new defuzzification method for fuzzy control of power converters

8/4/2019 New Defuzzification Method for Fuzzy Control of Power Converters

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A New Defu zzifica t ion Met h od for F u zzy Con t r o l ofP ow er Con ve r t e r s

Yiga ng Sh i a nd P .C. Sen , F ellow, IE EEDep ar tm en t of E lect rica l a nd Compu ter E ngin eer in gQueens University

Kin gst on , On ta rio, Ca na da , K7L 3N6

Abs t rac tIn t his p ap er , a n ew d effu ziflca tion m et hod is p rop ose dw ld ch ca n p r ov id e im p r oved p er for ma n ce in F u zzyCon tr ol for DC-DC con ve r te r s. A com pa ra tive st ud yof d iffe ren t d efu zzifica tion m et hod s a dop ted in F uzzyLogic Con tr ol (FLC), su ch a s Cen te r of Ar ea (COA),Cen te r of Su ms (COS), He igh t Met h od (HM), Mid d leof Ma xim a (MOM), Cen te r of La rges t Ar ea (COLA),a n d F ir st of Ma xim a (FM), for a p p lica t ion t o DC-DCBu ck -Con ver te r s i s p resen ted . Th e d ht in ct ion a mon gt he ch ar ac t er is t ics w hich le ad t o va ry in g p er for m an ceis ou t lin ed . A n ew met h od ca lle d He igh t Weigh t edSecon d Maxim a (HWSM) is p r op osed a nd it s p er for -m an ce is a sse sse d. Th e p ap er a lso p re sen ts sim ula tionr esu lt s of t he p er for ma nce of t he closed -loop con ver t -e r s fr om t he st eu dp oin t of st ar t-u p t ra ns ien t, loa d r eg-u la tion e nd lin e r egu la tion . Th e s im ula tion s sh ow t ha tCOA, COS, a nd HM d efu zzifica tion m et hod s h ave b et -t e r d yn am ic p er for ma n ce a n d le ss s t e a d y s t a t e e r ro r .Th e n ew HWSM d efu zzifica tion m et hod p rov id es fu r-t h er i mp r ov em e n t.

1 In tr od u ct ionTh er e h a s b een a r a p id gr ow th of r e sea r ch in fu zzycen t r ol a nd fu zzy m od elin g s in ce Za deh [1] fir st ga vem ath em at ica l fou n da tion of fu zzy syst em s. Mam-dani a nd h is collea gu es fir st a pp lied fu zzy logic in in -d us t ria l con tr ol a pp lica tion s [2]. R ece nt ly, t he in te r -e s t for p ra ct ica l a pp lica tion of fu zzy logic is gr ow in gr ap id ly. I t h as b ee n su ccessfu lly a pp lied in fa ct or y a u-t oma t i on , su ch as in du st ria l r obot a nd NC m ach in es[3, 13]. Norm ally, p owe r ele ct r on ics ba sed on con ven -t ion al con tr ol m et hods [4] fa iled t o per form sa tisfa c-t or ily u nd er p ar am et er va ria tion , n on lin ea rit y, loa ddisturbance, etc. Many effor t s have been made re-cen tly t o im pr ove t he per for ma nce of t he con tr ollerin p ow er con ver te r s. St at e feed ba ck con tr olle r s, se lf

tunningcon t rolle rs a nd mod el r efe re nce a da pt ive con -tr oller s, et c., wer e adopt ed t o th e cont rol of powerelect ron ics [5]. Bu t t hese cont roller s also need ac-cu ra te m at hem at ica l m od els a nd a re t her efor e sen si-t ive t o p ar amet er va r ia t ion . S lid in g mod e con t roller s(SLMC) [6, 7, 8, 11] wer e in t roduced since it doesn ot n eed an a ccur at e m athematical m odel. Th e dh i-advan tage of th is method is the drast ic changes oft he con tr ol variable whkh leads t o cha tt er in g. Thech at t er in g problem can be scdved by in tr odu cing abou nda ry layer ar oun d t he sliding plane [9]. H ow-ever , t he local non -linear it ies in t he st at e sp~e wen ot con sider ed in t he SLMC design .

