Intelligent Adaptation Mechanism of Full-Order Luenberger Observer

based on Fuzzy Logic Applied to Direct Control by Flux Orientation

without Speed Sensor for Doubly Fed Induction Motor

DJAMILA CHERIFI, YAHIA MILOUD

Department of Electrical Engineering,

University of Dr.Tahar Moulay,

Saida GACA Laboratory,

ALGERIA

Abstract: - The present work focused on the control by flow orientation without mechanical sensors on the

doubly fed induction motor by the use of artificial intelligence techniques. Our main contribution lies in the

development of control methodologies to improve control by flux orientation, for this, we acted on the type and

control of inverters by the use of multilevel inverters and then on the observation technique of rotor speed

based on the high gain Luenberger observer and improvements to the speed adaptation mechanism using fuzzy

logic. Simulation tests of the proposed improvement approach were carried out at the end of this work to verify

the behavior of this system in the face of different types of training.

Key-Words: - artificial intelligences, doubly fed induction motor, field-oriented control, fuzzy logic, multi-level

inverter, Luenberger observer.

Received: February 29, 2023. Revised: December 19, 2023. Accepted: February 23, 2024. Published: April 9, 2024.

1 Introduction

The three-phase asynchronous machine powered by

a voltage inverter is a drive system with many

advantages: a simple, robust, and inexpensive

machine structure, and control techniques that have

become efficient thanks to advances in

semiconductors. Power and digital technologies.

This converter-machine assembly, however, remains

restricted to the lower limit of the high power range

(up to a few MW), due to the electrical constraints

experienced by the semiconductors and their low

switching frequency, [1], [2], [3], [4]. In the field of

high-power drives, other solutions are using the

reciprocating machine operating in a somewhat

particular mode, these are double-fed induction

machines "DFIM": are three-phase asynchronous

machines with a wound rotor, which can be powered

by two voltage sources, one of the stator and the

other at the rotor, [5], [6].

Despite all these qualities mentioned above,

many problems remain. Its control, on the other

hand, that of the direct current machine, [7], is more

difficult given the non-linearity and the strong

coupling of its model due to the absence of natural

decoupling between the different input-output

variables, [8]. Considerable progress, both in the

fields of power electronics and micro-electronics,

has made it possible to implement adequate controls

for this machine, making it a machine that

guarantees performances similar to those obtained

by the control of a DC machine with separate

excitation. Along with these technological advances,

the scientific community has developed numerous

control strategies in the literature, the most popular

of which is vector control in its different versions to

control the flux and torque of the asynchronous

machine in real-time, [9], [10], [11].

Whether it is these commands, the control of the

speed and the position of the rotor requires the

presence of an incremental encoder (a sensor).

However, this sensor must be placed in its

environment of use and provide additional space for

its installation. Something which leads to an increase

in cost and a weakening of the drive system. In

addition, the introduction of this fragile device leads

to a reduction in the reliability of the system which

requires special care for itself. Something that is not

always desirable or possible, [12], [13].

For reasons of reliability and economy, the idea

of replacing the mechanical sensor with another of

the algorithm type was born and control without a

speed sensor has become a serious subject of study

for research in recent years. It is then necessary to

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have record estimation or observation techniques to

reconstruct the speed and position of the rotor from

the information collected by measuring the electrical

terminals of the machine's stator. It is therefore

important, when developing the control without a

speed sensor, to emphasize dynamic (pursuit) and

static (rejection) performance, [14], [15].

Currently, the study of DFIM powered by static

converters constitutes a vast research theme in

electrical engineering laboratories. This research

work has led to the appearance of new power

converter structures intended for high-voltage

applications called multilevel converters. The use of

multilevel converters in high power and high voltage

areas makes it possible to simultaneously resolve the

difficulties relating to the size and control of groups

of two-level inverters generally used in this type of

application. To satisfy certain optimization criteria,

namely the reduction of harmonics, [16], [17].

Improving these classic techniques is the concern of

several researchers. This improvement consists of a

compromise between performance and robustness on

the one hand, and simplicity and cost on the other

hand. Thus, intelligent controls that have appeared in

recent years, making it possible to reproduce human

reasoning and which are based on fuzzy logic,

currently occupy an important place in the field of

machine controls, [18], [19].

