Robust MRAS Speed Observer and an Improved Zero-speed Position Estimation Design for Surface Permanent Magnet Synchronous Motor

Robust MRAS Speed Observer and an Improved Zero-speed Position Estimation Design for Surface Permanent Magnet Synchronous Motor D. Zaltni and M. N. Abdelkrim National Engineering School of Gabes Research unit of Modelling, Analysis and Control Systems Rue Omar al Khattab, 69 Gabes TUNISIA zaltni dalila@yahoo.fr Abstract: Model Reference Adaptive System (MRAS) based techniques are one of the best methods to estimate the rotor speed and position due to its performances and straight forward stability approach. In this paper, we propose a new robust MRAS scheme based on sliding mode techniques to the rotor speed of a Surface Permanent Magnet Synchronous Motor (SPMSM). Furthermore, an Estimator/Observer swapping system is designed to overcame position observability problems at zero speed which is an unobservable state point. The stability of the proposed observer is also discussed. Various tests are carried out in simulation to highlight the effectiveness and the robustness of the proposed rotor speed and position estimations design. Key Words: Permanent magnet synchronous motor, Model reference adaptive system, Observer, Sliding mode Introduction Permanent Magnet Synchronous Motor (PMSM) is an ideal candidate for high-performance industrial drives since it features simple structure, high-energy efficiency, reliable operation and high power density. In the conventional PMSM drive systems, speed and torque control are achieved by obtaining the rotor position or speed informations through shaft sensors such as optical encoders, Hall-Effect sensors or resolvers. The use of such sensors will increase the complexity, weight and cost of the system and reduce the overall reliability of the controlled drive system. Speed and position sensorless control of motor drive systems overcome above shortcomings and improve the overall system reliability, ruggedness and dynamic performance. Several methods are available for rotor speed estimation in a sensorless PMSM drive and they have been extensively studied in the last few decades [,, 3,, 5, 6, 7, 8]. The speed estimate is mandatory if speed control (feedback) is employed. Also the speed estimate is needed if decoupling is intended in the current regulation loops of the rotational reference frame. In a sensorless PMSM, the speed can be by various techniques. A speed estimate can be directly obtained using the machine s model equations. However, the accuracy is not very good. Other techniques are available: the Extended Kalman Filter method and the Luenberger observer construct full order estimators based on the machine model. Both approaches tend to depend heavily on the machine parameters and are generally difficult to implement. Compared with other methods, MRAS based techniques are one of the best methods to estimate the rotor speed due to its performance and straight forward stability approach. The MRAS system is wellknown in the sensorless control of Induction Motors and has been proved to be effective and physically clear [9,, ]. However, this techniques still sensible to some parameter variation [, 3]. To overcome this problem, Sliding Mode (SM) techniques can be introduced to the MRAS structure to ensure robustness and accuracy of the observer. In fact, SM techniques has attractive advantages of robustness to disturbances and insensitivity to parameter variations when the SM happens [, 5, 6]. However, the chattering behavior, that is inherent in standard sliding mode control, is often an obstacle for practical application if neglected. Higher Order Sliding Mode (HOSM) [7, 9,, ] is one of the solutions which does not compromise robustness. In this paper, a Second Order Sliding Mode (SOSM) observer is used as reference model in the proposed MRAS speed observer. This reference model is a speed-independent observer which computes the Back Electro- Motive Forces (BEMFs). The proposed model reference based on HOSM techniques is designed to ensure ISSN: 79-57 ISBN: 978-96-7-85-

