International Journal of Vehicle Systems Modelling and Testing, Vol. 3(4): , P.K. Wong a *, C.M.Vong b, L.M.Tam a and K.

Transcription

1 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 Data Ppocssing and Modling of Elctonically- Contolld Automotiv Engin Po Pfomanc using Knl Pincipal Componnts Analysis and Last-Squas Suppot Vcto Machins PK Wong a *, CMVong b, LMam a and KLi a a Dpatmnt of Elctomchanical Engining, Faculty of Scinc & chnology, Univsity of Macau, Macau b Dpatmnt of Comput and Infomation Scinc, Faculty of Scinc & chnology, Univsity of Macau, Macau ABSRAC Modn automotiv ngins a contolld by th lctonic contol unit ECU h lctonically-contolld automotiv ngin po & toqu a significantly affctd ith ffctiv tun-up of ECU contol paamts Cunt pactic of ECU tun-up lis on th xpinc of th automotiv ngin h ngin tin-up is usually don by tial-and-o mthod, and thn th vhicl ngin is un on th dynamomt to tst th actual ngin output po and toqu Obviously th cunt pactic costs a lag amount of tim and mony, and may vn fail to tun up th ngin optimally bcaus a fomal po and toqu modl of th ngin has not bn dtmind yt With an mging tchniqu, Last Squas Suppot Vcto Machins LS-SVM, th appoximatd po and toqu modl of a vhicl ngin can b dtmind by taining th sampl data acquid fom th dynamomt h numb of dynamomt tsts fo an ngin tun-up can thfo b ducd bcaus th stimatd ngin po and toqu functions can plac th dynamomt tsts to a ctain xtnt Moov, Baysian infnc is also applid to automatically inf th hyppaamts usd in LS-SVM so as to liminat th o of coss-validation, and this lads to a significant duction in taining tim Bsids, th actual numb of adjustabl paamts in th ECU fo an automotiv ngin is vy hug Knl Pincipal Componnts Analysis KPCA is thfo poposd to duc th numb of unimpotant adjustabl paamts KPCA is a ll non data p-pocssing mthod und suppot vcto machins SVM fomulation KPCA can tansfom automotiv ngin adjustabl paamts into a mo compact subst, hil taining th infomation contnt as much as possibl With f input paamts, th taining tim fo modl constuction can b shotnd and also th pdiction accuacy of th modl can b impovd Expimntal sults sho that th intgation of KPCA and LS-SVM can ally impov th taining tim and accuacy of an automotiv ngin modl as compad ith th LS-SVM mthod ithout data ppocssing Kyods: Last squas suppot vcto machins; Knl pincipal componnts analysis; Automotiv po pfomanc modl IRODUCIO oadays, automotiv ngins a fou-sto lctonic ful injction ngins and thy a contolld by th lctonic contol unit ECU h automotiv ngin po pfomanc usually fs to output po and toqu, and thy a significantly affctd by th stup of contol paamts in th ECU omally, th output pfomanc data of a ca ngin is obtaind though dynamomt dyno tsts An xampl of po pfomanc *Cosponding autho -mail: fstp@umacmo

2 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 data of an ngin output hospo and toqu against spds is shon in Fig aditionally, th stup of ECU is don by th vhicl manufactu Hov, in cnt, th pogammabl ECU and ECU dito hav bn idly adoptd by many high pfomanc passng cas hs dvics allo th non-factoy ngins to tun up thi ngins accoding to diffnt add-on componnts and div s quimnts Fig Exampl of ngin output hospo and toqu cuvs Cunt pactic of ngin tun-up lis on th xpinc of th automotiv ngin ho ill handl a hug numb of combinations of ngin contol paamts h lationship btn th input and output paamts of an automotiv ngin is a complx multi-vaiabl nonlina function, hich is vy difficult to b dtmind, bcaus th automotiv ngin is an intgation of thmo-fluid, lctomchanical and comput contol systms Consquntly, ngin fin tun-up is usually don by tial-and-o mthod h ngin fist gusss an ECU stup basd on his/h xpinc and thn th ngin is un on a dynamomt to tst th actual ngin pfomanc h tm, ngin po pfomanc, alays fs to th ngin output po and toqu If th pfomanc is poo, th ngin adjusts th ECU paamts and pats th pocdu until th pfomanc is satisfactoy hat is hy vhicl manufactus nomally spnd many months to tun-up an ngin optimally fo a n ca modl Moov, th pfomanc function is ngin dpndnt as ll Evy ngin must undgo th simila tun-up pocdu By noing th pfomanc function/modl, th automotiv ngins can pdict if a tial ngin stup is gain o loss via a comput h ca ngin only quis going though a dynamomt tst fo vification aft stimating a satisfactoy stup fom th function Hnc th numb of unncssay dynamomt tsts fo th tail stup can b dastically ducd so as to sav a lag amount of tim and mony fo tsting Rcnt sachs hav dscibd th us of nual ntos fo modling th disl ngin mission Bac, 998; av t al, 999; Su t al, 00; Yan t al, 003; Liu t al, 004 and ful consumption Shayl, 995 basd on xpimntal data It is ll non that a nual nto is a univsal stimato Bishop, 995 It has in gnal, hov, to main dabacs fo its laning pocss Hayin, 999 h achitctu, including th numb of hiddn nuons, has to b dtmind a pioi o modifid hil taining by huistic, hich sults in a non-ncssaily optimal nto stuctu