Our work consists of proposing contributions to

improving the performance and robustness of the

sensorless control of the DFIM by integrating the

concept of fuzzy logic into the speed adaptation

mechanism of the Luenberger observer.

The present work devoted to the implementation

of an algorithm for observing the speed of a DFIM

powered by two five-level inverters using the high-

gain Luenberger observer based on fuzzy logic. At

the end of the work, we will present simulation

results to show the performance of this approach.

2 System Modeling

2.1 Mathematical Modeling of DFIM

For the double-fed induction machine, the control

variables are the stator and rotor voltages.

Considering the stator currents and the rotor fluxes

as state variables, then the DFIM model is described

by the following equation, [20]:





















J

C

J

f

ii

L

p

dt

d

v

T

i

T

L

dt

d

v

T

i

T

L

dt

d

vKv

LT

K

Kiii

dt

d

vKv

L

K

T

K

iii

dt

d

r

sdrqsqrd

r

m

rqrq

r

rdsq

r

m

rq

rdqrrd

r

sd

r

m

rd

sqsq

s

rq

r

rdsqsqssq

sdsd

s

rqrd

r

sqssdsd















)(

1

.

1

.

1

.

2

(1)

with

rs

m

rs

m

rs

s

r

rLL

L

LL

L

K

TR

L

T

R

L

T2

.

1;;

.

1

;; 







2.2 Model of a Five-Level Inverter Type

NPC

The configuration of a five-level NPC-type inverter

is illustrated in Figure 1, for this type of assembly

we use four pairs of complementary transistors each

one with an antiparallel diode, the number of locking

diodes is six, and four capacitors allow the input

voltage to be divided into four equal voltages, it

should be noted that each main switch is sized to

block a voltage level (E/4), [21], [22], [23]:

Fig. 1: Structure of a phase of a five-level NPC

inverter

E

VC1

o

N

VC2

VC3

VC4

Ka11

K'a11

Ka12

K'a12

Ka13

K'a13

Ka14

K'a14

4/E

a

N

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2.3 Switching Functions

For each switch Kxi (i= 11… 14, x= a, b and c), we

define a switching function as follows:







penisKif

isKif

F

xi

xi o0

closed1

The switch controls for the lower half-arms are

complementary to those for the upper half-arms:

)4(

1

 ixxi FF

2.4 Principle of Operation of a Five-Level

NPC Inverter

For this type of inverter, there are only five

functional sequences:

 Vao = +E/2 ⇒[1 1 1 1 0 0 0 0]: switches k11

k’11 k12 k’12 are ordered at closing, tends that

k13 k’13 k14 k’14 are ordered at the opening

 Vao = +E/4 ⇒[0 1 1 1 1 0 0 0]: switches k’11

k12 k’12 k13 are ordered at closing, tends that

k’13 k14 k’14 k11 are ordered at the opening.

 Vao = 0 ⇒[0 0 1 1 1 1 0 0]: the switches k12

k’12 k13 k’13 are ordered at closing, tends that

k14 k’14 k11 k’11 are ordered at the opening.

 Vao = −E/4 ⇒[0 0 0 1 1 1 1 0]: the switches

k’12 k13 k’13 k14 are ordered at closing, tends

that k’14 k11 k’11 k12 are ordered at the opening.



 Vao = −E/2 ⇒[0 0 0 0 1 1 1 1]: the switches

k11 k’11 k12 k’12 are ordered at opening, so that

k13 k’13 k14 k’14 are ordered at closing.

The vector control of the DFIM requires the

installation of an incremental encoder to be able to

measure the rotor speed or position. The

disadvantages inherent in the use of this mechanical

sensor, placed on the machine shaft, are multiple.

First, the presence of the sensor increases the

volume and overall cost of the system. Then, it

requires an available piece of shaft, which can

constitute a disadvantage for small machines.

