SLIDING MODE APPLICATION IN SPEED SENSORLESS TORQUE CONTROL OF AN INDUCTION MOTOR

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1 Copyght IFAC 15th ennal Wol Conge, Bacelona, Span SLIDING MODE APPLICAION IN SPEED SENSORLESS ORQUE CONROL OF AN INDUCION MOOR Kael Jezenk 1, Man Roč 1, Af Šabanovć 1 Unvety of Mabo, Faculty of Electcal Engneeng an Compute Scence Smetanova 17, SI- Mabo, Slovena Sabanc Unvety,Faculty of Engneeng an Natual Scence Ohanl, uzla-itanbul, ukey Abtact: Inucton moto pee enole toque contol, whch allow opeaton at low an zeo pee, optmzng both toque epone an effcency, popoe. he contol qute ffeent than the conventonal fel-oente o ect toque contol. he pouce toque explctly contnuou output vaable of contol. A new tato an oto flux contolle/obeve bae on contnuou lng moe an Lyapunov theoy ae evelope. A mooth tanton nto the fel weakenng egon an the full utlzaton of the nvete cuent an voltage capablty ae poble. he efeence tackng pefomance of toque an oto flux emontate n tem of tanent chaactetc by expemental eult. Keywo: Inucton moto, contol ytem, toque contol, lng moe contol, obeve. 1. INRODUCION Senole contol of nucton moto (IM) ve ecevng we attenton (Va, 1998). he man eaon that the ue of pee eno pol the uggene an mplcty of IM. Even moe, n a hotle envonment pee eno cannot be mounte. Howeve, ue to the hgh oe an nonlneaty of the ynamc of an IM, etmaton of the angle pee an oto flux wthout the meauement of mechancal vaable become a challengng poblem. o ovecome th, vaou pee enole contol algothm have been peente (Holtz, 1996; Boe an Smoe, 1995; Kubota an Nakano, 1993). Due to t oe eucton, tubance ejecton, tong obutne, an mple mplementaton by the mean of powe convete an gtal gnal poceo, lng moe contol (SMC) theoy one of the popectve contol methoologe fo electc machne (Utkn, 1993). In th pape, an SMC bae oto flux obeve egne an a lng moe appoach apple to the toque tackng contol of the IM. In pncple, the popoe metho bae on vng the tato flux towa the efeence tato flux vecto efne by the nput comman, the efeence toque an the efeence oto flux. h acton cae out by the obut lng moe flux contolle applyng a utable tato voltage vecto to the machne n oe to compenate the tato flux vecto eo.. MACHINE DYNAMICS AND PROBLEM SAEMEN he tate equaton of the IM vewe fom the tato fame (a, b fame) ven by voltage ouce nvete gven by R RLm t = σl σll u, (1) Lm R R jp t = ω L σl σl, () Lm = σ L, (3) L Lm e = p ( ab ba), (4) 3 σ LL ω J = e L, (5) t whee two menonal vecto = a, b,

2 = a, b, u = ua, u b, = a, b ae tato an oto flux, tato voltage an cuent, epectvely. e moto toque an L loa toque. J neta of the oto an p the numbe of pole pa, an ω the mechancal oto angle velocty. R an R ae the tato an oto etance, wheea L an L ae the tato an oto elf-nuctance. L m σ =1 L L L. the mutual nuctance an ( ) In oe to llutate the nonlnea behavo of nucton moto contol, a theoetcal eucton of toque evatve (4) wa cae out. h yel: e p Lm = ( u pω ) t 3 σ LL, (6) R R e σl σl whee ncate ot pouct (actve tem) an ncate cala pouct (eual tem). Equaton (6) how that the toque evatve compoe of two tem: an actve tem whch epenent on the voltage pace vecto an a eual tem whch a lnea functon of elvee toque an oto pee. he actve tem wll be nfluence by the tato flux contolle. he eual tem epen only on the opeaton pont on the toque/pee plane, an then uppote to vay n a vey we ange, epenng on the eman of the mechancal poce. he conventonal fel oente appoach an ect toque contol on't take full avantage an capablty of evo ve wth nucton moto, becaue they both on't cone the toque evatve (6) n the contol cheme. he popoe lng moe toque contol