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A Decoupling Dynamic Estimator for Online Parameters Indentification of Permanent Magnet Three-Phase Synchronous Motors
16th IFAC Symposium on System Identification The International Federation of Automatic Control Brussels, Belgium. July 11-13, 2012
A Decoupling Dynamic Estimator for Online Parameters Indentification of Permanent Magnet Three-Phase Synchronous Motors Paolo Mercorelli Leuphana University of L¨ uneburg Institute for Product and Process Innovation Volgershall 1, D-21339 L¨ uneburg, Germany Tel.: +49-(0)4131-677-5571, Fax: +49-(0)4131-677-5300 [email protected] Abstract: This paper deals with a dynamic estimator for fully automated parameters indentification of permanent magnet three-phase synchronous motors. High performance application of permanent magnet synchronous motors (PMSM) is increasing. PMSM models with accurate parameters are significant not only for precise control system designs but also in traction applications. Acquisition of these parameters during motor operations is a challenging task due to the inherent nonlinearity of motor dynamics. This paper proposes parameters estimator technique for PMSMs. The technique uses a decoupling procedure optimized by a minimum variance error to estimate the inductance and resistance of the motor. Moreover, a dynamic estimator is shown. The estimator uses the measurements of input voltage, current and mechanical angular velocity of the motor, the estimated winding inductance, and resistance to identify the amplitude of the linkage flux. The presented technique is generally applicable and could be used also for the estimation of mechanical load and for other types of electrical motors, as well as for other dynamic systems with nonlinear model structure. Through simulations of a synchronous motor used in automotive applications, this paper verifies the effectiveness of the proposed method in identification of PMSM model parameters and discusses the limits of the found theoretical and the simulation results. Keywords: Parameters identification, synchronous motor, dynamic systems, decoupling control 1. INTRODUCTION As the high field strength neodymium-iron-boron (NdFeB) magnets become commercially available with affordable prices, the sinusoidal back EMF permanent magnet synchronous motors (PMSMs) are receiving increasing attention due to their high speed, high power density and high efficiency. These characteristics are very favorable in some special high performance applications, e.g. robotics, aerospace, and electric ship propulsion systems [M.A. Rahman et al. 1996],[M. Ooshima et al. 2004]. Permanent magnet synchronous motors (PMSMs) as traction motors are common in electric or hybrid road vehicles. For rail vehicles, PMSMs as traction motors are not widely used yet. Although the traction PMSM can bring many advantages, just a few prototypes of vehicles were built and tested. The next two new prototypes of rail vehicles with traction PMSMs were presented on InnoTrans fair in Berlin 2008 Alstom AGV high speed train and skoda Transportation low floor tram 15T ForCity. Advantages of PMSM are well known. The greatest advantage is low volume of the PMSM in contrast with other types of motors. It makes possible a direct drive of wheels. On the other hand, the traction drive with PMSM has to meet special requirements typical for overhead line fed vehicles. 978-3-902823-06-9/12/$20.00 © 2012 IFAC
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To achieve desired system performance, advanced control systems are usually required to provide fast and accurate response, quick disturbance recovery, and parameter variations insensitivity [M.A. Rahman et al. 2003]. Acquiring accurate models for systems under investigation is usually the fundamental part in advanced control system designs. For instance, proper implementation of flux weakening control requires the knowledge of synchronous machine parameters. The most common parameters required for the implementation of such advanced control algorithms are the classical simplified model parameters: Ld - the direct axis self-inductance, Lq - the quadrature axis selfinductance, and Φ - the permanent magnet flux linkage. Prior knowledge of the previously mentioned parameters, and the number of pole pairs p, allow for the implementation of torque control through the use of current vector control. In [R. Dolecek 2009] it is shown how using a flux weakening control strategy for PMSM a prediction control structure improves the dynamic performance of traditional feedback control strategies in terms, for instance, of overshoot and rising time. To realize an effective prediction control, it is known that an accurate knowledge of the model and its parameters is necessary. Some techniques have been proposed for the parameter identification of PMSM from different perspectives. Some of them are fo10.3182/20120711-3-BE-2027.00131
16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012
cused on offline applications, e.g. [A. Kilthau et al. 2002]. Online identification of PMSM electrical parameters is analyzed and discussed in [B.N. Mobarakeh et al. 2001] and also see in [D.A. Khaburi et al. 2003], which are based on the decoupled control of linear systems when the motor mechanical dynamics are ignored. Using a decoupling control strategy, internal dynamics may be almost obscured, but it is useful to remember that, there are not limitations in the controllability and observability of the system. In [P. Mercorelli et al. 2003] a decoupling technique is used to control a Permanent Magnets Machine more efficiently in a sensorless way using an observer. The work in [L. Liu 2008] investigates the possibility of using a numerical approach Particle Swarm Optimization (PSO) as a promising alternative. PSO approach uses a system with known model structure but unknown parameters. The parameter identification problem can be treated as an optimization problem. The basic idea is well known and consists of comparing the system output with the model output. The discrepancy between the system and model outputs is minimized by optimization based on a fitness function. The fitness function is defined as a measure of how well the model output fits the measured system output. This approach utilizes numerical techniques for the optimization and it can