PEFOMANCE MPOEMEN OF HE NDUCON MOO DE BY USNG OBUS CONOE S. SEAA, PG Schola,. GEEHA, ctu, N. DEAAAN, Aitant Pofo Abtact -: h tanint pon of th induction oto i obtaind by uing it d-q fnc odl. h tanint pon i ipovd by uing optial contol tchniqu bcau of th popty of bt poibl contol. By olving icatti quation, a contoll gain ati i dvlopd uch that th pfoanc ind i iniu. hi gain ati will giv fdback contol law. h contoll will giv contol ignal accoding to thi law. h output i fd back and th pon i analyzd. hu th tanint pon i ipovd. hi contoll i obut againt ditubanc nd t nduction oto, Q, Stability NOMENCAUE, Stato voltag and cunt pac vcto Stato flu pac vcto, oto cunt and flu pac vcto d, q Stato voltag in d-q otating f. fa d, q Stato cunt in d-q otating f. fa ψ d, ψ q oto flu in d-q otating f. fa d, q Stato cunt in d-q oto flu f. fa ω oto pd (ad/), Stato itanc and lf inductanc oto lf inductanc τ oto lctical i contant Magntic inductanc P Nub of pol pai otal oto intia contant (Kg ) F Daping cofficint (N) l oad toqu (N) Elctoagntic toqu (N) σ Cofficint of dipion Synchonou pd (ad/) h autho a with th Dpatnt of Elctical Engining, Govnnt Collg of chnology, Coibato, ndia. NODUCON nduction achin hav bn th ot widly ud achin in fid-pd application fo aon of cot, iz, liability and fficincy. Howv, bcau of th involvd odl high nonlinaiti, thy qui uch o copl thod of contol, o pniv and high atd pow convt than DC and pannt agnt achin. Nowaday, a a conqunc of apid advanc in pow lctonic tchnology, vcto contol tatgy bad lctical ac div hav gd a a powful tool fo high pfoanc contol of nduction achin. hi contol tatgy can povid th a pfoanc fo an invt divn nduction achin a i achivd fo a paatly citd DC achin. n thi thi th autho intoduc a nw contoll calld ina Quadatic gulato which i obut againt tnal ditubanc and povid cllnt pfoanc ipovnt with th ipovnt of tability agin. h yt i iulatd uing Matlab and it chaactitic and fatu a tudid in thi thi. d-q MODE OF HE NDUCON MOO A two pha d-q odl of an nduction achin otating at th ynchonou pd i intoducd which will hlp to cay out th dcoupld contol concpt to th induction achin. hi odl can b uaizd by th following quation d j () d j () h tato and oto flu a givn by th following lation: (3)
(4) n quation to 4, th voltag, cunt and flu pac vcto a function of th coponding th-pha vaiabl [3]. A an apl, th tato cunt pac vcto i linkd to th coponding th pha cunt by th following lation: c b a a a 3 / (5) Wh a = j /3. h poducd lctoagntic toqu i givn by p 3 (6) Figu. fnc fa and pac vcto pntation Uing th d-q coodinat yt, a illutatd in Figu, and paating th achin vaiabl tat vcto into thi al and iaginay pat, th wll-known nduction oto odl pd in t of th tat vaiabl i obtaind fo quation to 6, and i givn by: q d l q d d q q d q q d d q d q d q d q d q d q d P F p d (7) n (7), th cofficint of dipion σ i givn by: A hown in Figu, th d-ai i alignd with th oto flu pac vcto. Und thi condition w hav; ψ q = and ψ d = ψ. Conquntly, th induction oto odl tablihd in th oto flu fild coodinat i thn givn by th quation 9 to. q d l q d q d q d q d P F p d q p () d () q () n odinay u, only tato voltag, cunt and oto pd a availabl fo aunt. n thi ca, th d-q tato voltag and cunt a obtaind fo th coponding α β tationay fnc fa vaiabl though an appopiat tanfoation involving oto flu pac vcto angl θ, a hown in Figu. hi tanfoation i givn by: q d co in in co... (3) n quation 3, "" i a voltag, a cunt o a flu. A ntiond bfo, θ i th oto flu pac vcto angl. n dict vcto contol, th oto flu i availabl fo aunt o i tiatd fo aud tato voltag and cunt. h oto flu angl i thn givn by: a tan h oto flu aplitud i obtaind by olving quation, and it patial poition i givn by:
q h ndict vcto contol tatgy can now atifactoily b achivd inc both aplitud of oto flu vcto and it patial poition a known. A in DC achin, th toqu and th flu a contolld indpndntly: h lctoagntic toqu i contolld by q (toqu poducing cunt), and th flu i contolld by d (flu poducing cunt). Q QUADAC EGUAO A ina Quadatic gulato (Q) i ud to in th SFB K uch that th Pfoanc nd i iniizd. t co und Optial Contol. t i calld o bcau in vy contol tp th pfoanc ind i ducd to a iniu. Futho, it ha a copaabl high obutn againt paat chang. A yt can b pd in tat vaiabl fo a. (4) A Bu n With ( t), u( t). h initial condition i (). W au h that all th tat a auabl and k to find a tat-vaiabl fdback (SFB) contol. (5) u K hat giv diabl clod-loop popti. h clod loop yt uing thi contol bco. (6) ( A BK) Bu Ac Bu With Ac th clod-loop plant ati and u(t) th nw coand input. h output atic C and D a not ud in SFB dign. f th i only on input o that =, thn Ackann foula giv a SFB K that plac th pol of th clod-loop yt a did. Howv, it i vy inconvnint to pcify all th clod-loop pol, and a tchniqu i ndd that wok fo any nub of input. h optial contoll qui lat contol ngy fo contol th yt. Sinc any natually occuing yt a optial, it ak n to dign an-ad contoll to b optial a wll. o dign a AFB that i optial, a t pfoanc ind (P) i to b conidd. Q v v Subtituting th SFB contol into thi yild ( Q k k) (7) (8) h objctiv in optial dign i to lct th SFB K that iniiz th pfoanc ind. h pfoanc ind can b intptd a an ngy function, o that aking it all kp all th total ngy of th clodloop yt. f both th tat (t) and th contol input u (t) a wightd in, o that if i all, thn nith (t) no u (t) can b too lag. f i iniizd, thn it i ctainly finit, and inc it i an infinit intgal of (t), thi ipli that (t) go to zo a t go to infinity. hi in tun guaant that th clod-loop yt will b tabl. h two atic Q (an n n ati) and (an ati) a lctd by th dign ngin. Dpnding on how th dign paat a lctd, th clod-loop yt will hibit a diffnt pon. Gnally paking, lcting Q lag an that, to kp all, th tat (t) ut b all. On th oth hand lcting lag an that th contol input u (t) ut b all to kp all. hi an that lag valu of Q gnally ult in th pol of th clod-loop yt ati Ac = (A BK) bing futh lft in th -plan o that th tat dcay fat to zo. On th oth hand, th lag an that l contol ffot i ud, o that th pol a gnally low, ulting in lag valu of th tat (t). On hould lct Q to b poitiv i-dfinit and to b poitiv dfinit. hi an that th cala quantity Q i alway poitiv o zo at ach ti t fo all function (t), and th cala quantity u u i alway poitiv at ach ti t fo all valu of u (t). hi guaant that i wll-dfind. n t of ignvalu, th ignvalu of Q hould b non-ngativ, whil tho of hould b poitiv. f both atic a lctd diagonal, thi an that all th nti of ut b poitiv whil tho of Q hould b poitiv, with poibly o zo on it diagonal. Not that thn i invtibl. h u of ina Quadatic gulato (Q) i to in th SFB K uch that it iniiz th 3