Finally, the reliability of the system decreases

because of this fragile device which requires special

care for itself. Under these conditions, it is necessary

to reconstruct the state of the machine from easily

measurable or estimable stator voltages and currents.

Several strategies have been proposed in the

literature to achieve this goal. The proposed methods

are based on estimators and observers which lead to

the implementation of simple and fast algorithms

depending on the model of the asynchronous

machine, [24]. This work proposes a Luenberger

observer-type observation technique for rotor speed

and flux for DFIM without a mechanical sensor.

3 Principle of an Observer

The structure of a state observer is shown in Figure

2. It firstly involves an estimator operating in an

open loop which is characterized by the same

dynamics as that of the system. The structure

operating in a closed loop obtained by the

introduction of a gain matrix "L" makes it possible

to impose the dynamics specific to this observer,

[25].

Fig. 2: Principle of a state observer

This observer principle diagram (Figure 2) makes

it possible to use all kinds of observers, their

difference being located only in the synthesis of the

gain matrix "L".

3.1 Determination of the Gain Matrix L

The determination of the "L" matrix uses the

conventional pole placement procedure. We proceed

by imposing the poles of the observer and

consequently its dynamics. We determine the

coefficients of "L" by comparing the characteristic

equation of the observer “

0))((det  LCAI



”

with the one we wish to impose, [26], [27].

The machine model equations are expressed by:











)()(

tCxty

tButAxx



(2)

The state model of the Luenberger observer used

for the estimation of the rotor flux and the

(measured) stator currents is given by:















)(

ˆ

)(

ˆ

))(

ˆ

)(()()(

ˆ

txCty

tytyLtButxAx



(3)

Processus

Cxy

BuAx

x





L

Modèle





















xCy

yyLBuxAx

u



y



x

+

-

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Or in the form















)(

ˆ

)(

ˆ

)()()(

ˆ

)(

ˆ

txCty

tLytButxLCAx



(4)

with

 

T

rqrdsqsd iix





;

 

T

rqrdsqsd iix



ˆˆ

ˆ

;

 

T

sqsd iiy 

;

 

T

rqrdsqsd vvvvu 

The estimation error is determined by the

difference:

)().()( teLCAte 



(5)

This error will converge towards zero by a

suitable choice of the gain matrix "L" to make the

matrix

)(

0LCAA 

stable, or the eigenvalues of

this matrix are negative real parts. The method of

imposing the poles consists of choosing the poles of

the observer to accelerate its dynamics about the

system (the poles of the observer are proportional to

those of the motor),

Let's define the matrix "L" in its specific form:

T

JLIL

L















43

21

(6)

L1, L2, L3, L4 sont données par :





























































r

rr

m

r

k

L

T

k

T

L

k

L

kL

T

kL

















ˆ

)1(

1)1()1(

ˆ

).1(

1

)1(

4

2

3

2

1

(7)

Or

k

: Positive constant

The poles of the observer are chosen to accelerate

its convergence about the dynamics of the open loop

system in general, but they must remain slow about

the measurement noise, which means that we choose

the constant k usually small.

4 Application of the Luenberger

Observer to the DFIM

4.1 State Model of the DFIM in the Reference

(α,β)

The DFIM model in the reference (α,β) is defined by

the following system of equations :







































rr

r

rs

r

m

r

rrr

r

s

r

m

r

ss

s

r

rssss

ss

s

rr

r

ssss

v

T

i

T

L

v

T

i

T

L

vKv

LT

K

Kiii

vKv

L

K

T

K

iii

1

.

1

.

1

.



(8)

with: Tr =

rs

m

rs

s

r

rLL

L

K

TR

L

T

R

L

.

;

.

1

;;







4.2 State Representation of the Luenberger

Observer

As the state is generally not accessible, the objective

of an observer consists of carrying out a command

by feedback of the state and estimating this state of a

variable which we will note

X

ˆ

Such as :

T

rrss iiX ]

ˆˆ

[

ˆ

 



According to equation (2), we can represent the

observer by the following system of equations:

  

































































ssssrr

r

rs

r

m

r

ssssrrr

r

s

r

m

r

ssssrs

s

r

rssss

ssssrs

s

rr

r

ssss

iiLiiLV

T

i

T

L

iiLiiLV

T

i

T

L

iiLiiLKVV

LT

K

Kiii

iiLiiLKVV

L

K

T

K

iii

ˆ

(

ˆ

1

ˆ

.

