Robust Speed Control of a Doubly Fed Induction Motor using State-Space Nonlinear Approach

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1 Leondo Jounl of Scence ISSN Iue, Jnuy-June Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE *, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS Electcl Engneeng Detment, Intellgent Contol nd Electcl Powe Sytem Lbotoy (ICEPS), Djlll Lbe Unvety, Sd-Bel-Abbe, 000, Alge. Electcl Engneeng Detment, Gene Electcl Lbotoy (LGE), Mouly Th Unvety, Sd, 0000, Alge. E-ml: * Coeondng utho: Phone: ; Fx: Abtct Th e eent comon between two contolle (fuzzy logc nd vble gn PI) of the one t nd the conventonl PI on the othe hnd, ued fo eed contol wth ndect oto flux oentton of doubly fed Inducton Moto (DFIM) fed by two PWM nvete wth ete DC bu lnk. By ntoducng new och fo decoulng the moto cuent n ottng (d-q) fme, bed on the tte ce nut-outut decoulng method, we obtn the me tnfe functon (/) fo ll fou decouled cuent. Theefte nd n ode to move the efomnce of the mchne contol, the VPGI nd fuzzy logc contolle wth fve ubet wee ued fo the egulton eed. The Reult obtned n Mtlb/Smulnk envonment how well the effectvene of the technque emloyed fo the decoulng nd the eed egulton of the mchne. eywod Doubly fed nducton moto (DFIM); Inut outut decoulng; Feld-oented contol; Modellng; Vble gn PI contolle; Fuzzy logc contolle, Conventonl PI contolle. htt://lj.cdemcdect.og 03

2 Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS Intoducton The oge ccomlhed, n the few t ye, n the owe electonc nd dgtl feld mke the Doubly Fed Inducton Mchne (DFIM) n ndutl tndd due to t low cot nd hgh elblty [, 5]. Doubly Fed nducton moto n electcl thee-he ynchonou mchne wth wound oto cceble fo contol. Snce the owe hndled by the oto de (l owe) ootonl to the l, the enegy eque oto-de owe convete whch hndle only mll fcton of the ovell ytem owe []. In ecent ye, thee h been get mount of ctvty on bck teng contol och n AC dve feld [].The non lne contol och h bette econ nd tblty. Howeve, t mjo oblem of t entvty to moto mete vton nd lod dtubnce. The DFIM contol ue e tdtonlly hndled by fxed gn ootonl ntegl (PI) contolle. Howeve, the fxed gn contolle e vey entve to mete vton, cnnot ovde good dynmc efomnce. So, the contolle mete hve to be contnully dted [3]. The vble gn PI nd fuzzy logc contolle gve bette eult to mete vton fo nonlne ytem. So, the DFIM n del cnddte to tet the efomnce of t egulto [5]. Fuzzy contol technque doe not need ccute ytem modellng. It emloy the ttegy doted by the humn oeto to contol comlex ocee nd gve ueo efomnce. The fuzzy lgothm bed on humn ntuton nd exeence, nd cn be egded et of heutc decon ule [8, 8]. The eent wok concen feld-oented contol wth vble gn PI nd fuzzy logc contolle of doubly fed nducton moto wth tte ce decoulng method.the vecto contol of the DFIM wth two ndeendent convete h been tuded ecently n evel wok. The lnezton of the nonlne model of the mchne cn be done n dffeent mnne wth vou tem of comenton. In th e nonlne tte ce ooed to enue the decoulng of the mult-vble ytem nut-outut tht conttute the DFIM. Dynmc Model of DFIM The dynmc model of the DFIM n (d-q) ynchonou ottng fme gven by the 04

