Sensorless Control of Induction Motor Drives Using

J Electr Eng Technol.2017; 12(4): 1456-1470 http://doi.org/10.5370/JEET.2017.12.4.1456

ISSN(Print) 1975-0102 ISSN(Online) 2093-7423

Sensorless Control of Induction Motor Drives Using an Improved MRAS Observer Zineb Kandoussi†, Zakaria Boulghasoul*, Abdelhadi Elbacha* and Abdelouahed Tajer* Abstract – This paper presents sensorless vector control of induction motor drives with an improved model reference adaptive system observer for rotor speed estimation and parameters identification from measured stator currents, stator voltages and estimated rotor fluxes. The aim of the proposed sensorless control method is to compensate simultaneously stator resistance and rotor time constant variations which are subject of large changes during operation. PI controllers have been used in the model reference adaptive system adaptation mechanism and in the closed loops of speed and currents regulation. The stability of the proposed observer is proved by the Lyapunov’s theorem and its feasibility is verified by experimentation. The experimental results are obtained with an 1 kW induction motor using Matlab/Simulink and a dSPACE system with DS1104 controller board showing the effectiveness of the proposed approach in terms of dynamic performance. Keywords: Induction motor, Vector control, Improved ‘model reference adaptive system’ observer, Lyapunov criterion, Parameters identification, dSPACE 1104

1. Introduction This article gives you guidelines for preparing papers that, after thoroughly reviewed by the referees, have been decided to be published for KIEE transactions on systems and control (SC). If you are using Microsoft Word 6.0 or later and reading a paper version of this document, please download the electronic file, J_KIEE.doc from the KIEE homepage so you can use this document as a template. Otherwise, you may use this as just an instruction set. It is remarked that you do not have to follow this style file when your works is submitted for the initial review stage. Asynchronous or Induction Motor (IM) is nowadays the most used motor in all industrial applications due to its low cost maintenance, its good performances and excellent reliability. Its main advantage lies on the absence of sliding electrical contacts, which leads to a simple and robust structure. However, induction motors are considered as nonlinear, multivariable and highly coupled systems [1, 2]. Therefore, efficient speed and torque control requires a simultaneous control of several variables. The reason why, it is necessary to artificially achieve a decoupling between the flux and the torque using Field Oriented Control (FOC) for better performance [3, 4]. The installation of an incremental encoder to measure speed and / or the rotor position needs an additional place, involves an extra cost that can be more important than the †

Corresponding Author: Laboratory of Electrical Engineering and Control Systems, Cadi Ayyad University, Morocco. ([email protected]) * Laboratory of Electrical Engineering and COntrol Systems, Cadi Ayyad University, Morocco. ([email protected], [email protected], [email protected]) Received: February 25, 2016; Accepted: January 28, 2017

1456

motor itself for low power and often leads to increase sensitivity to electromagnetic noise and system complexity. Thing that is not always desirable. That’s why, the idea of eliminating the physical speed sensor was born and researches about sensorless control of the IMs were begun. Several strategies have been proposed in literature to achieve this goal. Many of the proposed methods are based on observers that depend on the induction motor model. However, these structures require an accurate knowledge of the motor parameters especially stator and rotor resistances. Therefore, performance studies of sensorless IM drives concerning the effect of these parameters and their compensation were subject of several previous works such as; the Sliding Mode (SM) observer [5] which has the advantage of being insensitive to rotor time constant variations, this observer unfortunately requires an infinite switching rate and suffers from chattering phenomenon. Later, a fuzzy SM observer has been developed in [6] to overcome such limitations by introducing fuzzy logic techniques. However, the use of intelligent methods usually increases the complexity of the algorithm and its execution time; Luenberger observer [7, 8] that is known for its relative simplicity, is one of the common used observers, but despite its numerous advantages, its main weak point lies in the use of a constant gain contrary to the Extended Kalman Filter [9] that uses a non-constant gain updated over time for high system dynamic. However, this update requires high sampling frequency and therefore high computational capacities with very fast microprocessors. The high gain observer [10-12] that is also employed in IM variables estimation provides the advantage of using one parameter for the adjustment of the system dynamic. Yet,

Copyright ⓒ The Korean Institute of Electrical Engineers This is an Open-Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/ licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Zineb Kandoussi, Zakaria Boulghasoul, Abdelhadi Elbacha and Abdelouahed Tajer

no clear analytical tuning that has been reported concerning the choice of the right value of this parameter, only trial and error method that can be used for this issue. To overcome problems associated with schemes based on the induction motor model, estimators using motor saliency and high frequency signal injection have been proposed [13]. There major drawback is that they require a high precision measurement and need an external hardware for signal injection. Out of various approaches, Model Reference Adaptive System (MRAS) which was originally published by Schauder [14] and has received a lot of attention from other researchers is one of the most popular adaptive observers used in sensorless induction motor control applications due to its good performance, ease of implementation, good stability and low computational effort [15-20]. The MRAS structures that have been recently proposed use estimated rotor fluxes [14], estimated stator flux [21], estimated back-EMF [22, 23], estimated stator currents [24], estimated reactive power [25, 26] and estimated active power [27, 28] in the error expression for speed and motor parameters estimation. Another scheme of the MRAS observer (termed Closed Loop Flux Observer MRAS) has been developed in [29] to improve the estimated speed at low speeds; however this kind of structures includes the mechanical model of the IM that it is not easy to identify its parameters. The stability of the MRAS observer is guaranteed using Popov [30] or Lyapunov [31] criterion or Recursive Least-Square (RLS) algorithm [32]. Generally, the choice of the appropriate scheme depends on the application and the quantity to be estimated. Regarding the adaptation mechanism which is one of the main component of the adaptive MRAS observer, PI controllers will be used for all quantities estimation since they have a simple structure and can offer satisfactory performances over a wide range of operation [14, 15]. Fuzzy Logic Controller and Artificial Neural Networks Controller have been used as adaptation mechanism in [1619] to improve the observer performance but they require heavy computational burden especially when we increase the number of fuzzy membership functions or the number of neurons in hidden layer. In practice, considerable variations of the stator resistance and rotor resistance / rotor time constant (RTC) take place when the motor temperature changes at varying load or speed or air temperature surrounding the motor, thing that may lead to improper estimation of the rotor speed over the whole operating range of sensorless drives. It should not be forgotten that the slip speed is also function on rotor resistance / RTC, which means that any variation of this latter will affect the Indirect Field Oriented Control performance. Hence, continuous adaptation of these quantities is required to maintain stable operation and to improve performance of the MRAS sensorless drives especially at very low speeds. Numerous methods based on the MRAS observer for

speed estimation taking into account the effect of parameter variations have been developed. However, these methods estimate either speed & stator resistance [30, 33] or speed & rotor resistance/RTC [21, 25, 31, 34] unlike this paper that aims at providing a simultaneous estimation of rotor speed, stator resistance and RTC inverse using the MRAS scheme based on the Lyapunov stability theorem. The structure of the proposed MRAS observer uses the rotor fluxes which can provide the advantage of producing rotor flux angle for field orientation scheme but the weak point of this kind of structure is that it needs an open-loop integration. However, thanks to a LPF with a suitable cutoff frequency [35], this pure integral is no more a problem. This paper is organized as follows. Section 2 shows the induction motor dynamic model in the coordinate d-q and a theoretical background of field oriented motor control principle is explained. In sections 3, structure and stability of the proposed MRAS observer is implemented for motor speed estimation and simultaneous stator resistance and rotor time constant adaptation. The performances of this proposed sensorless induction motor control technique are illustrated by experimental results in section 4. Finally section 5 draws the final conclusion.

