2th WSEAS Intenational Confeence on SYSEMS, Healion, Geece, July 22-24, 28 Feedbac Signal Etimation of an Induction Machine Dive CONSANIN APOSOAIA, ZOAN SZEKEY, and DONAD GRAY Electical and Compute Engineeing Depatment Pudue Univeity Calumet 22 69 th Steet, Hammond, IN 46323, USA Abtact: - hi pape peent the tudy of flux, toque and peed etimation of an induction machine dive. he d-q model of the induction machine i ued to detemine the vaiou matice of the tate pace epeentation of the ytem. he modeling and imulation i caied out by uing the MAAB/Simulin oftwae. wo etimato uing the machine voltage and cuent meauement ae analyzed. Fit etimato i deived by diect ynthei fom tate equation, and then an optimal extended Kalman filte (EKF) i invetigated. he pefomance of both etimato ae peented and good eult wee obtained fo EKF even in the peence of the ytem and meauement noie. Key-Wod: - induction moto, feedbac ignal etimation, Kalman filte Intoduction he feedbac ignal etimation technique popoed in the liteatue fo induction machine dive ae abundant. A few elected method ae mainly baed on the following []: lip calculation, diect ynthei fom machine tate equation, model efeencing adaptive ytem, uenbege obeve, Kalman filte, lot hamonic, and injection of auxiliay ignal. Deciption and hitoical eview of thee method [2] ae beyond the goal of thi pape. he apid development of DSP baed ytem and the deceaing cot of powe electonic ha allowed the complex model and contol algoithm of induction machine to become popula fo a lage ange of application uing eithe the moto o the geneato egime of the machine [3]. In thi pape, the model of the induction machine, baed on fluxe a tate vaiable, i ued to deign and tudy two diffeent etimato that can povide the coect feedbac ignal fo an induction machine contol ytem. 2 Etimato U-I Model he goal i to deign an etimato that can be ued by a enole ac dive in ode to avoid the need of the peed meauement. A deign containt i the accuacy o that the dive mut achieve a coect etimation of the feedbac ignal equied by any cloed loop motion contol ytem. In ode to integate thi etimato model in the contol ytem of an induction machine, the tato voltage and cuent ae conideed hee a input, and the etimated output ae the magnitude and the angle of the oto flux and the electomagnetic toque. hi etimato i deived by diect ynthei fom the machine tate equation which ae witten in tem of the d-q axi component of the tato and oto flux a tate vaiable. hi choice i jutified by the fact that the ytem matix i imple than the d-q axi cuent tate pace model [3]. he machine equation ae deived fom it geneal model whee the peed of the otational d-q ytem of axe =, ince a fixed efeence fame (α-β) i conideed in thi pape. he oto voltage u d = u q = due to a quiel-cage type of oto conideed hee. he ymbol ued in thi pape ae explained in the it of Symbol in the Appendix. In matix fomulation, the machine tate pace model i [4] dx dt = A x B u, (tate equation), and y = C x (output). () whee the vecto of the tate vaiable, input, and output ae a follow: [ ψ ψ ψ ψ ] x =, (2) d q d [ ] q u = u d u q, (3) y ] = [ id iq id iq. (4) ISBN: 978-96-6766-83- 53 ISSN: 79-2769
2th WSEAS Intenational Confeence on SYSEMS, Healion, Geece, July 22-24, 28 he ytem matice A, B, and C, in equation () ae R σ A = R σ B = σ C = σ R σ R σ σ R σ R σ, and σ σ σ R σ, R σ σ σ. (5) he modeling equation of the etimato U -I wee obtained by diect ynthei fom the machine equation ()-(5), including the electomagnetic toque expeion in equation (6) witten in tem of tate vaiable 3 = z 2 e p m ( ψ i ψ i ) d q q d. (6) 3 Kalman Filte Model he induction machine model containing multiple coupling tem and vaiable paamete (uch a R and m ). hi i a nonlinea dynamic ytem and, in addition, the meaued input