Predictive Control of Permanent Magnet Synchronous Motor Based on Optimization Model Algorithmic Control

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1 Pedictive Contol of Pemanent Magnet Synchonous Moto Based on Optimization Model Algoithmic Contol 1 Chen Junshuo, 2 Liu Jinglin, 3 Zhang Ying *1, Chen Junshuo School of Automation, Nothwesten Ploytechnical Univesity, Xi an Shaanxi , China, jsch66@gmail.com 2, Liu Jinglin School of Automation, Nothwesten Ploytechnical Univesity, Xi an Shaanxi , China, liujinglinxian@sina.com 3, Zhang Ying School of Automation, Nothwesten Ploytechnical Univesity, Xi an Shaanxi , China, zhangying_1102@163.com Abstact This pape pesents an optimization model algoithmic contol (OMAC) fo pemanent magnet synchonous moto dive system. The OMAC contolle is designed based on the second ode ARMA model of pemanent magnet synchonous moto. It optimizes the coection coefficient of calculation eo and pediction eo coefficient of output as well as the amount of the weighting facto in the taditional model algoithmic contol. Then selecting only one futue time of pedictive values to calculate the optimal contol law simplifies the calculation pocess and so ovecomes the shotcomings of taditional contol needed to select the numbe of pediction hoizon and equied to choose a longe sampling peiod to calculate appopiate contol quantity. The algoithm is simple and obustness of the system can be guaanteed. Theefoe, the OMAC is well positioned to meet the dynamic and apid moto eal-time equiements. Simulation is conducted to veify the pefomance of the poposed contol scheme. Keywods: Pedictive Contol Algoithm, Optimization Model Algoithmic Contol, Pemanent Magnet Synchonous Moto 1. Intoduction The pemanent-magnet synchonous moto (PMSM) dive has emeged as a top competito fo a full ange of motion contol applications. Fo example, the PMSM is widely used in the moden Ac sevo systems such as industial obots, machine tools and is being consideed in high-powe applications such as vehicula populsion and industial dives. It is also becoming viable fo commecial/esidential applications. The PMSM is known fo having high efficiency, high-toque, low toque ipple, supeio dynamic pefomance, high powe density, and fee maintenance. These dives often ae the best choice fo high-pefomance applications and ae expected to see expanded use as manufactuing costs decease [1-3]. Howeve, Many of the contol systems of those dives set foth have paametedependent fomulations. With time, wea, and tempeatue, the machine paametes can vay significantly, negatively affecting system pefomance. It is a multivaiable, nonlinea and stong coupling system and the sevo system s unning situation is moe complex. The PI contol system is vulneable to moto paamete vaiations, load distubance and othe uncetainties. Theefoe, the taditional PI contol stategy is applied fo PMSM to ealize closed loop contol, which has dawback in high-pefomance diving system. It can't meet the system equiements both of dynamic esponse and anti-intefeence ability at the same time. In ode to ovecome the dawback of PI contol, many contol stategies eliminating the influence of uncetainties cuently have caused the extensive attention of people [4-6]. Pedictive Contolles, which allow us to ovecome the dawback of PI contol, ae known in dive contol fo quite a long time. In ecent yeas, thee ae successful examples of pedictive contol being applied in the complicated industial pocess [7-10]. Nevetheless all pedictive contol schemes poposed fo contolling electical dives until today only pecalculate the system behavio fo one single futue sampling cycle. Poweful contol stategies like model algoithmic contol (MAC), allow highe pediction hoizons, but these stategies have not yet extensive been used fo dive contol. In this pape, a new pedictive contolle fo electical dives based on OMAC is pesented. MAC was developed in Fance duing the late 1970s in the chemical pocess industy. Richalet established the oiginal concept [11]. In contast to taditional PI, MAC is based on the finite impulse esponse Intenational Jounal of Advancements in Computing Technology(IJACT) Volume5,Numbe8,Apil 2013 doi: /ijact.vol5.issue

