Adaptive Neuro-Fuzzy Inference Controllers for Smart Material Actuators

Transcription

1 Adapve Neuro-Fuzz Inference Conrollers for Smar Maeral Acuaors Teodor Lucan Grgore and Ruxandra Mhaela Boez École de Technologe Supéreure, Monréal, Quebec HC K Canada An nellgen approach for smar maeral acuaor modelng of he acuaon lnes n a morphng wng ssem s presened, based on adapve neuro-fuzz nference ssems. Four ndependen neuro-fuzz conrollers are creaed from he expermenal usng a hbrd mehod -- a combnaon of bac-propagaon and Leas-Mean-Suare LMS) mehods -- o ran he fuzz nference ssems. The conrollers obecve s o correlae each se of forces and elecrcal currens appled on he smar maeral acuaor o he acuaor s elongaon. The acuaor expermenal esng s performed for fve force cases, usng a varable elecrcal curren. An negraed conroller s creaed from four neuro-fuzz conrollers, developed wh Malab/Smuln sofware for elecrcal curren ncreases, consan elecrcal curren, elecrcal curren decreases, and for null elecrcal curren n he coolng phase of he acuaor, and s hen valdaed b comparson wh he expermenall obaned. Nomenclaure A assocaed ndvdual aneceden fuzz ses of each npu varable, N) a, b parameers of he generalzed bell membershp funcon a parameers of he lnear funcon,,, N) b scalar offse, N) C p pressure coeffcen c cluser cener c cluser cener,) Posdocoral fellow, Laboraor of Research n Acve Conrols, Avoncs and AeroServoElasc LARCASE, Nore-Dame Wes Sree, Monreal, Quebec, HC K, Canada, AIAA Member Professor, Laboraor of Research n Acve Conrols, Avoncs and AeroServoElasc LARCASE, Nore- Dame Wes Sree, Monreal, Quebec, HC K, Canada, AIAA Member

2 F force elecrcal curren varable for neuro-fuzz conroller selecon sep sze l dmenson of he vecors M number of pons N number of rules r m radus whn he fuzz neghborhood, conrbues o he dens measure Re Renolds number me T emperaure of he smar maeral acuaor T amb amben emperaure u c cener of he h cluser u vecors V speed w degree of fulfllmen of he aneceden,.e., he level of frng of he h rule x npu vecor x ndvdual npu varables,) oupu of he fuzz model frs-order polnomal funcon n he conseuen, N) α angle of aac ε cos funcon ρ dens measure σ dsperson σ cluser dsperson me varaon δ acuaor elongaon

3 I. Inroducon The am of hs paper s o oban a relable, eas-o-mplemen model for Smar Maeral Acuaors SMAs), wh drec applcaons n he morphng wng proec. Based on adapve neuro-fuzz nference ssems, an negraed conroller s bul o model he smar maeral acuaors used n he acuaon lnes of a wng. As shown n Fg., a complex ssem was obaned, whch modfes he arfol shape n order o opmze from he perspecve of he lamnar flow regon. For varous flgh condons angles of aac α, speeds V and Renolds numbers Re ), he loop conroller would receve he arfol upper surface surface pressure dsrbuon measured b he ule sensors. The C dsrbuon, deermned from he p C dsrbuon s compared wh a compuaonal p flud dnamcs CFD) base, whch s generaed so ha for dfferen arfol pes, he ranson pon s gven as a funcon of he C p dsrbuon. Once a mach s found, a ranson pon s offered o he loop conroller b he CFD base; he conroller wll be able o decde f he arfol shape mus be adused or no. The adusmen of he arfol shape s made n real me usng he SMA acuaors. The loop s closed b he arfol shape, whch offers he opcal sensors anoher surface pressure dsrbuon. Varable flow condons α, V, Re Opcal sensors Loop conroller SMA Acuaors Transon pon CFD Daabase Varable arfol shape Fg. Closed-loop morphng wng ssem In order o valdae he morphng wng ssem numercal smulaon versus es resuls), good numercal models for each of he phscal elemens n he ssem mus be obaned. The am of hs paper s o offer a good model for SMA acuaors, wh drec applcaon o our morphng wng ssem.

