Online Multivariable Identification of a MIMO Distillation Column Using Evolving Takagi-Sugeno Fuzzy Model

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1 Proceedngs of e 6 Chnese Control Conference July 6-3, 7, Zhangae, Hunan, Chna Onlne ultvarable Identfcaton of a IO Dstllaton Colun Usng Evolvng aag-sugeno Fuzzy odel olaze Sananda Borhan, Salahshoor Kar,. Autoaton and Instruentaton Engneerng Departent, Petroleu Unversty of echnology, ehran, Iran E-al: borhan_olazesananda@yahoo.co. E-al: salahshoor@put.ac.r Abstract: In s paper, an evolvng aag-sugeno (es fuzzy odel has been utlzed for onlne dentfcaton of a ult-nput, ult- output (IO dstllaton colun. In s approach, e rule-base structure and e odel paraeters of e consequent parts of fuzzy IF-HE rules gradually evolve durng e onlne dentfcaton process. In addton, an exponental te-varyng weght s ncluded n e orgnal rule generaton condton n order to control e rate of rule generaton at e start of e tranng process and consequently reduce e total nuber of generated rules n coparson w e orgnal IO es algor. Recursve-Least Squares (RLS algor s eployed to estate e consequent part of each rule. he results show at e odfed condton reduces e total nuber of generated rules for a certan data set w lower RSE error n coparson w e orgnal es eod. Key Words: Systen Identfcaton,Evolvng aag-sugeno,fuzzy Systes, Dstllaton Colun IRODUCIO Fuzzy odel dentfcaton s an effectve tool for e approxaton of nonlnear dynacal systes on e bass of easured data []. hs approach has been popular for e past years due to ts ablty to utlze heurstc nowledge to provde quanttatve odel whch can accurately represent coplex nonlnear systes. Aong dfferent fuzzy odelng technques, aag-sugeno (S odel [] has receved a great deal of attenton and has been eployed n any applcatons n nonlnear syste dentfcaton. hs s anly due to ts good results n dfferent applcatons and also because t eploys aeatcal functons as rule consequent parts. hs odel conssts of IF-HE rules w fuzzy antecedents and aeatcal functons n e consequent parts. hs structure gves e ablty to utlze e nput-output data n an effcent way. In e offlne ode all of e data are avalable at e start of e tranng process. In e onlne ode, however, we do not have e whole data at e start of e tranng process. hus learnng of e S fuzzy odels should be started w e frst data saple [4,9]. In s condton, e odel structure s not nown a pror, but nstead t evolves gradually durng e dentfcaton process. hen we coe to e concept of Evolvng aag-sugeno (es fuzzy odels [4,9].IneS,epotental of e new data saple s used as a trgger to update e rule-base. It s portant to note at learnng could start wout a pror nforaton and only w a sngle data saple. hs nterestng feature aes e approach potentally very useful n any sart adaptve systes [4,9]. In s paper, e orgnal rule generaton condton s odfed so as to effcently control e rate of rule generaton especally at e start of e tranng process. hs leads to a reducton n e total nuber of generated rules for a certan data hstory w a better accuracy, expressed by lower RSE error, n coparson w e orgnal es algor. hs technque s appled to a sulated nonlnear 38 IO dstllaton colun as a popular benchar proble n nonlnear dentfcaton [3,5,7]. he paper s organzed as follows. In secton, e IO aag-sugeno fuzzy odel s ntroduced. In secton 3, e Recursve-Least Squares (RLS algor s presented. he es eod s descrbed n secton 4. he dstllaton colun odel s gven n secton 5. In secton 6, our odfcaton for es s presented and en ese eods are appled to e dstllaton colun. Soe bref concluson rears are gven n secton 7. IO S FUZZY ODEL In s paper, a IO extenson of e S fuzzy odel s consdered [6, 9]. herefore, e rule of e IO S fuzzy odel