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Power Electronics and Drives
Power Electronics and Drives
Volume 6(41), 2021 DOI: 10.2478/pead-2021-0017
* Email: lukniewiara@umk.pl
Research Paper
1 Institute of Engineering and Technology, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University in Toruń, Toruń, Poland
2 Institute of Control and Industrial Electronics, Warsaw University of Technology, Warsaw, Poland
Tomasz Tarczewski1,*, Łukasz J. Niewiara1, Lech M. Grzesiak2
Articial Neural Network-Based
Gain-Scheduled State Feedback Speed
Controller for Synchronous Reluctance Motor
1. Introduction
High torque density, robust design and low manufacturing cost have caused synchronous reluctance motors
(SynRMs) to recently receive increased attention (Farhan et al., 2020). As stated in Boldea and Tutelea (2018),
SynRMs are even 20% cheaper and 4% more efcient than induction motors in the eld of variable-speed drives.
For this reason, this type of motor is recently applied in electric vehicles, elevators, chillers and HVAC systems
(Bianchi et al., 2016; Credo et al. 2020; Oliveira and Ukil 2019; Li et al., 2020).
The advantages of SynRM mentioned above come with non-linear inductance characteristics (Farhan et al.,
2020; Boldea and Tutelea 2018). Such shortcoming causes the need for advanced control algorithms to achieve
high dynamic performance. Several publications concerning non-linear control approaches are available in the
literature. In Senjyu et al. (2003), a high-efciency control strategy based on an extended Kalman lter (EKF) is
proposed to improve machine efciency. The EKF is used to estimate the inductance and resistance of the SynRM.
These are used to modify the parameters of controllers that operate in a cascade manner. Such a solution ensures
better efciency when compared with conventional control methods. However, the tuning process of the EKF and
cascade controllers with non-constant gains is not trivial. In Hadla and Cruz (2016), a control structure with nite
control set model predictive controller with the outer PI speed controller is proposed. The active ux predictive
control is developed to assure fast torque response and ripple minimisation. Reduced cross-coupling effects and
suitable dynamic responses are obtained for robust control based on linear matrix inequalities (Scalcon et al.,
2020). In this solution, expert knowledge is required as the synthesis process of the controller is based on the
Lyapunov approach.
276
Received: October 12, 2021; Accepted: November 10, 2021
Keywords: synchronous reluctance motor • state feedback controller • gain-scheduling • articial neural network • robustness analysis
Abstract: This paper focuses on designing a gain-scheduled (G-S) state feedback controller (SFC) for synchronous reluctance motor (SynRM)
speed control with non-linear inductance characteristics. The augmented model of the drive with additional state variables is introduced
to assure precise control of selected state variables (i.e. angular speed and d-axis current). Optimal, non-constant coefcients of the
controller are calculated using a linear-quadratic optimisation method. Non-constant coefcients are approximated using an ar ticial
neural network (ANN) to assure superior accuracy and relatively low usage of resources during implementation. To the best of our
knowledge, this is the rst time when ANN-based gain-scheduled state feedback controller (G-S SFC) is applied for speed control of
SynRM. Based on numerous simulation tests, including a comparison with a signum-based SFC, it is shown that the proposed solution
assures good dynamical behaviour of SynRM drive and robustness against q-axis inductance, the moment of inertia and viscous
friction uctuations.
Open Access. © 2021 Tarczewski et al., published by Sciendo. This work is licensed under the Creative
Commons Attribution NonCommercial-NoDerivatives 4.0 License.
ANN-based gain-scheduled controller for reluctance motor
Articial intelligence-based control methods can also be applied to cope with the non-linear and cross-coupled
behavior of electrical drives (Cvetkovski and Petkovska 2021, Ewert 2019). In Lin et al. (2019), an adaptive
backstepping speed controller is designed. In order to improve the transient dynamic response of SynRM under
maximum torque per ampere (MTPA) operating conditions, a recurrent Hermite fuzzy neural network is used. Thanks
to applying the articial intelligence-based approach, a higher current angle command for the transient torque
results in faster dynamic response of the SynRM. Due to this, drawbacks of the classical PI control structure have
been overcome. By contrast, designing a control system with a recurrent Hermite fuzzy neural network seems to be
not trivial as it is difcult to adjust the fuzzy rules and membership functions online. The presented results indicate
that the described solution assures robustness and satisfactory speed control performance. In Truong et al. (2016),
an adaptive approach based on articial neural networks (ANNs) is used to calculate the optimal stator currents of
SynRM. The Adaline with an online learning process (i.e. the Widrwo–Hoff algorithm) is utilised. Designed Adaline
controllers take the place of the conventional torque and speed ones. The presented results show the reduction of
torque and speed ripples and better convergence; the copper losses have also been reduced.
