Generalized predictive control robust for position control of induction motor using field-oriented control

Transcription

1 Elect Eng (25) 97:95 24 DOI.7/s x ORIGINAL PAPER Genealized pedictive contol obust fo position contol of induction moto using field-oiented contol Wellington A. Silva Antonio B. S. Junio Bismak C. Toico Dalton A. Honóio Tobias R. Fenandes Neto Lauinda L. N. dos Reis Luiz H. S. C. Baeto Received: 25 Febuay 24 / Accepted: 26 Decembe 24 / Published online: 9 Januay 25 Spinge-Velag Belin Heidelbeg 25 Abstact This pape pesents the study and implementation of a diffeent field-oiented contol stategy using a genealized pedictive contol (GPC) technique applied to the mechanic position loop aiming to obtain a system that acts in the factional hosepowe moto dive unning at nea zeo fequency. The position and speed loops wee identified to veify the system behavio and, fom the model found design the GPC contolle. Simulation and expeimental esults ae shown and discussed to demonstate the meit of the poposed appoach and the pefomance and obustness of the algoithms have been evaluated. Keywods Field-oiented contol Pedictive contol Position contol Robustness Intoduction Induction machines (IM) ae widely used in industy due to simplicity, lowe cost, educed need fo maintenance, and also geate obustness if compaed to othe types of electical machines. The main difficulty of using the IM fo position contol is the mathematical modeling of the contolle design. Typically, position contol of the moto shaft is pefomed by employing DC motos and sevomotos. In the last two decades, advances have occued in the study of pinciples that goven field oiented contol (FOC) applied to altenating cuent (AC) machines. Theefoe, the contol of induction machines can achieve pefomances simila to those of DC motos. The machine cuents and voltages using FOC allow the diect contol of the spatial oientation of electomagnetic W. A. Silva A. B. S. Junio (B) B. C. Toico D. A. Honóio T. R. F. Neto L. L. N. dos Reis L. H. S. C. Baeto Fotaleza, Bazil babosa@dee.ufc.b fields, esulting in the use of the tem field-oiented fo this type of contolle. In this type of contol, a diect analogy can be established with the contol of a DC moto with sepaate excitation []. Thee ae seveal techniques fo speed contol applied to IM dives, such as the use of pedictive stategies. The pedictive model was employed in [2] to contol both the speed and oto flux. Additionally, in [3] uses a stategy pedictive applied to the diect toque contol (DTC) to decease flux and toque ipple. Technical adaptations ae widely used in [4], which uses the stategy named model efeence adaptive contol (MRAC) to contol speed in the IM. Besides thee ae obust techniques of DTC to speed sensoless adjust using a pedictive contolle in a PI stuctue [5]. Recently, stuctues mixing vaious types of contolles known as hybid contolles have been poposed by seveal eseaches to achieve the best pefomance fo each stategy. In [6] was used a hybid PID contolle, which has the advantage of being easily tuned by a fuzzy contolle with the aim of impoving the system obustness. In [7] was poposed a new obust MRAC using a hybid stategy, whee fuzzy logic is then used to achieve the hybidization between sliding mode contol (SMC) and a PI contolle fo flux and speed contol in the IM. To contol the shaft position in IMs consideing that speed is almost zeo, a pope stategy is necessay, while studies egading this subject ae aely found in liteatue. A sensoless contol fo the IM using hamonic pulse width modulation (PWM) was poposed in [8]. The stategy named intenal model contol (IMC) to achieve zeo speed was used in [9], while in othe wok was used vaiable stuctue contol (VSC) with adaptive gain in the speed loop fo the positioning of IM []. The fuzzy logic to contol the IM was employed in []. The SMC stategy was utilized in position loop in [2]. Pevious woks have

