Sensorless control of a PMSM using an efficient extended Kalman filter Z. Boulbair(1) , M. Hilairet(2) , F. Auger(1) , L. Loron(1) (1) IREENA, Bd de l’Universit´e, BP 406, F-44602 Saint Nazaire cedex, France (2) LESTER (UBS), rue Saint Maud´e, BP 92116, F-56321 Lorient Cedex, France. E-mail: {boulbair,auger,loron}@ireena.univ-nantes.fr,
[email protected] Abstract—This paper deals with the sensorless control of permanent magnet synchronous motors (PMSM). To perform the estimation of the speed and position of the rotor, we present an efficient extended Kalman (EKF) filter using an exact discrete-time model. Compared to a classical EKF, the proposed algorithm reduces the number of arithmetic operations by a factor 3.6. Experimental results have been performed on a 4.4 kW machine to compare the performances of the proposed algorithm with an adaptive approach.
I. Introduction Permanent Magnet Synchronous Motors (PMSM) are used in a large variety of speed control applications, because of their high efficiency and large power density [13]. But the control of these motors requires the knowledge of the rotor shaft position. Electromechanical sensors can be used, but they increase the cost and the encumbrance of the motor, and reduce the system reliability. Therefore, many research efforts have been made in the last years to estimate the rotor speed and position from the stator voltages and currents [18]. The resulting methods can be classified in three classes of solution. One of them is based on the injection of a high frequency voltage [7], [4]. A demodulation of the stator current signals yields the rotor position, even at standstill. However, this approach requires a voltage source inverter with a high switching frequency. All the adaptive approaches, trying to cancel the difference between the real and the estimated rotor position [15], [5], [14], [2] can be brought together in a second class of solutions. These approaches often require to differentiate the measured stator current signals, and may perform poorly at low speed. The third class of solutions is based on the rotor Back Electromotive Force (EMF), which bears both the velocity and the position informations. Nonlinear State observers such as extended Kalman filters can be used to estimate this back EMF from the stator currents and voltages. This paper presents several original results which are organized as follows: After a presentation of the chosen motor model in §II-A, an exact discretization of the continuous-time state model is described in §II-B. The observability of the nonlinear state space model used for the estimation of the rotor flux and speed is studied in §II-C. Then an efficient implementation of an extended Kalman filter, based on the use of an additionnal virtual state [11], is presented in §III. Some experimental results are presented in §IV to compare the resulting sensorless control with an adaptive speed estimator. These comparisons use a benchmark which
has been defined to test the observers on and near the unobservability singularity. II. Modelization, discretization and state
observability A. Motor model The usual model of a PMSM is a fourth order state space model [18], with the stator voltages as inputs, the stator currents as measured outputs and four internal states which can be chosen as the stator current and the rotor flux components. To design the extended Kalman filters, we choosed to write the motor equations in the α-β stationary reference frame (fixed to the stator windings) ½