has e followng for: R : IF x s A AD x s A AD... AD xn s An HE y = Xe π =,,, (, x, x,..., xn are nput varables; n s e total nuber of nput varables; y = y, y,..., y s e output of rule; e =,,..., n w to defne a free bas paraeter for each rule [4,9]. As a IO extenson of e S odel, y n e consequent part of e fuzzy rule denotes e ultdensonal vector of e lnear sub-syste [9]. Slarly, s e total nuber of output varables; s e total nuber of rules; A( A... An are e lngustc ters of fuzzy sets; In (, R denotes e fuzzy rule; X e s e extended nput vector, X =, X, whch s fored by appendng e nput vector X [ x x x ]

2 e paraeters n e consequent parts wll be denoted by [4], [9]: a a... a a a... a π = ( a a... a n n n Syetrcal Gaussan functons are used as antecedent fuzzyebershpfunctonsasfollows: * x x μ ( x = exp 4,=, (3 r r s a postve constant whch defnes e radus of e antecedent and e zone of nfluence of e * odel; x s e focal pont assocated w e center of e rule antecedent. A value of r n e range of [.3;.5] has been recoended [9]. In order to calculate e crsp output, for e fuzzfer, adan I operator can be used, and defuzzfcton ay be obtaned usng weghted average eod as follows: y = y = (Crsp Output (4 = n = A ( x = μ (5 s e fulfllent degree of e rule. Equaton (4 can be rewrtten as follows. Defnng: ξ = = (6 Equaton (4 becoes: y = y ξ (7 = hen, by substtutng ( n (7, t gves: ( e = (8 = y X π ξ 3 RECURSIVE-LEAS SQUARES (RLS ALGORIH Recursve-Least Squares (RLS algor s used to estate e paraeters n atrx π, defned n (, whch are dependent on e values of e ebershp functons. In order to use RLS dentfcaton algor, (8 can be expanded as follows [9] : y = ξ Xe π + ξ Xe π ξ Xe π (9 whchcanbedefnedneforoftwonewters ϕ and θ as : ϕ = ξ X, ξ X,..., ξ X ( e e e ( π,( π,...,( π θ = ( Consequently, (8 can be rewrtten as follows: y = ϕ θ ( For a gven set of nput-output data ( X, y = [l, ] and s e nuber of tranng data,θ should nze e followng globally optal obectve functon [8]: = ( J = y ϕ θ (3 4 EVOLVIG AKAGI-SUGEO ODEL In Onlne ode, e tranng data are collected contnuously, raer an beng a fxed set and e dentfcaton process starts w e frst data pont. Onlne learnng of es odels ncludes onlne clusterng under assupton of a gradual change of e rule-base. herefore, e nubers of fuzzy rules grow durng e dentfcaton process and are not fxed beforehand. So a odfed Recursve-Least Squares (RLS algor wll be requred [4, 9]. 4. Onlne Potental Clusterng Approach he onlne clusterng procedure starts w e frst data pont. hs pont s also consdered as e center of e frst cluster. Its coordnates are used to for e antecedent part of e fuzzy rule ( usng Gaussan ebershp functons (3. Its potental s assued to. Startng fro e next data pont onwards e potental of new data ponts s calculated recursvely. As a easure of potental, we use a Cauchy type functon of frst order as follows [4,9] : P ( z = ; =,3,... n+ + ( d ( (4 = = P z denotes e potental of e data pont ( z calculated at te ; d = z z,denotes e dstance between two data ponts at dfferent saple tes, for dfferent nputs( x for =,,,n and outputs( y for =n+,n+,, n+. After e new data are avalable n onlne ode, ey nfluence e potentals of e centers of e prevous * clusters ( z, =,,...,.hereasonsatbydefnton e potental depends on e dstance to all data ponts, ncludng e new ones. So e potental of e exstng rule centers should be updated when a new data pont s added. hs can be done by usng e followng equaton recursvely: * ( P * ( z P ( z = n+ * * + P ( z + P ( z ( d( (5 * P ( z s e potental of e exstng rule centers whch s updated at e saple te. = 39