2 models and these models ae non-paametic models. Theefoe, MAC can dastically educe the dependence on paametes. Howeve, on-line computation of convolutions is complicated and it needs to set moe paametes in MAC. Thus, the sampling peiod is difficult to choose smalle to adapt to the equiements of the moto s faste dynamic. Geneal MAC which u(k) is contolling quantity is still popotional contol in essence. When the contolling quantity has the constaints, even closed-loop pediction will also lead to static diffeence. Theefoe, this pape pesents an OMAC fo PMSM dive system. It integates the advantages of MAC such as online eal-time pediction, optimization and feedback coection. By intoducing the second ode ARMA model of PMSM as pedict model of OMAC, the numbe of optimization vaiables at evey time is geatly educed so that the on-line computation is simplified. Unlike MAC, the incemental Δu(k) is contolling quantity in OMAC contolle. Thus it contains an integal element which can effectively eliminate the static diffeence of the PI dive system. The est of the pape is oganized as follows. The mathematical model of PMSM is pesented in Section 2. We outline basic steps of MAC and illustate its application in a PMSM dive system in Section 3. Designing an OMAC contolle fo PMSM dive system is intoduced in Section 4. We demonstate that the poposed contolle pefoms well by conducting seveal compute simulations in Section 5, and povide the conclusion of this study in Section A time-continuous model of PMSM As aleady mentioned, a model of contol object is one of the essential components of a MAC contolle. Hence, fo dive contol a mathematical model suitable fo PMSM has to be designed. Fo high fequency signals the stato esistance and the effects of the pemanent magnet flux linkages can be neglected. We also neglect the effects of magnetic satuation. A pemanent-magnet synchonous moto is chaacteised by the following equations in a efeence fame, otating with the speed ω e. The stato voltage equations of a PMSM in the synchonously otating efeence fame ae descibed as follows: d u L i Ri dt d u L i Ri dt d d e q q s d q q e d d s q (1) Whee u d, u q, i d, i q, ψ d, ψ q, and L d, L q ae the voltage, cuent, flux linkage and constant inductance, espectively in d-q axis, R s epesents the esistance of the amatue winding, and ω e is an angula velocity. The stato flux linkage equations ae given by: L i q Li q q L i d d d md f (2) Whee L md is the mutual inductance between the stato winding and the oto magnet, and i f denotes an equivalent field cuent. It is impotant to note that fo the suface-mounted PMSM, L d = L q because the stato inductances ae independent of oto position. Theefoe, the toque equation can be expessed as: T p( i i ) pl i i (3) em d q q d md f q Whee p is the numbe of pole pais. By using (1) and (2), the mechanical motion equation can be deived as follows: 1028

3 T J d p dt e em TL (4) Whee J and T L epesent the moment of inetia and a cogging toque, espectively. Cuent-based PMSM dives ae typically used in PMSM contol systems whee a commanded toque is geneated. Let the commanded toque be designated as T em. By using the machine equations that wee tansfomed into the oto efeence fame, othe useful fomulas can be developed. This method of building the cuent commands elies heavily on machine paameteization. Advanced methods such as MAC and OMAC, utilize closed-loop methods to eliminate o dastically educe the dependence on paametes. Indeed, advanced contol methods fo moto dives is still an active eseach aea [12-13]. 3. Model algoithmic contol fo PMSM dives The typical stuctue of a Model algoithmic contolle is depicted in Figue. 1. It basically involves: (1) an impulse esponse model fo system epesentation and pediction, (2) a efeence tajectoy, (3) an optimality citeion, and (4) the consideation of state and contol constaints [14]. The main idea of the MAC stategy is to pedict the deviation of the futue system outputs fom the efeence path based on the model, define an optimality citeion which eflects the deviations, and obtain the optimal contol stategy to minimize the citeion ove a cetain hoizon in the futue. k i iq k k m k i k i q k e k Figue 1. Diagam of conceptual dive system The MAC contolle is designed based on the sampled-data epesentation of the PMSM. In MAC method the contol law is obtained fom the minimization of the output eo at time k+i. The pediction output has two main components: * the eo esponse, being the expected behaviou of the output eo vecto e(k) between measued value and pedicted value * the foced esponse, being the additional component of output due to the pecalculated set of futue actuating values ω m (k) The total futue system behavio is taken to be the addition of the eo esponse and the foced esponse. This sum is calculated until the pediction hoizon N p. A cost function is then used to valuate the esulting total esponse of the system. Finally, an optimise detemines the best set of futue contol values i q (k+i), i.e. contol actions to minimise the esult of the cost function. The choice of this function can be made accoding to the demands of the pocess and is not dependent on the contolle itself. So it is possible to use linea o quadatic cost functions and penalise e. g. the contol eo o the contol effot with appopiate weighting factos. Hence, the contol enginee can design the contolle in an optimal way to fit the equiements of the contolled system (pocess). An open loop-stategy now would simply asset the set of contols calculated by the optimise in sequence. The use of the eceding hoizon appoach makes MAC a eal closed-loop feedback contol. Only the fist membe of the set, i. e. i q (k), is tansmitted to the contolled system, and the whole pocess of pediction, optimisation and contol is epeated at each sample. 1029

