Optimal torque control of permanent magnet synchronous machines using magnetic equivalent circuits

Transcription

1 This oument ontains a post-print version of the paper Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits authore by W. Kemmetmüller, D. Faustner, an A. Kugi an publishe in Mehatronis. The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting. Please, sroll own for the artile. Cite this artile as: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 BibTex author = {Kemmetmüller, W. an Faustner, D. an Kugi, A.}, title = {{O}ptimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits}, journal = {Mehatronis}, year = {5}, volume = {3}, pages = {-33}, oi = {.6/j.mehatronis.5..7} } Link to original paper: Rea more ACN papers or get this oument: Contat: Automation an Control nstitute (ACN nternet: Vienna University of Tehnology offie@ain.tuwien.a.at Gusshausstrasse 7-9/E376 Phone: Vienna, Austria Fax: Copyright notie: This is the authors version of a work that was aepte for publiation in Mehatronis. Changes resulting from the publishing proess, suh as peer review, eiting, orretions, strutural formatting, an other quality ontrol mehanisms may not be reflete in this oument. Changes may have been mae to this work sine it was submitte for publiation. A efinitive version was subsequently publishe in W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7

2 Optimal Torque Control of Permanent Magnet Synhronous Mahines Using Magneti Equivalent Ciruits Wolfgang Kemmetmüller a,, Davi Faustner, Anreas Kugi a a Automation an Control nstitute, Vienna University of Tehnology, Gusshausstr. 7-9, Vienna, Austria Abstrat n reent years, permanent magnet synhronous mahines (PSMs are often esigne in a mehatroni way to obtain e.g. speial torque harateristis at zero urrents or maximum effiieny. These esigns are often haraterize by a pronoune magneti saturation an non-sinusoial properties. This paper esribes the optimal torque ontrol of suh PSMs utilizing a magneti equivalent iruit (MEC moel. n ontrast to approahes base on funamental wave moels (q-moels, whih utilize the Blonel-Park transformation an typially onsier saturation an non-sinusoial harateristis only in a heuristi way, MEC moels allow to systematially aount for these effets. Given the MEC moel, optimal values for the oil urrents are obtaine from a onstraine, nonlinear optimization problem, whih an be effiiently solve by exploiting the speial mathematial struture of the moel. The results of the optimization are use in a flatness-base torque ontrol strategy. The performane an pratial feasibility of the propose torque ontrol onept are emonstrate by experiments on a test stan. Finally, it is shown that using this torque ontrol in an outer angular spee ontrol loop also proves to be benefiial. Keywors: ontrol Optimal torque ontrol, magneti equivalent iruit, permanent magnet synhronous motor, flatness base 5 5. ntroution The aurate ontrol of the torque is essential in many appliations of permanent magnet synhronous mahines (PSMs, whih makes this topi an ative fiel of researh in reent years. The inustrial stanar to ontrol PSMs is fiel oriente ontrol (FOC, whih is base on a funamental wave moel an the appliation of the Blonel- Park transformation, see, e.g., [, ]. A number of researh papers have isusse the evelopment of avane (nonlinear ontrol strategies base on this moel. E.g., [3 5] propose exat feebak linearization, [6 8] use bakstepping ontrol an passivity base methos are applie in [9 ] to the ontrol of PSMs. Furthermore, sliing moe ontrol is examine in [ 4], moel preitive ontrol is use in [5 7] an iret torque ontrol onepts an be foun in [8 ]. These ontrol strategies in general exhibit a goo performane for PSMs an operating regions, whih an be aurately esribe by a (magnetially linear funamental wave moel (q-moel. For appliations with high emans on torque, spee or position auray, it happens more often in reent years that motor esigns are employe whih o not satisfy the assumptions that have to be mae for the erivation of Corresponing author aresses: kemmetmueller@ain.tuwien.a.at (Wolfgang Kemmetmüller, (faustner,kugi@ain.tuwien.a.at (Davi Faustner, Anreas Kugi lassial q-moels. n partiular, frational slot onentrate winings an rotors with interior permanent magnets are preferre by inustry ue to the simpler an heaper onstrution. Moreover, to shape the torque harateristis espeially for zero urrents, inhomogeneous air gap geometries are frequently use in a mehatroni esign approah. These onstrutions often yiel pronoune nonsinusoial (non-funamental wave harateristis of the bak-emf an the inutanes of the motor. Moreover, PSMs are often operate in a region, where signifiant saturation of the iron parts ours. Heuristi extensions of the q-moel are typially propose in literature to aount for saturation an non-funamental wave harateristis. E.g., the ontrol strategies in [ 3] are base on an extension of the q-moel by higher harmonis in the bak emf, the inutanes or the resulting torque. n these works, however, the influene of saturation an the resulting ogging or relutane torque are not onsiere. Saturation is again inorporate into the q-moel in a heuristi manner, see, e.g., [3 37]. The limitations of these approahes learly result from the unerlying q-moel suh that a more rigorous moeling approah is preferable. A magneti equivalent iruit approah was esribe in [38, 39] for the moeling of PSMs with internal or surfae mounte magnets. t was shown that an aurate esription of the behavior of PSMs with non-funamental wave harateristis an signifiant saturation an be ahieve with this approah. Sine the resulting moels feature a limite moel om- Preprint submitte to Mehatronis Deember 9, 5 Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