ˆ

(

ˆ

.

ˆ

1

ˆ

(

ˆ

1

ˆˆ

.

ˆˆ

ˆ

(

ˆ

1

ˆ

.

ˆ

ˆˆ

ˆ

34

43

12

21



(9)

This leads to the equation:

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





















r

s

i

ˆ



=





















rr

m

rr

m

r

s

r

s

TT

LTT

LT

K

T

K

1

ˆ

0

ˆ

1

0

ˆ





























r

s

i

ˆ

+













1000

0100

0

1

0

00

1

K

L

K

L

s























r

s

v

+

















34

43

12

21

LL



















ss

ii

ii ˆ

(10)

This presentation then takes the following form:

)

ˆ

(

ˆ

)(

ˆssr IILBUXAX 





(11)

with:

)

ˆ

,

ˆ

()

ˆ

(



ssssss iiiiII 

=

)(



ss ee

4.3 Speed Estimation by Adaptive

Luenberger Observer

Now suppose that the speed is an unknown constant

parameter. This involves finding an adaptation law

that allows us to estimate it. The state equation of

this observer is given by (11)

with:

)(



A





















rr

m

rr

m

r

s

r

s

TT

LTT

LT

K

T

K

1

ˆ

0

ˆ

1

0

.

ˆ

.

ˆ







The speed adaptation mechanism will be deduced

from Lyapunov theory, [28], [29], by choosing an

adequate candidate function. The estimation error on

the stator current and the rotor flux, which is none

other than the difference between the observer and

the motor model, and given by (5), can be

reformulated by:

XAeLCAe ˆ

)()( 



(12)

Or





















000

)

ˆ

()(





K

AAA

(13)

with:

 

T

rrisis eeeeXXe



 )

ˆ

(

and





=



ˆ



Now consider the following Lyapunov function:





2

)(

 eeV T

(14)

Or

:



Positive constant

The derivative of this function concerning time is:

)().(

2

...

)(





































dt

d

dt

de

ee

dt

ed

dt

dV T

T

(15)

We know from (5) that,

eLCAe )( 



, replacing

this expression in (15), we obtain:





 dt

d

eLCAeeeLCA

dt

dV TTT )(

2

)(.)(

(16)

 





 





dt

d

eeeLCALCAe

dt

dV risris

TT

)(

2

)

ˆˆ

.(.2)()(

(17)

A sufficient condition to have uniform asymptotic

stability is that

0

dt

dV

, which amounts to cancel

the last two terms knowing that the first term is

negative (imposed by the gain matrix), which

implies:





 

ˆ

)(

2

)

ˆˆ

.(2dt

d

ee risris 

, and

from this equation, we obtain:

dtee risr

t

is )

ˆ

.

ˆ

.(

ˆ

0

 

 

(18)

However, this adaptation law is established for a

constant speed, and to improve the response of this

algorithm, the speed is estimated by a PI regulator,

hence the new expression for the speed:

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dteeKeeK risrisirisrispr .)

ˆˆ

()

ˆˆ

(

ˆ

 

(19)

With

p

K

, and

i

K

: positive constants.

4.4 Application of Fuzzy Logic to the

Leunberger Observer Adaptation

Mechanism

In this part, fuzzy control was proposed to improve

the performance of sensorless control of DFIM. This

method proposes to replace the classic PI speed

adaptation mechanism with a fuzzy controller, [30],

[31].

4.5 Design of a Fuzzy Adaptation Mechanism

of the Luenberger Observer

The input variables of the fuzzy controller are

subjected to a defuzzification operation and

consequently converted to fuzzy sets. The universe

of discourse of each variable of the regulator is

subdivided into five fuzzy sets, the latter are

represented by membership functions of triangular

shape, except for the ends where the trapezoidal

shape is used as shown in the Figure 3, [32], [33].