ignal ae alo coupted by noie. All thee facto equie the ue of an obeve that can povide a good filteing and unnown paamete etimation. he extended Kalman filte (EKF), which i a full-ode tochatic obeve, can be the olution to thee iue. he EKF i applied hee to etimate the machine oto flux, the electomagnetic toque, and the peed by uing the machine dynamic model, whee the peed i conideed a joint tate a well a a paamete [5], [6]. he Kalman filte equation ae deived fom the augmented machine model in the following fom dx = A x B u w, (7) dt y = C x v. (8) whee, w and v ae the poce and the meauement white noie with a Gauian ditibution with zeo mean and covaiance matice Q and R. he vecto of the tate vaiable, input, and output ae given a [ ψ ψ ψ ] x =, (9) ψ d u u u ] d q q d q = [, () y i i ] d q = [. () he etimato U -I bloc diagam implemented in Matlab-Simulin i hown in Fig.. [F_alfa] Goto2 2 F_alfa F_alfa2 G3 i_alfa G4 G u_alfa ia P3 F_alfa [i_alfa] Goto [F_alfa] 4 em G8 P4 [i_beta] i_beta F_beta [F_beta] lambda ambda_e f(u) G u_beta G7 2 ua [i_alfa] i_alfa G G9 3 ub F_beta 3 [F_beta] Goto3 F_beta G5 G6 i_beta [i_beta] Goto G2 4 ib Fig. Flux and toque etimation uing the etimato U -I ISBN: 978-96-6766-83- 54 ISSN: 79-2769
2th WSEAS Intenational Confeence on SYSEMS, Healion, Geece, July 22-24, 28 he ytem matix, the input and output matice of the ytem ae deived a R σ A= R σ B = R σ R σ, R σ R σ C = R σ, (2) R σ σ σ σ σ (3) K = P H ( H P H R ), (2) whee, K i the Kalman gain, z i the meauement at time tep, h(.) i a function of the tate vecto, H i the obevation matix, R i the covaiance of the meauement noie v. 4 Simulation Reult he etimato U I i applied to a ytem uing PWM thee phae invete and induction moto, and the Simulin bloc diagam i hown in Fig.2. he induction machine paamete ued in the imulation ae given in the Appendix. he expended Simulin bloc diagam of the U I etimato ubytem i epeented in Fig.. Fo the MAAB code digital implementation of the EKF, the filte model i witten in dicetized fom a x A x B u w, (4) = d d z = C x v, (5) d whee, z i the meauement vecto. Fit the etimated tate ae calculated (pediction tep) and then, uing the meauement, thee tate ae updated a a function of the Kalman gain, K. he etimation (pediction) equation ae: x ˆ, (6) ˆ ˆ ˆ = f ( x, u ) w P, (7) = A P A B U B Q whee, x ˆ i the peviou tate etimate, with ˆ covaiance matix P, u i the contol input with covaiance matix U, f (, ) i the ytem dynamic function and wˆ i the ytem noie with covaiance Q. he update equation following a meauement: Fig.2 PWM induction moto ytem with etimato U I In Fig.3, Fig.4 and Fig.5, imulation eult ae hown fo the etimato peented in Fig., duing a tat up of the quiel cage induction moto. A ated peed command unde full toque load wa ued in the imulation. he magnitude and the angle of the oto flux etimate ae hown in Fig.3 and Fig.4, and the electomagnetic toque etimation i een in Fig.5. x ˆ xˆ K ( z h( xˆ )), (8) = P ( ), (9) K = I K H P Fig.3 he etimated magnitude of the oto flux ISBN: 978-96-6766-83- 55 ISSN: 79-2769
2th WSEAS Intenational Confeence on SYSEMS, Healion, Geece, July 22-24, 28 Fig.4 he etimated angle of the oto flux Fig.7 PWM induction moto ytem with EKF etimato Fig.5 he etimated electomagnetic toque With the etimato U I good output etimate wee obtained when the machine wa upplied with voltage in the ange of ated value. At low peed ange the acquiition of the voltage i difficult o the peed etimation i not accuate. Fo a enole dive the peed etimation can be achieved uing the extended Kalman filte. In the EKF algoithm the peed i aumed to be contant duing the tate etimate time update computation but it i included in covaiance time update computation. he peed will be etimated in the tate etimate meauement update tep [7]. he EKF i applied hee to a ytem uing a PWM thee phae invete and an induction moto. he bloc diagam of the ytem i hown in Fig.7. In the imulation the eo covaiance matix P of the Kalman filte, the tate noie covaiance matix Q and the meauement noie covaiance matix R ae 6 P= 6 6 = 6 R Q = 6 6 6 6 6 6 (2) In Fig.8, Fig.9, Fig., Fig. imulation eult ae hown fo the extended Kalman filte duing the tat up of the induction moto with full load toque. he oto flux magnitude and angle etimate ae hown in Fig.8 and Fig.9, the etimated toque in Fig., and the etimated peed in Fig.. Fig.8 he etimated magnitude of the oto flux Fig.9 he etimated angle of the oto flux ISBN: 978-96-6766-83- 56 ISSN: 79-2769
2th WSEAS Intenational Confeence on SYSEMS, Healion, Geece, July 22-24, 28 APPENDIX Induction Machine: R =7.782Ω; =.487H; m =.45H; R =7.Ω; =.47H; z p =2; U N =22V(m), P N =.W, N =5Nm, J=.6 gm 2. Fig. he etimated electomagnetic toque Fig. he etimated peed uing EKF In the tate noie covaiance matix Q, the coeponding numeical value fo the peed etimation hould be adjuted in ode to obtain good etimation eult. Depite the fact that the induction machine paamete uch a R and m, ae vaiable paamete, the Kalman filte handle them a noie and emain inenitive to thei vaiation. 5 Concluion In thi pape, two etimato uing voltage and cuent meauement ae peented fo flux, toque and peed etimation of an induction machine dive. he model of both etimato wee implemented in Matlab-Simulin envionment. Even though the U I etimato model eulted in a faily imple tuctue it emain enitive to paamete vaiation. he tochatic Kalman filte obeve ued in the imulation poved that the paamete vaiation and vaiou modeling uncetaintie can be uccefully ovecome. A coect etimation of the oto flux phao angle it wa poved by the good imulation eult of both etimato. hi i of main impotance, ince the integation of thi dive into a futue vecto contol ytem i intended. it of Symbol, : ubcipt, denoting tato and oto quantitie ^ : etimated value ψ, ψ, ψ, ψ : d-q component of tato and oto d q d q tanient flux linage (Wb) u, : d-q component of tato and oto tanient d u q voltage (V) i, i, i, i : d-q component of tato and oto d q d q tanient cuent (A), : electical angula peed of the induction machine oto, and d-q efeence fame peed (ad/) R, : tato, and oto eitance(ω) R,, :tato, oto and magnetization inductance(h), σ = 2 m : leaage coefficient z p : numbe of induction machine pole pai e : electomagnetic toque (Nm) Refeence: [] Bimal K. Boe, Moden Powe Electonic and AC Dive, Pentice Hall PR, 22. [2] K. Rajaheaa, A. Kawamua, and K. Matue, Senole Contol of AC Dive, IEEE Pe, 996. [3] C. Apotoaia, D. Kozel, and D. Gay, Simulation of a Wind Enegy Conveion Embedded Sytem, Poceeding of 27 IEEE Intenational Confeence on Electo/Infomation echnology, Chicago, May 7-2, 27, pp. 64-69. [4] C. Apotoaia, Gh. Scutau, A Dynamic Model o a Wind ubine Sytem, Poceeding of the th Intenational Confeence OPIM 6, Baov, vol. 2, May 8-9, 26, pp. 26-266. [5] M. S. Gewal, A.P. Andew, Kalman Filteing heoy and Pactice Uing MAAB, 2 nd edition, John Wiley & Son, Inc., 2. [6] G. Welch, G. Bihop, An intoduction to the Kalman Filte, UNC-Chapel Hill, R 95-4, July 24, 26. [7] B. Ain, U. Ogune, A. Ea, M. Ehani, A compaative tudy on non-linea tate etimato applied to enole AC dive: MRAS and Kalman ISBN: 978-96-6766-83- 57 ISSN: 79-2769
2th WSEAS Intenational Confeence on SYSEMS, Healion, Geece, July 22-24, 28 Filte, IECON-24, 3 th Annual Confeence of the IEEE Indutial electonic ociety, Buan, South Koea, 2-6 Novembe, 24, pp. 248-253 [8] K.. Shi,.F. Chan, Y.K. Wong, and S.. Ho, Speed Etimation of an Induction Moto Dive Uing Extended Kalman Filte, 2 IEEE Powe Society Winte Meeting, Singapoe,23-27 Januay 2, pp. 243-248 ISBN: 978-96-6766-83- 58 ISSN: 79-2769