4 3.1. Intenal model Fo MAC uses a model to pecalculate the futue system states, a time-discete model has to be defined at fist. Based on the assumption that the sampling value of velocity s unit impulse esponse is ω 1, ω 2, ω N, intenal model is descibed by using the convolution equation as follows: k i k i k1 i kn k 1 q 2 q N q 1 z iq k 1 k (5) Whee ω(k) is the velocity of PMSM, i q (k) is incoming cuent, and ξ(k) is an uncetain sequence of impulse esponse at t = t k = kt with T being the sampling time. N denotes the sequence length of the impulse esponse. By utilizing diffeence method fo Eq. (5), the incement model could be expessed as: ^ ^ ^ k i k i k 1 i k N m 1 q 2 q N q z iq k 1 ^ 1 (6) Whee the subscipt m is added to indicate estimates of ω obtained in the model simulation and diffeentiate the simulated ω m fom the measued output ω. ω 1^,ω 2^, ω N^ ae sampling value of velocity s unit impulse esponse Feedback compensation Since the dive systems ae time-vaying nonlinea systems, eos must exist in the pactical application. The model (6) can t esponse the system pecisely. Based on the feedback coection law of the auto-contol pinciple, e(k) is added to the pediction output of the model fo getting closed-loop pediction output. Afte coection, the pediction output of PMSM in the futue P step can be ewitten as: kp kp h k k P m P m N i q i1 i kpi h e k P (7) Whee e(k) is the speed eo between pedictive value and the actual value at time k, and h p is an eo coection coefficient Refeence tajectoy Then, the question is the choice of i q (k) to obtain a desiable output esponse afte i time steps. One can equest ω(k+i) to be in the ight diection and cove a faction of the distance between ω(k+i-1) and the setpoint value. In othe wods, one can define a desiable value ω of the output at the (k+i) th time step by: k i k k1 it / e (8) The expectations output ω(k) of the system gadually each on the set value ω along a peodained cuve. ω is the cuve and efeence tajectoy. Whee α epesents a tunable paamete limited by the bound values: 0<α<1, and T is a sampling peiod. Clealy, α 0 coesponds to ω (k+i) ω, and theefoe, will ty to foce the output to go to the set-point as soon as possible, 1030

5 wheeas α 1 coesponds to ω (k+i) ω(k), leaving the output unaffected. An intemediate choice of a coesponds to a desiable value of the output in between ω and ω(k) that ties to bidge the gap to a cetain extent. Thee is not a citeion to choose α. It depends on the chaacteistic of the system to be contolled and the contol equiement. If the taget of contol system is to make the system output follow the efeence output quickly α should be chosen as a small value; on the othe hand if the taget of contol system is to make the system output follow the efeence output gently with a small input a big α can be used. Eq. (8) is efeed to as the efeence tajectoy in the MAC liteatue Optimality citeion Once the efeence tajectoy has been specified, the question then becomes how to choose the contol input i q (k) so that ω p (k) will match ω (k). This can be fomulated as an optimization poblem: 2 2 J P min q p kp kp iq k Whee q is the pediction eo coefficient of output, λ is weighting coefficient of contol quantity, and ω (k+p) is the efeence value of input velocity at k+p. Consideing Eqs. (7) and (8), this becomes: 2 N JP min q i iq kpi hp k m k kp iq k i1 2 (10) In the absence of the input constaints, this minimization poblem is tivially solvable. Contol input i q (k) is the solution of the nonlinea algebaic equation, and Δi q (t) is the incement of contol cuent. Hence, MAC seems to be a pomising appoach to be applied to dive contol. Unfotunately, solving the optimization poblem needs a vey lage amount of calculation powe. The time of online calculation has been found to be too long fo applying MAC to an online contol of electical dives. 4. Optimization model algoithmic contol contolle fo PMSM In this section, fo simplifying calculation we optimize MAC fame stuctue by choosing only one pedicted value in the futue to calculate the optimal contol law. The pediction hoizon is involved in the optimization. Since the model used fo MAC has a time-discete stuctue (see (5) and (6)), the time-continuous model fom Eq. (4) has to be discetized. Using i d = 0 vecto contol, Eq. (4) can be epesented in the fom: de pndiq Be J (11) dt Using Laplace tansfom and zeo ode hold (ZOH), discete z-tansfe function of PMSM can be witten as: ts 1 1 e k T bz z 1 s Js B (12) 1az Whee k T = p n Ψ d,a = -e -TB/J, b = k T (1-e -TB/J )/B with T being the sampling time. Velocity diffeence equation of PMSM can be deived fom Eq. (12): t a t1 bi t 1 (13) e e q (9) 1031

