Passivity-Base Contol of Satuate Inuction otos Levent U. Gökee, embe, IEEE, awan A. Simaan, Fellow, IEEE, an Chales W. Bice, Senio embe, IEEE Depatment of Electical Engineeing Univesity of South Caolina Sweaingen 3A80 Columbia, SC 29208 Depatment of Electical Engineeing Univesity of Pittsbugh Pittsbugh, PA 15261 ABSTRACT A passivity-base contolle, which takes into account satuation of the magnetic mateial in the main flux path of the inuction moto, is evelope to povie close tacking of time-vaying spee an flux tajectoies in the high magnetic satuation egions. The popose passivity-base contolle is expeimentally veifie. Also, a compaison between the contolles base on the satuate an nonsatuate magnetics is pesente to emonstate the benefit of the contolle base on the satuate magnetics. Key wos: Inuction moto, magnetic satuation, passivity-base contol. Coesponing autho. Tel: (803) 777-9314; Fax: (803) 777-8045; E-mail: gokee@ece.sc.eu.
I. INTRODUCTION In [1], following the wok in [2, 3, 4], a passivity-base contolle fo an inuction moto was evelope, an also expeimentally veifie, to povie close tacking of time-vaying spee, position an flux tajectoies une the assumption of linea (nonsatuate) magnetics. In this pape, the esults in [1] ae extene to incopoate the magnetic satuation effects. Heinemann et al. [5] have evelope a fiel-oiente contolle base on the satuate magnetic moel of the inuction moto. An input-output lineaization contolle, which takes the satuation effects into account, was implemente in [6]. In [5, 6], the satuation is assume to be entiely in the main flux path of the inuction moto. That is, the change in the mutual inuctance ue to the satuation in the magnetic mateial is consiee an the changes in the stato an oto leakage factos ae neglecte. In this wok, the same appoach is use to incopoate the magnetic satuation effects into the passivity-base contol of inuction moto. II. INCORPORATION of AGNETIC SATURATION EFFECTS into PASSIVITY-BASED CONTROLLER In the cuent-comman passivity-base contol of inuction motos an une the assumption of linea magnetics, the asymptotic stability of spee/position tacking eos ae achieve by making the flux eo ynamics given by (1) exponentially stable at the oigin (see [1]). ~ l t q ~ R ~ R q R + ωsjl q + l q i q ω sjl q l q (1) L L t L l In [1], it is shown that this can be achieve by making the ight han sie of (1) equal to zeo; that is, by efining the efeence stato cuent vecto an the slip fequency as [1] 1
i q λ L λ + R t n τ phl 2np λ (2) an ω s n ph 2n R p τ 2 λ. (3) The inuctances an L in (1) will not bet at thei nominal values when satuation occus. As a esult, the contolle (2), which is base on the nominal inuctance values, oes not guaantee the exponential tacking of the flux eo. Taking this into account, an also using the appoach in [5, 6], we can then eaange (2) as i q f 1 m L ( λ ) + R p nphl τ 2n λ λ t (4) whee f 1 m ( ) is the invese of the magnetization cuve function of the inuction moto. In (4), the nominal values of L an ae use. This is easonable since L ( 1 + σ ), whee σ is the oto leakage facto, an the change in σ ue to satuation is neglecte. Fo the values of λ which emain in the linea magnetic egion, f m 1 ( λ ) λ with constant so that (4) euces to (2). Note hee that this is an a hoc moification to the passivity-base contolle base on the linea magnetics [1-4, 7-8] an the following section shows that it impoves the pefomance significantly when the moto is opeate in the high magnetic satuation egions. 2
III. EXPERIENTAL RESULTS The passivity-base contolle (4) was teste on the same expeimental setup as in [6]. The expeimental setup consiste of (i) a 3-phase, 6-pole, 1-Hp, squiel cage inuction moto, (ii) a otoola DSP96002 (floating point pocesso) ADS system, (iii) a ata acquisition boa, an (iv) thee 20 khz PW amplifies ( ± 150 Volts an ± 10 Amps). The position measuements wee obtaine though a 2880 pulses/evolution (esolution of 2π / 2880 aians) line encoe. The inuction moto paametes ae 0.225 H, L 0.244 H, L s 0.244 H, R 2.1 Ω, R s 1.85 Ω, f 0.0 N-m/aians/secon, an J.0185 N-m-s 2 [6]. Fig. 1 shows the magnetization cuve of the inuction moto, which was expeimentally etemine in [6]. Fig. 1. agnetization cuve of the inuction moto. 3
