A Functional-Link-Based Fuzzy Neural Network for Temperature Control

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1 Proceedngs of he 7 IEEE Symposm on Fondaons of Compaonal Inellgence (FOCI 7) A Fnconal-Ln-Based Fzzy Neral Neor for emperare Conrol Cheng-Hng Chen *, Chn-eng Ln, Fello, IEEE, and Cheng-Jan Ln, ember, IEEE Absrac hs sdy presens a fnconal-ln-based fzzy neral neor (FLFNN) srcre for emperare conrol he proposed FLFNN conroller ses fnconal ln neral neors (FLNN) ha can generae a nonlnear combnaon of he np varables as he conseqen par of he fzzy rles An onlne learnng algorhm, hch consss of srcre learnng and parameer learnng, s also presened he srcre learnng depends on he enropy measre o deermne he nmber of fzzy rles he parameer learnng, based on he graden descen mehod, can ads he shape of he membershp fncon and he correspondng eghs of he FLNN Smlaon resl of emperare conrol has been gven o llsrae he performance and effecveness of he proposed model I INODUCION HE concep of fzzy neral neor (FNN) for conrol problem has been gron no a poplar research opc n recen years []-[4] he reason s ha classcal conrol heory sally reqres a mahemacal model for desgnng he conroller he naccracy of mahemacal modelng of plans sally degrades he performance of he conroller, especally for nonlnear and complex conrol problems [5] On he conrary, he FNN conroller offers a ey advanage over radonal adapve conrol sysems he FNN do no reqre mahemacal models of plans he FNN brng he lo-level learnng and compaonal poer of neral neors no fzzy sysems and gve he hgh-level hman-le hnng and reasonng of fzzy sysems o neral neors hs sdy presens a fnconal-ln-based fzzy neral neor (FLFNN) srcre for emperare conrol he FLFNN conroller, hch combnes a fzzy neral neor (FNN) h fnconal ln neral neor (FLNN) [6]-[7], s desgned mprove he accracy of fnconal approxmaon Each fzzy rle ha corresponds o a FLNN consss of fnconal expanson of he np varables he orhogonal polynomals and lnearly ndependen fncons are adoped as fnconal ln neral neor bases An onlne learnng algorhm, conssng of srcre learnng and parameer learnng, s proposed o consrc he FLFNN model aomacally he srcre learnng algorhm deermnes heher or no o add a ne node hch sasfes he fzzy C H Chen and C Ln are h he Dep of Elecrcal and Conrol Engneerng, Naonal Chao-ng Unversy, Hsnch, aan, OC C J Ln s h he Dep of Comper Scence and Informaon Engneerng, Chaoyang Unversy of echnology, No68, Jfong E d, Wfong onshp, achng Cony 449, aan, OC * Correspondng ahor (E-mal: chchenece9g@nced) paron of he np varables Inally, he FLFNN model has no rles he rles are creaed aomacally by enropy measre he parameer learnng algorhm s based on bacpropagaon o ne he free parameers n he FLFNN model smlaneosly o mnmze an op error fncon he characerscs of he proposed FLFNN model are explaned as follos Frs, he conseqen of he fzzy rles s a nonlnear combnaon of he np varables hs sdy ses he fnconal ln neral neor o he conseqen par of he fzzy rles he fnconal expanson n FLFNN model can yeld he conseqen par of a nonlnear combnaon of np varables Second, he onlne learnng algorhm can aomacally consrc he FLFNN model No rles or membershps exs nally hey are creaed aomacally as learnng proceeds, as onlne ncomng ranng daa are receved and as srcre and parameer learnng are performed hrd, as demonsraed n secon IV, he FLFNN model s a more adapve and effecve conroller han he oher mehods II SUCUE OF FLFNN hs secon descrbes he srcre of fnconal ln neral neors and he srcre of he FLFNN model In fnconal ln neral neors, he np daa sally ncorporae hgh order effecs