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A comparative study of three different sensorless vector control strategies for a Flywheel Energy Storage System Ghada Boukettaya, Lotﬁ Krichen*, Abderrazak Ouali University of Sfax, National School of Engineering, Department of Electrical Engineering, BP W, 3038, Sfax, Tunisia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 4 March 2009 Received in revised form 31 August 2009 Accepted 1 September 2009 Available online 23 October 2009

The aim of this paper is to ensure the sensorless control of an inertial storage system associated to an isolated Hybrid Energy Production Unit (HEPU). The Flywheel Energy Storage System (FESS) is used as energy buffers in order to store or retrieve energy into a stand-alone load. A comparative study of three different techniques based on a sensorless vector-controlled induction motor (IM) driving a ﬂywheel are presented. First, a speed estimation algorithm based on model reference adaptive system (MRAS) theory is proposed. Then, a model reference adaptive speed observers is introduced in this paper with an accurate stability study. This observer strategy is then ameliorated with a new reduced adaptive speed observer. The observer parameters are adapted during ﬂux weakening in order to obtain close tracking of the ﬂywheel speed. The accuracy of the presented models is conﬁrmed by simulation results. Ó 2009 Elsevier Ltd. All rights reserved.

Keywords: Flywheel Energy Storage System Power control MRAS Adaptive observer Full and reduced order

1. Introduction In the current context of electricity production increasing from renewable energies, accompanied by the decentralization of the production, the storage energy system has got a new ﬁeld of favourable application with their development and became necessary [1]. In fact, the renewable sources do not provide a regular and adjustable energy according to the needs. Their variations impose a storage system for which we use in general batteries. However those cannot support the number of required cycles nor to store a signiﬁcant quantity of energy in a restricted volume [2]. This is why the ﬂywheel energy storage system (FESS) offers an interesting solution to adapt the production to the consumption [3,4]. The studied system is a hybrid diesel-photovoltaic production installation as represented in Fig. 1. These electric generation hybrid systems are usually more reliable than the systems that use a single source of energy. When designing a hybrid system, both the sizing of the elements and the most adequate control strategy must be obtained [5]. This stand-alone systems must, therefore, have some means of storing energy, which can be used later to supply the load during the periods of low or no power output. Flywheels are used as

* Corresponding author. Tel.: þ216 74 274 088; fax: þ216 74 275 595. E-mail addresses: [email protected] (G. Boukettaya), lotﬁ[email protected] enis.rnu.tn (L. Krichen), [email protected] (A. Ouali). 0360-5442/$ – see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2009.09.003

energy buffers in order to store or retrieve energy into a standalone load [6]. This inertial storage system is then associated to the structure to control the power provided by the two sources. In fact, the excess energy with respect to the load requirement has been stored as ﬂywheel to generate electricity during low sunshine periods. The exceeded generated power than required by the load is diverted towards a pump. Thus, there is no more a loss or lack of energy. In our application the wheel is coupled mechanically with an induction motor controlled by a voltage inverter as represented in Fig. 1. The control of the induction motor requires precise speed information. In most cases, a shaft encoder is required in vector control to compare the speed of induction motors with the commanded speed. This has some disadvantages such as extra cost, reduced reliability, added mounting space, etc. To eliminate the speed sensor, speed estimation techniques may be used [7]. In these techniques, speed is estimated using stator voltages and currents of the induction motor and this speed is used to compare the commanded speed. The most effective sensorless control techniques mentioned in the literature are MRAS [7–9] and the Luenberger observer [8,11]. In MRAS, speed is estimated using the difference between the reference model’s output and the adjustable model’s output which includes the estimated speed as a system parameter [12,13]. The error vector is driven to zero through an adaptation law and the estimated speed then converges to its true value. In [14] Schauder

G. Boukettaya et al. / Energy 35 (2010) 132–139

DC C

DC

Chopper

Load AC

DC PV

133

Inverter

AC IG Flywheel storage

DC

Diesel engine

Rectifier/Inverter

Pump

Fig. 1. Conﬁguration of the studied hybrid generation system.

