A review on MRAS-type speed estimators for reliable and efficient induction motor drives

Accepted Manuscript A review on MRAS-type speed estimators for reliable and efficient induction motor drives Mateusz Korzonek, Grzegorz Tarchala, Teresa Orlowska-Kowalska

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S0019-0578(19)30153-3 https://doi.org/10.1016/j.isatra.2019.03.022 ISATRA 3151

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ISA Transactions

Received date : 28 October 2018 Revised date : 4 March 2019 Accepted date : 21 March 2019 Please cite this article as: M. Korzonek, G. Tarchala and T. Orlowska-Kowalska, A review on MRAS-type speed estimators for reliable and efficient induction motor drives. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.03.022 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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A review on MRAS-type speed estimators for reliable and efficient induction motor drives Mateusz Korzonek, Grzegorz Tarchala, Teresa Orlowska-Kowalska Department of Electrical Machines, Drives and Measurements Wroclaw University of Science and Technology, Wroclaw, Poland [email protected], [email protected], [email protected]

Corresponding author: Teresa Orlowska-Kowalska Department of Electrical Machines, Drives and Measurements Wroclaw University of Science and Technology ul. Wybrzeze Wyspianskiego 27 50-370 Wroclaw Poland [email protected] Acknowledgments: (given here to avoid identification of authors in article text) This research was partly supported by the National Science Centre Poland under grant number 2015/17/B/ST7/03846 and by statutory funds of the Faculty of Electrical Engineering of the Wroclaw University of Science and Technology (2018-2019).

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Revision_ISATRANS-D-18-01350 Highlights:

   

A comprehensive review on speed estimation methods for induction motor with focus on MRAS techniques. Analysis of speed adaptation algorithms for different algorithmic MRAS estimators based on Lyapunov theory. Consideration and comparison of stability regions of selected MRAS-based techniques; overview of the stability improvement methods for algorithmic speed estimators. Consideration and comparison of robustness of different algorithmic speed estimators to induction motor parameters mismatch.

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A review on MRAS-type speed estimators for reliable and efficient induction motor drives Abstract — Induction machines have recently been very popular in variable-speed drives, because of their robust construction and relatively low manufacturing costs (brushless), maintenance-free and well-matured control methods. However, for high-precision control and efficiency optimization one needs the information on the rotor speed which can be measured using different speed sensors. All sensors require a mounting space and cabling, they also generate extra costs and reduce system reliability. Therefore, many of the recent research efforts have been dedicated to sensorless or encoderless electrical drives offering such considerable advantages as: lower cost, reduced size and hardware complexity of the drive system, elimination of sensor cables, lower maintenance requirements, possible operation in aggressive environment, higher noise immunity, reliable and user friendly operation. In this article all well-known sensorless techniques are shortly addressed, but the main focus is on the solutions based on the Model Reference Adaptive System (MRAS) concept. The mathematical models and schemes of all types of MRAS-type speed estimators known from the literature are gathered in this article. The comparative analysis of these speed estimators is done from the following points of view: the speed adaptation mechanism derivation based on the Lyapunov theory, stability problems near zero speed and in the regenerating operation mode, and the sensitivity of MRAS estimators to induction machine parameter changes. Keywords—Induction machine, sensorless control, MRAS-type estimator, stability, parameter sensitivity Nomenclature Variables and parameters: A, B, C

state, input and output matrices,

e

estimation error of the state vector,

fsN

nominal frequency of the motor,

isA, isB, isC

stator phase currents,

is=isα+jisβ

stator current vector in stationary frame,

is=isx+jisy

stator current vector in synchronous frame,

G, gs, gr

gain matrix and its components,

Ki, Kp

speed adaptation parameters,

lr=lm+lrσ

rotor inductance,

ls=lm+lsσ

stator inductance,

lrσ, lsσ

rotor and stator leakage inductance,

lm

main inductance,

me, mL

electromagnetic and load torques,

rr , rs

rotor and stator winding resistance,

TM

mechanical time constant of a drive,

TN=1/fsN

nominal time constant (appears after the per unit system is introduced),

u

control vector,

us=usα+jusβ stator voltage vector in stationary frame, x, xˆ

state vector of the reference and adaptive model,

y

output vector,

V

Lyapunov function,



error function in a speed adaptation mechanism,

ψr=ψrα+jψrβ rotor flux vector in stationary frame, ψr

amplitude of the rotor flux vector,

ψs=ψsα+jψsβ stator flux vector in stationary frame, σ=1-lm2/(lslr) total leakage factor, ωk

arbitrary reference frame velocity,

ωm

rotor speed,

ωr=ωs-ωm

slip angular velocity,

ωs

synchronous velocity (rotor flux synchronous speed).

All variables and parameters are in per unit system, according to [1-3]. Main abbreviations: AFO – Adaptive Full-order Observer, ANN-MRAS – Artificial Neural Network-Model Reference Adaptive System, DC – Direct Current (motors), DFOC – Direct Field Oriented Control, DTC – Direct Torque Control, FL-MRAS – Fuzzy Logic-Model Reference Adaptive System, FOC – Field Oriented Control, IFOC – Indirect Field Oriented Control, IM – Induction Motor, LPF – Low-Pass Filter, MRAS – Model Reference Adaptive System, MRASCC – Model Reference Adaptive System (Current, Current), MRASCV – Model Reference Adaptive System (Current, Voltage), MRASEMF – Model Reference Adaptive System (Electro-Motive Force), MRASF – Model Reference Adaptive System (Flux), MRASRP – Model Reference Adaptive System (Reactive Power), P-, Q-MRAS – Model Reference Adaptive System (P – active power, Q – reactive power), PWM – Pulse Width Modulation, X-, Y-MRAS – Model Reference Adaptive System (X and Y are virtual variables), SM-MRAS – Sliding Mode-Model Reference Adaptive System.

