he sysem n ollng pedcon me ange as: = τ τdτ 8 J e e 6

3 Yang Zhao/Jounal o Engneeng Scence and echnology Revew y τ = Γ τ Y Fomula8 shows ha he sysem akes he ackng eo as a peomance ndcao and can also be consdeed ha he sysem s peomance ndcao s a uncon o ackng eo. Suppose he conol sep npued by he sysem s τ τ m m Γ τ = I mτ I m I m R n he omula m m and τ = dag{τ τ } R and I m s he un max o m m s. and make pedcve oupu y o he sysem a momen deved o mes hen y h x y L h x m Y = y = L h x m y L h x H u y L h x y = L h x # = L h x y % y = L h x Lg L h x u 9 n he omula L hx s he L h x = h x x x L h x = h x x x x x M Lg L h x s smla o ha o L hx y τ = Γ τ Y Lg L h x = L h x x x x g x Γ = Γ τ Γ τ dτ R m m make s ound y τ = Γ τ Y om ha he elemens o hs max ae as ollows: make pedcve oupu y o he sysem a momen connue o deve me and you wll oban: y = L h x p u x L L h x u n he omula g p u x = Lg L h x u Γ. j = d Lg L h x d u. J= x L = L hx p u L u - x Lg L hxu p u 7 Accodng o omulas 6and7 peomance ndex omula 8 can be ewen as: deved o mes: y j j j m m n he omula = dag{ } R j =. Smlaly make he pedcve oupu y o he sysem 6 y y& L y n he omula Y =. Make he conol sep o conol npu u ake and 5 H u = p u x Lg L hx u % Lg L hx u n he omula p u x p u - u Lg L hx u # & y τ Smlaly make eeence ajecoy a momen τ usng he aylo sees appoxmaely expessed as: ode devave o y o x and h x = h x h x hm x. he denon o Y Y Γ Y Y dτ 8 In ode o make he above-menoned peomance ndex mnmum make 3 J = u and n he omula u = u u u p u x p u u - x s a complex nonlnea uncon on u x. I s known om he above heoecal devaon ha n 9 he ollowng heoems can be obaned: heoem. Fo a class o nonlnea sysems composed o omula on sasyng assumpons o 3 use he aylo omula o make he sysem oupu expand o ollng pedcve me doman pedcve oupu y τ a momen τ usng he aylo sees can be appoxmaely expessed as: seps. Among hem > s he conol sep. he 7

4 Yang Zhao/Jounal o Engneeng Scence and echnology Revew ollowng pedcve conol law can ensue ha pedcve conol peomance omula8 s opmal: u = Lg L hx KM L hx y whee he non-sngula max: Lg L h x Lgm L h x Lg L h x Lgm L h x Lg L h x = Lg L hm x Lgm L hm x L h x L h x = L hm x y sysem s apdy s guaaneed has poo obusness and sably; you selec a geae pedcon me doman he obusness o he sysem wll be songe bu s dynamc esponse s slowe whch nceases he compuaon me and educes he sysem s eal-me popey. In acual selecon you may ake a value beween he wo. I he apdy s bad educe ; he sably s bad ncease. Only he wo coodnang each ohe can make a closed loop sysem have equed obusness and desed apdy. Use aylos omula o make he pedcve values o jon angles θ s and θ s a momen yx = ym x s appoxmaely expanded as ollows: s θ s θ = H s y = s θ θ θ = H s y s = hx y L hx y M = L hx y n he omula K R m mp s a new max composed o he s m lnes o max Γ Γ and Γ Γ Γ max Γ Γ Γ = Γ Γ s 3 3 H s = I s I s I s I s I H s = I s I s I s I 6 6 s obaned by paonng ha y = θ θ θ θ 3 θ s θ s θ = H s y = s θ θ θ = H s y s = he specc devaon pocess can be ound n leaue. Pedcon equaon and conolle solvng he pedcon eo 3 3 n he omula y = θ θ θ θ θ Accodng o omula and omula3 peomance ndcao omula can be appoxmaely expessed as: A s wh gven desed eeence ajecoy θ dene he pedced value o jon angle θ s a momen s as θ s he eeence ajecoy pedcon a me s as and s devave a momen s appoxmaed as:. I especvely ae ode zeo max and -ode un max. Smlaly we can ge he aylo expanson o θ and Γ Γ. Γ = Γ Γ θ s J = y y R y y as: θ ε s = θ s s. he peomance ndex uncon s aken as: R = H H λh H d s n he omula s a consanelaed symmec max and conssen wh he oma o omula8. n he omula λ s a weghng aco and a consan geae Fo jon angle veco θ especvely oban s 3ode devaves and we wll ge: J= ε s ε s λε s ε s d s han zeo; s he pedced me doman. ε and ε among he peomance ndcaos elec he opmzaon equemens o a closed-loop sysem sae. In ode o make he ollng opmzaon meanngul you should make nclude he dynamc pa o a conolled objec ha s you should nclude all he eecs wh moe conol nluence. Unde a nomal ccumsance make close o he se me o a sysem. In acual ndusal applcaon choosng a geae s moe appopae so ha wll exceeds he evese pa caused by he non-mnmum phase chaacescs o he me delay pa o he pulse esponse o he sysem and wll cove he man dynamc esponse o a conolled objec. he selecon o has a geae mpac on he sably and apdy o a sysem. Alhough ae selecng a smalle pedcon me doman a θ = m θ θ M θ τ 5 θ 3 = m θ θ pθ θ θ M θ τ 6 θ = m3θ θ p θ θ θ θ 3 τ τ M θ τ 7 Whee m θ θ = M θ C θ θ θ Gθ m θ θ m θ θ = θ 8 m θ θ m3 θ θ = θ