dX dt (t)
= Y (t) =
£
vα
vβ
£ ¤t Y = ıα ıβ ¤t Φα Φβ ¸t · 0 0 0 1 0 C= 1 0 0 0 1
U
=
X
ıα ıβ · 1 1 = 0 Lc −R/Lc 0 = 0 0
B
A(ω)
=
£
A(ω) X(t) + B U (t) C X(t)
¤t
0 −R/Lc 0 0
0 −ω/Lc 0 ω
0 0
0 0
¸
ω/Lc 0 , −ω 0
where ıα , ıβ , vα , vβ , Φα and Φβ are the coordinates of the stator current, stator voltage and rotor flux, respectively. The motor is supposed to be controlled by the stator voltage, and the stator current is supposed to be measured. The parameter ω is the electrical velocity, linked to the mechanical rotor velocity Ω by the relation Ω = ω/p, 2 p being the number of poles. B. Exact discrete-time modelization For the implementation of a flux estimator on a microcontroller or a DSP, a discrete-time state space model is required. Provided that the rotor velocity is slowly varying compared to the sampling period and to the electrical quantities, the previous continuous-time state model leads to the following discrete-time state space model [16]: ½
X[k + 1] = Ad (ωm ) X[k] + Bd U [k] Y [k] = C X[k]
¡ ¢ with Ad (ω) = eA(ω) Ts and Bd = A(ω)−1 eA(ω) Ts − I . To compute these matrices, we first define complex ¯ = Φα + Φβ and v¯ = vα + vβ quantities ¯ı = ıα + ıβ , Φ to obtain a second order model with complex valued coefficients and states. This allows to use the expression of the matrix exponential of a 2 × 2 matrix [1] ¡ ¢ ¢ ¡ ¯ ¯ − λ1 I eλ1 Ts A(ω) − λ2 I − eλ2 Ts A(ω) eA(ω) Ts = , λ1 − λ2 where λ1 =µ−R/Lc and λ2 =¶ ω are the eigenvalues −R/Lc − Lωc ¯ . This complex valued of A(ω) = 0 ω matrix exponential is finally expanded to a 4 × 4 realvalued matrix Ad (ω) and associated to a real-valued matrix Bd a11 0 a12 −b12 0 a11 b12 a12 Ad (ω) = 0 0 a22 −b22 0 0 b22 a22 · ¸t b1 0 0 0 Bd = 0 b1 0 0 with a11 a12 b12 b1
− TTs
, a22 = cos(ωTs ), b22 = sin(ωTs ) = e 2 2 w T (a11 − a22 ) + ωT b22 = Lc (1 + ω 2 T 2 ) ωT (a11 − a22 ) − w2 T 2 b22 = Lc (1 + ω 2 T 2 ) 1 − a11 = R
where Ts is the sampling period and T = Lc /R. This exact discretization avoids the use of a second order series expansion of a matrix exponential [18], reducing therefore the uncertainties caused by the discretization process. As for an induction machine [11], the matrices Ad and Bd contain many symmetries and antisymmetries which reduce the computational cost of the transition matrix Ad (ω) by a factor 2. Moreover, a11 and b1 do not depend on ω and therefore do not need to be computed at each time sample. Finally, major simplifications in the forthcoming Kalman filter equations will also be allowed since a222 + b222 = 1. C. Observability of the extended state When electromechanical position encoders are not used, the rotor speed can be considered as an additionnal state which varies slowly and randomly (ω˙ = v5 , where v5 is a random variable with a low variance). To study the observability of this continuous-time extended state space, we can first consider the case when ω = 0 rad/s. The motor equations become dıα dt (t) dıβ dt (t)
= − LRc ıα , = − LRc ıβ ,
dΦα dt (t) dΦβ dt (t)
= =
0 0
These equations show that the rotor flux is constant and not linked to the stator currents. This means that the rotor flux can not be deduced from measurements
State and parameters prediction X[k + 1|k] Θ[k + 1|k]
= =
Ad [k] X[k|k] + Bd [k] U [k] Θ[k|k]
A priori covariance matrix computation P [k + 1|k]
=
F [k]
=
F [k] P [k|k] F [k]t + Q
h