Th e a pp lica t ion of fu zzy t h eor y in p owe r elect r on icsis r ela t ively n ew [3, 10, 12] a ncl h as r ece ived a t ten t ionof a num ber of r esear ch er s in r ecen t yea rs. A powerelect ron ics syst em , in gen er al, h as com plex n on lin -ea r m odel wit h pa ra met er va ria tion pr oblem , a nd t hecon tr ol n eeds t o be ver y fa st . Th e oper at ion of fu zzylogic con tr oller (F LC) does n c~t r ely on h ow a ccu ra tet he m odel is, bu t on h ow effect ive t he lin gu ist ic r ulesof t he fu zzy con tr oller a re. F uzzy con tr ol t her efor es im plifies t h e d es ign of op tim a l compen sa t ion for DC-DC con ver ter s. Unlike SLMC, it is possible t o ta kea ccou n t for loca l n on -lin ea r it ies in FLC. Many p ap er ssh owed t he pot en tia l a nd fea sibilit y of FLC con tr olfor P ower E lect ron ic cir cu it s [11, 12]. F uzzy con tr olca n p rovid e be tt er p er form an ce t h an t h e con ven t ion a lP I-con tr oller for DC-t o-DC Bu ck con ver t er [11]. Th eF hzzy Con t roller a ls o s how Slid in g Mod e ch a ra ct er is -t ics r esu lt in g in r obu st con tr ol [11].

A typical Fuzzy Logic Controller (FLC) has thefoll ow ing componen t s: fum ifica t ion , know ledge bas e,decision making and defuzzifica tion . The per for -m an ce oft h e F LC depen ds ver y m uch on t he defu zzi-ficat ion process. Th is is beca use t he over all p er for -m ance of t he system u nder cont rol is deter mined byt he con tr ollin g sign al (t he defu zzified ou tpu t of t heFLC) t he sys t em r ece ive s .

In t h is p a p er , som e u sefu l r e su lt s w ill b e fir st r e-vie we d, in p ar t ic ula r t h e e ffe ct s of d iffe re nt d efu zzifi-

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cat ion met hods as applied t o volt age con tr ol of DC-DC Con ver ter s will be pr esen ted. Th en a n ew m et hodcalled Hight Weight ed Second Maximum (HWSM)is in vest iga teed , wh ich ca n p rovid e im pr oved p er for -ma nce su ch as r edu ct ion of t he high star ting cur ren tin F uzzy Con tr ol for DC-DC con ver ter s.

2 Review of fu zzy logic con t r olThe FLC has evolved over ahnost twenty years.Three classes of FLC have been recogn ized by in-dust ry. They a re Direct FLC, Self-organized FLCand FLC based on Fuzzy Model [13]. However , thesim plest con ven tion al F LC is t he ba sis for t he a for e-mentioned three types of FLCS and st ill play animpor tant role in pract ice. The conven tional FLCsh own in F ig. 1 con sis ts of t h e follow in g compon en t s:F uzzifica tion , Ru le Ba se, In fer en ce Mech an ism , a ndDefuzzification.I n or de r t o con n ect lin gu is tic con t rol s tr a tegies wit hspeciilc con tr ol a ct ion s, lin gu ist ic va ria bles a nd t he

cor respon din g fu zzy set s or fu zzy m em ber sh ip fu nc-t ions should fist be defin ed. The simplest and m ostefficien t form of m em ber sh ip fu nct ion is t he t ria ngu -lar on e shown in Fig. 2(a).

The main pa rt of FLC is Rule base and InferenceMech nism . Ru le ba se is n or ma lly expr essed in a set ofF uzzy Lin gu ist ic r ules, wit h ea ch r ule t rigger ed wit hva ryin g belief or su ppor t. Th e i+% l ingu i st ic con t rolr ule can be expr essed as:

~: IF e~ is Ai AND dea is Bi THEN Ui is C~where Ai and Bi (a nt eceden t), Ci (con sequ en t) a refu zzy va ria bles ch ar act er ized by fu zzy m em ber sh ipfu nct ion . Th e set of fu zzy r ules n or ma lly ca n be sum-mar ized aa a table as shown in Fig. 2(b).