3 Leondo Jounl of Scence ISSN Iue, Jnuy-June equton of the voltge: wth: ω dφ V = R I + + jωφ dt dφ V = + + ω φ R I j dt Exeon of the fluxe e gven by: φ = LI + MI φ = LI + MI Fom () nd () the ll cuent tte model wtten follow: d I R MR MR (3) = I + I + V V dt σl σll σl σll di R MR MR = I + I + V V dt σl σll σl σll The mechncl equton exeed by (4): J =. dω dt Ω = T ƒω em T And the electomgnetc toque gven by: ( I I ) T em = MI (5) m So, the equton fo the eed vton become: J dω = M dt I m ƒω ( I I ) T (6) () () (4) Vecto Contol Sttegy of DFIM by Decoulng Stte Sce. Roto Flux Oented The ncle fo th tye of contol cont n oentng the flux nto the mchne, to the oto, to the tto o n the g. Conventonlly, we wok wth n oentng on the d x. The n qudtue x wll theefoe cy the cuent tht wll tcte n the ceton of the electomgnetc toque n the mchne [5], [9]. To elze the contol lw, the oto flux oentton choen long the d x (fgue.). Theefoe, we obtn: φ q = 0 ; φ = φ (7) d Then t come: 05

4 Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS M (8) Iq = Iq L The mgnetzton of the mchne, llow, to moe the oto flux module, o we dtnguh two ttege [5]: Wokng wth unty owe fcto to tto o to oto, whch mle tht one of the two cuent I d φ = M d o I d I d wll be null,whethe: Slt the mgnetzng cuent eqully between the two convete, e Id = Id = whethe: φ Id I ( L M ) d = + The choce of I d = 0 g. In ddton, the exeon deend only on oto. q d, gve the me exeon fo the flux to the tto nd to the q O I d φ = φ d θ Fgue. Roto Flux Oented on the d Ax M θ (9), nd wth unty owe fcto t the θ d Roto x Stto x b. Cuent Decoulng by Stte Sce b. Pncle of the method Conde the followng multvble ytem: T [ x x... x n ] T [ u u... u m ] [ y y... y ] T x x = x& = Ax + Bu m wth y R u = y = Cx m u R y = m The objectve to detemne tte ce of the fom: u = d x + Lv, wth v n R (0) m R () v denote the new nut vecto, whch decoule the ytem, n wy tht the outut y (= 06

5 Leondo Jounl of Scence ISSN Iue, Jnuy-June to m) deend only on the nut v. The outut y wtten: whee y = C x C the th ow of the mtx C. Let u deve y few tme n ode to bng u the commnd. We cll chctetc ndex noted δ, the numbe of devton t tke n ode to bng u the commnd. We then hve uccevely fo ech outut : wth y& = C x& = C (Ax + Bu) = CAx wth : CBu = 0 && y = C Ax& = CA(Ax + Bu) = CA x wth : CABu = 0 (3) 3 && y = C A x& = CA (Ax + Bu) = CA x wth : CA Bu = 0 M (δ ) δ δ δ y = CA x + CA Bu wth : (CA Bu 0) Tht we cn tll wte n mtx fom: y (δ y M (δm y (δ ) Tht : ) δ CA δ CA = M δ CmA ) y * = A x + B u m m m y R, A R nd m δ CA δ CA x + M δ CmA B m B B u B m m R. We eek contol lw u d x + Ldv () (3) (4) = uch y = v. The looed ytem wtten: ( x + L v ) = ( A B ) x + B L v y = A x + B (5) d d d d To obtn y = v we mut hve B L d = nd A B d = 0. If the mtx B nvetble, the choce of : d = ( B ) A nd L = ( B ) Gve: y = v Tht : Y () = δ + V () (6) (7) 07

6 Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS b. Alcton to the DFIM We ech to exlot th method fo decoulng the cuent of the mchne ojected on (d-q) ottng fme [5, 0, 6]. Sttng fom the exeon (3) nd choong tte vecto equl to the outut vecto, fomed of fou cuent of the mchne. The nut vecto fomed of uly voltge. Then we obtn the followng exeon: x& = Ax + Bu y = Cx wth: x = [ I I I I ] T the tte vecto (fo ll cunt) nd v = [ V V V V ] T d q the nut vecto voltge. whee: d ω ω A = 4 6ω b 0 0 b B = b3 0 0 b σ = ; σ M 6 = ; σl The choce of 3 q = ; σl ω + ω 3 5ω 5ω 3 ω + 6ω ω σ ω 4 ω σ b b C = b b R b = ; σl y R = ; σl R M 3 = ; σll R M 4 = ; σll d q M d 5 = ; σl (8) q (9) (0) M M b = ; b 3 = ; σ = σl σll σll x = mke the ytem comletely contollble nd obevble. In lyng the decoulng method on th ytem, t follow tht: y = v ; δ = 0 nd L = B () d d = B A, theefoe: Y () () = V () The fou cuent e decouled nd thu govened by the me tnfe functon n oen loo G () =. c. Degn the Contol Loo c. Cuent contol 08