1. Introduction A relatively short time ago most electrical drives in different industrial applications were equipped with DC motors due to the natural decoupling between the flux and electromagnetic torque in the DC machine which enables a very easy speed and torque control. Recently induction machines, especially their squirrel-cagerotor type counterparts, have become very popular in such drives, because of their robust construction and

relatively low manufacturing costs (brushless), the fact that they are practically maintenance-free, and at last well-matured control methods, which ensure as good dynamical performance as in the case of DC motor drives, are applied for the speed control [1-6]. These control methods, based mainly on the FOC or DTC strategies, require the knowledge of the internal state variables of the IM, such as stator or rotor flux vectors. Thus, in the last decades, the research effort has been directed to the estimation methods of the IM state variables. In most industrial applications also the speed and/or position identification is needed for robust and highprecision control of IM drives. However, the use of speed/position sensors (mechanical or optical) in the electrical drive is associated with problems, such as extra cost, reduced reliability, added mounting space, cabling. Therefore, many of the recent research efforts have been dedicated to sensorless or encoderless electrical drives. The main advantages of sensorless controlled drives are: lower cost, reduced size and hardware complexity, elimination of sensor cables, lower maintenance requirements, possible operation in aggressive environment, higher noise immunity, reliable and user friendly operation [1, 3, 4, 6]. Rotor speed of IM is estimated indirectly, using motor terminal variables (voltages and currents), but detailed algorithms require the internal variables of the machine, e.g. stator and/or rotor flux space vectors, to be estimated first. It will be shown in the further parts of this paper. Numerous methods and estimation techniques for sensorless control of IM have been proposed in the literature [1-6]. The main three groups of methods used to estimate the rotor speed encompass: signal injection methods, neural methods and algorithmic methods (Fig. 1.) [4-6]. The signal injection methods use the physical phenomena of the IM, caused by different motor saliences. These techniques are insensitive to changes in machine parameters, they give good accuracy in the near zero speed range and do not cause stability problems. They typically use a carrier signal that is superimposed on the PWM signal of the inverter. Two types of signals are most often used for the estimation of the rotor speed: negative-sequence carrier-signal and zero-sequence carrier-signal components [5]. These methods, however, require complex signal processing in a wide range of changes in the basic frequency of the motor and are sensitive to the noise ratio.

Also neural networks were used as speed estimators. These applications are based on neural modelling or neural identification methods [3]. Contrary to the algorithmic methods neural network-based methods do not need a machine mathematical model, and thus they are also robust to motor parameter changes. However, especially neural estimators based on identification methods, which use the multilayer neural networks trained off-line, require big training data sets to obtain good estimation accuracy. The third group – algorithmic methods – is the most widespread and developed method in recent decades. All types of estimators belonging to this group: simulators (open-loop estimators), observers [7] and Kalman filters [8,9], are based on the fundamental wave model of the induction machine. Thus, they are more or less sensitive to changes in machine (and mathematical model) parameters, which can significantly affect the accuracy of speed estimation and also have worse properties in the near zero speed range. In the closed-loop-type estimators, such as state variable observers and Kalman filters, the parameter sensitivity can be controlled using proper design of feedback gains. However, their values have to be limited due to stability problems in the speed-closed loop operation of the vector controlled IM drive system and usually have to be changed according to the reference speed value of the drive, which complicates the observer algorithms. Moreover, particularly Kalman filters algorithms are computationally complex [1, 3, 6]. This problem becomes especially important in practical implementation, when extended observers or extended Kalman filters, enabling the simultaneous estimation of selected motor parameters in order to reduce errors in the rotor flux and speed estimation, are used [7-9]. Another important problem of the algorithmic speed estimators is their instability in the regenerating mode of the drive system. This issue will be also discussed later in the paper. Among different algorithmic speed estimators those based on the concept of Model Reference Adaptive System have recently enjoyed increasing popularity [10-29] because of their simple algorithms, therefore the comparative analysis of these estimators is the main purpose of this work. The general idea of MRAS-type estimators was proposed in 1987 by S. Tamai [10] and was developed in 1992 by C. Schauder [11]. As is shown in Fig. 1, quite a lot of different MRAS-type estimators have been developed, based on simulators (mathematical models) of different state variables of the induction motor: MRASF – based on the simulators of the rotor flux vector [10, 11], MRASEMF – based on the

simulators of the electromotive force [12], MRASRP – based on the simulators of the reactive power [12, 13], MRASCV – based on the simulators of the stator current and voltage model of the rotor flux vectors [14], MRASCC – based on the simulators of the stator current and current model of the rotor flux vectors [15], P-, Q-MRAS – based on the simulators of the active and reactive power respectively, and other estimators which use virtual variables in the error function introduced into the adaptation mechanism (X-, YMRAS, where X and Y are virtual variables) [16] and SM-MRAS – using sliding-mode-based adaptation algorithm and simulators of different motor variables [17-22]. In some works a well-known Adaptive Full-order Observer [23] is classified as a MRAS-type estimator [24, 25] because it uses an induction motor as a reference model. Therefore, it has several features in common with MRASCC and MRASCV estimators [14, 15], which will be addressed in more detail later. Thus in this paper AFO is considered as a MRAS-type speed estimator (see the dashed line in Fig. 1) and its features are compared with other algorithmic MRASs. All MRAS-type speed estimators mentioned above will be discussed in details in the next sections of this article and compared from the point of view of the speed adaptation mechanism derivation based on the Lyapunov theory, stability problems near zero speed and in the regenerating operation mode, and the sensitivity to IM parameter changes. The algorithmic MRAS-type estimators which are presented in Fig. 1, can be also gathered into one group because of the method of deriving the value of an error function in the speed adaptation algorithm, which is the Lyapunov stability theory or the Popov hyperstability theory [1]. MRAS-type speed estimators which use artificial neural networks (ANN-MRAS) [26-30] or fuzzy logic (FL-MRAS) [31-33] were also proposed in the literature. These estimators differ in the mathematical models of suitable state variables used as a reference and adaptive models, and also in the adaptation mechanism used for rotor speed calculation. ANN-MRASs can be considered as estimators belonging to neural modelling methods [3, 26-28]. Both ANN-MRAS and FL-MRAS are be discussed in this paper, because their speed estimation algorithm is not derived using the Lyapunov or Popov theory and also due to the lack of the stability analysis methods of these nonlinear estimators. The review of different MRAS-type estimators was shown in [34, 35], however, they present only basic schematic diagrams of the main estimator types based on source articles, and address merely the general

problems of speed estimation in high and low speed regions. In the paper [35], besides relevant mathematical models of selected MRAS estimators, also some simulation and experimental results are shown. However, the authors do not make a theoretical analysis of the speed adaptation mechanisms or stability problems of these estimators, especially in the regenerating mode. In this article, apart from the discussion and comparison of the characteristic features of all algorithmic MRAS techniques known from the literature, special attention was devoted to the generalization of the method of deriving a speed adaptation mechanism for different MRAS-type speed estimators for IM using the Lyapunov theory as well as to stability issues in the regenerating mode and to the robustness of MRAS estimators to IM parameter changes. Due to the analysis presented in this work it was also possible to gather the speed adaptation mechanism equations used in different MRAS-type estimators in one paper and show the reader how these equations were obtained based on the Lyapunov function analysis. Moreover, based on the literature analysis, the limitation in the stable operation of these estimators are shown. So the main contributions of this survey are listed as follows: –

a comprehensive review on speed estimation methods for IM with focus on MRAS techniques,

–

analysis of speed adaptation algorithms for different algorithmic MRAS estimators based on the Lyapunov theory,

–

consideration and comparison of stability regions of selected MRAS-based techniques; overview of the stability improvement methods for these speed estimators,

–

consideration and comparison of robustness of different algorithmic speed estimators to IM parameters mismatch.

