Torque Ripple Minimization in Switched Reluctance Motor Using Passivity-Based Robust Adaptive Control

Transcription

1 Intrnational Journal of Elctrical and Elctronics Enginring 5: orqu Rippl Minimization in Switchd Rluctanc Motor Using Passivity-Basd Control M.M. Namazi, S.M. Saghaiannjad, and A. Rashidi Abstract In this papr by using th port-controlld Hamiltonian (PCH) systms thory, a full-ordr nonlinar controlld modl is first dvlopd. hn a nonlinar passivity-basd robust adaptiv control (PBRAC) of switchd rluctanc motor in th prsnc of xtrnal disturbancs for th purpos of torqu rippl rduction and charactristic improvmnt is prsntd. h proposd controllr dsign is sparatd into th innr loop and th outr loop controllr. In th innr loop, passivity-basd control is mployd by using nrgy shaping tchniqus to produc th propr switching function. h outr loop control is mployd by robust adaptiv controllr to dtrmin th appropriat orqu command. It can also ovrcom th inhrnt nonlinar charactristics of th systm and ma th whol systm robust to uncrtaintis and boundd disturbancs. A 4KW 8/6 SRM with xprimntal charactristics that tas magntic saturation into account is modld, simulation rsults show that th proposd schm has good prformanc and practical application prospcts. Kywords Switchd Rluctanc Motor, Port Hamiltonian Systm, Passivity-Basd Control, orqu Rippl Minimization I I. INRODUCION N rcnt yars, thr is a growing concrn about th us of switchd rluctanc motor. h major rasons for SRM ar robustnss, high fficincy, low cost, high spd, simpl structur, asy to maintain, high controllability, high torqu to inrtia ratio, simpl powr convrtr circuits with rducd numbr of switchs and smallr dimnsion of th motor in comparison to th othr motors. High torqu rippl and acoustic nois ar most disadvantags of th motor. Main rason of ths problms is th stpping natur and inhrnt nonlinar charactristics of th motor, which causs an undsird ffct on baring. If th problms of th SRM can b solvd, it can b an altrnativ to th othr motors []. M.M. Namazi is with th Dpartmnt of Elctrical and Computr Enginring, Isfahan Univrsity of chnology, Isfahan, Iran (-mail: mm.namazisfahani@ c.iut.ac.ir). S.M. Saghaiannjad, is with th Dpartmnt of Elctrical and Computr Enginring, Isfahan Univrsity of chnology, Isfahan, Iran (-mail: saghaian@cc.iut.ac.ir). A. Rashidi is with th Dpartmnt of Elctrical and Computr Enginring, Isfahan Univrsity of chnology, Isfahan, Iran (-mail: 35.amir@gmail.com). h nonlinar charactristics includ th nonlinar torqu function of position and currnt and th magntic saturation at crtain opration rgions. hr ar two approachs followd for torqu rippl minimization. On is improv th magntic dsign of th motor whil th othr is us of advancd control tchniqus. Svral control mthods and schms hav bn proposd to ovrcom ths problms. For xampl, variabl structur controllr mad th SRM driv systm insnsitiv to paramtr variations and load disturbanc []. Artificial nural ntwor and fuzzy controllr nds a lot of dsignr xprinc [3]. Fdbac linarization controllr has high snsitivity to modl uncrtaintis [4]. Nonlinar intrnal modl control for SRM driv rquird vry complicatd computations and implmntation of th systm is vry difficult [5]. In th mthods mntiond abov, for th control stratgis of th SRM it is assumd that its paramtrs ar nown xactly or th unnown paramtrs can b idntifid by th adaptiv tchniqu. Howvr th paramtrs of th SRM ar not xactly nown and always vary with currnt and position. Actually, control is difficult to implmnt owing