As shown in Tarczewski et al. (2021) and Hannoun et al. (2011), a gain-scheduled (G-S) approach can also be
applied to cope with the non-linear and cross-coupled behaviour of SynRM. In Hannoun et al. (2011), a PI current
controller with variable gains is proposed, while in Tarczewski et al. (2021), a cascade-free state feedback controller
(SFC) is applied for simultaneous control of motor’s current and speed. However, in Hannoun et al. (2011), a state
feedback approach is also utilised to synthesise the self-tuned PI current controller. Its parameters are adjusted online
in relation to the current and position change. In addition, a back-EMF compensation scheme has been implemented
to reduce the bandwidth requirements placed upon the controller. As a result, the controller limits the loop bandwidth
variations due to the gain changing. The results prove the good performance of this type of regulation.
The non-linearity tolerance and robustness shows SFC to be a good alternative for complex control schemes
developed for SynRM (Tarczewski et al., 2021; Brasel 2014; Safonov and Athans 1977; Shyu et al., 2001; Tarczewski
and Grzesiak 2009). The provisional results shown in Tarczewski et al. (2021) indicate that high-performance speed
control of SynRM can be obtained if a gain-scheduled state feedback controller (G-S SFC) is used. For this reason, it
was decided to perform further investigations of this solution. In Tarczewski et al. (2021), the non-constant coefcients
of the controller are implemented using the lookup table (LUT)-based approach, where a relatively large amount of
the memory resources is used to assure satisfactory accuracy. As ANNs can be applied to approximate non-constant
relationships with superior accuracy (Grzesiak and Tarczewski 2015), it was decided to design and investigate the
behaviour of ANN G-S SFC. Numerical tests were performed in terms of (i) precise control of angular velocity and
d-axis current and (ii) robustness against q-axis inductance, the moment of inertia, and viscous friction uncertainties.
This paper is organised as follows. Section II describes a model of the SynRM drive with respect to simplifying
assumptions. In Section III, ANN-based G-S SFC is presented, and the training process of the ANN gain approximator
is shown. Section IV discusses numerical tests, including the behavior of angular velocity and d-axis current control
and robustness against q-axis inductance and mechanical parameters uctuations. A comparison of the proposed
ANN-based G-S SFC with signum-based SFC is also included. Section V concludes this paper.
2. SynRM Drive’s Model
In this section, a model of the SynRM drive is introduced. Since an SFC responsible for cascade-free control of motor
currents and angular velocity is to be designed, it was decided to express the model of the plant in a state equation
form. For convenient development of vector control, it is described in the d-q reference frame (Kazmierkowski et al.,
2001). A schematic diagram of the considered control structure is shown in Figure 1.
Because of the complexity of the considered model, a few assumptions are made to simplify the designing
process of the controller and assure high-performance operation of the drive (Tarczewski et al., 2021) as follows:
• Magnetic saturation of inductance is considered for d-axis, that is, Ld(id) is taken into account (Boldea and
Tutelea 2018; Awan et al., 2019);
• A constant value of Lq is assumed (Boldea and Tutelea 2018; Kazmierkowski et al. 2011; Youse-Talouki et al.,
2017);
• The decoupling procedure is applied to remove cross-coupling between the d- and q-axes introduced by the
back-EMFs (Kazmierkowski et al. 2011; Tarczewski et al., 2021);
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Tarczewski et al.
• Additional state variables are introduced to provide steady-state error-free control of the d-axis current and the
angular velocity (Tarczewski et al., 2021);
• The load torque is omitted during the synthesis process of the controller.