4 Ths model uses he numercal values from he SMAs expermenal esng and aes advanage of he ousandng properes of fuzz logc, whch allow he sgnal s emprcal processng whou he use of mahemacal analcal models. Fuzz logc ssems can emulae human decson-mang more closel han man oher classfers hrough he processng of exper ssem nowledge, formulaed lnguscall n fuzz rules n an IF-THEN form. Fuzz logc s recommended for ver complex processes, when no smple mahemacal model exss, for hghl nonlnear processes, and for mul-dmensonal ssems. The npu varables n a fuzz conrol ssem are usuall mapped no place b ses of membershp funcons mf) nown as fuzz ses ; he mappng process s called fuzzfcaon. The conrol ssem s decsons are made on he bass of a fuzz rules se, and are nvoed usng he membershp funcons and he ruh values obaned from he npus; a process called nference. These decsons are mapped no a membershp funcon and ruh value ha conrollng he oupu varable. The resuls are combned o gve a specfc answer n a procedure called defuzzfcaon. Elaboraon of he model hus reures a fuzz rules se and he membershp funcons mf) assocaed wh each of he npus [], []). The abl and he experence of a desgner n evaluang he rules and he membershp funcons of all of he npus are decsve n obanng a good fuzz model. However, a relavel new desgn mehod allows a compeve model o be bul usng a combnaon of fuzz logc and Neural-Newor echnues. Moreover, hs mehod allows he possbl o generae and opmze he fuzz rules se and he parameers of he membershp funcons b means of fuzz nference ssems ranng. To hs end, a hbrd mehod -- a combnaon of bac-propagaon and Leas-Mean-Suare LMS) mehods s used, n whch expermenall obaned are consdered. Alread mplemened n Malab [], []), he mehod s eas o use, and gves excellen resuls n a ver shor me. II. Acuaor expermenal esng The SMA esng was performed usng he bench es n Fg. a T o amb C, for fve load cases wh forces of N, N, N, 8 N and 9 N. The elecrcal currens followng he ncreasng-consan-decreasng-zero values evoluon were appled on he SMA acuaor n each of he fve cases consdered for load forces. In each of he cases o be analzed, he followng parameers were recorded: me, he elecrcal curren appled o he SMA, he load force, he maeral emperaure and he acuaor elongaon measured usng a Lnear Varable Dfferenal Transformer LVDT)).

5 wheel racon wheel racon LVDT load cell load cell fxed aws SMA sample moble aws fxed aws SMA sample moble aws free wegh sprng embeddng Fg. The SMA bench es To model he SMA we bul an negraed conroller based on Adapve Neuro-Fuzz Inference Ssems. The expermenal elongaon-curren curves obaned n he fve load cases are shown n Fg.. One can observe ha all fve of he obaned curves have four dsnc zones: elecrcal curren ncrease, consan elecrcal curren, elecrcal curren decrease and null elecrcal curren n he coolng phase of he SMA. Four Fuzz Inference Ssems FIS s) are used o oban four neuro-fuzz conrollers: one for he curren ncrease, one for he consan curren, one for he curren decrease, and one conroller for he null curren afer s decrease). For he frs and he hrd conrollers, npus such as he force and he curren are used, whle for he second and he fourh, npus such as he force and he me values reflecng he SMA s hermal nera are used he me values reured for he SMA o recover s nal emperaure value approxmael C) are used for he four conroller). Fnall, he four obaned conrollers mus be negraed no a sngle conroller. The reasonng behnd he desgn of he frs and he hrd conrollers s ha, from he avalable expermenal, wo elongaons for he same values of forces and currens are used see Fg. ). Due o he expermenal values, hs canno be represened as algebrac funcons; herefore, s mpossble o use he same FIS represenaon. Malab produces an nerpolaon beween he wo elongaon values obaned for he same values of forces and currens, whch canno be vald for our applcaon. The consan values, namel he null values of he curren before and afer he curren decrease phase, should no be consdered as npus n he second and fourh conrollers because he are no suggesve for he characerzaon of he SMA elongaon. The values of he acuaor emperaures ma appear o be ver suggesve n hese phases,