2 96 Elect Eng (25) 97:95 24 also pesented a compaison between SMC and the use of FOC applied to the speed contol fo the IM opeating at low speeds, pesented in [3]. Theefoe, this pape poposes the design of the contolle that acts on the position loop to achieve the positioning of the IM shaft. Substituting loop speed contol by the position contol stategy using GPC with RST polynomial stuctue of contol [4], i.e., instead of having two contolles, one fo position and one fo speed, as shown in classic stategy, thee is only a block position contolle. The technology contibution of this study lies in the possible applications to obotics. When induction motos (IMs) ae used, inexpensive, ugged, and easy to maintain units can be used in the joint degees of feedom of the obot am, fo example. The main scientific contibution consists in the study GPC applied to position contol of IM. The use of the afoementioned contol techniques GPC is justified by simplicity and ease of implementation in an embedded system. Fo instance, a DSC (digital signal contolle) is used in the poposed appoach. Finally, this wok pesents simulations tests and expeimental esults to demonstate the main featues of the developed system, thus validating the employed methodology. The pape is oganized as follows. Sections 2 and 3 descibes the induction moto modeling and the contolles design, espectively. Afte, in Sect. 4 a design scheme of obust GPC contolle is poposed. The Sect. 5 pesents the discussion of both simulation and expeimental esults, espectively. Finally, the pope conclusions ae given in Sect Dynamic modelling of the indiect field-oiented contol of an induction machine The block diagam of the indiect field-oiented induction moto dive is shown in Fig.. The state equations of the induction moto in the synchonously otating efeence fame can be descibed by []as: d [A] =[B][C]+ [D], () dt σ L s whee i ds i ds v ds [A] = i qs λ d ; [C] = i qs λ d ; [D] = v qs ; λ q λ q R s σ L R ( σ) L s σ L ω m R Pω L m e σ L s L 2 2σ L s L 2 ω e R s σ L R ( σ) Pω L m L m R [B]= s σ L 2σ L s L 2 σ L s L 2 L m R L R L ω e P 2 ω ; ( ) L m R L ω e P 2 ω R L (2) The toque is given by: T e = 3P 4 L m L (i qs λ d i ds λ q ), (3) While expessions (4), (5), (6) give the following paametes: σ = L m 2, (4) L s L λ q = L m i qs + L i d, (5) λ d = L m i ds + L i q, (6) whee, T e = Electomagnetic toque (Nm) T L = Load toque (Nm) R s = Stato esistance ( ) L s = Stato magnetizing inductance (H) L s = Stato tansient inductance (H) R = Roto esistance ( ) L = Roto magnetizing inductance (H) L m = Magnetizing inductance (H) P = Numbe of poles, adimensional ω e = Electical angula speed (ad/s) ω = Roto angula speed (ad/s) ω sl = Slip angula speed (ad/s) v ds = d-axis stato voltage (V) v qs = q-axis stato voltage (V) i ds = d-axis stato cuent (A) i qs = q-axis stato cuent (A) ids = d-axis stato cuent command (A) iqs = q-axis stato cuent command (A) λ q = Roto flux align with the q-axis (Wb) λ d = Roto flux align with the d-axis (Wb) σ = Magnetic coupling constant, adimensional Accodingly, the flux linkage and its deivative in the q- axis ae set to zeo as: λ q = and dλ q dt =, (7) In an ideal field-oientated induction moto, decoupling between d and q-axis can be achieved, while the total oto flux linkage is foced to align with the d-axis. Then esulting in (8) and (9) v qs = (R s + L s )i qs + ω e L s i ds, (8) v ds = s i ds ω e L s i qs, (9) whee, L s = L s L2 m L, ()