and hs arfcally ncrease he dmensons of he np space Accordngly, he np represenaon s enhanced and lnear separably s acheved n he exended space he FLFNN model adoped he fnconal ln neral neor generang complex nonlnear combnaon of he np varables as he conseqen par of he fzzy rles he res of hs secon deals hese srcres A Fnconal Ln Neral Neors he fnconal ln neral neor s a sngle layer neor n hch he need for hdden layers s elmnaed Whle he np varables generaed by he lnear lns of neral neors are lnearly eghed, he fnconal ln acs on an elemen of np varables by generang a se of lnearly ndependen fncons, hch are sable orhogonal polynomals for a fnconal expanson, and hen evalang hese fncons h he varables as he argmens herefore, he FLNN srcre consders rgonomerc fncons For example, for a o-dmensonal np X = [ x x ], enhanced daa are obaned sng rgonomerc fncons as Φ =, x,sn(,cos(,, x,sn(,cos( x ),] [ π /7/$ 7 IEEE 5

2 Proceedngs of he 7 IEEE Symposm on Fondaons of Compaonal Inellgence (FOCI 7) hs, he np varables can be separaed n he enhanced space [8] In he FLNN srcre h reference o Fg, a se of bass fncons Φ and a fxed nmber of egh parameers W represen f W (x) he heory behnd he FLNN for mldmensonal fncon approxmaon has been dscssed elsehere [6] and s analyzed belo X x x FE x N φ φ φ Fg Srcre of FLNN Consder a se of bass fncons Β = {φ Φ( A)}, W ŷ ŷ ŷ m yˆ' yˆ' yˆm' Ŷ Κ Κ = {,, } h he follong properes; ) φ, ) he = sbse Β = φ Β } s a lnearly ndependen se, { = meanng ha f = φ =, hen = for all = / =,,,, and ) sp [ φ ] < Le Β = { φ } be a se of bass fncons o be = consdered, as shon n Fg he FLNN comprses bass fncons { φ, φ,, φ } Β he lnear sm of he h node s gven by ŷ = ( X ) = = [ x, x,, x N ] A φ () N here X Α, X s he np vecor and W = [,,, ] s he egh vecor assocaed h he h op of he FLNN ŷ denoes he local op of he FLNN srcre and he conseqen par of he h fzzy rle n he FLFNN model hs, Eq() can be expressed n marx form as yˆ = W Φ, here Φ = [ φ( x ), φ ( x),, φ ( x)] s he bass fncon vecor, hch s he op of he fnconal expanson bloc he m-dmensonal lnear op may be gven by ŷ = WΦ, here ˆ y = [ ŷ, ŷ,, ŷ m ], m denoes he nmber of fnconal ln bases, hch eqals he nmber of fzzy rles n he FLFNN model, and W s a (m )-dmensonal egh marx of he FLNN gven by W = [,,, m ] he h op of he FLNN s gven by ŷ ' = ρ( ŷ ), here he nonlnear fncon ρ () = anh() hs, he m-dmensonal op vecor s gven by Y ˆ = ρ (ˆ) y = f W ( x) () here Ŷ denoes he op of he FLNN In he FLFNN model, he fnconal ln bases do no exs n he nal sae, and he amon of fnconal ln bases generaed by he onlne learnng algorhm s conssen h he nmber of fzzy rles Secon III deals he onlne learnng algorhm B Srcre of FLFNN Conroller hs sbsecon descrbes he FLFNN model, hch ses a nonlnear combnaon of np varables (FLNN) Each fzzy rle corresponds o a sb-flnn, comprsng a fnconal ln Fgre presens he srcre of he proposed FLFNN model Nodes n layer are np nodes, hch represen np varables Nodes n layer are called membershp fncon nodes and ac as membershp fncons, hch express he np fzzy lngsc varables Nodes n hs layer are adoped o deermne Gassan membershp vales Each node n layer s called a rle node Nodes n layer are eqal o he nmber of fzzy ses ha correspond o each exernal lngsc np varable Lns before layer represen he precondons of he rles, and lns afer layer represen he conseqences of he rle nodes Nodes n layer 4 are called conseqen nodes, each of hch s a nonlnear combnaon of he np varables he node n layer 5 s called he op node; s recommended by layers and 4, and acs as a defzzfer he FLFNN realzes a fzzy f-hen rle n he follong form