3 ow after calculatng e potental of e new data pont and also updatng e potental of e exstng rule centers, e potental of e new data pont should be copared to e updated potental of e exstng rule centers wheer to add a new rule or to odfy e exstng rule-base by replaceent. he evoluton of e rule-base s conducted by e followng two basc prncples: CASE (REPLACE: IF e potental of e new data pont s hgher an e axu of e updated potental of e exstng rule centers: AD = * ( ax ( P z > P z (6 = z sclosetoanexstngrulecenter: P ( z δ n > ax P r (7 * ( z * n δ n = z z denotes e dstance between e new data pont and e closest rule center = of e exstng rule-base. HE e new data pont z replaces s closest center whch we assue ts ndex h as follows: h* h* z = z ; P z = P z (8 ( ( CASE (ADD: IF only (6 s satsfed but not (7 HE e new data pont s added to e rule-base as a new rule center as follows: * * = + ; z = z ; P z = P z (9 ( ( 4. Onlne Recursve Estaton of Consequent Paraeters of es Because n es, e rule-base gradually evolves, e straghtforward applcaton of e RLS s not applcable. A resettng of e covarance atrces and paraeters of e RLS should be done each te a new rule s added to e rule-base. In e case when e globally optal obectve functon s nzed (3, e RLS algor s conducted n e followng steps [8] : For e new ϕ at each saple te and en evaluate e followng gan vector naed as Kalan gan vector: C ϕ K = Cϕ = ( + ϕc ϕ Update e paraeter atrxθ : θ = θ + Ke ( e = y ϕθ 3 Update e covarance atrx C : C ϕϕ C C = C = Ι K ϕ C + ϕ C ϕ (,,..., ; C θ ( π ( π ( π = = =ΩΙ In e case when a new rule s added to e rule-base, e RLS wll be reset n e followng way [4,9] : + Paraeters of e new rule ( π are deterned by e weghted average of e paraeters of e oer exstng rules. he weghts are e noralzed frng levels of e exstng rules. Paraeters of e oer rules are nherted fro e prevous step [4] : + θ π, π,..., π, π = (3 + π = ξ π = Covarance atrces are reset as follows: ρζ, ρζ, Rn ( + ρζ R( n+, ρζ R( n+, R( n+ C = (4 Ω Ω ζ, =,,,R(n+; =,,,R(n+ s an eleent of e covarance atrx at prevous saple R + te; ρ = [4]. R In e case when a rule s replaced w anoer one, e covarance atrces and paraeters are nherted fro e prevous te step [4,9].Fnally,nesaeloopand after estatng and updatng e paraeters, e next value of e outputs can be predcted onlne as follows: y ϕ θ + = =,3 (5 5 DISILLAIO COLU CASE SUDY he process to be dentfed s a frst-prncple odel of a bnary dstllaton colun (see Fg..he colun s referred to as colun A whch has been studed n several papers [3]. he sulated syste covers e ost portant effects for e dynac of a real dstllaton colun. Furer detals of e sulated process odel are descrbed n [3], [5], and [7]. o dentfy e sulated process, an approprate odel should be selected. It s assued at e process under study can be represented by e followng one-step predcton odel [5] : ( = (, (, ( y f y y u (6 hs structure s selected due to ts ntrnsc nonlnear characterstcs and ts generalty. 33

4 In Fg. 3 and Fg. 4, e correspondng outputs B and obtaned usng orgnal es fuzzy odel, are depcted. he absolute error for each of e outputs s also depcted n Fg. 5 and Fg.6 he evoluton of e generated rules durng e onlne dentfcaton process s also shown n Fg. 7. As a easure of perforance, e Root ean Square Error s used as follows: RSE y = = y (7 s e total nuber of tranng data saples, y s e actual output at e saple te and estated output at e saple te. y s e.9 Output changes for B,reboler holdup Actual Output Estated Output.8.7 Fg. Dstllaton Colun In e dentfcaton procedure, L(reflux flow rate and V (bolup flow rate are used as nputs whle B (reboler holdup and D (condenser holdup are used as outputs. Fg. shows e bloc dagra of e assued nput-output varables for e fuzzy odel dentfcaton of e sulated syste. As shown, by consderng (6, ere are 6 nputs and outputs for e dentfcaton schee Dta Saple(n Fg.3.4. Output