The rest of paper is organized as follows: the general methodology of the Lyapunov stability theory application to deriving the equation of the estimator error function is discussed in detail in Section 2. Next the mathematical models of the induction motor and the above mentioned MRAS-type estimators are presented in Section 3. In Section 4 the steps of calculation of the error function of all analysed MRASs with PI controllers and adaptation mechanisms are briefly discussed and collated in a table. Next, Section 5 focuses on the stability problems of selected estimators near zero speed and in a regenerating operation

mode. In the last Section the sensitivity of MRAS estimators to the IM parameter changes is addressed. The last part is a short summary. 2. General method of the Lyapunov stability theory application for MRAS-type estimators MRAS-type estimators consist of three main parts: reference model, adaptive model for a selected state variable and a speed adaptation mechanism, as shown in Fig. 2. The reference model does not depend on the calculated speed value, contrary to the adaptive model. Speed is estimated based on the error function, calculated from the outputs of the two models, respectively. The speed is the output variable and it is used in the adaptive model through a closed loop, simultaneously. The Lyapunov stability theory [36] can be used to determine the value of the error function  which is introduced into the adaptation mechanism of the MRAS-type estimator. The procedure of the adaptation mechanism derivation consists of five steps. a) Writing the state equations of reference and adaptive models, respectively: for reference model:

TN

dx  A m  x  Bu , dt

(1a)

y  Cx , for the adaptive model:

TN

(1b)

dxˆ ˆ ˆ  Ge ,  A ˆ m  xˆ  Bu i dt

(2a)

yˆ  Cxˆ ,

(2b)

where u in equation (2) can include stator voltage and/or stator current, and matrix C form depends on the MRAS type and selected components of the state vector x. Matrix G   g s

g r  (gs=gsx+jgsy, T

gr=grx+jgry) is defined only in case of AFO (in other MRAS-type estimators it is equal to 0). Error of stator current estimation ei  i s  ˆi s . b) Calculating the differential equation for the state estimation error: After suitable calculations the state equation for the estimation error e  x  xˆ takes the following form: TN

where:

d e   A m   GC  e  ΔA  Δm  xˆ  ΔBu , dt

(3a)

ˆ ˆ  , ΔB  B  B ˆ, ΔA  Δm   A m   A m

(3b)

and Δ  m  ˆ m is the speed estimation error. c) Writing the Lyapunov function and its requirements The Lyapunov stability theory allows to derive the speed estimation algorithm to ensure that the state variable xˆ converges to real value x, and the estimation error e disappears to zero. Simultaneously, the calculated motor speed converges to its real value. However, a certain scalar Lyapunov function V(e,t) must be positively determined for the whole area of error e variation and the Lyapunov function derivative in regard to time must exist and must be negatively determined [36]. The function which can be used in case of all speed estimators analysed in this paper takes the following form [10, 11, 14, 15, 23]:

V (e, t )  eT e 

m  ˆ m 

2



,

(4)

where λ is a positive constant. d) Calculation of the derivative of the Lyapunov function: The derivative of the selected Lyapunov function is: d V (e, t )  eT dt

 A    GC m

T



  A m   GC  e   u T ΔB Te  eT ΔBu  

 xˆ T ΔA  Δm  e  eT ΔA  Δm  xˆ  T

2Δm

d ˆ m dt

(5)



where B=0, because it is assumed that the motor parameters are known and constant in time. In order to fulfil the condition of negative-semidefinite value of the Lyapunov function, the following inequalities should be granted:





eT  A m   GC    A m   GC  e  0 , T

xˆ T ΔA  Δm  e  eT ΔA  Δm  xˆ  T

2Δm

d ˆ m dt 0.



(6)

(7)

Condition (6) can be fulfilled by an appropriate choice of the matrix (A(m)-GC) eigenvalues – if the real parts of them are negative, the estimator is stable. Because the real value of matrix A(m) is unavailable, the ˆ ˆ  . From the second condition (7) the value of the error following assumption can be used: Am  ~ A m

function, which is introduced into the speed adaptation mechanism, can be determined.

e)

Calculation of the value of the error function for the speed adaptation loop

The value of the error function is calculated using the formula for the estimated value of the rotor speed obtained directly from (7) is: t

ˆ m  K I   dt  ˆ m 0 ,

(8)

0







1 T xˆ T A  m  e  eT A  m  xˆ , 2m

(9)

where KI = λ and ˆ m 0 is the initial speed value at t0=0. Usually, the PI controller is used to improve the dynamics of the speed estimate [10, 11, 14, 15, 23]: t

ˆ m  K I   dt  K P ,

(10)

0

where: KI, KP – gain coefficients of the PI controller. Equation (10) describes the adaptation mechanism of all MRAS-type estimators that are discussed in this paper. This algorithm ensures the asymptotic stability of the estimator. By modifying the controller gains the dynamics of the estimated speed can be adjusted, and consequently the dynamical properties of the adaptation process determined. 3. Mathematical models The following Section presents only mathematical models of induction motor and MRAS speed estimators, taken into consideration in this paper. Their comparison and detailed analysis of stability in regenerating mode and dependence on IM parameter mismatch is shown in Sections 5 and 6, respectively. 3.1. Mathematical model of the induction motor The mathematical model of the IM is presented below, taking into account the commonly used assumptions (lumped and constant parameters of the windings, symmetrical windings, neglected higher harmonics in the air gap, hysteresis and eddy-current). Moreover, it is written in normalized units (per unit system) with usage of generalized space vectors, in a stationary reference frame α-β (k = 0) [1-3]. All estimators compared in this paper are based on this model.