5 Yang Zhao/Jounal o Engneeng Scence and echnology Revew m θ θ M θ p θ θ θ τ = θ θτ m θ θ M θ p θ θ θθ 3 τ τ = θ p θτ 3 dene y = = Accodng o he denons o K K hey can be calculaed as: k K = k = k / k / k 3 / k y y y = m θ m τ k = λ 68λ k = λ 3 368λ 56λ λ 78λ n he omula τ = τ τ τ m = m θ θ mθ θ mθ θ M = M m p 3 M # p k3 = λ 368λ 56λ λ 756λ % & k = λ3 368λ 56λ k = λ 68λ R R R = make R R whee R R R R R R. k = λ 3 368λ 56λ λ/ λ 68 λ 6 k3 = / λ368 λ / 3λ5 / 6λ λ Deve om omula8 we wll see: R R J y y = R y y = m τ τ τ R R τ y m y m τ m τ = R τ 756λ k = λ 3 368λ 56λ 5 6 / 6 / λ y R R m τ m θ 5. Closed-loop sysem peomance analyss Unde a ccumsance ha he modelng o a obo sysem s accuae and s neeence s zeo a convenonal PD conolle can also be used o mplemen he ajecoy ackng o s jon angle. Is conol law s omula33: 9 τ n omula9 oban he paal devave Fo m o τ and we wll see: M θ m τ = * M θ τ * * M θ τ = M θ θ K v e K P e C θ θ θ G θ 3 J coecens K v and K P has a gea deal o andomness and he enave selecon o paamees does no ensue ha a desgned conolle s globally opmum. Pedcve conol usng ollng opmzaon o subsue global one-me opmzaon ha s he opmzaon pocess s no peomed olne a a me bu undegoes onlne epeaed opmzaon calculaon and ollng mplemenaon hus makng he unceany caused by model msmach me vayng neeence and he lke made up n me always makng new opmzaon bul on an acual oundaon and makng conol keep acually opmal. Fo an acual complex ndusal pocess hee nevably s unceany heeoe a ollng opmzaon saegy bul on a ne me doman wll be moe eecve. Fomula 3 K K s exacly he embodmen o hs chaacesc. M θ s a evesble max so ae makng obus pedcve conol law 3 subsued no equaon o a conolled objec we can oban a closed-loop sysem equaon as ollows: = evesble max. Fom he mnmum condon τ o peomance ndcaos make omula 9 equal o and we wll have: m τ = R R y m θ 3 Dene e = θ θ K as a squae max composed o he s wo lnes and he s wo columns o R R and K as a squae max composed o he s wo lnes and he las wo columns o R R. Fom he above omulas he conol law makng he peomance ndcaos mnmum he ollowng devaons all om me can be obaned as ollows: τ = M θ θ K e Ke C θ θθ Gθ 33 n he omula K v = ai K P = b I a > b >. Alhough hs omula33 and he conol law omula 3 obaned om peomance ndex omula ollng opmzaon n hs pape ae conssen n om he wo ae essenally deen. In he PD conolle he selecon o m τ heeoe τ s a evesble max. R s also a k k = k / k / k3 / k om omula7 we can see: k K = k 3 9