cheme alo nclue the evatve (6) thu lghtly mpovng the contol of nucton moto. It known that the value of oto flux neee fo the mplementaton of toque o pee contol. Unfotunately, t cannot be meaue ectly. If the oto pee avalable, the oto flux can be etmate wth a mple obeve explotng the wellknown cuent moel. Howeve, f no nfomaton about the mechancal vaable acque, the egn of the obeve not a tval poblem. Ou objectve n th pape to egn a toque tackng contol fo the IM gven by (1)-(5) wthout the meauement of mechancal vaable. heefoe, we have the followng poblem tatement: egn a flux/pee obeve to etmate the flux an pee multaneouly bae on the meauement of the tato cuent an voltage an then egn a coeponng contolle to guaantee that the eal toque tack the ee one. 3. PROPOSED CONROL SYSEM he contol objectve n th ecton to egn a toque tackng contolle fo the electomechancal ytem gven by (1)-(6). Specfcally, bae on the oto flux an pee etmaton tategy ecbe n th ecton, we egn a coeponng lng moe toque contolle to guaantee the aymptotc tablty of the lng moe obeve an toque tackng m contolle. he atonal goal of the contol ctate by technologcal equement to make the flux tack the comman flux nput. 3.1 oque contol Dee value ef. val. q/ab σ ab - C () Θ ω Roto flux obeve DSP u Conv. IM Fg.1. Popoe contol cheme wth the nclue cloe-loop obeve he popoe contol ytem ummaze n the cheme on Fg. 1. he ee tato flux component n a oto flux efeence fame =, q ae calculate fom the toque an oto flux comman L L = σ, (7) Lm R t 3 σ L L e q =, (8) plm an aftewa tanfome nto a tato efeence fame. he tanfomaton angle θ the phae angle of the etmate oto flux vecto n a tato efeence fame an can be efne a θ actan ( / = b a). (9) By compang the efeence value of the tato flux pace vecto wth the etmate one, the eo n the tato flux pace vecto ( σ = ) ealy obtane. he knowlege of th eo allow the etemnaton of the appopate voltage pace vecto u, whch the PWM nvete ha to apply to IM ung next amplng peo. he ee voltage pace vecto of PWM nvete calculate wth the ue of contnuou lng moe contol law n cete fom (Sabanovc, et al., 1994; Roc, et al., ) an wll be explane n next ecton. 3. Chatteng fee lng moe contol he VSS theoy (Sabanovc, 1993; Sabanovc, et al., 1994; Utkn, 1981) ha been apple to nonlnea pocee. One of the man featue of th appoach that one only nee to ve the eo to a "wtchng uface", afte whch the ytem n "lng moe" an wll not be affecte by any moelng uncetante an/o tubance. Let u cone the plant x = f( x, t) B( x) u, (1) n m wth ank( B) = m, x, u. In VSS contol, the goal to keep the ytem moton on the manfol S, whch efne a

3 { : (, t) } S = x σ x =. (11) he oluton to acheve th goal can be calculate fom the equement that σ( x, t ) = table. he contol houl be choen uch that the canate Lyapunov functon atfe the Lyapunov tablty ctea. he am to foce the ytem tate to the lng uface efne by σ = G( x x ). (1) Ftly, a canate Lyapunov functon electe a σ σ v = > an v = σ σ <. (13) It ame that the evatve of the Lyapunov functon negatve efnte. h can be aue f we can omehow make ue that v = σ Dσ <, (14) D alway potve efnte. Fom (13) an (14) σ = Dσ (15) can be wtten. Equalzng (15) to zeo eult n what known a "equvalent contol". In othe wo, the contol that make the evatve of the lng functon equal to zeo calle equvalent contol. Devatve of (1) Gx G( f( x, t) Bueq ) =. (16) A a eult, the equvalent contol can be wtten n the followng fom: ueq = ( GB) G( f( x, t) x ). (17) Fom evatve of (1) an ung (17) σ ( )( eq ) t = GB u u (18) obtane. hen, anothe equaton fo equvalent contol can be wtten a gven below. σ ueq () t = u() t ( GB ) (19) t By ung the efnton gven by (1) an (1) n (15) G( x x ) = G( x f( x, t) B( x) u) = Dσ () the contol obtane a: u= ( GB( x)) ( G( x f( x, t) ) Dσ ). (1) Ung (11) fo the equvalent contol can be wtten a: u() t = ueq () t ( GB) Dσ. () By lookng at (19) etmaton fo u eq can be mae ung the popety that u( t) contnuou an can not change too much n a hot tme a gven below: 1 () ( ) ( ) σ ueq t = u t δ t GB, (3) t whee δt a hot elay tme. h etmaton alo content wth the logc that u eq electe a the aveage of u. By puttng (3) nto (), we get the lat fom fo the contolle u() t = u( t δ t) ( GB) ( Dσ σ ). (4) By ung Eule ntepolaton ( GB) u() t = u( t δt) (( Dδt 1 ) σ() t σ ( t δt) ). δ t (5) he next tep n contolle egn the cetzaton of (5) nto the tate equaton on the aumpton that the voltage vecto u ( k) an u ( k ) ae contant wthn mall k-th an (k1)-th amplng nteval. he tato flux contol poblem can be fomulate a follow: let u efne the nput voltage vecto n (1) o that each eo of the vecto σ = aymptotcally appoache zeo a quckly a poble at vayng loa paamete an opeatng conton. Fo the elatvely mall amplng tme acheve n avance PWM contolle nvete, the magntue of upply can be aume to eman contant, o the new value of the voltage vecto can be pecte fom the pevou value by otaton n the plane: u ( k) = C( θ) u ( k ); co θ -n θ. (6) C( θ ) = n θ co θ Dee voltage u calculate a n cete tme expeon: u ( k 1) = C( θ) u ( k) ( GB). (7) (( I D) C( θ) σ( k) C( θ) σ( k )) Ou ytem wll each the lng moe manfol S n nfnte tme, but we can efne the ε vcnty of the manfol, whch eache n fnte tme. he matce C( θ), C ( θ) ae ue n algothm fo compenaton of computatonal elay, namely the tato flux eo σ ( k), σ ( k ) ae meaue an compute fom actual an efeence tato flux n pevou amplng tme one tep ( θ ) an two tep ( θ ) backwa. he tablty conton of algothm fulflle f D electe o that all egenvalue of (I-D) ae wthn the unty ccle of z-plane. Snce n aton to the tato flux contol eo evatve futhe employe to etmate the changng tenency of flux eo, the wtchng fequency of PWM nvete euce. he ac phae voltage of PWM nvete unquely etemne fom u ( k 1 ) ung the mple coonate tanfomaton: u13 = Kab u, K ab = / 3/. (8) / 3/ 4. SLIDING MODE FLUX AND SPEED OBSERVER In the obeve egn utable fo the enole ve the obeve contol nput houl be a known functon of the moto pee o that, afte etablhng lng moe n toque tackng loop, the pee can be etemne a unque oluton. h lea to the followng electon of the tuctue of the tato cuent obeve wth the combnaton of voltage an cuent moel

4 1 L m Lm R = u R e (9) t σl L L σl R wth e = jpω. σ L Etmaton eo etemne a ε 1 Lm Lm R ( = εe Rε ), t σl L L σl wth ε ε =, t ε e e. e = (3) = ( ) t, an Slng moe algothm (5) coul be ue to calculate contol, whch ha meanng of electomotve foce σ L Lm e ( ) k = e( k 1) (( I D) ε( k) ε( k ) ) L (31) Roto flux obeve coul be electe fom () havng the tuctue of flux moel wth atonal convegence tem Lm R = e. (3) t L σ L Now flux etmaton eo can be calculate a ( ε ) Lm = = εe R ε.(33) t t L o enue convegence coul be electe n the fom = K ε. (34) Degn paamete K coul be electe fom (34) o that etmate oto flux ten to t eal value. o analyze convegence of the etmate to eal value fo the popoe obeve tuctue, we ft nee to analyze the oto flux popety. he oto flux eo yel fom (33) εα a b µ ; ε = b b a η, (35) x x ωω x ω ωx µ = ; η = ω x ω x an ω 1 b a ea x = a b e, (36) b ω σ L L a b ε x = Lm b D ε a t. (37) If lng moe n cuent contol (3) ext both ω = an x =, then µ = an η = an conequently ε =. x ha a meanng of x /( = R σ L) an etmate nvee value of leakage oto tme contant. u ^ L m L R ( ) R σl 1-1 σl - - L m L ^ e C e () Fg.. Cloe-loop EMF ( ê ) obeve. In ou popoe