3 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 h taining pocss i, th minimization of th sidual squad o cost function in nual ntos can asily b stuc by local minima Vaious ays of pvnting local minima, li aly stopping, ight dcay, tc, a mployd Hov, thos mthods gatly affct th gnalization of th stimatd function, i, th capacity of handling n input cass aditional mathmatical mthods fo nonlina gssion Rnch, 00 may b applid to stimat th ngin po and toqu modls It os by tansfoming th nonlina data spac into lina data spac, i, moving th nonlinaity, and thn pfoms lina gssion ov th tansfomd data spac Hov, th a to main dabacs of nonlina gssion mthods: hs nonlina tansfomations a not guaantd to tain th infomation of th tansfomd data Usually aft th tansfomation, th taining data ould b distotd and hnc affcting th pdictability of th gssd modl fom th tansfomd taining data hs nonlina tansfomations can o ll only fo lo dimnsional data st In th cunt application, an ngin stup involvs hug numb of adjustabl paamts Constucting th pdiction modls in such a high-dimnsional and nonlina data spac is vy difficult fo taditional gssion mthods So, it is not commndd to apply th taditional nonlina gssion mthods fo high dimnsional data st With an mging tchniqu of Suppot Vcto Machins SVM Cistianini and Sha- aylo, 000; Vapni, 998 combining th advantags of nual ntos handling lag amount of highly nonlina data and nonlina gssion high gnalization, th issus of high dimnsionality as ll as th pvious dabacs fom nual ntos a ovcom In vi of th abov asons, th class of SVM mthod is poposd to stimat th ngin po and toqu modls basd on th xpimntal data such that it can b usd fo pcision pfomanc pdiction EMPLOYED ECHIQUES Last Squas Suppot Vcto Machins SVM is an intdisciplinay fild of machin laning, optimization, statistical laning and gnalization thoy It is also anoth catgoy of fd-foad ntos Hayin, 999 Basically it can b usd fo pattn classification and function stimation Sinc th pap focuss on function stimation, th discussion is only latd to function stimation issus SVM is a vy nic famo o mthodology to fomulat th mathmatical pogam fo th taining o function usd in any application o matt hich application, SVM fomulats th taining pocss i, minimization of squad sidual o function as a Quadatic Pogamming QP poblm fo th ights ith gulaization facto includd Sinc QP poblm is a convx function, th solution tund is global o vn uniqu instad of many local ons, unli nual ntos his sult nsus th high gnalization of th taind SVM modls ov nual ntos Last squas suppot vcto machins LS-SVM Suyns t al, 00 is a vaiant of SVM, hich mploys last squa os in th taining o function SVM solvs nonlina gssion poblms by mans of convx quadatic pogam QP and th spasnss is obtaind as a sult of this QP poblm Hov, QP poblms a inhntly difficult to b solvd Although many commcial pacags xist in th old fo solving QP poblms, it is still pfd to hav a simpl fomulation LS-SVM solvs a st of lina quations that is asi to us o solv than QP poblms In addition, LS-SVM quis only to hyp- 3

4 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, paamts fo Radial Basis Function nl has SVM quis th hyp-paamts Moov, th thshold b is tund as pat of th LS-SVM solution hil on th contay SVM must calculat th thshold b spaatly In vi of th abov advantags, LS-SVM is poposd in this pap h dtail dsciption of th LS-SVM mthodology is shon blo Consid th datast, D {x, y,, x, y }, ith ngin stups i data points fo taining h x R n psnts th ngin input paamts including ECU adjustabl and nvionmntal paamts in th th sampl point,, i th th ngin stup, y R is th ngin output toqu in th th sampl data point basd on th ngin stup x, to LS-SVM dals ith th folloing optimization poblm in th pimal ight spac + + b y J P b,, ], [ such that, min,, x ϕ γ h h n R is th ight vcto of th tagt function, [ ;, ] is th sidual vcto, and h n n R R : ϕ is a nonlina mapping, n is th dimnsion of x, and n h is th dimnsion of th unnon fatu spac h modl of th stimatd function is considd as b M y + x x ϕ Hov, may b in vy high o vn infinit dimnsions that cannot b solvd dictly In od to solv th poblm, th Lagangian of Eq is constuctd to div th dual poblm and th Lagangian is as follos: + + p y b J b L } {,,,, x α ϕ α h α α a Lagang multiplis h conditions fo optimality a givn by + + y b L L b L L,, 0, 0,,, x x ϕ α γ α α ϕ α Aft limination of th vaiabls and in Eq 3 using th sults fom Eq 4, th LS- SVM dual fomulation of nonlina function stimation is thn xpssd as follos Suyns t al, 00: + y α I Ω α 0 0 :, Solv in b b v v γ 5 3 4