3 Elect Eng (25) 97: Fig. Block diagam epesenting field-oiented induction moto dive G () s C Position Contol Gid 38 V 6 Hz - Vaiac G () C s Toque Contol V 6 Hz Limite L C i qs PWM Invete T a T b T c Cuent Contol Ramp Compaison i b i c Tigge tansisto M Stato i b Encode Digital Filte d/dt Limite i ds cos e sin e qs Geneato of sines ds and cosines Ri Li sl e The oto flux linkage can be found fom the thid ow in () and by using (4) as: λ d = L mi ds + s L, () R Accoding to (), since the electical time constant (L /R ) is much smalle (nea to zeo) than the mechanical one, it can be neglected. As a esult, the cuent i ds is constant (i ds = ids ) and the desied oto flux will be constant eithe. Thus, the Eq. () becomes: λ d = L m i ds, (2) Fom (4) and (6), expession (3) is simplified to: T e = 3P 4 L 2 m L i qs (3) whee i qs denotes the toque cuent command geneated fom the toque contolle G c (s). When using indiect field oientation, the slip angula speed is necessay to calculate the unit vecto fo coodinate tanslation. By employing the fouth ow of () and also (4), the slip angula fequency ω sl can be estimated as: ω sl = L m R i qs L λ d = R i qs L i ds (4) The geneated toque T e, oto speed ω, and oto angula position θ ae elated by: ω = sθ = /J s + B/J [T e(s) T L (s)] (5) whee B is the viscous damping coefficient, J is the inetia constant, and T L is the load toque applied to the shaft. 3 GPC appoach to model pedictive contol The pedictive contol stategy equies a system model unde study to compute the pediction inside the contol hoizons to be used. So it is needed a peliminay study to find the model that best suit the system. Theeby, the pedictive contolle can be implemented. This section is subdivided into two subsections, one elating to the system modeling and anothe to the pedictive contol used. 3. System modeling In the identification of the model system thee ae vaious types of methodologies. Fo this pocess epesented in Fig., the stuctue of the FOC scheme has been changed and pesented in Fig. 2. Then was pefomed some steps in toque efeence system being analyzed the output of oto speed. This behavio can be obseved in Fig. 3. This data was used to identify the model, whee half of the data was used fo the identification pocess and the othe half used fo validation. The pocess dynamics can be epesented using the Contolled Auto-Regessive and Integated Moving Aveage (CARIMA) model [4,5]: A(q )y(t) = B(q )u(t) + C(q ) e(t), (6)

4 98 Elect Eng (25) 97:95 24 Gid 38 V 6 Hz - Vaiac V 6 Hz Toque Refeence L C i qs PWM Invete T a T b T c Cuent Contol Ramp Compaison i b i c Tigge tansisto i b M Stato Encode G(q ) = B(q ) A(q ) = q.9775q.2285q 2 (7) The polynomial C(q ) coesponds the model noise dynamics is modeled by a filte in following section. As the poposed contolle is in the loop position, so just add an integato in the system model. 3.2 Genealized pedictive contol i ds cos e qs Geneato of sines ds and cosines Ri Li sl e sin e The GPC algoithm consists in applying a contol sequence that minimizes a multistage cost function of the fom [5]: J = N 2 j=n [y(t + j t) ω(t + j)] 2 + Nu j= λ[ u(t + j )] 2, (8) Fig. 2 Stuctue used fo identification whee e(t) is uncoelated (white) noise with zeo mean value, A(q ), B(q ) and C(q ) ae polynomials in the backwad shift opeato q in the fom and = q : A(q ) = + a q + a 2 q 2 + +a na q na, B(q ) = b + b q + b 2 q 2 + +b nb q nb, C(q ) = + c q + c 2 q 2 + +c nc q nc. Fom the data obtained in Fig. 3, the least squaes method was used [4], and consideing Eq. (6) with n a = 2 and n b =, the following discete tansfe function elating oto speed output and efeence toque is given by: whee N and N 2 ae the minimum and maximum costing hoizons, espectively, N u is the contol hoizon, λ is the contol weight, ω(t + j) is a futue