Fig. 3: Internal structure of the Luenberger

observer's fuzzy adaptation mechanism

This mechanism takes as input:

- The error E,

- The variation of this error dE,

And outputs the compensation signal u2.

After processing and integration of the fuzzy

inference system, the compensator generates the

adequate signal corresponding to its two inputs

which are determined by the relation:

)()1()( *

2

*

2

*

2kdukuku 

The block diagram of the fuzzy Luenberger type

observer for estimating the flux and speed is shown

in Figure 4.

Fig. 4: Block diagram of the fuzzy Luenberger

observer

Figure 5 illustrates the general structure of

sensorless control of DFIM connected by two 5-

level voltage NPC inverters. The rotation speed is

estimated by a high gain Luenberger observer

improved by fuzzy logic, and the desired speed is

compared with a reference, the error of this

comparison passes through a PI type regulator to

construct the torque reference.

To highlight the performance and robustness of

the fuzzy Luenberger observer several cases will be

treated, namely, the empty start followed by the

introduction of a load torque and the reversal of the

direction of rotation, the influence of parametric

variations (stator and rotor resistance) on this

control. Thus, several dynamic responses will be

presented and discussed to validate the control

algorithm used. Remember that the speed is

regulated by a classic PI and the study is devoted to

this one only. The simulation results obtained are

represented by the Figure 7 and Figure 8.

)(



ss iiY 

L

C



B

DFIM

Fuzzy Correcteur



ˆ

+

r



ˆ

s

Iˆ

-

x

ˆ

A

)(

 

risris ee 



ˆ

x

ˆ



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Fig. 5: Global scheme of sensorless direct control by

flux orientation of DFIM with intelligent adaptation

mechanism of full-order Luenberger observer based

on fuzzy logic

5 Simulation Results and

Interpretations

To evaluate the performance of the DFIM machine

control without a speed and flux sensor, we carried

out a series of simulations in a MATLAB/Simulink

environment.

The simulations presented in this section are

carried out on DFIM powered by a five-level

inverter with an NPC voltage structure controlled by

PWM, and driven by direct vector control. To carry

out this simulation, we took a sampling period of

ms

3

10

, the different parameters used are indicated

in Table 1:

Table 1. DFIM Parameters

The simulations are carried out for a simulation

time of 3 seconds, and their objectives are:

- Application of a load torque of 10 Nm at time t =

1 sec.

- Elimination of the charge at time t = 2 s.

- Reversal of direction of rotation of time t = 2.5 s.

To be able to test the robustness of the proposed

fuzzy Leunberger observer, we applied a parametric

variation of the machine up to 50% for the

resistances Rr, Rs. The tests carried out are:

- Pursuit test

- A no-load start with a reference scale of 250

rad/s at t=0s

- setpoint change to -250 rad/s at t =2.5s.

Figure 6 shows the simulation results of direct

vector control without a velocity and flux sensor

with a fuzzy Luenberger observer. We notice an

improvement in the overall performance of the

system with the insertion of the fuzzy Luenberger

observer compared to the PI Luenberger. The curves

show that during the empty start, all the quantities

stabilize after a response time that lasts 0.2 s, the

observed speed is indeed the rotation speed and the

reference speed with almost zero static error.

When starting and reversing the direction of

rotation, the speed reaches its set value with

practically no overshoot. Good rejection of the

disturbance due to the application of the load.

Decoupling is ensured by this type of observer and

regulation. The current is well maintained at its

admissible value, and the flux has a very rapid

dynamic to reach its reference value. Also, note

some additional ripples in torque and current caused

by the PWM.

Item

Data

DSIM Mechanical Power

Nominal speed

Pole pairs number

Stator resistance

Rotor resistance

Stator self-inductance

Rotor self-inductance

Mutual inductance

Moment of inertia

friction coefficient

Nominal Frequency

1.5 Kw

1450 rpm

2

1.68

1.75

295 mH

104 mH

165 mH

0.01 kg.m2

0.0027kg.m2/s

50 Hz

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Figure 7 and Figure 8 present the simulation

results of direct vector control with a fuzzy

Luenberger observer concerning the variation of the

rotor and stator resistance. We note that at each

instant of variation of rotor resistance, all the

quantities of the machine, namely the speed, the

flux, and the electromagnetic torque present a small

disturbance especially during the changes in the

instructions, and in particular during the reversal of

rotation, but this variation in resistance does not

degrade the orientation of the flow.