3 plexity, these moels an be onsiere a goo basis for the ontroller esign. n this work, the optimal torque ontrol of PSMs whih show both signifiant magneti saturation an non-sinusoial harateristis is onsiere. As a test ase, a PSM with interior permanent magnets is use, whih was esigne for a rear steering system of a ar. n this appliation, a motion of the PSM has to prevente in ase of a failure of the power eletronis. This is ahieve by esigning an inhomogeneous air gap geometry whih results in a large ogging torque. This in turn prevents unesire rotation of the motor at zero urrents. However, the signifiant non-sinusoial behavior an saturation ompliates the aurate torque ontrol of suh PSMs. The orresponing mathematial moel is esribe in [38], whih will serve as a basis for the ontroller esign. The ontrol strategy is base on the solution of a nonlinear, onstraine optimization problem in ombination with a flatness-base feebak ontrol. n [4], also the optimal torque ontrol of a PSM, whih exhibits signifiant saturation, base on an MEC moel is onsiere. For this surfae magnet PSM it is, however, possible to aurately approximate the harateristi quantities like the flux linkage of the oils by means of funamental wave omponents, with only their amplitues an phase angles being nonlinear funtions of the oil urrents. t is emonstrate in [4] how the funamental wave harateristis an be benefiially utilize to solve the resulting optimal ontrol problem. The present work eals with the more general ase ontaining both magneti saturation an non-funamental wave harateristis. The paper is organize as follows: The mathematial moel of [38] is briefly summarize in Setion. n Setion 3, the alulation of optimal urrents is esribe, whih is use in the flatness base ontrol strategy outline in Setion 4. Measurement results of a test stan presente in Setion 5 emonstrate the goo ontrol performane an the pratial feasibility of the propose ontrol strategy. Finally, Setion 6 elaborates the benefits of using the op- timal torque ontrol strategy in an outer ontrol loop for the angular spee.. Mathematial Moel n [38], a general framework for the mathematial moeling of permanent magnet synhronous mahines base on a magneti equivalent iruit approah was erive. This approah was suessfully applie to the moeling of both a surfae-mounte PSM [39] an a PSM with internal magnets [38]. n this work, the same motor as in [38] will be use an therefore, the mathematial moel erive in [38] serves as the basis for the evelopment of the optimal torque ontrol strategy. The onsiere PSM with internal magnets omprises oils an eight NFeB-magnets. Figure epits the ross setion of a quarter of the motor. As alreay briefly isusse in the introution, the PSM is use in an automotive appliation, where it is absolutely important that magnet rotor oil 3 magnet oil stator oil Figure : Cross setion of the onsiere PSM [38]. no motion ours in the ase of e.g. a failure of the power eletronis. This behavior is ahieve by esigning an inhomogeneous air gap whih yiels a large ogging torque, see Fig.. Moreover, this esign also results in a nonsinusoial bak emf an a signifiant influene of saturation in the stator an rotor. As shown in [38], the motor an be aurately esribe by a (magnetially nonlinear MEC, omprising magneto-motive fore (mmf soures esribing the oils an the permanent magnets, an magnetially nonlinear or position epenent permeanes esribing the stator, the rotor, the air gap an the leakages. The mathematial equations of this MEC are erive using network theory, well establishe in the moeling of eletri networks, see, e.g., [4 43]. For this purpose, a tree being ompose of elements of the MEC an onneting all noes of the network without forming a mesh is efine. The hoie of this tree is arbitrary exept for the fat that all mmf soures of the MEC have to be part of the tree. The remaining elements of the network form the orresponing o-tree of the network. The interonnetion of the tree an o-tree elements, i.e. the topology of the MEC, is esribe by the iniene matrix D T = [ DT, D T m, ] DT g, where D = N D, with the wining matrix N = iag[n, N, N ] an the number N of winings per oil. Therein, D is the part of the iniene matrix whih is relate to the oils, D m is relate to the permanent magnets an D g is relate to the permeanes of the tree. The permeanes of the tree an o-tree are ombine in the (iagonal permeane matries G t an G, respetively. Both, G t an G, are nonlinear funtions of the rotor angle an the orresponing mmfs. The mathematial moel of the MEC an then be formulate Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