le-: IF x s A and x s A and x s A and x s A N N HEN ŷ = φ = () = φ + φ + + φ here x and ŷ are he np and local op varables, respecvely; A s he lngsc erm of he precondon par h Gassan membershp fncon; N s he nmber of np varables; s he ln egh of he local op; φ s he bass rgonomerc fncon of he np varables; s he nmber of bass fncon, and le- s he h fzzy rle he operaon fncons of he nodes n each layer of he FLFNN model are no descrbed In he follong descrpon, (l) denoes he op of a node n he lh layer x x x x Normalzaon FE φ φ φ Layer Layer Layer Layer 4 Layer 5 ŷ ŷ ŷ Fg Srcre of proposed FLFNN model Layer (Inp node): No compaon s performed n hs layer Each node n hs layer s an np node, hch y 54

3 Proceedngs of he 7 IEEE Symposm on Fondaons of Compaonal Inellgence (FOCI 7) corresponds o one np varable, and only ransms np vales o he nex layer drecly: = x (4) Layer (embershp fncon node): Nodes n hs layer correspond o a sngle lngsc label of he np varables n layer herefore, he calclaed membershp vale specfes he degree o hch an np vale belongs o a fzzy se n layer he mplemened Gassan membershp fncon n layer s ( ) [ m ] = exp (5) here m and are he mean and varance of he Gassan membershp fncon, respecvely, of he h erm of he h np varable x Layer (le Node): Nodes n hs layer represen he precondoned par of a fzzy logc rle hey receve one-dmensonal membershp degrees of he assocaed rle from he nodes of a se n layer Here, he prodc operaor descrbed above s adoped o perform he IF-condon machng of he fzzy rles As a resl, he op fncon of each nference node s ( ) = (6) here he of a rle node represens he frng srengh of s correspondng rle Layer 4 (Conseqen Node): Nodes n hs layer are called conseqen nodes he np o a node n layer 4 s he op from layer, and he oher nps are nonlnear combnaons of np varables from a fnconal ln neral neor, here he nonlnear combnaon fncon has no sed he fncon anh (), as shon n Fg For sch a node, ( 4 ) = ( ) = φ (7) here s he correspondng ln egh of fnconal ln neral neor and φ s he fnconal expanson of np varables he fnconal expanson ses a rgonomerc polynomal bass fncon, gven by [ x sn( cos( x sn( cos( ] for o-dmensonal np varables herefore, s he nmber of bass fncons, = N, here N s he nmber of np varables Layer 5 (Op Node): Each node n hs layer corresponds o a sngle op varable he node negraes all of he acons recommended by layers and 4 and acs as a defzzfer h, y φ ( 4 ) ( 5 ) = = = = = = = = = = = here s he nmber of fzzy rles, and y s he op of he FLFNN model ŷ (8) III Learnng Algorhms of he FLFNN Conroller hs secon presens an onlne learnng algorhm for consrcng he FLFNN model he proposed learnng algorhm comprses a srcre learnng phase and a parameer learnng phase A Srcre Learnng Phase he frs sep n srcre learnng s o deermne heher a ne rle from shold be exraced he ranng daa and o deermne he nmber of fzzy ses n he nversal of dscorse of each np varable, snce one clser n he np space corresponds o one poenal fzzy logc rle, n hch m and represen he mean and varance of ha clser, respecvely For each ncomng paern x, he rle frng srengh can be regarded as he degree o hch he ncomng paern belongs o he correspondng clser For compaonal effcency, he measre creron calclaed from s adoped as he enropy measre E ( ) here = exp( ) N = = D log D (9) D and E [, ] Accordng o Eq(9), he creron for generang a ne fzzy rle and ne fnconal ln bases for ne ncomng daa s descrbed as follos he maxmm enropy measre E = max E () max ( ) s deermned, here ( s he nmber of exsng rles a me If E max E hen a ne rle s generaed, here E [, ] s a prespecfed hreshold ha decays drng he learnng process In he srcre learnng phase, he hreshold parameer E s an mporan parameer he hreshold s se o beeen zero and one A lo hreshold leads o he learnng of coarse clsers (e, feer rles are generaed), hereas a hgh hreshold leads o he learnng of fne clsers (e, more rles are generaed) If he hreshold vale eqals zero, hen all he ranng daa belong o he same clser n he np space herefore, he selecon of he hreshold vale E ll crcally affec he smlaon resls, and hs vale ll be based on praccal expermenaon or on ral-and-error ess E s defned as 6- mes he np varance Once a ne rle has been generaed, he nex sep s o assgn he nal mean and varance for he ne membershp fncon and he correspondng ln egh for he conseqen par Snce he goal s o mnmze an obecve fncon, he mean, varance and egh are all adsable laer n he parameer learnng phase Hence, he mean, varance and egh for he ne rle are se as follos; ( ) ( + ) m = x () = ( ( + ) ) () n = random[ ( ) ( + ), () ] 55

4 Proceedngs of he 7 IEEE Symposm on Fondaons of Compaonal Inellgence (FOCI 7) here x s he ne np and s a prespecfed consan n he hole algorhm for he generaon of ne fzzy rles and fzzy ses n each np varable s as follos No rle s assmed o exs nally exs: Sep : IF x s he frs ncomng paern HEN do {Generae a ne rle h mean m =x, varance =, n egh = random[, ] here s a prespecfed consan n } Sep : ELSE for each nely ncomng x, do {Fnd E = max E max ( ) IF E max E do nohng ELSE { (+) = ( + generae a ne rle ( + ) ( + ) h mean m = x, varance =, n egh ( + ) = random[, ] here s a prespecfed consan} n } B Parameer Learnng Phase Afer he neor srcre has been adsed accordng o he crren ranng daa, he neor eners he parameer learnng phase o ads he parameers of he neor, opmzed accordng o he same ranng daa he learnng process nvolves deermnng he mnmm of a gven cos fncon he graden of he cos fncon s comped and he parameers are adsed along he negave graden he bacpagaon algorhm s adoped for hs spervsed learnng mehod When he sngle op case s consdered for clary, he goal o mnmze he cos fncon E s defned as d E( = [ y( y ] = e (4) here y d ( s he desred op and y( s he model op for each dscree me In each ranng cycle, sarng a he np varables, a forard pass s adoped o calclae he acvy of he model op y( When he bacpropagaon learnng algorhm s adoped, he eghng vecor of he FLFNN model s adsed sch ha he error defned n Eq(4) s less han he desred hreshold vale afer a gven nmber of ranng cycles he ell-non bacpropagaon learnng algorhm may be ren brefly as E( W ( + ) = W + ΔW = W + η (5) W here, n hs case, η and W represen he learnng rae and he nng parameers of he FLFNN model, respecvely Le W = [ m,,] be he eghng vecor of he FLFNN model hen, he graden of error E ( ) n Eq(4) h respec o an arbrary eghng vecor W s E( ) y( = e( ) (6) W W ecrsve applcaons of he chan rle yeld he error erm for each layer hen he parameers n he correspondng layers are adsed Wh he FLFNN model and he cos fncon as defned n Eq(4), he pdae rle for can be derved as follos; ( + ) = + Δ (7) here E Δ φ e = Smlarly, he pdae las for m, and are m ( + ) = m + Δm (8) ( + ) = + Δ (9) here E Δm m m ( 4 ) ( ) ( m ) e m = E Δ ( 4 ) ( ) ( m ) e = here η, η and η are he learnng rae parameers of he m egh, he mean, and he varance, respecvely IV Conrol of Waer Bah emperare Sysem he goal of hs secon s o elcdae he conrol of he emperare of a aer bah sysem accordng o, dy( ) ( Y y( = + () d C C here y( s he op emperare of he sysem n C ; ( s he hea flong no he sysem; Y s room emperare; C s he eqvalen hermal capacy of he sysem, and s he eqvalen hermal ressance beeen he borders of he sysem and he srrondngs and C are assmed o be essenally consan, and he sysem n Eq() s reren n dscree-me form o some reasonable approxmaon he sysem δ αs ( e ) αs y e y( ) α αs + = + ( ) + [ e ] y () 5 y( ) 4 + e s obaned, here α and δ are some consan vales of and C he sysem parameers sed n hs example are 56