For the assumptions listed above, the following model of the SynRM drive is obtained (Tarczewski et al., 2021):
()
( )
()
( ) ( )
= ++
,
dd d
d
Li t L t t
dt
x
A x B u Fr (1)
with:
()
()
()
( )
()
()
( )
( )
( )
( )
( )
( )
0 0 00 000
1 0 0 00 00 10
0 0 00 00
, , , ,
000
301
00
00 0
200
0 0 0 10
s
p
dd
d
dd
i
s
p
dd d q
q
qm
dd qd m
mm
RK
Li it
Li
et
RK
Li L t it
L
Lt
pL i L i Bet
JJ
−
−
−
= = ==
−
−
−
A B Fx
ω
ω
,
( )
( )
( ) ( )
( )
( )
,
ref
dc d
ref
qc m
ut
it
tt
ut t
==
ur
ω
where
s
R
– stator resistance,
( )
,
dd q
Li L
– stator inductances,
p
– number of pole pairs,
m
B
– viscous friction,
m
J
–
moment of inertia,
p
K
– converter gain,
( ) ( )
,
dq
itit
– space vector current components,
( )
mtω
– angular velocity,
Fig. 1. Schematic diagram of SynRM control structure with ANN-based G-S SFC. ANN, articial neural network; G-S SFC, gain-scheduled state
feedback controller; SynRM, synchronous reluctance motor.
278
ANN-based gain-scheduled controller for reluctance motor
( ) ( )
,
dc qc
u tu t
– decoupled space vector voltage components and
( )
i
et
– state variable corresponds to the integral
of the d-axis current error:
( ) ( ) ( )
=−
∫0
tref
i dd
et i i dτ ττ
(2)
( )
et
ω
– state variable corresponds to the integral of the angular velocity error:
( ) ( ) ( )
0
tref
mm
et d
=−
∫
ωωτ ω τ τ
(3)
( )
ref
d
iτ
– the reference value of d-axis current,
( )
ref
m
ωτ
– the reference value of angular velocity. From Eq. (1), it can
be seen that cross-couplings between d and q axes do not exist. These were removed using a feedback decoupling
method. In this approach, additional voltage components consisting of cross-coupled back-EMFs are introduced
with respective signs to eliminate cross-coupled terms. A detailed explanation can be found in Tarczewski et al.
(2021).
After analysis of the state and input matrices from Eq. (1), one can see that regardless of the FDM procedure
used, dependence between Ld and id(t) is necessary to calculate the respective components. In this paper, the
following notation has been adopted:
()
()
( )
()
−
=− = =
11 43 11
3
, ,
2
dd q p
s
dd m dd
pL i L K
R
aa b
Li J Li (4)
and the coefcients mentioned above were calculated for SynRM from ABB (type M3AL 90LA 4 IMB3/IM1001) with
the parameters listed in Table 1.
Based on the motor parameters, the relation between Ld and id(t) and the shape of Eq. (4) have been obtained,
and these are shown in Figure 2.
Table 1. Parameters of SynRM drive
Parameter Symbol Value Unit
Nominal power PN1.1 kW
Nominal current IN4.1 A
Stator resistance Rs6
q-axis inductance Lq40 mH
Moment of inertia Jm2×10−3 kgm2
Viscous friction Bm1.4×10−2 Nms/rad
Number of pole pairs p2
Converter gain Kp282
SynRM, synchronous reluctance motor.
Fig. 2. Inductance and matrix coefcients versus d-axis current: (a) Ld, (b) a11, (c) a43 and (d) b11.
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Tarczewski et al.
As highly non-linear relationships have been obtained, it was decided to propose a non-linear control strategy.
In Figure 2(a), it can be seen the constant value of d-axis inductance. It was calculated as a mean value of the
presented relationship and will be used to design SFC for comparison. Both are described in the following
section.
3. ANN-based G-S SFC and Signum-based SFC
In this section, the design process of state feedback speed controller with non-constant coefcients is presented.