4 in the form, see [38], t ψ = R i + ( T D v (a [ ] [ ] i = K ψ [( T ] + D D G D T m D (b g with K = [ ( D T D G DT H ( D T D G D T g D g G DT H G t + D g G D T g ]. ( Therein, ψ is the vetor of inepenent flux linkages, i is the orresponing vetor of inepenent oil urrents an R is the eletrial resistane of a oil. The influene of the oil voltages v on the flux linkage is esribe by the matrix ( T D, whih reflets the magneti onnetion of the oils. The set of algebrai equations (b esribes the inepenent oil urrents i an the mmfs of the permeanes of the tree of the magneti network as a funtion of the flux linkage ψ of the oils an the mmfs u T tm = [u ms, u ms ] of the permanent magnets, f. [38]. Finally, H results from the inverse of a transformation matrix T, whih has been use in [38] to eliminate the reunanies of the nonlinear algebrai equations, in the form T = [H, H ]. The torque proue by the motor is given by τ = (u p T G t tg ϕ + [ (H i T, u T tm, ut tg] D G ϕ D T H i, with p = 4 being the number of pole-pairs of the motor. A etaile erivation of this moel an a evaluation of the moel auray is given in [38], where a slightly ifferent notation is use. t shoul be note that the optimal ontrol strategy evelope in this manusript an be applie to any motor onstrution whih an be esribe by an MEC moel of the form (-(3. 3. Calulation of optimal oil urrents The main goal of this work is to erive a ontrol strategy whih alulates the ontrol inputs v in a way that the torque τ traks a esire torque τ. As an intermeiate step to this goal, the urrents i are etermine suh that the resulting torque is equal to the esire torque for a given angle ϕ an the opper losses of the motor are minimal. n the following subsetions, the alulation of optimal oil urrents is isusse for the general magnetially nonlinear ase, the magnetially linear ase an the magnetially linear funamental wave ase. ( Optimal urrents: Magnetially nonlinear ase Calulating optimal urrents for the magnetially an geometrially nonlinear ase iretly leas to a nonlinear optimization problem of the form min i, (i T ( i = min i T ( i H T H i (4,utg }{{} Q subjet to the nonlinear equality onstraints g = τ(i, τ = h = [ D g G DT H, G t + D g G D T g + D g G D T m =, ] [ ] i (5a (5b with the positive efinite matrix Q >. Please note that by means of the onstraint (5a it is ensure that a solution is foun whih yiels the esire torque τ. Moreover, (5b, whih results from the seon row of (b, guarantees that the solution is ompatible with the MEC-moel of the motor. To solve this optimization problem, the Lagrange funtion L is introue in the form L = ( i T Qi + λg + µ T h, (6 with the Lagrange multipliers λ an µ. The first orer neessary optimality onition states that the partial erivatives of L with respet to i,, λ an µ must be equal to zero. n [44] an [45], a similar approah is hosen for the magnetially linear ase, i.e. without taking into aount the ogging torque, the relutane torque an saturation. The partial erivative of L with respet to i reas as ( L i T = Qi + λp [ (H T,, ] D G ϕ D T +λ p [ (H i T, u T tm, utg] T D G ϕ i D T H i H i [ (H + i T ], u T tm, u T D G tg D T g µ + ( H T D G D T g µ i an the partial erivative with respet to an be formulate as given in (8. The partial erivatives with respet to λ an µ of ourse yiel the nonlinear equality onstraints (5. The solution of the onstraine optimization problem (4, (5 is thus trae bak to the solution of a system of nonlinear equations (5, (7 an (8. For a real-time implementation of the optimal ontrol strategy, this set of equations must be solve in eah sampling interval with sampling time T s. Typially, T s is in the orer of 5 µs to µs aoring to pulse-with-moulation (pwm frequenies of khz to 5 khz. Thus, high numeri effiieny (7 Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

5 ( T L = λp G t ϕ + [,, ] D G ϕ D T + λp u T tg H i G t + [ (H ϕ u i T, u T tm, u T tg tg + ( G t + D g G D T g µ ] D G ϕ DT H i ([ G + D g DT u H, G ] [ ] t G + D g D T i G g + D tg u g D T m tg u tg T µ ( is inispensable for a pratial implementation. For this purpose, two assumptions are mae, whih have proven to be pratially feasible: First, the partial erivatives of G t an G with respet to i an are neglete in (7 an (8. This signifiantly simplifies the omplexity of the set of nonlinear equations to be solve. Seon, it is assume that a goo initial guess of the solution at sampling interval k is given by the solution of the previous sampling interval k. This assumption is obviously vali if the angle ϕ an the esire torque τ o not signifiantly hange from one step to the next. The angle ϕ is of ourse suffiiently smooth ue to physis an the esire torque τ is efine in suffiiently smooth manner, i.e. step-like esire torques are not onsiere. f these prerequisites are met, then the set of nonlinear equations ( T L i ( T f nl, L ( L T = f nl, f nl,3 = f nl (x =, (9 ( λ L µ T f nl,4 with the vetor of unknowns x T = [ (i T, (utg T, λ, µ T ], an be solve by applying Newton iteration in the form x j+ k = x j k J nl (xj k f nl(x j k, for j =,..., n i, ( with the initial onition x k = xni k an the number of iterations n i per sampling interval. The Jaobian J nl of f nl an be alulate analytially from (5, (7 an (8. Here again, negleting the erivatives of the permeane matries with respet to an i is meaningful an signifiantly simplifies the alulation. The entries of J nl are summarize in Appenix A. Summarizing, the propose approah allows to alulate optimal oil urrents i for a given esire torque τ in real-time for a magnetially nonlinear, non-funamental Of ourse, the Jaobian J nl if f nl is not inverte in ( but the orresponing set of linear equations is solve for x j+ k. For this purpose, effiient solvers for linear equations are use wave PSM base on a magneti equivalent iruit moel. Before proeeing with the alulation of the real ontrol inputs, i.e. the voltages v, two simplifiations frequently use in literature are analyze from an optimal torque ontrol point of view. First, magneti saturation is neglete whih yiels a magnetially linear moel. Still the non-sinusoial flux harateristis is present. Afterwars, by assuming only funamental wave omponents, the optimal torque ontrol problem is isusse for the well-known q-moel yieling the well-known results from literature. 3.. Optimal urrents: Magnetially linear ase f the iron is not saturate, then the permeane matries G t an G are inepenent of the mmfs an (b is a set of linear equations whih an be solve analytially for ψ as a funtion of i by utilizing the matrix inversion lemma with ψ = ( D T D T l (ϕ ( DT H i + DT m, ( T l (ϕ = ( G + D T g G t D g. ( Using ( an ( in (a, the transforme oil urrents i are given by L t i = ψ ϕ ω R i + ( T D v, (3 with the inutane matrix L L (ϕ = ( D T D T l (ϕ D T H, (4 whih is, as was expete, non-singular. Furthermore, the partial erivative of the flux linkages with respet to ϕ an be written as ψ ϕ = ( T T l ( D D DT ϕ H i + D T m. (5 The mathematial moel in the magnetially linear ase is omplete by the torque equation τ = [ (H p i T ] [ ], u T D Tl [ tm DT D m ϕ, D T ] [ ] H i m. (6 Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