the robustness of the observer and to reduce the chattering phenomenon. Therefore, the outputs signals of the reference model are smooth enough to be used directly as a reference BEMFs. A speed-dependent model is designed as adjustable model which computes BEMFs. Outputs of the reference and adjustable model are then fed into an adaptive sliding mode mechanism ensuring the convergence of the speed to the speed. The proposed MRAS observer is of high robustness and accuracy compared to other MRAS based techniques proposed in the literature [, 3]. The stability of the proposed observer is presented and discussed. The rotor position is obtained from the phase of the BEMFs. Since the position can not be calculated at very low frequencies, because the BEMFs are practically non existent, an Estimator/Observer swapping system is proposed to ensure rotor position estimation in all frequencies range. Tests are carried out at various operating conditions to illustrate the effectiveness and the high robustness of the proposed estimation design This paper is organized as follows: In section two, the mathematical model of the used PMSM is presented. In section three, a robust MRAS speed observer based SM techniques is proposed. Simulation results are illustrated in section four. Finally, some concluding remarks are given in the last section. T l is the load torque [i α i β ] T, [u α u β ] T are stator current and voltage vector respectively K e is the BEMF constant. θ e is the rotor position. 3 Robust MRAS Speed Observer The structure of the proposed MRAS speed observer is shown in Fig.. This structure is made up of a reference model, an adjustable model and an adaptation mechanism. In this structure the mechanical speed is considered slowly variable with respect to electrical dynamics. This assumption is a usual one for synchronous motor. Mathematical Model Of SPMSM The mathematical model of the used Surface Permanent Magnet Synchronous Motor in the (α-β) fixed coordinate is given by equation () and () [8, ]. ( ) i α = [( ) ( ) uα iα R i β L u β i β ω e K e ( sin(θe ) cos(θ e ) )] () ω e = P J φ m( sin(θ e )i α + cos(θ e )i β ) f v J ω e T l () J where ω e = Pω is the electric rotor speed ω is the rotor speed P is the pair pole number R is the stator resistance L is the stator inductance J is the moment of inertia φ m is the rotor flux f v is the viscous friction Figure : Structure of the MRAS Speed Observer based HOSM 3. The reference model The reference model consists in designing a second order sliding mode observer (Super Twisting Algorithm) which computes the reference BEMFs ê α,β = [ê α ê β ] T using only measured stator currents and voltages. This reference model does not depend on the velocity. The general form of the Super Twisting Algorithm is defined as follows [9]: u(e ) = u + λ e sgn(e ) u = α sgn(e ) (3) with e = x ˆx, λ, α > are the observer parameters, u is the output of the observer, x is the variable and: ISSN: 79-57 3 ISBN: 978-96-7-85-

if e > sgn(e ) = if e < [ ] if e = Let e α and e β be the BEMFs and x = [i α i β ]. Consider only current dynamic equations of the SPMSM, we can write: ẋ = ax be α + cu α ẋ = ax be β + cu β () with Let eα = ω e sin(θ e ) e β = ω e cos(θ e ) (5) [x a x b ] = b[e α e β ] (6) be the vector of unknown variables. Using (6), equation () becomes: ẋ = ax + x a + cu α ẋ = ax + x b + cu β (7) Currents and voltages are assumed to be measurable. Applying the STA (3) to system (7), we obtain systems (8) and (9): ˆx = x a + ax + cu α + λ e sgn(e ) (8) x a = α sgn(e ) Following the results proposed in [] and [] with respect to the super twisting algorithm (3) dedicated to the observer design given by equations (8) and (9), we set: Corollary: For any initial conditions x(), ˆx(), there exists a choice of λ i and α i such that the observer state ˆx converges in finite time to x, i.e. ˆx x and ˆx x then e, e, ė and ė converges to zero and by consequence x a x a and x b x b. Proof : Consider system (). To show the convergence of (ˆx, x a ) to (x, x a ) (ie., (e, e a ) (,)) let consider the system s dynamic ë ë = f (x b ) α sign(e ) λ e ė () with d x dt = ẋsign(x) Equation () leads to ë where [ ] f +, f+ α sign(e ) λ e ė (3) f + = max(f (x b )), ˆx = x b + ax + cu β + λ e sgn(e ) x b = α sgn(e ) (9) Where e = x ˆx, e = x ˆx and λ, λ, α, α are positive constants that will be given later. x a and x b are the values of the unknown variables x a and x b. According to equations (7), (8) and (9), error dynamics of the observer are given by: ė = e a λ e sgn(e ) () ė a = f (x b ) α sgn(e ) ė = e b λ e sgn(e ) ė b = f (x a ) α sgn(e ) () With e a = x a x a, e b = x b x b, f (x b ) = ω e x b and f (x a ) = ω e x a Figure : Finite time convergence behavior of the proposed observer : The majoring curve for the finite time convergence. ISSN: 79-57 ISBN: 978-96-7-85-

