High Gain Observer for Induction Motor in Presence of Magnetic Hysteresis

Transcription

1 Preprint of the 8th IFAC World Congre Milano (Italy Augut 8 - September, High Gain Oberver for Indution Motor in Preene of Magneti Hyterei H. Ouadi, F. Giri,. Dugard, Ph-Dorléan, A. Elfadili 4, J.F.Maieu MACS ab, Univerity Haan II, faulty of iene, Caablana, Moroo GREYC ab, UMR CNRS, Univerity of Caen, Caen, Frane GIPSA ab, UMR CNRS, INPG, Grenoble, Frane EMI, Univerity of Rabat, Moroo (Correponding author: fouad.giri@uniaen.fr Abtrat: he problem of tate etimation in indution motor i onidered in thi paper. In mot previou work, motor oberver have been deigned uing tandard model, negleting the aturation and hytereti effet in the mahine magneti iruit. A a matter of fat, thee effet annot be ignored epeially when onidering (peed, torque ontrol trategie involving wide range flu variation, neeary to meet optimal operation ondition when faing ignifiant load torque hange. In thi paper, a high gain oberver i deigned for indution motor baed on a model that aount for the nonlinear hytereti nature of the magneti iruit. he oberver provide etimate of the magnetizing urrent and magneti variable uing available meaurement i.e. tator urrent and voltage, rotor peed. he oberver i formally hown to enjoy nie onvergene propertie. he theoretial reult i onfirmed by imulation.. INRODUCION Controlling indution mahine require meaurement of eletromagneti and mehanial variable (voltage, urrent, flu, peed, poition, et. For ome variable (e.g. tator voltage and urrent, there eit reliable and not too epenive enor providing uffiiently aurate meaure. hi i not the ae with other variable uh a the rotor flu. hen, oberver are reorted to get on-line etimate of all variable whih are not aeible to meaurement. he firt oberver (ee e.g. ubineau et al, 999 were developed baed on implified aumption namely, linear magneti harateriti and ontant (or lowly varying rotor peed. Under thee aumption, the model of the indution motor beome linear and, therefore, obervability analyi and oberver deign may be dealt with, uing tandard linear theory tool (e.g. pole plaement and optimal deign, uenberger and Kalman oberver. Intereting ontribution ame later, propoing nonlinear oberver developed without uppoing a ontant rotor peed (e.g. De eon et al,. he propoed oberver have been deigned uing different approahe uh a high gain, liding mode and dynami tate feedbak. However, even in thee ontribution, the harateriti of the mahine magneti iruit are till uppoed to be linear. he point i that thi aumption i only valid when the mahine remain all time in the viinity of it nominal flu value. But, a ontant-flu operation mode annot be optimal when wide range peed variation are needed with a mahine faing large load torque hange. o ahieve high obervation performane, regardle the mahine operation mode, the oberver deign hould be baed on a model that aount for the nonlinearity of the mahine magneti iruit. hi nonlinearity i haraterized both by aturation and hyterei. Few previou work have attempted to deal with the mahine magneti iruit nonlinearity in oberver deign (Krzeminki et al., 99, (Ouadi et al., 5. Furthermore, the model baed upon in thee work only aounted for the magneti iruit aturation (negleting the hytereti effe. In thi paper, a new tate oberver i propoed for etimating the rotor flu (and tator urrent in indution motor faing large peed and load torque variation. It i deigned by the high gain tehnique, uing a reently developed model that properly aount for hyterei in the magneti iruit (Ouadi et al.,. In thi model, the hyterei effet i aptured through the Coleman-Hodgdon model that proved to be quite onvenient for hyterei modelling in ferromagneti material (Coleman & Hodgdon, 987; Maki et al., 99. In the oordinate, the new model turn out to be a 7 th order, there where the tandard model are 5 th order. he two new tate variable are the magnetizing urrent omponent. For the purpoe of oberver deign, a Park model tranformation i firt performed, yielding a more tratable model fitting the required truture for the high-gain oberver deign tehnique to be appliable. herefore, the obtained oberver i formally proved, under tandard aumption, to be globally aymptoti onvergent. hi reult i onfirmed by a numerial imulation involving ignifiant hange of the peed and load torque, making the mahine operate both in linear and nonlinear part of it magneti harateriti. he oberver thu developed an be ued a an element of a more global ontrol trategy eeking (in addition to peed regulation further objetive e.g. power effiieny improvement, flu optimiation, power fator orretion, et. hee annot be ahieved with oberver obtained from tandard model beaue thee are only repreentative of the mahine within a limited flu range. he paper i organied a follow: the model of the indution motor i preented in Setion ; the high-gain oberver i Copyright by the International Federation of Automati Control (IFAC 5

2 Preprint of the 8th IFAC World Congre Milano (Italy Augut 8 - September, deigned and analyed in Setion, the oberver performane are illutrated by imulation in Setion 4; a onluion and referene lit end the paper.. INDUCION MOOR MODEING A model that aount for the magneti iruit hyterei ha reently been developed and validated in (Ouadi et al.,. he hyterei phenomenon (Fig. i aptured through the Coleman-Hodgdon (C-H model (Coleman & Hodgdon, 987, (Maki et al., 99. he omplete mahine model ha been given the following tate-pae repreentation: di R r i i p u dt ( di R r i i p u dt ( dr. i. i pr dt ( dr. i. i pr dt (4 d p f r i r i dt J J J (5 di (4h h h (. i. i r dt ( h h (. i. i r ( di ( h h (. i. i r dt ( h h (. i. i r (7 where: - ( r, r, ( i, i, ( i, i and u [ u, u ] are the -omponent of the rotor flu, tator urrent, magnetizing urrent and tator voltage, repetively; - i the motor peed; - R, R r denote the tator and rotor reitane; - i the load torque; - p deignate the number of pole pair; - i the equivalent indutane (of both tator and rotor; - he h k are funtion of the magnetizing urrent: hk ign( i k f ( ik rk g ( ik (k =,, (8 - f (. and g (. are two funtion: f ( i tanh( i ai (9 i g( i f ( i be ( he five firt equation (-5 are tandard, eept for the dependene, now made epliit, on the magnetizing urrent ( i, i. he remaining equation (-7 aount for the hyterei effet in the magneti iruit. hey involve the funtion f (. and g (. whih are a part of the C-H hyterei model (Ouadi et al.,. All the parameter an be identified via peifi eperimental meaurement on real-life mahine. hi alo onern the hyterei parameter ( a, b,, that an be obtained from the mahine magneti harateriti (Fig., (Ouadi et al.,. M agnetizi ng Flu (Wb Magnetizing Current (A Fig : Hyterei urve given by C-H model.. SAE OBSERVER DESIGN. Motor model reformulation he oberver i reorted to etimate online the mahine tate variable that are not aeible to meaurement i.e. the magnetizing urrent and the flu. It will be obtained from the tranformed model (-7 uing the high gain deign tehnique, e.g. (Beançon, 7. o redue the number of equation, the Park tranformation i applied to the model (- 7. he obtained dq-oordinate model i oriented along the magnetizing urrent. Doing o, one get the following form: F( G( y C ( with: where: i i (a d q rd rq i d 4 5 u d u q, ontrol input output vetor (b u y i d i q, g ( G ( g ( R id i R ( iq i f J J q g d u u d q ( (4 5

3 Preprint of the 8th IFAC World Congre Milano (Italy Augut 8 - September, R r. i d. id ( p rq g (, u. iq ( p rd A(. id. id p rq A(. iq p rd where: A H o ( H in( H in (.5H in( H o(.5h in( H in ( H in( H o ( A A where H H H H 4h ( h h h ( h ( h h h ( h (5 F ( F ( ( p ( p F (7 p p J iq J id C (8 he notation tand for the zero matri. Note that, the above required dq-frame orientation an be enured by making a uitable hoie of the input tator frequeny. Indeed, by letting i q i q, it follow that mut atify the following ondition: ( A.( id id A. iq ( Ard Arq (9 i d In the ret of the tudy, the ondition (9 i uppoed to be fulfilled. he oberver ynthei and analyi neeitate the following aumption: A. he tate vetor and the ontrol vetor u are bounded. Remark. he above aumption are natural in the preent open-loop ontrol ontet. Furthermore, it i readily een from (7 that the funtion F ( i linear and o i iphitz.. Model tranformation Conider the following tate tranformation: : IR IR ( z ( F ( For onveniene, the tranformation i given the following ompat form: z ( ( ( with ( I F ( ( where I denote the identity matri. Remark. a Uing Aumption A., one get from ( that the new tate vetor z i alo bounded. b Alo, uing A, it follow that (. i iphitz. Furthermore, it i lear that the tranformation (. i invertible A F ( i iphitz (Remark, equation ( implie that i in turn iphitz Uing ( and (, it i readily hown that: z ( ( ( F( G( ( whih, together with (, yield: z ( F( ( ( F( ( G( (4 Uing the notation I A (5 it an be eaily heked, uing ( and (7, that: ( F( A( ( or, equivalently: F( ( A( (7 It i eaily een from ( that: I ( ( F ( Furthermore, it an be heked uing (7 that: p F ( R i r d J pid pj pr i r d (8 Remark. It i well known that, in normal operation ondition, the mahine tator urrent an never be in phae or quadrature phae with the magnetizing urrent (eonard,. Bearing in mind the (d,q-frame orientation (alo ng the urrent i, it follow that the tator urrent ompo nent are d neearily nonzero i q (9 i, i. hen, F ( d involve no ingularity whenever the mahine i rotating i.e. for a non zero rotor peed et u introdue the following notation: z z ( ( inv ( ( inv ( I Az ( ( ( inv ( G( ( Introduing (7 in (4 lead to the new tate-pae repreentation of the indution mahine: q 5

4 Preprint of the 8th IFAC World Congre Milano (Italy Augut 8 - September, z Az ( ( ( Note that thi alo aount, in the z-oordinate, for the magneti iruit hyterei. Remark 4. a From ( one get: p 5 I p with A ( A ( F ( 4 p p J 5 J 4 ( From Remark and the fat that the lat matri i linear in tate variable, it follow from ( that i iphitz. Uing (, (, (8 and (5, one obtain: p z z 4 rq ( p (4 A ( z5 z rd z p p rq z4 rd z J J 5 Referring again to Aumption A and to the onideration deribed in Remark, the funtion ( turn out to be iphitz whenever the mahine i rotating (non zero peed. Similarly, uing the fat that: z z z z. it follow from (, (4 and (5 that: I g (, u ( z, u ( A ( ( F 5 g (,, u From (4 and (5, it i readily een that A ( and g (, u are bounded, beaue u and ( z u are o. Furthermore, the funtion g ( z, and F are learly iphitz due their linearity. hen, equation (5 implie that that ( z, u i in turn iphitz. Mahine oberver in z-oordinate et be the blok diagonal matri defined by: I I diag ( where i any real poitive ontant. et S be the olution of the algebrai yapunov equation: S A S SA C C (7 where A and C are repetively given by equation (5 and (8. It i hown in many work (e.g. Beançon, 7 that the olution S of (7 i unique, ymmetri, poitive definite and atifie: I S C (8 With the above notation, a andidate oberver for the ytem ( i the following: ẑ ˆ Az ( ( S C ( Czˆ y ( inv( ( inv( I S C ( Czˆ y (9 Propoition. he tate oberver (9, when applied to the nonlinear ytem ( (where all ignal ( z, y are bounded, i globally eponen tially onvergent. More peifial ly, there eit a uh that for all, there i a o that: zˆ ( z( S ep( zˆ( z( (4 whatever zˆ ( IR where S denote the ondition number of the matri S. Furthermore, one ha: lim Proof. et e( zˆ( z(. hen, e A S C Ce ( ( (4 ( ( ( S C Ce with ( ( inv ( ( inv ( I (4 Subtituting (, ( and (8 in (4, one obtain from (4: ( (4 A ( inv ( It i eaily heked that: A A and C C C C Hene, e A S C C e ( ( z ( u, ( ( Introdue the variable hange: e e (45 It follow uing (4 that: e A S C C e ( ( S ( ( ( S C C Ce (44 Conider the yapunov funtion andidate V (e e S e. It time derivative along the trajetory of (4 i: V e Se V e e S Ce (4 Ce e S ( ( ( ( e S ( S C Ce C whih together with (8 implie: V V e ( ( e ( ( (47 e ( S C Ce (48 etting, it follow uing the triangular truture and iphitz nature of, and (Remark one get: ( ˆ ( (49 e z 54

5 Preprint of the 8th IFAC World Congre Milano (Italy Augut 8 - September, (, ( e u (5 for ome poitive ontant and independent on. Furthermore, the iphitz property of and (Remark and 4 and the boundedne of all ignal imply that Γ i bounded. Moreover, ine Γ i lower triangular with zero on the main diagonal (4, one get: ( for (5 for ome ontant independent on. In view of (49- (5, inequality (48 yield: V V ( e V V (uing (4 (5 It readily follow from (5 that: V ( ep ( tv ( (5 Now, letting ma, and, one obtain from (5, uing the fat that V ( e e S e : ( e( S ep t e( (54 On the other ha nd, for, it follow uing (45 and (: e e e (55 Combining (54 and (55, one get: e( S ep e( ( t whih end the proof of heorem.4 Mahine oberver in -oordinate (5 he tranformation ( ugget that etimate an be obtained from z etimate by olving the following equation: zˆ. (57 Differentiating both ide of (57 with repet to time yield: ( ˆ ˆ z ˆ (58 ˆ Uing (9 and (, it follow from (58: ˆ A I A ˆ ( S C ( Czˆ y whih, together with ( and (, give: ˆ A ˆ S C G( ˆ ( Czˆ y I I S C Uing (7, equation ( implifie to: S C ( Czˆ y (59 ( Czˆ y ( ˆ F ˆ G( ˆ I I S C ( Czˆ y whih give the final form of the oordinate oberver: ˆ F ˆ G( ˆ S C ( Cˆ y ( heorem. he tate oberver (, when applied to the indution mahine model ( ubjet to Aumption (A- A, i globally eponentially onvergent. More peifially, there eit real ontant, uh that for any real there i a real o that: ˆ ( ( ep t ˆ( ( ( whatever ˆ( ; where ( denote the unknown trajetory of ( aoiated to the input ignal u(. Furthermore: lim Proof. Introdue the etimation error ~ ( ˆ( (. From ( and (57, one ha: ~ ( inv ( inv ( ( A inv (. i iphitz, it follow from ( that: ~ ( zˆ z (4 where denote the iphitz ontant of inv (.. Uing Propoition, one get: ~ ( t ep ~ S z ( (8 hi etablihe heorem, uing the fat that z ( i a iphitz funtion (Remark, part b 4. SIMUAION he indution mahine harateriti are deribed in (Ouadi et al., 5. he funtion f and g, that haraterize the model hyterei part (8-, are given the following value determined in (Ouadi et al., : a., b.,. 995, d. and. 4 he imulation protool i oneived in a way that make the mahine operate around different point of it magneti harateriti, different rotor peed and different load torque value. Doing o, the hyterei effet on the mahine behavior i meaningful. o thi end, the tator voltage and load torque are let to be varying a hown by Fig.. he mahine i open-loop ontrolled uing a inuoidal tator voltage input. he remaining imulation detail are the following: the oberver gain i = and the initial ondition of the oberver and the true ytem model are quite different ( ˆ r (.75( wb, r (.5( wb. Fig. to 4 how that the tate etimate onverge to their true value after a tranient period that lat le than.5 eond. 55

6 Preprint of the 8th IFAC World Congre Milano (Italy Augut 8 - September, oad torque (Nm tator voltage norm (v Flu etmation error (WB ime ( ime ( Fig. : Upper : oad torque (Nm ower : Stator voltage norm (v Rotor flu (WB ime ( ime ( Fig : Rotor flu norm (Wb etimation. Upper: real and oberved flu norm; lower: etimation error Stator urrent (A Stator urrent etmation error (A 8 4 Real Etimated ime ( ime ( imulation uing the numerial parameter value of a 7.5kW indution motor. REFERENCES Beançon G., (7. Nonlinear Oberver and Appliation. Springer Berlin / Heidelberg. Coleman B. D., M.. Hodgdon, (987. On a la of ontitutive relation for ferromagneti hyterei'. Arh. Rational Meh. Anal., pp De eon J., Alvarez-eal J. G., Catro-inare R., and Alvarez-Gallego J. (. Control of a fleible joint robot manipulator via a nonlinear ontrol-oberver heme. International Journal of Control, 74(: 9-. Heinemann G. and W. eonhard (99. Self-tuning fieldoriented ontrol of an indution motor drive. International Power Eletroni Conferene, okyo, Japan. Khalil H. (. Nonlinear ytem. Prentie-Hall. Kim K., Ortega R., Charara A. and Vilain J. P. (997. heoretial and eperimental omparion of two nonlinear ontroller for indution motor. IEEE ranation on Control Sytem ehnology, vol. 5, no, pp Krzeminki Z, A. Jaderko (99. Main magneti path aturation effet in oberver ytem of the indution motor. IEEE International Sympoium on Indutrial Eletroni, Budapet, Hungry, eonard W., (. Control of Eletrial Drive. Springer. ubineau D., J.M. Dion,. Dugard, and D. Roye. (. Deign of an advaned nonlinear ontroller for indution motor and eperimental validation on an indutrial benhmark, EPJ Applied Phyi, vol. 9, pp Maki J. W., P. Nitri, P. Zea, (99 Mathematial model for hyterei. SIAM Review, vol. 5, no., pp Ouadi, H., F. Giri, J. De eon-morale, and. Dugard (5 High gain oberver deign for indution motor with non linear magneti harateriti. IFAC World Congre, Prague, Czeh Republi. Ouadi H., Giri F., Elfadili A., Dorléan P. and Maieu J.F. (. Indution Mahine Control in Preene of Magneti hyterei: Modelling and Speed referene traking IFAC International Workhop on Adaptation and earning in Control and Signal Proeing ACOSP, -8, Antalya, urkey. Fig. 4: Stator urrent norm etimation. Upper: meaured and etimated urrent norm; lower: etimation error 5. CONCUSION he problem of tate obervation i addreed for indution mahine. he originality lie in the fat that the problem i dealt with, baed a model that aount for the magneti iruit hyterei. A tate oberver i thu developed uing the high-gain deign tehnique and it i formally hown to be eponentially onvergent. hi reult i onfirmed by 5