Ad [k] 0
∂ ∂Θ (Ad [k] X[k|k]
+ Bd U [k])Θ[k|k] I
i
Kalman gain computation
h
K[k + 1]
=
H
=
P [k + 1|k] H t (H P [k + 1|k] H t + R)−1
£
C
0
¤
State and parameters correction X[k + 1|k + 1] Θ[k + 1|k + 1]
i
h
= +
X[k + 1|k] Θ[k + 1|k]
i
K[k + 1] (Y [k + 1] − C X[k + 1|k])
A posteriori covariance matrix computation P [k + 1|k + 1] = P [k + 1|k] − K[k + 1] H P [k + 1|k]
TABLE I Conventional extended Kalman filter equations
of the stator currents. The state space is therefore unobservable. When ω is not equal to zero, the local observability of this non-linear state model can be studied [6]. For this, we compute ¶ µ ω 3 Φα ∂ıα ∂ıβ ∂˙ıα ∂˙ıβ ∂¨ıα , , , , = D1 = det ∂Xe ∂Xe ∂Xe ∂Xe ∂Xe L3c µ ¶ ∂ıα ∂ıβ ∂˙ıα ∂˙ıβ ∂¨ıβ ω 3 Φβ D2 = det , , , , = ∂Xe ∂Xe ∂Xe ∂Xe ∂Xe L3c T
where Xe = [ıα ıβ Φα Φβ ω] is the extended state. Since Φ2α + Φ2β = Φ2f is a constant, D1 and D2 can not be both equal to zero. This shows that there is always at least one of these systems which is full rank, showing therefore the local observability of the system. III. Sensorless speed estimators A. Classical extended Kalman filter So as to perform a vectorial control of the machine, the rotor flux can be estimated by a extended Kalman filter [8], [10] as done in [9], [3]. Its conventional equations are recalled in Table I. When implemented in a straightforward manner, these equations require one matrix inversion and several matrix sums and products and therefore have a cumbersome computational complexity. If we focus more closely on velocity estimation, the parameter vector Θ can be chosen as Θ[k] = ω[k] T , and the matrices F [k] and H are a11 0 a12 −b12 f1 0 a11 b12 a12 f2 0 a22 −b22 f3 F [k] = 0 0 0 b22 a22 f4 0 0 0 0 1
· H
with f1
=
f2
=
f3
=
f4
=
d1
= +
d2
= −
=
1 0
0 1
0 0
0 0
0 0
¸
∂ıα [k + 1|k] = d1 Φα [k|k] − d2 Φβ [k|k] ∂Θ ∂ıβ [k + 1|k] = d2 Φα [k|k] + d1 Φβ [k|k] ∂Θ ∂Φα Ts [k + 1|k] = − Φβ [k + 1|k] ∂Θ T ∂Φβ Ts [k + 1|k] = Φα [k + 1|k] ∂Θ T (Θ[k|k]2 TTs − 1) b22 + Θ[k|k] TTs a22 ∂a12 = ∂Θ Lc (1 + Θ[k|k]2 ) 2(a11 − a22 )Θ[k|k] + b22 Lc (1 + Θ[k|k]2 )2 2Θ[k|k] b22 + 2(a22 − a11 ) ∂b12 =− ∂Θ Lc (1 + Θ[k|k]2 )2 (Θ[k|k]2 TTs − 1) a22 − TTs Θ[k|k] b22 + a11 Lc (1 + Θ[k|k]2 )
But the resulting extended Kalmna filter has a heavy computational cost, because the Kalman gain and the covariance matrices have many distinct values. This makes this algorithm impossible to implement on cheap microcontrollers or DSPs. B. Efficient extended Kalman filter So as to force the computational burden of the speed estimation algorithm, a virtual parameter ω v [k] T is added to the parameter vector: Θ[k]t = [ω[k] T ω v [k] T ]. This new parameter is supposed to be nearly constant (ω v [k + 1] T = ω v [k] T + v6 [k]) and to be virtually bound to the plant model so that a11 0 a12 −b12 f1 −f2 0 a11 b12 a12 f2 f1 0 0 a22 −b22 f3 −f4 F [k] = 0 0 b22 a22 f4 f3 0 0 0 0 1 0 0 0 0 0 0 1 This matrix now satisfies the same structural properties as Ad (ω), so if Q and R are chosen so as to share the same structure, for example α1 0 α4 0 0 0 0 α1 0 α4 0 0 · ¸ 1 0 α4 0 α2 0 0 0 R= Q= 0 α4 0 1 0 α2 0 0 0 0 0 0 α3 0 0 0 0 0 0 α3 then the two covariance matrices and the correction gain have the following structures: P11 0 P13 P14 P15 P16 0 P11 −P14 P13 −P16 P15 P13 −P14 P33 0 P35 P36 P = P14 P13 0 P33 −P36 P35 P15 −P16 P35 −P36 P55 0 P16 P15 P36 P35 0 P55
· t
K =
K11 0
0 K11
K13 −K14
K14 K13
K15 −K16
K16 K15
¸