Figures and 3 illustrate the finite time convergence behavior of the reference model. In what follows we will give the error trajectory for each quadrant in the worst cases. de/dt 6 5 3 3.....6.8....6 e Figure 3: The trajectory ė = f(e ): Finite time convergence. First quadrant: e > and ė > Starting from point A of Fig. the trajectory of ė = f(e ) is in the first quadrant e and ė. The rising trajectory is given by ë = (α f + ) By choosing α > f + we ensure that ë < and hence ė decreases and tends towards the y-axis, corresponding to ė = (point B in Fig. ). Let e () be the intersection of this trajectory with ė = ; thus e () = (α f + )ė () () Then the rising trajectory for e > and ė > can be given by the following expression: ė = (α f + )(e () e ) (5) Second quadrant: e > and ė < In this case, ë = f + α sign(e ) λ e ė becomes negative (ë < ) on making a good choice of α which leads to (α + f + ) > λ e ė So, the rising trajectories as illustrated in Fig. are given as e = e () with ė λ (α + f + )e/ and ė = ė () = λ (α + f + )e/ () with ė > λ (α + f + )e/ (where ė () corresponds to the intersection of ė = λ (α + f + )e/ and e = e () (denoted by point C in Fig. )). Hence, if ė () ė () < which means that λ > (α + f + ), we can state that α f + α > f + and λ > (f + + α ) α f + (6) are sufficient conditions guaranteeing the state convergence (i.e. the states (e, ė ) tend towards e = ė = and in consequence the full convergence of ė (i) ). i= Now we will consider the system (). Following the same procedure, we can state that α > f + and λ > (f + + α ) α f + (7) are sufficient conditions guaranteeing the state convergence (i.e. the states (e, ė ) tend towards e = ė = and in consequence the full convergence of ė (i) ). i= Where f + = max(f (x a )), Now, in order to prove the finite time convergence, it is necessary to know the time passed in following the trajectory in each quadrant. In the first quadrant, the rising trajectory is given by ë = (α f + ). So, by integration, we can find that ė = (α f + )t+cte with cte = ė (). Hence the necessary time for going from A to B is t () = (α f + )ė() (8) In the second quadrant, we can find the rising trajectory leading to a much longer time. This is given by ë = f + α λ e ė. Then, as previously, we can write that ė = ( α f + )t λ e / + cte with cte = λ e / () because at B we have t = and ė =. Now in C we have ė () = ( α f + )t () + λ e / () ; then the necessary time for going from B to C is t () = ė ()+λ e / () which can be expressed as a function of ė () α +f + as: t () = (α + f + ) + λ λ (α + f + ) (α f + ) ė () (9) Following the same procedure and using the property of symmetry of the first and the third quadrant, we can find the time for going from C to D which can be given by t () = (α f + )ė(), and expressed also as function of ė () as t () = λ (α f + ) ė () () ISSN: 79-57 5 ISBN: 978-96-7-85-

and for the second and fourth quadrants, the time for going from D to E is t (3) = ė()+λ e () / ; since ė () = (α f + ) λ α +f + λ α f + α +f + α f + ė (), one easily obtains t (3) = (α f + ) + λ λ (α f + ) α f + ė () and e () / = α + f + α f + ė () () Let T d () be the necessary time interval for going from A to E; then T d () = t () + t () + t () + t (3) = Kė () () where K = + (α f + ) + (α + f + ) + λ (3) λ (α + f + ) (α f + ) + ) + λ + (α f λ (α f + ) λ (α f + ) α + f + α f + where ˆω e is the velocity (the output of the MRAS observer (show fig. )) and ( ) J = (6) For convergence, a feedback loop is introduced and the feedback gain is: ( ) G = g (7) where g is a positive constant. 