inserts a linear transfer function in the form of high pass ﬁlter in both reference and adjustable models. In [13] Tajima and Hori improved Schauder’s work by proposing a robust ﬂux observer of which the poles are designed in function of rotor speed and rotor time constant. However, in the low speed range, the observer is very sensitive to the variation of stator resistance and integrator drift [12]. Moreover, the accuracy of speed estimation is affected by the parameter variations of the induction motor (IM) [13]. Adaptive observers introduced in [11] and [15] have been a powerful prolongation of initial sensor based observers [16]. However, some operation limits of conventional observers were quickly highlighted [17]. The drive stability cannot be guaranteed when this type of observer is associated with a ﬁeld oriented control. The objective of this study is to compare three different speed observer techniques which are the speed observer based on the MRAS theory and the full and reduced order Luenberger observers. In addition, the model reference adaptive speed observer is developed with a new strategy of computation of the gain matrix observer. This speed estimation algorithm is then sophisticated, without the use of stator currents. The new design observer with new reduced technique is implemented to increase the accuracy of the sensorless strategy. 2. Control strategies of the HEPU The purpose of the proposed system is to supply an isolated load. In [5], a fuzzy logic supervisor is designed as a power management system for a PV/diesel/ﬂywheel storage hybrid power system to guarantee the required load power under irradiation level variations. In this case, the ﬂywheel inverter must control the DC bus voltage [3] i.e. maintaining constant the continuous bus voltage following any change of the transited power. If we neglect the power losses, the power assessment is deﬁned by:

PPV þ Pdiesel þ Pflywheel ¼ Pload þ Ppump

(1)

where PPV is the produced PV power, Pdiesel is the produced diesel power, Pﬂywheel is the storage or generate ﬂywheel power, Pload is the load demand power, and Ppump is the pump power. The management of the ﬂywheel power ensures the control of the DC bus voltage. The control strategies of the HEPU are represented in Fig. 2. In order to ensure the PV-diesel production control and to maintain the storage system in this correct operating zone, a supervisor should be considered [5,18]. The supervisor will determine the pump consumed power and the diesel produced power according to the available instantaneous photovoltaic energy EiPV and the ﬂywheel speedUf. This speed is considered as a reference which will be applied to the control of the ﬂywheel motor. 3. Modeling and controlling the ﬂywheel energy storage system The ﬂywheel subsystem, as shown in Fig. 3, comprises a ﬂywheel, an induction motor and an AC/DC converter (converter 1: rectiﬁer/inverter), which controls the speed of the ﬂywheel and therefore the exchanged power. This storage system has been used as an energy buffer in a PV/diesel system. The converter 2 is associated to the AC load and the converter 3 is the DC/DC converter, which controls the operating point of the PV arrays. The model of the induction motor is developed in Park model. The voltages on the AC side of the associated converter are related to the mid-point of the DC voltage Vbus, by reference waves uwd and uwq which are deduced from the three phase reference voltages by a Park transformation [19,20]:

usd usq

¼

Vbus uwd 2 uwq

Fig. 2. Block diagram of the storage system control loop.

(2)

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G. Boukettaya et al. / Energy 35 (2010) 132–139

The current modulated by the converter has the following expression:

i1f ¼

1 þ uwq isq1 u i 2 wd sd1

(3)

The DC voltage varies in function of the power exchanged between the induction motor and the load. Its evolution is obtained by:

dVbus 1 ¼ ði3 i2 i1 Þ C dt

(4)

The state equation of induction motor associated to the FESS in the stationary reference frame (a,b) can be described as:

Ec ¼

8 > <_ X ¼ A Uf X þ BU > : Y ¼ CX

(5)

where X¼[isa isb jra jrb]T is the state vector, U¼[vsa vsb]T is the controlled vector of known inputs, Y is the vector of outputs

0

1 g2 g2 pUf 0 g1 sr B g2 C B 0 C g1 g2 pUf sr B C A Uf ¼ B L C; 1 B m C U 0 p f sr @ sr A Lm 0 pUf s1r sr 1 0 1 0 sLs C B 1 C B 0 1 0 0 sLs C; C ¼ B ¼ B C B 0 1 0 @ 0 0 A 0

g1 ¼

L2m R L2r r

þ Rs

sLs

and g2 ¼

0 0

8 pﬃﬃﬃ > > < 3 Lr vsn

Lm usn Pn L r > > : pLm isqmax Uf

if Uf Un if Uf > Un

(7)

where vsn is the nominal stator voltage, usn is the nominal stator pulsation, Pn is the nominal power, isqmax is the maximum stator q current and Un is the nominal speed of the ﬂywheel IM. To calculate the wheel inertia, we consider a power required during Dt time. In fact, to store the power PN during Dt, the energy DEis then necessary such as:

J ¼

(8)

2PN Dt

DU2

¼

2PN Dt U2max U2min

(9)

Umax andUmin represent the minimal speed limit and the maximal speed limit of the ﬂywheel, respectively. Dt is then the storage period. This limit must be respected otherwise we risk to deteriorate the ﬂywheel energy storage operation [6].