7 Leondo Jounl of Scence ISSN Iue, Jnuy-June The cuent e decouled, nd then we cn conde tte ce coecton wth the method of lcement of ole. The ncl chemtc dgm of th coecton gven by the fgue. I éf L + - Decouled Sytem I Coecto cuent Fgue. Cuent Regulton by Stte Sce To enue the me eone fo the cuent loo, the next choce cn be doted. k k 0 0 L = = 0 0 k k So the tnfe functon of ech cuent cloed loo wll be of the fom: () H k = + k c. Seed Contol The mechncl equton gven by: dω (5) J = Tem ƒω T dt The oentton of the oto flux on the d x, nd the hyothe to wokng wth I d = 0, confe on the electomgnetc toque the followng exeon: Tem = MIqId = φdi (6) q A we oceed to the mgnetzton of the mchne befoe lyng eed efeence, φ d cn be elced by t efeence φ def n the elton (6), theefoe: nd T = φ I = I (7) em dω J = dt Such em def I q q em ƒω T em the contnt toque. q (3) (4) (8) 09

8 Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS Thu, the tnfe functon of the eed wll be exeed by: em (9) Ω () = Iq () T () ƒ + J ƒ + J The mgntude T () ly the ole of dtubnce nut fo eed, the ncl nut beng I q (). The block dgm of the egulton wll be n confomty wth tht of fgue 3. Ω + - C Ω () I q H () I q -Φdef T em + - T /ƒ+j Ω Fgue 3. Seed Contol Chn VPGI nd Fuzzy Logc Contolle n Seed Contol of DFIM. VPGI Contolle. VPGI Contolle Stuctue A vble gn PI (VPGI) contolle genelzton of clcl PI contolle whee the ootonl nd ntegto gn vy long tunng cuve gven by fgue 4. Ech gn of the ooed contolle h fou tunng mete [4]: Gn ntl vlue o tt u ettng whch emt ovehoot elmnton. Gn fnl vlue o tedy tte mode ettng whch emt d lod dtubnce ejecton. Gn tnent mode functon whch olynoml cuve tht jont the gn ntl vlue to the gn fnl vlue. Stuton tme whch the tme t whch the gn ech t fnl vlue. The degee n of the gn tnent mode olynoml functon defned the degee of the vble gn PI contolle. 0

9 Leondo Jounl of Scence ISSN Iue, Jnuy-June Vble Gn V f V 0 Polynoml Cuve Intl Vlue Fnl Vlue 0 Tme (ec) t Fgue 4. Vble PI Gn Tunng Cuve wth: whee It e(t) the gnl nut to the VPGI contolle the outut gven by : y(t) = e(t) + t 0 e(τ)dτ n (3) t ( ) + < f t t = t f t t n (3) t < = f f t t t f f t t nd e the ntl nd fnl vlue of the ootonl gn, nd f the (30) fnl vlue of the ntegto gn. The ntl vlue of clc PI contolle VPGI contolle of degee zeo. tken to be zeo. It noted tht The VPGI unt te eone gven by: y (t) = f + + t n + n + f t t n + t t n f f t < t t t (33) Fgue 5 gve the unt te eone of VPGI contolle of dffeent vlue of the degee n.