setpoint o efeence sequence, u(t) is the incemental contol action and y(t + j t) is the optimum j-step ahead pediction of the system output y(t) on data up to time t. The solution of this optimization poblem is a cucial step in MPC (model-based contol) algoithms. The numeical complexity depends on the chaacteistics of the models in tems of lineaity, constaints, numbe of manipulated and contolled vaiables, etc. Fo linea models without constaints, the MPC optimization can be pefomed analytically [6]. The futue outputs can be computed using filteing techniques o Diophantine equations [6] while this wok uses the second appoach. To compute the futue outputs y(t + j) fo j = N,...,N 2, the following Diophantine equation must be solved: Fig. 3 Input and output fo system identification 2 Toque Refeence (N.m) Roto Speed (ad/s) Time (s)

5 Elect Eng (25) 97: C(q ) = E j (q ) A(q ) + q F j (q ), (9) whee E j (q ) and F j (q ) ae uniquely defined polynomials with degees j and n a, espectively. Using Eqs. (6) and (9) the futue pocess output can be descibed by: y(t + j) = F j(q ) C(q ) y(t) + E j(q )B(q ) C(q u(t + j ) ) +E j (q )e(t + j). (2) ut () t () et () qt () yt () Tq ( ) ZOH G(s) - Rq ( ) Fig. 4 Classical RST stuctue Sq ( ) + + () t As the degee of E j (q ) is j, then all the noise tems ae in the futue, and theefoe the optimal pediction can be obtained eplacing e(t + j) fo its expected value (zeo) as: I (Ω) y(t+ j t) = F j(q ) C(q ) y(t)+ E j(q )B(q ) C(q u(t+ j t). ) (2) Fom Eq. (2), the past contol inputs can be sepaated solving a new Diophantine equation: E j (q )B j (q ) = H j (q )C(q ) + q j I j (q ), (22) 2 Fig. 5 Robustness index fequency (Hz) σ=.4 σ=.3 σ=.2 σ=. σ=.5 whee H j (q ) has degee j and I j (q ) has degee n i = max(n a, n b j ). Using Eqs. (2) and (22) the pediction output can be witten as: y(t + j t) = F j(q ) C(q ) y(t) + I j(q ) C(q u(t ) ) +H j (q ) u(t + j t), (23) which can be expessed in a vecto fom as: y = F j (q y(t) ) C(q ) + I u(t ) (q ) C(q ) whee + G u, (24) Gid 38 V 6 Hz G () C s i ds -Vaiac GPC Position Contol Limite V 6 Hz Limite L C i qs PWM Invete T a T b T c Cuent Contol Ramp Compaison i b i c Tigge tansisto cos e sin e qs Geneato of sines ds and cosines Ri Li M Stato i b Encode y =[y(t + N t) y(t + N + t) y(t + N 2 t)] T, u =[ u(t t) u(t + t) u(t + N u t)] T, sl e and G is a N N u constant matix based on the coefficients of H j (q ), while F(q ) and I (q ) ae polynomial vectos. Fom contolle implementation standpoint, an analytical solution with low computational cost is impotant. Thus, this wok is concened with the investigation of a especial case whee N u =, N =, N 2 = N and λ =, which epesents the best tadeoff between the computational cost and close loop pefomance [7], then the optimal input is [8]: u(t) = (G T G) G T (w f ) = k(w f ), (25) Fig. 6 Contol stuctue poposed position using GPC whee k is a constant vecto with dimension N, w is a vecto which contains the futue efeence and fee esponse given by: f = F(q y(t) ) C(q ) + I u(t ) (q ) C(q ), (26)

6 2 Elect Eng (25) 97:95 24 Fig. 7 Simulation esults fo the system model using the GPC contol stategy Output Signal (y) Input Signal (u) Time (s) Since N u =, it is impotant to notice that the constained contolle is equivalent to clipping, a case valid only fo monovaiable systems [9]. The tem clipping assumes that the pedictive contolle does not take into account constaints while computing the optimal input, but only aftewads, pefoming had limitations if constaints ae violated. Though some manipulations, Eq. (25) can be witten in the RST fom: u(t) = R(q ) (T (q )(t) S(q )y(t)), (27) whee (t) = w(t + j) is the setpoint, T (q ) = C(q ) N i= k(i), S(q ) = N i= k(i)f i (q ), R(q ) = C(q ) + q N i= k(i)i i (q ). The RST stuctue is impotant fom contol analysis standpoint because it can be deived popeties such as stability and obustness. 