6 bu he emperaure mus be a model oupu. For hese phases he me values are ver suggesve, as he represen a measure of he acuaor hermal nera. Tme s he second npu of he hrd conroller, and so me s also he second npu of he second and he fourh conrollers snce force was consdered as he frs npu he me values mus be consdered when he curren becomes consan or null). Elongaon [mm] These curves are no algebrc funcons force 9 N) force 8 N) force N) force N) force N) Curren [A] Two dferen values of he elongaon for he same value of he curren and of he force Fg. Elongaon versus he curren values for dfferen force values for four cases III. The proposed mehod Fuzz conrollers are ver smple concepuall and are based on fuzz nference ssems FIS s). Three seps are consdered n a fuzz nference ssem desgn: an npu, he processng, and hen an oupu sep. In he npu sep, he conroller npus are mapped no he approprae membershp funcons mf). Nex, a collecon of IF-THEN logc rules s creaed; he IF par s called he aneceden and he THEN par s called he conseuen. In hs sep, each approprae rule s nvoed and a resul s generaed. The resuls of all of he rules are hen combned. In he las sep, he combned resul s convered no a specfc conrol oupu value. Consderng he numercal values resulng from he SMA expermenal esng, an emprcal model can be developed, whch s based on a neuro-fuzz newor. The model can learn he process behavor based on he npuoupu process b usng an FIS, whch should model he expermenal. Usng mehods alread mplemened n commercal sofware, an FIS can be generaed smpl wh he Malab genfs or genfs funcons. The genfs funcon generaes a sngle-oupu Sugeno-pe fuzz nference ssem FIS) usng a grd paron on he no cluserng). Ths FIS s used o provde nal condons for ANFIS ranng. The genfs funcon uses generalzed Bell-pe membershp funcons for each npu. Each rule

7 generaed b he genfs funcon has one oupu membershp funcon, whch s, b defaul, of a lnear pe. I s also possble o creae an FIS usng he Malab genfs funcon. Ths funcon generaes an nal Sugeno-pe FIS b decomposng he operaon doman no dfferen regons usng he fuzz subracve cluserng mehod. For each regon, a low-order lnear model can descrbe he local process parameers. Thus, he non-lnear process s locall lnearzed around a funconng pon b use of he Leas Suares LS) mehod. The obaned model s hen consdered vald n he enre regon around hs pon. The lmaon of he operang regons mples he exsence of overlappng among hese dfferen regons; her defnon s gven n a fuzz manner. Thus, for each model npu, several fuzz ses are assocaed o her membershp funcons correspondng defnons. B combnng hese fuzz npus, he npu space s dvded no fuzz regons. A local lnear model s used for each of hese regons, whle he global model s obaned b defuzzfcaon wh he grav cener mehod Sugeno), whch performs he nerpolaon of he local models oupus [], []). Based on he goal of fndng regons wh a hgh dens of pons n he feaured space, he subracve cluserng mehod s used o dvde he space no a number of clusers. All of he pons wh he hghes number of neghbors are seleced as ceners of clusers. The clusers are denfed one b one, as he pons whn a prespecfed fuzz radus are removed subraced) for each cluser. Followng he denfcaon of each cluser, he algorhm locaes a new cluser unl all of he pons have been checed. If a collecon of M pons, specfed b l-dmensonal vecors u,,..., M, s consdered, a dens measure a pon u can be defned as follows: ρ M exp u u, rm / ) ) where r m s a posve consan ha defnes he radus whn he fuzz neghborhood and conrbues o he dens measure. The pon wh he hghes dens s seleced as he frs cluser cener. Le u c be he seleced pon and ρ c s dens measure. Nex, he dens measure for each pon u s revsed b he formula: exp u, / ) u c ρ ρ ρ ) c rn where r n s a posve consan, greaer han r m, ha defnes a neghborhood where dens measures wll be reduced n order o preven closel spaced cluser ceners. In hs wa, he pons near he frs cluser cener u c wll have sgnfcanl reduced dens measures, and herefore canno be seleced as subseuen cluser ceners. Afer

8 he dens measures for each pon have been revsed, hen he nex cluser cener u c s seleced and all he dens measures are agan revsed. The process s repeaed unl all he pons have been checed and a suffcen number of cluser ceners generaed. When he subracve cluserng mehod s appled o an npu-oupu se, each of he cluser ceners are used as he ceners for he premse ses n a sngleon pe of rule base []). The Malab genfs funcon generaes membershp funcons of he generalzed Bell pe, defned as follows [], []): where b ) x c A x +, ) a c s he cluser cener defnng he poson of he membershp funcon, a and b are wo parameers whch defne he membershp funcon shape, and A, N) are he assocaed ndvdual aneceden fuzz ses of each npu varable N number of rules). Malab s genfs funcon generaes Gaussan-pe membershp funcons, defned wh he followng expresson [], []): where A x) x c exp. σ c s he cluser cener and σ s he dsperson of he cluser., ) The Sugeno fuzz model was proposed b Taag, Sugeno and Kang o generae fuzz rules from a gven npuoupu se []). In our ssem, for each of he four FIS s wo npus and one oupu), a frs-order model s consdered, whch for N rules s gven b [], []): Rule : If x Rule : If x Rule N : If s A s A x s A N and x and x and x s A, s A, s A N, M M hen hen hen x, x ) b x, x ) b + a x N x, x ) b + a x + a x, N N + a x, + a x + a x, N ) where x,) are he ndvdual npu varables and, N) s he frs-order polnomal funcon n he conseuen. a,,, N) are parameers of he lnear funcon and b, ) denoes a scalar offse. The parameers a, b,,, N) are opmzed b he Leas Suare mehod. For an npu vecor, x T [ x, x ] N, f he sngleon fuzzfer, he produc fuzz nference and he cener average