5 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 h I is an -dimnsional idntity matix, y [y,, y ], v is an -dimnsional vcto [,,], α [α,, α ], and γ R is gulaization facto hich is a hyp-paamt fo tuning h nl tic is mployd as follos: Ω, l ϕ x ϕ x K x, x l l, l, K, h K is a pdfind nl function h sulting LS-SVM modl fo function stimation is constuctd by substituting th sult of Eq 4, i, and bcoms y M x α ϕ x α K x ϕ x + b, x + b xp x α x σ + b α ϕx into Eq h α, b R a th solution of Eq 5, x is th n input stup fo ngin po pfomanc pdiction, and Radial Basis Function RBF is chosn as th nl function K, hich is th common choic fo modling In th RBF, σ spcifis th nl sampl vaianc hich is also a hyppaamt fo tuning Basd on Eq 7, th ngin output pfomanc modl y can b asily obtaind h output pfomanc is only th ngin toqu at a ctain ngin spd bcaus th po of an ngin at th ctain spd is calculatd using Eq 8Pulab, h HP π 98 HP : Engin po at th cosponding ngin spd Hp : Engin toqu at th cosponding ngin spd gm : Engin spd RPM : Rvolution p minut Evn though LS-SVM sms to b a good modling tool, if gabag data is passd in, gabag sults a tund, this is a natual la - Gabag In Gabag Out Pyl, 999 In pactical automotiv ngin tun-up, many adjustabl paamts o input vaiabls a involvd Fo a complt ECU stup, th numb of input vaiabls is usually mo than 40, vn th ngin full-load condition is assumd It may b imagind that th combination of 40 paamts is vy massiv Moov, th vaiabls a nonlinaly colatd and ECU dpndnt Fo tuning up a n ECU, vn an xpincd automotiv ngin also dos not no claly hich adjustabl paamts ill hav significant contibution to th output po and toqu at th paticula ngin spd fo a spcific ngin Hnc data ppocssing is ncssay to filt out som unimpotant adjustabl paamts to th ngin and output pfomanc, sulting in high accuacy of th modl constuctd aditional statistical mthods Rnch, 00 a not capabl to handl high dimnsionality and nonlina data spac o ovcom this poblm, dimnsionality duction is usually applid Hov, ducing som input vaiabls may caus infomation loss bcaus th input vaiabls thmslvs may b highly colatd and phaps nonlinaly Und this situation, 5

6 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 on of mostly usd and ll non data ppocssing mthods, nl pincipal componnts analysis KPCA Suyns t al, 00; Schölopf & Smola, 00, is poposd to ffctivly duc th nonlina-colatd adjustabl ngin paamts in th input spac Knl Pincipal Componnts Analysis A ll-non and fquntly usd tchniqu fo dimnsionality duction fo input spac is Pincipal Componnts Analysis PCA Shlns, 005 Consid a taining datast D {x, y,, x, y } PCA os on th input spac of D only, i, D x {x i }, i to, gadlss of th output y i h main concpt of PCA is to obtain a minimum psntation of x i hil taining its maximal vaianc his is don by finding a tansfomation vcto as th follos: max va x cov x, x C 9 h C E[ x x] By divation, Eq9 is quivalnt to solving th folloing ignvalu poblm: C λ 0 h th matix C is symmtic and positiv smidfinit Fig dpicts th concpt of dimnsion duction though PCA h ignvcto cosponding to th lagst ignvalu λ dtmins th pojctd vaiabl having maximal vaianc By solving Eq 0, a squnc of pai j, λ j a tund, j to m, in dscnding od of ignvalu λ j h vctos j constuct th tansfomation matix W ; ; m hich maps th oiginal input x to a lo dimnsional spac, i, W:R n R m, m < n h valu of m is simply dtmind by λ j j ε, h ε is a us-dfind valu fom 0 to Fo xampl, ε 095 mans that th tansfomd psntation x captus at last 95% of vaianc of x m m λ j j ε Fig Gnal Concpt of Dimnsion Rduction of PCA 6