The state equation of the electromagnetic circuit of IM is as follows:

 r1  d  i s   l TN   dt  ψ r    rr kr 

kr  m  1 l  r l i   s    l  u s ,   1  ψ   jm   r   0  r  kr

j

(11)

where: kr = lm/lr, l = ls, r1 = rs + rrkr2, r = lr/rr. In case of MRAS-type speed estimators only electromagnetic circuit equations of IM are taken into account; the equation of motion is neglected. The superscripts (F, EMF, RP, AFO, CV, CC) and prefixes (P-, Q-, X-, Y-, SN-, FL-, ANN-) will determine the MRAS-type estimator analysed in the following sections. 3.2.Mathematical model of MRASF The first MRAS-type estimator which has been proposed is MRASF estimator [10, 11]. It consists of two well-known models of the rotor flux space vector: voltage simulator used as a reference model, and current simulator taken as an adaptive model. The equation of the voltage simulator is as follows:

TN

r l d d u 1 ψ r   s i s  TN  is  us . dt kr kr dt kr

(12)

The equation of the current simulator is as follows: TN

 i d i  1 ˆ r     jˆ m  ψ ˆ r  rr kr i s . ψ dt  r 

(13)

The value of the error function F is derived using above models (11), (12) and equation (9):

 F   r ˆ r  rˆ r 

(14)

The speed value is calculated by a PI controller (10). The block diagram of the above estimator is presented in Fig. 3. 3.3.Mathematical models of MRASEMF and MRASRP Due to the interest in the technique of adaptive systems in subsequent years, estimators using the electromotive force vector em (MRASEMF) [12] and reactive power qm (MRASRP) [12, 13] have been

proposed. The equations of the reference model and adaptive model of the first of the above mentioned estimators are as follows: d u d ψ r  u s  rs i s  l TN i s , dt dt

(15)

 d i 1 l  ˆ r  kr  jm ψ ˆr ψ ˆ r  m is  . ψ dt r r  

(16)

em  krTN

eˆ m  krTN

The block diagram of MRASEMF is presented in Fig. 4a. The speed adaptation mechanism has the same form (10) as the MRASF estimator (PI controller), but the error function is different and can be obtained by combining (15)-(16) and general equation (9). After suitable calculations it is based on the reference and the estimated values of the electromotive force vectors as follows:

 EMF  em eˆm  em eˆm .

(17)

The MRASRP estimator is an extended version of the MRASEMF estimator [12, 13]. The reactive power which is used for the calculation of the error function in MRASRP is obtained from the equations describing the electromotive force (15)-(16) and the measured stator current value: qm  i s  e m  is em  is em ,

(18)

qˆ m  i s  eˆ m  is eˆm  is eˆm .

(19)

The block diagram of MRASRP is presented in Fig. 4b. Obviously, the PI controller (10) composes the adaptation mechanism of MRASRP, however, in this case the error function is different and depends on the reactive power obtained from the reference and adaptive models, respectively:

 RP  qm  qˆm  i s   em  eˆ m  .

(20)

It can be proved (see Section IV), that MRASRP maintains the same stability performance as MRASEMF. 3.4.Mathematical models of AFO, MRASCV and MRASCC The next group of the MRAS-type estimators analysed in this paper are AFO (Fig. 5) [23], MRASCV (Fig. 6a) [14] and MRASCC (Fig. 6b) [15]. These three estimators can be included into one group because of two important common features, namely:  the induction motor is taken as a reference model,  the same formula of error function in the adaptation mechanism is obtained.

The state equation of AFO (Fig. 5) can be written in a similar way to the IM model and takes directly the form from equation (2) with u=us. State vector and state matrix are as follows:  r1  l  ˆi s  ˆ  xˆ    , A ˆ m     ˆ ψ  r  rr kr 

kr  ˆ m  l  r l   1   jˆ m  r  kr

j

(21)

According to (9) the estimator’s error function is calculated as follows:

 AFO  ˆ r  ei ˆ r ei .

(22)

The next estimator from the analysed group is the voltage speed estimator MRASCV [14]. This estimator consists of a stator current estimator and a voltage simulator (12). The equation of the stator current estimator is as follows: TN

 k  r k dˆ 1 ˆ r  us . i s   1 ˆi s   r  j r ˆ m  ψ dt l l  l l     r 

(23)

The error function value of the MRASCV estimator is the same as in AFO: CV = AFO (22). The last estimator from this group is the current speed estimator MRASCC [15]. This version of the estimator is similar to MRASCV and the main difference is the usage of the current simulator (13) instead of the voltage simulator (12). In this estimator the error function is the same as for AFO and MRASCV (CC = CV = AFO (22)) and the adaptation mechanism is also the same (PI controller). 3.5.Mathematical models of P-, Q-, X-, Y-MRAS The next group of MRAS-type speed estimators has been proposed in [16]. It consists of four estimators which are based on four different quantities: –

active power (P-MRAS),

–

reactive power (Q-MRAS),

–

virtual variable (X- and Y- MRAS).

The equations which describe the reference models are: a) Prm  usy* isy*  usx* isx* b) Qrm  usy* isx*  usy* isx*  Vrm c) X rm  usy* isx*  usy* isx* d ) Yrm  usy* isy*  usx* isx*

(24)

and the adaptive models:

a) Pam  usy* isy  usx* isx b) Qam  usy* isx  usy* isx  Vam , c) X am  usy* isx  usy* isx d ) Yam  usy* isy  usx* isx

(25)

where x-y indicate the rotor flux vector oriented synchronous frame. The superscript * refers to the values of stator voltage and stator current which are generated by PI controllers in the FOC structure (reference values). The values of stator current components without superscript * (in adaptive models) are calculated in the current feedback loops from measured stator currents (isA, isB, isC). The equation of error function  constitutes a simple subtraction between the reference model and the adaptive model and, obviously, it is not obtained from the Lyapunov theory:

 K  Vrm  Vam ,

(26)

where subscript K refers to one of the estimators from the analysed group (P, Q, X or Y). The PI controller is used in the speed adaptation loop, similarly to the previous MRAS estimators. The general block diagram of P-, Q-, X-,Y-MRASs is presented in Fig. 7. 3.6. Other MRAS-type estimators The last group of MRAS-type speed estimators consists of Sliding Mode-based MRAS (SM-MRAS) [17-22], Artificial Neural Network-based MRAS (ANN-MRAS) [26-30] and Fuzzy Logic basedMRASs (FL-MRAS) [31-33]. This group is based on early proposed models of MRAS-type speed estimators, which are modified using a sliding–mode or artificial intelligence concepts. The most popular idea is to replace the PI controller in the speed adaptation mechanism with another type of controller or to change the state variable models used in estimators (reference, adaptive or both) into neural models. The main idea of SM-MRAS is based on the application of a sliding-mode controller instead of a PI controller in the speed adaptation loop as it was done for classical MRASF [18] or MRASCC [19, 20]. Moreover, it is necessary to calculate the switching functions and the low-pass filter (LPF) must be applied, because of the oscillations in the estimated speed.