6 Consideing diffeence opeato Δ = 1-z -1, we can get the following time-discete velocity model fo the pediction pefomance of PMSM. t a t a t b i t em, 1 1 e e 1 (14) q Whee ω e (t) and Δi q (t) epesent an actual velocity and the incement of contol cuent at time t, espectively. Fo getting each output of the contolled object in MAC, geneal accoding to multiple pedicted values to calculate the optimal value, it needs to select the pediction step numbe and the length of the pediction hoizon. Accoding to the above optimization idea, fo the each output of contolled pocess, we only choose one pedicted value in the futue to calculate the optimal contol law in OMAC. Fo example, a pedicted value at k+1 time is used to calculate contol quantity at k time and contol quantity emain unchanged at k+1 time and late, such as i q (t+1) = i q (t+2) = i q (t+p). That is the size of the contol quantity which has nothing to do with pediction steps. Eqs. (7) and (8) can be modified to: t1 t1 h t t y t 1 t 1 ep, em, P em, (15) (16) Whee ω e,p (t+1) is the pediction output of a contolled object in the futue one step. Hee y denotes a efeence tajectoy. Applying the model (15) and (16) to the optimisation poblem (10), the quadatic pefomance index is:, 2 2 e e q p em q Jmin q1 a t a t 1 b i t h t t y t 1 i t (17) Fo simplicity, the weighting coefficient λ, eo coefficient q and eo coection coefficient h p ae set to zeo, one and one. It is ecommended, but not equied. Now Eq. (17) has to be minimised, dj/dδi q (t) = 0. The contol quantity can be obtained: iq t t t t m, t y t b (18) Accoding to the above analysis, in ode to validate the contol pefomance of the optimization model algoithm contol the whole dive system simulation model based on vecto contol fo the pemanent magnet synchonous is established in the MATLAB/SIMULINK envionment. The system block diagam is shown in Figue. 2. Figue 2. The system block diagam of MAC contolle fo dive system 1032

7 5. Simulation esults In this section, we evaluate the pefomance of the poposed contol method by applying it to dive system. To assess the pefomance of OMAC, OMAC is compaed with PI. The main paametes of PMSM ae shown in Table 1. The esults unde the OMAC velocity contolle and the PI velocity contolle ae shown in Figue. 3 and Figue. 4, espectively. The sampling peiod T = 1ms, efeence tajectoy time constant τ = 6ms, load T L = 2Nm. Table 1. Basic moto data Moto paamete d-axis inductances L d q-axis inductances L q Amatue winding esistance R s Numbe of pole pais P Moment of inetia J 0 Fiction coefficient B 0 Value 7 mh 7 mh 1.5 Ω kg m (a) (b) Figue 3. Simulation esults with OMAC scheme: 500pm, 2Nm, model matching (a) Output velocity, (b) phase cuent. (a) (b) Figue 4. Simulation esults with PI: 500pm, 2Nm, model matching (a) Output velocity, (b) phase cuent. Figue. 3 indicates that the system dynamic esponse quickly, velocity can be quickly up to expectation and thee is no oveshoot and steady-state velocity eo. In ode to compae the esults with conventional contol, the same expeiment has been caied out using PI cuent 1033

8 contolle. Figue. 4 shows that a PI contolle esults in a much bigge cuent distotion when the moto stats up, a few velocity oveshot and steady-state velocity eo. To show the obustness of the system, a seious mismatch in the paametes of PMSM model and pactical paametes has been simulated. The load is suddenly inceased fom 2Nm to 4Nm at 0.2s, and educed fom 4Nm to 2Nm at 0.6s. (a) (b) Figue 5. Simulation esults with OMAC scheme: 500pm, model mismatching (a) Output velocity, (b) phase cuent. (a) (b) Figue 6. Simulation esults with PI scheme: 500pm, model mismatching (a) Output velocity, (b) phase cuent. Figue. 5 pesents the tansient esponse of the OMAC system when the moment inetia J = 8J 0 and the fiction coefficient B = 13B 0. The simulation esults show that velocity is the constant when load toque changes and the defeent sine wave distotion is smalle than PI system. Although model paametes ae seious mismatching, stong obustness is achieved by the effective distubance estimation. The OMAC guaantees that the model fom the uncetainty set leads to a contolle with stabilizing the tue system. The system is stable and its tansient popeties ae nice. Using PI contolle in the same expeiment, Figue. 6 shows that thee is velocity fluctuation when load toque changes and the defeent sine wave distotion is much bigge. 6. Conclusion The ideas of MAC in combination with systems, aleady well know in the field of contol engineeing and applied to pocess engineeing poblems, have been shown to be easonably applicable to dive contol as well. The advantages of MAC can be utilized especially fo PMSM contol. Since the second ode ARMA model of the PMSM is consideed, an optimal 1034