to its complx algorithm whn considring th structural information of SRM in dsign. Improving th applicability of th SRM on th basis of th simplifid modl and taing th structural charactristics into account is a significant stp in dsigning th controllr of th SRM. In this papr, a nonlinar fdbac controllr, which ffctivly us th natural nrgy dissipation proprtis of th SRM is proposd. Passivity basd Control (PBC), introducd in [6], to dfin a controllr dsign mthodology which achivs stabilization by passivation [7]. Papr is organizd as follows. At first, in Sction II, th SRM nonlinar modling is prsntd. In Sction III th concpts of Port Controlld Hamiltonian Systm and th thory of PCH modling of SRM ar dscribd. In Sction IV a passivity-basd robust adaptiv controllr is dsignd basd on combination of passivity-basd currnt control (PBC) and robust adaptiv tchniqu. In Sction V, simulation rsults obtaind confirm that th torqu rippl rducd in spit of xtrnal load disturbancs. Finally, conclusions ar givn in Sction VI. II. NONLINEAR CONSRUCION AND MODELING OF SRM A. SRM Configuration and Basic Principl Opration h switchd rluctanc motor has a simpl dsign with a 7

2 Intrnational Journal of Elctrical and Elctronics Enginring 5: rotor without windings and stator with windings locatd at th pols [8]. h principl of opration is basd on th tndncy of an lctromagntic systm to obtain a stabl quilibrium position minimizing magntic rluctanc. Whnvr diamtrically opposit stator pols ar xcitd, th closst rotor pols ar attractd, rsulting in torqu production. B. Nonlinar Mathmatical Modl of SRM An important stp in any control systm dsign is to dvlop a good mathmatical modl, which rprsnts th plant undr various oprating conditions. h basic sts of diffrntial quations usd to dynamic modling of th SRM ar as follows. First, th lctrical stat quations of th SRM can b xprssd as []: λ ( θ, ) d λ ( θ, ) dθ v = R + + () i θ Whr u is th phas voltag, i is th phas currnt, r is th phas rsistanc, = a, b, c, d is th activ phas and λ ( θ, ) is th flux linag. h flux linag in an activ phas is givn by product of slf inductanc and th instantanous phas currnt as: λ ( θ, ) = L ( θ, ) () Whil th simplifid modl for SRM is obtaind by substitution of () into (), which rsults in: di dl( θ, i ) L( θ, ) = v r ω (3) dθ Scond, th mchanical stat quation of th SRM can b xprssd as follows: dω d = θ i Bω j = a (, ) L (4) H & ( ) = [ J ( ) R( )] + g( ) u H ( ) y = h( ) = g ( ) Whr g( ) dnots th input matrix, u and y dnot th systm input and output. h intrconnction structur is capturd in th matrix J ( ) with J ( ) = J( ). h dissipation ffcts ar capturd by th matrix R ( ) which is a smi-positiv dfinit or positiv dfinit matrix. h total nrgy of th systm is dfind by th positiv dfinit function H ( ). h port controlld Hamiltonian systms with dissipation (6) satisfy th following powr-balanc quation: dh ( ) H ( ) H ( ) = u y R( ) h port-controlld Hamiltonian systm is passiv if th Hamiltonian H ( ) is boundd from blow []. o simplify controllr dsign, a complx systm can b dcomposd into subsystms by using som tchniqus of ordr rduction, such as multi-tim-scal analysis. Considr two subsystms in th form of (7) with ngativ fdbac intrconnction by th powr port p as shown in Fig. []. Aftr th intrconnction, th closd-loop rprsntd as: H x& J R x = x& J R H x g( u py ) + g ( u + p y) (6) (7) (8) Whr j and B ar th momnt of inrtia and viscous friction cofficint, rspctivly and L is th load torqu. Also, th instantanous torqu ( θ, i ) can b writtn as: dl ( θ, ) ( θ, ) = (5) dθ III. PCH MODELING OF SRM Enrgy consrving dynamical systms with indpndnt storag lmnts can b dscribd by th form of port controlld Hamiltonian (PCH) systms as follows [9]: Fig. Ngativ fdbac intrconnction of subsystms For th closd-loop modl, th nrgy function is th total nrgy of th two subsystms; H ( ) = H( ) + H ( ) thrfor, Using (7) and suppos that y py = y p y, th drivativ of this nrgy function along tim is givn by: 8