9 defuzzfer are appled, hen he oupu of he fuzz model s nferred as follows weghed average): where N N w x ) / w x), ) w x x ) A x ) A ). 7) w x) represens he degree of fulfllmen of he aneceden,.e., he level of frng of he h rule. The adapve neuro-fuzz nference ssem calculaes he Sugeno-pe fuzz nference ssem parameers usng Neural Newors. A ver smple wa o ran hese FISs s o use Malab s ANFIS funcon, whch uses a learnng algorhm o denf he membershp funcon parameers of a Sugeno-pe fuzz nference ssem wh wo oupus and one npu. As a sarng pon, he npu-oupu and he s generaed wh he genfs or genfs funcons are consdered. ANFIS opmzes he membershp funcons parameers for a number of ranng epochs, deermned b he user. Wh hs opmzaon, he neuro-fuzz model can produce a beer process approxmaon b means of a ual parameer n he ranng algorhm []). Afer hs ranng, he models ma be used o generae he elongaon values correspondng o he npu parameers. To ran he fuzz ssems, ANFIS emplos a bac-propagaon algorhm for he parameers assocaed wh he npu membershp funcons, and Leas-Mean-Suare esmaons for he parameers assocaed wh he oupu membershp funcons. For he FIS s generaed usng he genfs or genfs funcons, he membershp funcons are generalzed Bell pe or Gaussan pe, respecvel. Accordng o euaons ) and ), n hese pes of membershp funcons, a, b and c, respecvel σ, and c, are consdered varables and mus be adused. The bacpropagaon algorhm ma, herefore, be used o ran hese parameers. The goal s o mnmze a cos funcon of he followng form des ), ε 8) where des s he desred oupu. The oupu of each rule x, x ) s defned b: ε + ) ), 9) where s he sep sze. Sarng from he Sugeno-ssem s oupu e. )), modfng wh e. 9) resuls n:

10 , ε ε ) wh. ) ), N des w w x x ε ) Therefore, he oupu of each rule s obaned wh he euaon:. ) ) ) ) ) + N des w w x x ) If a generalzed bell-pe membershp funcon s used, he parameers for he h membershp funcon of he h fuzz rule are deermned wh he followng euaons:. ) ), ) ), ) ) c b a c c c b b b a a a ε ε ε ) For a Gaussan-pe membershp funcon, he parameers of he h membershp funcon of he h fuzz rule are calculaed wh:. ) ), ) ) c c c c + + ε σ ε σ σ σ ) Afer he four conrollers Conroller for ncreasng curren, Conroller for consan curren, Conroller for decreasng curren and Conroller for null curren) have been obaned, he mus be negraed, resulng n he logcal scheme n Fg.. The decson o use one of he four conrollers depends on he curren vecor pe ncreasng, decreasng, consan or zero) and on he varable value. Dependng on he value of, we ma decde f a consan curren value s par of an ncreasng vecor or par of a decreasng vecor. The nal value s eual o when Conroller s used, and s eual o when Conrollers, or are used.