3.3 The adaptation mecanism If the velocity estimation error exist, it will lead to the BEMFs estimation error : ε = ê α,β ẽ α,β (8) Then, this error together with the estimation model s output ẽ α,β is used to construct the manifold S as : and thus the convergence time for this step can be, in the worst case, given as T = T d (i) = Kė () + KWė () + KW ė () i= + KW 3 ė () +... () which gives With W = (α f + ) λ T = W Kė () α +f + α f + < and where λ is given by (6), and by consequence, we can finally conclude on the finite time convergence of (ˆx, x a ) towards (x, x a ). The same procedure is followed to demonstrate the finite time convergence of (ˆx, x b ) towards (x, x b ). Having x a and x b we can easily deduce the BEMFs ê α,β using (6). 3. The adjustable model The adjustable model is tunable by the velocity and it computes the BEMFs ẽ α,β = [ẽ α ẽ β ] T from the following equation: ẽ α,β = ˆω e Jẽ α,β + G(ẽ α,β ê α,β ) (5) The velocity is : S = ε T Jẽ α,β (9) ˆω e = Msgn(S) (3) Note that the speed estimate is a discontinuous function of the manifold and M is a positive constant. The BEMFs ê α,β computed by the reference model will converge in finite time T to e α,β. After this time (t > T ), the BEMFs used in the reference model are also satisfied the following equation: ê α,β = ω e Jê α,β (3) To show that the sliding mode can be enforced in the manifold S =, we need to show that there exist M sufficiently high such that the manifold is attractive: SṠ < (3) After differentiating (9) and replacing the derivative of the BEMFs from (5) and (3), the following expression is obtained: Ṡ = f(ω e, ê α,β, ẽ α,β ) M(ẽ T α,βê α,β )sgn(s) (33) where f is a function of the reference and BEMFs and speed. Since this term is greater than zero when the motor is exited and f has a positive upper value, it s clear from (33) that sufficiently high M can ISSN: 79-57 6 ISBN: 978-96-7-85-

be selected such that condition (3) is fulfilled. Thus, sliding mode is enforced in the manifold S and after sliding mode begins, we have S =. The boundary layer method described in [] is used to find the equivalent control ω e,eq. Once sliding mode occurs, we can also assume Ṡ = along with S =. The expression of the equivalent control becomes: ω e,eq [ẽ T α,βê α,β ] = ω e [ẽ T α,βê α,β ] + gε T Jẽ α,β (3) From (3) when the manifold converge to zero (S = ε T Jẽ α,β = ), the equivalent speed tends to the speed. The equivalent speed represents the lowfrequency component of the discontinuous term (3). Thus, while the high-frequency switching function is fed into the observer, it s low-frequency component can be obtained by Low-Pass Filtering (LPF) and represents the speed estimate. Simulation Results The used motor in the simulation testing is a threephase SPMSM. The specifications and parameters are listed in Table. Parameters of the MRAS observer are α =.5, α =.8, λ =.6, λ =., g = and M = 5. The observer is tested in open loop. Using current and voltage signals, the BEMFs are using the proposed SOSM BEMFs observer which is used as the reference model of the MRAS speed estimator. This observer does not depend on the speed. The adjustable model is tunable by the speed and it is implemented using equation (5) ( the output of the MRAS observer (ˆω e ) is used as a feedback to adjust the adjustable model as shown in Fig. ). The rotor position is obtained simply from the phase of the BEMFs as follows: ˆθ e = arctan( ê α ê β ) (35) However, it is shown in [] that the PMSM is not observable at zero speed. To overcome this problem, un Estimator/Observer swapping system is proposed to ensure position estimation in all speed range and to overcome position observability problems at very low frequencies. The estimator is obtained by integrating the speed as: ˆθ e = t ˆω e dt + cte (36) The initial value of the position (ˆθ e () = cte) is equal to the last value computed by the observer (35) before swapping to the estimator. Thus, since the speed is always observable, there is no problem of observability using this position estimator. The position is equal to the observer (35) when the motor operate at high frequencies and swap to the estimator (36) since the speed becomes less than a defined very low value. Reference rotor speed (rad/s) Load torque(n.m) 5 5 5 3 5 6 7 8.5.5.5 3 5 6 7 8 Figure : Benchmark trajectories: Reference rotor speed (rad/s), Load torque (N.m) speed (rad/s) rotor position (rad) 5 5 5 3 5 6 7 8 3 5 6 7 8 Figure 5: Nominal Case: rotor speed (rad/s), rotor position (rad) Dedicated Benchmark. The proposed observer is tested to the benchmark trajectories [3] presented in Fig.. In this benchmark, two reference trajectories are defined: The reference rotor speed (Fig. ) and the load torque (Fig. ). Initially, the motor is started to run from zero and increase to rad/s and still constant until t=.5s. The load torque is applied at t=.5s and removed at t=s. This first phase permit to test and evaluate the performance and the ISSN: 79-57 7 ISBN: 978-96-7-85-