ytem hown n Fg., the EMF etmate by (31). he oto flux etmate by (3), whee the etmate tato flux ( ) eplace by the ee tato flux. he oto pee an oto leakage tme contant ae etmate by (36). he popoe lng moe EMF obeve hown n Fg. obut agant moto paamete vaaton, uch a tato etance themal vaaton an atuaton of nuctance paamete. he ynchonou pee can be etmate by the evatve of tanfomaton angle θ ba ab ω = =. (38) t a b Wth the etmate oto pee (36) the actual lp pee wll be ω = ω ω. (39) l he tato an oto etance ae alo of geat mpotance to ytem tablty of evo ve. heefoe, an entfcaton algothm ung the nfomaton on EMF (31) an etmate oto flux wth (36) can be ue fo oto etance etmaton: aea beb R L = σ. (4) Stato an oto etance ae explctly expee wth (6). Fo nteval of opeaton of nucton moto wth contant toque, the evatve of toque wll be zeo. Fom that the entfcaton algothm fo etance can be obtane: u pω R R = σ L. (41) σ L In the popoe lng moe contol cheme of the nucton moto only the nuctance σ L an L / L eman contant value. m 5. FLUX CONROL FOR EFFICIENCY IMPROVEMEN oque contol kllfully obtane by contollng the ntantaneou lp fequency, wheea mpovement of the effcency can be acheve by contollng the ampltue of lke a felweakenng opeaton. In the cae of the fequent change of the toque comman, t neceay to keep the flux ampltue at maxmum. Une the fel-weakenng tate, the uffcent toque o quck toque epone woul not be expecte becaue of the low epone of the flux nceae. On the othe han, n the teay-tate opeaton, epecally at mall loa, the maxmum effcency can be obtane at a low flux level. heefoe, n oe to obtan the ^ ε

5 maxmum effcency the flux level ajute n accoance wth the toque comman. he maxmal allowe oto flux woul epen on actual tato fequency an c-lnk voltage: / 3 Uc R max =, (4) ω an toque epenent effcency optmal oto flux wll be (Stefank an Kay, 1996): e R R opt =, (43) p 4 Lm R ω R R gl Rgl the etance of on loe an houl be etemne by expemental entfcaton. he actual oto flux comman wll be ue. (44) opt max 6. EXPERIMENAL RESULS PC - efeence an paamete ettng, ata acquton, oftwae evelopment ] m - [N -Ț e < e, t [] Fg. 4. Locke oto - ee, etmate, an actual value of IM toque e. /]- [a ,ω ṗ -1 / < ω - DC lnk -IM Contolle-IM (DSP) Invete-IM SM/ωSM cuent meauement out Contolle/ Invete-BLM DC lnk -SM t [] Fg. 5. Locke oto - etmate value of tato fequency an actual value of pee ω Dve Machne (IM) Fg.3. Expemental ytem. e L Loa Machne (BLM) Expement wee cae out to vefy the feablty of the evelope contol cheme ung the 1.5 kw thee-phae IM. Contolle bae on exa Intument DSP MS3C31 an the PWM nvete bult wth IGB moule pouce by Semcon. he contolle ha the agnotc featue that ae neceay fo ve ntallaton tet poblem etecton an elmnaton. he contol algothm calculate evey 16 µ. Buhle AC evomoto mechancal connecte wth the IM une tet wa ue a the Loa Machne. he contol of both, pee an apple toque, poble, thu hawae-n-the-loop opeaton can be pefome (Fg. 3). In the peente wok the apple toque of the Loa Machne meaue an thu the value of the toque apple by the Dve Machne obtane. Dee, actual an obeve value of toque ae featue on Fg. 4, wheea Fg. 5 how actual oto pee an etmate tato fequency. Fom the fgue the pefomance of the algothm n the low pee aea can be evaluate. Some poblem occu ung the tanton though the tato fequency zeo, howeve they ae eolve by the peente contol cheme, povng t convegence. oque tackng pefomance of the ytem atfactoy even at vey low pee nclung pee zeo..., e --, L - [Nm] ^ t [] Fg. 6. Step loa - ee an etmate value of IM toque e an loa toque L. ω/p..,ω - [a/] ^ t[] Fg. 7. Step loa - etmate value of tato fequency an actual value of pee.