10 Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS y(t) + t n=0 n= n=5 n=0 T Tme (ec) T Fgue 5. VGPI Ste Reone fo Dffeent Vlue of the Degee n t If t t (tnent egon) the clcl PI unt te eone lne cuve begnnng nd fnhng t t +, whee the VPGI unt te eone ( 0) n ve long olynoml cuve of degee n + begnnng t nd fnhng t + t (n ). f + If t t (emnent egon), the unt te eone of PI nd VPGI contolle e both lne wth loe. f Fom thee eult, one cn y tht VPGI contolle h the me oete thn clcl PI contolle n the emnent egon wth dmed te eone n the tnent. A VPGI contolle could then be ued to elce PI contolle when we need to olve the lod dtubnce ejecton nd ovehoot oblem multneouly.. Settng method of the VPGI contolle Unlke the clcl PI contolle, tunng of the VPGI contolle doe not need comomng. Seed ovehoot cued by hgh ntegto gn could be elmnted by nceng ethe the tuton tme o the degee of the contolle. One cn chooe the fnl vlue of the ntegto gn needed fo the lcton nd then tune the othe contolle mete o to elmnte eed ovehoot. Hee ooed method of tunng VPGI contolle.. Chooe ft degee VPGI contolle wth hgh vlue of f (d lod dtubnce ejecton).. Chooe n ntl vlue of the tuton tme t.

11 Leondo Jounl of Scence ISSN Iue, Jnuy-June Detemne nd fo eed ovehoot elmnton by ung the followng te: Conde to be contnt nd multe the contolled ytem fo mll ntl vlue of. Incee gdully nd multe the contolled ytem gn untl eed ovehoot get to t otmum. Chooe to be the vlue of tht gve otml ovehoot. Smulte the contolled ytem fo n ntl vlue of of. Incee gdully the vlue of equl to the choen vlue nd multe the contolled ytem gn untl eed ovehoot totlly elmnted o get to t otml vlue. If ovehoot totlly elmnted y then obtned nd the contolle tuned. 4. If ovehoot not elmnted, then the vlue of the tuton tme t not uffcency hgh, ncee t gdully wthout exceedng lmtng vlue nd eet te 3 untl ovehoot totlly elmnted. 5. If t the lmtng vlue of t ovehoot tll not elmnted, then the degee of the contolle not hgh enough. Incee t nd eet the contolle tunng gn. Ung th tunng method wth f = 4, the tuned VPGI contolled gven by:.5 t f t < =.9 f t 4 t f t < = 4 f t (34) b. Fuzzy Logc Contolle The tuctue of comlete fuzzy contol ytem comoed fom the followng bloc: Fuzzfcton, nowledge be, Infeence engne, Defuzzfcton. Fgue 6 how the tuctue of fuzzy contolle. nowledge Bed ef + - E de Fuzzfcton Infeence Engne Defuzzfcton ouut Sytem Fgue 6. The Stuctue of Fuzzy Logc Contolle 3

12 Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS The fuzzfcton module convet the c vlue of the contol nut nto fuzzy vlue. A fuzzy vble h vlue, whch e defned by lngutc vble (fuzzy et o ubet) uch low, medum, hgh, low whee ech defned by gdully vyng membeh functon. In fuzzy et temnology, ll the oble vlue tht vble cn ume e nmed unvee of dcoue, nd fuzzy et (chctezed by membeh functon) cove whole unvee of dcoue. The he fuzzy et cn be tngul, tezodl, etc [6,, ]. A fuzzy contol eentlly embed the ntuton nd exeence of humn oeto, nd ometme thoe of degne nd eeche. The dt be nd the ule fom the knowledge be whch ued to obtn the nfeence elton R. The dt be contn decton of nut nd outut vble ung fuzzy et. The ule be eentlly the contol ttegy of the ytem. It uully obtned fom exet knowledge o heutc; t contn collecton of fuzzy condtonl ttement exeed et of IF-THEN ule, uch : R () :IF x F nd x THEN Y G () F...nd x n, =,..., M wee: ( x,x,...,x n ) the nut vble vecto, Y the contol vble, M the numbe of ule, n the numbe fuzzy vble ( F,F,...Fn ) e the fuzzy et. Fo gven ule be of contol ytem, the fuzzy contolle detemne the ule be to be fed fo the ecfc nut gnl condton nd then comute the effectve contol cton ( the outut fuzzy vble) [7, 8, 7]. The comoton oeton the method by whch uch contol outut cn be geneted ung the ule be. Sevel comoton method, uch mx-mn o u-mn nd mx-dot hve been ooed n the ltetue. The mthemtcl ocedue of convetng fuzzy vlue nto c vlue known defuzzfcton. A numbe of defuzzfcton method hve been uggeted. The choce of defuzzfcton method uully deend on the lcton nd the vlble oceng owe. Th oeton cn be efomed by evel method of whch cente of gvty (o centod) nd heght method e common [7, 3, 9]. F n (35) b. Fuzzy-PI contolle The fuzzy PI contolle bclly n nut/ outut ttc non-lne mng, the 4