4 Robust GPC-based contol (GPCBC) applied to IM position loop Fom model Eq. (6) the polynomial C(q ) is a monic that can be teated as a filte [6]. The selection of C(q ) is not a tivial matte, some guidelines fo open-loop stable pocesses and some case studies can be found in liteatue [9]. In this study, the pocess is of second-ode, then the C(q ) polynomial, it is enough to use a filte with degee n c = 2 to attenuate the noise, consideing that it is popely tuned [6]. Thus, the poposed filte is given by: C(q ) = + c q + c 2 q 2, (28) whee c and c 2 ae constants which must be tuned consideing noise attenuation, distubance ejection, and obustness. Consideing Eqs. (6) and (28), the contol input u(t) in Eq. (27) can be calculated explicitly by pefoming some mathematical manipulation. Thus, the contol polynomials Table Moto paametes Paametes Value Rated powe.25 HP Rated speed,725 pm Rated voltage 22 V Rated cuent.26 A Numbe of poles 4 Roto esistance (efeed to the stato) Stato esistance Roto inductance (efeed to the stato).6 H Stato inductance.6 H Mutual inductance.884 H Inetia moment 5 4 kg m 2 Viscous fiction coefficient kg m 2 /s R, S, and T ae given by: T (q ) = t + t q + t 2 q 2, (29) R(q ) = + q + 2 q 2, (3) S(q ) = s + s q + s 2 q 2, (3) α = N N 2. (32) It is impotant to notice that polynomials R, S and T contain paamete a, which on the othe hand depends on N. Fom Eq. (32), it can be seen that a vaies fom to when the pediction hoizon N vaies fom to inf. Theefoe, the diect use of alpha is poposed as a tuning paamete, assuming an adjustable value between and. It makes the tuning pocess moe pecise than the use of discete values of the N. The block diagam of the poposed obust GPC-based contol (GPCBC) can be posed in the classical RST stuctue as illustated in Fig. 4. To undestand the nominal behavio, the tansfe functions elating the efeence with the output, the input distubance with the output, the noise with the contol

7 Elect Eng (25) 97: Fig. 8 Schematics of the system Compute Powe Supply TMS32F282 Dive Invete IM Potenciomete I abc,, Position input and the noise with the output ae calculated as follows: { } y(t) T (z)g(z) H y (z) = Z = (t) R(z) + S(z)G(z), (33) { } y(t) R(z)G(z) H yq (z) = Z = q(t) R(z) + S(z)G(z), (34) { } u(t) S(z) H un (z) = Z = n(t) R(z) + S(z)G(z), (35) { } y(t) S(z)G(z) H yn (z) = Z = n(t) R(z) + S(z)G(z), (36) whee Z{.} denotes z-tansfom, q(t) is an input distubance, and n(t) is a measuement noise signal at a given discete instants t. Equation (33) shows that H y (z) is a fist ode tansfe function which depends just on a. Theefoe, faste o slowe setpoint esponses can be achieved by deceasing o inceasing a, espectively. On the othe hand, C(z) can be used to change distubance ejection esponse (Eq. (34)) and as a measuement noise filte to attenuate noise effect in both contol and output signals [(Eqs. (35), (36), espectively]. Assuming that C(z) has oots with the same eal pat, in manne that Eq. (28) can be ewitten as: C(q ) = ( e σ +iβ q )( e σ iβ q ), (37) whee σ and β ae tuning paametes, and i is the imaginay opeato. The atio β/σ imposes cetain chaacteistics to the filte C(z), and theefoe, a set of filtes with diffeent atios β/σ. In [2] is shown that the optimal second ode filte has a damping ξ = / 2 to attenuate the noisy sensibility which implicates in β = σ tan(π/4) = σ. Theefoe, the filte has only one tuning paamete to distubance ejection, noisy attenuation and obustness. The modeling eos can be epesented by G(z) = G n (z)( + δg(z)), whee G n is the nominal model. Consideing