6 In the peente wok the apple toque of the Loa Machne meaue an thu the value of the toque apple by the Dve Machne obtane. Dee, actual an obeve value of toque ae featue on Fg. 4, wheea Fg. 5 how actual oto pee an etmate tato fequency. Fom the fgue the pefomance of the algothm n the low pee aea can be evaluate. Some poblem occu ung the tanton though the tato fequency zeo, howeve they ae eolve by the peente contol cheme, povng t convegence. oque tackng pefomance of the ytem atfactoy even at vey low pee nclung pee zeo. Secon tet nvetgate the pefomance at the loa toque changng wth nealy tep functon epenant on pee, a peente n (45): gn( ) L = c3 ω. (45) he eult ae peente n Fg. 6 an Fg. 7. Fg. 6 featue the ee an actual value of apple toque e togethe wth the loa toque L. Agan actual toque not peente. he actual pee meaue on the oto ax togethe wth the etmate value of tato fequency peente n Fg. 7. By th expement t poven that the hape of loa toque ha no effect on the toque tackng pefomance of the peente contol algothm. 7. CONCLUSION Senole toque an flux contol of an IM an emegng new technology, though n the ealy tate of evelopment. A lng moe algothm fo toque an flux tackng contol peente n the pape. he algothm ame to olve pactcal hghly nonlnea poblem of the opeaton of IM at hgh an low pee nclung zeo pee. h algothm epecally utable fo the applcaton whee ee toque an oto flux ae vayng ung the opeaton. Fo example when effcency of the IM opeaton an mpotant ue, lke n electc vehcle. Smulaton an expement vefe the pefomance of the popoe algothm. 8. REFERENCES Boe, B.K. an M.G Smoe (1995). Spee enole hyb vecto contolle nucton moto ve. In: Conf. Rec. IEEE-IAS Ann. Meetng, pp Holtz, J., (1996). Metho fo pee enole contol of ac ve. In: Senole contol of ac moto, e. K. Rajahekaa, IEEE Pe Book. Kubota, K.M.H. an. Nakano, (1993). DSP-bae pee aaptve flux obeve of nucton moto. IEEE an. In. Appl. 9, pp Roc, M., K. Jezenk, an A. Sabanovc, (). Spee enole lng moe toque contol of nucton moto. In Conf. Rec. IEEE-IAS Ann. Meetng, pp Sabanovc, A., (1993). Slng moe n obotc manpulato contol ytem. Electotechncal Revew, Ljubljana (SLO), 6, No.-3, pp Sabanovc, A., K. Jezenk, N. Sabanovc, an K. Waa, (1994). Chatteng fee lng moe. In: Poc. Wokhop on the Robut Contol va Lyapunov echnque - VSL'94, Benevento, Italy, pp Stefank,., an S. Kay, (1996). Lo mnmaton contol of nucton moto ve fo electc vehcle. In Poc. IEEE Int. Symp. on Inutal Electonc, ISIE'96, Waaw, Polan, pp Utkn, V.I. (1981). Slng moe n contol optmzaton. Spnge-Velag. Utkn, V.I. (1993). Slng moe contol egn pncple an applcaton to electc ve. IEEE an. In. Elec., 4, pp Va, P., (1998). Senole vecto an ect toque contol, Oxfo Unvety Pe.