6 95 Base on this mathematial moel for the magnetially linear ase, the optimal inepenent oil urrents an be foun by solving the optimization problem min i ( i T Qi (7 subjet to the (nonlinear salar equality onstraint g = τ ( i τ. (8 ntrouing the Lagrange funtion L = /(i T Qi + λg, the first orer neessary optimality onitions take the form ( T L = Qi + λp(h T T l [ D DT ϕ, D T ] [ ] H i m i (9a ( T L = τ τ, (9b λ whih still onstitutes a set of nonlinear equations. Thus, again Newton iteration is employe, as propose in ( for the general magnetially nonlinear ase. The elements of the Jaobian J l of (9 rea as J l, = Q + λp ( H T T l D ϕ D T H J l, = Jl, T = p ( H T T l [ D DT ϕ, D T ] [ ] H i m J l, =. (a (b ( Thus, the metho for the alulation of the optimal urrents i for a given esire torque τ in the magnetially linear ase is similar to the magnetially nonlinear ase but signifiantly less omplex Optimal urrents: Funamental wave ase The inutane matrix L an T l in (4 an ( epen on the angle ϕ. f only funamental wave omponents, i.e. omponents multiplie by sin(pϕ or os(pϕ, are onsiere, then the well-known q-transformation (Blonel Park transformation, see, e.g., [, ] an be applie using the transformation matrix T q T q (ϕ = os(pϕ os(pϕ π 3 os(pϕ 4π 3 sin(pϕ sin(pϕ π 3 sin(pϕ 4π 3. ( The resulting transforme magnetially linear funamental wave moel is given by, see also [,, 38] t i = ( 3 3 L L mpωi q R i + v (a m t i q = ( 3 3 L mpωi 3 Ĵω R i q + v q, (b L m with the transforme urrents an voltages i = T q i, v = T q v (3 i q i i i 3 v q v v v 3 an the inutane L m. Sine the eletrial interonnetion fores v =, the zero omponent i also vanishes, i.e. i =. The orresponing transforme torque reas as, f. [,, 38] τ = p ˆMu ms N i q. (4 The onstant oeffiients L m, Ĵ an ˆM an be obtaine from the magnetially linear moel (-(6, e.g., by applying a Fourier analysis, see, also [38]. The etermination of optimal urrents for a given esire torque τ is trivial, sine (4 implies i q = τ /(p ˆMu ms N an minimal losses are obtaine by setting i = Simulation results n this setion, the optimal urrents are evaluate base on the magnetially nonlinear moel presente in [38], whih was alibrate an valiate by measurement results on a test stan. Basially, two major points shoul be isusse: What is the improvement by using the magnetially nonlinear or the magnetially linear moel in omparison to the funamental wave moel typially use in literature? How oes the number of Newton iterations n i use in ( influene the auray of the optimal solution? As alreay shortly isusse in Setion 3., besies the number of iterations n i, the quality of the initial guess x k use in the Newton iteration has an important influene on the auray. Obviously, it an be expete that the quality of the initial guess inreases if the solution of the nonlinear equations (9 for the magnetially nonlinear ase an (9 for the magnetially linear ase only slightly hanges from one sampling time (k T s to the next kt s. Clearly, the solution of the nonlinear equations hanges ue to hanges in the rotor position ϕ an the esire torque τ. For the subsequent isussions, it is assume that the esire torque τ is hosen onstant. Aitionally assuming a onstant angular spee ω = ϕ, ϕ = ϕ k ϕ k = ωt s hols. The largest value of ϕ arises at maximum spee ω max, whih is given by ω max = π ra s ( rpm. n the subsequent simulation results, an angular spee of 83 rpm is hosen, whih orrespons to a typial operating point of 7% of the maximum spee. Using a sampling time T s = µs, this results in ϕ =.5. Fig. shows the results of the optimal urrents obtaine from the magnetially nonlinear moel for ifferent values of the esire torque τ from N m to 3 N m, whih orrespons to the maximum torque of the motor. n the left olumn, the three oil urrents i, i an i 3 are epite, whih result from ( with (9 for n i = iterations. The right olumn shows the resulting errors τ τ Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

7 urrent in ma urrent in A urrent in A urrent in A τ = N m τ = N m τ = N m τ =3 N m τ τ in mn m.5..5 i 5. i 7.5 i 3. τ τ in mn m τ τ in mn m τ τ in mn m τ = N m τ = N m τ = N m τ =3 N m Figure : Optimal urrents an torque error τ τ for the magnetially nonlinear ase for ifferent esire torques τ. 6 Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