7 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 PCA alays pfoms ll in dimnsionality duction hn th input vaiabls a linaly distibutd Hov, fo nonlina cas, PCA cannot giv good pfomanc Hnc PCA is xtndd to a nonlina vsion und suppot vcto machins SVM fomulation Suyns t al, 00; ining and aylo, 00; ining and aylo, 003 his nonlina vsion is calld Knl Pincipal Componnts Analysis KPCA h basic ida of KPCA mains th sam as PCA xcpt nl tic is applid Radial Basis Function as shon in Eq is slctd fo nl function, hich is a common ul of thumb,, xp p q K p q σ h σ is th us pdfind standad dviation, and p, q R n h LS-SVM fomulation of KPCA can b found in Suyns t al, 00, and Eq lists out th dual poblm fomulation ith gulaization fo KPCA In od to obtain th dimnsions of maximal vaianc, th us slcts th ignvctos α j R cosponding to th lagst ignvalus λ j : Ωα λα c h Ω M Ω I v c c l MΩM I c K x, x fo, l,, c v v is an x idntity matix is a - vcto l is a cntd nl matix / R is th numb of taining data in D Aft solving Eq, th squnc of pais α j, λ j, j to m, a usd to tansfom all taining data x i D x, i to, into a simpl psntation x i {z x,j }, j to m, and m is slctd such that λ j : hα j,l m j is th l ε zx, j α j,l K xl, x - K x, x - K x, xl + K x, xs l s th lmnt ofα j 3 APPLICAIO OF LS-SVM O MODEL PEROL EGIE POWER PERFORMACE h sction discusss th applications of LS-SVM ithout data ppocssing and LS- SVM ith data ppocssing fo modling an lctonically-contolld ptol ngin po pfomanc h sults obtaind by both mthods a thn compad and discussd in Sction 4 3 Modling ithout data ppocssing h taining datast is xpssd as D {x i, y i }, i to Pactically, th a many input contol paamts hich a also ECU and ngin dpndnt Moov, th ngin 7 3

8 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 hospo and toqu cuvs a nomally obtaind at full-load condition and on atmosph pssu Fo dmonstating th LS-SVM mthodology, th folloing common adjustabl ECU and nvionmntal paamts a slctd to b th input i, ngin stup at ngin full-load condition: x< I, V, f, J, d, a t, t, b v, b f, R h, S, λ, i, U L, U h, F +, F -, L f, F a, F a, F a3, F a4, F a5, F a6, F a7, F, F, F 3, F 4, F 5, F 6, F 7, I a, I a, I a3, I a4, I a5, I a6, I a7, I, I, I 3, I 4, I 5, I 6, I 7 > and y< > h is ngin spd pm and {000, 500, 000, 500,, 8000}; I, ignition spa advanc at th cosponding ngin spd dg bfo top dad cnt; V, stat of unning high spd valv timing & lift at spcific ngin spd pm; f, ful injction tim at th cosponding ngin spd milliscond; J, timing fo stopping th ful injction at th cosponding ngin spd dg bfo top dad cnt; d, ignition dll tim at 5 V milliscond; a t, ai tmpatu o C; t, ngin tmpatu o C; b v, batty voltag V; b f, injcto batty voltag compnsation facto; R h, lativ humidity %; S, pcntag coction p ach Lambda contol updat piod %; λ, tagt Lambda contol valu at th cosponding ngin spd ; i, nomal position of idl spd valv %; U L, Lambda contol calculation updat at at 5% duty cycl s; U h, Lambda contol calculation updat at at 50% duty cycl s; F +, ich tim limit % F -, lan tim limit % L f, long tm filt in Lambda contol; F a, F a, F a3, F a4, F a5, F a6, F a7, cofficints fo dfining ful tim function fo ai tmpatu compnsation; F, F, F 3, F 4, F 5, F 6, F 7, cofficints fo dfining ful tim function fo ngin tmpatu compnsation; I a, I a, I a3, I a4, I a5, I a6, I a7, cofficints fo dfining ignition tim function fo ai tmpatu compnsation; I, I, I 3, I 4, I 5, I 6, I 7, cofficints fo dfining ignition tim function fo ngin tmpatu compnsation;, ngin output toqu gm at th cosponding ngin spd h ngin spd ang fo this sach has bn slctd fom 000 pm to 8000 pm Although th ngin spd is a continuous vaiabl, in pactical ECU stup, th ngin nomally fills th stup paamts fo ach catgoy of ngin spd in a map fomat h map is usually dividd th spd ang disctly ith intval 500, i {000, 500, 000, 500,8000} hfo, it is unncssay to build a function acoss all spds Figu shos on xampl map in an ECU stup So, is manually dividd ith a spcifid intval of 500 instad of any intg anging fom 0 to