In case of ANN-MRASs the adaptive model, the reference model or both models of classical MRAS estimators are replaced by an adequate ANN-model. If only one model is replaced by ANN, the method is based on neural modelling. In such a case the differential equation for the state variable (e.g. stator or rotor flux) is modelled using a simple linear NN [26-29], and a Back-Propagation algorithm is used for NN model adaptation. This algorithm replaces the PI controller in the speed adaptation loop of the classical MRAS estimator. However, it is different when the IM is used as a reference model (as in AFO or MRASCC,CV), and the adaptive model is replaced by a Multilayer Perceptron network [30], then the neural identification method is used. In such a case it is necessary to select the NN structure and a training method. Such an ANN-MRAS also requires samples for training and testing. In the case of FL-MRAS [31-33] the PI controller in the speed adaptation loop is replaced by a fuzzy logic controller. It is worth noting that the most important issue in case of these estimators is the selection of the input vector and the application of appropriate inference rules. The stability analysis of this group will be ignored in this paper because these estimators are some modifications of classical MRAS estimators, and the application of a different controller in the speed adaptation loop (SM of FL controllers) or the usage of the ANN model in the MRAS structure is often applied to correct the stability properties or minimize the parameter sensitivity of the estimators. MRAS estimators from this group are more complicated than classical algorithmic MRAS estimators, especially when the comparison is made in terms of their computational burden. 4. Error functions of MRAS-type estimators All steps of the error function calculation for the analysed MRAS-type speed estimators (except MRASRP estimator) with the usage of the Lyapunov stability theory are presented in Table I. The analysis of each step of these calculations is made according to the methodology discussed in Section 2 and allows to obtain the appropriate value of the error function in the speed adaptation loop independently on the selected estimator. Moreover, the values of the error function validate the classification of the MRAS-type estimators into groups described in Section 3 (marked with red/blue/yellow/green colours in Fig. 1). As it was mentioned in Section 3, the MRASRP estimator is an extended version of the MRASEMF estimator. Because of this close relation between MRASRP and MRASEMF, the stability of MRASRP and

the validity of error function choice (20) can be explained by comparing the values of the error function of both estimators. Based on the values of EMF, RP and the vector graph from Fig. 8, the following relationships can be obtained [13, 35]:  EMF  eˆ m  em

 RP  is  em  eˆ m 

~ sin  m  ˆ m  ~ sin  m   sin ˆ m 

(27)

By linearizing the both sides of relationships (27), for very small  deviations and assuming that  m0  ˆ m0 the following relationships can be calculated:

Δ EMF ~  Δ  m  Δˆ m  cos   m0  ˆ m 0  Δ RP ~

Δ

 m0  ˆ m 0    Δ  m  Δˆ m 

m

cos   m0   Δˆ m cos  ˆ m0  

 m0  ˆ m 0    Δ  m  Δˆ m  cos   m0 

(28)

 Δ RP ~ Δ EMF

Equation (28) confirms that the dynamic performances of both estimators are similar. Moreover, as MRASEMF is asymptotically stable, which results from the Lyapunov theory, MRASRP estimator is also stable when the proposed adaptation algorithm is used.

ΔA

3.

5.

T



e ΔAxˆ

xˆ T ΔA Te 

e

2.

4.

xˆ

1.

variable

estimator

 0 ΔA    m

Δ r   T Δem  T

eˆm  T

 2 m eˆm em   2 m eˆm em

  m  0 

 Δem

eˆm

MRASEMF

 F   rˆ r  rˆ r  EMF  em eˆm  em eˆm

 2 mˆ r  r   2 mˆ r  r

Δ r

ˆ r ˆ r   T

MRASF

 0   ΔA  0  0  0

AFO

0 0 Δm

0

kr Δm l

0

0 0

0 

0

0 kr  m l 0 0

Δ r   T

 0  ΔA  0  0  0

Δ r

kr k ˆ r ei  2m r ˆ r  ei l l

kr  Δ m  l   0   Δm   0 

Δis

iˆs ˆ r ˆ r   T

MRASCV

 AFO   CC   CV  ˆ r ei ˆ r ei

2m

0 

0

Δis

iˆs 

MRASCC

TABLE I. ERROR FUNCTIONS OF THE ANALYSED MRAS-TYPE SPEED ESTIMATORS FOR INDUCTION MOTOR.

kr   m  l  0   0   0 

5. Stability problems of selected MRAS-type speed estimators The most important problems of MRAS-type speed estimators are: the stability in regenerating mode and low speed range operation. In this section these problems are discussed for all the analysed estimators and the solutions proposed in the literature are briefly discussed. 5.1.Stability of MRASF In the case of MRASF, apart from the need to use pure integration for flux calculation, the stability problem occurs in the low speed region in both: motoring and regenerating modes [37-40]. Many authors proposed some methods to improve the properties of the MRASF speed estimator [39-44]. In [39, 41, 42] the authors proposed an additional component in the adaptation mechanism which depends on electromagnetic torque (reference and estimated values). The additional component – electromagnetic torque error – is filtered by a LPF and this value is added to the calculated speed which is obtained from the same PI controller as in the classical MRASF. In this solution the adaptation law is changed and the stability around the zero speed is improved, in particular, the greater accuracy of the estimation is obtained. In [40, 43] the measured voltage is replaced by its reference value obtained from the outputs of current PI controllers in the DFOC structure. This approach improves the stability of classical MRASF. The authors claim that the proposed solution is stable in the entire operating range, but they prove it only in simulation results. Another advantage is that voltage sensors are not required in this solution. In [44] the authors proposed two additional estimators for stator resistance and rotor time constant which cooperate with reference and adaptive models of the speed estimator. This solution improves the dynamic performance in high (flux weakening region) and low speed, too. 5.2.Stability of MRASEMF, MRASRP Although MRASEMF has better properties than MRASF (pure integration is not necessary), it is also unstable in the motoring and regenerating mode in the low speed region [45-47]. Unfortunately, this estimator requires a differentiation operation which is noise sensitive. In addition, the adaptation gain constants are very difficult to select [45], because these values have a non-linear influence on the estimated speed. In [46, 48, 49] the authors proposed a new adaptation law for MRASEMF. Because the new adaptation law did not provide the stability in the regenerating mode, they proposed an additional

component with an auxiliary variable, which improved the stability properties. The auxiliary variable in the adaptation mechanism is calculated from the equation which depends on the rotor speed value. MRASEMF with this solution is stable in four quadrants of the drive system operation. Because MRASRP requires the differentiation operation and is unstable in almost all operation points in the regenerating mode, as is shown in [48], the authors have started to improve this estimator. In [68] the authors disposed of the differentiation operation by forming MRASRP in a synchronous reference frame, but the stability was not improved. In another work [47] they improved the stability properties (the first version of Q-MRAS), nevertheless this MRASRP was still unstable in the regenerating mode – in a smaller range than its previous version. Moreover, this version involves the full machine model and is dependent on all parameters of IM. Therefore, the authors proposed a new MRAS based on MRASRP – the first version of X-MRAS [69]. This solution is stable in the regenerating mode; however, it is sensitive to stator resistance. 5.3.Stability of AFO, MRASCC and MRASCV AFO is the most known speed estimator analysed in the literature [23, 50-64]. Unfortunately, this estimator is not stable in whole speed-torque operation range; some unstable points occur in the regenerating mode for low speed [62]. For the improvement of the AFO stability, two main methods can be distinguished: – the first method, which is the most widespread method, is based on appropriate selection of the coefficients of a gain matrix of the state observer [50-60], –

the second method, which is proposed and discussed in [54, 55, 58, 59, 61-63], consists in the modification of the speed adaptation algorithm by a shift angle between the stator current error and the estimated rotor flux vector [54, 55, 61-63] or by adding an additional component which is based on a rotor flux error [58, 59].