incur in difficult non convex optimization problems because of the nonlinearity of the motor model. Despite some limitation on the frequency rage of identification, this paper proposes a dynamic observer based on an optimized decoupling technique to estimate Ldq and Rs parameters. The proposed optimization technique is similar to that presented in [P. Mercorelli 2009] and applies a procedure based on minimum variance error to minimize effects of non exact cancelation due to the decoupling controller. In the meantime, the paper proposes a particular observer that, using the estimated Ldq and Rs parameters, identifies the permanent magnet flux. The limit of this observer for the estimation of the permanent magnet flux is given by the range of work frequency. In fact, examining the theoretical structure of the observer, these limits appear evidently. Moreover, these conjectures on the limits are validated with simulated data. The estimation becomes inaccurate for low and high velocity of the motor. Because of the coupled nonlinear system structure, a general expression of such things as limits is not easy to find. The paper is organized in the following way. In Section 2 a sketch of the model of the synchronous motor and its behaviour are given. Section 3 is devoted to deriving a property, proposing and discussing the dynamic estimator. Section 4 shows simulation results using real data. The conclusions close the paper. 2. MODEL AND BEHAVIOR OF A SYNCHRONOUS MOTOR To aid advanced controller design for PMSM, it is very important to obtain an appropriate model of the motor. A good model should not only be an accurate representation of system dynamics but also facilitate the application of existing control techniques. Among a variety of models presented in literature, since the introduction of PMSM, the two axis dq-model obtained using Parks transformation is the most widely used in variable speed PMSM drive control applications, see [M.A. Rahman et al. 2003] and
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[D.A. Khaburi et al. 2003]. The Park’s dq-transformation is a coordinate transformation that converts the threephase stationary variables into variables in a rotating coordinate system. In dq-transformation, the rotating coordinate is defined relative to a stationary reference angle as illustrated in Fig. 1. The dq-model is considered in this work. The dynamic model of the synchronous motor in
Fig. 1. Park’s transformation for the motor d-q-coordinates can be represented as follows: R L s q did (t) − ωel (t) dt Ld Ld id (t) di (t) = Rs Ld i (t) + q q − − ωel (t) Lq Lq dt 1 0 0 Ld ud (t) 1 uq (t) − Φωel (t) , (1) 0 Lq and 3 Mm = p{Φiq (t) + (Ld − Lq )id (t)iq (t)}. (2) 2 In (1) and (2), id (t), iq (t), ud (t) and uq (t) are the dqcomponents of the stator currents and voltages in synchronously rotating rotor reference frame; ωel (t) is the rotor electrical angular speed; parameters Rs , Ld , Lq , Φ and p are the stator resistance, d-axis and q-axis inductance, the amplitude of the permanent magnet flux linkage, and p the number of couples of permanent magnets, respectively. At the end with Mm the motor torque is indicated. Considering a isotropic motor for that Ld ' Lq = Ldq , it follows: R s did (t) − ωel (t) id (t) dt Ldq di (t) = Rs iq (t) + q − ωel (t) Ldq dt 1 0 0 Ldq ud (t) − , (3) 1 uq (t) Φωel (t) 0 Ldq and 3 Mm = pΦiq (t) (4) 2 with the following movement equation:
16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012
dωmec (t) , (5) dt where pωmech (t) = ωel (t) and Mw is an unknown mechanical load. Mm − Mw = J
3. STRUCTURE OF THE DECOUPLING DYNAMIC ESTIMATOR The present estimator uses the measurements of input voltages, currents and angular velocity of the motor to estimate the ”d-q” winding inductance, the rotor resistance and amplitude of the linkage flux. The structure of the estimator is described in Fig. 2. In this diagram is visible how the estimator works. In particular, after having decoupled the system described in (3), the stator resistance Rs and the inductance Ldq are estimate through a minimum error variance approach. The estimated values ˆs and L ˆ qd ) are of rotor resistance and Ldq inductance (R used for the estimation of the amplitude of the linkage flux ˆ (Φ).
Rs − ω (t) el Ldq + im Rs − ωel (t) Ldq 1 0 0 0 Ldq F11 F12 im ⊆ im , (8) 1 F21 F22 1 1 0 Ldq where parameters F11 , F12 , F21 , and F22 are to be calculated in order to guarantee condition (8) and a suitable dynamics for sake of estimation, as it will be explained in the next. Condition (8) is guaranteed if F12 = −ωel (t)Ldq . (9) After decoupling the first equations of the system represented in (3) becomes as follows: did (t) Rs ud (t) =− id (t) + , (10) dt Ldq Ldq An exact decoupling is just a mathematical abstraction. In fact, because of the uncertainties in the model and noise effects, the decoupling is never exact. For that it follows that: dˆid (t) Rs ˆ ud (t) ˆ dq ))), =− id (t) + + n(∆(ωel (t)(Ldq − L dt Ldq Ldq (11) ˆ dq )) is the disturbance due to the where n(∆(Ldq − L inexact cancelation. ˆ dq )) Proposition 1. Considering disturbance n(∆(Ldq − L of Eq. (10) as a white noise, then the current minimum variance error σ eid (t) = σ id (t) − ˆid (t) is obtained by minimising the estimation error of parameters Ldq and Rs . Proof 1. If Eqs. (10) and (11) are discretizated using Implicit Euler with a sampling frequency equal to ts , then it follows: ˆ ts ˆid (k) = id (k − 1) + ud (k). (12) Rs (1 + ts Ldq ) Ldq (1 + ts LRdqs )
Fig. 2. Conceptual structure of the whole estimator
3.1 Decoupling structure and minimum error variance algorithm
ˆ ts ˆid (k) = id (k − 1) + ud (k) + n(k). (13) (1 + ts LRdqs ) Ldq (1 + ts LRdqs )
To achieve a decoupled structure of the system described in Eq. (3) a matrix F is to be calculated such that, (A + BF)V ⊆ V, (6) T where u = Fx is a state feedback with u = [ud , uq ] and x = [id , iq ]T , 1 Rs − ω (t) 0 el Ldq L , B = A= (7) dq 1 , Rs 0 − ωel (t) Ldq Ldq
It is possible to assume an ARMAX model for the system represented by (13) and thus
and V = im([0, 1]T ) is a controlled invariant subspace. More explicitly it follows: F11 F12 ud (t) id (t) F= , and =F , F21 F22 uq (t) iq (t) then, according to [G. Basile et al. 1992], the decoupling of the dynamics is obtained considering the following relationship:
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id (k) = ˆid (k) + a1ˆid (k − 1) + a2ˆid (k − 2) + b1 ud (k − 1)+ b2 ud (k − 2) + n(k) + c1u n(k − 1) + c2u n(k − 2), (14) letting eid (k) = id (k)−ˆid (k) as mentioned above, it follows that: eid (k) = a1ˆid (k−1)+a2ˆid (k−2)+b1 ud (k−1)+b2 ud (k−2)+ n(k) + c1 n(k − 1) + c2 n(k − 2), (15) where coefficients a, b, c1 , c2 , are to be estimated, n(k) is assumed as white noises. The next sample is: eid (k+1) = a1ˆid (k)+a2ˆid (k−1)+b1 ud (k)+b2 ud (k−1)+ n(k + 1) + c1 n(k) + c2 n(k − 1). (16) The prediction at time ”k” is: eˆid (k+1/k) = a1ˆid (k)+a2ˆid (k−1)+b1 ud (k)+b2 ud (k−1) + c1 n(k) + c2 n(k − 1). (17)
16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012
Considering that: J = E{e2id (k + 1/k)} = E{[ˆ eid (k + 1/k) + n(k + 1)]2 }, assuming that the noise is not correlated to the signal eid (k) it follows: E{[ˆ eid (k + 1/k) + n(k + 1)]2 } = E{[ˆ eid (k + 1/k)]2 }+ E{[n(k + 1)]2 } = E{[ˆ eid (k + 1/k)]2 } + σn2 , (18) 2 where σn is defined as the variance of the white noises. The goal is to find ˆid (k) such that: eˆid (k + 1/k) = 0. (19) It is possible to write (15) as n(k) = eid (k) − a1ˆid (k − 1) − a2ˆid (k − 2) − b1 ud (k − 1)− b2 ud (k − 2) − c1 n(k − 1) − c2 n(k − 2). (20) Considering the effect of the noise on the system as follows c1 n(k − 1) + c2 n(k − 2) ≈ c1 n(k − 1), (21) and using the Z-transform, then: N (z) = Iˆd (z) − a1 z −1 Iˆd (z) − a2 z −2 Iˆu (z) − b1 z −1 Ud (z) − b2 z −2 Ud (z) − c1 z −1 N (z) (22) and (1 − a1 z −1 − a2 z −2 ) ˆ (b1 z −1 + b2 z −2 ) I (z)− Ud (z). d 1 + c1 z −1 1 + c1 z −1 (23) Inserting equation (23) into equation (17) after its Ztransform, and considering position (21) and position (19), the following expression is obtained: (a1 + c1 + b1 z −1 ) Iˆd (z) = − Ud (z). (24) b1 (1 + c1 z −1 ) + b2 (1 + c1 z −1 ) Using the Z-transform for Eq. (12) it follows: ts Id (z) = Ud (z) (25) Ldq + ts Rs − Ldq z −1 Comparing (24) with (25), it is left with a straightforward diophantine equation to solve. The diophantine equation gives the relationship between the parameters Θ = [a1 , b1 , b2 , c1 ] as follows: N (z) =
−b1 = 0
(26)
a 1 + c 1 = ts
(27)
b1 + b2 = Ldq + ts Rs
(28)
b1 c1 + b2 c1 = −Ldq . (29) Guess initial values for parameters Ldq , Rs are given. This yields initial values for the parameters Θ = [a1 , b1 , b2 , c1 ]. New values for the vector Θ are calculated using the recursive least squares method. 2 Remark 1. The approximation in equation (21) is equivalent to consider kc2 k << kc1 k. In other words this position means that a noise model of the first order is assumed. An indirect validation of this assumption is given by the results. In fact, the final measurements show in general good results with the proposed method. 2 Remark 2. At the end, the recursive least squares method gives the estimation of parameters Ldq and Rs , which, in the meantime, minimize the current minimum variance error σ(eid (t)) = σ(id (t) − ˆid (t)). 2
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3.2 The dynamic estimator of Φ If the electrical part of the system ”q” and ”d” axes is considered, then, assuming that ωel (t) 6= 0, iq (t) 6= 0, and id (t) 6= 0, the following equation can be considered: Ldq
diq (t) dt
+ Rs id (t) + Ldq ωel (t)iq (t) + uq (t) (30) ωel (t) Consider the following dynamic system: ˆ dΦ(t) ˆ = −KΦ(t)− dt L ˆ dq diq (t) + R ˆ s id (t) dt K + ωel (t) ˆ dq ωel (t)iq (t) + uq (t) L , (31) ωel (t) where K is a function to be calculated. Eq. (31) represents the estimators of Φ. If the error functions are defined as the differences between the true and the observed values, then: ˆ eΦ (t) = Φ(t) − Φ(t), (32) and ˆ deΦ (t) dΦ(t) dΦ(t) = − . (33) dt dt dt If the following assumption is given: ˆ dΦ(t) dΦ(t) k k << k k, (34) dt dt then in Eq. (33), the term dΦ(t) is negligible. Using dt equation (31), Eq. (33) becomes Φ=−
deΦ (t) ˆ = KΦ(t)+ dt L ˆ dq diq (t) + R ˆ s id (t) dt K + ωel (t) ˆ dq ωel (t)iq (t) + uq (t) L . (35) ωel (t) Because of Eq. (31), (35) can be written as follows: deΦ(t) ˆ − KΦ(t) = KΦ(t) dt and considering (32), then deΦ (t) + KeΦ (t) = 0 (36) dt K can be chosen to make Eq. (36) exponentially stable. To guarantee exponential stability, K must be K > 0. ˆ
To guarantee << k dΦ(t) dt k, then K >> 0. The observer defined in (31) suffers from the presence of the derivative of the measured current. In fact, if measurement noise is present in the measured current, then undesirable spikes are generated by the differentiation. The proposed algorithm needs to cancel the contribution from the measured current derivative. This is possible by correcting the observed velocity with a function of the measured current, using a supplementary variable defined as ˆ + N (iq (t)), η(t) = Φ(t) (37) k dΦ(t) dt k
where N (iq (t)) is the function to be designed.
16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012
dη(t) dΦ(t) dN (iq (t)) = + dt dt dt
and let ˆ dq diq (t) dN (iq (t)) dN (iq ) diq (t) KL = = . (39) dt diq (t) dt ωel (t) dt The purpose of (39) is to cancel the differential contribution from (31). In fact, (37) and (38) yield, respectively, ˆ Φ(t) = η(t) − N (iq (t)) and (40) ˆ dΦ(t) dη(t) dN (iq (t)) = − . (41) dt dt dt Substituting (39) in (41) results in ˆ ˆ dq diq (t) dΦ(t) dη(t) KL = − . (42) dt dt ωel (t) dt Inserting Eq. (42) into Eq. (31) the following expression is obtained 1 : ˆ dq diq (t) dη(t) KL ˆ − = −KΦ(t)− dt ωel (t) dt L ˆ dq diq (t) + R ˆ s id (t) dt K + ωel (t) ˆ dq ωel (t)iq (t) + uq (t) L ; (43) ωel (t) then ˆ s id (t) + L ˆ dq ωel (t)iq (t) + uq (t) R dη(t) ˆ −K = −KΦ(t) . dt ωel (t) (44) Letting N (iq (t)) = kapp iq (t), where with kapp a parameter k ω (t) has been indicated, then, from (39) ⇒ K = appLˆ el , Eq. dq
(40) becomes ˆ Φ(t) = η(t) − kapp iq (t). (45) Finally, substituting (45) into (44) results in the following equation dη(t) kapp ωel (t) =− η(t) − kapp iq (t) + ˆ dt Ldq kapp ˆ ˆ dq ωel (t)iq (t) + uq (t) Rs id (t) + L ˆ dq L ˆ Φ(t) = η(t) − kapp iq (t). (46) Using the implicit Euler method, the following velocity observer structure is obtained: η(k − 1) η(k) = + k ω (k) 1 + ts appLˆ el dq
k2 ωel (k)iq (k) ts app Lˆ dq
+ kapp ωel (k)iq (k) + 1 + ts
ˆ s kapp id (k) ts R ˆ dq L
kapp ωel (k) ˆ dq L kapp ˆ dq L kapp ωel (k) ts Lˆ dq
iq (k)+
ts
1+
ˆ Φ(k) = η(k) − kapp iq (k),
0.25
(38)
uq (k) (47)
1
Expression (31) works under the assumption (34): fast observer dynamics.