For a nominal value of Rr, the stator resistance Rs

is increased by +50% of its nominal value. We also

notice that this variation presents a small disturbance

during the load application, but it does not degrade

the orientation of the flux.

According to the simulation results obtained, we

can say that our control without a speed sensor

achieves good performance.

Speed (rad/s)

0 1 2 3

-200

0

200

ref

mes

obs

Time (s)

(a)

Torque (N.m)

0 1 2 3

-40

-20

0

20

40

Cem

Cr

Time (s)

(b)

Rotor flux (Web)

0 1 2 3

-0.5

0

0.5

1

1.5

rd

rq

Time (s)

(c)

Stator flux (Web)

0 1 2 3

-1

0

1

2

sd

sq

Time (s)

(d)

Rotor carrent (A)

0 1 2 3

-20

-10

0

10

20

ird

irq

Time (s)

(e)

Stator currents (Web)

0 1 2 3

-20

-10

0

10

20

isd

isq

Time (s)

(f)

Fig. 6: Simulation results of DFOC without speed

sensor based on fuzzy Luenberger observer

during an empty start followed by an introduction

of a load torque then a reversal of direction of

rotation

Speed (rad/s)

0 1 2 3

-200

0

200

ref

mes

obs

Time (s)

(a)

Torque (N.m)

0 1 2 3

-40

-20

0

20

40

Cem

Cr

Time (s)

(b)

Rotor flux (Web)

0 1 2 3

-0.5

0

0.5

1

1.5

rd

rq

Time (s)

(c)

Stator flux (Web)

0 1 2 3

-1

0

1

2

sd

sq

Time (s)

(d)

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Speed (rad/s)

0 1 2 3

-200

0

200

ref

mes

obs

Time (s)

(a)

Torque (N.m)

0 1 2 3

-60

-40

-20

0

20

40

Cem

Cr

Time (s)

(b)

Rotor flux (Web)

0 1 2 3

-0.5

0

0.5

1

1.5

rd

rq

Time (s)

(c)

Stator flux (Web)

0 1 2 3

-1

0

1

2

sd

sq

Time (s)

(d)

Fig. 7: Simulation results of DFOC without speed

sensor based on fuzzy Luenberger observer during

variation of +50% of Rr

Speed (rad/s)

0 1 2 3

-200

0

200

ref

mes

obs

Time (s)

(a)

Torque (N.m)

0 1 2 3

-50

0

50

Cem

Cr

Time (s)

(b)

Rotor flux (Web)

0 1 2 3

-0.5

0

0.5

1

1.5

rd

rq

Time (s)

(c)

Stator flux (Web)

0 1 2 3

-1

0

1

2

3

sd

sq

Time (s)

(d)

Fig. 8: Simulation results of DFOC without speed

sensor based on fuzzy Luenberger observer during

variation of +50% of Rs

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6 Conclusion

Control without mechanical speed sensors is in full

development. It aims to eliminate sensors with their

disadvantages such as fragility, cost, and noise.

In this work, we studied the fuzzy Luenberger

observer applied to DFIM. This technique is

exploited in direct vector control to improve the

performance of the sensorless control of the DFIM,

powered by two five-level inverters

According to the simulation results obtained, it

can be concluded that the proposed estimation

technique is valid for nominal conditions, even

satisfying basic speed operations, stopping, and even

when the machine is loaded. On the other hand, the

proposed observer has good robustness concerning

the variation of the load and the tracking, making it

possible to achieve good functional performances

with a low cost and reduced volume installation, this

gives a minimal structure to our order.

The work carried out in this paper directs us

towards several research perspectives which it seems

useful to cite:

1. The application of artificial intelligence

regulators instead of traditional

regulators to increase the performance of

the applied control.

2. Application of other control techniques,

such as Backstepping control.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

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