13 Leondo Jounl of Scence ISSN Iue, Jnuy-June contolle cton cn be wtten n the fom [3]: whee: u = e.e + de.de (36) The Fuzzy-PI outut : y =.u.u (37) + e the gn of the eed eo, de the gn of the chnge of eed eo, the ootonl fcto, the ntegl fcto, E the eed eo, de the chnge of the eed eo, u the fuzzy outut. The Fuzzy-PI contolle n vecto contol of Dfm ued eented n Fgue 7. Fgue 7. The Stuctue of Fuzzy PI Contolle n Vecto Contol of Dfm b. nowledge Be Pooed Fgue 8 nd 9 how eectvely the tngle-hed membeh functon of eo E nd Chnge of eo de.the fuzzy et e dengnted by the lbel: Negtve bg (NB), Negtve medum (NM), Negtve mll (NS), Zeo (Z), Potve mll (PS), Potve medum (PM), Potve bg (PB) NB NM NS Z PP PM PG 0.8 µ (En), µ (den) E n,de n Fgue 8. Membeh Functon Dtbuton fo Inut Vble 5

14 Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS NB NM NS Z PS PM PB 0.8 µ (dun) du n Fgue 9. Membeh Functon Dtbuton fo Outut Vble In th e, the tngul membeh functon, the mx-mn eonng method, nd the cente of gvty defuzzfcton method e ued, thoe method e mot fequently ued n mny ltetue [4, 0]. The nfeence ttegy ued n th ytem the Mmdn lgothm. Tble. Lngutc Rule Tble E NB NM NS Z PS PM PB de NB NG NG NG NG NM NP EZ NM NG NG NG NM NP EZ PP NS NG NG NM NP EZ PP PM Z NG NM NP EZ PP PM PG PS NM NP EZ PP PM PG PG PM NP EZ PP PM PG PG PG PB EZ PP PM PG PG PG PG All the membeh functon (MF) e ymmetcl becue ne the ogn (tedy tte), the gnl eque moe econ. Seven MF e choen fo E, de gnl nd fo ouut. All the MF e ymmetcl fo otve nd negtve vlue of the vble. Thu, mxmum 7 7 = 49 ule cn be fomed tbulted n Tble [7, 4]. 6

15 Leondo Jounl of Scence ISSN Iue, Jnuy-June Smulton Reult The DFIM ued n th wok.5 w-50hz, whoe mete e eoted n endx. The globl chem of the Stte-Sce Nonlne contol of doubly fed Inducton moto Ung vble gn PI nd fuzzy logc contolle eented n fgue 0. A. Fuzzy Seed Revel of Rted Vlue In ode to mke comon between the behvo of the conventonl PI contolle nd tht of the fuzzy logc nd VPGI contolle, tuded unde dffeent oetng condton, dect tt of the moto unde no lod elzed wth et ont of 57 d / followed by n nveon of the otton decton t tme t = 3, the extenl etubton e ntoduced by udden lcton of 0N.m nomnl chge t t = nd emoved t t =. The eult gven by fgue ( ) how excellent efomnce n egulton fo the vble-gn nd the fuzzy logc contolle wth vey good montong of the efeence eed. Fgue 0. Block Dgm of Seed Contol of Dfm ung Stte Sce Nonlne Aoch 7