In [5], the management and the supervision of the ﬂywheel power are done in the aim of ensuring the control of the DC bus voltage. Fig. 4 shows the control structure proposed in this paper, without including the DC-link voltage regulation, and without the

Conv.1 Rectifier/Inverter

i1

AC

C

DC

Conv.3 PWM generation

Ω *f

with Jf (kg m2) and Uf (rad/s) are the inertia moment and the speed of the ﬂywheel, respectively. Considering equation (6), ﬂux-weakening operation is then appropriate for this control application. Flux-weakening control of the IM is usually accomplished by regulating the rotor ﬂux according to the following control law [22,23]:

Combining equations (6) and (8), we deﬁne the necessary value of the wheel inertia as:

Lr and Ls: rotor and stator inductances Lm: mutual inductance Rr and Rs: rotor and stator resistances p: number of pole pairs J: rotor moment of inertia sr ¼ RLrr : rotor speed constant 2 s ¼ 1 LLsmLr : leakage coefﬁcient uf is the electric speed Uf ¼ puf is the mechanical speed

Flywheel storage

(6)

DE ¼ PN Dt

Lm Lr Ls s

i1 f i2 f

1 2 JU 2 f f

jr ref ¼

0

IM

observer techniques. The control system is based on a direct rotorﬂux-oriented (DRFO) vector control of the IM driving the ﬂywheel. The current and voltage values are referred to the reference frame aligned to the rotor ﬂux and take DC values in steady state. The vector control principle provides great ﬂexibility for IM drives. However, it is costly to implement because of the need for a shaft speed or shaft position encoder. The use of a position encoder has several drawbacks in terms of robustness, cost, cabling, and maintenance. In the small power range, the cost of a position encoder is almost the same than that of an IM [7]. The energy stored in the ﬂywheel is dependent on the square of the rotational speed [21], for a given inertia, as represented as follow:

Control strategy

Fig. 3. Conﬁguration of the FESS.

i3

i2

U Conv.2

4. Model reference adaptive system (MRAS) Many methods have already been proposed for speed estimation, the MRAS approach is the most attractive method because in this method, the models are simple and very easy to implement [24,25]. Fig. 5 illustrates an alternative way of calculating the motor speed by means of MRAS techniques. Two independent observers are constructed to estimate the components of the rotor ﬂux vector: one based on (1) and the other based on (2). The reference model, which is independent of the rotor speed, calculates the state variable,x, from the terminal voltages and currents, and it is deﬁned by:

8 > > > djra ref ¼ Lr v R i sL disa > sa s sa s > < dt Lm dt > djrb ref disb > Lr > > s v R i L ¼ > s sb s : dt Lm sb dt

(10)

G. Boukettaya et al. / Energy 35 (2010) 132–139

135

Fig. 4. DRFO vector control of the IM driving the ﬂywheel.

Then the adjustable model, which is dependent on the rotor speed, estimates the state variableb x:

8 > > djra adp 1 Lm > b > ¼ jra adp þ isa p Uj > rb adp < sr sr dt > > djrb adp 1 Lm > b > ¼ jrb adp þ isb þ p Uj > ra adp : sr sr dt

þKi

ref

jr

!

(12)

¼

!

ea eb

!