9 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 Fig 3 Exampl of ful map in a typical ECU stup h th ngin spd RPM is disctly dividd As th taining data is ngin spd dpndnt, anoth notation D is usd to futh spcify a datast containing th data ith spct to a spcific Fo xampl, D 000 contains th folloing paamts: <I 000, V, f 000, J 000, d, a t, t, b v, b f, R h, S, λ 000, i, U L, U h, F +, F -, L f, F a, F a, F a3, F a4, F a5, F a6, F a7, F, F, F 3, F 4, F 5, F 6, F 7, I a, I a, I a3, I a4, I a5, I a6, I a7, I, I, I 3, I 4, I 5, I 6, I 7, 000 >, hil D 8000 contains <I 8000, V, f 8000, J 8000, d, a t, t, b v, b f, R h, S, λ 8000, i, U L, U h, F +, F -, L f, F a, F a, F a3, F a4, F a5, F a6, F a7, F, F, F 3, F 4, F 5, F 6, F 7, I a, I a, I a3, I a4, I a5, I a6, I a7, I, I, I 3, I 4, I 5, I 6, I 7, 8000 > Consquntly, D is spaatd into fiftn substs namly D 000, D 500,, D 8000 An xampl of th taining data ECU stup fo D 000 is shon in abl It may b notd th numb of input vaiabls in this application is qual to 46 input vaiabls; th numb is vy hug indd Fo a subst D, it is passd to th LS-SVM gssion modul, Eq5, in od to constuct fiftn toqu modls M Eq7 ith spct to ngin spd Accoding to th division of taining data, th a totally 5 toqu functions, i, M xm {M 000, M 500,, M 8000 } In this ay, th LS-SVM modul is un fo fiftn tims In vy un, a diffnt subst D is usd as taining st to stimat its cosponding toqu function An ngin toqu against ngin spd cuv is thfo obtaind by fitting a cuv that passs though all data points gnatd by M 000, M 500,, M 8000 abl Exampl of taining data d i in data st D 000 I 000 V f 000 J 000 d a t t I I I 3 I 4 I 5 I 6 I d d d

10 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, Modling ith data ppocssing Fiftn taining substs a dnotd by D x, h {000, 500, 000, 500,8000} Each of D x, is qual to D \{ }, hich mans th data sts just includ all input vaiabls xcpt th output vaiabl ogth a total of 5 tims of KPCA pocdu a un so that all D x, a tansfomd to D KPCA, spctivly h sam modling pocdu of LS-SVM and schma dscibd in Sction 3 a thn applid to D KPCA, in od to constuct fiftn toqu functions M KPCA, ith spct to th ngin spd 33 Data Sampling and Implmntation In pactical ngin stup, th automotiv ngin dtmins an initial stup, hich can basically stat th ngin, and thn th ngin is fin-tund by adjusting th paamts about th initial stup valus hfo, th input paamts a sampld basd on th data points about an initial stup supplid by th ngin build In th xpimnt, a sampl datast D of 000 diffnt ngin stups along ith po pfomanc has bn acquid fom a Honda B8C DOHC VEC ngin contolld by a pogammabl ECU, MoC M400 Fig 4, unning on a chassis dynamomt Fig5 at id opn thottl ith Lambda contol Aft collction of sampl datast D, fo vy data subst D D, it is andomly dividd into to sts: RAI fo taining and ES fo tsting, such that D RAI ES, h RAI contains 80% of D and ES holds th maining 0% Fig6 hn vy RAI is snt to th LS-SVM modul and KPCA+LS-SVM modul fo taining, and hnc to diffnt sts of 5 toqu functions constuctd by th to mthods a obtaind Both moduls hav bn implmntd using LS-SVMlab Plcmans t al, 003, a MALAB toolbox und MS Windos XP Implmntation and oth impotant issus a discussd in th folloing sub-sctions Fig 4 Adjustmnt of ngin contol paamts using a pogammabl ECU 0

11 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 Fig5 Ca ngin pfomanc data acquisition on a chassis dynamomt D % RAI % R 000 0% VALID 000 0% ES 000 D % RAI % R % VALID % ES 8000 Fig 6 Futh spaation of data andomly into taining sts RAI and tst sts ES 33 Data nomalization and d-nomalization In od to hav a mo accuat gssion sult, th datast is convntionally nomalizd bfo taining and data tansfomation Pyl, 999 his pvnts any paamt fom dominating th output valu Fo all input and output valus, it is ncssay to b nomalizd ithin th ang [0,], i unit vaianc, though th folloing tansfomation fomula: v v v v v * min max min 4 h, v min and v max a th minimum and maximum domain valus of th input o output paamt v spctivly Fo xampl, v [5, 38], v min 5 and v max 38 h limits fo ach input and output paamt of th tst ngin should b pdtmind via a numb of xpimnts o xpt noldg o manufactu data shts As all input and output valus a nomalizd, th output toqu valu v * poducd by th ngin pfomanc modls is not th actual valu It must b -substitutd into Eq 4 in od to obtain th actual output toqu valu v