an uppe limit to the nom of δg(e jo ) given by δg( ) in the inteval = <p. The closed loop obust stability is eached if [2]: δg( ) I ( ) = R(z) + S(z)G(z), (38) S(z)G(z) Fig. 9 Expeimental setup whee I ( ) is defined as the obustness index of the contolle. Figue 6 shows the obustness index I ( ) to the contolle fo seveal values of σ, consideing a eo of 2 % in the gain and thee sample delays. Fom Fig. 5, inlowfequencies, the obustness index fo all evaluated values of s is vitually the same and in the middle fequencies the obustness index can be easily tuned by σ. 5 Simulation and expeimental esults The stuctue poposed has been seen in Fig. 6. Fo implement the GPC contolle was used the use-specified paametes included in the poposed method that ae N 2 = 3, N u = and α =.95. The filte polynomial used C(q ) =.9q +.925q 2 which coesponds to a σ =.5. Using (29) to(3) the paametes RST found wee: s =.65, s =.586, s 2 =.3, =.848, 2 =.85, t = 4., t = 7.8, t 2 = 3.7. The poposed system was simulated consideing the analysis of the following paametes: oto position, oto speed, diect-axis cuent (esponsible fo the field geneation in the machine), and quadatue axis cuent (esponsible fo the electical toque of the machine). Based on such data, the behavio of the oto position efeence is analyzed, coesponding to the adian and afte a petubation to the

8 22 Elect Eng (25) 97: Roto Position (ad) Real Position Ref Position Roto Speed (ad/s) d axis Cuent (A) q axis Cuent (A) Time (s) Fig. Expeimental esults fo the system using the GPC contol stategy.2 duing 2.5 s, in Fig. 7. Thus, it is possible to veify the capability of the petubation ejection of the positioning oto. Fo the expeimental implementation of the system, a kit consisting of a DSC fom Texas Instuments TMS32F282 was used. The machine is a factional hosepowe thee-phase squiel cage IM, whose paametes ae given in Table. The emaining instuments ae Hall-effect cuent sensos, the auxiliay voltage souces, a thee-phase voltage invete module by Semikon with a switching fequency of 2.5 khz, a multi-tun pecision potentiomete coupled to the moto shaft, with a sampling time of.4 ms and fo the contolle was used a sampling time of ms fo being a slow loop. The schematics and expeimental setup ae shown in Figs. 8 and 9, espectively. The same contol techniques consideed in the simulation tests ae studied. The conditions fo the efeence position steps ae fom 6 to adian afte fom adian to 6 adians again, each step change occus afte few seconds. Thus, it is possible to veify the capability of accuacy and epeatability of the positioning oto. Figue shows the evaluation of the GPC contol stategies poposed. By obseving the tacking position is about.5 seconds. Analyzing the behavio of the oto speed thee is a small oscillation afte eaches efeence and duing changes efeence ae about 2 and 2 ad/s. The cuents id in the efeence changes has peaks about.8 A and steady state oscillation with peaks about.6 A. And the iq oscillates with peaks of.5 A in steady state and peaks about 2 A in efeence changes. An assembly was pefomed to veify the ejection of system distubances implemented. Being this assembly consisting of a DC moto model manufactue PHYWE 6., Fig. Mounting fo load test which has a toque of appoximately,.2 N m. The DC moto engine, which emulates the chage to distubing the system, is connected to the ea axle of the IM though a shaft sleeve and is positioned by a mounting backet fo the coect alignment of engines fo no slips axes when the dive DC moto, as shown in Fig.. Then, caying out the same pocedue of the simulation, that is, by adding a load of.2 N at a given instant 8 s of the test, tests wee made to examine the behavio of the contolle unde study.