8 in the torque. t an be seen that the propose metho to alulate the optimal urrents for the magnetially nonlinear moel results in very small errors in the torque over the entire operating range of the motor. Taking a loser look at the optimal urrents also reveals that the motor uner onsieration shows a signifiant nonlinear behavior ue to magneti saturation an non-funamental wave harateristis of the oil fluxes. Espeially, the ase τ = N m emphasizes the nee to atively ontrol the urrents in orer to suppress the pronoune ogging torque of the motor. Sine the alulation of the optimal urrents given in Fig. is base on the same magnetially nonlinear moel whih was use for the simulations, the torque errors given in Fig. result solely from the non-exat solution of the onstraine optimization problem (4, (5. Although two Newton iterations alreay yiel a goo auray in the torque, in Fig. 3 the influene of the number of iterations n i on the auray of the torque is examine in more etail. As expete, a higher number of iterations improves the auray. For n i = almost perfet traking of the esire torque is ahieve. For the pratial appliation, however, a ompromise between auray an omputation time must be foun. As it will be shown in the measurement results isusse later on, a number of n i = iterations turns out to be a goo hoie. Sine the magnetially linear ase is numerially less expensive, a higher number of iterations oul be use. However, the errors resulting from the numeri solution of the optimization problem are very small alreay after iterations suh that n i = is also a goo hoie for the magnetially linear ase. τ τ in mn m it. it. 5 it. it. it Figure 3: nfluene of the number of Newton iterations on the auray of the optimal solution for the magnetially nonlinear ase for τ = N m. Finally, the results given in Fig. 4 show the avantages gaine from the usage of the magnetially nonlinear moel for the alulation of optimal oil urrents. Taking a look at the ase τ = N m first, it turns out that the optimal urrents obtaine from the magnetially nonlinear an the magnetially linear moel exhibit a similar behavior. Even though the results of the magnetially nonlinear moel show better torque auray, the results of the magnetially linear moel are still omparably goo. This is ue to the fat that for the resulting small values of the urrents the influene of the magneti saturation of the ore is negligible. The results base on the funamental wave moel (q-moel, however, are signifiantly worse. As a matter of fat, it is not possible to systematially inlue the ogging torque in the q-moel, whih results in i = i = i 3 = as the optimal values. Then, the resulting torque τ is equal to the ogging torque of the motor. The avantages of using the magnetially nonlinear moel instea of the magnetially linear moel an be reognize for large values of τ. The results for τ = N m in Fig. 4 show that negleting magneti saturation entails an error in the torque of approximately 5 mn m. n onlusion, these first simulation results emonstrate that onsiering the magneti nonlinearities in the alulation of the optimal urrents yiels a signifiant improvement of the auray of the torque in omparison to methos base on a magnetially linear moel or the q-moel. 4. Flatness-base urrent ontrol n the previous setion, optimal values of the urrents i have been alulate suh that the torque τ traks a esire torque τ an the opper losses are minimize. These results are the basis for a flatness-base feeforwar an feebak ontrol strategy to be evelope in this setion. 4.. Magnetially nonlinear ase The solution of the optimization problem (4, (5 results in optimal urrents i an optimal values u tg of the mmfs of the tree permeanes. Consiering (b with (, the orresponing optimal values of the flux linkages ψ are given by ψ = ( D T D G ( DT H i + D T g u tg + DT m. (5 Using this result in (a, i.e. in t ψ = R i + ( T D v, (6 with ( D T v =, yiels the feeforwar part v of the ontrol input v in the form ( v = H t ψ + R i. (7 The time erivative of the optimal oil flux linkage ψ an be alulate from (5 t ψ = ( ( T D D G D T H ( D T D G ϕ t i + D T g t u tg + ( DT H i + D T g u tg + D T m ω, (8 Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

9 in ma i i in A τ = N m τ = N m nonlinear linear q τ τ in mn m τ τ in mn m τ = N m 75 τ = N m Figure 4: Optimal urrent i an torque error τ τ for the magnetially nonlinear, the magnetially linear an the funamental wave ase for ifferent esire torques τ where again the partial erivatives of G with respet to i an have been neglete. To obtain the time erivatives of the optimal urrent i an the optimal mmfs of the tree permeanes u tg, the first orer optimality onitions (9 are utilize. The total time erivative of f nl reas as t f nl = J nl t i u tg λ µ + f nl ϕ ω + f nl τ t τ =, (9 with the Jaobian J nl an the partial erivatives f nl / ϕ an f nl / τ summarize in Appenix A. The time erivatives an be easily obtaine from this set of linear equations, sine J nl is non-singular. Remark. The alulation of ψ /t as esribe in (8 an (9 is omputationally expensive, even if the Jaobian J nl neee in (9 has alreay been alulate in the Newton iteration (. Given the fat that the ontrol strategy is implemente using a fixe sampling time T s, the approximation t ψ (kt s ψ,k ψ,k T s (3 an be obtaine. Using this approximation in (7 signifiantly simplifies the alulation of the feeforwar ontrol part an is therefore preferable for the pratial appliation. The errors resulting from this approximation are typially small ue to the small sampling time T s. 3 8 With (7 an optimal feeforwar ontrol law v is given. To ope with moel inauraies an external isturbanes, in aition a feebak ontroller has to be esigne. n a pratial appliation, measurement of the oil flux linkage is not reasonable. Thus, a ontrol strategy base on the measure oil urrents has to be evelope. Using the feeforwar ontrol (7 in (a, the following ifferential equation for the error in the oil flux linkage an be formulate ( ψ t ψ = R e i + ( T D v, (3 with the urrent error e i = i i an the feebak part v of the input voltage v = v + v. The oil flux linkage error an be foun as a funtion of the urrent error e i an the error e u in the mmfs of the tree permeanes, e u = u tg, from the following set of equations [ ] ( [( T ] ψ ψ = D D G DT H D g G DT H e i [ ( ] (3 D + T D G D T g G t + D g G D T e u. g This equation results from (b assuming G ( D T ū t, ϕ G ( D T ū t, ϕ an G t (, ϕ G t (u tg, ϕ. The solution of (3 for the oil flux linkage error reas as ψ ψ = ( D T D T l DT H e i = L (ū t, ϕ e i, (33 Note that this assumption is basially equal to assuming that the ontrol errors are small. Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