6 Yang Zhao/Jounal o Engneeng Scence and echnology Revew e K e Ke = 3 Fom he om o K and K we can know s a posve dene max. When a conolled objec akes and he conol law akes omula 33 equaon 3 o a closedloop sysem wll be globally asympocally sable. 6. Smulaon Consde ha he knec model o a wo-jon obo sysem s whee v q q cos θ q q cos θ M θ = q q cos θ q θ sn θ θ sn q q θ θ C θ θ = qθ sn θ 5g cos θ 8.75g cos θ θ G θ = 8.75g cos θ θ smulaon paamees ae v = 3.33 q = 8.98 q = 8.75 g = 9. 8 θ.sn.5π = and he desed ackng sgnal s θ.cos.5π. In peomance ndcaos selec λ = pedcve me doman = s and he soluon wll be K =. 8I K = I. Ae calculaon he elave sep and conol sep o hee obo model used n hs secon ae. Ae usng Smulnk and S uncon o cay ou conol sysem smulaon he smulaon esuls ae shown n Fgues a o d. Fgues a o b show he jon angle oupu and s eeence ajecoy Fgue c shows he eo cuve and Fgue d shows he conol oque. Fom Fg.a o Fg.b can be seen ha he oupu cuve can quckly ack he eeence cuve. Fom Fgue c can be seen ha he ackng eo s and he ackng eo convegence s elavely apd. Fom Fgue d can be seen ha he conol oque and s desed sgnal ae bascally he same wh he conolle havng a good eec. jon /ad eeence ajecoy acual ajecoy /s Fg. a Jon and s eeence ajecoy jon /ad eeence ajecoy.6 acual ajecoy /s Fg. b Jon and s eeence ajecoy ackng eo /ad jon jon /s Fg. ceo beween jon and jon conol oque/n m jon jon /s Fg. d Conol oque o jon and jon 7. Conclusons A s hs pape gves a pedcve conolle desgn mehod o a mul-npu and mul-oupu nonlnea sysem based on aylo expanson unde a nomnal condon. akng a obo as a conolled objec hs pape mpoves he peomance ndex uncon n he mehod we popose and hen uses o obo conol. Fo he above mehods he desgn pocess s smple whou onlne calculaon hus gealy educng he heavy compuaon buden bough by pedcve conol ollng opmzaon. Reeences. W. H. CHEN D. J. BALANCE P. J. Gawhop Opmal Conol o Nonlnea Sysem: a Pedcve Conol Appoach Auomaca 39 3 pp AYEBI A Adapve eave leanng conol o obo Manpulaos Auomaca7 pp LIUZZO S OMEI P A global adapve leanng conol o oboc manpulaos Auomaca5.8pp LIUZZO S MARINO R OMEI P Adapve leanng conolo nonlnea sysems by oupu eo eedback IEEE ansacons on Auomac Conol57pp SHI J LIU H BAJCINCA N Robus conol o oboc manpulaos based on negal sldng mode Inenaonal Jounal o Conol8.8 pp LI H S HUANG Y C MIMO adapve uzzy emnal Sldngmode conolle o oboc manpulaos Inomaon Scences38. pp ZUO Y WANG Y N LIU X Z SIMON X Neual newok obus H ackng conol saegy o obo manpulaos Appled Mahemacal Modellng3 7.pp

7 Yang Zhao/Jounal o Engneeng Scence and echnology Revew LI Z JGE S ZWANG Z P Robus adapve conol o coonae mulple moble manpulaos Mechaoncs858pp YILDIRIM S ESKI I Nose analyss o obo manpulaousng neual newoks Robocs and Compue-Inegaed Manuacung6 pp KUMAR N PANWAR V SUKAVANAM S P Neual newokbased nonlnea ackng conol o knemacally edundan obo manpulaos Mahemacal and Compue Modellng pp. 3.. ALAM M SOKHI M O Hybd uzzy logc conol wh genec opmsaon o a sngle-lnk lexble manpulao Engneeng Applcaons o Acal Inellgence 6 8pp A. Kovansov R. Kumbegs Ceaon o gaphs o uncons wh use o heoems o elemenay geomey Compue Modellng and New echnologes6 pp