In the opinion of authors of this paper, the second method is more convenient, as it requires the selection of one parameter only (value of the shift angle), not four or five (four gain coefficients and an additional variable k) as it is required in the other methods. However, this stabilisation method requires changing the correction parameter (shift angle) under transition from regenerating to motoring mode

and vice versa, as some new unstable operating points occur [61-63]. Nevertheless, this problem can be easily avoided by switching off the correction factor during motoring operation. In recent years yet another solution has been proposed where both of the mentioned methods are used simultaneously [58, 59], however it is more complicated than above described methods. In [64] a new method with an additional variable in the AFO model is proposed; this variable is obtained from an additional PI controller and it is introduced in the speed estimator model. Obviously, in all the analysed methods their authors obtained the enhancement of the stability region of the AFO estimator. The stability problem of the MRASCV estimator is analysed only in one paper [63]. The authors presented the stability analysis of this estimator in detail. It is shown that MRAS CV is stable in the full operating range, except the unobservability line defined by s=0. However, as it uses the voltage simulator of the rotor flux (12), the problem of pure integration exists and a LPF has to be used [14, 15]. The last estimator from the MRASs in the blue group in Fig.1 is MRASCC speed estimator. It is also unstable in regenerating mode as AFO, but in a much wider range [62, 63, 65-67]. The literature is not extensive for this MRAS, there are only few works where authors proposed some solutions for stability improvement. Similarly to AFO, two methods were proposed: appropriate selection of the coefficients of additional gain matrix introduced to the estimator structure in the same way as the gain matrix in AFO and the modification of the speed adaptation algorithm, using the shift angle between the stator current error and the estimated rotor flux vector [62, 63, 65, 66]. When the drive system changes its operating mode from regenerating to motoring, the correction factor (shift angle) in the speed adaptation mechanism or the additional gain matrix should be switched off [62, 63, 66]. Moreover, recently a new solution with an additional variable obtained from an additional PI controller and introduced into adaptive model has been proposed in [67]. All these methods improve the stability properties of MRASCC. However, the last method seems to be the most advantageous, because it allows to obtain stable operation of the MRASCC estimator in the whole operating range (except the line defined by

s=0) by introducing only one additional variable and without any changes in the estimator structure under transition from the regenerating to motoring modes.

5.4.Stability of P-, Q-, X-, Y-MRAS Based on the experience with MRASRP estimators, the authors proposed a new series of four MRAS-type speed estimators: P-, Q-, X-, Y-MRASs [16]. All these estimator models were designed in synchronously rotating reference frames. Having analysed them, it is worth noting that P- and Y-MRAS are unstable in both: motoring and regenerating mode, in contrary to Q- and X-MRAS, which are stable in the full operating range. In [70] the authors proposed new versions of these four MRAS estimators. The reference model was replaced by the same model, written in a stationary reference frame. This approach improved the properties of two of these estimators significantly. However, it is very important to note that X- and YMRAS are not reference frame independent, so both the reference and adjustable models should operate in the same reference frame. Hence, X- and Y-MRAS from [70] cannot be used to estimate the motor speed. As a result, Q-MRAS seems to be the best solution for speed estimation in sensorless systems, because it is stable in four quadrant operation and is reference frame independent. Table II presents the most significant properties and the information about the stability of all the analysed estimators according to the above discussion, it also addresses the methods of stability improvement.

MRASF

Unstable practically in full range of the regenerating mode [48, 70]

Unstable in small range in regenerating mode [51]

Stable in full operating range [63]

MRASRP

AFO

MRASCV

MRAS

EMF

Unstable in motoring and regenerating mode in low speed region [37-40]

Unstable in motoring and regenerating mode in low speed region [45, 46, 70]

Stability

Estimator

isx

 r isy

if r  0; s r  0

isx if r  0  r isy

rs 

r

l

rs  rr kr2

if m  0

- pure integration – LPF required [14, 15]

s  0 or s  m

r

- stability boundaries in steady-state [62]: rs s  0 or s  m if m  0 l rs    rr kr2

The above two curves and the zero stator frequency line divide the stable and unstable region.

m  ~ 

m  ~ 

- requires differentiation operation [13] - stability boundaries in steady-state [48]:

- requires differentiation operation [13, 70] - this MRAS is complicated in terms of its design because of the non-linear influence of the adaptation gain constants on speed estimation [45]

- pure integration  LPF required [12, 70]

Additional important information

Stability improvement methods

not necessary

- modification in speed adaptation law:  usage of shift angle between rotor flux and stator current estimation error [54, 55, 61, 62, 63]  additional component in adaptation law which is based on a rotor flux error [58, 59] - appropriate selection of gain matrix coefficients of the state observer [50-60] - application of both of the above proposed methods [58, 59] - additional auxiliary variable obtained from the adaptation mechanism with a PI controller and introduced in AFO [64]

- forming the estimator in a synchronous reference frame (the estimator is free from differentiation operation, but it is still unstable) [68] - changing the estimator into the first version of Q-MRAS, which is, however, unstable partially in the regenerating mode [47] - changing the estimator into the first version of X-MRAS, which is stable, but sensitive to stator resistance changes [69]

- new adaptation law and additional component with an auxiliary variable which improve the stability [46, 48, 49]; stator resistance estimator was proposed simultaneously [48]

- additional component in adaptation mechanism which depends on electromagnetic torque [39, 41, 42] - replacing a measured stator voltage by its reference value obtained from the outputs of PI controllers in DFOC structure [40, 43] - usage of two additional estimators of stator resistance and rotor time constant [44]

TABLE II. STABILITY OF MRAS-TYPE ESTIMATORS

Partially unstable (in motoring and regenerating mode) [16]

Stable in full operating range [16]