11 START no I)>I-) es go o Conroller and ) I)I-) no I)<I-) es go o Conroller and ) no I) es go o Conroller and ) -) no -) es go o Conroller and ) go o Conroller and ) Fg. The logcal scheme for he four conroller s negraon IV. The negraed conroller desgn and evaluaon In he frs phase, he genfs Malab funcon []) was used o generae and ran he FISs assocaed wh he four conrollers n Fg. : ElongaonFs for he curren ncrease phase), celongaonfs for he consan phase of he curren), delongaonfs for he decrease phase of he curren) and delongaonfs for he null values of he curren obaned afer he decrease phase). The FISs are raned for dfferen epochs. for he frs FIS,. epochs for he second and he las FISs, and. epochs for he hrd) usng he ANFIS Malab funcon. Fgure dsplas he devaon beween he neuro-fuzz models and he expermenall obaned for dfferen ranng epochs, defnng he ual parameer from he ranng algorhm. A rapd decrease n he devaon beween he expermenal and he neuro-fuzz model s apparen for all four FISs n erms of he ual parameer whn he ranng algorhm over he frs ranng epochs. Evaluang each of he four FISs for he expermenal usng he evalfs command, he characerscs shown n Fg. were obaned. The means of he relave absolue values of he errors for all four FISs are:.8%,.978%,.7% and.8% for ElongaonFs, celongaonfs, delongaonfs and delongaonfs, respecvel. The error obaned for he hrd FIS delongaonfs ) s ver good, and so hs FIS wll be consdered for mplemenaon n he Smuln negraed conroller. The frs, second and fourh FISs have large error values and so he generang mehod mus be changed.

12 Devaon "ElongaonFs" Number of ranng epochs x x - "delongaonfs" Devaon "celongaonfs" Number of ranng epochs x. "delongaonfs". Devaon Devaon Number of ranng epochs x Number of ranng epochs x Fg. Tranng errors for he FISs generaed and raned n he frs phase Elongaon [mm].... FN "ElongaonFs" afer ranng FN FN F8N F9N.. Number of expermenal pons "delongaonfs" afer ranng F9N Elongaon [mm] FN "celongaonfs" afer ranng FN FN F8N Number of expermenal pons "delongaonfs" afer ranng F9N F9N Elongaon [mm].. FN FN F8N Elongaon [mm] FN FN FN F8N FN Number of expermenal pons 8 Number of expermenal pons Fg. FISs evaluaon as a funcon of he number of expermenal pons n he frs phase

13 Durng he second phase, he genfs Malab funcon []) can be used o buld and ran he remanng hree fuzz nference ssems: ElongaonFs, celongaonfs, and delongaonfs. The number of he membershp funcons consdered for each of hese s for he frs npu and for he second npu. The number of he ranng epochs consdered for he hree FISs are. for he frs and second FISs, and for delongaonfs. Followng he evaluaon of hese hree raned FISs for expermenal, he characerscs depced n Fg. were obaned. The evoluon of he ranng errors s represened n Fg. 7. Evaluaon of hese hree FISs gves he followng values of he mean of he relave absolue errors:.99%,.9%, and.%. for ElongaonFs, celongaonfs, and delongaonfs, respecvel. Elongaon [mm].... FN "ElongaonFs" afer ranng FN FN F8N F9N. Number of expermenal pons Elongaon [mm] FN "delongaonfs" afer ranng "celongaonfs" afer ranng FN FN F8N F9N Number of expermenal pons F9N Elongaon [mm] FN FN FN F8N 8 Number of expermenal pons Fg. FISs evaluaon as a funcon of he number of expermenal pons n he second phase The errors obaned n he second phase for he frs and he second FISs are ver good, and so hese FISs can be mplemened n he Smuln negraed conroller. For he las FIS delongaonfs ), he error values are sll oo large, and so he number of he membershp funcons used o generae mus be adused. Therefore, a hrd phase of FISs buldng and ranng s reserved o oban a beer soluon for he delongaonfs fuzz nference ssem. In hs phase, wo cases were consdered for he numbers of he membershp funcons. In he frs case, he

14 mf numbers are for he frs npu and for he second npu, and n he second case he mf number s for he frs npu and for he second. A number of ranng epochs were consdered n he frs case, and n he second. The ranng errors for boh cases, afer ranng wh he ANFIS funcon, are presened n Fg. 8, and he evaluaon as a funcon of he number of expermenal pons s shown n Fg. 9. The means of he relave absolue error values for he wo cases are:.78% and.7 %, respecvel. Snce he errors n he second case are lower, ha s he confguraon ha was chosen o be mplemened n a Smuln negraed conroller..8 "ElongaonFs".7 "celongaonfs".7... Devaon.. Devaon Number of ranng epochs x Number of ranng epochs x "delongaonfs".... Devaon Number of ranng epochs x Fg. 7 Tranng errors for he hree FISs generaed and raned n he second phase The fnal values of he relave absolue errors for he four generaed and raned FISs are:.99% for ElongaonFs,.9% for celongaonfs,.7% for delongaonfs, and.7% for delongaonfs. Represenng he elongaons hose obaned expermenall and b usng he four s) as funcons of elecrcal curren for he frs and hrd FISs, and as a funcon of me for he oher wo FISs, produces he graphcs n Fg.. The curves are represened for all fve cases of he SMA load. One can easl observe ha, hrough ranng, he FISs model he expermenal ver well, and he SMA has dfferen hermal consans, dependng on he force value.