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0.2 Stator resistance (Ohm)
Consider
0.15
0.1
0.05 Estimated values of the stator resistance Real value of the stator resistance 0 0
1
2 Time (sec.)
3
4 −3
x 10
Fig. 3. Estimated and real values of Rs stator resistance where ts is the sampling period. Remark 3. Assumption (34) states that the dynamics of the approximating observer should be faster than the dynamics of the physical system. This assumption is typical for the design of observers. 2 Remark 4. The estimator of Eq. (47) presents the following limitations. For low velocity of the motor (ωmec. (t) << ωmecn (t)), where ωmecn (t) represents the nominal velocity of the motor) the estimation of Φ becomes inaccurate. In fact, because of ωel (t) dividing the state variable η, the observer described by (47) becomes hyperdynamic. Critical phases of the estimation are the starting and ending of the movement. Another critical phase is represented by high velocity regime. In fact, it has been proven through simulations that if ωmec (t) >> ωmecn (t), then the observer described by (47)becomes hypodynamic. According to the simulation results, within some rage of frequency, this hypo-dynamicity can be compensated by suitable choice of kapp . 2
4. SIMULATION RESULTS Simulations have been carried out using a special stand with a 58 kW traction PMSM. The stand consists of PMSM, tram wheel and continuous rail. The PMSM is a prototype for low floor trams. PMSM parameters: nominal power 58 kW, nominal torque 852 Nm, nominal speed 650 rpm, nominal phase current 122 A and number of poles 44. Model parameters: R = 0.08723 Ohm, Ldq = Ld = Lq = 0.8 mH, Φ = 0.167 Wb. Surface mounted NdBFe magnets are used in PMSM. Advantage of these magnets is inductance up to 1.2 T, but theirs disadvantage is corrosion. The PMSM was designed to meet B curve requirements. The stand was loaded by an asynchronous motor. The engine has parameters as follows: nominal power 55 kW, nominal voltage 380 V and nominal speed 589 rpm. Figures 3, 4, and 5 show the estimation of Rs stator resistance, Ldq inductance, and Φ magnet flux, respectively. From these figures, at the beginning of the estimation, it is visible the effect of the limit of the procedure discussed in remark 4. In the present simulations at t = 0 corresponds ωel (t) = 0.
16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012
−3
1
x 10
0.9
Ldq−Inductance (H)
0.8 0.7 0.6 0.5 0.4
Estimated values of the Lqd inductance Real value of Lqd−Inductance
0.3 0.2 0.1 0 0
0.5
1 Time (sec.)
1.5
2 −3
x 10
Fig. 4. Estimated and real values of Ldq inductance
10 Permanent magnet flux linkage (Vs)
8 6 4 2 0 −2
Estimated values of the permanent magnet flux linkage Real value of the permanent magnet flux linkage
−4 −6 −8 −10 0.5
1 Time (sec.)
1.5
2 −3
x 10
Fig. 5. Estimated and real values of the permanent magnet flux linkage 5. CONCLUSIONS AND FUTURE WORK This paper deals with a dynamic estimator for fully automated parameters indentification for three-phase synchronous motors.The technique uses a decoupling procedure optimized by a minimum variance error to estimate the inductance and resistance of the motor. Moreover, a dynamic estimator is shown to identify the amplitude of the linkage flux using the estimated inductance and resistance. It is generally applicable and could be used also for the estimation of mechanical load and for other types of electrical motors, as well as for other dynamic systems with similar nonlinear model structure. Through simulations of a synchronous motor used in automotive applications, this paper verifies the effectiveness of the proposed method in identification of PMSM model parameters and discusses the limits of the found theoretical and the simulation results. Future work includes the estimation of a mechanical load and the general test of the present algorithm using a real motor. REFERENCES [M.A. Rahman et al. 1996] M.A. Rahman and P. Zhou. ”Analysis of brushless permanent magnet syn-
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chronous motors”, IEEE Transactions on Industrial Electronics, vol. 43, no. 2, 1996, pages 256-267. [M. Ooshima et al. 2004] M. Ooshima, A. Chiba, A. Rahman and T. Fukao. ”An improved control method of buried-type IPM bearingless motors considering magnetic saturation and magnetic pull variation”, IEEE Transactions on Energy Conversion, vol. 19, no. 3, Sept. 2004, pages 569-575. [R. Dolecek 2009] R. Dolecek, J. Novak, and O. Cerny. ”Traction Permanent Magnet Synchronous Motor Torque Control with Flux Weakening”, Radioengineering, Vol. 18, No. 4, December 2009. [M.A. Rahman et al. 2003] , M.A. Rahman, D.M. Vilathgamuwa, M.N. Uddin and T. King-Jet. ”Nonlinear control of interior permanent magnet synchronous motor”, IEEE Transactions on Industry Applications, vol. 39, no. 2, 2003, pages 408-416. [D.A. Khaburi et al. 2003] D.A. Khaburi and M. Shahnazari. ”Parameters Identification of Permanent Magnet Synchronous Machine in Vector Control”, in Proceedngs of th 10th European Conference on Power Electronics and Applications, EPE 2003, Toulouse, France, 2-4 September 2003. [A. Kilthau et al. 2002] A. Kilthau and J. Pacas. ”Appropriate models for the controls of the synchronous reluctance machine”, in Proceedings of IEEE IAS Annual Meeting, 2002, pages 2289-2295. [B.N. Mobarakeh et al. 2001] B.N. Mobarakeh, F. Meibody-Tabar and F. M.Sargos. ”On-line identification of PMSM electrical parameters based on decoupling control”, in Proceeding of the Conf. Rec. IEEE-IAS Annual Meeting, vol. 1, Chicago, IL, 2001, pages 266-273. [L. Liu 2008] L. Liu, W. Liu and D. A. Cartes. ”Permanent Magnet Synchronous Motor Parameter Identification using Particle Swarm Optimization”, International Journal of Computational Intelligence Research Vol.4, No.2 (2008), pages 211-218. [P. Mercorelli 2009] P. Mercorelli. ”Robust Feedback Linearization Using an Adaptive PD Regulator for a Sensorless Control of a Throttle Valve”, Regular paper in ”Mechatronics”, a Journal of IFAC, Elsevier Science publishing. DOI: 10.1016/j.mechatronics.2009.08.008, Volume 19, Issue 8, pages 1334-1345, December 2009. [P. Mercorelli et al. 2003] P. Mercorelli, K. Lehmann and S. Liu. ”On Robustness Properties in Permanent Magnet Machine Control Using Decoupling Controller”, In Proceedings of the 4th IFAC International Symposium on Robust Control Design, 25th27th June 2003, Milan (Italy). [G. Basile et al. 1992] G. Basile and G. Marro. ”Controlled and conditioned invariants in linear system theory”, New Jersey-USA, Prentice Hall, 1992.