2. Indirect Field Oriented Control 2.1 Induction motor dynamic model in the d-q coordinate The electrical equations of the induction motor are presented and described using the space vector notation written in the d-q reference frame rotating with the synchronous speed ωs as follows [36]: ks 1 ⎧ dI sd ⎪ dt = −λ I sd + ωs I sq + T ϕrd + ω ksϕrq + σ L Vsd ⎪ s s ⎨ dI k 1 sq s ⎪ V = −ωs I sd − λ I sq − ω k sϕrd + ϕrq + ⎪⎩ dt σ Ls sq Ts 1 ⎧ dϕrd M ⎪ dt = T I sd − T ϕrd + (ωs − ω ) ϕrq ⎪ r r ⎨ dϕ 1 M rq ⎪ = I sq − (ωs − ω ) ϕrd − ϕrq ⎪⎩ dt Tr Tr

(1)

(2)

where, (Vsd, Vsq) and (Isd, Isq) denote the d-q stator voltages and currents, respectively. (φrd, φrq) denote the d-q rotor fluxes. ωs and ω = pΩ are the synchronous speed and rotor speed, respectively. Ts = Ls/Rs and Tr = Lr/Rr denote the stator and the rotor time constants, respectively. Ls and Lr are the stator and the rotor inductances, respectively. Rs and Rr are the stator and the rotor http://www.jeet.or.kr │ 1457

Sensorless Control of Induction Motor Drives Using an Improved MRAS Observer

resistances, respectively. M is the mutual inductance and p is the number of pole pairs. M2 M 1−σ 1 1−σ ; ks = = ;λ = + And: σ = 1 − Ls Lr σ Ls Lr M σ Tsσ Tr σ The electromagnetic torque and the mechanical equations in the d-q reference frame are given by:

Ce = p

M (ϕrd I sq − ϕrq I sd ) Lr

dΩ = (Ce − Cr − f r Ω) / J dt

(4)

θs = ∫

The Field Oriented Control is mainly based on the orientation of the rotating frame as the d-axis coincides with the direction of φr to have a decoupling between the flux and the torque of the induction motor which is one of the intrinsic characteristic of the DC machine [37]. Under this condition we get: (5)

(6)

We then obtain a new system of equations completely decoupled: dI sd + λσ Ls I sd dt dI sq + λσ Ls I sq dt

M ϕr I sq Lr

And (2) becomes:

1458 │ J Electr Eng Technol.2017; 12(4): 1456 -1470

+θ

(10)

3.1 MRAS structure based on rotor fluxes estimation The MRAS observer analyzed in this paper employs two independent expressions for the time derivative of rotor fluxes, obtained from equations of the IM model in the stationary reference frame α-β. They are usually referred to as the “Voltage model” for the “Reference model” and “Current model” for the “adaptive model”. They are given, respectively, by: ⎧ d (ϕrα )ref L ⎛ ⎪ = r ⎜ Vsα − Rs I sα − σ Ls dt M ⎝ ⎪⎪ ⎨ ⎪ d ϕr β ref L ⎛ = r ⎜ Vsβ − Rs I sβ − σ Ls ⎪ dt M⎝ ⎪⎩

(

(7)

(8)

)

dI sα ⎞ dt ⎟⎠ dI sβ ⎞ ⎟ dt ⎠

(11)

and ⎧ d (ϕrα )ad = − β (ϕrα )ad − ω ϕr β + M β I sα ⎪ ad dt ⎪ ⎨ ⎪ d ϕr β ad = − β ϕr β + ω (ϕrα )ad + M β I s β ⎪ ad dt ⎩

(

(

where Ed and Eq represent the compensation terms. Furthermore, from (3) and (5), the torque expression becomes: Ce = p

Tr I sd

where θs and θ denote the position of the d-q reference frame and the rotor, respectively. Later in this article, the superscript ‘^’ will represent estimated quantities.

From (1), we can notice that the voltages (Vsd, Vsq) act both on the currents (Isd, Isq) and consequently on both the flux and the torque. In this case, a decoupling by compensation will be used. Equation (1) can be arranged as follows:

⎧ ' ⎪⎪Vsd = σ Ls ⎨ ⎪V ' = σ L s ⎪⎩ sq

I sq

3. Sensorless IFOC Using the MRAS Observer

2.2 Principle of the indirect field oriented control

' ⎧⎪Vsd = Vsd − Ed ⎨ ' ⎪⎩Vsq = Vsq − Eq

(9)

Finally, we have:

(3)

where: Cr is the load torque, J is the total inertia and fr is the friction coefficient.

ϕrd = ϕr , ϕrq = 0

⎧ϕr = MI sd ⎪ MI sq ⎨ ⎪ (ωs − ω ) = T ϕ ⎩ r r

)

(

)

(12)

)

where, (Vsα, Vsβ) and (Isα, Isβ) denote the α-β stator voltages and currents, respectively. ((φrα)ref, (φrβ)ref) and ((φrα)ad, (φrβ)ad) denote the α-β rotor fluxes of the reference and the adaptive models, respectively. β is the inverse of the RTC. As shown in Fig. 1, the MRAS observer makes use of two redundant motor models of different structures that estimate the same state variables on the basis of different sets of input variables and an adaptation mechanism. The output of the adaptation mechanism is the estimated quantity, which is used for the tuning in adjustable model and/or for feedback in the case of the estimated speed. Its


Variable reference and measurement

Reference model

⎡ Δβ Δω ⎢ ⎢ −Δω Δβ ⎢ W =⎢ 0 0 ⎢ ⎢ 0 ⎢⎣ 0

Error

Adaptive model Adaptation mechanism

Fig. 1. Principle of the MRAS observer input is the error between the adaptive model and the reference model.