ð14Þ

5. Full order Luenberger observer

adp

s1r u u s1r

jrb ref jra adp jra ref jrb adp ds

(11)

In designing the adaptation mechanism for a MRAS, it is important to take account of the overall stability of the system and to ensure that the estimated quantity will converge to the desired value with suitable dynamic characteristics. In [26] Landau has described practical synthesis techniques for MRAS structures based on the concept of hyperstability. When designed according to these rules, the state error equations of the MRAS represented in equation (13) are guaranteed to be globally asymptotically stable.

e_ a e_ b

Zt

0

The error between the states of the two models is then used to drive a suitable adaptation mechanism that generates the estimate, for the adjustable model, as deﬁned by:

e ¼ jr

b ¼ Kp jrb ref jra adp jra ref jrb adp u

jra adp 0 1 b þ u u jrb adp 1 0

The Luenberger observer (full order adaptive state observer) is constructed using the equations of the induction motor in stationary reference frame [7] by adding an error compensator. Using the motor model deﬁne in (5), the motor observer model is represented by:

8 > >

b is the estimated state vector, Y b the estimated outputs where X vector and G is the matrix gain observer where:

0

g11 B g21 G ¼ B @ g31 g41

1 g12 g22 C C g32 A g42

The estimation error vector is deﬁned by :

(13)

iT h b X ¼ e e e e e ¼ X isa isb jra jrb

In this method, the rotor speed is estimated as:

(15)

(16)

The differential equation representing the state estimation error is then obtained from the two equations (5) and (15):

vsaβ

isaβ Induction machine

Reference model (1)

x + -

Adjustable model (2) ∧

ωf

Adaptative mechanism Fig. 5. Rotor speed estimation structure using MRAS.

e ∧

( e_ ¼

b A U b þ GC e þ A U b 3 ¼ Ce X A U f f f

(17)

x The error dynamics are described by the eigenvalues of the state matrix and these are usually used to design a stable observer. In this paper the strategy of computation of the matrix observer is based on the search of a Lyapunov particular function [10,11,13,27]. To determine the stability of the error dynamics of the observer, we

136

G. Boukettaya et al. / Energy 35 (2010) 132–139

b ÞþGCÞ is stable, then there are two matrixes P ¼ PT>0 If ðAð U f and Q ¼ QT>0such as:

Table 1 Parameters of IM used in simulation. Pn ¼ 3 KW

T b Þ þ GC P þ P A U b þ GC ¼ Q < 0 Að U f f

Vn ¼ 220 V Ls ¼ 0.1164H Lr ¼ 0.1164H Lm ¼ 0.1113H fn ¼ 50 Hz Jv ¼ 0.34 kg.m2

Uvn ¼ 1460 tr/min Rr ¼ 1.045U Rs ¼ 1.411U p¼2 JMAS ¼ 0.011 kg.m2 fv ¼ 6.46 103 Nm/rad/s

The condition (21) is then also veriﬁed. The state matrix developed observer AO is:

b þ GC ¼ AO1 AO ¼ A U f AO3 AO1 ¼ "

(18)

AO2 ¼

where l is a constant andP is a symmetrical positive deﬁnite matrix. This function must be continuous, differentiable, and positive deﬁnite. The adaptive observer construction objective is in the determination of the matrix G coefﬁcients and in the determination of b to have a system with asymptotic stability. the law deﬁning U f The stability condition of system (17) is veriﬁed if the time derivative Lyapunov function is negative deﬁnite. Then, we have:

b <0 b U dU 2eT P e_ þ 2l U f f dt f

AO2 AO4

(24)

where

can use Lyapunov’s stability theorem, which gives a sufﬁcient condition for the form asymptotic stability of a non-linear system by using a Lyapunov function V [5,27]:

2 b U V ¼ eT Pe þ l U f f

(23)

AO3 ¼

g1 þ g11 g21 g2 sr

g12 g1 þ g22 # g2 pUf g2 sr

g2 pUf "L m sr þ g31 "

AO4 ¼

#

g32 Lm

g41

sr þ g42

s1r

pUf

pUf

s1r

and

#

As AO4 is stable, then to guaranty the stability of AO, two necessary and sufﬁcient conditions are established: ﬁrst to guarantee that AO1 is stable and second to cancel AO3. A judicious choice of the parameters which guarantee the stability of AO1 is such as:

(19)

The development of the last equation (19) gives:

g11 ¼ g22 < 0 and g12 ¼ g21 ¼ 0

T b ÞþGC P þP A U b þGC e 2 p U b U eT Að U 1 f f f f b b ra e j b b ra e j b ejrb j eisb j jra rb isa rb þ 22 p U f Uf b <0 b U d U þ2l U f f f dt

The cancellation of AO3 gives:

(20)

g31 ¼ g42 ¼ Lsmr g32 ¼ g41 ¼ 0

db br U f ¼ k1 f1 X; Xb þk2 f2 jr ; j dt 1 1 b ra e j b b ra e j b p ejrb j ¼ 21 p eisb j jra rb isa rb 22 l l 2 2

