Interval Type-1 Non-Singleton Type-2 TSK Fuzzy Logic Systems Using the Kalman Filter - Back Propagation Hybrid Learning Mechanism

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1 Inerval Type- Non-Sngleon Type- TSK Fuzzy Logc Sysems Usng he Kalman Fler - Bac Propagaon Hybrd Learnng Mechansm Gerardo M Mendez, Angeles Hernández, Marcela Casllo-Leal, Danel Loras, Insuo Tecnologco de Nuevo Leon, Deparmen of Elecrcal and Elecroncs Engneerng, Av Eloy Cavazos 00, Cd Guadalupe, NL, CP 6770, Mexco gmm_paper@yahoocommx Insuo Tecnologco de Nuevo Leon, Deparmen of Bussness Admnsraon, Av Eloy Cavazos 00, Cd Guadalupe, NL, CP 6770, Mexco hernandez@yurracommx, Absrac Ths arcle presens a novel learnng mehodology based on he hybrd mechansm for ranng of nerval ype- non-sngleon ype- Taag- Sugeno-Kang fuzzy logc sysems (IT TSK NSFLS) The process of combnng mulple compuaonal nellgence echnques o buld a hybrd model has become ncreasngly popular As repored n he leraure, he performance ndces of hese hybrd models have proved o be beer han he ndvdual ranng mechansm when used alone In hs wor usng nonsngleon npu-oupu daa pars durng he forward pass of he ranng process, he oupu s calculaed and he consequen parameers are uned by recursve fler mehod (REFIL), a Kalman fler ype In he bacward pass, he error propagaes bacward, and he aneceden parameers are uned by bacpropagaon mehod (BP) The proposed hybrd mehodology was esed hough he modelng and predcon of he seel srp emperaure as s beng rolled n an ndusral ho srp mll, and for comparave purposes, under he same conrolled condons mananed on prevous wor relaed o IT TSK hybrd ranng Resuls show he performance of he hybrd learnng mehod (REFIL-BP) boh, usng sngleon, and ype- non-sngleon npu-oupus daa pars The laer s capable of compensae he IT TSK predcor s unng for unceran measuremens, whls he former canno Also, he resuls show ha he hybrd models perform beer han he ndvdual echnques when used alone for he same daases Keywords: IT TSK fuzzy logc sysems, ANFIS, hybrd learnng Inroducon In [] boh, one-pass and bac-propagaon (BP) mehods are presened as IT Mamdan FLS learnng mehods, bu only BP s presened for IT Taag-Sugeno- Kang (TSK) FLS sysems The one-pass mehod generaes a se of IF-THEN rules by

2 usng he gven ranng daa one me, and combnes he rules o consruc he fnal FLS When BP mehod s used n boh Mamdan and TSK FLSs, none of aneceden and consequen parameers of he IT FLS s fxed a sarng of ranng process; hey are uned usng exclusvely seepes descen mehod In [] recursve leas squares (RLS) and recursve Kalman fler (REFIL) algorhms are no presened as IT FLS learnng mehods The am of hs wor s o presen and dscuss he hybrd-learnng algorhm for aneceden and consequen parameers unng durng ranng process for nerval ype- non-sngleon ype- TSK FLS (IT TSK NSFLS- or IT NS ANFIS) sysems, usng durng he forward pass of ranng REFIL mehod, whle durng he bacward pass, he BP mehod The hybrd algorhm for IT Mamdan FLS has been already presened elsewhere [], [3], [4], [5] and [6] wh hree combnaons of he learnng mehod: RLS-BP, REFIL-BP and orhogonal leas-squares-bp (OLS-BP) The hybrd algorhm for sngleon IT TSK SFLS (IT ANFIS) has been presened elsewhere [7] and [8] wh wo combnaons of he learnng mehod: RLS-BP and REFIL-BP, whls he hybrd algorhm for nerval non-sngleon ype- IT TSK NSFLS- (IT NS ANFIS) has been presened n [9] and [0] only wh he hybrd learnng mechansm RLS-BP There does no seem o be any oher menon of ype- or ype- non-sngleon IT TSK FLS n he leraure [], usng he REFIL-BP learnng mechansms I has been never presened before In hs wor, he IT TSK NSFLS- sysem ha uses he hybrd learnng mechansm (REFIL-BP) has been developed and mplemened for emperaure predcon of he ransfer bar a ho srp mll (HSM) The same daa-se used n prevous wors [7], [8], [9], and [0] n order o serve as comparson of funconaly and sably of he novel hybrd mechansm and for comparaon resuls The nenon of hs paper s o show he mplemenaon n a real ndusral applcaon of he (REFIL-BP) hybrd mechansm, ranng a non-sngleon ype- IT TSK FLS Proposed Mehodology Mos of he ho srp mll processes are hghly unceran, non-lnear, me varyng and non-saonary [,], havng very complex mahemacal represenaons IT NS ANFIS aes easly he random and sysemac componens of ype A or B sandard uncerany [] of ndusral measuremens The non-lneares are handled by FLS as denfers and unversal approxmaors of nonlnear dynamc sysems [3,4,5,6,7] Saonary and non-saonary addve nose s modeled as a Gaussan funcon cenered a he measuremen value [] In saonary addve nose, he sandard devaon aes a sngle value, whereas n non-saonary addve nose he sandard devaon vares over an nerval of values [] The BP learnng mehod for IT TSK SFLS has been used as a benchmar algorhm for parameer esmaon or sysems denfcaon [] To he bes nowledge of he auhors, IT NS ANFIS approach has no been repored n he leraure [,8]