3.2 Stability analyses using Lyapunov criterion It should be noted that the proposed parallel rotor speed and stator resistance/rotor time constant estimation scheme is designed based on the concept of hyperstability [14] in order to make the system asymptotically stable. We consider the following estimation error vector: ⎡ (ϕrα )ad ⎢ ⎢ ϕr β ad eαβ = ⎢ ⎢ (ϕˆrα ) ad ⎢ ⎢ ϕˆr β ad ⎣

( (

) )

− (ϕˆrα )ad ⎤ ⎥ ⎥ − ϕˆr β ad ⎥ , − (ϕrα )ad ⎥ ⎥ ⎥ − ϕr β ad ⎦

( (

dI sα ⎞ dt ⎟⎠ dI sβ ⎞ ⎟ dt ⎠

⎧ d (ϕˆrα )ad = − βˆ (ϕˆrα )ad − ωˆ ϕˆr β + M βˆ I sα ⎪ ad dt ⎪ ⎨ ⎪ d ϕˆr β ad = − βˆ ϕˆr β + ωˆ (ϕˆrα )ad + M βˆ I s β ⎪ ad dt ⎩

(

(14)

( Rs − Rˆ s )2

+

+

δ Rs

( β − βˆ )2

δβ

(17)

deαβ T dt

eαβ + eαβ T

deαβ dt

−2

1

δω

Δω

( )

d (ωˆ ) dt

d βˆ d ( Rˆ s ) 1 −2 −2 ΔRs Δβ δ Rs δβ dt dt 1

(18)

)

−ω 0 0 ⎤ − β 0 0 ⎥⎥ , 0 0 0⎥ ⎥ 0 0 0⎦

)

where:

)ad ( (ϕrα )ad − (ϕˆrα )ad ) , + 2Δω (ϕˆrα )ad ( (ϕr β ) − (ϕˆr β ) ) ad ad (

P1 = −2Δω ϕˆr β

Lr 2Δ Rs I sα (ϕˆrα )ad − (ϕrα )ad M , L − r 2Δ Rs I s β ϕˆr β − ϕr β ad ad M

( (( )

P2 = −

(15)

)

eαβ = AMRAS eαβ − W

Where: AMRAS

δω

(

The expression of the error derivative is as follow:

⎡−β ⎢ω =⎢ ⎢0 ⎢ ⎣0

(ω − ωˆ ) 2

T V = eαβ T AMRAS + AMRAS eαβ + P1 + P 2 + P3 + Q (19)

and

(

0

)

By developing the expression of V , we obtain:

)

)

(

where, δω, δRs and δβ are positive constants. The Lyapunov function is positive definite, thus, the sufficient condition for the uniform asymptotic stability is that the time derivative of the Lyapunov function should be negative definite. The time derivative of V is defined as:

)

⎧ d (ϕˆrα )ref L ⎛ ⎪ = r ⎜ Vsα − Rˆ s I sα − σ Ls dt M ⎝ ⎪⎪ ⎨ ⎪ d ϕˆr β ref L ⎛ = r ⎜ Vsβ − Rˆ s I sβ − σ Ls ⎪ dt M⎝ ⎪⎩

(

V = eαβ T eαβ +

(13)

From (11) and (12), we obtain:

(

0 ⎤ ⎡(ϕˆ ) ⎤ − M Δβ ⎥⎥ ⎢ rα ad ⎥ ⎥ ⎢ ϕˆr β ad ⎥ ⎥ 0 ⎥⎢ ⎢ ⎥ I ⎥ sα ⎥ Lr ⎥⎢ I ΔRs ⎥ ⎢⎣ sβ ⎥⎦ M ⎦

and Δω = ω − ωˆ , ΔRs = Rs − Rˆ s , Δβ = β − βˆ We consider the following Lyapunov function candidate:

V =

)

− M Δβ 0 Lr ΔRs M

(16)

)

(

)

)

( ) ( (ϕrα )ad − (ϕˆrα )ad ) and −2Δβ ( (ϕˆr β ) − MI sβ ) ( (ϕr β ) − (ϕˆr β ) ) ad ad ad d ( βˆ ) d ( Rˆ s ) 1 1 1 d (ωˆ ) Δω ΔR Δβ Q = −2 −2 −2 P3 = −2Δβ (ϕˆrα )ad − MI sα

δω

δ Rs

dt

s

δβ

dt

dt

The derivative V must be negative in order to satisfy the stability criteria of Lyapunov. T T Since the term eαβ AMRAS + AMRAS eαβ is negative as proved in the appendix, then the other terms can be set to zero or to values less than zero.

(

)

http://www.jeet.or.kr │ 1459


3.3 Rotor speed estimation In order to obtain the expression of the error between the reference and the adjustable model for rotor speed estimation from Eq. (19), the term below should be set to zero:

(

)

(

−2Δω ⎡ ϕˆr β (ϕrα )ad − (ϕˆrα )ad ϕr β ad ⎣ d (ωˆ ) 1 Δω −2 =0 dt δω

⎣

ref

(

− ϕˆr β

(

)ref − (ϕˆr β )ad (ϕˆrα )ref

(22)

The adaptation law has a pure integration in open loop. To overcome this problem and improve the estimation accuracy, many researchers have proposed using a PI adaptation mechanism. The final expression of the estimated rotor speed is given by:

⎛

ωˆ = ⎜ K pω + ⎝

Kiω s

(ϕˆr )ad

⎞ ⎟ eω ⎠

(23)

where, Kpω and Kiω, are the adaptation gains that can be adjusted and s is the Laplace operator. Fig. 2 illustrates the structure of the speed MRAS observer with online parameters tuning.

)

)

((ϕˆrα )ad ) + ((ϕˆrβ )ad ) 2

=

(

G(s)

)

The classical MRAS observer is designed to estimate the rotor speed only. As mentioned above, the main idea behind the MRAS technique is that there is a reference model and an adjustable model that both estimate the rotor fluxes. The output of the reference model is compared to the adjustable model. The error obtained between the reference

K pω +

K iω s

ωˆ

Imaginary Axis (seconds-1)

eω

(25)

3.4 Parameters sensitivity analysis of the MRAS speed observer without online parameters identification

0.76

0.86

Δω

2

Fig. 4 shows the root locus of the forward transfer function for Kiω K pω 〉 1 Tr From the root locus of the forward transfer function (as shown in Fig. 4), we can notice that the closed loop first pole (equal to zero) moves to the first zero (equal to - βˆ ) and the two other conjugated poles (one is equal to - βˆ + iω and the other is equal to - βˆ - iω) move to the second zero (-Kiω/ Kpω) and infinity respectively. Therefore, good selection of (-Kiω/ Kpω) and Kpω will help the estimated speed to correctly track the actual speed and essentially to maintain the stability of the whole system.

0.64

0.5

0.34

0.16

ω

150

ω

(24)

K ⎞ ωˆ ⎛ = G ( s ) ⎜ K pω + iω ⎟ Δω s ⎠ ⎝

200

Fig. 2. Block diagram of the MRAS observer for rotor speed estimation

2

Using the error transfer function (24), the forward transfer function of the speed MRAS observer is given by:

(21)

In the case of rotor speed estimation, the expression of the error between the two models is: eω = (ϕˆrα )ad ϕˆr β

(

where:

)ad (ϕˆrα )ref ⎤⎦⎥

(

s + βˆ e G (s) = ω = (ϕˆr )ad 2 Δω s + βˆ + ω 2

(20)

(ϕr β )ad = (ϕˆrβ )ref [30],

Since: (ϕrα )ad = (ϕˆrα )ref and Finally, we get:

ωˆ = ∫δω ⎡⎢(ϕˆrα )ad (ϕˆr β )

)ad ⎤⎦

In the steady state, we have: Rs = Rˆ s and β = βˆ The closed loop block diagram of the dynamic response of the speed MRAS observer is shown in Fig. 3. The error transfer function is given by:

100 0.94

-β

-β ω

50 0.985 0

250

200

150

- K

pω

-50 0.985

100

50

K

-β

iω

-100 0.94 -150 0.86 -300 -250

0.76 -200

0.64 -150

0.5 -100

0.34 -50

0.16 0

50

Real Axis (seconds -1)

Fig. 3. Synthesis of the PI controller 1460 │ J Electr Eng Technol.2017; 12(4): 1456 -1470

Fig. 4. Root locus of the forward transfer function


and adjustable model is given to an adaptation mechanism, which adjusts the adaptive model by generating the adequate estimated value. However, if parameters of the reference model are not well calculated (are not online identified), this will distort the estimated quantity. In the case of the stator resistance variations, when this parameter varies inside the IM, this will not affect the stability of the observer. However, according to the Eq. (11), when the stator resistance increases by δRs (0 → 50% Rs), the new estimated rotor fluxes of the reference model becomes: ⎧ (ϕrα )ref = (ϕrα )ref ± δ (ϕrα )ref ⎪ ⎨ ⎪ ϕr β ref = ϕr β ref ± δ ϕr β ref ⎩

(

)

(

)

(

(26)

)

Consequently, the PI adaptation mechanism will try to generate an incorrect estimated speed in order to have rotor fluxes of the adaptive model that stick with those of the reference model. This means that the new estimated speed will be: ωˆ + δω Where: δω is the speed estimation error that becomes very important especially at very low speeds. Fig. 5 shows the speed estimation error caused by the stator resistance error when there is no online identification of this parameter. Concerning the RTC inverse variations, when (-β) increases or decreases (never ≥ zero) inside the induction motor, this will not affect the stability of the observer either as shown in Fig. 4 since both the conjugated poles remain with negative real parts. From Eqs. (8) and (9), we get the following new expression of the estimated slip speed: ⎛ 1 ⎞ , Ce ⎟ = ωˆ sl ⎝ Tr ⎠

ωˆ s − ωˆ = f ⎜

(27)

where ωsl is the slip speed.

We can remark from the Eq. (27), that the slip speed depends on the RTC inverse and the electromagnetic torque, this means that when ωsl is too small (ωs ≈ ω : in the case when the induction motor is not loaded) and whatever are the RTC inverse variations, the overall sensorless FOC system is not affected. Now, in the case when the IM is loaded, we have:

ωˆ s = ωs ,

Δω = ω − ωˆ =

−δβ ω β nom + δβ sl

(29)

3.5 Stator resistance estimation The same steps used for rotor speed estimation will be followed for stator resistance and RTC estimations; hence, to obtain the expression of the error between the reference and the adjustable model for stator resistance estimation from Eq. (19), the term below should be set to zero: Lr ΔRs ⎡ I sα ⎢⎣ M

+ I sβ ⎛⎜ ϕˆr β ⎝

(

((ϕˆ α ) r

ad

− (ϕˆrα )ref

)ad − (ϕˆr β )ref ⎞⎟⎠⎤⎥⎦ − 2 δ 1

)

Rs

ΔRs

( ) = 0 (30)

d Rˆ s dt

50 * Ω = 50 rpm

80

* Ω = 150 rpm

0

* Ω = 300 rpm

60

* Ω = 500 rpm

δΩ [rpm]

δΩ [rpm]

(28)

Fig. 6 shows the speed estimation error caused by the RTC inverse error that becomes very important when we apply an important load torque. From this analysis, we can touch the effect of the stator resistance and the RTC inverse on the whole sensorless FOC system, hence the interest of simultaneous estimation of these parameters.

100

40 20

-50 Cr = 1 N.m

-100

Cr = 2 N.m

0

Cr = 3 N.m

-150

-20 -40

= β nom ωsl

where βnom is the inverse of the RTC used in the MRAS observer when we have only rotor speed estimation, whereas β= βnom+δβ denotes the actual RTC inverse of the induction motor. Thus, even if δβ ≠ 0, the vector control condition is always verified (complete field orientation is achieved). Therefore, this error produces an error in the speed feedback by affecting the accuracy of the speed control as follows:

−2

⎛ 1 ⎞ Lr 1 Ce And f ⎜ , Ce ⎟ = 2 ⎝ Tr ⎠ p ( MI sd ) Tr

( β nom + δβ ) ωˆ sl

Cr = 4 N.m Cr = 5 N.m

6

6.5

7

7.5

8

8.5

9

9.5

Rs + δ Rs [Ω ]

Fig. 5. Speed estimation error caused by δRs

10

-200 14

15

16

17

18

19

20

21

22

23

β + δβ [Hz]

Fig. 6. Speed estimation error caused by δβ http://www.jeet.or.kr │ 1461


Fig. 7. The MRAS observer for stator resistance estimation

Ω*

PI speed controller

ˆ Ω

Eq

I*sq

Lr

1 M

* Vsq

Vsq

Q-axis current controller

p M φ*r

φ*r

Fig. 8. The MRAS observer for RTC inverse estimation

Ed

I*sd

Vsd

D-axis current controller

* Vsa

dq

* Vsb

* Vsd

abc

* Vsc

PWM generator

PWM1 PWM2 PWM3 PWM4 PWM5 PWM6

Three phase inverter

Field Weakening

θˆ s

Isq dq

∫

Isa, Isb, Isc Isd

ωˆ s

abc

I sq Tˆr I sd

ωˆ 1 Tˆr

1 p

Improved MRAS observer

Isα

ωˆ = f (Vs , is , Rˆs , 1 ˆ ) Tr

Isβ

Rˆs

1 = f (V , i , ωˆ , Rˆ ) s s s Tˆr

Vsα

ωˆ

Rˆs = f (Vs , is , ωˆ , 1 ˆ ) Tr

Vsβ

αβ abc αβ

Vsa, Vsb, Vsc abc

Fig. 9. Block diagram of sensorless FOC using MRAS observer for IM drive with online parameters identification

Finally, we get: L Rˆ s = ∫δ Rs r M

⎡I ⎢⎣ sα

((ϕˆ α )

ref

(

)

r

− (ϕˆrα )ad

K ⎞ ⎛ Rˆ s = ⎜ K pRs + iRs ⎟ eRs s ⎠ ⎝

)

(31)

⎞⎤ + I sβ ⎛⎜ ϕˆr β − ϕˆr β ⎟ ref ad ⎠ ⎥⎦ ⎝

(

)

In the case of stator resistance estimation, the expression of the error between the two models is:

(

)

⎞ − ϕˆr β eRs = I sα (ϕˆrα )ref − (ϕˆrα )ad + I sβ ⎛⎜ ϕˆr β ⎟ ref ad ⎠ ⎝ (32)

(

)

(

)

The final expression of the estimated stator resistance is given by: 1462 │ J Electr Eng Technol.2017; 12(4): 1456 -1470

(33)

where, KpRs and KiRs, are the adaptation gains that can be adjusted. Fig. 7 illustrates the structure of the MRAS observer with stator resistance estimation.