5 Proceedngs of he 7 IEEE Symposm on Fondaons of Compaonal Inellgence (FOCI 7) 4 α = 5e, δ = e and Y =5 ( C ), hch ere obaned from a real aer bah plan consdered elsehere [9] he np () s lmed o, and 5V represen volage n he samplng perod s s= he convenonal onlne ranng scheme s adoped for onlne ranng Fgre 4 presens a bloc dagram for he convenonal onlne ranng scheme hs scheme has o phases - he ranng phase and he conrol phase In he ranng phase, he sches S and S are conneced o nodes and, respecvely, o form a ranng loop In hs loop, ranng daa h np vecor I = [ y ( + ) y ] and p p desred op can be defned, here he np vecor of he FLFNN conroller s he same as ha sed n he general nverse modelng [] ranng scheme In he conrol phase, he sches S and S are conneced o nodes and 4, respecvely, formng a conrol loop In hs loop, he conrol sgnal û ( ) s generaed accordng o he np vecor I' = [ y ( + ) y ], here y s he plan op and y ref p p ref s he reference model op A seqence of random np sgnals rd () lmed o and 5V s neced drecly no he smlaed sysem descrbed n Eq(), sng he onlne ranng scheme for he FLFNN conroller he ranng paerns are seleced based on he np-ops characerscs o cover he enre reference op he emperare of he aer s nally 5 c, and rses progressvely hen random np sgnals are neced Afer ranng eraons, for fzzy rles are generaed he obaned fzzy rles are as follos le-: IF x s μ(46,65) and x s μ(74,7 49) HEN ŷ = 95x sn( 4 546cos( 7 6x 4799 sn( + 54cos( le-: IF x s μ(496,967) and x s μ(468,977) HEN ŷ = 447x + 766sn( 7775cos( 59x 687 sn( cos( le-: IF x s μ(677,69) and x s μ(6499,5864) HEN ŷ = 575x 97 sn( 4659cos( 4x + 675sn( + cos( le-4: IF x s μ(7965,8769) and x s μ(64654,9 97) HEN ŷ = 4655x 7sn( 57759cos( 4 585x + 665sn( + 488cos( yp(+) yref(+) S Z - FLFNN Conroller S 4 () + ˆ Z - Plan yp(+) Fg Convenonal onlne ranng scheme hs sdy compares he FLFNN conroller o he PID conroller [], he manally desgned fzzy conroller (FC) [], he fnconal ln neral neor (FLNN) and he SK-ype fzzy neral neor (SK-ype FNN) Each of hese conrollers s appled o he aer bah emperare conrol sysem he performance measres nclde he se-pons reglaon, he nflence of mplse nose and a large parameer varaon n he sysem he frs as s o conrol he smlaed sysem o follo hree se-pons 5 c, for 4 y ref = 55 c, for 4 < 8 () 75 c, for 8 < Fgre 4 presens he reglaon performance of he FLFNN conroller We also es he reglaon performance by sng he FLNN conroller and he SK-ype FNN conroller o es her reglaon performance, a performance ndex, he sm of absole error (SAE), s defned by here () SAE = y y( ) () ref y ref and y () are he reference op and he acal op of he smlaed sysem, respecvely he SAE vales of he FLFNN conroller, he PID conroller, he fzzy conroller, he FLNN conroller and he KS-ype NFN conroller are 58, 485, 45, 79 and 69, hch are gven n he second ro of able he proposed FLFNN conroller yelds a mch beer SAE vale of reglaon performance han he oher conrollers he second se of smlaons s performed o elcdae he nose-reecon ably of he fve conrollers hen some nnon mplse nose s mposed on he process One mplse nose vale 5 C s added o he plan op a he sxeh samplng nsan A se-pon of 5 C s adoped n hs se of smlaons For he FLFNN conroller, he same ranng scheme, ranng daa and learnng parameers as ere sed n he frs