The last com pon en ts of FLC is defu zzificat ion .Sever al d e fu zzifica tion m et hods h ave been pr oposed[10, 14,15, 16], t hey a re Cen ter of Ar ea (COA), Cen -ter of Sums (COS), Heigh t Method (HM), Mean ofMaxima (MOM), Center of Largest Area (COLA),and Fir st of Maxima (FM).

3 Defu zzifica t ion m eth od sBa sica lly, d efu zzifica t ion is a m appin g fr om a s pa ce offu zzy con tr ol a ct ion defin ed over a n ou tpu t u niver seof discou rse in to a spa ce of n on fu zzy (cr isp) con tr olact ions. A de fuzzificat ion str at egy is aimed at pr o-du cin g a cr isp con tr ol a ct ion t ha t best r epr esen ts t hepossibilit y dist ribu tion of a n in fer red fu zzy con tr ola ct ion [14]. Th e va riou s st ra tegies t ha t h ave been r e-por ted in lit er at ur e ar e descr ibed as follows and t hegr ap hica l r ep resen ta tion s of t hem a re sh own in F ig. 4.

(a ) Cen t e r of Ar e a (COA). Th e cen t r oid d e-fu zz ifi ca t io n m e th od select s t he ou tpu t cr ispy va lu ecor respon ding t o t he cen ter of gra vit y of t he ou tputmem ber ship fun ct ion which is given by th e expr es-sion: ~.= J wp(w)dw

SAw)dw(1)

(b ) Center of Sums (COS). A similar to COAbut faster de fuzzificat ion method is the center ofsums. This method avoids the computa t ion of theu nion of t he fuzzy set s, an d consider s t he con tr ibu-t ion of t he a rea of ea ch fu zzy set in dividu ally, wh ichis given by t he expr ession :

~.= J-wZ;=lP(w)dw.(X;+ Aw)dw

(c) Heigh t Met h od (HM). In

(2)

the height. . method, the cent roid of ea& outpu t membershipfunct ion for each ru le is fir st evaluated. The &alou tput is then calcu la ted as the average of the indi-vidu al cen tr oid s, weigh ted by t heir h eigh ts (d egr ee ofmembe rs hip ) a a foll ows:

~.= Z;=l %4%)Z;=l /-4%) (3)(d) Middle of Maxima (MOM). The MOM

st ra tegy gen er at es a con tr ol a ct ion wh kh r epr esen tst he m ea n va lu e of a ll loca l con tr ol a ct ion s wh ose m em -bership funct ions reach the maximum and may beexp r es sed a s: 1

(4)j=l

(e ) Cen t e r of La r ge st Ar ea (COLA). Th eCOLA method is used in the case when universe ofdkcourse W i s n on -con vex, i.e., it con sist s of a t lea sttwo convex fuzzy subsets. Then the method deter -mines t he con vex fuzz y subset with t he lar gest ar eaa nd defin es t he cr isp ou tpu t va lu e U. t o be t he Cen terof Ar ea of t his pa rt icu la r fu zzy su bset . It is difficu ltt o r ep res en t t his d e fu zzifica tion m et h od form ally.(f) F ir st of Ma xim a (FM). Th e FM m et hod

uses the union of the fuzzy sets and takes the small-est value of t he dom ain wit h maximum member sh ipd egr ee, wh ich is exp ressed a s:

Ull = inf{w e w Ip(w) = hgt(w) (5)where hgt(W)is t he highest mem bership degr ee ofw.Wh en t he MOM st ra tegy is used, t he per for ma nce

of an FLC is similar to that of a mult ilevel relaysystem [17], while t he COA str ategy yields r esu lts

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wh ich a re sim ila r t o t hose obt ain able wit h a con ven -t ion al PI cont roller [18]. It can be expect ed th at t heCOA st ra tegy ca n yield su per ior r esu lt s, especia llyt h e s t eady-s t at e per formance .From the equat ions of HM and COA, it can be

noted tha t HM gives m ore considerat ion on the lo-ca l con tr ol a ct ion s wit h la rger m em ber sh ip fu nct ion sthan COA does. Tha t means FLC with HM methodhas la rger effect ive gain than th at of FLC wit h COAeven if they have the same va lue of gain factor &(equ a tion 8).

(g) Height Weighted Second Maxima(HWSM). This is a new method propsed in this pa-per . In this method, the second maximum of eachou tpu t m em ber sh ip fu nct ion for ea ch r ule is fir st eva l-u at ed. Th e final ou tpu t is calcu la t ed as t he aver ageof t he in dividu al m axim a, weigh ted by t heir h eigh ts(d egr ee of m ember sh ip ) a a follow s:

(6)wher e wj takes the largest value of the domain withm axim al m ember sh ip d egr ee.

4 Sim u la t ion r e su lt sTh e fu zzy con tr ol a lgor it hm a nd t he a for e m en tion edd efu zzifica tion m et hod s a re n ow ver ified by sim ula -t ion s. F ig. 3 sh ows t he a rr an gem en t a nd pa ra met er sof a bu ck con ver ter wit h t he fu zzy logic con tr oller .F or t he su bsequ en t discu ssion s, t he lin ea r differ -

en t ia l equat ions are used by represen t ing the buckcon ver ter as a sim plified equ iva len t cir cu it sh own inFig. 3. We can express them in the usual sta te va ri-a ble m at rix form:[H=[l-3rJkl+[w()7)whe re vOi=V8 whe n t h e cir cu it is on , ot h erwis e v~~=0.The par am et er s of bu ck con ver ter a re L=1OOPH,

C=200pF , &=2.5fl, V,=20V. Th e sim ula tion r esu lt sa re for star t up of the buck conver ter horn the zeroin it ia l s ta t e. Basically, the rule table of the fuzzycon t rol le r a s socia ted with the d iffe ren t defuzzi fi ca t ionmethods a re the same. And, the gain factor &, thenorma liza t ion fa ct or Ke of the er r or e and the nor -ma liza t i on fa ct or KCe of the change of er ror ce haveto be adjusted to fit the operat ing condit ion of theconver te r [12].Th ese t h ree sca lin g fa ct or s wh ich d escr ibe t he pa r-

t icu la r in pu t n orma liza tion a nd ou tpu t r en orma liza -t ion pla y a r ole sim ila r t o t ha t of t he ga in coefficien tsin a con ven tion al con tr oller . Th e PI-like fu zzy con -t roller [15] u sed in th is paper can be r epr esen ted as

Kd . As(k ) = F(K. . e(k), K.. . Ae(k)) (8 )

wh er e F is a n on lin ea x fu nct ion r ep resen tin g t he fu zzycontroller.WMle ca rr yin g on t he sim ula tion s, t he p ar am et er s

Kd, K ., and K(ce) a re a dju st ed by t ria l a nd er ror a p-pr oa ch t o pr ovide a good com pr ise bet ween t he t ra n-sient an d st ea dy sta te per for ma nce. Alt hough th ism et hod is r at her t im e-con sumin g, it h as been widelyu sed for Fu zzy Logic Con tr ol in ma ny in dustr ia l ap-p lica t ion s [3 , 13].Sim ula tion r esu lt s for defu zzifica tion m et hod s a re

shown in Fig. 5 to Fig. 11. Results are obtained forsupply voltage change of 20V t o 15V and for resis-t an ce ch an ge of Ml t o 2.50F or COA defu zzifica tion m et hod, for su pply volt -

a ge va ria t ion, r egu lat ed volt age in Fig. 5(a) showssm all over sh oot a nd it set tles down qu ick ly wit h ver ybr ief an d small oscilla t ion. For load ch an ge, r egu -la ted volt age in F ig. 5(b) sh ows sm all over sh oot a ndit set t les down quickly to a steady sta te with smalloscilla t ion . It a lso can be n ot ed t hat t her e exists n ost ea dy st at e er ror for bot h volt age ch an ge a nd r esis-t an ce ch an ge, bu t t her e exist s a h igh st ar tin g cu rr en t.F or COS d efu zzifica t ion m et h od , r egu la t ed volt a ge

for bot h su pply ch an ge a nd r esist an ce ch an ge in F ig. 6shows almost the same resu lt as that in Fig. 5 forCOA defuzzifica t ion method .F or HM d efu zzifica tion m et hod , for su pp ly volt age

va ria tion , r egu la ted volt age in Fig. 7(a ) sh ows ver ysm all over sh oot a nd u nder sh oot a t t he t ra nsit ion a l-m ost wit hou t oscilla tion . F or loa d ch an ge, r egu la tedvolta ge in Fig. 7(b) shows ver y small over sh oot an dit set t les down quickly. There exist s a high init ia lcu rr en t and n o st ea dy st at e er ror with t he r egu la tedvolt age r espon se in bot h figu res.F or MOM d efu zzifica tion m et hod , for su pply volt -

a ge var ia t ion, r egu lat ed volt age in Fig. 8(a) showsla rge over sh oot a nd u nder damped oscilla tion . Mor e-over , it haa a steady sta te er ror and a h igh star t -ing cur rent . For load change, regulated voltage inF ig. 8(b) sh ows a ppr ecia ble over sh oot a nd it set tlesdown slowly to a steady sta te with a steady sta teer r or compared t o both COA and HM methods.F or COLA d efu zzifica tion m et hod, r egu la ted volt -

age for bot h supply cha nge an d r esist a nt e ch an ge inFig. 9 shows almost the same resu lt as that in Fig. 8for MOM defu zzifica t ion m et h od .F or FM det izzifica tion m et hod, for su pply volt age

va ria tion , a nd for loa d ch an ge, r egu la ted volt age inFig. 10 sh ows la rge over sh oot an d u nder shoot and ishighly oscilla tory. As well, ther e is a steady sta teerror.F or pr oposed HWSM de fu zzifica tion m et hod, for

supply volt age var ia t ion , r egu la t ed vo lt a ge inF ig. 1 l(a ) sh ows n o over sh oot at t he st ar t-up, n o os-cilla t ion in t he st ea dy sta te and t he sta rt -u p cu rr en t

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r ed uces s ign ifica nt ly. F or loa d ch an ge, r egu la t ed volt -a ge in Fig. 11(b) sh ows n o over sh oot a nd t he st ar t-u pcu rr en t goes d own gr ea tly. As well, t her e is n o st ea dyst ate er ror for both su pply volt age chan ge and loaddisturbance.

5 Com p a r a t ive Eva lu a t ionF or su pply volt age ch an ge a nd loa d dist ur ba nce fu zzycon tr oller wit h COA, COS, or HM met hods r espondin a highly damped manner whereas FLCS withMOM, COLA, or FM methods respond in an nnder -damped manner . In the steady sta te FLCS withCOA, COS, or HM m et hods have almost zer o st eadyst ate er ror wher eas FLCS with MOM, COLA, or FMmethods have a n on zero steady sta te er r or . TheFLC with HWSM method has ahnost the same per -for man ce as COA but with bet ter star t up tr ansienta nd less in it ia l cu r ren t .It is evident that MOM and FM are the simpleston es for com pu ta tion an d im plem en ta tion . Bu t t hey

have poor syst em r esponses. COA met hod has m or ecomputat ional intensity than that of MOM or FMwhile it can yield a sat isfactory r espon se during th et ransient aa well as in th e steady sta te for both loada nd su pply dist ur ba nce. HM, COS,, or HWSM m et h-ods h ave a lm ost t he sa me com pu ta tion al in ten sit y a sthat of MOM or FM. However , they have almost thesame good perfor ma nce as that of COA.From the simulat ion result s, it can also be noted

that the FLCS with MOM or COLA methods haveless oscilla t ion in the steady sta te whereas otherm et h od s, s uch a s COA, or COS, h a ve sm all os cilla tionin t he st ea dy st at e.Sin ce a fa st r espon se a nd a sm aIl st ea dy st at e er ror

ar e r equired, COA, COS, HM, or HWSM can be u sedin t he defu zzifica tion pr oced ur e of t he fu zzy con tr olof DC-DC Con ver ter s. Bu t t he fu zzy con tr ol syst em swit h COA, COS, or HM m et hod h ave h igh init ial cu r-r en t . S in ce HWSM shows good s ta rt in g p er form an ce,such as r edu ced init ia l cur rent , and ver y smoot h r e-spon se in t he st ea dy st at e, it is t he best ch oice for t hefu zzy con t rol of DC-DC Conver t er s.

6 Con c lu s ionIn t his pa per , differ en t defu zzifica tion m et hods t ha tcan be adopted by FLC for applicat ion to DC-DCcon ver ter s h ave been st udied. It is obser ved t ha t ea chd e fu zzifica tion t ech riqu e h as cer ta in a dva nt ages a ndcer t ain d is ad va nt ages . A n ew HWSM defu zzifica tionm et hod is pr oposed. F rom com pu ter sim ula tion st ud-ies it is observed that the new proposed HWSM de-fuzzifica t ion t echn ique p rov id es t h e bes t p er formance .

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D. Dr ia nk ov, H . H ellen door n, a nd M. Rein fr an k.An In troduct ion t o Fuz zy Con t rol . Second Edi-t ion , S pr in ger P ublica t ion , p p. 115-126, 1996.D. H. Rao and S. S. Saraf, Study of Defuzzi-fica tion Met hods of F uzzy Logic Con tr oller forSpeed Con tr ol of a DC Mot or : IE EE fian s. onIndustry Applicat ion, vol . 1,n o. 1, pp. 782-787,1995.W. J . M. Kich er t a nd E . H. Ma rn da ni, An alysisof a fu zzy logi c con t r ol le r, Fuzzy Sets Sys t., vol.1, no. 1, pp. 29-44, 1978.W. J . M. Kicher t , F ur th er a na lysis a nd a p-p lica tion of fu zzy logic con t rol. In ternal Rep.F / WK2 / 75, Que en Ma r y College , Lon don , 1975.

, ----- -----. .- .1 FLC III IEE l 1 Vo

?

:1Oefuification Plant-.. - ---- ---,

F igu r e 1: Con ve nt ion a l F uzzy Logic Con tr olle r

w

eun iv e r s e f d i s cou r s e(a ) Me mb er sh ip fu n ct ion

ErrorE~/ NB NM NS Z PS PM PBNB PB PB PB PB PM PS Z

~ NM PB PB PB PM PS Z NS4-,0 NS PB PB PM Ps z NS NMLa,c+

z PB PM PS Z NS NM NBe Ps PM PS Z NS NM NB NBPM Ps Z NW NM NB NB NBPB Z NS NM NB NB NB NB

(b) Ru le Ta bleF igu re 2: Com pon en ts of F uzzy Logic Con tr oller


6/8

23

K1------o10-0 o.ocz2 o,m4 o.m6 o.tm O.OIW(a) ~S ChangaX3

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Figure5: Fuzzy Con trol wit h COADefuzz ifi ca t ion Method

VIATlbsm1510 Vo5~~o O.OCQ o.m4 o.m6 0.ux3 0,01 W(a) Vs Change

o ~~o O.cm 0,034 0.006 O.ox 0,01 w

(b] ROCharige

Figu re 6: F uzzy Con tr ol wit h COS Defu zzifica tionMethod

-o o.mz 0,024 0 ,m 6 o.cm(b] ROGhange

Figure 7: Fuzzy Cont rol with HMMethod

0.01(4

Defuzzification

0-7803-6404-X/00/$10.00 (C) 2000


7/8

Esxl1510 Voo 0.002 o.m4 o.ms o.ma 0.01(s)20

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10

ol- 10 o.m O.CXM o.m6 0.028 0.0!(s)(b) ROChancy

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m15

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n-o 0.0(Y2 O.(X)4 o.a16 O.ma 0,01 w

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15

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o.m2 O.CCM o.m s o.m a 0.0((s)(b) .qOChange

F igu re 10: F uzzy Con tr ol wit h FM Defu zzifica tionMethod

P420t510 Vo0 ~-~o o.ow o.m4 o,m6 O,OM o.01 (.s)(a) Vs ChangeWA25m

15

5

00 0.032 o.m4 o.m6 o.m9 o(b) R(3Change 11(4

Figu re 11: Fuzzy Cont rol wkh HWSM Defuzzifica -t ion Met h od

0 7803 6404 X/00/$10 00 (C) 2000

new defuzzification method for fuzzy control of power converters

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