A Decoupling Dynamic Estimator for Online Parameters Indentification of Permanent Magnet Three-Phase Synchronous Motors Paolo Mercorelli Leuphana University of L¨ uneburg Institute for Product and Process Innovation Volgershall 1, D-21339 L¨ uneburg, Germany Tel.: +49-(0)4131-677-5571, Fax: +49-(0)4131-677-5300 [email protected] Abstract: This paper deals with a dynamic estimator for fully automated parameters indentification of permanent magnet three-phase synchronous motors. High performance application of permanent magnet synchronous motors (PMSM) is increasing. PMSM models with accurate parameters are significant not only for precise control system designs but also in traction applications. Acquisition of these parameters during motor operations is a challenging task due to the inherent nonlinearity of motor dynamics. This paper proposes parameters estimator technique for PMSMs. The technique uses a decoupling procedure optimized by a minimum variance error to estimate the inductance and resistance of the motor. Moreover, a dynamic estimator is shown. The estimator uses the measurements of input voltage, current and mechanical angular velocity of the motor, the estimated winding inductance, and resistance to identify the amplitude of the linkage flux. The presented technique is generally applicable and could be used also for the estimation of mechanical load and for other types of electrical motors, as well as for other dynamic systems with nonlinear model structure. Through simulations of a synchronous motor used in automotive applications, this paper verifies the effectiveness of the proposed method in identification of PMSM model parameters and discusses the limits of the found theoretical and the simulation results. Keywords: Parameters identification, synchronous motor, dynamic systems, decoupling control 1. INTRODUCTION As the high field strength neodymium-iron-boron (NdFeB) magnets become commercially available with affordable prices, the sinusoidal back EMF permanent magnet synchronous motors (PMSMs) are receiving increasing attention due to their high speed, high power density and high efficiency. These characteristics are very favorable in some special high performance applications, e.g. robotics, aerospace, and electric ship propulsion systems [M.A. Rahman et al. 1996],[M. Ooshima et al. 2004]. Permanent magnet synchronous motors (PMSMs) as traction motors are common in electric or hybrid road vehicles. For rail vehicles, PMSMs as traction motors are not widely used yet. Although the traction PMSM can bring many advantages, just a few prototypes of vehicles were built and tested. The next two new prototypes of rail vehicles with traction PMSMs were presented on InnoTrans fair in Berlin 2008 Alstom AGV high speed train and skoda Transportation low floor tram 15T ForCity. Advantages of PMSM are well known. The greatest advantage is low volume of the PMSM in contrast with other types of motors. It makes possible a direct drive of wheels. On the other hand, the traction drive with PMSM has to meet special requirements typical for overhead line fed vehicles. 978-3-902823-06-9/12/$20.00 © 2012 IFAC
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To achieve desired system performance, advanced control systems are usually required to provide fast and accurate response, quick disturbance recovery, and parameter variations insensitivity [M.A. Rahman et al. 2003]. Acquiring accurate models for systems under investigation is usually the fundamental part in advanced control system designs. For instance, proper implementation of flux weakening control requires the knowledge of synchronous machine parameters. The most common parameters required for the implementation of such advanced control algorithms are the classical simplified model parameters: Ld - the direct axis self-inductance, Lq - the quadrature axis selfinductance, and Φ - the permanent magnet flux linkage. Prior knowledge of the previously mentioned parameters, and the number of pole pairs p, allow for the implementation of torque control through the use of current vector control. In [R. Dolecek 2009] it is shown how using a flux weakening control strategy for PMSM a prediction control structure improves the dynamic performance of traditional feedback control strategies in terms, for instance, of overshoot and rising time. To realize an effective prediction control, it is known that an accurate knowledge of the model and its parameters is necessary. Some techniques have been proposed for the parameter identification of PMSM from different perspectives. Some of them are fo10.3182/20120711-3-BE-2027.00131
16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012
cused on offline applications, e.g. [A. Kilthau et al. 2002]. Online identification of PMSM electrical parameters is analyzed and discussed in [B.N. Mobarakeh et al. 2001] and also see in [D.A. Khaburi et al. 2003], which are based on the decoupled control of linear systems when the motor mechanical dynamics are ignored. Using a decoupling control strategy, internal dynamics may be almost obscured, but it is useful to remember that, there are not limitations in the controllability and observability of the system. In [P. Mercorelli et al. 2003] a decoupling technique is used to control a Permanent Magnets Machine more efficiently in a sensorless way using an observer. The work in [L. Liu 2008] investigates the possibility of using a numerical approach Particle Swarm Optimization (PSO) as a promising alternative. PSO approach uses a system with known model structure but unknown parameters. The parameter identification problem can be treated as an optimization problem. The basic idea is well known and consists of comparing the system