16 Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS Th wll eult n much lowe tckng eo thn tht obtned ung the conventonl PI tuctue. Note lo tht the oentton of the oto flux fully elzed; futhemoe, the develoed electomgnetc toque eoduce t efeence tfctoly. It cn lo be noted tht the low entvty nd dtubnce ejecton e excellent fo the two tuctue; both Fuzzy logc nd VPGI contolle lo ovde bette efomnce n tem of eed nd tme dtubnce ejecton. B. Robut Contol fo Dffeent Vlue of Roto Retnce In ode to vefe the obutne of VPGI nd Fuzzy-PI egulto unde moto mete vton, we hve multed the ytem wth dffeent vlue of the mete condeed nd comed to nomnl vlue (el vlue), one ce condeed: The oto etnce vton (ncee t 50% of nomnl vlue oto etnce). Fgue (3-4) how the eone eed, toque nd oto flux n the tet of obutne fo dffeent vlue of oto etnce. The eult ndcte tht the VPGI nd Fuzzy-PI egulto e nentve to the etnce chnge, whch eult n the no nfluence on the toque nd oto flux. Fo the obutne of contol, n ncee of the etnce doe not hve ny effect on the efomnce of the ooed contolle. Concluon In th e, we eented the ncle of eed contol of double-fed nducton moto ung vble gn PI nd fuzzy logc eed contolle. Tkng dvntge of the cceblty of the cuent meuement of the moto, new och w dcued to llow the decoulng of t cuent n ottng (dq) fme. Th ncle bed on n nut-outut decoulng by tte ce feedbck tht wll led to obtn vey mle cuent tnfe functon, nd theefoe, mlfed clculton of the coecton. Subequently, we demontted the movement mde by the vble gn PI nd fuzzy logc eed contolle on the efomnce of the DFIM comed to the conventonl 8

17 Leondo Jounl of Scence ISSN Iue, Jnuy-June PI contolle. Smulton eult demontte tht VPGI nd fuzzy-pi contolle outefom the clcl PI contolle n eed contol. The multon eult howed emkble behvou of the fuzzy-pi nd vble gn PI contolle dung egulton nd tckng, wth gnfcntly bette dtubnce ejecton thn the clc PI contolle nd good efomnce towd obutne. Aendx DFIM Doubly Fed Inducton Moto. VPGI Vble Gn PI Contolle., Stto nd Roto ndce, d,q Indce of the othogonl comonent dect nd qudtue. X Comlex vble uch : X = Re[ X ] + j Im[ X ]. R, R Stto nd Roto etnce. L, L Stto nd Roto nductnce. T, T Stto nd oto tme contnt. σ Lekge fcto ( σ = M L L ). M Mutul nductnce. θ The electcl oto oton. θ,θ Sttoc flux oton, Rotoc flux oton. ω The mechncl oto fequency. Ω Mechncl eed. ω The electcl tto fequency. P Numbe of ole. T em The electomgnetc toque. T The lod toque. J The moment of net. ƒ The fcton coeffcent. Rted dt of the multed doubly fed nducton moto: Rted vlue:.5w; 0/380V-50Hz; Rted mete: R =.75Ω R =.68Ω L = 0.95 H L = 0.04 H M = 0.65 H 9

18 P =.0 Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS Mechncl contnt J = 0.0g.m Refeence. Slloum G., Mbyed R., M. Petezk-Dvd M., De Fome B., Mxed Sentvty H Contol of Doubly Fed Inducton Moto, /07/$0.00, IEEE 007, 3, Yun L, Feng-You He, Zong-Bn Ye, Study on Sldng Mode Seed Contol wth Fuzzy Aoch fo Doubly-Fed Inducton Moto, IEEE.Intntonl Confeence on Confeence on Contol nd Automton,Chtchuch, New Zelnd, Decembe 9-, Chb A., Abdeemed R., Bend, M.L., Dendoug A., Pefomnce of Toque Tckng Contol Fo Doubly Fed Aychonou Moto Ung PI nd Fuzzy Logc Contolle, Jounl of Electcl Engneeng IEE, 005, 5(). 4. Mloud A., Dou A., Gn PI Contolle Degn fo Seed Contolled Inducton Mchne Dve, Conf. Rec. IEEE/IECON, Sevll, Sn, 5-8 Novembe, Slloum G., Contbuton t the Robut Contol of Doubly Fed Inducton Moto, Doctot The of Ntonl Inttue Polytechnc of Toulou, Fnce Pt M. A. M., Cco J. M., Glvn E., Snchez J. A., Fnquelo L. G., Btt C., Imovng Tnton between Powe Otmton nd Lmtton of Vble Ptch Wnd Tubne ung Fuzzy Contol, In Poc, IEEE Indutl Electonc Socety, Conf, 000, 3, Attou D. B., Bekk Y., Seed Contol of Doubly Fed Inducton Moto ung Fuzzy Logc Technque, Intentonl Jounl on Electcl Engneeng nd Infomtc, 00, (3), Alloue B., Abdehmn A., Gbou B., N A., The Effcency of Ptcle Swm Otmton Aled on Fuzzy Logc DC Moto Seed Contol, Sebn Jounl on Electcl nd Engneeng, 008, 5(), Dd S., Tdjne M., nd Nt-Sd M.S., Robut bckcteng vecto contol fo doubly 0