changes for B (t, reboler holdup Output changes for D,condenser holdup Actual Output Estated Output Data Saple(n Fg.4 Output changes for D (t, condenser holdup.6 Absolute error for B,reboler holdup Fg. Bloc dagra of dstllaton colun fuzzy odel APPLICAIO O DISLLAIO COLU Sgnals at have appled as nput changes n bol up flow V and reflux flow L are chosen such at ey wll generate data w good enough sgnal to nose rato but wll not dsturb e product qualty. he steady state values of e proposed dstllaton colun are gven n [3]. hese values are used for e process ntal condtons. Fro 6 saples of data, 3 are used for dentfcaton of fuzzy paraeters and e rest wll be used for evaluaton purpose. he only pre-specfed paraeters n e algor are r =.5 and Ω = Applcaton of e Orgnal es eod D Dat Saple(n Fg Absolute error for B (t, reboler holdup Dta Saple(n Fg.6 Absolute error for D,condenser holdup Absolute error for D (t, condenser holdup 33

5 uber of rules otal uber of genereated rules=3 RSE= Data Saple(n Fg.7 Evoluton of e rule-base and rule generaton 6. Applcaton of e odfed es eod he followng exponental te-varyng weght (9 s ncluded n (6 n order to control e rate of rule generaton effcently. hs odfcaton aes e algor add new rules w ore cauton n e ntal dentfcaton phases. hen, as e dentfcaton algor progresses and ore dynac nowledge are captured, e rule generaton condton gets bac exponentally to ts orgnal lower decson level. So (6 changesasfollows: * P ( z > ( ax P ( z (8 = ( = + ( exp τ (9 and τ are chosen as follows: =.5 ; τ = ( s e total nuber of tranng data 5 saples. Of course, s strategy can be resued whenever a sgnfcant dynac change, detected by e resdual error, s occurred. he te evoluton of e generated rules has been llustrated n Fgure 8. uber of rules Fg otal nuber of generated rules=6 RSE= Data Saple(n Evoluton of rule-base by exertng e odfed condton As shown, e generaton rate becoes ore effcent leadngtoareductonnetotalnuberofgenerated rules. ab. shows a coparson between e orgnal and e odfed eod. he results deonstrate at e obtaned reducton n e nuber of generated rules does not attenuate e accuracy. ab. RSE and otal uber of Generated Rules Coparson otal uber of RSE Generated Rules Orgnal eod odfed eod COCLUSIO hs paper presents e applcaton of e es fuzzy odel for onlne dentfcaton of a IO dstllaton colun benchar proble. An exponental te-varyng weght has been proposed to enhance e rule generaton echans. he odfed approach adds rules cautously at e start of dentfcaton and gradually becoes ore flexble when ore dynac nowledge s captured. he sulaton results ndcate e effcent perforance of e resultng dentfcaton eod w lower total generated rules. REFERECES [] aag, Sugeno. Fuzzy dentfcaton of systes and ts applcaton to odelng and control[j]. IEEE ransactons on Systes, an and Cybernetcs,985, 5(:6-3. [] Hellendoorn H, Dranov D, Fuzzy odel Identfcaton: Selected Approaches[]. Berln: Sprnger, Gerany, 997. [3] Sogestad S, ALAB Dstllaton Colun odel ("ColunA".Avalable: htl. [4] Angelov P P, Flev D P. An approach to onlne dentfcaton of aag-sugeno odels[j]. IEEE ransactons on Systes, an and Cybernetcs, Part B: Cybernetcs, 4, 34(. [5] Sogestad S, orar. Understandng e dynac behavor of dstllaton coluns[j]. Industral and Engneerng Chestry Research, 988, 7: [6] ollov S, Babusa R, Verbruggen H B. Analyss of Interactons n IO aag-sugeno Fuzzy odels[j]. In Proc. FUZZ- IEEE, : [7] Sogestad S.Dynacs and control of dstllaton coluns. Che. Eng. Res. Des. (rans ICheE, 997, 75: [8] Lung L, Syste Identfcaton: heory for e User[]. Prentce-Hall, Englewood Clffs,.J, 987. [9] Angelov P, Xydeas C S, Flev D. Onlne dentfcaton of IO evolvng aag-sugeno fuzzy odels. Proceedngs of e IJC-FUZZ-IEEE, Budapest, Hungary, July, 4: