Theoretical comparison between Field Oriented and Generalized Predictive Control for an Induction Motor

Transcription

1 Int. Jounal on Sciences and echniques of Automatic contol Vol., N heoetical compaison between Field Oiented and Genealized Pedictive Contol fo an Induction Moto aek Gallah (, Adel Khedhe (, Mohamed Faouzi Mimouni ( and Faouzi M sahli ( ( Institut Supéieu des Sciences Appliquées et de echnologie de Sousse, cité Ibn-khaldoun 43 Sousse. ( École Nationale d ingénieus de Monasti, Rue Ibn El Jazza, 59 Monasti, unisie. taek.gallah@topnet.tn Abstact: In this wok, we pesent an induction moto flux and speed contol stategy. Initially, we used the field oiented contol (FOC pinciple to geneate two stuctues of this machine contol. he fist stuctue is ealized aound two popotional integal egulatos (PI, the second consists of the intoduction of two Genealized pedictive contolles (GPC. he poposed stuctue is based on the synthesis of the flux and speed egulatos and on the conception of the oto flux sliding mode obseve. In ode to impove the obustness of the two contol stategies face to the paametes vaiations and the extenal petubations, we have developed a oto esistance adaptation algoithm based on the Lyapunov theoy. A compaative study between the two suggested stuctues shows good pefomances of the genealized pedictive contol. Keywods: Induction moto, PI contolle, genealized pedictive contol, Sliding mode obseve, paamete adapte, Robustness. List of symbols V sαβ : stato voltages in efeence fame αβ, i sαβ : stato cuents in efeence fame αβ, αβ : oto fluxes in efeence fame αβ, Rs, R : stato and oto esistances, l s, l : stato and oto cyclic inductances, M : stato-oto cyclic mutual inductance, e : the estimate of oto time constant. σ : Blondel coefficient, ω s : angula speed of the otating field efeed to the stato, ω : angula speed of the otating field efeed to the oto, p : Laplace opeato, J : moment of inetia, , Ref. 8, June 7, pp Academic publication cente 7

2 44 IJ-SA, Vol., N, June 7 f : damping coefficient, em, L : electomagnetic and load toque, np : pais pole numbe.. Intoduction he development of the contol techniques at a vaiable speed makes necessay the modelling of the induction moto in a pecise way to be able to simulate its behaviou in a dynamic mode [3]. he induction moto saw its application field of stongly inceased thanks to the implementation of the FOC which makes it possible to decouple the basic vaiables fom its opeation to knowing the electomagnetic toque and oto flux. hus, the equested pefomances fom the vaiatos becoming inceasingly aised, the sophistication of the contolles made only gow. Sight that the egulation of the electical and mechanical paametes machine by PI egulatos has sensitivity with espect to the paamete vaiations; the genealized pedictive contol constitutes one of the new most inteesting solutions to implant duing this last decade [8,9,]. In this wok, we pesent two appoaches of diect oto field oiented contol (DRFOC by oto flux and speed egulatos. he fist contol is aound two PI egulatos wheeas the second contol uses two pedictive contolles. Knowing that oto flux is a not measuable, its estimate o its obsevation appeas of capital impotance. Although the DRFOC gives good esults fo the speed contol, it is pactically penalized fo its paamete vaiations sensitivity [,3]. Fo that, we popose in this aticle to intoduce a oto flux obseve based on the sliding mode pinciple, and including a oto esistance adaptation algoithm. his aticle is oganized in six sections. he second section elates to the modelling of an induction moto. he two stuctues of the suggested contol ae detailed in the thid section. hen, we develop a sliding mode flux obseve in the fouth section. In the fifth section we set up a oto esistance adaptation algoithm based on the theoy of Lyapunov stability wheeas the simulations esults ae illustated in the last section. We close the pape by a conclusion.. Mathematical model of the induction moto he induction moto can be descibed by five non-linea diffeential equations with fou electical vaiables, one mechanical vaiable and two contol vaiables such as []:

3 IJ-SA, Vol., N, June 7 45 d k = + + β ω i sα γ i sα α n p k + V σl sα s d k = + + α ω i sβ γ i sβ β n p k V σl sβ s d M α = i sα α n p ω β d M = i + n β sβ pω α β with k M σ l sl = and γ M R R = s + σ l sl σ ls ( he mechanical equation is given by: d Ω J m + f Ω m = em L ( he expession of the electomagnetic toque is witten as follows: 3n p M em = ( i d sq l q i sd ( 3 3. Induction moto contol 3.. FOC Pinciple he pinciple of the FOC is based on the adaptation of the d-axis Pak's fame accoding to the oto flux level [,3], that allows to wite: d = = q ( 4 In these conditions, one can wite: d = R i d + φ d = R + ω i q φ d ( 5

4 46 IJ-SA, Vol., N, June 7 while the expession of the electomagnetic toque is given by: 3 M em = n p φ d i sq ( 6 l Afte some easy development, elations 5 and 6 show that the dynamics of the flux and the toque is diectly connected to the contol of d-axis and q-axis stato cuents. he concept of the poposed FOC stuctue is based on the detemination of two tansfe functions connected, espectively, the oto flux in the d-axis stato voltage, and the mechanical speed in the q-axis stato voltage. o detemine these tansfe functions, we conside the following system: d sd V sd = Rsisd + ωs sq ( 7 d sq V = + + sq Rsisq ωs sd whee d-axis and q-axis flux components ae expessed by: sd sq = σ l s i = σ l s i sd sq M + l ( 8 Duing this development we used the speed and oto flux expessed as follows: M isq ω s = ω + ( 9 d = + Misd ( he load toque is consideed popotional to the speed = ω ( L K c ( Afte some development using the Laplace tansfome and elations ( 6 to (, we ewite the stato voltage system given by equation ( : V V sd sq = R = R s s M p + M + pσ l s + l M l M Jp + K + pσ l s + l K ef c ωsσ l si sq M ω + + ω ωsσ l si l sd (

5 IJ-SA, Vol., N, June 7 47 Finally, afte some simplification, the two tansfe functions can be expessed as follows: φ M/R F (p = = s V sd ( p σ p s s ω Kφ ef F (p = = ω V K K (JK K σ p Jσ p sq l l c c s s with 3 M M V sd = V sd + V sd K = n, K R p s l = +, l V sq = V sq + V sq and ( 3 V sd = ω s σl si sq M V sq = ω + ω s σ l s i sd l he voltages V sd and V sq epesent the voltage compensation tems. he discete lineaized tansfe function fo the oto flux magnitude including a zeo-ode hold is then given by [8] F φ (q = B (q = Z[ B (pf (p] φ A (q ( 4 It esults in the following tansfe function A ( q B ( q q = + a + a q = b + b q he same appoach is consideed to obtain the discete lineaized tansfe function of the oto speed B (q (q ω ω = = Z[ B (pf (p] ω A (q ( 5 ω F whee A ( q ω B ( q ω + aωq + ω + bω ω = a q = b q e p ( e Whee Z[.] is the Z tansfom, B (p = p is the zeo-ode hold tansfe function, and e is the sampling peiod. he coefficients of A and B polynomials ae given in the appendix. A, B, ω ω

6 48 IJ-SA, Vol., N, June Popotional Integal Regulato he method that we popose consists in intoducing two PI egulatos. he fist egulato concens the speed and the second is fo the oto flux egulation: [ y ( k y( k ] + k [ y ( k y( ] u( k = u( k + k p c i c k (6 he detemination of the egulatos paametes depends on the consideed loop and the induction moto load type. In ou case we calculated these paametes by poles compensation technique []. he popotional action is used on the measuement and the integal action is used on the tacking eo Pedictive contolle he pediction digital model is defined by an input/output tansfe function and it is epesented as a contolled auto egessive integated moving aveage fom (CARIMA [8,9,]. ξ ( k A( q y( k = B( q u( k + D( q (7 whee u(k and y (k ae the pocess input and output. A and B ae two polynomials in the backwad shift opeato q. A( q B( q = + a q = b + b q + L + a n q a b n a + L + b n q n b ξ (k is an uncoelated andom sequence with zeo mean and unit vaiance. he opeato D( q = q ensues an integal contol law. By assuming that the system output is affected by noise, the two tansfe functions given in equations ( 4 and ( 5, can be easie witten as a CARIMA model. he contol law is obtained by minimization of a quadatic cost function on the futue tacking eos with a weighting contol tem. he quadatic citeion is expessed by: N j= N N u J = y( k + j yc( k + j + λ Du( k + j (8 j= with, y ( k + j, y c ( k + j and D u( k + j epesent espectively, the pedicted output, the efeence output and the contol incement.

7 IJ-SA, Vol., N, June 7 49 N, N, N u and λ epesent espectively, the initialization hoizon, the pediction hoizon, the contol hoizon and the contol weighting coefficient. he optimal sequence of futue contol is given by: ~ u = M[ y c if ( q y( t ih( q Du( t ] (9 with, M = [ G G + I ] m '. λ = N u G and. m N u if ( q ih( q = [ F N = [ H ( q N ( q ( q ' ] ' y c = [ y c ( t + N... y c ( t + N ] ~ ' u = [ Du( t... Du( t + N u ]... F... H N N ( q ' ] In pedictive contol, only the fist value of the sequence of futue contol is actually applied to the system. he polynomial epesentation of the equivalent egulato in RS fom is given by the following equation: S( q Du( t = R( q y( t + ( q y c ( t ( Wheeas, the thee polynomials ae given by: S( q R( q = + m ih( q q = m if ( q ( ( N N ( q m [ q... q ] = (3 With the polynomial contolle, the stability of the contolled loop may be examined fo specific paamete values. he selection pocess fo ( the RS paametes is accomplished by geneating a lage paamete set N,,, λ N Nu assuing closed-loop stability []. - he e N poduct is chosen equal to the system dead time. Geneally N can be selected equal to one fo a taditional configuation without paticula dead time. - he e N poduct is limited by the esponse time value. When N is lage, the coected system is stable and slow. he impovements bought on stability ae minimal compaed to the incease in the esponse time.. - Fo an open loop stable pocess, N u is chosen equal to one due to its fai influence on the stability magins and fo simplicity of calculations.

8 5 IJ-SA, Vol., N, June 7 - he contol weighting coefficient λ is attached to the system static gain. he optimal value of λ assuing a bette stability magins, can be given by : λ = ( G G (4 whee (. is the tace of the matix. able. Choice of the GPC paametes N N N λ u Flux 4 Speed, 5 4. Adaptive Sliding mode obseve he majo advantage of a sliding mode obseve is its insensitivity to paamete vaiations and extenal load distubances once on the switching suface. he sliding mode obseve design pocedue consists of pefoming the following two steps: Design the manifold (the intesection of the sliding mode suface S. he obseve gain is detemined to dive the estimation eo tajectoies to S and maintain it on the set [,5,6,7]. he poposed adaptive obseve has the following fom: d k i sα = γ i sα + + n p k + V + A Is + A I e α β ω σ l sα s s e d k i sβ = γ i sβ + n p k + V + A Is + A I e β α ω σ l s s s e β d α = d β = M i sα α n p ω + A Is + A I β 3 3 s e e M i sβ + n p ω + A Is + A I α β 4 4 s e e (5 he vaiable x is the estimates of paamete x. with, l M R =, e R s e γ = + e Re σ l sl σ ls, I s Is = I s

9 IJ-SA, Vol., N, June 7 5 is the estimate of oto time constant and A ( i =,,3, 4 et j =, epesent the e obseve gains, I s is the vecto signs of the chosen sliding suface. he eos of stato cuents ae diectly connected to the sliding suface by: ij S S i s α i sα = = D S i s β i sβ ( 6 with k D = n k p ω n p kω k he matix D depends on induction moto electical and mechanical paametes and it allows fixing the obseve dynamics. If we conside the numeical teatment of the stato cuents is pefect, one can conside that the obsevation of these sizes gives a zeo eo. Unde these conditions, the cuents matix gains is given by: A A δ AI = = D ( 7 A A δ with δ and δ ae positive constants satisfied the Lyapunov appoach stability condition. [5,6]. ( q δ n ω A δ 3 A3 p A Φ = = ( 8 A4 A4 n p ω δ ( q δ with, q, q, δ and δ ae positive constants. he sliding mode obseve gains ae chosen in such away that Lyapunov stability conditions ae satisfied [5].his yields:

10 5 IJ-SA, Vol., N, June 7 km Re δ (( ( R Re( α i + kωe4 + ke3 l M sα l km β Re δ (( ( R Re( i + kωe3 + ke4 l M sβ l (9 whee e 3, e 4 ae flux eos. It is well known that sliding mode techniques geneate undesiable chatteing. his poblem can be discaded by eplacing the switching function by a smooth continuous function. o this end, we popose to eplace the Sign function by the following function: if Si > γ if S < γ Sat( S = i i (3 Si if Si < γ γ whee γ epesents the thickness of the bounday laye. he estimate of the oto esistance is given as follows: dr e = q (A I K ( 3 3, s with, q 3 > and e M K = is l l It should be noted that the sliding mode obseve is asymptotically stable while being based on the conditions given by [5], and that the pedictive contol law, pesented in this wok, is pactically stabilizing and limited by a good choice of the synthesis paametes. hese conditions enable us to peseve the sepaation pinciple, and to lead theeafte to a global stabilization via output feedback contol. 5. Contol Stuctue he FOC stuctue suggested in this wok intoducing a oto flux obseve based on the sliding mode techniques is given by figue. he contol though PI egulatos is illustated by figue.a wheeas that constucted aound the RS fom pedictive egulatos is given by figue.b.

11 IJ-SA, Vol., N, June 7 53 (a (b Figue : FOC Stuctue fo an induction moto including a sliding mode flux obseve : (a Stuctue with PI egulatos, (b Stuctue with pedictive contolle. 6. Simulation esults In ou simulations, we consideate a two hose powe teta-pola induction moto. he moto paametes ae given in the appendix. he simulated mode elates to a case of a load stating. he load toque is consideed popotional to speed (the popotionality constant is equal to.67.

12 54 IJ-SA, Vol., N, June 7 he efeence speed is taken as being a slope equation ω = t beginning with ef t = s and limited to d/s. In the same way, the efeence flux is a slope equation =,48 t and limited to.89wb (nominal flux. ef Fom the pemanant established state, we intoduced an incease of 5% of the initial oto esistance value, that is to say a change of esistance of 4.8 Ω to 6.4 Ω With an aim of seeking of moto obustness advantage, we simulated the machine esponse by intoducing one oto esistance adapte. he esponse of the vaiation of oto esistance is illustated by figue. One illustates on figues 3, 4 and 5 the moto esponses opeating unde FOC without paametic vaiations, with paametic vaiations and with oto esistance adapte espectively. 6 (b 5.5 (a Figue : Response of the vaiation of oto esistance following an incease of 5% of its efeence value: (a Actual Resistance (b Estimated Resistance.

13 IJ-SA, Vol., N, June 7 55 φ GPC 8 ω PI ω GPC.8 φ PI V sq GPC V sd PI V sq PI - V sd GPC Figue 3: Simulation esults fo an induction moto opeating unde DFOC with PI/GPC egulatos including a sliding mode obseve without oto esistance vaiations. Legend: (3.a: Angula speed vs time, (3.b: oto flux vs time, (3.c: q-axis contol stato voltage vs time and (3.d: d-axis contol stato voltage vs time. φ GPC 8 ω GPC.8 φ PI φ ef 6 ω PI V sd PI V sq PI Vsq GPC V sd GPC Figue 4: Simulation esults fo an induction moto opeating unde DFOC with PI/GPC egulatos including a sliding mode obseve without oto esistance adapte. Legend: same as in figue 3.

14 56 IJ-SA, Vol., N, June ω PI ω GPC φ GPC φ PI V sq GPC V sd PI V 6 sq PI V sd GPC Figue 5: Simulation esults fo an induction moto opeating unde DFOC with PI/GPC egulatos including a sliding mode obseve with oto esistance adapte. Legend: same as in figue 3. Consideed simulations esults, we emak that the GPC esponse time contol is evaluated at,94s fo the mechanical speed and at.47s fo the oto flux, these PI contol time is equal to,99s and.5s fo the mechanical speed and the oto flux espectively. he GPC oveshoot is evaluated at % fo the mechanical speed and at.3% fo the oto flux, these PI oveshoot is equal to 7% and.44% fo the mechanical speed and the oto flux espectively. One notes accoding to the simulation esults that the vaiations of the eal and efeence speed pesent a simila dynamic in tems of continuation and establishment.his dynamic is insensitive fo the paametes vaiations. he dynamics of oto flux depends on the paametes vaiations. he simulation esults show that an incease of 5% of oto esistance esults in a tacking eo of 8% fo PI egulato and 3% fo GPC egulato. he intoduction of the algoithm adapte made it possible to coect the impefections intoduced by the oto esistance vaiations. It makes it possible to educe the tacking eo to 3% fo PI egulato and of less than % fo the GPC contolle.

15 IJ-SA, Vol., N, June Conclusion In this pape, we could highlight the FOC pefomances of induction moto by using two contol stategies. he fist uses two PI egulatos and the second uses two pedictive contolle fo mechanical speed and oto flux. he two poposed stategies wee applied to an induction moto opeating unde FOC technique. In these contol stategies, a oto flux obseve is developed. he poposed obseve is based on the sliding mode technique. In ode to deal with paametes vaiations and the extenals petubations, we have developed a oto esistance adaptation algoithm based on the Lyapunov theoy. he obtained simulation esults show a clea impovement on the level of the machine esponse by use of the pedictive contolle. High obustness with espect to the paamete vaiations is checked duing integation of oto esistance algoithm adaptation. Actually, we oiented ou wok to the implementation the eal time contol stategy devoted fo induction machine. Bibliogaphy [] R. W. De Doncke, «Paamete sensitivity of indiect univesal field-oiented contolles», IEEE ans. On Powe Electonics, vol. 9, no. 4, pp , July 994. [] Khedhe A., Mimouni M. F., Debel N., et Masmoudi A., «Robust Field Oiented Contol Analysis of an Induction Moto Using an Adaptive Flux Obseve Based on Sliding Mode Methodology». IEEE-MESM 3, pp.86-9, Januay 4, Shajah, UAE. [3] A Khedhe., Mimouni M. F., Debel N. et Masmoudi A., «A Suvey on Modelling, Estimation and On-Line Adaptation of Induction Moto Paametes Unde R.F.O.C.», Accepted in ansactions On Systems Signals and Devices (Powe Electical Systems (PES Issues, 5. [4] R. E. Aaujo, D. S. Feitas «Non-linea contol of an induction moto: sliding mode theoy leads to obust and simple solution», Intenational Jounal of Adaptive Contol and Signal Pocessing, vol. 4, n -3, pp , Mach-May. [5] Benchaib A., Rachid A., and Audezt E. «Sliding Mode Input-Output Lineaization and Field Oientation Fo Real-ime Contol of Induction Motos», IEEE ans. Powe Electonics., vol. 4, no., pp. 3-3, Januay 999. [6] Zheng Y., Fattah H. A. A. and Lopao K. A. «Non-linea adaptive sliding mode obseve- contolle scheme fo induction motos», Intenational Jounal of Adaptive Contol and Signal Pocessing, vol. 4, n -3, pp , Mach-May.

16 58 IJ-SA, Vol., N, June 7 [7] Chen F. and Dunnigan M.W., «Compaative study of a sliding-mode obseve and Kalman filtes fo full state estimation in an induction machine» IEE Poc.- Elect. Powe Appl., vol. 49, no., pp.53-64, Januay. [8] M. K. Maaziz, P. Bouche and D. Dumu «A new contol stategy fo induction moto based on non-linea pedictive contol and feedback lineaization», Intenational Jounal of Adaptive Contol and Signal Pocessing, vol. 4, n -3, pp , Mach-May. [9] M. K. Maaziz, P. Bouche and D. Dumu «Flux and speed tacking of an induction moto based on nonlinea pedictive contol and feedback lineaization», Poceedings of the Ameican Contol Confeence, pp. 48-5, 999. [] L. Zhang, R. Noman, and W. Shephed «Long-Range Pedictive Contol of Cuents Regulated PWM fo Induction Moto Using the Synchonous Refeence Fame», IEEE ans. On contol systems technology, Vol 5, NO., pp. 9-6, Januay 997. [] D. Dumu, P. Bouche, K. M. Muphy and F. Déqué «Comfot Contol in Residential Housing using Pedictive Contolles», Poceedings of the 997 IEEE Intenational Confeence on Contol Applications Hatfod, C. Octobe 5-7, 997. Appendix A. Moto Paametes - Roto esistance R = 4.8 Ω, - Stato esistance R s = 5.77 Ω, - Stato inductance l s =. 464H, - Roto inductance l =. 464H, - Mutual inductance M =. 447H, - otal leakage facto σ =.9838, - Moment of inetia J =.49kgm², - Mechanical viscous damping f =. 9, - Numbe of pole pais np =. A. Discete tansfe functions paametes a -. 8 a = 834 =. b = 4. 5e-5 b = 386. e-5 a ω = aω =.79 b ω = bω =

17 IJ-SA, Vol., N, June 7 59 aek Gallah was bon in M Saken, unisia in 98. He eceived the engineeing and Maste degees in industial computing sciences and automatics fom the INSA of unis, in 4 and 5 espectively. Fom 6 he egisteed in Ph.D. in Monasti Engineeing School. His specific eseach inteests ae in the aea Pedictive Contol of Linea and Nonlinea Systems applied to Moto Dives. Adel Khedhe, was bon in Mahdia, unisia in 967. He eceived his Mastey of Science and DEA fom ENSE of unis, unisia, in 99 and 994, espectively, the Ph.D. in Electical Engineeing fom the Sfax Engineeing School in 6. Fom 995 to, he has been a taining teache in the pofessional taining centes. Fom 3 to 6, he has been an Aassistant pofesso in the Electonic Engineeing Depatment of High Institute of Applied Sciences and echnology. He has been pomoted to the associate pofesso gade in the same depatment since Mach 6. His main eseach inteests cove seveal aspects elated to the Contol of the Static Invetes, the Electic Machine Dives and the Renewable Enegy Systems. Mohamed Faouzi Mimouni, eceived the Ph.D and Univesity habilitation degees in Electical Engineeing Depatment of Monasti Engineeing School, unisia, in 997 and 4 espectively and is cuently a Pofesso. His specific eseach inteests ae in the aea Powe Electonics, Moto Dives, Sola and Wind Powe Geneation. Faouzi M'Sahli, was bon in Beja, unisia in 963. He eceived his Mastey of Science and DEA fom ENSE, unis, unisia in 987 and 989, espectively. In 995 he obtained his Doctoate Degee in Electical Engineeing fom ENI, unisia. He is cuently Pofesso of Electical Engineeing at National School of Enginees, Monasti, unisia. His eseach inteests include Modeling, Identification, Pedictive and Adaptive Contol of Linea and Nonlinea Systems. He has published ove 8 technical papes and co-autho of a book "Identification et Commande Numéique des Pocédés Industiels" echnip editions, Pais, Fance.