9 Elect Eng (25) 97: Roto Position (ad) Real Position Ref Position Roto Speed (ad/s) d axis Cuent (A) q axis Cuent (A) Time (s) Fig. 2 Behavio GPC contolle to distubance duing the expeimental test The behavio of the contolle can be seen in Fig. 2. It is obseved that the oto speed has a small oscillation in petubation instant and has a value of 4. ad/s in steady state. The cuent id and befoe the petubation was about.62 A and afte has peaks about.98 A and i q have peaks of about 2.43 A and befoe petubation was 2.4 A. 6 Conclusions Contolling the position of an induction machine is paticulaly difficult due to existing inetia moments and lowviscous fiction coefficient, which bings complexity to the contol of oto position. To analyze the pefomance of the contolles, the stato cuents can be measued, thus epesenting the contol system effot. The GPC stategy was then used in the studied plant poviding the expected esults, with good esponse to the efeence and low swings in steady state, besides pesenting a good pefomance of the distubance ejection. The use of a linea model simplifies the model stuctue and consequently the contolle to be used. The pupose of this study is to use a elatively simple contolle with the stength chaacteistics of the pedictive stategy. In a pevious wok [22] can be seen the use of classical models, whee wee shown that with the use of PID is achieved a longe time tacking position and a geate oscillation aound the efeence, as well as lowe values of peak to i d and i q. Finally, such techniques can be applied to a obotic am, while the actuation egading of othe moto and contolle types is supposed to be investigated. Refeences. Bose BK (2) AC machines fo dives in moden powe electonics and AC dives. Pentice Hall PTR, Englewood Cliffs 2. de Santana ES, Bim E, do Amaal WC (28) A pedictive algoithm fo contolling speed and oto flux of induction moto. IEEE Tans Ind Electon 55(2): Beeten J, Veveckken J, Diesen J (2) Pedictive diect toque contol fo flux and toque ipple eduction. IEEE Tans Ind Electon 57(): Jacobina CB, Ribeio LAS, Filho JBM, Salvadoi F, Lima AMN (23) Sistema de acionamento com moto de indução oientado indietamente pelo campo com adaptação MRAC da velocidade. Rev Contol Autom 4(): Aab Makadeh GR, Soltani J (26) Robust diect toque and flux contol of adjustable speed sensoless induction machine dive based on space vecto modulation using a PI pedictive contolle. Elect Eng 88(6): Ho T-J, Yeh L-Y (2) Design of a hybid PID plus fuzzy contolle fo speed contol of induction motos. In: IEEE confeence on industial electonics and applications, pp Halbaoui K, Boukhetala D, Boudjema F (28) A new obust model efeence adaptative contol fo induction moto dives using a hybid contolle. In: IEEE intenational symposium on powe eletonics, electical dives, automation and motion, pp Raute R, Cauana C, Staines CS, Cilia J, Sumne M, Ashe GM (2) Sensoless contol of induction machines at low and zeo speed by using PWM hamonics fo Roto-Ba slotting detection. In: IEEE intenational confeence on tansations electonics, vol 46, issue 5, pp Ying-Yu T (996) DSP-based obust contol of an AC induction sevo dive fo motion contol. IEEE Tans Contol Syst Technol 4(6): Egiguen PA, Osca BC (2) Robust position contol of induction moto dives. In: IEEE intenational symposium on industial electonics, pp Chung-Yuen W, Sei-Chan K, Bose BK (992) Robust position contol of induction moto using fuzzy logic contol. In: IEEE/IAS annual meeting confeence ecod, vol, pp

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