robustness of the observer at low frequencies with application of the load torque. At t=.5s, the motor is accelerated until high frequency (57 rad/s). Then, the load torque is applied again at t=3s. This second phase permit to test the robustness of the observer at high frequency operating conditions. After that, the motor is decelerated to rad/s and still constant (zero speed and acceleration) until t=6s. This last phase permit to show the loss of observability at zero speed and acceleration. Finally, the motor is controlled out of the unobservable conditions. So the motor is tested at nominal case (Fig. 5 and 6). Robustness to internal disturbances is then tested by variation of +5% of stator resistance (Fig. 7), +5% of stator inductance (Fig. 8) and +5% of rotor flux (Fig. 9). Speed Estimation. From these tests, we remark that the estimate speed mach the one with very small steady-state error and good dynamics. The proposed observer is of high accuracy and robustness against internal disturbances (parameter variations) and external disturbances (load torque). Nevertheless an error occurs at fast changed speed time (for example in Fig. 5 at t=.6 s) because in this case the speed is not slowly variable compared to the electrical dynamic. Obviously this behavior is not a physically one. It is done only to show the limit performances of the proposed control. speed (rad/s) rotor position (rad) 5 5 5 3 5 6 7 8 3 5 6 7 8 Figure 6: Estimator/Observer Swapping: Nominal Case: rotor speed (rad/s), rotor position (rad) Rotor position Estimation. For rotor position estimation, two tests are carried out. In the first test, we use only the observer ( Fig. 5). Fig. 5 shows the position which reach the one with good accuracy and robustness. However, at zero speed and acceleration, the rotor position is not observable. In the second test, we use the Estimator/Observer swapping system (Fig. 6, Fig. 7 Fig. 8 and Fig. 9). Thus, speed (rad/s) rotor position (rad) 5 5 5 3 5 6 7 8 3 5 6 7 8 Figure 7: Estimator/Observer Swapping: +5% variation of stator resistance: rotor speed (rad/s), rotor position (rad) in these figures, we show that the rotor position can be obtained at all range of frequencies. However, in the unobservable region, we remark in Fig. 6 that the position is sensitive only to stator resistance variation. This is due to the use of the estimator in this range and not the observer. speed(rad/s) rotor speed(rad) 5 5 5 3 5 6 7 8 3 5 6 7 8 Figure 8: Estimator/Observer Swapping: +5% variation of stator inductance: rotor speed (rad/s), rotor position (rad) 5 Conclusion In this paper, a new robust MRAS scheme based HOSM techniques is proposed for high accuracy and ISSN: 79-57 8 ISBN: 978-96-7-85-

speed (rad/s) rotor position (rad) 5 5 5 3 5 6 7 8 3 5 6 7 8 Figure 9: Estimator/Observer Swapping: +5% variation of rotor flux: rotor speed (rad/s), rotor position (rad) robustness rotor speed estimation of a SPMSM. The stability of the proposed MRAS system has been studied and discussed. High performance estimation of the rotor position has been obtained using an Estimator/Observer swapping system and permit to overcome the problem of observability at low speed. Selected simulation results has been presented to illustrate the performance and the robustness of the proposed speed and position estimations design. Acknowledgements: The authors thank Doctor Malek GHANES and Professor Jean Pierre BARBOT for their helpful insight and support.. References: [] A. Consoli, S. Musumeci, A. Ractiti and A. Testa, Sensorless Vector and Speed Control of Brushless Motor Drives, IEEE Trans. Ind. Elect., Vol., 