output with the model output. The discrepancy between the system and model outputs is minimized by optimization based on a fitness function. The fitness function is defined as a measure of how well the model output fits the measured system output. This approach utilizes numerical techniques for the optimization and it can incur in difficult non convex optimization problems because of the nonlinearity of the motor model. Despite some limitation on the frequency rage of identification, this paper proposes a dynamic observer based on an optimized decoupling technique to estimate Ldq and Rs parameters. The proposed optimization technique is similar to that presented in [P. Mercorelli 2009] and applies a procedure based on minimum variance error to minimize effects of non exact cancelation due to the decoupling controller. In the meantime, the paper proposes a particular observer that, using the estimated Ldq and Rs parameters, identifies the permanent magnet flux. The limit of this observer for the estimation of the permanent magnet flux is given by the range of work frequency. In fact, examining the theoretical structure of the observer, these limits appear evidently. Moreover, these conjectures on the limits are validated with simulated data. The estimation becomes inaccurate for low and high velocity of the motor. Because of the coupled nonlinear system structure, a general expression of such things as limits is not easy to find. The paper is organized in the following way. In Section 2 a sketch of the model of the synchronous motor and its behaviour are given. Section 3 is devoted to deriving a property, proposing and discussing the dynamic estimator. Section 4 shows simulation results using real data. The conclusions close the paper. 2. MODEL AND BEHAVIOR OF A SYNCHRONOUS MOTOR To aid advanced controller design for PMSM, it is very important to obtain an appropriate model of the motor. A good model should not only be an accurate representation of system dynamics but also facilitate the application of existing control techniques. Among a variety of models presented in literature, since the introduction of PMSM, the two axis dq-model obtained using Parks transformation is the most widely used in variable speed PMSM drive control applications, see [M.A. Rahman et al. 2003] and
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[D.A. Khaburi et al. 2003]. The Park’s dq-transformation is a coordinate transformation that converts the threephase stationary variables into variables in a rotating coordinate system. In dq-transformation, the rotating coordinate is defined relative to a stationary reference angle as illustrated in Fig. 1. The dq-model is considered in this work. The dynamic model of the synchronous motor in
Fig. 1. Park’s transformation for the motor d-q-coordinates can be represented as follows: R L s q did (t) − ωel (t) dt Ld Ld id (t) di (t) = Rs Ld i (t) + q q − − ωel (t) Lq Lq dt 1 0 0 Ld ud (t) 1 uq (t) − Φωel (t) , (1) 0 Lq and 3 Mm = p{Φiq (t) + (Ld − Lq )id (t)iq (t)}. (2) 2 In (1) and (2), id (t), iq (t), ud (t) and uq (t) are the dqcomponents of the stator currents and voltages in synchronously rotating rotor reference frame; ωel (t) is the rotor electrical angular speed; parameters Rs , Ld , Lq , Φ and p are the stator resistance, d-axis and q-axis inductance, the amplitude of the permanent magnet flux linkage, and p the number of couples of permanent magnets, respectively. At the end with Mm the motor torque is indicated. Considering a isotropic motor for that Ld ' Lq = Ldq , it follows: R s did (t) − ωel (t) id (t) dt Ldq di (t) = Rs iq (t) + q − ωel (t) Ldq dt 1 0 0 Ldq ud (t) − , (3) 1 uq (t) Φωel (t) 0 Ldq and 3 Mm = pΦiq (t) (4) 2 with the following movement equation:
16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012
dωmec (t) , (5) dt where pωmech (t) = ωel (t) and Mw is an unknown mechanical load. Mm − Mw = J
3. STRUCTURE OF THE DECOUPLING DYNAMIC ESTIMATOR The present estimator uses the measurements of input voltages, currents and angular velocity of the motor to estimate the ”d-q” winding inductance, the rotor resistance and amplitude of the linkage flux. The structure of the estimator is described in Fig. 2. In this diagram is visible how the estimator works. In particular, after having decoupled the system described in (3), the stator resistance Rs and the inductance Ldq are estimate through a minimum error variance approach. The estimated values ˆs and L ˆ qd ) are of rotor resistance and Ldq inductance (R used for the estimation of the amplitude of the linkage flux ˆ (Φ).
Rs − ω (t) el Ldq + im Rs − ωel (t) Ldq 1 0 0 0 Ldq F11 F12 im ⊆ im , (8) 1 F21 F22 1 1 0 Ldq where parameters F11 , F12 , F21 , and F22 are to be calculated in order to guarantee condition (8) and a suitable dynamics for sake of estimation, as it will be explained in the next. Condition (8) is guaranteed if F12 = −ωel (t)Ldq . (9) After decoupling the first equations of the system represented in (3) becomes as follows: did (t) Rs ud (t) =− id (t) + , (10) dt Ldq Ldq An exact decoupling is just a mathematical abstraction. In fact, because of the uncertainties in the model and noise effects, the decoupling is never exact. For that it follows that: dˆid (t) Rs ˆ ud (t) ˆ dq ))), =− id (t) + + n(∆(ωel (t)(Ldq − L dt Ldq Ldq (11) ˆ dq )) is the disturbance due to the where n(∆(Ldq − L inexact cancelation. ˆ dq )) Proposition 1. Considering disturbance n(∆(Ldq − L of Eq. (10) as a white noise, then the current minimum variance error σ eid (t) = σ id (t) − ˆid (t) is obtained by minimising the estimation error of parameters Ldq and Rs . Proof 1. If Eqs. (10) and (11) are discretizated using Implicit Euler with a sampling frequency equal to ts , then it follows: ˆ ts ˆid (k) = id (k − 1) + ud (k). (12) Rs (1 + ts Ldq ) Ldq (1 + ts LRdqs )