3.6 Rotor time constant estimation To obtain the expression of the error between the reference and the adjustable model for the inverse of the rotor time constant estimation from Eq. (19), the term below should be set to zero:


( ) ((ϕˆrα )ref − (ϕˆrα )ad ) + ((ϕˆrβ )ad − MI sβ ) ⎛⎜⎝ (ϕˆrβ )ref − (ϕˆrβ )ad ⎞⎟⎠⎤⎦⎥ d ( βˆ ) 1 −2 =0 Δβ

−2Δβ ⎡ (ϕˆrα )ad − MI sα ⎢⎣

δβ

(34)

dt

Finally, we get:

) ( (ϕˆrα )ad − (ϕˆrα )ref ) ⎤ + ( (ϕˆr β ) − MI sβ ) ⎛⎜ (ϕˆr β ) − (ϕˆr β ) ⎞⎟ ⎥ ad ad ref ⎠ ⎦ ⎝ (

βˆ = ∫δ β ⎡ (ϕˆrα )ad − MI sα ⎣

(35)

In the case of rotor constant time estimation, the expression of the error between the two models is:

( ) ((ϕˆrα )ad − (ϕˆrα )ref ) + ( (ϕˆr β ) − MI sβ ) ⎛⎜ (ϕˆr β ) − (ϕˆr β ) ⎞⎟ ad ad ref ⎠ ⎝

eβ = (ϕˆrα )ad − MI sα

⎛ ⎝

4. Experimental Results and Discussions

(36)

The final expression of the estimated rotor time constant inverse is given by:

βˆ = ⎜ K pβ +

is shown in Fig. 9. The vector control scheme for an IM utilized in this paper in the experimental investigation is illustrated in Fig. 9. It includes, apart from a PI speed controller, two PI d-q axis stator currents controllers as well. The required feedback quantities (Isa, Isb and Isc) for the currents closed loop control are obtained from two currents sensors (Isc may be deducted from: Isa + Isb + Isc = 0) and that for the speed closed loop control is obtained from the MRAS observer (where the superscript ‘*’ represents quantities references). The estimated speed is also used in ωs whereas the estimated RTC inverse is used in the slip speed. Fig. 9 includes also a simplified schematic of three phase IM drive fed by a Three phase inverter (with six IGBTs) controlled by a PWM technique (where PWM1...6 are the six PWM signals and Vsa, Vsb and Vsc are the output voltages applied to a star-connected IM windings).

Ki β ⎞ ⎟ eβ s ⎠

(37)

where, Kpβ and Kiβ, are the adaptation gains that can be adjusted. Fig. 8 illustrates the structure of the MRAS observer with RTC inverse estimation. Finally, the complete structure of the Indirect FOC using the improved MRAS observer (with online estimation of stator resistance and RTC inverse) for sensorless IM drive

To validate the performance of the proposed method, a prototype implementation of the sensorless Indirect FOC of an IM drives is carried out. Experimental tests were done based on the scheme proposed in Fig. 9. The experimentation has been achieved using Matlab/ Simulink and dSPACE DS1104 real time controller board shown in the bloc diagram of the experimental bench in Fig. 10. From Fig. 11, the experimental setup is composed by an 1 kW squirrel-cage induction motor and pulse width modulation signals with 6 kHz switching frequency that are generated by dSPACE system to control the power modules through a Voltage Source Inverter. This latter is connected to a DC voltage of 400 V. There is also a dSPACE panel for stator currents and rotor speed acquisition. Resistive bank

PWM Unit DS1104

DSP TMS 320F240

Improved MRAS observer

DC Voltage DS 1104 Controller Board

DC

PWM

ADC Unit

DAC Unit

Real Time Interface (RTI) Real Time Workshop (RTW)

8 I/O TTL (0-5V)

I/O Bit

ia ib

IM Interface card

ω

Fig. 10. Block diagram of the experimental test bench http://www.jeet.or.kr │ 1463


Fig. 11. Photograph of the experimental setup Table 1. Motor specifications Rated power, rated voltage Rated current, rated frequency Rated speed, number of pole pairs

1 kW, 400 V 2.65 A, 50 Hz 1425 rpm, 2

Table 2. Motor parameters Stator and rotor inductances Stator and rotor resistances Friction coefficient Total inertia, mutual inductance

Speed[rpm]

1500

0.3973 H – 0.3558 H 6.8 Ω – 5.43 Ω 0.0025 Nm.s.rad-1 0.02 Kg.m², 0.3558 H

Real Speed Estimated Speed Speed reference

1000

500

0

0

5

10

15

20

25

Time[s]

(a) 50

Error[rpm]

Speed Estimation Error

0

-50

0

5

10

15

20

25

Time[s]

(b) Fig. 12. Experimental responses of: (a) Actual and estimated speed; (b) Speed estimation error

Table 3. Gains of PI controllers used to estimate the speed and the other parameters and for speed and currents regulation PI speed controller PI d-q axis currents controller

(0.06, 0.04) (40, 300)

Kpω , Kiω KpRs , KiRs Kpβ , Kiβ

Motor used in this research are listed in Table 1 and Table 2 in the appendix. Load conditions for the Induction Motor are adjusted by changing the external resistances through a DC generator connected to the IM. Between the two motors, there is a tachymeter which is placed to provide the actual motor speed to compare it with its estimated value. Gains of PI controllers used for regulation and estimation are listed in Table 3 in the appendix. In order to verify the performances of the proposed observer with sensorless Indirect FOC, many ranges of speed frequency are tested under or without load torque.

4.1 Ramp speed response at rated speed The performance of the proposed sensorless drives (without stator resistance and RTC estimation) is observed, it is shown in Fig. 12(a) for a ramp speed command. They show sensorless IM drives under high (1425 rpm) / medium (700 rpm) and low (100 rpm) speed applications respectively. Rotor reference speed is gradually increased from 100 rpm to 700 rpm during 5 s. Thereafter, the command speed is maintained constant at 700 rpm up to 5 s and etcetera until it reaches the rated speed (1425 rpm). In these tests the estimated speed is used as feedback signal for the closed-loop control. The error between the measured speed and the estimated one as shown in Fig. 12(b) is very small especially in steady state.

4.2 Speed reversal (zero crossing) The specifications and the parameters of the Induction 1464 │ J Electr Eng Technol.2017; 12(4): 1456 -1470

(200, 30000) (800, 7000) (5, 300)


300

100

600

Speed[rpm]

Speed[rpm]

800


200

0 -100

400

-300


200

-200 0

5

10

15

20

25

0

30

0

5

10

15

Time[s]

20

25

30

35

40

Time[s]

(a)

(a) 200

Speed Estimation Error

Speed Estimation Error Error[rpm]

Error[rpm]

200 100 0

100 0 -100

-100

-200 -200

0

5

10

15

20

25

30

35

40

Time[s] 0

5

10

15

20

25

30

(b)

Time[s]

(b)

4.3 Effect of step speed and rated load The IM is operated at 400 rpm and loaded with 4.5 N.m. A step command of 200 rpm is applied at 6 s and removed at 33 s. The loading performance of the sensorless drives is observed in Fig. 14(a). The motor is operated at 600 rpm and it is star connected when a load torque (1.5 N.m) is suddenly applied at 15 s to reach 6 N.m (the rated load) then withdrawn at 27 s. As one can see, the estimated speed follows the measured one in transient and steady states. The undershoot and the overshoot appear in the actual and estimated speed during loading and unloading respectively but the estimation error in these two cases remains negligible as observed in Fig. 14(b) whereas Fig. 14(c) shows the profile of the developed electromagnetic torque.