Among other adaptive control schemes, the G-S SFC is relatively simple for design and implementation and
assures robustness and high-performance operation of the AC motors (Tarczewski et al., 2021; Brasel 2014;
Tarczewski et al., 2017). In this solution, a non-stationary model of the plant, as in Eq. (1), is applied to obtain
the controller’s coefcients for the actual value of Ld and id. In such a case, a set of SFC coefcients at the
operating points dened by an actual value of d-axis current will be calculated and the following control law is
introduced:
( )
( )
( )
=d
t it
u Kx (5)
with:
()
()()()()()
()()()()()
=
123 45
123 45
dddddddddd
d
qdqdqdqdqd
kikikikiki
i
kikikikiki
K
where
( )
d
i
K
is the non-constant gain matrix of SFC controller. In this approach, a linear-quadratic optimisation
method has been applied to calculate the coefcients of the controller. These are selected during minimisation of
the following performance index:
( ) ( ) ( ) ( )
=+
∫
TT TT
0
t
LQR
Id
x Qx u Ruτ τ τ ττ (6)
where
( )
( )
==
1 2 345 12
diag , diag q q qq q rrQR
– manually selected penalty matrices. Values of Q and R
have been selected to provide steady-state error-free control of the angular velocity and d-axis current and good
dynamical behaviour of the drive. According to the information presented in Tarczewski et al. (2021), the following
coefcients were selected:
= = === = =
1 3 412 2 5
1, 1000, 100q q q rr q q
. Gain coefcients have been calculated for
operating points dened by the d-axis current in a range of
[ ]
10;10∈−
d
i A with 10 mA resolution. The non-constant
coefcients obtained using the lqrd MATLAB’s function are shown in Figure 3, while the rest from Eq. (5) are equal
to zero.
From Figure 3., one can see that coefcients of SFC are highly non-linear, and therefore its approximation and
implementation seem to be non-trivial. For the sake of comparison, a constant approximation of the coefcients also
has been made. Since kq4 and kq5 coefcients are discontinuous, it was decided to apply signum-based approximation,
as shown in Figure 3 (d) and (e). Therefore, the obtained controller was named a signum-based SFC. In the
case of non-constant coefcients, a gain-scheduling task can be made using lookup tables or a polnomials-based
approach. However, such solutions require large hardware resources for good accuracy (Tarczewski et al., 2021;
Hannoun et al., 2011; Kumar et al., 2016). In the proposed approach, an ANN gain approximator is designed. As
an input, the actual value of d-axis current in a range of
[ ]
10;10
d
i
∈− A is used, while the output should approximate
non-constant coefcients of SFC shown in Figure 3 and the Ld(id) relationship from Figure 2(a). For this reason, a
neural network with one input and six outputs has been used. The structure of ANN used is shown in Figure 4(a),
while the training process is presented in Figure 4(b).
From Figure 4(a), it can be seen that a feedforward neural network with one hidden layer has been used.
A hyperbolic tangent activation function has been used in the hidden layer, while a linear function has been applied
in an output one. The samples were divided for training, validation and testing sets in the following proportions:
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ANN-based gain-scheduled controller for reluctance motor
70%, 15% and 15%, to provide an efcient learning process. The Levenberg–Marquardt backpropagation algorithm
has been used to learn ANN. From the recorded training process, it can be seen that a superior approximation
level has been obtained for a relatively small ANN after 40 iterations. The considered task was made using nftool
from MATLAB R2021a. The overall time required for neural tting made on PC with Intel Core i7-4720 HQ CPU @
2.6 GHz and 8 GB ram is less than 1 s. Numerical validation of the designed ANN-based G-S SFC is presented in
the following section.
Fig. 3. Non-constant coefcients of SFC versus d-axis current and constant approximation: (a) kd1, (b) kd2, (c) kq3, (d) kq4 and (e) kq5.
Fig. 4. Training stage of ANN gain approximator: (a) structure of ANN and training progress and (b) epochs. ANN, articial neural network.
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Tarczewski et al.
4. Numerical Experiments
The proposed control scheme has been implemented in MATLAB/Simulink, and the designed control structure is
shown in Figure 5(a), the block diagram of ANN-based G-S SFC is presented in Figure 5(b), while the block diagram
of signum-based SFC is depicted in Figure 5(c).
As was stated before, an ANN is applied to approximate the non-linear coefcients of SFC. Moreover, the d-axis
inductance value necessary for calculating the decoupling components in FDM is also provided. From Figure 5(a),
one can see that the proposed SFC allows controlling the d-axis current and the angular velocity. Due to this,
various control strategies for SynRM can be implemented.