Stable in full operating range [16]

Partially unstable (in motoring and regenerating mode) [16]

MRASCC

P-MRAS

Q-MRAS

X-MRAS

Y-MRAS

Stability

Unstable almost in a full range of regenerating mode [62, 65-67]

Estimator

r

rs  rr kr2 if m  0 l rs    rr kr2

- if reference model operates in stationary reference frame and adjustable model operate in synchronous reference frame this estimator works incorrectly [70] estimator is reference frame dependent;

- if reference model and adjustable model operate in synchronous reference frame this estimator works correctly [70] - estimator is reference frame independent; - if reference model operates in stationary reference frame and adjustable model operates in synchronous reference frame this estimator works incorrectly [70] estimator is reference frame dependent;

- if reference model and adjustable model operate in synchronous reference frame this estimator works correctly [70] - estimator is reference frame independent;

s  0 or s  m

- stability boundaries in steady-state [62]: rs  rr kr2 s  0 or s  m if m  0 l rs    rr kr2 r

Additional important information

no stability improvement methods in literature

not necessary

not necessary

no stability improvement methods in literature

- modification in speed adaptation law – shift angle between rotor flux and stator current estimation error [62, 63, 65, 66] - appropriate selection of the additional gain matrix coefficients of the estimator [62, 65, 66] - additional auxiliary variable obtained from the adaptation mechanism with a PI controller and introduced in the reference and the adaptive model [67]

Stability improvement methods

6. Parameter sensitivity of the MRAS-type estimators The second problem of MRAS-type estimators is the influence of the IM parameter mismatch on the speed estimation accuracy and the stable operation of the estimator. This problem appears in every proposed solution, because MRAS-type speed estimators are based on the mathematical model of the IM. 6.1. Influence of parameter mismatch on MRASF In [15] the influence of four parameters mismatch (stator and rotor resistance and stator and rotor inductance) on the four estimators: MRASF, MRASCV, AFO and MRASCC, in steady-states with three different speed values and nominal load torque in the motoring mode, is discussed. According to the authors, MRASF has the worst properties, especially due to its sensitivity to stator resistance changes. If stator resistance changes its value of +/-20% or more of its nominal value, MRASF estimator loses the stability. Both inductances (stator and rotor) influence this estimator in a similar way - when they are higher than the nominal values, MRASF loses stability (when the reference speed is bigger, the estimator loses its stability earlier). If the inductances are lower than their nominal values, the estimator works in a stable manner, but the constant speed error appears. The advantage of MRASF is its robustness over rotor resistance mismatch. Because of the above mentioned properties, the authors of [39-43] proposed some improvement methods to solve the stability problems of MRASF (as it was described in the previous section) and proved by simulation analysis that the dependence on stator resistance mismatch became marginal, when compared to the classical version. In [44] the authors proposed the estimators of two IM parameters (stator resistance and rotor time constant) not only for stability improvement, but also to increase the robustness. The stator resistance estimator is proposed especially to improve rotor flux estimation in the voltage. The rotor time constant estimator is proffered mainly for a better calculation of slip angular velocity in IFOC (i.e. for the better operation of the whole system) and for improving the estimation of fluxes in the current simulator. 6.2.Influence of parameter mismatch on MRASEMF and MRASRP In the case of MRASEMF [12] the authors assumed theoretically that the stator resistance thermal variation affects the stability of this estimator, especially in the low speed region. In [48] apart from the stability improvement method, the authors proposed also an additional stator resistance estimator, because the

influence of this parameter on the MRASEMF stability is essential. The estimator should work simultaneously with the discussed MRASEMF [48]. Nevertheless, the usage of this estimator meant that the auxiliary variable in the new adaptation mechanism should be calculated from one of four different equations. The choice of the correct equation depends on the operation point. Hence, the stability improvement method [48] is more complicated and computationally complex with additional rs estimator. Contrary to MRASEMF, its improved version - MRASRP [12, 13] – is completely robust to stator resistance changes. As discussed in [70], all estimators from the whole group, from classical MRASRP to the first versions of Q- and X-MRAS [13, 45, 68, 69], are sensitive to the IM parameters, especially to stator resistance changes. Unfortunately, both estimators in this group (MRASEMF, MRASRP) were not analysed from the point of view of the influence of other parameters mismatch on their stability. 6.3. Influence of parameter mismatch on AFO, MRASCV and MRASCC In the case of MRASCV the influence of the stator resistance is very high, because even a small variation (about 10%) can lead to the unstable work of this estimator, especially for high speeds [15]. The increase in the rotor inductance (more than +20% of its nominal value) can result in the unstable work of MRASCV as well. If this inductance decreases or if stator inductance varies, this estimator works in a stable way, but the constant speed error can occur. However, MRASCV is robust to rotor resistance mismatch. In the case of AFO, many authors analysed the influence of the parameter mismatch [15, 51, 52, 56, 5860, 71-73]. In [15] the authors presented the influence of four IM parameters mismatch. Similarly, the rotor resistance has no negative impact on the stability of AFO. When the reference speed has the nominal value, the increase in the remaining parameters leads to unstable work. In the case of stator resistance, it happens also for a smaller speed value. In all other conditions analysed in [15], for all four parameters, AFO works stably, but in the case of some parameter changes, the constant speed error appears. In [52] the sensitivity of the main parameters of the AFO estimator with a new gain matrix is discussed. It is shown that the most serious influence on speed estimation accuracy (and on the estimator stability) is still exerted by stator resistance. The other analysed parameters (rotor resistance, main inductance, stator leakage inductance) do not cause the instability problem of the proposed improvement method. The authors only mentioned the need for the additional estimation of rs in the

proposed AFO with new gain matrix. In [58, 59] the authors proposed a new solution and analysed the influence of stator and rotor resistance. They showed that the speed estimation error is much smaller than in case of the modified AFO [52], but it is still noticeable in the low speed region (especially in the regenerating mode). In [72] the authors analysed only the influence of all parameters on the stability of classical AFO [23] and the modified AFO proposed in [52]. They concluded that the classical AFO (with zero feedback gains) has better robustness to motor parameter changes than the one proposed in [52] in the high speed region. The authors of [72] also proved that the method proposed in [52] can lead to a static speed error or even to instability in the low speed region, when the parameters are mismatched. In [51] the sensitivity of the estimator model to three main parameters mismatch (stator and rotor resistances, main inductance) is analysed and the gain which depends on the actual operation point and the appropriate tuning of the parameters is proposed. The author of the method proposed in [56] presented the influence of stator resistance on AFO stability. The estimator of rs is proposed there and it is successfully used in the whole estimator structure. This extended speed and stator resistance estimator improves the accuracy and robustness of the discussed method, especially in the low speed region. The papers [60] and [71] presented methods with two additional PI estimators – one for stator resistance estimation and the other one for rotor resistance estimation. This solution minimises the influence of parameter mismatch and improves the speed estimation stability of the AFO speed estimator. The authors of [73] analysed the parameter sensitivity of the method proposed in [62]. They analysed the influence of stator and rotor resistance and the main inductance. The authors concluded that the estimated values of stator resistance and the main inductance estimators are needed only in the regenerating mode in a low speed region and high torque. In the other operating points the parameter estimators are not necessary. MRASCC is more robust to parameter mismatch than AFO, MRASCV and MRASF. This estimator works stably for quite large changes of the four motor parameters (±50% of their nominal values): stator and rotor resistance and stator and rotor inductance, but in some conditions a small static speed estimation error appears [15]. Unfortunately, in some operating points, when the stability improvement methods in the regenerating mode are applied [62], the main inductance and stator resistance estimators