12 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, Eo function o vify th accuacy of ach function of M, an o function has bn stablishd Fo a ctain function M, th cosponding validation o is: E y M y h {000, 500, 000, 500,8000}, and x R n is th ngin input paamts of th data point in a tst st o a validation st, y is th tu toqu valu in th data point d x, y psnting th th data point and is th numb of data points in th tst st o validation st h o E is a oot-man-squa of th diffnc btn th tu toqu valu y of a tst point d and its cosponding stimatd toqu valu M x h diffnc is also dividd by th tu toqu y, so that th sult is nomalizd ithin th ang [0, ] It can nsu th o E also lis in that ang Hnc th accuacy at fo ach toqu function of M is calculatd using th folloing fomula: x 5 Accuacy E 00% Pocdus fo tuning hyp-paamts in LS-SVM and KPCA Accoding to Eqs 5, 7 and, it can b notd that th us has to adjust on hyppaamt σ in KPCA and to hyp-paamts γ, σ in LS-SVM Without noing thi bst valus fo ths hyppaamts, all stimatd toqu functions could not achiv high gnalization In od to slct th bst valus fo ths hyppaamts, 0-fold coss validation is usually applid Suyns t al, 00, but it tas a vy long tim Rcntly, th is a mo sophisticatd Baysian famo Suyns t al, 00 that can automatically inf th hyppaamt valus fo LS-SVM Givn a st of taining xampls, Baysian infnc is a vy obust famo to comput th distibution of th stimatd modl paamts basd on th taining xampls Basd on th distibution of th modl paamts computd, th optimal modl paamts valus can b pdictd As th thoy using Baysian infnc to pdict th hyppaamts γ and σ is out of th scop of this sach, it is not discussd in dtail h basic ida of th hyppaamts infnc pocdu using Baysian famo Macay, 995; Van Gstl t al, 00 is basd on a modifid vsion of LS-SVM pogam in Eq 7, h µ is no th gulaization facto instad of γ, and ζ is th vaianc of th nois fo sidual assuming constant vaianc: min J P, µ E + ζed, b, such that y + b [ ϕ x ],,, 7 ith E E D y [ ϕ x + b] 8

13 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 h hos dual pogam is th sam as Eq 5, h R n is th ight vcto of th tagt function and [ ;, ] is th sidual vcto h lationship of γ ith µ and ζ is γ ζ / µ It should b notd that aft substituting Eq 8 and th lationship of γ into Eq 7, it dictly bcoms Eq Fig 7 bifly illustats th algoithm fo Baysian infnc fo ths to hyppaamts basd on a ctain data st RAI, and this figu is dan by fing to Van Gstl t al, 00 Although th infnc pocdu is thotically vy complicatd, Plcmans t al, 003 has povidd a MALAB/C toolbox to handl this infnc pocdu Unfotunatly, up to no th is no automatic mthod fo finding th hyppaamt fo KPCA vthlss, th sach objctiv is to compa th accuacy of th gssd modls ith and ithout data tansfomation, it is still fai to compa modl accuacy vn und th simplst hyp-paamt Hnc, th hyp-paamt σ fo KPCA is st to 334 aining Fist of all, th taining data is nomalizd; if KPCA is applid, it should b tansfomd aft nomalization Aft that, th hyppaamts γ,σ fo th tagt toqu functions a infd at this point Sinc th a fiftn tagt toqu functions, thn fiftn individual sts of hyppaamts γ, σ a infd ith spct to h dtaild infnc pocdu fo a ctain taining data st RAI is listd in Fig 7 Aft obtaining th fiftn pais of infd hyppaamts γ MP,, σ MP,, h th subscipt MP stands fo maximum postio, th taining data st RAI is usd fo calculating th suppot valus α and thshold b in Eq 5 Finally, th tagt function M can b constuctd using Eq 7 3

14 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 h objctiv of Baysian infnc is to find all σ fo disct ngin spds, such that Max P RAI M σ Usually a lin sach on-vaiabl optimization ov σ is σ applid Each itation of th lin sach involvs th folloing pocdu to calculat th postio P RAI M σ : Guss a valu fo σ, calculat th optimal µ MP,,ζ MP,, and γ MP, as follos: A St M I v v Us RAI {x, y } and nl paamt σ to ppa matix Ω, h Ω l Kx, x l ; x, x l RAI,and G MΩM B Solv th ignvalu poblm Gv λ v i,,, gtting G,i G, i G,i ff λ G,i,v G,i, and ff ff is dfind as th numb of non-zo ignvalus C St DG diag[ λg, K, λg, ] ; V G v ; K; v ] ; [ G G,, ff, ff mˆ y y i ; i ˆ EW + γ ED y myv VG DG + I V y mˆ v ff G y γ D Find th optimal valu γ MP, in min J γ log λg, i + + log EW + γed using anoth lin sach γ i γ E Givn γ MP,, calculat µ MP, using µ MP E + γ E F Givn th lationship that γ MP, ζ MP, /µ MP,, calculat ζ MP, ff γ MPλG, i G Givn γ MP,, and λ G,,i, calculat γ ff + γ λ H h postio P RAI M σ Fig 7 Baysian infnc pocdu fo hyppaamts γ, σ Van Gstl t al, 00 γ W MP i + MP G, i ff γ D µ ff MP ζ MP ff ff i µ MP + ζ MPλ G, i calculatd I Rpat th pocdu until th postio P RAI M σ is maximizd hn th cosponding σ and γ MP, a tund as solution is 4 RESULS 4 LS-SVM sults ithout data ppocssing Aft obtaining all toqu functions fo an ngin, thi accuacis a valuatd on by on against thi on tst sts ES using Eqs 5 and 6 Accoding to th accuacy shon in abl, th pdictd sults a in good agmnt ith th actual tst sults und thi hyppaamts γ MP,, σ MP, infd using th pocdu dscibd in Fig 7 4