This structure reduces the number of distinct values in these matrices, and allows the Kalman filter to be implemented by the following scalar recurrence equations: P11 [k + 1|k] = α1 + a211 P11 [k|k] + (a212 + b212 ) P33 [k|k] + 2 a11 (a12 P13 [k|k] − b12 P14 [k|k]) + 2 a11 (f1 P15 [k|k] − f2 P16 [k|k]) + 2 (a12 f1 + b12 f2 ) P35 [k|k] + 2 (b12 f1 − a12 f2 ) P36 [k|k] + (f12 + f22 ) P55 [k|k] P13 [k + 1|k] = α4 + a11 (a22 P13 [k|k] − b22 P14 [k|k]) + (a12 a22 + b12 b22 ) P33 [k|k] + a11 (f3 P15 [k|k] − f4 P16 [k|k]) + (f1 a22 + f2 b22 + f3 a12 + f4 b12 ) P35 [k|k] {z } | {z } | tmp1
tmp2
+
(f1 b22 − f2 a22 + f3 b12 − f4 a12 ) P36 [k|k] | {z } | {z }
+
(f1 f3 + f2 f4 ) P55 [k|k]
tmp3
tmp4
P14 [k + 1|k] = a11 (b22 P13 [k|k] + a22 P14 [k|k]) + (a12 b22 − a22 b12 ) P33 [k|k] + a11 (f4 P15 [k|k] + f3 P16 [k|k]) + (b22 f1 − a22 f2 +a12 f4 − b12 f3 ) P35 [k|k] | {z }| {z } tmp3
−tmp4
+
(a12 f3 + b12 f4 −a22 f1 − b22 f2 ) P36 [k|k] | {z }| {z }
+
(f1 f4 − f2 f3 ) P55 [k|k]
tmp2
−tmp1
P33 [k + 1|k] = α2 + P33 [k|k] + 2 (a22 f3 + b22 f4 ) P35 [k|k] + 2 (b22 f3 − a22 f4 ) P36 [k|k] + (f32 + f42 ) P55 [k|k] P15 [k + 1|k] = a11 P15 [k|k] + a12 P35 [k|k] + b12 P36 [k|k] + f1 P55 [k|k] P16 [k + 1|k] = a11 P16 [k|k] + a12 P36 [k|k] − b12 P35 [k|k] − f2 P55 [k|k] P35 [k + 1|k] = a22 P35 [k|k] + b22 P36 [k|k] + f3 P55 [k|k] P36 [k + 1|k] = a22 P36 [k|k] − b22 P35 [k|k] − f4 P55 [k|k] P55 [k + 1|k] = P55 [k|k] + α3 K11 [k + 1] = P11 [k + 1|k]/(P11 [k + 1|k] + 1) K13 [k + 1] = P13 [k + 1|k]/(P11 [k + 1|k] + 1) K14 [k + 1] = P14 [k + 1|k]/(P11 [k + 1|k] + 1) K15 [k + 1] = P15 [k + 1|k]/(P11 [k + 1|k] + 1) K16 [k + 1] = P16 [k + 1|k]/(P11 [k + 1|k] + 1) P11 [k + 1|k + 1] = K11 [k + 1] P13 [k + 1|k + 1] = K13 [k + 1] P14 [k + 1|k + 1] = K14 [k + 1]
D. Computational cost of the algorithms
P15 [k + 1|k + 1] = K15 [k + 1] P16 [k + 1|k + 1] = K16 [k + 1] P33 [k + 1|k + 1] = P33 [k + 1|k] − (K13 [k + 1] P13 [k + 1|k] + K14 [k + 1] P14 [k + 1|k]) P35 [k + 1|k + 1] = P35 [k + 1|k] − (K13 [k + 1] P15 [k + 1|k] + K14 [k + 1] P16 [k + 1|k]) P36 [k + 1|k + 1] = P36 [k + 1|k] − (K13 [k + 1] P16 [k + 1|k] − K14 [k + 1] P15 [k + 1|k]) P55 [k + 1|k + 1] = P55 [k + 1|k] − (K15 [k + 1] P15 [k + 1|k] + K16 [k + 1] P16 [k + 1|k]) C. Adaptive speed estimator The proposed extended Kalman filter is compared to the adaptive approach proposed by Matsui and Shigyo [15]. When expressed in the rotating frame fixed to the ˆ the motor equations become estimated rotor position θ, vd = R id + Lc didtd − Lc ω iq + Φf ω sin(∆θ) and vq = di R iq + Lc dtq + Lc ω id + Φf ω cos(∆θ), where ∆θ = θˆ − θ is the difference between the estimated and the real rotor position. The voltage references obtained at the outputs of the current regulators of the vector control are therefore vd∗ = R id +Lc didtd +Φf ω sin(∆θ) and vq∗ = di R iq + Lc dtq + Φf ω cos(∆θ). If ∆θ can be considered as small, sin(∆θ) ≈ ∆θ and cos(∆θ) ≈ 1, and these equations become did dt diq dt
= =
-
£
R 1 1 ∗ − id − ∆v + v Lc Lc Lc d R Φf 1 ∗ − iq − ω+ v , Lc Lc Lc q
£ - Luenberger £ id £ - observers ω ˆ £ iq £ £ £ ¾ vq∗
R −P n×+6 - sign 6
number of multiplications and inversions
Bd F X[k + 1|k] P [k + 1|k]
££± ∆ω ∆v - PI -
Computation of
Ad