Torque[N.m]

The motor reference speed is changed from 200 rpm to 200 rpm at 5 s and then it is set to 200 rpm at 20 s. The performance of the proposed technique for such kind of speed reference is shown in Fig. 13(a). The dynamics around 5 s and 20 s can be analyzed from Fig. 13(b) that represents the error speed estimation. It can be noticed that at very low speeds (around zero speed), the actual speed does not fit the estimated speed because of the reference voltages (used as inputs of the MRAS observer) that deviate substantially from the actual motor voltages. This problem which remains a challenge is caused by the inverter dead time effects and inverter nonlinearities [38, 39].

6 4 2 0

Torque Torque reference 0

5

10

15

20

25

30

35

40

Time[s]

(b) Fig. 14. Experimental responses of: (a) Actual and estimated speed, (b) Speed estimation error, (c) Electromagnetic torque 20

RTC inverse[Hz]

Fig. 13. Experimental responses of: (a) Actual and estimated speed; (b) Speed estimation error

8

15 10 5 0

Real RTC inverse Estimated RTC inverse 0

5

10

15

20

Time[s]

Fig. 15. Experimental response of the actual and the estimated rotor time constant inverse

4.4 Response of the estimated rotor time constant inverse In the rest of the tests, the rotor speed, the stator

http://www.jeet.or.kr │ 1465


resistance and the rotor time constant inverse are estimated simultaneously. The test objective is to show the response of the estimated RTC inverse. The estimated RTC inverse is used as input of the MRAS to estimate both the rotor speed and the stator resistance according to the adaptive model equations. From Eq. (10), as illustrated in Fig. 9, the inverse of RTC is also used in slip speed expression. Fig. 15 shows then a good convergence of β.

4.5 Comparative study of the classical and the improved MRAS under stator resistance variations at low speeds In order to show the stator resistance response and its effect on the estimated rotor speed, three rheostats in parallel with a three phase circuit breaker have been put in series with the stator resistances as shown in Figs. 10 and 11. Fig. 16(a) shows the actual and the estimated

rotor speed with the classical MRAS (when there is no online parameters identification). We can notice that when the resistances values are increased sharply at 5 s by 1.5 Ω (from 6.5 Ω to 8 Ω) and decreased sharply by 1.5 Ω at 21 s, an error between the actual and estimated speed is occurred. On the other hand, Figs. 16(b) and 16(c) show the performance of the improved MRAS observer when the resistances values are increased sharply at 5 s by 2.5 Ω (from 6.5 Ω to 9 Ω) and decreased sharply by 2.5 Ω at 18 s. It can be noticed that the estimated stator resistance follows the actual stator resistance and only small variations of the estimated rotor speed are observed at 5 s and 18 s when the stator resistance has been increased by more than 25% from its nominal value.

4.6 Effect of step speed on parameter identification at very low speeds The motor reference speed is changed from 50 rpm to

150

300

50

0

Speed[rpm]

Speed[rpm]

Real Speed Estimated Speed

200

100

Real Speed Estimated Speed Speed reference 0

5

10

15

20

Speed reference

100 0

25

30

-100

Time[s]

0

5

10

15

20

25

30

Time[s]

(a)

(a)

150

15 Estimated stator resistance

100

50

0

Resistance[Ohm]

Speed[rpm]

Real stator resistance


5

10

15

20

25

10

5

0

30

Time[s]

0

5

10

(b)

25

30

(b) Real inverse of the RTC

8

Inverse of RTC[Hz]

Stator Resistance[ Ω]

20

25

10

6 4

Real Stator Resistance Estimated Stator Resistance

2 0

15 Time[s]

0

5

10

15

20

20

Estimated inverse of the RTC

15 10 5

25

30

Time[s]

(c) Fig. 16. Experimental response of: (a) Actual and estimated speed with the classical MRAS; (b) Actual and estimated speed with the improved MRAS; (c) Actual and estimated stator resistance 1466 │ J Electr Eng Technol.2017; 12(4): 1456 -1470

0

0

5

10

15

20

25

30

Time[s]

(c) Fig. 17. Experimental response of: (a) Actual and estimated speed; (b) Actual and estimated stator resistance; (c) Actual and estimated inverse of rotor time constant


75 rpm at 2.5 s as shown in Fig. 17(a). At the same time, the stator resistance reference is changed from 6.5 Ω to 9 Ω as shown in Fig. 17(b). The test objective is to show the performance of the proposed sensorless drives with parallel parameters identification during the step change of the rotor speed and the stator resistance simultaneously. In 600

Speed[rpm]

500 400 300 200


100 0

0

5

10

15

20

25

30

Time[s]

(a) 200

Speed estimation Error Error[rpm]

100 0 -100 -200

0

5

10

15

20

25

30

Time[s]

(b)

order to apply a step change on the reference of RTC inverse, we need to act on rotor resistance by adding three rheostats in series with the rotor resistances of the induction motor or increasing motor temperature. However, there is no access to the rotor of the used squirrel cage motor and to observe the temperature variations; we need a long time vector. A longer execution time will lead to a lower sampling rate for the controller and as a result the control performance will deteriorate. Furthermore, experimental results show that the estimated RTC inverse is not affected by the speed variation as shown in Fig. 17(c), which means that the proposed algorithm has good speed estimation and adequate vector control characteristics.

4.7 Effect of loading (rated load) on parameter identification The loading performance of the sensorless drives with parameters identification is observed in Figs. 18(a) and 18(b). The motor is operated at 400 rpm. A load torque (0.5 N.m) is suddenly applied at 3 s then withdrawn at 17.5 s. Fig. 18(d) illustrates the evolution of the electromagnetic torque. The test objective is to show the effect of loading on the estimated parameters. As shown in Fig. 18(c), undershoot and overshoot appear in the estimated stator resistance but this latter eventually converges to its nominal value and since the induction motor is loaded for some hours, this has increased the motor temperature and consequently the stator resistance.

8

2000

6

1500

Speed[rpm]

Stator Resistance[ Ω]

10

4 2 0

Real Stator Resistance Estimated Stator Resistance 0

5

10

15

20

1000 500

25

30

0

Time[s]

Nominal speed

0


10

(c)

6

0.8

4 2

Flux[Wb]

Torque[N.m]

20

25

30

25

30

(a)

8

0

15

Time[s]

Torque Torque reference 0

5

10

15

20

Flux-rd Flux-rd reference Flux-rq

0.6 0.4 0.2

25

30

Time[s]

(d) Fig. 18. Experimental response of: (a) Actual and estimated speed, (b) Speed estimation error, (c) Actual and estimated stator resistance, (d) Electromagnetic torque

0 0

5

10

15

20

Time[s]

(b) Fig. 19. Experimental responses with field weakening of: (a) Actual and estimated speed; (b) d- and q-axis flux http://www.jeet.or.kr │ 1467


Therefore, the nominal value of this parameter has become 7.5 Ω instead of 6.5 Ω.