15 Devaon "delongaonfs" case ). Number of ranng epochs x Devaon "delongaonfs" case ) Number of ranng epochs x Fg. 8 Tranng errors for he delongaonfis generaed and raned n he hrd phase "delongaonfs" afer ranng F9N "delongaonfs" afer ranng F9N Elongaon [mm] FN FN FN F8N Elongaon [mm] FN FN FN F8N 8 Number of expermenal pons 8 Number of expermenal pons Fg. 9 FIS s evaluaon as a funcon of he number of expermenal pons for he hrd phase A good overlappng of he s elongaons wh he elongaon expermenal s clearl vsble n Fg.. Ths superposon s dependen on he number of ranng epochs, and mproves as he number of ranng epochs s hgher. Because he ranng errors of all of he raned FISs ulmael ae consan values, an mproved approxmaon of he real model can be acheved wh he neuro-fuzz mehods onl when a hgher uan of expermenal s used. To vsualze he FIS s feaures, he Malab anfsed command []) s used, followed b he FIS s mporaon on he nerface level. The resulng surfaces for all four fnal, raned FISs are presened n Fg.. The parameers of he npu s membershp funcons for each of he four FIS s before and afer ranng are shown n Tables and, respecvel. For he generalzed bell-pe membershp funcons, produced wh he genfs funcon, he parameers are he membershp funcon cener c) defnng her poson, and a, b whch defne her shape. For he Gaussan-pe membershp funcons, generaed wh he genfs funcon, he parameers are one-half of he dsperson σ/) and he cener of he membershp funcon c). For our ssem, a se of 7 rules for ElongaonFs

16 and anoher 7 for celongaonfs, rules for delongaonfs and 8 rules for delongaonfs are generaed. Elongaon [mm].... FN FN F9N F8N FN "ElongaonFs" afer ranng. Curren [A] Elongaon [mm] FN "celongaonfs" afer ranng F9N F8N FN FN Tme [s]. "delongaonfs" afer ranng F9N "delongaonfs" afer ranng F9N Elongaon [mm]. F8N FN Elongaon [mm] F8N FN. FN FN FN FN Curren [A] Tme [s] Fg. FIS evaluaons as funcons of curren or me Table Parameers of he FIS npu s membershp funcons before ranng ElongaonFs celongaonfs delongaonfs delongaonfs Force [N] Curren [A] Force [N] Tme [s] Force [N] Curren [A] Force [N] Tme [s] a b c a b c a b c a b c σ/ c σ/ c a b c a b c mf mf mf mf mf mf mf mf mf mf mf mf mf mf

17 Table Parameers of he FIS npu s membershp funcon afer ranng ElongaonFs celongaonfs delongaonfs delongaonfs Force [N] Curren [A] Force [N] Tme [s] Force [N] Curren [A] Force [N] Tme [s] a b c a b c a b c a b c σ/ c σ/ c a b c a b c mf mf mf mf e mf e mf mf mf mf mf mf mf mf mf ElongaonFs celongaonfs delongaonfs delongaonfs Fg. The surfaces produced for all four of he fnal raned FISs

18 Comparson of he FISs characerscs and membershp funcons parameers before and afer ranng, from Tables and, ndcaes a redsrbuon of he membershp funcons n he worng doman and a change n her shapes, b modfcaon of he a, b, and σ parameers. Accordng o he parameer values from Table, generang FISs wh he genfs and genfs funcons prmarl resuls n he same values for he a, b, and σ/ parameers for all of he membershp funcons ha characerze an npu. A secondar resul s he separaon of he worng space for he respecve npu usng a grd paron on he no cluserng) f he genfs funcon s used, or usng he fuzz subracve cluserng mehod f generang wh he genfs funcon. For he delongaonfs fuzz nference ssem nall generaed b usng he genfs funcon) he rules are of he pe: f n s ncluser ) and n s ncluser ) hen ou s oucluser ). For boh of he npus of hs FIS, sx Gaussan-pe membershp funcons mf) were generaed; whn he se of rules he are noed b: n cluser ; s he npu number ), and s he number of he membershp funcon ). The delongaonfs fuzz nference ssem has he srucure shown n Fg., whle he correspondng conroller Conroller ) has he srucure presened n Fg.. Inpus Inpu mf Rules Oupu mf Oupu ncluser n Force n ncluser ncluser ncluser ncluser oucluser oucluser oucluser Agregaed Oupu ou Elongaon Curren ncluser Fg. Srucure of he delongaonfs fuzz nference ssem n n Force Curren Sugeno FIS ou Elongaon Fg. The srucure of Conroller