Fig. 2. Conceptual structure of the whole estimator
3.1 Decoupling structure and minimum error variance algorithm
ˆ ts ˆid (k) = id (k − 1) + ud (k) + n(k). (13) (1 + ts LRdqs ) Ldq (1 + ts LRdqs )
To achieve a decoupled structure of the system described in Eq. (3) a matrix F is to be calculated such that, (A + BF)V ⊆ V, (6) T where u = Fx is a state feedback with u = [ud , uq ] and x = [id , iq ]T , 1 Rs − ω (t) 0 el Ldq L , B = A= (7) dq 1 , Rs 0 − ωel (t) Ldq Ldq
It is possible to assume an ARMAX model for the system represented by (13) and thus
and V = im([0, 1]T ) is a controlled invariant subspace. More explicitly it follows: F11 F12 ud (t) id (t) F= , and =F , F21 F22 uq (t) iq (t) then, according to [G. Basile et al. 1992], the decoupling of the dynamics is obtained considering the following relationship:
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id (k) = ˆid (k) + a1ˆid (k − 1) + a2ˆid (k − 2) + b1 ud (k − 1)+ b2 ud (k − 2) + n(k) + c1u n(k − 1) + c2u n(k − 2), (14) letting eid (k) = id (k)−ˆid (k) as mentioned above, it follows that: eid (k) = a1ˆid (k−1)+a2ˆid (k−2)+b1 ud (k−1)+b2 ud (k−2)+ n(k) + c1 n(k − 1) + c2 n(k − 2), (15) where coefficients a, b, c1 , c2 , are to be estimated, n(k) is assumed as white noises. The next sample is: eid (k+1) = a1ˆid (k)+a2ˆid (k−1)+b1 ud (k)+b2 ud (k−1)+ n(k + 1) + c1 n(k) + c2 n(k − 1). (16) The prediction at time ”k” is: eˆid (k+1/k) = a1ˆid (k)+a2ˆid (k−1)+b1 ud (k)+b2 ud (k−1) + c1 n(k) + c2 n(k − 1). (17)
16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012
Considering that: J = E{e2id (k + 1/k)} = E{[ˆ eid (k + 1/k) + n(k + 1)]2 }, assuming that the noise is not correlated to the signal eid (k) it follows: E{[ˆ eid (k + 1/k) + n(k + 1)]2 } = E{[ˆ eid (k + 1/k)]2 }+ E{[n(k + 1)]2 } = E{[ˆ eid (k + 1/k)]2 } + σn2 , (18) 2 where σn is defined as the variance of the white noises. The goal is to find ˆid (k) such that: eˆid (k + 1/k) = 0. (19) It is possible to write (15) as n(k) = eid (k) − a1ˆid (k − 1) − a2ˆid (k − 2) − b1 ud (k − 1)− b2 ud (k − 2) − c1 n(k − 1) − c2 n(k − 2). (20) Considering the effect of the noise on the system as follows c1 n(k − 1) + c2 n(k − 2) ≈ c1 n(k − 1), (21) and using the Z-transform, then: N (z) = Iˆd (z) − a1 z −1 Iˆd (z) − a2 z −2 Iˆu (z) − b1 z −1 Ud (z) − b2 z −2 Ud (z) − c1 z −1 N (z) (22) and (1 − a1 z −1 − a2 z −2 ) ˆ (b1 z −1 + b2 z −2 ) I (z)− Ud (z). d 1 + c1 z −1 1 + c1 z −1 (23) Inserting equation (23) into equation (17) after its Ztransform, and considering position (21) and position (19), the following expression is obtained: (a1 + c1 + b1 z −1 ) Iˆd (z) = − Ud (z). (24) b1 (1 + c1 z −1 ) + b2 (1 + c1 z −1 ) Using the Z-transform for Eq. (12) it follows: ts Id (z) = Ud (z) (25) Ldq + ts Rs − Ldq z −1 Comparing (24) with (25), it is left with a straightforward diophantine equation to solve. The diophantine equation gives the relationship between the parameters Θ = [a1 , b1 , b2 , c1 ] as follows: N (z) =
−b1 = 0
(26)
a 1 + c 1 = ts
(27)
b1 + b2 = Ldq + ts Rs
(28)
b1 c1 + b2 c1 = −Ldq . (29) Guess initial values for parameters Ldq , Rs are given. This yields initial values for the parameters Θ = [a1 , b1 , b2 , c1 ]. New values for the vector Θ are calculated using the recursive least squares method. 2 Remark 1. The approximation in equation (21) is equivalent to consider kc2 k << kc1 k. In other words this position means that a noise model of the first order is assumed. An indirect validation of this assumption is given by the results. In fact, the final measurements show in general good results with the proposed method. 2 Remark 2. At the end, the recursive least squares method gives the estimation of parameters Ldq and Rs , which, in the meantime, minimize the current minimum variance error σ(eid (t)) = σ(id (t) − ˆid (t)). 2
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3.2 The dynamic estimator of Φ If the electrical part of the system ”q” and ”d” axes is considered, then, assuming that ωel (t) 6= 0, iq (t) 6= 0, and id (t) 6= 0, the following equation can be considered: Ldq
diq (t) dt
+ Rs id (t) + Ldq ωel (t)iq (t) + uq (t) (30) ωel (t) Consider the following dynamic system: ˆ dΦ(t) ˆ = −KΦ(t)− dt L ˆ dq diq (t) + R ˆ s id (t) dt K + ωel (t) ˆ dq ωel (t)iq (t) + uq (t) L , (31) ωel (t) where K is a function to be calculated. Eq. (31) represents the estimators of Φ. If the error functions are defined as the differences between the true and the observed values, then: ˆ eΦ (t) = Φ(t) − Φ(t), (32) and ˆ deΦ (t) dΦ(t) dΦ(t) = − . (33) dt dt dt If the following assumption is given: ˆ dΦ(t) dΦ(t) k k << k k, (34) dt dt then in Eq. (33), the term dΦ(t) is negligible. Using dt equation (31), Eq. (33) becomes Φ=−
deΦ (t) ˆ = KΦ(t)+ dt L ˆ dq diq (t) + R ˆ s id (t) dt K + ωel (t) ˆ dq ωel (t)iq (t) + uq (t) L . (35) ωel (t) Because of Eq. (31), (35) can be written as follows: deΦ(t) ˆ − KΦ(t) = KΦ(t) dt and considering (32), then deΦ (t) + KeΦ (t) = 0 (36) dt K can be chosen to make Eq. (36) exponentially stable. To guarantee exponential stability, K must be K > 0. ˆ
To guarantee << k dΦ(t) dt k, then K >> 0. The observer defined in (31) suffers from the presence of the derivative of the measured current. In fact, if measurement noise is present in the measured current, then undesirable spikes are generated by the differentiation. The proposed algorithm needs to cancel the contribution from the measured current derivative. This is possible by correcting the observed velocity with a function of the measured current, using a supplementary variable defined as ˆ + N (iq (t)), η(t) = Φ(t) (37) k dΦ(t) dt k
where N (iq (t)) is the function to be designed.