with ∆v = ω Φf sin(∆θ) ≈ ω Φf ∆θ. The adaptive approach consists in estimating the speed and the angular difference ∆θ from id , iq , vd∗ , vq∗ . This difference is then cancelled by a PI regulator, whose output is combined ˆˆ . This to ω ˆ to build an improved speed estimation ω speed estimation is integrated to obtain the estimated ˆ used by the current loop. position θ, In the original paper [15], the stator currents were numerically differentiated, yielding an amplification of the measurement noise. To avoid this problem, we have used two Luenberger observers (see fig. 1). vd∗
Table II shows the number of arithmetic operations required at each time sample by our modified EKF algorithm, compared to a straightforward matrix-based implementation of a classical EKF. Since ω v [k] T is neither useful nor observable, its estimation does not need to be computed. The total number of arithmetic operations of our algorithm is 268 compared to 968 for a rough algorithm, which means a cost reduction of 3.6. This shows that a thorough study of both the plant model and the Kalman filter equations can halve the computational cost and add a parameter tracking capability. It should also be outlined that the algebraic inversion of the H P [k + 1|k] H t + R matrix reduces the divergence risk of this extended Kalman filter (see [10], p 278). For the evaluation of the total number of arithmetic operations, the trigonometric functions (a22 and b22 ) are supposed to be computed by a fifth degree polynomials, as done on DSP’s [17]. The two trigonometric functions are therefore equivalent to 10 multiplications and 10 additions. The 268 arithmetic operations of our EKF algorithm should also be compared to the 30 arithmetic operations required by the adaptive speed estimator (14 additions and 16 multiplications).
θˆ ω ˆˆ -
?
Fig. 1. Block diagram of the adaptive speed and position estimator.
K[k + 1] P [k + 1|k + 1]
h
X[k + 1|k + 1] Θ[k + 1|k + 1]
Total
i
Number of additions
20 0 23 12 79 6 8 8
(20) (0) (23) (24) (250) (146) (50) (10)
15 0 12 8 58 1 8 10
(15) (0) (12) (20) (225) (111) (50) (12)
114
(507)
154
(461)
TABLE II Number of operations by iteration for an efficient or a rough implementation of the extended Kalman filter.
IV. Experimental results To evaluate the performances of the described methods, a set of experiments have been performed on a test setup shown in figure (3). The tests are carried out on a 4.4 kW PMSM (Yaskawa SGMGH-44DCA6F) whose shaft is coupled to an IM. The numerical values of the parameters of our machine are given in the appendix. The motor is fed by a power converter which receives three control signals from a classical vector control (see fig. 2) implemented on a dSPACE board (DS 1103) with a sampling period of Ts = 100 µs. The stator phase currents are measured by two LEM current sensors and sent to the control system through 12-bit A/D con-
400
400
300
300
200
200
velocity [rpm]
velocity [rpm]
verters. An incremental encoder mounted on the rotor shaft provides the rotor angle. The real rotor speed is obtained by a simple position derivation (using a firstorder backward difference).
100
0
−100
0
−100
−200
−200
−300
−300
−400
100
−400
0
2
4
6
8
10
12
14
16
18
20
22
0
2
4
6
8
10
12
14
16
18
20
22
time [s]
time [s]
(a) proposed EKF
(b) Adaptative speed estimator
Fig. 4. Comparison between the proposed EKF and the adaptive speed estimator. Solid line : measured speed, dotted lines : estimated speed. 350
300
Fig. 2. Block diagram of the overall vector control scheme. velocity [rpm]
250
200
150
100
50
0
3
4
5
6
7
8
9
10
time [s]
Fig. 5. Comparison between the proposed and the classical EKF. Solid line : proposed EKF, dotted lines : classical EKF.