III-A. SPEED REVERSAL OVE To emonstate the benefit of the contolle base on the satuate magnetics ove the contolle base on the nonsatuate magnetics, a emaning spee evesal expeiment, which equies the opeation of the moto in the high magnetic satuation egions, was conucte. In this move, the moto was equie to acceleate fom a spee of -104 a/sec to a spee of 104 a/sec in 0.186 secon. The magnitue of the flux efeence was chosen as the solution of the iffeential equation λ t R 1 1 f m λ L ( ) + µδsopt ( ω ) λ ω t (5) whee δsopt( ω ) is the solution to the satuate magnetics optimal toque poblem [6]. Figues 2(a)-(b) show the spee an flux tacking pefomance of the contolle base on the satuate magnetics. Fom Fig. 2(a), it is seen that an excellent spee tacking was accomplishe. Fig. 2(b) shows the magnitues of estimate oto flux vecto l $ q an efeence oto flux vectol q. The oto flux vecto was estimate off-line by solving the following equation t λ$ + ω λ$ q s J λ$ R + $ L λ q f ( $ $ $ λ λ ) λ + λ$ $ 2 2 + λq λ$ q 1 2 2 m q R L i iq (6) whee ω s nphr τ 2 2n λ p R L iq λ. The values of i, i an i q q wee collecte fom the expeiment. Fo compaison puposes, passivity-base contolle (2), which is base on the linea magnetic moel, was also implemente to contol the same moto along the same mechanical tajectoy. The magnitue of the flux efeence was chosen as the solution of iffeential equation 4
λ t R R λ + L µ L δ ( ω ) λ lopt ω t (7) whee δlopt( ω ) is the solution to the linea magnetics optimal toque poblem [9]. Figue 3(a)-(b) show the esults. Fom these figues, it is seen that lage spee an flux tacking eos occu. In bief, the contolle base on the linea magnetics was not able to povie close tacking of the same mechanical tajectoy. In the passivity-base contol of inuction motos, the spee tacking can only be guaantee if the flux tacking is achieve. Othewise, lage flux tacking eos act as istubance input on the system, causing lage spee tacking eos. Anothe consieation is that the magnetic satuation effects ae incopoate assuming that the magnetic satuation cuve is a single-value function. Futhemoe, the moifie contolle is base on the nominal value of the oto esistance. It is clea that the oto esistance might vay fom its nominal value significantly with a consieable impact on the system pefomance. Taking this into account, Chang et al. [10] popose, an also expeimentally valiate, tuning ules fo the PI feeback gains to achieve goo tacking pefomance une a wie ange of vaiations of the moto paametes. IV. CONCLUSIONS A passivity-base contolle, which takes into account satuation of the magnetic mateial in the main flux path of the inuction moto, is evelope to povie close tacking of time-vaying spee an flux tajectoies in the high magnetic satuation egions. The expeimental esults with a emaning spee evesal move show that the popose passivity-base contolle exhibits an excellent spee tacking pefomance while the pefomance of contolle base on the linea magnetics eteioates consieably in the high magnetic satuation egions. ACKNOWLEDGENT This wok was suppote in pat by the Office of Naval Reseach une Gant N00014-96-1-0926. 5
REFERENCES [1] L. U. Gökee an. A. Simaan, A passivity-base metho fo inuction moto contol, IEEE Tans. Inustial Electonics, vol. 44, no. 5, pp. 688-695, Octobe 1997. [2] G. Espinosa an R. Otega, State obseves ae unnecessay fo inuction moto contol, Syst. an Contol Lettes, vol. 23, pp. 315-323, 1994. [3] R. Otega, P. J. Nicklasson an G. Espinosa, On spee contol of inuction motos, Automatica, vol. 32, no. 3, pp. 455-460, ach 1996. [4] G. Espinosa, P. J. Nicklasson an R. Otega, Contol of inuction motos in the fiel weakening egion, IEEE CDC, New Oleans, LA, 1995. [5] G. Heinemann an W. Leonha, Self-tuning fiel-oiente contol of an inuction moto ive, in Poceeings of the Intenational Powe Electonics Confeence, pp. 465-472, Apil 1990, Tokyo, Japan. [6] R. T. Novotnak, J. Chiasson, an. Boson, High pefomance motion contol of an inuction moto with magnetic satuation, in Poceeings of the 34 th IEEE Confeence on Decision & Contol, pp. 2145-2150, Decembe 1995, New Oleans, LA. [7] K. Kim, R. Otega, A. Chaaa an J. Vilain, Theoetical an expeimental compaison of two nonlinea contolles fo cuent-fe inuction motos, IEEE Tans. Cont. Syst. Tech., vol. 5, no. 3, pp. 338-348, ay 1997. [8] P. J. Nicklasson, R. Otega, an G. Espinosa-Peez, Passivity-base contol of a class of Blonel- Pak tansfomable machines, IEEE Tans. Automat. Cont., vol. 42, no. 5, pp. 629-647, ay 1997. 6
[9]. Boson, J. Chiasson, an R. Novotnak, A systematic appoach to selecting flux efeences fo toque maximization in inuction motos, IEEE Tans. Contol System Technology, vol. 3, pp. 388-397, Decembe 1995. [10] G. W. Chang, G. Espinosa-Peez, E. enes an R. Otega, Tuning ules fo the PI gains of fieloiente contolles of inuction motos, IEEE Tans. Inustial Electonics, vol. 47, no. 3, pp. 592-602, June 2000. (a) (b) Fig. 2. Spee evesal move with the contolle base on nonlinea (satuate) magnetics. (a) Estimate (soli) an efeence (ashe) spees in aians pe secon vesus time in secons. (b) $ l q (soli) an l q (ashe) in Webes vesus time in secons. 7
(a) (b) Fig. 3. Spee evesal move with the contolle base on linea (nonsatuate) magnetics. (a) Estimate (soli) an efeence (ashe) spees in aians pe secon vesus time in secons. (b) l $ q (soli) an l q (ashe) in Webes vesus time in secons. 8