se of smlaons Fgre 5 presens he behavors of he FLFNN conroller nder he nflence of mplse nose he SAE vales of he FLFNN conroller, he PID conroller, he fzzy conroller, he FLNN conroller and he SK-ype FNN conroller are 74, 5, 758, 45 and 7475, hch are shon n he hrd ro of able he FLFNN conroller performs qe ell I recovers very qcly and seadly afer he presenaon of he mplse nose One common characersc of many ndsral-conrol processes s ha her parameers end o change n an npredcable ay he vale of 7 ( ) s added o he plan np afer he sxeh sample n he hrd se of smlaons o es he robsness of he fve conrollers A se-pon of 5 C s adoped n hs se of smlaons Fgre 6 presens he behavors of he FLFNN conroller hen n he plan dynamcs change he SAE vales of he FLFNN conroller, he PID conroller, he fzzy conroller, he FLNN conroller and he SK-ype FNN conroller are 6,, 75, 5 and 654, hch vales are shon n he forh ro of able he resls presen he favorable conrol and dsrbance reecon capables of he raned FLFNN conroller n he aer bah sysem he resls presen he favorable conrol and dsrbance reecon capables of he raned FLFNN conroller n he 57

6 Proceedngs of he 7 IEEE Symposm on Fondaons of Compaonal Inellgence (FOCI 7) aer bah sysem he aforemenoned smlaon resls, presened n able, demonsrae ha he proposed FLFNN conroller operforms oher conrollers performng onlne srcre/parameer learnng schemes concrrenly Fnally, comper smlaon resls have shon ha he proposed FLFNN conroller has beer performance han ha of oher models able : Comparson of performance of varos conrollers FLFNN PID [] FC [] FLNN SK-ype FNN eglaon Performance Inflence of Implse Nose Effec of Change n Plan Dynamcs Fg 4 Fnal reglaon performance of he FLFNN conroller for aer bah sysem ACKNOWLEDGEN hs research as sponsored by Deparmen of Indsral echnology, nsry of Economc Affars, OC nder he gran 95-EC-7-A--S-9 Fg 5 Behavor of he FLFNN conroller nder he mplse nose for aer bah sysem Fg 6 Behavor of he FLFNN conroller hen a change occrs n he aer bah sysem EFEENCES [] aag and Sgeno, Fzzy denfcaon of sysems and s applcaons o modelng and conrol, IEEE rans on Sys, an, Cybern, vol 5, pp 6-, 985 [] J-S Jang, ANFIS: Adapve-neor-based fzzy nference sysem, IEEE rans on Sys, an, and Cybern, vol, pp , 99 [] C Ln and C S G Lee, Neral Fzzy Sysems: A Nero-Fzzy Synergsm o Inellgen Sysem, NJ: Prence-Hall, 996 [4] C F Jang and C Ln, An on-lne self-consrcng neral fzzy nference neor and s applcaons, IEEE rans Fzzy Sysems, vol 6, no, pp -, Feb 998 [5] K J Asrom and B Wenmar, Adapve Conrol, A: Addson-Wesley, 989 [6] J C Para and N Pal, A fnconal ln arfcal neral neor for adapve channel eqalzaon, Sgnal Process, vol 4, pp 8-95, ay 995 [7] J C Para, N Pal, B N Chaer, and G Panda, Idenfcaon of nonlnear dynamc sysems sng fnconal ln arfcal neral neors, IEEE rans on Sys, an, and Cybern, vol 9, Apr 999 [8] Y H Pao, Adapve Paern ecognon and Neral Neors, A: Addson-Wesley, 989 [9] J anomar and S Oma, Process conrol by on-lne raned neral conrollers, IEEE rans on Ind Elecron, vol 9, pp 5-5, 99 [] D Psals, A Sders, and A Yamamra, A mllayered neral neor conroller, IEEE Conr Sys, vol 8, pp 7, 988 [] C L Phllps and H Nagle, Dgal Conrol Sysem Analyss and Desgn, Prence Hall, 995 V Conclson hs sdy proposes a fnconal-ln-based nero-fzzy neor (FLFNN) srcre for emperare conrol he FLFNN model ses he fnconal ln neral neor (FLNN) as he conseqen par of he fzzy rles herefore, he FLFNN model can generae he conseqen par of a nonlnear combnaon of he np varables o be approxmaed more effecvely he FLFNN model can aomacally consrc and ads free parameers by 58