150 400
100
300
200
velocity [rpm]
Fig.(4.a) and (4.b) show the experimental speed responses obtained by the proposed EKF and by the adaptive approach for a succession of transient and steady states. This benchmark has been defined to evaluate the ability of the sensorless control algorithms to follow speed reference steps and ramps at high, medium or zero levels. We underline that the speed used in the control loop is the estimated one. These figures show that both methods give generally satisfying results. However, it should be noted that if the speed remains equal to zero during a long time, the proposed EKF algorithm diverges, because at this value of speed the system model is not observable. Moreover, unlike the adaptive algorithm, the proposed EKF is not able to follow the last speed reference step. To compare the performances of the proposed EKF to those of the classical one, we have applied the same current measurements on both estimators. The results presented on Fig.(5) show that the main difference between the two algorithms can be observed during the transient response. This difference probably lies in the unconstrained structure of the covariance matrices of the classical EKF. After convergence, the estimated velocity obtained by the classical ekf (dotted lines) is nearly the same as the one obtained by the proposed algorithm (solid line). Figures (6.a) and (6.b) show the experimental veloc-
ity responses obtained when low speed reference steps are applied. Any one of the two methods yields satisfying results. For the proposed EKF, a better tuning of the state noise covariance matrix could give more acceptable results at low speed. Thus, it is necessary to find a method to determine the optimal covariance parameters at low and high speed.
velocity [rpm]
Fig. 3. experimental test setup.
100
0
50
0
−100
−50
−200 −1
0
1
2
3
4
5
time [s]
(a) proposed EKF
6
7
−100 −1
0
1
2
3
4
5
6
7
time [s]
(b) Adaptative speed estimator
Fig. 6. response to a low-amplitude speed reference step from standstill. Solid line : measured speed, dotted lines : estimated speed.
Figure (7.a), (7.b), (7.c), (7.d), (7.e) and (7.f) show the experimental speed responses obtained when considering three parameter variations : the stator resistance is increased by 20% of the nominal value, the rotor flux is increased by 20% and the stator inductance is decreased by 20%. The obtained experimental results show that both algorithms are robust to realist parameter variations. In fig. (8), the estimated position deduced from the
250 250
6
200 200
150
velocity [rpm]
50
50
0
−50 −1
5 100
position [rad]
velocity [rpm]
150
100
0
0
1
2
3
4
5
6
−50 −1
7
0
1
2
3
4
5
6
7
time [s]
time [s]
(a) proposed EKF
(b) Adaptative speed estimator
250
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150
4
3
velocity [rpm]
velocity [rpm]
2
100
50
50
0
0
−50 −1
0
1
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3
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6
−50 −1
7
1
100
4
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5
temps[s] 0
1
2
time [s]
3
4
5
6
7
time [s]
(c) proposed EKF
4.1
(d) Adaptative speed estimator
Fig. 8. Comparison between the real rotor position and the position estimated by the proposed extended Kalman filter. Solid line : measured position, dotted lines : estimated position.
ekf with virtual state (∆R = 20%) 250
250
200
200
References [1] 150
velocity [rpm]
velocity [rpm]
150
100
100
50
50
0
0
−50 −1
0
1
2
3
4
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6
−50 −1
7
[2]
0
1
2
3
4
5
6
7
time [s]
time [s]
(e) proposed EKF
(f) Adaptative speed estimator
Fig. 7. Evaluation of the robustness of the algorithms against parameter variations. Solid line : measured speed, dotted lines : estimated speed.
estimated stator flux components is compared to the real position. The highest estimation error is found at the start-up and quickly goes to zero.
[3]
[4]
[5]
[6]
V. Conclusion In this paper, different methods for position and speed estimation have been implemented and tested experimentally. The obtained results show that both methods (adaptive approach and modified ekf) give good results at medium and high speed, but have a poor performance at low speed. The proposed ekf has a low cost compared to the classical ekf, but the adaptive approach still has a lower one. The robustness study shows that both algorithms have a good performance towards parameter variations. The knowledge of initial position is not necessary for the convergence of both algorithms. The drawback of these sensorless control algorithms is therefore the inability to hold a nonzero torque at zero speed. Further research will include a finer study of the proposed EKF at zero speed.
[7] [8] [9] [10] [11] [12] [13] [14] [15]
Appendix : Model parameters Rated flux Rated output power Rated torque Rated speed Rated voltage Rated current Stator resistance Stator inductance Number of pole pairs
Φr = 0.32 Wb Pn = 4.4 kW Cn = 28.4 N m N = 1500 rpm Vn = 400 V In = 16.50 A R = 0.25 Ω Lc = 7.0 mH p=4
[16] [17] [18]
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