[6]

4.8 Field weakening test [7] As shown in Fig. 19(a), the rotor reference speed is gradually increased from 1425 rpm to 2000 rpm during 6 s. Thereafter, the command speed is maintained constant at 2000 rpm up to 10 s. We can remark that the estimated speed tracks the actual speed very well. Fig. 19(b) illustrates the waveforms of the rotor fluxes, unlike the rotor speed; the d-axis flux is gradually decreased from 0.7 Wb to 0.4 Wb during 6 s. Thereafter, it is maintained constant at 0.4 Wb up to 10 s according to the d-axis flux reference command that we have applied. Results in Fig. 19 show the performance of sensorless induction motor drive using the MRAS observer beyond the nominal speed (field weakening area).

[8]

[9]

[10]

5. Conclusion This paper proposes a speed estimation method for sensorless IM drives using an improved model reference adaptive system observer with parameters identification. The stability of the proposed sensorless Indirect FOC with stator resistance and rotor time constant inverse tuning has been demonstrated by Lyapunov criterion and its validity has been proved by experimentation applied to an 1 kW squirrel-cage IM for a wide range of speed under load and no load. Results confirm the good dynamic performances of the developed drives system during transient and steady state conditions and show the validity of the suggested method. It is concluded from the results presented in this paper that the proposed scheme performs well for both high and low speed.

References [1] [2] [3]

[4]

[5]

B. K. Bose, Modern Power Electronics and AC Drives, Prentice-Hall, chap. 8, 1986. W. Leonhard, Control of Electrical Drives, SpringerVerlag, chap. 12, 1990. K. Hasse, “Zum dynamischen Verhalten der Asynchronmaschine bei Betrieb mit variabler Ständerfrequenz und Ständerspannung,” ETZ-A 89, pp. 387-391, 1968. F. Blaschke, “The principles of field orientation as applied to the new transvector closed loop control system for rotating field machines,” Siemens Review, vol. 39, no. 4, pp. 217-220, May 1972. A. Derdiyok, M. K. Güven, H. Rehman, N. Inanc, and L. Xu, “Design and implementation of a new sliding-mode observer for speed-sensorless control of induction machine,” IEEE Transactions on Industrial


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Electronics, vol. 49, no. 5, pp. 1177-1182, Oct. 2002. Z. Kandoussi, Z. Boulghasoul, A. Elbacha and A. Tajer, “Fuzzy Sliding Mode observer based sensorless Indirect FOC for IM drives,” in Proc. WCCS, 2015. Z. Kandoussi, Z. Boulghasoul, A. Elbacha and A. Tajer, “Luenberger observer based sensorless Indirect FOC with stator resistance adaptation,” in Proc. WCCS, pp. 367-373, 2014. S. Huh, S. J. Seo, I. Choy and G. T. Park, “Design of a Robust Stable Flux Observer for Induction Motors,” Journal of Electrical Engineering & Technology, vol. 2, no. 2, pp. 280-285, 2007. G. Garcia Soto, E. Mendes and A. Razek, “Reducedorder observers for rotor flux, rotor resistance and speed estimation for vector controlled induction motor drives using the Extended Kalman Filter technique,” IEE Proceedings - Electric Power Applications, vol. 146, no. 3, pp. 282-288, May 1999. A. Benheniche and B. Bensaker, “A high gain observer based sensorless nonlinear control of induction machine,” International Journal of Power Electronics and Drive System, vol. 5, no. 3, pp. 305-314, Feb. 2015. S. Hadj Saïd, M. F. Mimouni, F. M’Sahli and M. Farza, “High gain observer based on-line rotor and stator resistances estimation for IMs,” Simulation Modelling Practice and Theory, vol. 19, no. 7, p. 1518-1529, Aug. 2011. Z. Boulghasoul, A. Elbacha, E. Elwarraki, “A Comparative Study of Rotor Time Constant Online Identification of an Induction Motor Using High Gain Observer and Fuzzy Compensator,” WSEAS Transactions on Systems and Control, vol. 7, no. 2, pp. 37-53, April 2012. J. I. Ha and S. K. Sul, “Sensorless field-orientation control of an induction machine by high-frequency signal injection,” IEEE Transactions on Industry Applications, vol. 35, no. 1, pp. 45-51, Jan/Feb. 1999. C. Schauder, “Adaptive speed identification for vector control of induction motors without rotational transducers,” IEEE Transactions on Industry Applications, vol. 28, no. 5, pp. 1054-1061, Sept/Oct. 1992. Y. A. Zorgani, Y. Koubaa and M. Boussak, “MRAS state estimator for speed sensorless ISFOC induction motor drives with Luenberger load torque estimation,” ISA Transactions, Jan. 2016. T. Ramesh, A. K. Panda, and S. S. Kumar, “Type-2 fuzzy logic control based MRAS speed estimator for speed sensorless direct torque and flux control of an induction motor drive,” ISA Transactions, vol. 57, pp. 262-275, July 2015. Z. Boulghasoul, A. Elbacha, E. Elwarraki, “Real Time Implementation of Fuzzy Adaptation Mechanism for MRAS Sensorless Indirect Vector Control of Induction Motor,” International Review of Electrical Engineering, vol. 6, no. 4, pp. 1636-1653, July/Aug.


2011. [18] P. Brandstetter, R. Cajka and O. Skuta, “Rotor time constant adaptation with ANN application,” in Proc. EPE, pp. 1-10, 2007. [19] Y. Sayouti, A. Abbou, M. Akherraz and H. Mahmoudi, “Sensorless low speed control with ANN MRAS for direct torque controlled induction motor drive,” in Proc. POWERENG, pp. 1-5, 2011. [20] S. M. Gadoue, D. Giaouris and J. W. Finch, “MRAS Sensorless Vector Control of an Induction Motor Using New Sliding-Mode and Fuzzy-Logic Adaptation Mechanisms,” IEEE Transactions on Energy Conversion, vol. 25, no. 2, pp. 394-402, 2010. [21] Y. A. Zorgani, Y. Koubaa and M. Boussak, “Sensorless speed control with MRAS for induction motor drive,” in Proc. ICEM, pp. 2259-2265, 2012. [22] N. Bensiali, E. Etien and N. Benalia, “Convergence analysis of back-EMF MRAS observers used in sensorless control of induction motor drives,” Mathematics and Computers in Simulation, vol. 115, pp. 12-23, Sept. 2015. [23] M. Rashed and A.F. Stronach, “A stable back-EMF MRAS-based sensorless low-speed induction motor drive insensitive to stator resistance variation,” IEE Proceedings - Electric Power Applications, vol. 151, no. 6, pp. 685-693, 2004. [24] T. Orlowska-Kowalska, and M. Dybkowski, “StatorCurrent-Based MRAS Estimator for a Wide Range Speed-Sensorless Induction-Motor Drive,” IEEE Transactions on Industrial Electronics, vol. 57, no. 4, pp. 1296-1308, April 2010. [25] S. Maiti, C. Chakraborty, Y. Hori and M. C. Ta, “Model reference adaptive controller-based rotor resistance and speed estimation techniques for vector controlled induction motor drive utilizing reactive power,” IEEE Transactions on Industrial Electronics, vol. 55, no. 2, pp. 594-601, Feb. 2008. [26] A. V. Ravi Teja, V. Verma, and C. Chakraborty, “A New Formulation of Reactive Power Based Model Reference Adaptive System for Sensorless Induction Motor Drive,” IEEE Transactions On Industrial Electronics, vol. 62, no. 11, pp. 6797-6808, May 2015. [27] W. Rao and H. Wan, “Parameter sensitivity of rotor time constant estimation based on MRAS for induction motors,” in Proc. ICIEA, pp. 391-394, 2014. [28] H. M. Kojabadi, “Active power and MRAS based rotor resistance identification of an IM drive,” Simulation Modelling Practice and Theory, vol. 17, no. 2, pp. 376-386, Feb. 2009. [29] R. Blasco-Gimenez, G. M. Asher, M. Sumner, K. J. Bradley, “Dynamic performance limitations for MRAS based sensorless induction motor drives. Part 1: Stability analysis for the closed loop drive,” IEE Proceedings - Electric Power Applications, vol. 143, no. 2, pp. 113-122, 1996. [30] V. Vasic, S. N. Vukosavic and E. Levi, “A stator