3 Hybrd REFIL_BP Mehod n IT ANFIS Tranng The IT NS ANFIS s raned usng he hybrd mechansm: uses REFIL durng forward pass for unng of consequen parameers as well as he BP mehod for unng of aneceden parameers, as shown n Table I has he same ranng mechansm as he ype- ANFIS [8, 9], he RLS-BP hybrd combnaon Table Two passes n hybrd learnng procedure for IT NS ANFIS (REFIL-BP) Forward Pass Bacward Pass Aneceden Parameers Fxed BP Consequen Parameers REFIL Fxed The ranng mehod s presened as n []: Gven N npu-oupu ranng daa pars, he ranng algorhm for E ranng epochs, should mnmze he error funcon: where f IT FLS y e x e s he error funcon a me, x () f IT FLS x s he oupu of he IT FLS usng he npu vecor from he non-sngleon ype- npu-oupu daa pars, and y s he oupu from he non-sngleon ype- npu-oupu daa pars 3 Applcaon o Transfer Bar Surface Temperaure Predcon 3 Ho Srp Mll Because of he complexes and unceranes nvolved n rollng operaons, he developmen of mahemacal heores has been largely resrced o wo-dmensonal models applcable o hea losng n fla rollng operaons Fg shows a smplfed dagram of a HSM, from he nal pon of he process a he rehea furnace enry o s end a he colers Besdes he mechancal, elecrcal and elecronc equpmen, a bg poenal for ensurng good qualy les n he auomaon sysems and he used conrol echnques The mos crcal process n he HSM occurs n he Fnshng Mll (FM) There are several mahemacal model based sysems for seng up he FM

4 Fg Typcal ho srp mll A model-based se-up sysem [0] calculaes he FM worng references needed o oban gauge, wdh and emperaure a he FM ex sands 3 Desgn of he IT NS ANFIS The archecure of he IT NS ANFIS s esablshed n such a way ha s parameers are connuously opmzed The number of rule-anecedens s fxed o wo, one for he roughng mll (RM) ex surface emperaure, and one for ransfer bar head ravelng me Each aneceden-npu space s dvded n hree fuzzy ses (FSs), fxng he number of rules o nne Gaussan prmary membershp funcons (MFs) of unceran means are chosen for he anecedens Each rule of he IT NS ANFIS s characerzed by sx aneceden MFs parameers (wo for lef-hand and rgh-hand bounds of he mean, and one for sandard devaon, for each of he wo aneceden Gaussan MFs), and sx consequen parameers (one for lef-hand and one for rghhand end pons of each of he hree consequen ype- FSs) Each npu value has one sandard devaon parameer, gvng foureen parameers per rule 33 Inpu-Oupu Daa Pars From an ndusral HSM, nosy non-sngleon ype- npu-oupu pars of hree dfferen produc ypes were colleced and used as ranng and checng daa The npus are he nosy measured RM ex surface emperaure and he measured RM ex o SB enry ransfer bar ravelng me The oupu s he nosy measured SB enry surface emperaure 34 Fuzzy Rule Base The IT NS ANFIS fuzzy rule base consss of a se of IF-THEN rules ha represens he model of he sysem The IT NS ANFIS sysem has wo npus x X, x X and one oupu y Y The rule base has M = 9 rules of he form:

5 F ~ ~ F, R : IF x s and x s THEN Y C0 C x Cx () where Y he oupu of he h rule s a fuzzy ype- se, and he parameers wh =,,3,,9 and j = 0,,, are he consequen ype- FSs C j, 35 Inpu Membershp Funcons The prmary MFs for each npu of he IT NS ANFIS are Gaussans of he form: X x exp where: [, ] =, (he number of ype- non-sngleon npus), and X(x ) cenered a he measured npu x =x The unceran sandard devaon of RM ex surface emperaure measuremen was nally se as 30 C and he unceran sandard devaon of head-end ravelng me measuremen was nally se o 9s x x ' (3) 36 Aneceden Membershp Funcons The prmary MFs for each aneceden are FSs descrbed by Gaussan wh unceran means: m m m x m x exp where, s he unceran mean, wh =, (he number of anecedens) and =,,9 (he number of M rules), and s he sandard devaon The means of he aneceden fuzzy ses are unformly dsrbued over he enre npu space (4) 36 Consequen Membershp Funcons Each consequen s an unnormalzed nerval ype- TSK FLS wh where p p yl c j jx j c x j j s j s 0 0 Y yl, y r (5) and

6 RMSE p p yr c j j x j c x j j s j s 0 0 (6) boh, are unnormalzed ype- TSK FLS, where C j and s j denoes he spread of yr are he consequen parameers c j denoes he cener (mean) of C j, wh,,3,,9 and j 0,, Then y l and 4 Applcaon Resuls The IT NS ANFIS (REFIL-BP) sysem was raned and used o predc he SB enry emperaure, applyng he RM ex measured ransfer bar surface emperaure and RM ex o SB enry zone ravelng me as npus We ran ffy epochs of ranng; one hundred and en parameers were uned usng eghy seven, sxy-egh and weny-egh npu-oupu ranng daa pars per epoch, for ype A, ype B and ype C producs respecvely The performance evaluaon for he hybrd IT NS ANFIS (REFIL-BP) sysem was based on roo mean-squared error (RMSE) benchmarng crera as n [] Fg shows he RMSEs of hree non-hybrd IT TSK ANFIS sysems raned usng only he BP algorhm for boh, aneceden and consequen parameers; all of hem for ffy epochs of ranng for he case of producs of ype C Epoch Fg (*) RMSE IT TSK SFLS (BP-BP) (+) RMSE IT TSK NSFLS (BP-BP) Fg 3 shows he RMSEs of wo IT TSK ANFIS sysems raned usng he proposed hybrd REFIL-BP algorhm, for producs of ype C For hs expermen, sarng a epoch, he IT NS ANFIS has beer performance han he sngleon IT

7 RMSE ANFIS When compared o he IT TSK NSFLS (BP) sysems, he proposed hybrd approach IT TSK NSFLS (REFIL-BP) proved o be beer n erms of boh, he emperaure predcon wh only one epoch and several epochs of ranng Epoch Fg 3 (*) RMSE IT TSK SFLS (REFIL-BP) (o) RMSE IT TSK NSFLS (REFIL-BP) 5 Conclusons An IT NS ANFIS usng he hybrd REFIL-BP ranng mehod was esed and compared for predcng he surface emperaure of he ransfer bar a SB enry The aneceden MFs and consequen cenrods of he IT NS ANFIS absorbed he uncerany nroduced by all he facors: he aneceden and consequen nally values, he nosy emperaure measuremens, and he naccurae ravelng me esmaon The non-sngleon ype- fuzzy npus are able o compensae he unceran measuremens, expandng he applcably of IT NS ANFIS sysems I has been shown ha he proposed IT NS ANFIS sysem can be appled n modelng of he seel col emperaure I has also been envsaged s applcaon n any unceran and non-lnear sysem predcon and conrol, as n furnace emperaure conrol, aerospace sably conrol, urbne rus conrol and especally n hose applcaons where here are measuremens uncerany The proposed hybrd IT NS ANFIS (REFIL-BP) sysem has he good performance and sably afer only one epoch of ranng: an mporan characersc for compuaonal nellgen sysems when here s a chance of only one epoch of ranng I s requred o emphasze ha he used IT ANFIS sysems are very sensve o he values of learnng parameer s gan

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