9 contol of PMSM is possible. Due to the high pediction hoizon, the poposed OMAC contolle is supeio pefomance. In addition to that, OMAC allows simplifying calculating method in a vey simple way just by modifying the coesponding weighting factos in the cost function. The esults obtained by an emulational eal-time OMAC encouage fo futhe eseach. 7. Acknowledgments This pape was suppoted by National Natual Science Foundation of China unde Gant NO , Science and Technology Pogam of Beilin Distict, Xi an unde Gant NO.GX1110 and Foundation fo Basic Reseach fom Nothwesten Ploytechnical Univesity unde Gant NO.JC Refeences [1] B. K. Bose, Ed., Powe Electonics and Vaiable Fequency Dives, IEEE Pess, USA, [2] Singh. J., Singh. B., Singh. S.P., Naim. M, Investigation of pefomance paametes of PMSM dives using DTC-SVPWM technique, In Poceedings of Engineeing and Systems(SCES) 2012 Students Confeence, pp.1-6, [3] Chou. M.C., Liaw. C.M., Chien. S.B., Shieh. F.H., Tsai. J.R., Chang, H.C., Robust Cuent and Toque Contols fo PMSM Diven Satellite Reaction Wheel, Aeospace and Electonic Systems, vol. 47, no. 1, pp.58-74, [4] Xue-liang Wang, An Application on Thee-Level System with Dynamic Matix Contol Based on Modified Genetic Algoithm, AISS: Advances in Infomation Sciences and Sevice Sciences, vol. 4, no. 22, pp , [5] Wenyong Xiao, Multi-facto pedication of diesel engine by using atificial neual netwoks, JDCTA: Intenational Jounal of Digital Content Technology and its Applications, vol. 4, no. 6, pp , [6] Shi Weiguo, Shao Cheng, Sun Zhengyang, The Algoithms of Impoved GPC Netwok Time- Delay Compensation Based on Online Time-Delay Estimation, JCIT: Jounal of Convegence Infomation Technology, vol. 6, no. 3, pp.46-54, [7] Bempoad, A., Moai, M., Dua, V., Pistikopoulos, The explicit linea quadatic egulato fo constained systems, Automatica, vol 38, no. 1, pp.3-20, [8] Gimble Cike J., Odys Adzejw, Pedictive contol fo industial applications, Annual Reviews in Contol, vol. 25, no. 5, pp.13-24, [9] Canale, M., Fagiano, L.,Milanese, M., Set membeship appoximation theoy fo fast implementation of model pedictive contol laws, Automatica, vol 45, no. 1, [10] Delanoue, N., Jaulin, L., Hadouin, L.,Lhmmeau, Guaanteed Chaacteization of Captue Basins of Nonlinea State-Space Systems, In Poceedings of the Intenational Confeence on Infomatics in Contol, Automation and Robotics, pp , [11] J. Richalet, A. Rault, J. L. Testud, J. Papon, Model pedictive heuistic contol: application to industial pocesses, Automatica, vol. 14, no. 3, pp , [12] Ran xuenong, Study on the Modeling Technology based on the Extended Kalman Filteing Sevo System, JCIT: Jounal of Convegence Infomation Technology, vol. 7, no. 11, pp , 2012 [13] Xiaohui Yang, Shaoping Xu, Hesheng Liu,Guoping Liu, Chaos Analysis and extended adaptive chaos contol of Pemanent Magnet Synchonous Motos, JCIT: Jounal of Convegence Infomation Technology, vol. 7, no. 18, pp.58-65, [14] Qijun Xia, Ming Rao, Jixin Qian, Model algoithm contol fo pape machines, In Poceedings of the Second IEEE Confeence on Contol Applications, vol. 1, pp ,