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resistance estimation scheme for speed sensorless rotor flux oriented induction motor drives,” IEEE Transactions on Energy Conversion, vol. 18, no. 4, pp. 476-483, Dec. 2003. R. Beguenane, M. El Hachemi Benbouzid, M. Tadjine and A. Tayebi, “Speed and rotor time constant estimation via MRAS strategy for induction motor drives,” in Proc. IEMDC, pp. TB3/5.1-TB3/5.3, 1997. Q. Yang, Y. Xue, S. X. Yang, Q. Li, R. Li and M. Q. H. Meng, “An embedded structure of model reference adaptive system,” in Proc. ICARCV, pp. 256-261, 2008. C. Lu, Y. Liu, X. Bao, Y. Zhang and J. Ying, “Considerations of Stator Resistance Online-tuning Method for MRAS-based Speed Sensorless Induction Motor Drive,” in Proc. IEMDC, pp. 1142-1147, 2007. F. J. Lin, R. J. Wai and H. J. Shieh, “Robust control of induction motor drive with rotor time-constant adaptation,” Electric Power Systems Research, vol. 47, no. 1, pp. 1-9, Oct. 1998. M. Hinkkanen, J. Luomi, “Modified integrator for voltage model flux estimation of induction motors,” IEEE Transactions On Industrial Electronics, vol. 50, no. 4, pp. 818-820, Aug. 2003. Z. Boulghasoul, A. Elbacha, E. Elwarraki, “AdaptivePredictive Controller based on Continuous-Time Poisson-Laguerre Models for Induction Motor Speed Control Improvement,” Journal of Electrical Engineering & Technology, vol. 9, no. 3, pp. 742-759, May. 2014. Z. Boulghasoul, A. Elbacha, E. Elwarraki, “Intelligent Control for Torque Ripple Minimization in Combined Vector and Direct Controls for High Performance of IM Drive,” Journal of Electrical Engineering & Technology, vol. 7, no. 4, pp. 546-557, Feb. 2012. C. Silva and R. Araya, “Sensorless Vector Control of Induction Machine with Low Speed Capability using MRAS with Drift and Inverter Nonlinearities Compensation,” in Proc. “Computer as a Tool”, pp. 19221928, 2007. I. Vicente, M. Brown, A. Renfrew, A. Endemano and X. Garin, “Stable MRAS-based sensorless scheme design strategy for high power traction drives,” in Proc. PEMD, pp. 562-567, 2008.

Appendix

(

)

T T Sign of the term: eαβ AMRAS + AMRAS eαβ T

We put: eαβ = [x1 x2 x3 x4] in order to simplify the analysis.

(

)

T T Let’s prove that eαβ AMRAS + AMRAS eαβ is negative:

We have: http://www.jeet.or.kr │ 1469


(

)

T eαβ T AMRAS + AMRAS eαβ =

⎡ −2β ⎢ 0 [ x1 x2 x3 x4] ⎢⎢ 0 ⎢ ⎣ 0

We obtain:

(

0 −2 β 0 0

)

0 ⎤ ⎡ x1 ⎤ 0 ⎥⎥ ⎢⎢ x 2 ⎥⎥ 0 ⎥ ⎢ x3 ⎥ ⎥⎢ ⎥ 0⎦ ⎣ x4⎦

0 0 0 0

(

T eαβ T AMRAS + AMRAS eαβ = −2 β x12 + x 22

)

As β is strictly positive, finally, we get:

(

)

T eαβ T AMRAS + AMRAS eαβ ≤ 0

Zineb Kandoussi was born in Settat in Morocco, on April 24, 1988. In 2011, she got her engineer’s degree in Electrical Engineering from the National School of Applied Sciences, Cadi Ayyad University, Morocco. In October 2013, she embarked on the Ph.D. research in the Ph.D. Center of engineering sciences in the same university. Her employment experience included the Zodiac Aerospace international company, she worked there as a software development engineer for two years and a half. Her special fields of interest included sensorless control strategies of AC Drives and embedded software. Zakaria Boulghasoul was born in Essaouira, Morocco, on November 11, 1986. He received the M.S. degree in electrical engineering in 2009 and the Ph.D. degree in electrical engineering in 2014 from Cadi Ayyad University, Morocco. Currently he is an Assistant professor at the National School of Applied Sciences, Cadi Ayyad University. His area of interest is related to the innovative control strategies for AC Drives, especially Induction Motor Drives, Predictive Control, Neural Network, Fuzzy logic, RFOC, DTC, and Sensorless Control


Abdelhadi Elbacha was born in Zagora, Morocco, in 1975. He received the B.S. degree and the Aggregation in electrical engineering from ENSET Rabat, Morocco in 1995 and 1999 respectively, and then he received the M.S. degree in industrial informatics and the Ph.D. degree in electrical engineering from Cadi Ayyad University, Morocco, in 2001 and 2006 respectively. Currently, he is an Assistant Professor of Automatics and Control at the National School of Applied Sciences, Cadi Ayyad University. His current area of interest is related to the innovative control strategies for AC Drives, especially Induction Motor Drives, DTC, RFOC and Sensorless Control. Abdelouahed Tajer was born in Morocco in 1977. After a Master degree in Systems Optimization and Safety at the University of Reims ChampagneArdenne / University of technology of Troyes in France, he achieved a Ph.D. degree in Control of Discrete-events Systems at the University of Reims Champagne-Ardenne in France in 2005. His research interests are: Discrete Event Systems, Fault diagnosis, Modeling, Supervisory Control Theory, Optimal Control, Manufacturing systems. Currently, he is an HDR professor at the National School of Applied Sciences of Marrakech, Cadi Ayyad University, Morocco.