10 35 with T l from (. Please note that, in ontrast to the magnetially linear ase, the inutane matrix L is both a funtion of ū t an ϕ in the magnetially nonlinear ase. Using (33 in (3, the error ynamis is given by L t e i = ( ( T Tl D D ϕ ω + T l ū t D T ū t t H e i R e i + ( D T v. The feebak ontrol law v [ ( ( v = H T Tl D D ϕ ω + T l ū t ū t t ( t + H L λ i e i λ i e i t, (34 D T H + R ] e i (35 with λ i, λ i >, finally yiels an exponentially stable urrent error ynamis. Remark. The feebak ontrol law (35 again inlues some parts whih are omputationally expensive. However, in a pratial implementation, negleting the first part on the right han sie of (35 turns out to be a feasible simplifiation if the urrent error e i is small. Then, the simplifie feebak ontrol law reas as t v = H L ( λ i e i λ i e i t. (36 The inutane matrix L is, in the magnetially nonlinear ase, both a funtion of ϕ an ū t. Thus, it reflets the hanges ue to the rotor position ϕ an saturation of the iron. Fig. 5(a shows the entries L,kj, k, j =, of L for τ = N m of the motor uner onsieration. t an be seen that the inutanes show a signifiant variation with respet to a hange in the rotor position. A omparison of the self inutane L, for τ = N m with L, for τ = 3 N m in Fig. 5(b reveals that the influene of saturation on the inutane matrix L is rather small an an therefore be neglete in a pratial appliation, i.e. L (ū t, ϕ L (ū t, ϕ using a nominal operating point ū t. As will be isusse in Setion 5 a further simplifiation using the average value L, L = π π L (ū t, ϕϕ, (37 oes not signifiantly eteriorate the urrent ontrol auray for the onsiere motor. Of ourse, it is require to hek if the above prerequisites are fulfille for a speifi motor before these simplifiations an be mae. inutane in mh inutane in mh (a (b L, L, L, L,.75 τ = N m τ = 3 N m Figure 5: (a Entries of the inutane matrix L for τ = N m. (b Comparison of L, for τ = N m an τ = 3 N m. the optimal values i of the oil urrents, the orresponing voltage an be obtaine from (3 in the form ( v = H L t i + ψ ϕ ω + R i, (38 with ψ / ϕ resulting from using the optimal urrents i in (5. The time erivative of i in (38 an be obtaine as a solution of [ ] f l t = J i l t λ + f l ϕ ω + f l τ t τ =, (39 using f l, being the right-han sie of (9, an J l from (. The partial erivatives of f l with respet to ϕ are given by f l, ϕ = λp ( H T T l D ϕ f l, ϕ [ ] DT H i D T m = [ (H p i T ] [ ], u T D T l [ tm DT D m ϕ, D T m (4a ] [ ] H i (4b an f l, / τ =, f l, / τ = hols. For the pratial implementation, it again proves to be useful to approximate i /t by Magnetially linear ase The alulation of the feeforwar ontrol v is signifiantly simplifie for the magnetially linear ase. Given f. Remark. i t (kt s i,k i,k, (4 T s 9 Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

11 The ynamis of the urrent error e i = i i is given by L t e i = ( T T l D D ϕ D T H e i ω R e i + ( T D v (4 an the feebak ontrol law v ( ( v T T l =H D D ϕ D T H e i ω + R e i ( t + H λ i e i λ i e i t, (43 with λ i, λ i >, reners the error ynamis exponentially stable. As isusse in Remark for the magnetially nonlinear ase, also in the magnetially linear ase the first part of the right han sie of (43 an be neglete an L (ϕ an be approximate by the average L for the onsiere motor without introuing signifiant errors Funamental wave ase For the funamental wave moel, again a flatness-base ontrol strategy in the form of v = v +v an v q = vq +vq is evelope. n this ase, the feeforwar part is simply given by v = 3 L mpωi q vq = R i q + 3 Ĵω + 3 L m t i q, (44a (44b where i = an i q = τ /(p ˆM are use. t an be easily seen that the feebak ontrol law v = R e i + 3 L mpωe iq + 3 t L m ( λ i e i λ i e i t (45a vq = R e iq 3 L mpωe i + 3 t L m ( λ i e iq λ i e iq t, (45b in ombination with the feeforwar ontrol (44 yiels an exponentially stable ynamis for the errors e i = i i an e iq = i q i q with λ i, λ i >. A simplifie version of (45 in the form ( λ i e i λ i t v = 3 L m vq = 3 t L m ( λ i e iq λ i e i t (46a e iq t, (46b whih is frequently use in pratial appliations, only yiels a small reution of the ontrol auray Measurement results To ompare an to evaluate the performane of the propose torque ontrol onepts, measurement results on the test stan epite in Fig. 6 are presente. The test stan omprises the PSM uner onsieration whih is ouple to a harmoni rive (loa motor via a torque sensor an a high resolution enoer. By means of the harmoni rive it is possible to fix the angular spee of the motor at a esire value. The torque generate by the PSM is measure by the torque sensor. The PSM is ontrolle by a threephase brige using MOSFETs as power swithes. A fixe pulse-with moulation frequeny of khz is use an the uty yles of the iniviual half-briges are utilize to ajust the voltages v k, k =,, 3, applie to the oils. The ontrol strategies esribe in the Setions 3 an 4 were implemente on a SPACE 3 real-time harware equippe with a GHz PowerPC proessor. The funamental wave ontrol strategy was implemente at a sampling time of µs, while for the magnetially linear an the magnetially nonlinear ontrol strategy an inrease sampling time of µs ha to be use ue to the higher numerial omplexity. Moreover, average values L of the inutane matrix aoring to (37 an n i = Newton iterations have been use. For the subsequent measurements, a fixe angular spee of 4 rpm was hosen. Please note that this slow spee prevents exitation of resonanes of the mehanial setup whih woul eteriorate the auray of the measure torque. n the next setion, it will be proven that the propose ontrol onept also works well for fast angular spees. PSM torque sensor angle sensor fly wheel Figure 6: Setup of the test stan. harmoni rive Fig. 7 shows the measurement results of the oil urrent i an the torque τ for the three ontrol strategies of Setions 3 an 4, an for τ =, an N m. n aorane with the simulation results in Fig. 4, the funamental wave moel performs worst, espeially for τ = N m. This is, as has been alreay isusse, ue to the fat that the ogging torque is not inlue in the funamental wave moel. The magnetially linear an the magnetially nonlinear ontrol strategies, however, show almost iential behavior for τ = N m. The benefits of the nonlinear Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