Pfoanc nd. h wod gulato f to tack pobl, wh th objctiv i to ak th output follow a pcibd (uually nonzo) fnc coand. o find th optial fdback K it i pocdd a follow. Suppo th it a contant ati P uch that d( P) ( Q k k) hn, ubtituting into quation (7) yild, (9) d( P) () () P() Wh it i aud that th clod-loop yt i tabl o that (t) go to zo a ti t go to infinity. Equation () now ipli that i now indpndnt of K. t i a contant that dpnd only on th auiliay ati P and th initial condition. Diffntiating (7) and thn ubtituting fo th clodloop tat quation (4) it i n that (7) i quivalnt to ( A P PA Q k k) () c c inial valu of th P uing thi gain i givn by (), which only dpnd on th initial condition. hi an that th cot of uing th SFB (4) can b coputd fo th initial condition bfo th contol i v applid to th yt. h dign pocdu fo finding Q fdback K i: Slct dign paat atic Q and Solv th algbaic iccati quation fo P Find th SFB uing h ati Q and can b found out by tial and o thod o uing GA tchniqu. h a vy good nuical pocdu fo olving th AE. h MAAB outin that pfo thi i nad lq (A, B, Q, ).h Q dign pocdu i guaantd to poduc a fdback that tabiliz th yt a long a o baic popti hold. Q HEOEM: t th yt (A, B) b achabl. t b poitiv dfinit and Q b poitiv i-dfinit. hn th clod loop yt (A-BK) i ayptotically tabl. Not that thi hold gadl of th tability of th opn-loop yt. call that achability can b vifid by chcking that th achability ati ha full ank n. t ha bn aud that th tnal contol v(t) i qual to zo. Now not that th lat quation ha to hold fo vy (t). hfo, th t in backt ut b idntically qual to zo. hu, pocding it i n that. X=AX+BU C Y=CX A P PA Q k k k B P PBk () hi i a ati quadatic quation. Eactly a fo th cala ca, on ay coplt th qua. hough thi pocdu i a bit coplicatd fo atic, uppo if -K k B P (3). Figu. Syt Block Diaga hn, it ult in A P PA Q PB B P (4) hi ult i of t ipotanc in odn contol thoy. Equation (4) i known a th algbaic iccati quation (AE). t i a ati quadatic quation that can b olvd fo th auiliay ati P givn (A, B, Q, ). hn, th optial SFB gain i givn by (3). h SMUAON ESUS Siulation, uing Matlab-Siulink oftwa packag, hav bn caid out to vify th ffctivn of th popod contol thod. h ult a hown in figu 3. 4., & 5. Figu 3 how th unit tp pon bfo and aft applying contoll. Figu 4 how th cunt cuv id and iq of th oto div. Figu 5 how th location 4
of pol bfo and aft applying contoll. t alo how how th tability i nhancd by odifying th pol location. Figu 5. Pol location bfo and aft applying Q Figu 3. pon of yt bfo and aft applying Q t i alo obvd that tability i alo analyzd aft applying th contoll. h ult how that th agin of tability alo inca by incopoating th contoll. h tability tt a caid out uing h-infinity dfinition and yapunov tt fo poitiv dfinitn. CONCUSON h iulation of Q contolld induction oto div i uccfully iplntd in thi pap. h application of th contoll and it pon ipovnt contibuting to th tability nhancnt i tudid. hi pap can b futh tndd by copaing thi contoll pfoanc with th iting contolling thod lik P and o on. hi pap can b futh tndd to oth typ of div alo nhancing th pfoanc. EFEENCES Figu 4. pon cuv of id, iq and phid. Fata Gubuz, Eyup Akpina, Stability Analyi of a clod-loop contol fo a Pul Wih Modulatd DC Moto Div", ukih ounal of Elctical Engining, vol., No.3,.. Bial K. Bo," Modn Pow Elctonic & AC Div, Paon Education,. 3. Katuhiko Ogata, Modn Contol Engining, Pntic-Hall of ndia Pvt. td,. 4. Dign and Analyi of Contol Syt, Athu G.O. Mutabaa, CC P, ondon, 999. 5. ina Contol Syt Engining, Moi Dil, McGaw Hill intnational dition, 995. 5
6. Modn Contol Dign with Matlab and Siulink, Ahih wai, ohn Wily and Son td.,. APPENDX nduction Moto Paat Un () = 44 (Stato lin voltag) P n (hp) = (Noinal output pow) (Ω) =.95 (Ω) =.75 (H) = 6.5 (H) = 6.4 (H) = 6 f (Hz) = 6 p = (kg. ) = 5 6