are necessary, as it is presented in [74]. In [74] the authors showed that MRASCC with a modification in the speed adaptation law becomes unstable when stator resistance or main inductance mismatch occur, the reference speed is small and the estimator works in the regenerating mode. 6.4. Influence of parameter mismatch on P-, Q-, X- ,Y-MRAS The P-, Q-, X-, Y-MRAS estimators (the green group of estimators in Fig.1) are dependent on rotor parameters (rotor resistance and rotor inductance), because the reference model depends on the position angle of the coordinate system, which is obtained, among others, from the calculation of the slip angular velocity that depends on these parameters significantly [16]. The most important advantage of this group of estimators is robustness to stator resistance variations. As it is discussed in this Section and in many cited articles, the parameter sensitivity of all MRAS-type speed estimators is very important and should be analysed in every proposed method. The application of additional estimators for the IM parameters estimation complicates the MRAS-type speed estimator algorithms and requires stability analysis of the whole extended system [75]. In a vast majority of presented papers this analysis is undone or simplified. 7. Conclusions In this paper all analysed MRAS-type speed estimators were presented in detail. As the literature on the IM speed reconstruction based on MRAS techniques is very extensive and dispersed, the contribution of this paper is the discussion and comparison of these techniques by analysis of their features and properties from few different points of view. The application of the Lyapunov stability theory is discussed as it allows to determine the general methodology for the speed adaptation mechanism and error function calculation. For every MRAS-type estimator which is discussed in this paper, all steps for obtaining the error functions are presented. The whole analysis is summarized in the form of a table in order to easily compare the results obtained for all considered estimators. The form of the error function  can be one of the basis for dividing MRAS-type estimators into groups (as it is done in Fig. 1). MRAS-type speed estimators are the most widespread group in the sensorless control of the induction motor drive. Taking this into consideration, two important disadvantages of MRAS-type speed estimators, namely the stability in the regenerating operation mode and IM parameter sensitivity, are

discussed in this paper as well. Many papers related to these drawbacks and their improvement are briefly discussed. The summary of the stability problems of MRAS estimators, especially in the regenerating mode, is presented in the form of a table, which – in the opinion of authors – helps to gain an overview of all solutions presented in the literature. It results from this summary that from all the proposed solutions only MRAS CV and Q-MRAS are stable in the whole operating range of the speed-load torque changes. For AFO and MRASCC three methods for stability improvement are proposed in literature, however, the last one with an auxiliary variable introduction [64, 67] seems to be more convenient for practical use. However, all proposed MRAS-type estimators are to some extent dependent on IM parameter changes, which affects the speed estimation accuracy as well as the estimator stability. The effective stator resistance and magnetizing inductance estimators should be applied for the most existing MRASs, particularly for low speed and very high speed regions, respectively. Although there are numerous methods of parameter estimation, the stability of most of them is not verified theoretically. Hence, there is still open space for further research as the speed estimation, especially in the low and zero speed region, in the regenerating mode as well as in a field-weakening region, with changes of IM motor parameters taken into account still constitutes a hot scientific and technical problem [21, 22]. References [1] Vas P, Sensorless Vector and Direct Torque Control. Oxford University Press New York USA 1998. [2] Kazmierkowski M. P. Blaabjerg F. Krishnan R. Control in Power Electronic – Selected Problems. Academic Press New York USA 2002. [3] Orlowska-Kowalska T. Sensorless Induction Motor Drives. Wroclaw University of Technology Press, Poland, 2003. [4] Holtz J. Sensorless Control of Induction Machines–With or Without Signal Injection? IEEE Trans Ind Electron 2006;53:7–30. [5] Finch J. W. Giaouris D. Controlled AC Electrical Drives. IEEE Trans Ind Electron 2008;55:481–91. [6] Pacas M. Sensorless Drives in Industrial Applications. IEEE Ind Electron Mag 2011;5:16-23. [7] Orlowska-Kowalska T. Application of extended Luenberger observer for flux and rotor time constant estimation in induction motor drives. IEE Proc Contr Theory Appl-D 1989;136:324-30. [8] Barut M. Bogosyan S. Gokasan M. Speed-Sensorless Estimation for Induction Motors Using Extended Kalman Filters. IEEE Trans Ind Electron 2007;54: 272-80. [9] Zerdali E. Barut M. Extended Kalman Filter Based Speed-Sensorless Load Torque and Inertia Estimations with Observability Analysis for Induction Motors. Power Electronics and Drives 2018;3:115-27. [10] Tamai S. Sugimoto H. Yano M. Speed sensorless vector control of induction motor with model reference adaptive system. In: Proceedings of the IEEE 1987 Industry Applications Society Annual Meeting; 1987. p.189-95. [11]Schauder C. Adaptive speed identification for vector control of induction motors without rotational transducers. IEEE Trans Ind Appl 1992;28:1054-61.

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List of figure labels

Revision_ISATRANS-D-18-01350_Labels for Figures Fig. 1. The general classification of the speed estimation methods and detailed division of MRAS-type estimators. Fig. 2. The general bock scheme of MRAS- type estimators. Fig. 3. The block diagram of MRASF [10, 11]. Fig. 4. Block diagrams of: a) MRASEMF [12], b) MRASRP [13] Fig. 5. The block diagram of AFO [23]. Fig. 6. Block diagram of: a) MRASCV [14], b) MRASCC [15]. Fig. 7. Block diagram of Q-,P-,X-,Y-MRAS [16]. Fig. 8. Vectors graph in α- coordinates for MRASEMF and MRASRP estimators.

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*Conflict of Interest

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Teresa Orlowska-Kowalska (corresponding author) Mateusz Korzonek Grzegorz Tarchala