15 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 An xampl of compaison btn th pdictd and actual ngin toqu and hospo und th sam ECU configuation is shon in Fig 8 abl Accuacy of diffnt functions M and its cosponding hyppaamt valus using LS-SVM oqu function M γ MP, σ MP, Avag accuacy ith tst st ES M % M % M % M % M % M % M % M % M % M % M % M % M % M % M % Ovall avag 890% 5

16 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, m g 0 u q 8 o 6 4 u toqu acquid fom dyno Estimatd toqu fom LS-SVM Engin Spd RPM p H o P u po acquid fom dyno Estimatd po fom LS-SVM Engin Spd RPM Fig 8 Exampl of compaison btn pdictd and actual ngin toqu and po 4 LS-SVM sults ith data ppocssing o xamin th pfomanc of data ppocssing tchniqu, th input data fo ngin modl constuction is tansfomd by KPCA bfo applying LS-SVM h sults obtaind a thn compad ith th LS-SVM mthod ithout data ppocssing, in od to chc hich mthod is th bst fit to this application Figu 8 shos ho th sults a obtaind und th intgation of LS-SVM and KPCA 6

17 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 Fig 9 Pdiction pocdu fo ngin toqu und KPCA and LS-SVM mthodology With fnc to Sction 3, th a totally 46 contol vaiabls in this application Aft applying KPCA, th input vaiabls a tansfomd into anoth st of vaiabls, h som of thm can b ignod basd on thi vaiancs Expimntal sult shos that th numb of input vaiabls is ducd fom 46 in th oiginal data spac to 35 in a n fatu spac Aft obtaining all toqu modls fo an ngin, thi accuacis a valuatd on by on against thi on tst sts ES using Eqs 5 and 6 as ll Accoding to th sults shon in abl 3, th pdictd sults a in good agmnt ith th actual tst sults abl 3 Accuacy of diffnt functions M and its cosponding hyppaamt valus using KPCA+LS-SVM oqu function M γ MP, σ MP, Avag accuacy ith tst st ES M % M % M % M % M % M % M % M % M % M % M % M % M % M % M % Ovall avag 903% 43 Compaison of sults With fnc to abls and 3, LS-SVM+KPCA outpfoms LS-SVM about 30% in ovall avag accuacy und th sam tst sts ES h ason is that KPCA is suitabl fo handling nonlina data, and th taining data st in this application is intinsically highly nonlina i, th lationship of th input vcto x and output valu y is highly nonlina It is vifid that KPCA os ll in dimnsion duction It is also blivd that th accuacy of KPCA+LS-SVM could b impovd by incasing th numb of taining data 7

18 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 Anoth issu is about th tim quid fo taining Und a 34GHz Pntium 4 PC ith GB RAM on boad, LS-SVM tas about 575 minuts fo taining 000 data points of 46 attibuts fo on tim h Baysian infnc fo to hyppaamts tas about 96 minuts In oth ods, fiftn ngin toqu functions qui minuts i 699 days taining tim Fo KPCA+LS-SVM, it tas about 500 minuts fo data tansfomation and taining 000 data points of 35 attibuts fo on tim h Baysian infnc fo to hyppaamts tas about 74 minuts In oth ods, fiftn ngin toqu functions qui minuts i 598 days taining tim Accoding to this stimation, KPCA+LS-SVM only tas 855% of taining tim of LS-SVM ithout data tansfomation bcaus of limination of numb of input vaiabls fo pocssing Evn th LS-SVM mthods compa ith standad SVM; th LS-SVM mthods supplmntd ith Baysian infnc qui lss taining tim bcaus of limination of taditional 0-fold coss-validation fo gussing hyppaamts 5 COCLUSIOS LS-SVM plus Baysian infnc is fistly applid to poduc a st of toqu function fo an lctonically-contolld automotiv ngin accoding to diffnt ngin spds Accoding to Eq 8, th ngin po is calculatd basd on th ngin toqu In this sach, th toqu functions a spaatly gssd basd on fiftn sts of sampl data acquid fom an automotiv ngin though th chassis dynamomt h ngin toqu functions dvlopd a vy usful fo vhicl fin tun-up bcaus th ffct of any tial ECU can b pdictd to b gain o loss bfo unning th vhicl ngin on a dynamomt o oad tst If th ngin pfomanc ith a tial ECU stup can b pdictd to b gain, th vhicl ngin is thn un on a dynamomt fo vification If th ngin pfomanc is pdictd to b loss, th dynamomt tst is unncssay and anoth ngin stup should b mad Hnc, th function fo pdiction can gatly duc th numb of xpnsiv dynamomt tsts, hich savs not only th tim tan fo optimal tun-up, but also th lag amount of xpnditu on ful, spa pats and lubicants, tc It is also blivd that th function can lt th automotiv ngin pdict if his/h n ngin stup is gain o loss duing oad tsts, h th dynamomt is unavailabl Evn though LS-SVM is a pomising modling tchniqu, its pfomanc can b futh impovd by pfoming data ppocssing mthod KPCA is also fistly applid to b a data ppocssing tchniqu fo this application KPCA can tansfom ilvant adjustabl vaiabls into a compact subst, hil taining th infomation contnt as much as possibl Consquntly, th modl complxity and taining tim fo modl constuction can b ducd bcaus of f vaiabls lo data dimnsion Moov, xpimnts hav bn don to indicat th accuacy of th toqu functions built by KPCA+LS-SVM, and th sults a highly satisfactoy In compaison to th taditional LS-SVM mthod, th LS-SVM plus KPCA outpfoms about 30% in ovall accuacy und th sam tst st and its taining tim is appoximatly 45% lss than that using LS-SVM It is also blivd that th modl accuacy and th taining tim could b impovd by incasing th numb of taining data, and fin tun-up of hyp-paamts in KPCA Fom th pspctiv of automotiv ngining, th constuction of lctonicallycontolld automotiv ngin po and toqu functions using KPCA + LS-SVM is a n attmpt and this mthodology can also b applid to diffnt inds of vhicl ngin stup poblms 8