19 For he oher hree FISs nall generaed b usng he genfs funcon) he rules are of he pe: f n s nmf ) and n s nmf p ) hen ou s oumf r ). The number of he oupu membershp funcons mf) s p r p)) and s eual o he number of rules. For hese hree FISs, generalzed bell-pe membershp funcons were generaed; whn he ses of rules he are noed b: n mf n ; s he npu number ), and n s he number of he membershp funcons. For ElongaonFs and celongaonfs, sx membershp funcons for he frs npu ) and membershp funcons for he second npu p r7) are produced. The delongaonfs resuls n membershp funcons for he frs npu ), and for he second npu p r8). For example, he ElongaonFs fuzz nference ssem has he srucure shown n Fg., whle he correspondng conroller Conroller ) has he same srucure as Conroller see Fg. ). Inpus Inpu mf Rules Oupu mf Oupu nmf oumf n Force n Curren nmf nmf nmf nmfp nmf p r... 7 r p oumfp oumf oumfr oumf7 Agregaed Oupu ou Elongaon Fg. Srucure of he ElongaonFs fuzz nference ssem Each of he four FISs s mpored a he fuzz conroller level, resulng n four conrollers: Conroller ElongaonFs ), Conroller celongaonfs ), Conroller delongaonfs ), and Conroller delongaonfs ). These four conrollers are negraed usng he logcal scheme gven n Fg. ; he Malab/Smuln model n Fg. s he resul. In he Malab/Smuln model shown n Fg., he second npu of Conroller and ha of Conroller Tme) are generaed b usng negraors, sarng from he momen ha hese npus are used n Conroller or Conroller he npu of he Gan bloc s f he schema decdes no o wor wh one of he Conrollers or ). I s possble ha he smulaon sample me ma be dfferen han he sample me used n he expermenal acuson process, and herefore we use he Gan bloc ha gves her rao; Te s he sample me n he expermenal

20 and T s he smulaon sample me. In he schema, he consan C represens he maxmum me consdered for he acuaor o recover s nal emperaure approxmael o C) when he curren becomes A. Force Force Fuzz Logc Conroller Consan Consan varable " " z Swch Sgnal From Worspace Curren Curren Deec Increase U > U/z Elongaon El Sgnal From Worspace Fuzz Logc Conroller Swch To Worspace Consan Te s he sample me n he expermenal T s he value of he sample me for smulaon Deec Decrease U < U/z Swch AND Logcal Operaor Consan NOT Logcal Operaor Swch7 Consan C s he maxmum me for he acuaor o recover s nal emperaure f he curren becomes null Consan AND Logcal Operaor Consan T/Te Gan Swch s Inegraor T/Te Gan C s Inegraor Swch Compare To Zero Fuzz Logc Conroller Compare To Zero Fuzz Logc Conroller Swch Swch Fg. The negraon model schema n Malab/Smuln Evaluang he negraed conroller model see Fg. ) for all fve expermenal cases produces he resuls shown n Fgs. and 7. These graphcs show he elongaons versus he number of expermenal pons and versus he appled elecrcal curren, respecvel, usng he expermenal and he negraed neuro-fuzz conroller model for he SMA. A good overlappng of he oupus of he negraed neuro-fuzz conroller wh he expermenal can be easl observed. The same observaon can be made from he D characerscs of he expermenal and he neuro-fuzz modeled n erms of emperaure, elongaon and force, depced n Fg. 8 a, and n erms of curren, elongaon and force, depced n Fg. 8 b. The mean values of he relave absolue errors of he negraed conroller for he fve load cases of he SMA acuaor, based on adapve neuro-fuzz nference ssems, are:.997% for N,.9% for N,