19 Leondo Jounl of Scence ISSN Iue, Jnuy-June fed nducton moto, IET Contol Al, 007, (4), Vcto M. S., Tegooulo J. A., Doubly-Fed Inducton Mchne Dffeentl Dve Model Fo Automoble, IEEE Tncton on Enegy Conveon. Contol Theoy Al., 003, 8(), hojet S., Slm-Belkhogj E. I., Petzk-Dvd M., de Fomel B., A Fult Tolent Oetng Sytem n Doubly Fed Inducton Mchne unde Invete Shot-ccut Fult Dffeentl Dve Model, IEEE, 006, Dd S., Tdjne M., Nt-Sd M. S., Nonlne Feedbck contol nd toque otmzton of doubly fed nducton moto, Jounl of Electcl Engneeng, 005, 56(3-4), Ch A., Soltn M., Go M., Comtve tudy between the conventonl egulto nd fuzzy logc contolle: lcton of nducton mchne, Intentonl Jounl of Scence nd Technque of Automtc contol nd comute engneeng IJ-STA, 007, (), Aou M., Abd H., Abd A., Zeblh A., A Fuzzy Logc Contolle fo Synchonou Mchne, Jounl of Electcl Engneeng, 007, 58(5), Nemmou A. L., Abdemed R., hezz A., Louze L. nd Bouchem M., The Inut- Outut Lnezton Contol Scheme fo Doubly-Fed Inducton Moto Dve, /08/$5.00, IEEE, 008, Vdl P-E., Dvd M-P., Bonnet V., Mxed contol ttegy of doubly fed nducton mchne, Snge-Vegl, 007, Zekt M., Chekoun S., Mechenene A., A.Develoment nd Imlementton of Hgh- Pefomnce Vble Stuctue Tckng fo Inducton Moto Ung Fuzzy-Logc Contolle, Intentonl Revew on Modellng nd Smulton, 00, 5(), Negd., Mnou A., htem B., Rel Tme Imlementton of Fuzzy Logc Bed MRAS Obeve fo Seed Senole Vecto Contol of Inducton Moto, Intentonl Revew on Modellng nd Smulton, 00, 5(4), Abedn O., Nde Mohmmd S., Jll A., Mokhtou A., A Novel Hybd GA-PSO Technque fo Otml Tunng of Fuzzy Contolle to Imove Mult-Mchne Powe Sytem Stblty, Intentonl Revew on Modellng nd Smulton, 0, 6(), Ben Hmed Moun, Abd Ach, Sbt Ld, Neul Netwok Seed Senole Dect

20 Robut Seed Contol of Doubly Fed Inducton Moto ung Stte-Sce Nonlne Aoch Tk MOHAMMED CHIOUCHE, Sm HADJRI, Abdelkde MEZOUAR, nd Th TERRAS Vecto Contol of Inducton Moto ung Fuzzy Logc n Seed Contol Loo, Intentonl Revew on Modellng nd Smulton, 0, 6(5),

E-Companion: Mathematical Proofs

E-Companion: Mathematical Proofs E-omnon: Mthemtcl Poo Poo o emm : Pt DS Sytem y denton o t ey to vey tht t ncee n wth d ncee n We dene } ] : [ { M whee / We let the ttegy et o ech etle n DS e ]} [ ] [ : { M w whee M lge otve nume oth