16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012
dη(t) dΦ(t) dN (iq (t)) = + dt dt dt
and let ˆ dq diq (t) dN (iq (t)) dN (iq ) diq (t) KL = = . (39) dt diq (t) dt ωel (t) dt The purpose of (39) is to cancel the differential contribution from (31). In fact, (37) and (38) yield, respectively, ˆ Φ(t) = η(t) − N (iq (t)) and (40) ˆ dΦ(t) dη(t) dN (iq (t)) = − . (41) dt dt dt Substituting (39) in (41) results in ˆ ˆ dq diq (t) dΦ(t) dη(t) KL = − . (42) dt dt ωel (t) dt Inserting Eq. (42) into Eq. (31) the following expression is obtained 1 : ˆ dq diq (t) dη(t) KL ˆ − = −KΦ(t)− dt ωel (t) dt L ˆ dq diq (t) + R ˆ s id (t) dt K + ωel (t) ˆ dq ωel (t)iq (t) + uq (t) L ; (43) ωel (t) then ˆ s id (t) + L ˆ dq ωel (t)iq (t) + uq (t) R dη(t) ˆ −K = −KΦ(t) . dt ωel (t) (44) Letting N (iq (t)) = kapp iq (t), where with kapp a parameter k ω (t) has been indicated, then, from (39) ⇒ K = appLˆ el , Eq. dq
(40) becomes ˆ Φ(t) = η(t) − kapp iq (t). (45) Finally, substituting (45) into (44) results in the following equation dη(t) kapp ωel (t) =− η(t) − kapp iq (t) + ˆ dt Ldq kapp ˆ ˆ dq ωel (t)iq (t) + uq (t) Rs id (t) + L ˆ dq L ˆ Φ(t) = η(t) − kapp iq (t). (46) Using the implicit Euler method, the following velocity observer structure is obtained: η(k − 1) η(k) = + k ω (k) 1 + ts appLˆ el dq
k2 ωel (k)iq (k) ts app Lˆ dq
+ kapp ωel (k)iq (k) + 1 + ts
ˆ s kapp id (k) ts R ˆ dq L
kapp ωel (k) ˆ dq L kapp ˆ dq L kapp ωel (k) ts Lˆ dq
iq (k)+
ts
1+
ˆ Φ(k) = η(k) − kapp iq (k),
0.25
(38)
uq (k) (47)
1
Expression (31) works under the assumption (34): fast observer dynamics.
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0.2 Stator resistance (Ohm)
Consider
0.15
0.1
0.05 Estimated values of the stator resistance Real value of the stator resistance 0 0
1
2 Time (sec.)
3
4 −3
x 10
Fig. 3. Estimated and real values of Rs stator resistance where ts is the sampling period. Remark 3. Assumption (34) states that the dynamics of the approximating observer should be faster than the dynamics of the physical system. This assumption is typical for the design of observers. 2 Remark 4. The estimator of Eq. (47) presents the following limitations. For low velocity of the motor (ωmec. (t) << ωmecn (t)), where ωmecn (t) represents the nominal velocity of the motor) the estimation of Φ becomes inaccurate. In fact, because of ωel (t) dividing the state variable η, the observer described by (47) becomes hyperdynamic. Critical phases of the estimation are the starting and ending of the movement. Another critical phase is represented by high velocity regime. In fact, it has been proven through simulations that if ωmec (t) >> ωmecn (t), then the observer described by (47)becomes hypodynamic. According to the simulation results, within some rage of frequency, this hypo-dynamicity can be compensated by suitable choice of kapp . 2
4. SIMULATION RESULTS Simulations have been carried out using a special stand with a 58 kW traction PMSM. The stand consists of PMSM, tram wheel and continuous rail. The PMSM is a prototype for low floor trams. PMSM parameters: nominal power 58 kW, nominal torque 852 Nm, nominal speed 650 rpm, nominal phase current 122 A and number of poles 44. Model parameters: R = 0.08723 Ohm, Ldq = Ld = Lq = 0.8 mH, Φ = 0.167 Wb. Surface mounted NdBFe magnets are used in PMSM. Advantage of these magnets is inductance up to 1.2 T, but theirs disadvantage is corrosion. The PMSM was designed to meet B curve requirements. The stand was loaded by an asynchronous motor. The engine has parameters as follows: nominal power 55 kW, nominal voltage 380 V and nominal speed 589 rpm. Figures 3, 4, and 5 show the estimation of Rs stator resistance, Ldq inductance, and Φ magnet flux, respectively. From these figures, at the beginning of the estimation, it is visible the effect of the limit of the procedure discussed in remark 4. In the present simulations at t = 0 corresponds ωel (t) = 0.
16th IFAC Symposium on System Identification Brussels, Belgium. July 11-13, 2012
−3
1
x 10
0.9
Ldq−Inductance (H)
0.8 0.7 0.6 0.5 0.4
Estimated values of the Lqd inductance Real value of Lqd−Inductance
0.3 0.2 0.1 0 0
0.5
1 Time (sec.)
1.5
2 −3
x 10
Fig. 4. Estimated and real values of Ldq inductance
10 Permanent magnet flux linkage (Vs)
8 6 4 2 0 −2
Estimated values of the permanent magnet flux linkage Real value of the permanent magnet flux linkage
−4 −6 −8 −10 0.5
1 Time (sec.)
1.5
2 −3
x 10
Fig. 5. Estimated and real values of the permanent magnet flux linkage 5. CONCLUSIONS AND FUTURE WORK This paper deals with a dynamic estimator for fully automated parameters indentification for three-phase synchronous motors.The technique uses a decoupling procedure optimized by a minimum variance error to estimate the inductance and resistance of the motor. Moreover, a dynamic estimator is shown to identify the amplitude of the linkage flux using the estimated inductance and resistance. It is generally applicable and could be used also for the estimation of mechanical load and for other types of electrical motors, as well as for other dynamic systems with similar nonlinear model structure. Through simulations of a synchronous motor used in automotive applications, this paper verifies the effectiveness of the proposed method in identification of PMSM model parameters and discusses the limits of the found theoretical and the simulation results. Future work includes the estimation of a mechanical load and the general test of the present algorithm using a real motor. REFERENCES [M.A. Rahman et al. 1996] M.A. Rahman and P. Zhou. ”Analysis of brushless permanent magnet syn-
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