3 Intrnational Journal of Elctrical and Elctronics Enginring 5: dh H H H H = R R + u y + u y hrfor, th stability of th ovrall closd-loop systm can b guarantd suppos both of th two subsystms ar stabl. In [7] it is prov that th complt modl of a SRM can b dcomposd as th fdbac intrconnction of th following two passiv lind subsystms (lctrical and mchanical passiv subsystm): lctrical subsystm : mchanical subsystm : v & a θ ( ) a & θ L (9) () For th intrconnctd systm, using (), (6) and a = [ x x x3 x4 ] = [ ia ib ic id ] as th stat vctor, th lctrical subsystm can b rprsntd as: D & = ) = g Whr: [ J ( x R ] + g ( v pg ) j + ξ D = diag( La, Lb, Lc, Ld ), R = diag( Ra, Rb, Rc, Rd ) g = diag(,,, ), u = u u u u J =, [ ] a b c d dl = dl ξ ω, p = diag dθ dθ () Hr g dnots th input matrix, u th control vctor and ξ th disturbanc. a = ω as th stat variabl for mchanical subsystm, th dynamic proprty dscribd as: D & = ω = g Substituting [ J ( x R ] ) dl p = diag dθ writtn as (4). + g p D = j, R = B, = L g + ξ (), = [ ] g, ξ and J () can b = IV. PASSIVIY-BASED ROBUS ADAPIVE CONROL In this sction, w prsnt a PBRAC mthod for dsigning SRM controllr. h PBRAC bloc diagram, shown in Fig. 3, is sparatd into th innr loop and th outr loop controllr. Basd on th assumption that stator currnt i as wll as rotor spd ω ar availabl for masurmnt, th controllr dsign procdurs can b dividd into thr stps. h first stp is Passivity-basd currnt control of th lctrical subsystm by injcting a nonlinar lctrical damping trm. hn a st of rfrnc currnt vctors i ar found out to achiv currnt tracing. h scond stp is aimd at rfrnc currnt gnration. h final stp is to dsign a spd controllr for spd tracing of th ovrall systm. h sa of th outr loop is to gnrat th appropriat orqu command fd for th innr loop. Finally, passivity-basd innr loop currnt controllr will produc th switching functions. A. Passivity-Basd Innr Loop Control Dsign Passivity-basd control aimd at achiving th signal rgulation is obtaind by controlling th suitabl nrgy of th closd-loop systm, and add th rquird damping. h dynamic modl () can b rwrittn as: ( v pg ) + & + R = j ξ (3) D h objctiv of PBC dsign is to guarant tracing of signal i toward to its dsird stat i []. his mthod proposs to ma a copy of (5), whr th stat is rplacd by th dsird stat and th injction damping trm is addd, that is a dsird systm: ~ ( v pg ) +ξ & + R K = j (4) D Whr th injction damping trm is K = diag{,, 3, 4}, rfrrd as damping injction matrix, is a positiv dfinit diagonal matrix and ~ =. Subtracting (4) from (3) yilds th following rror dynamic modl: ~ & ~ ~ ~ D + R + K + pg = (5) hus, w can dfin dsird storag nrgy function ~ ~ H d =.5 D as a Lyapunov function. h dsird quilibrium point is ralizd if th Hamiltonian H has its minimum at th quilibrium point. Asymptotic stability is provd using LaSall s invarianc principl for th closdloop systm (5) whr: d ~ ~ H d ( R + K) = (6) Following LaSall s thorm, th tim drivativ of storag nrgy function is forcd to b ngativ smidfinit, i.. H & d, bing qual to zro only for th quilibrium 9

4 Intrnational Journal of Elctrical and Elctronics Enginring 5: points. hrfor (6) is qual to zro only whn ~ =. h Lyapunov stability and convrgnc can b provn that (3) is passiv and input-output stabl. Finally, th abov mthod procds to rstrict = and solv from (4), which yilds th switching functions as: d L v = R + L + ω θ (7) L + ω K ( ) θ h rfrnc currnt is calculatd by using th dsird torqu: = (8) dl ( θ, i ) dθ Whr dl ( θ, ) dθ is obtaind by using th loo-up tabl that shown in Fig. A. Outr Loop Spd Controllr Aftr innr-loop controllr gnrats suitabl rfrnc currnt, th final stp is to dsign th outr-loop spd controllr to dtrmin th appropriat rfrnc torqu. For spd control dsign purposs, th dynamic modl of th SRM can b writtn as: dω = { L Bω} = β j d dα λ λ = v R ω J i = a θ d + ω u Bα J θ = a =, Lt ( ) ( ) (9) Whr u & L x = x x = β ω, Also lt th output of th SRM b y = ω. hus, th modl of th SRM systm can b writtn in compact form as & y = f ( x) + g( x) u, whr f and g ar dtrmind from (9). W considr { α } () t Ω = () t () t u α u : u as an xtrnal disturbanc and control law as: Partial Drivativ Of Flux to Position Currnt (Ampr) Rotor Position (Dgr) Fig. Partial drivativ of flux to position + ˆ α g ( f & ω ) u = K + g ( f & ω ) + δ xp ˆ α + ( δ / p)xp ( σ t) ( σ t) () Whr K >, is th stat fdbac gain and αˆ dnots th stimatd valu of th unnown paramtrα. h constants δ i andσ i i =, ar (small) positiv constants spcifid by th dsignr to avoid discontinuity and control chattring. h control law () with adaptation mchanism &ˆ α = γ, whr γ > is adaptation gain, guarants that all th closd loop signals ar boundd and tracing rror = ω ω is asymptotically convrgd to zro. Clarly, th smallr valu of δ, giv lss smoothnss to th control law [3], [4]. i V. SIMULAION RESULS h SRM systm is simulatd using th MALAB softwar. h modl tas magntic saturation into account. h paramtrs of th motor usd for th simulation studis ar givn in abl. h motor is commandd to acclrat from rst to RPM undr a suddn load chang from 3 Nm to 6 Nm. Fig. 4 shows spd rspons whn corrsponding to robust adaptiv spd control law dscribd by () and th proposd PBRAC algorithm for th prcis plant modl. It can b sn that thr is a stady rror on th rspons of robust adaptiv spd control without using of PBC, whras thr is no stady rror for th proposd algorithm. Also, transint rspons of th proposd PBRAC du to an xtrnal disturbanc quicly dampd. Hnc, thr is a considrabl improvmnt of th stady stat and transint rsponss. h proposd controllr is applid to implmnt th torqu rippl minimization. h torqu rippl is dfind as [5]:

5 Intrnational Journal of Elctrical and Elctronics Enginring 5: Fig. 3 Bloc diagram of th ovrall spd control systm ( max) ( min) inst inst orqu Rippl = () avg orqu profils of th convntional robust adaptiv and proposd PBRAC ar bing displayd in Fig. 5. hr ar significant amount of torqu rippls whn only th convntional robust adaptiv is usd. But, th instantanous torqu of th proposd PBRAC is highly improvd with much lowr torqu rippls. As can b sn from abl, whn th load torqu is 3 Nm, th prcntag pa-to-pa torqu rippl of robust adaptiv passivity-basd control rducs.94% compard with that of th robust adaptiv control. Whn th load torqu is 6 Nm, th torqu rippl factor rducs.3%. Paramtrs ABLE I PARAMEERS OF HE SRM Rating valus Output Powr 4 KW Ratd Spd 5 RPM Numbr of Stator Pols 8 Inrtia(J).8 N.ms Damping Factor(B).78 N. ms Phas Rsistanc.75 Ω Dc Voltag Supply 8 V ABLE II ORQUE RIPPLE PERCENAGE Load orqu Control stratgy 3 Nm 6 Nm Convntional 6.7 %.8% 3.76 %.5% Spd (RPM ) Spd (RPM ) Spd (RPM) Rfrnc im (Scond) (a) Rfrnc Spd im (Scond) (b) Rfrnc Spd im (Scond) (c) Fig. 4 (a) h spd profil upon th variation of load torqu at t=.5 s. (b) and (c) ar zoom of (a)