First, the complexity of the developed control algorithm has been investigated using the Simulink Proler Tool.
It was found that the execution time of the proposed approach is 60% longer compared with the LUTs-based SFC
described in Tarczewski et al. (2021). On the other hand, a LUT-based solution requires a relatively large amount
of memory resources to assure satisfactory accuracy. As the considered control schemes are implemented in a
microcontroller with ARM Cortex 32-bit core, the complexity of the ANN-based approach seems not to be an issue.
The operation of SynRM with ANN-based G-S SFC is presented in Figure 6. An analysis of angular velocity
reversal transients with 3 Nm load torque step changes in Figure 6(a) and (f) illustrates the satisfactory performance
(i.e. good dynamic behaviour, zero steady-state error, and fast load torque compensation). From Figure 6(b) and (g),
one can see proper d-axis current control in both directions, allowing various control strategies to be implemented.
It can be seen that the rise time and the maximum uctuation of angular speed caused by load torque step changes
are shorter for the increased value of d-axis current. The same observation applies to the electromagnetic torque
produced by SynRM, as shown in Figure 6(e) and (j). Finally, the sinusoidal shape of phase currents recorded
Fig. 5. Block diagram of (a) proposed control structure, (b) ANN-based G-S SFC and (c) signum-based SFC. ANN, articial neural network; G-S SFC,
gain-scheduled state feedback controller.
282
ANN-based gain-scheduled controller for reluctance motor
during start-up, velocity reversal, and the load torque compensation indicates a high-performance operation of the
investigated drive.
In the next step, the robustness of the proposed ANN-based G-S SFC is investigated. As the constant value of
Lq has been assumed during synthesis, it was decided to investigate its impact on control performance.
From Figure 7, it can be seen that the proposed control system is robust against inductance uctuations in the
range of [Lq/2; 2Lq]. The impact of the Lq value on the angular speed and the d-axis current control is negligible.
Shown in Figure 7(c) and (f), waveforms of the q-axis current indicate slight differences caused by the Lq uctuations,
especially when the load torque is imposed.
The impact of mechanical parameters uctuation on the control system performance has been investigated
in the next stage. Since the SynRM drive can be applied in an autonomous electric vehicle, robustness against
mechanical parameters uctuation (i.e. the moment of inertia and friction) was also investigated, and the respective
waveforms are shown in Figure 8.
Fig. 6. Angular velocity reversal transients of SynRM with ANN-based G-S SFC with 3 Nm load torque perturbation for idref = 2 A (left column) and
idref = −0.5 A (right column): (a) and (f) angular velocity, (b) and (g) direct current, (c) and (h) quadrature current, (d) and (i) phase currents, (e) and
(j) electromagnetic torque. ANN, articial neural network; G-S SFC, gain-scheduled state feedback controller; SynRM, synchronous reluctance motor.
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Tarczewski et al.
Fig. 7. Angular velocity reversal transients of SynRM with ANN-based G-S SFC with 3 Nm load torque perturbation for idref = 2 A (left column) and
idref = −0.5 A (right column) for Lq uctuation: (a) and (d) angular velocity, (b) and (e) direct current, (c) and (f) quadrature current. ANN, articial neural
network; G-S SFC, gain-scheduled state feedback controller; SynRM, synchronous reluctance motor.
Fig. 8. Angular velocity reversal transients of SynRM with ANN-based G-S SFC with 3 Nm load torque perturbation for idref = 2 A (left column) and
idref = −0.5 A (right column) for Jm and Bm uctuation: (a) and (d) angular velocity, (b) and (e) direct current, (c) and (f) quadrature current. ANN, articial
neural network; G-S SFC, gain-scheduled state feedback controller; SynRM, synchronous reluctance motor.
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ANN-based gain-scheduled controller for reluctance motor
In order to quantify the robustness of the proposed ANN-based G-S SFC, the IAE integral indicator has been
analysed for waveforms shown in Figure 7(a) and (d), and Figure 8(a) and (d), respectively. The obtained values
are summarised in Table 2.
The obtained results show that the impact of the q-axis inductance is negligible in both the scenarios considered.