99, pp. 9-95. [] C. French and P. Acarnley, Control of Permanent Motor Drives using New Position Estimation Technique, IEEE Trans. Ind. Appl., Vol. 3, 996, pp. 89-97. [3] O. Ostlund and M. Brokemper, Sensorless Rotor Position Detection from Zero to Rated Speed for an Integrated PM Synchronous Motor Drive, IEEE Trans. Ind. Appl., Vol. 3, 996, pp. 58-65. [] J. H. Jang, S. K. Sul, J. I. Ha, et al, Sensorless Drive of Surface-Mounted Permanent Magnet Motor by High Frequency Signal Injection based on Magnetic Saliency, IEEE Trans. Industry Appl., Vol. 39, No., 3, pp. 3-39. [5] H. R. Lie, J. H. Wang, S. S. Gu, et al, A Neural- Network-based Adaptative Estimator of Rotor Position and Speed for Permanant Magnet Synchronous Motor, Proc. IEEE ICEMS., Vol.,, pp. 735-738. [6] J. Solsona, M. I. Valla and C. Muravchik, A Non linear Reduced Order Observer for Permanent Magnet Synchronous Motors, IEEE trans. Ind. Electr., Vol. 3, No., 996, pp. 9-97. [7] S. Bolognani, L. Tubiana and M. Zigliotto, Extented Kalman Filter tuning in Sensorless PMSM Drives, IEEE trans. Ind. Appl., Vol. 39, No. 6, 3, pp. 7-77. [8] K. Y. Lian, C. H. Chiang, and H. W. Tu, LMIbased Sensorless Control of Permanent Magnet Synchronous Motors, IEEE trans. Ind. Elect., Vol. 5, No. 5, 7, pp. 5-6. [9] C. Schauder, Adaptative Speed Identification for Vector Control of Induction Motors without Rotational Transducers, IEEE trans. Ind. Appl., Vol. 8, No. 5, 99, pp. 769-778. [] S. Maiti, C. Chakraborty, Y. Hori and M. C. Ta, Model Reference Adaptative Controller-Based Rotor Resistance and Speed Estimation Techniques for Vector Controlled Induction Motor Drive Utilizing Reactive Power, IEEE trans. Ind. Elect., Vol. 55, No., 8, pp. 59-6. [] C. M. Ta, T. Uchida and Y. Hori, MRAS-based Speed Sensorless Control for Indution Motor Drives using Instantaneous Reactive Power, Proc. 7 th IEEE Conf. on Ind. Elect. Soc.,, pp. 7-. [] Y. S. Kim, S. K. Kim and Y. A. Kwon, MRAS based Sensorless Control of Permanent Magnet Synchronous Motor, SICE Annual Conf., Fukui, August -6, 3, pp. 63-638. [3] M. M. Kojabadi and C. Liuchen, Sensorless PMSM drive with MRAS-based adaptive speed estimator, Proc. 37th IEEE Spec. conf. on Pow. Elect., 8- June 6, Jeju, 6, pp. -5. [] V. Utkin, J. Guldner and J. Shi, Sliding mode control in electromechanical systems, London: st Edition, Taylor et Francis, 999. [5] L. Sbita, D. Zaltni and M. N. Abdelkrim, Adaptative Variable Structure Control for an Online ISSN: 79-57 9 ISBN: 978-96-7-85-

Tuning Direct Vector Controlled induction motor drive, Int. J. Appl. Sienc. Vol. 7, No.,, 7, pp. 377-386. [6] M. Ghanes and G. Zheng, On Sensorless Induction Motor Drives: Sliding-Mode Observer and Output Feedback Controller IEEE Transactions on Industrial Electronics (TIE), Vol.56, N9, p. 3-33, September 9. [7] G. Bartolini, A. Ferrara and E. Usani, Chattering Avoidance by Second-Order Sliding Mode Control, IEEE trans. Autom. Cont., Vol. 3, No., 998, pp. -6. Table : PMSM characteristics Rated Power P n.7kw Rated speed ω n 57rad.s Rated voltage U n 38V Rated current I n 3.8A Number of pole pairs P 3 Stator inductance L.7H Stator resistance R 3.3 Ω Rotor flux φ m.3 Rotor inertia J.6kg.m Viscous friction f v.3kg.m.s [8] Z. Chen, M. Tomita, S. Doki and S. Okuma, An Extended Electromotive Force Model for Sensorless Control of Interior Permanent-Magnet Synchronous Motors, IEEE trans. Ind. Elect., Vol. 5, No., 3, pp. 88-95. [9] L. Fridman, A. Levant, Sliding modes of higher order as a natural phenomenon in control theory. In Robust control via variable structure and Lyapunov techniques, Lecture notes in control and information Science 7. F. Garofalo, L. Glielmo Ed. Springer Verlag London, 996, pp. 7-33. [] A. Levant, Robust exact differentiation via sliding mode technique, Automatica, Vol. 3, No. 3, 998, pp. 379-38. [] W. Perruquetti, J. P. Barbot, Sliding mode Control in engineering, New York: Marcel Dekker,. [] D. Zaltni, M.N. Abdelkrim, M. Ghanes and J.P. Barbot, Observability Analysis of PMSM, In proc. of IEEE Int. Conf. Signal Circuit and System, 6-8 nov. 9, Djerba, Tunisie, pp. -6. [3] A. Glumineau, R. Boiliveau (IRCcyN), L. Loron (IREENA), www.irccyn.ecnantes.fr/hebergement/bancessai, 8. ISSN: 79-57 ISBN: 978-96-7-85-