(21)

b Þ must For experimental validation, the system state matrix Að U f be recomputed at every sampling moment. In order to ensure an implementation in real time and functioning at sufﬁcient frequency, we must reduce the complexity of the observer. Thus we

db b ra e j b b b 22 p ejrb j jra rb 21 p eisb j ra eisa j rb þ2l U f ¼ 0 dt (22)

300

1.1

Reference Sensor

Reference Sensor

Rotor Flux (Wb)

1

250

200

0.9 0.8 0.7 0.6

150

0

10

(27)

6. Reduced order Luenberger observer

The second Lyapunov condition is:

Speed (rd/s)

(26)

The resolution of the Lyapunov second condition (22), gives the speed adapted law:

where21 and22 are positive constants. To ensure stability, two conditions are then selected. The ﬁrst Lyapunov condition is:

T b þ GC e < 0 b Þ þ GC P þ P A U eT Að U f f

(25)

20

30

40

0.5

0

Time (s)

10

20

Time (s) Fig. 6. Response of sensor vector control.

30

40

G. Boukettaya et al. / Energy 35 (2010) 132–139

1.1

30 0 Ref erence MRAS

Re fe rence Se ns MR AS or

1

Rotor Flux (Wb)

Speed (rd/s)

137

25 0

20 0

0.9 0.8 0.7 0.6

15 0

0

10

20

30

0.5

40

0

10

Time (s)

20

30

40

Time (s)

Fig. 7. Response of sensorless vector control with MRAS.

proceed to the observer reduction by regarding the stator currents as inputs. The observer state vector is then completely in rotor ﬂux. We consider the following distribution of the IM state model:

Using equation (28), the development of the reduced observer model (29) is:

8 > > > i_s < ¼ j_r > > > : Y ¼i

c _

A11 A21

! A12 Uf BO is þ ðUÞ O4 A22 Uf jr

(28)

c _ b c jr ¼A22 U f jr þA21 is þG A11 is þA12 Uf jr þBO Uis

(30)

With the variable changing according to:

s

c b z ¼ jr þ Gis

where

A11 ¼ A21 ¼

g1 0 Lm

0

0

Lm

sLs

1

0

0

sLs

sr

0 g1 !

A12 Uf

A22 Uf

sr

¼

g2 sr

¼

g2 pUf

g2 pUf

1

pUf

pUf

s1r

sr

!

(31)

the new reduced observer model is:

!

g2 sr

b_ z ¼ Mb z MGis þ ðA21 þ GA11 Þis þ GBO U

(32)

b Þ þ GA ð U b Þ is the state matrix of the modiﬁed where M ¼ A22 ð U 12 f f reduced observer. The estimation error vector is deﬁned by:

and

BO ¼

!

iT h c e ¼ jr jr ¼ ejra ejrb

1

The differential equation representing the state estimation error is:

The reduced ﬂux observer is deﬁned by:

b _ _ b c jr ¼ A22 U f jr þ A21 is þ Gr is is

c _

(29)

The new reduced matrix gain observer Gr is:

g12 g22

1.1

300 Ref erence Full obser ver

Re fe rence Full obser ver

1

Rotor Flux (Wb)

g11 g21

b A br j e_ ¼ A22 Uf þGA12 Uf eþ A22 U 22 Uf f

250

200

0.9 0.8 0.7 0.6

150

0

10

(34)

To ensure the reduced observer convergence, we must ensure the asymptotic stability of system (34) by using the same function of Lyapunov deﬁned by equation (18). The stability condition is then deﬁned by the equation:

Speed (rd/s)

Gr ¼

(33)

20

Time (s)

30

40

0.5

0

10

20

Time (s)

Fig. 8. Response of sensorless vector control with full order Luenberger observer.