6 Intrnational Journal of Elctrical and Elctronics Enginring 5: orqu (Nm) orqu (Nm) orqu (Nm) im (Scond) (a) im (Scond) (b) im (Scond) (c) Fig. 5 (a) h SRM producd torqu profil. (b) and (c) ar zoom of (a) [3] A. D. Cho, and Z. Wang, Fuzzy Logic Rotor Position Estimation Basd Switchd Rluctanc Motor DSP Driv With Accuracy Enhancmnt, IEEE rans. on Powr Elctronics, vol., no. 4, pp. 98-9, July 5. [4] S. K. Panda and P. K. Dash, Application of nonlinar control to switchd rluctanc motors: A fdbac linarization approach, IEE Proc.-Elctric Powr App.s, vo. 43, no. 5, pp , Spt [5] B. G,. Wang, P. Su, and J. Jiang, Nonlinar intrnal-modl control for switchd rluctanc drivs, IEEE rans. on Powr Elctronics, vol. 7, no. 3, pp , May. [6] R. Ortga, A. van dr Schaft, I. Marls, and B. Masch, Putting nrgy bac in control, IEEE Contr. Syst. Mag., vol., no., pp. 8 33, Apr.. [7] G. Espinosa-P rz, P. Maya-Ortiz, M. Vlasco-Villa, and H. Sira- Ramírz, Passivity-basd control of switchd rluctanc motors with nonlinar magntic circuits, IEEE rans. On Control Systm chnology, Vol., no. 3, pp , May 4. [8] V. Ramanarayanan and D.Panda, Mutual Coupling and Its Effct on Stady-Stat Prformanc and Position Estimation of Evn and Odd Numbr Phas Switchd Rluctanc Motor Driv, IEEE rans. On Magntics, vol. 43, no. 8, pp , August 7. [9] S. Stramigioli and H. Bruynincx, Gomtric Ntwor Modling and Control of Complx Physical Systms. Brlin, Grmany: Springr- Vrlag, 8. [] R. Ortga, A. van dr Schaft, B. Masch, and G. Escobar, Intrconnction and damping assignmnt passivity-basd control of port-controlld hamiltonian systms, Automatica, vol. 38, no. 4, pp ,. [] J.E. Slotin and W. Li, Applid Nonlinar Control, Prntic Hall,. [] R. Ortga, A. Loria, P. J. Niclasson, and H. Sira-Ramirz, Passivitybasd Control of Eulr-Lagrang Systms. Mchanical Elctrical and Elctromchanical Applications: Springr-Vrlag, 998. [3] M. M. Namazi, S. M. Saghaiannjad, and A. Rashidi, Spd control of Switchd Rluctanc Motor Considring orqu Rippl Minimization, 8th Annual Conf. Elctric. Eng. Iran, pp , May. [4] H.R. Koofigar, S. Hossinnia, and F. Shholslam, Robust adaptiv nonlinar control for uncrtain control-affin systms and its applications, Nonlinar Dynamics, Springr Scinc Publishing, vol. 56, no., pp. 3-, Jun 8. [5] Iqbal Husain, Minimization of orqu rippl in SRM drivs, IEEE rans. on Industrial Elctronics, vol. 49, no., pp. 8-39, Fb.. VI. CONCLUSION In this papr, a robust adaptiv PBC algorithm is proposd for torqu rippl rduction and prcis spd tracing. By dcomposing th SRM driving systm into an lctrical subsystm and a mchanical subsystm, th ordr of th systm is rducd. h passivity-basd controllr is dsignd for th lctrical subsystm basd on th two-tim-scal and passivity proprtis. Bcaus of th strict passivity of th two subsystms and thir ngativ fdbac intrconnction, stability of th proposd controllr for th whol driving systm can b achivd. h rsults dmonstrat that th proposd robust adaptiv PBC for SRM posssss xcllnt prformanc and torqu rippl minimization compard with convntional robust adaptiv control. REFERENCES [] R. Krishnan, Switchd Rluctanc Motor Drivs, Boca Raton, FL:CRC Prss,. []. S. Chuang, and C. Polloc, Robust spd control of a switchd rluctanc vctor driv using variabl structur approach, IEEE rans. on Industrial Elctronics, vol. 44, no. 6, pp. 8 88, Dc. 997.