21 .9% for N,.79% for 8 N and.77% for 9 N. The mean value of he relave absolue error beween he expermenal and he oupus of he negraed conroller s.%. Elongaon [mm] F N. Number of expermenal pons. Elongaon [mm] F 8 N F N.8 Number of expermenal pons. Elongaon [mm] F 9 N F N. Number of expermenal pons Elongaon [mm].. Elongaon [mm]... Number of expermenal pons. Number of expermenal pons Fg. Elongaons versus he number of expermenal pons Elongaon [mm] F N. Curren [A] Elongaon [mm] F N.8 Curren [A] Elongaon [mm] F N. Curren [A]. F 8 N. F 9 N Elongaon [mm].. Elongaon [mm]... Curren [A]. Curren [A] Fg. 7 Elongaons versus he appled elecrcal curren

22 neuro-fuzz conroller neuro-fuzz conroller 8 8 Force [N] Force [N] Elongaon [mm] a. Temperaure [ o C] b. Fg. 8 D evaluaon of he negraed neuro-fuzz conroller Elongaon [mm] Curren [A] V. Conclusons In hs paper, an negraed conroller based on adapve neuro-fuzz nference ssems for modelng smar maeral acuaors was obaned. The drec applcaon of hs conroller s n a morphng wng ssem. The general am of he smar maeral acuaors desred model s o calculae he elongaon of he acuaor under he applcaon of a hermo-elecro-mechancal load for a ceran me. Therefore, he smar maeral acuaors were expermenall esed n condons close o hose n whch he wll be used. Tesng was performed for fve load cases, wh forces of N, N, N, 8 N and 9 N. Usng he expermenal, four Fuzz Inference Ssems were generaed and raned o oban four neuro-fuzz conrollers: one conroller for he curren ncrease ElongaonFs ), one for a consan curren celongaonfs ), one for he curren decrease delongaonfs ), and one conroller for he null curren, afer s decrease delongaonfs ). The genfs and genfs Malab funcons were used o generae he nal FISs, and he adapve neuro-fuzz nference ssem echnue was hen used o ran hem. The fnal values of he relave absolue errors for he four generaed and raned FISs were:.99% for ElongaonFs,.9% for celongaonfs,.7% for delongaonfs, and.7 % for delongaonfs. Each of he four obaned and raned FISs were mpored a he fuzz conroller level, resulng n four conrollers. Fnall, hese four conrollers were negraed b usng he logcal scheme gven n Fg. ; resulng n he Malab/Smuln model for he negraed conroller shown n Fg.. The negraed conroller performances were evaluaed for all fve load cases; he values obaned for he mean relave absolue errors were:.997% for N,.9% for N,.9% for N,.79% for 8 N and.77% for 9 N. Thus, he mean

23 value of he relave absolue error beween he expermenal and he oupus of he negraed conroller was.%. A parcular advanage of hs new model s s rapd generaon, hans o he genfs, genfs and ANFIS funcons alread mplemened n Malab. The user need onl assume he four FIS s ranng performances usng he anfsed nerface generaed wh Malab. Acnowledgmens We would le o han for he receved funds whch mae possble hs research o CRIAQ Consorum of Research n he Aerospaal Indusr n Quebec), Thales Avoncs, Bombarder Aerospace and he NSERC Naonal Scences and Engneerng Research Councl). Man hans are dues o Professors Vladmr Bralovs and Parc Terraul who gave us he opporun o use her SMA bench es eupmen for he realsaon of hs new conroller mehod. We manl han o George Henr Smon for nang he CRIAQ 7. proec as well as o Phlppe Molare from Thales Avoncs and o Erc Laurendeau from Bombarder Aeronaucs for her collaboraon on hs wor. References [] Svanandam, S.N., Sumah, S., Deepa, S.N., Inroducon o Fuzz Logc usng MATLAB, Sprnger, Berln Hedelberg, 7. [] Koso, B., Neural newors and fuzz ssems A dnamcal ssems approach o machne nellgence, Prence Hall, New Jerse, 99. [] *** Malab Fuzz Logc and Neural Newor Toolboxes - Help. [] Khezr, M., Jahed, M., Real-me nellgen paern recognon algorhm for surface EMG sgnals, BoMedcal Engneerng OnLne 7, :, do:.8/7-9x--. [] Kung, C.C., Su, J.Y., Affne Taag-Sugeno fuzz modellng algorhm b fuzz c-regresson models cluserng wh a novel cluser vald creron, IET Conrol Theor and Applcaons, Vol., Issue, 7, pp. -. [] Mahfouf, M., Lnens, D. A., Kandah, S., Fuzz Taag-Sugeno Kang model predcve conrol for process engneerng, Prned and publshed b he IEE, Savo place, London WCPR OBL. UK, pp., 999.