12 i in ma i in A i in A τ = N m τ = N m τ = N m q linear nonlinear τ in mn m τ in N m τ in N m τ = N m τ = N m τ = N m Figure 7: Comparison of the measure oil urrent i an torque τ of the funamental wave (q, the magnetially linear an the magnetially nonlinear ontrol strategy for τ =, an N m. ontrol strategy in omparison to the linear an the funamental wave strategy is evient for inrease esire torques τ = N m an τ = N m, see Fig. 7. n Fig. 8 the orresponing ontrol input v an the urrent ontrol error i i for the nonlinear ontrol strategy for τ = N m are epite. Taking a loser look at the ontrol input, harateristi steps an be seen in the measurements. These steps result from the ompensation of the nonlinearities of the three phase brige in the 4 39 viinity of zero urrents. The measurement of the urrent ontrol error shows that the urrent is ontrolle to the esire value within the measurement auray, whih is in the orer of ±7 ma in the present experimental setup. t has to be mentione that in omparison to the simulation results an inrease eviation from the esire torque values is present in the measurements. This an be attribute to two fats: 4. The mehanial setup, in partiular the flexible ouplings, introues small perioi isturbanes whih are also measure by the torque sensor.. The moel, as a matter of fat, oes not perfetly 4 represent the real system behavior, see also [38]. These moel errors are then reflete in the measure torque. t is worth mentioning that the measurements of the torque sensor are only use to evaluate the ontrol quality but are, of ourse, not part of the feebak loop. Thus, even if the urrent perfetly traks the esire urrent, f. Fig. 8, the errors in the moel from urrent to torque are still present. However, it is well oumente that the propose nonlinear ontrol onept whih is base on the magenti equivalent iruit moel of [38] brings along a signifiant improvement of the torque ontrol auray. n onlusion, the measurement results show that the propose nonlinear ontrol onept is pratially feasible an yiels goo traking results of the esire torque. 6. Control of angular spee There are a number of appliations where the PSM is use in a torque-ontrolle moe as e.g. in eletrial power steering systems or tration appliations. Therein, it is evient that an improvement of the torque ontrol Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

13 45 43 in ma i i v in V Figure 8: Control input v an urrent ontrol error i i of the nonlinear ontrol strategy for τ = N m. auray iretly yiels an improve system performane. n most ases, however, the torque (or urrent ontroller is use as a suborinate ontrol loop for an outer spee or position ontrol. n this setion, it will be outline that the improve auray of the propose nonlinear torque ontrol strategy also entails an improvement of a asae spee ontrol loop. To o so, the test stan epite in Fig. 6 is aapte by removing the torque sensor an the harmoni rive, iretly oupling the fly wheel an the angle sensor to the PSM. The angular spee ω = ϕ of the resulting system an be esribe in the form t ω = ( ( ω v ω tanh + τ, (47 θ m ω with the moment of inertia θ m of the PSM inluing the fly wheel an the visous amping oeffiient v. The Coulomb frition of the setup is approximate by the term tanh(ω/ω, where ω is use to parameterize the steepness at ω =. n a asae ontroller esign it is usually assume that the torque ontroller is very fast an thus the error between the esire torque τ an the real torque τ generate by the PSM is negligible. Setting τ = τ in (47, τ an be onsiere as a virtual ontrol input to the system. A two egrees-of-freeom flatness-base ontrol onept of the form τ = τ + τ is frequently employe for the spee ontrol of suh a system in literature. The feeforwar part τ an the feebak part τ are given by ( ω τ = v ω + tanh + θ m ω t ω (48a ( t τ = θ m λ ω e ω λ ω e ω t, (48b with the at least twie ontinuously ifferentiable esire angular spee ω, the ontroller parameters λ ω, λ ω > an the spee error e ω = ω ω. Remark 3. For the ontrol strategies evelope in Setions 3 an 4, the time erivative of the esire torque τ is neessary, f. (9 for the magnetially nonlinear ase, (39 for the magnetially linear ase an (44 for the funamental wave ase. The ontroller part τ inlues the measure spee ω, whih an ause problems when performing this ifferentiation. Thus, for the pratial appliation it is often useful to set τ τ, with τ from (48a. A similar iea an of ourse also be use for the simplifie erivations given by (3 an (4. Fig. 9 presents the measurement results of the spee ontroller for onstant esire spee ω. Therein, the spee ontroller (48 was use with the three torque ontrol strategies of Setions 3 an 4 in the suborinate ontrol loop. t an be seen that for the fast angular spee of rpm only small ifferenes between the three torque ontrol strategies are visible, while for slow angular spees the magnetially nonlinear an the magnetially linear ontrol strategies yiel signifiantly better results ompare to the funamental wave ase. The main reason for the errors in the spee are the errors in the suborinate torque ontrol, where the frequeny of the resulting isturbane is proportional to the angular spee. The high frequeny isturbanes for rpm are well suppresse by the mehanial inertia, while for lower angular spee an inrease influene an be seen. Only small ifferenes an be seen between the magnetially linear an the magnetially nonlinear ase sine the average torque neessary to rive the mehanial setup is rather small. As epite in Fig. 7, the magnetially linear an magnetially nonlinear ontrol strategies show similar performane in this torque range. This result proves that the propose torque ontrol strategy is benefiial also for spee ontrol, in partiular for slow angular spees as they typially our for preision position tasks in robotis. Remark 4. A number of papers ealing with the spee an the position ontrol of PSMs with pronoune ogging torque (as the PSM uner onsieration has been reporte in literature, see, e.g., [46 49]. A frequent approah to takle this problem is to onsier the torque ripple in the form of a (perioi isturbane for the spee ontroller, whih is ompensate by more or less involve ontrol strategies. n omparison to these approahes, using a torque ontroller as presente in Setions 3 an 4 alreay ensures that only small torque ripples are proue Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