19 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 Rfncs Bac C, 998 Pdiction of Disl Engin Exhaust Emission using Atificial ual tos, IMchE Smina S59, ual tos in Systms Dsign, UK Cistianini,, Sha-aylo, J, 000 An Intoduction to Suppot Vcto Machins and Oth Knl-basd Laning Mthods, Cambidg Univsity Pss Hayin, S, 999 ual tos: A comphnsiv foundation, Pntic Hall, nd dition, USA Liu Z, Fi S, 004 Study of CG/disl dual ful ngin s missions by mans of RBF nual nto, Jounal of Zhjiang Univsity SCIECE, 58: MacKay, D, 995 Pobabl tos and Plausibl Pdictions A Rvi of Pactical Baysian Mthods fo Supvisd ual tos, to Computation in ual Systms, 6, Plcmans, K, Suyns, J, Van Gstl,, D Babant, J, Luas, L, Hams, B, D Moo, B, and Vandall, J, 003 LS-SVMlab: a MALAB/C toolbox fo Last Squas Suppot Vcto Machins Availabl at SVMlab Pyl D, 999 Data Ppaation fo Data Mining, Mogan Kaufmann Rnch A C, 00 Mthods of Multivaiat Analysis Wily sis in Pobability and Statistics, Wily-Intscinc, nd dition Schölopf, B, Smola, A, 00 Laning ith Knls: Suppot Vcto Machins, Rgulaization, Optimization, and Byond, MI Pss Shayl P J, Danton J, Ma, 995 Pdicting th Ful Consumption of Vhicls fo Div Cycls Stating Fom Cold Ambint Conditions, Pocdings of th EAEC 5th Intnational Congss, SIA9506A7 Shlns J, 005 A tutoial of Pincipal Componnts Analysis Availabl at Su S, Yan Z, Yuan G, Cao Y, Zhou C, 00 A Mthod fo Pdiction In-Cylind Compound Combustion Emissions, Jounal of Zhjiang Univsity SCIECE, 35: Suyns, J, Gstl,, D Babant, J, D Moo, B, and Vandall, J, 00 Last Squas Suppot Vcto Machins, Wold Scintific av M, Atinson R and Atinson C, 999 ual to-basd Disl Engin Emissions Pdiction Using In-cylind Combustion Pssu, SAE Pap ining CJ and aylo CJ, 00 Knl Pincipal Componnt Analysis and th Constuction of on-lina Activ Shap Modls Availabl at 9

20 Intnational Jounal of Vhicl Systms Modlling and sting, Vol 34:3-330, 008 ining CJ and aylo CJ, 003 h us of nl pincipal componnt analysis to modl data distibutions Pattn Rcognition, 36: 7-7 Vapni V, 998 Statistical Laning hoy, John Wily & Sons, Yo Van Gstl,, Suyns, J, Lambchts, D, Lancit, A, Vandal, G, D Moo, B, Vandall, J, 00 Pdicting Financial im Sis using Last Squas Suppot Vcto Machins ithin th Evidnc Famo, IEEE ans on ual tos, Spcial Issu on Financial Engining, 4: Yan Z, Zhou C, Su S, Liu Z, Wang X, 003 Application of ual to in th study of Combustion Rat of ual Gas/Disl Dual Ful Engin, Jounal of Zhjiang Univsity SCIECE, 4 :