30

40

138

G. Boukettaya et al. / Energy 35 (2010) 132–139

1.1 Reference Reduced observer

300

Rotor Flux (Wb)

Speed (rd/s)

1

250

200

0.9 0.8 0.7 0.6

Reference Reduced observer 150 0

10

20

30

0.5

40

0

10

Time (s)

20

30

40

Time (s)

Fig. 9. Response of sensorless vector control with reduced order Luenberger observer.

eT

b ÞþGA ð U b Þ A22 ð U 12 f f

T

b þGA b e PþP A22 U 12 U f f

d b <0 b b U b ra e j b U þ22p U ejrb j jra rb þ2l U f Uf f f dt f

T b b b Þ þ GA ð U b Þ PþP A A22 ð U <0 12 22 U f þ GA12 U f f f

(37)

The second Lyapunov condition is:

ð35Þ

db b ra e j b 22p ejrb j jra rb þ 2l U f ¼ 0 dt

where2 is a positive constant. The ﬁrst Lyapunov condition is:

g12 ¼ g21 ¼ 0 g22 ¼ g11 < 0

(36)

The veriﬁcation of Lyapunov ﬁrst condition will ensure the determination of the observability matrix coefﬁcients such as:

(38)

The resolution of the Lyapunov second condition (38) gives the adaptation law associated to the speed:

db p b ra e j b U ¼ 2 ejrb j jra rb l dt f

(39)

20

0.5 Sensor vector control system

Model reference adaptive system

0.4

15

Speed error (rd/s)

Speed error (rd/s)

0.3 0.2 0.1 0 -0.1 -0.2

10 5 0 -5 -10

-0.3

-15

-0.4 -0.5

0

5

10

15

20

25

30

35

-20

40

0

10

Time (s)

20

30

40

Time (s)

0.5 Reduced order Luenberger observer

0.6

Speed erreur (rd/s)

Speed error (rd/s)

Full order Luenberger observer

0

0.4 0.2 0 -0.2 -0.4 -0.6

-0.5

0

10

20

Time (s)

30

40

0

10

20

30

40

Time (s)

Fig. 10. Speed estimation error associated to the sensor vector control system, the MRAS, the full and reduced order Luenberger observers.

G. Boukettaya et al. / Energy 35 (2010) 132–139

7. Simulation results The studied methods were veriﬁed by MATLAB SIMULINK. Table 1 shows the parameters of the IM associated to the Flywheel Energy Storage System (FESS), used in the simulation. Operating conditions and sensing variables are considered to be ideal for all three systems and the same tests were simulated. The objective is to compare the three sensorless control schemes for the wheel IM, drives with vector control. Indices used for the comparison are the steady state error, the dynamic response to the speed input and the dynamic response to a torque input, delivered both from the supervisor control. Also speed sensor and sensorless vector control schemes are studied and compared with a view to examining the inﬂuence of control strategies on the drive’s performance. Initially, the speed of the ﬂywheel is 260 rd/s, his evolution during 40 s is represented in Figs. 6–9. From the results of Fig. 6 associated to the DRFO vector control of the IM, we note the good performance of this technique. In fact, speed and ﬂux follow perfectly their references imposed by the supervisor. Fig. 7 shows the real and estimated speed and ﬂux, when sensorless control was performed using the MRAS control. We note that the rotor speed was correctly estimated during the time, except in the starting of the motor. This problem is caused by the right choose of the adaptation mechanism coefﬁcients. Fig. 8 shows the response of sensorless vector control with full order Luenberger observer. We noticed that the performances of this observer in transitory and established mode are very acceptable. Fig. 9 shows the response of sensorless vector control with reduced order Luenberger observer. The performances of this observer are similar to the full order Luenberger observer. 8. Error analysis As validation with respect to reality and overall calculation of the three proposed observers in this paper, an error analysis is included. In fact, waveforms in Fig. 10 represent the speed estimation error associated to each sensorless control strategy. These results translate the performance of the three different sensorless vector control developed strategies. 9. Conclusions The objective is to compare the three sensorless control schemes for the IM drives with vector control. Indices used for the comparison are the steady state error, the dynamic response to the speed input and the dynamic response to a torque input, delivered both from the supervisor control. Speed sensorless control and vector control techniques for IM were simulated and the ﬂux and speed characteristics obtained from these methods were compared. In the tests simulated, it is evident that the sensorless control methods work just as well as the vector control method. Particularly, the full order and the reduced Luenberger observers present a very good speed and ﬂux estimation technique. In fact, the proposed adaptive speed observers are simple and according to the simulation results are effective. The new proposed design of the gain matrix guaranties the stability of the adaptive observer, without complex computation us in linearization or in eigenvalues

139

determination. Simulation results show the good performances of the controller associated to the studied observers.

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