14 ω in rpm ω = rpm v in V ω in rpm ω in rpm ω = rpm ω = rpm q 99.5 linear nonlinear time t in s Figure 9: Comparison of the spee ontrol for onstant esire spee using the torque ontrol strategies of Setions 3 an 4. by the PSM. Thus, the aurate ontrol of the spee is signifiantly simplifie. Espeially for rive trains with small inertia or low angular stiffness, this approah an be onsiere avantageous. To analyze the urrent ontrol auray, the esire an the measure oil urrent i an i, respetively, are presente in Fig.. The orresponing ontrol input, i.e. the oil voltage v, is also epite there. t an be seen that, although the urrent has to trak a rather omplex esire urrent, a very goo traking auray an be ahieve. Finally, in Fig. the results of an experiment, whih rives the PSM to its operational limits, are given. Here, the esire spee is hange from 6 rpm to 6 rpm (half of the rate spee within.6 s using approx..5- times the rate torque of the PSM. An almost perfet traking auray of the esire spee is obtaine using the spee ontrol strategy from (48 an the magnetially nonlinear torque ontrol strategy of Setion 4.. The or urrent in A i i time t in ms Figure : Control input v an urrent ontrol auray for ω = rpm. responing oil urrent i an oil voltage v are also epite in Fig.. n onlusion, the experiments performe in this setion an in Setion 3.4 prove the pratial feasibility an emonstrate the improvements in ontrol auray ompare to lassial ontrol strategies base on a funamental wave moel. 7. Conlusion n this work, an optimal torque ontrol for PSMs esribe by MEC moels was presente an teste on an experimental setup. t was shown that by systematially inorporating the magneti saturation an the non-funamental wave harateristis into the moel an the ontroller esign, signifiant improvement of the ontrol auray an the performane an be obtaine. Moreover, the pratial feasibility was emonstrate by means of measurements on an experimental setup. Up to now, limitation of the ontrol input, i.e. the oil voltages, has not been taken into aount. Thus, future researh will be evote to the question how to exten the propose nonlinear ontrol strategy to the fiel weakening range of the motor. Moreover, sine the MEC moeling approah is not limite to PSMs, the appliation of this metho to other motor esigns as swithe relutane, synhronous relutane or inution mahines will be examine. Finally, the usage of the MEC moel in a moel preitive ontrol setup is a further topi of urrent researh. Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.

15 ω in rpm τ in N m ω ω time t in s i in A v in V time t in s Figure : Change of esire spee using the spee ontroller of (48 an the magnetially nonlinear ontrol strategy of Setion 4.. Appenix A. Entries of the Jaobian J nl of f nl This appenix summarizes the entries of the Jaobian J nl for the magnetially nonlinear ase 3. Given (f nl, T = L/ i, the orresponing partial erivatives are given by f nl, i = Q + λp [ (H T,, ] D G ϕ D T H (A.a f nl, f nl, λ = λp [ (H T,, ] D G ϕ D T = p [ (H T,, ] D G ϕ D T H i (A.b (A. 53 an the partial erivatives of (f nl,3 T = L/ λ an (f nl,4 T = L/ µ are given by f nl,3 λ = (A.3a f nl,3 µ = (A.3b f nl,4 =. µ (A.3 The remaining entries of J nl result from the symmetry of this matrix. f nl, µ = ( H T D G D T g (A. The partial erivatives of (f nl, T = L/ result in f nl, = λp G t ϕ + [,, ] D G ϕ D T (A.a f nl, λ = p G t ϕ + [,, ] D G ϕ D T H i (A.b f nl, µ = G t + D g G D T g (A. 3 Note that the erivatives of the permeane matries G t an G with respet to an i are assume to be negligible. 4 Post-print version of the artile: W. Kemmetmüller, D. Faustner, an A. Kugi, Optimal torque ontrol of permanent magnet synhronous mahines using magneti equivalent iruits, Mehatronis, vol. 3, pp. 33, 5. oi:.6/j.mehatronis.5..7 The ontent of this post-print version is iential to the publishe paper but without the publisher s final layout or opy eiting.