In the case of moment of inertia and friction, a higher impact on control system behaviour is observed for id
ref = − 0.5 A,
which results in a higher difference in the integral absolute error (IAE) index. It is caused by the higher value of the
angular rise time and the much worse load torque compensation. This is in line with the results presented earlier
and with expectations, as the dynamics of electromagnetic torque generation is lower in this case, as shown in
Figure 6(e) and (j). Regardless of IAE uctuations, it can be concluded that the proposed control scheme assures
good performance and robustness against q-axis inductance, the moment of inertia and viscous friction, in the
investigated ranges.
Finally, the robustness of signum-based SFC has been investigated. As in the case of ANN-based G-S SFC, an
impact of Lq, Bm, and Jm uctuations on the control performance has been analysed, and the obtained results are
shown in Figures 9 and 10.
From Figure 9, it can be seen that the robustness of signum-based SFC against q-axis inductance variation is
similar to those observed for ANN-based SFC. By contrast, the overall control performance (e.g. IAE performance
index listed in Table 3) is slightly worse for the signum-based approach. A similar conclusion can be drawn from
Figure 10, where an investigation against mechanical parameters uctuation is shown. It should be noted that the
Fig. 9. Angular velocity reversal transients of SynRM with signum-based SFC with 3 Nm load torque perturbation for idref = 2 A (left column) and
idref = −0.5 A (right column) for Lq uctuation: (a) and (d) angular velocity, (b) and (e) direct current, (c) and (f) quadrature current. SynRM, synchronous
reluctance motor.
Table 2. Comparison of the IAE performance for ANN-based G-S SFC index for parameters uctuation
Lq, Jm, BmLq/2, Jm, Bm2Lq, Jm, BmLq, 10Jm, BmLq, Jm, 3Bm
IAE for id
ref = 2 A 3.822 3.798 3.881 3.833 3.885
IAE for id
ref = −0.5 A 5.531 5.453 5.731 6.401 5.949
ANN, articial neural network; G-S SFC, gain-scheduled state feedback controller.
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Tarczewski et al.
greater difference in the IAE performance index between ANN-based G-S SFC (Table 2) and signum-based SFC
(Table 3) is observed for id
ref = 2 A. The higher value of the d-axis current, the greater deviation of the control system
from operating conditions are established by the mean value of Ld (Figure 2(a)) and constant coefcients of SFC
(Figure 3).
5. Conclusion
In this paper, a G-S SFC has been applied to high-performance control of SynRM with non-linear inductance
characteristics. Non-linear coefcients of the G-S SFC are approximated using an ANN. Such a solution
assures superior accuracy and relatively low usage of resources during implementation compared with the LUT-
based approach. For the sake of comparison, a signum-based SFC has also been developed and investigated.
It was proven that applying an augmented drive model with additional state variables assures precise control
of angular velocity and d-axis current in both the considered controllers. The obtained results indicate that the
ANN-based G-S SFC assures satisfactory dynamical behaviour of SynRM drive and robustness against q-axis
inductance, the moment of inertia and viscous friction. In the case of signum-based SFC, slightly worse control
performance is observed, especially for greater values of the d-axis current. By contrast, its implementation is
much more simplied. Further investigation of the proposed control scheme, including experimental tests, is
planned.
Fig. 10. Angular velocity reversal transients of SynRM with signum-based SFC with 3 Nm load torque perturbation for idref = 2 A (left column) and
idref = −0.5 A (right column) for Jm and Bm uctuation: (a) and (d) angular velocity, (b) and (e) direct current, (c) and (f) quadrature current. SynRM,
synchronous reluctance motor.
Table 3. Comparison of the IAE performance index for signum-based SFC for parameters uctuation
Lq, Jm, BmLq/2, Jm, Bm2Lq, Jm, BmLq, 10Jm, BmLq, Jm, 3Bm
IAE for id
ref = 2 A 4.187 4.173 4.219 4.198 4.248
IAE for id
ref=−0.5 A 5.567 5.517 5.685 6.141 5.811
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ANN-based gain-scheduled controller for reluctance motor
Acknowledgments
This research was supported by the ‘Excellence Initiative—Research University’ programme of Warsaw University
of Technology under grant ‘ENERGYTECH-1 Power’ and by the ‘Excellence Initiative—Research University’
programme of Nicolaus Copernicus University.
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