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Comparative Study of Sensorless Control Methods of PMSM
Drives
Arafa S. Mohamed, Mohamed S. Zaky, Ashraf S. Zein El Din and Hussain A. Yasin
Electrical Engineering Dept., Faculty of Engineering, Minoufiya University,
Shebin El-Kom (32511), Egypt
E-mail: arafamnsr@yahoo.com

Abstract
Recently, permanent magnet synchronous motors (PMSMs) are increasingly used in high performance
variable speed drives of many industrial applications. This is because the PMSM has many features, like
high efficiency, compactness, high torque to inertia ratio, rapid dynamic response, simple modeling and
control, and maintenance-free operation. In most applications, the presence of such a position sensor
presents several disadvantages, such as reduced reliability, susceptibility to noise, additional cost and
weight and increased complexity of the drive system. For these reasons, the development of alternative
indirect methods for speed and position control becomes an important research topic. Many advantages of
sensorless control such as reduced hardware complexity, low cost, reduced size, cable elimination,
increased noise immunity, increased reliability and decreased maintenance. The key problem in sensorless
vector control of ac drives is the accurate dynamic estimation of the stator flux vector over a wide speed
range using only terminal variables (currents and voltages). The difficulty comprises state estimation at
very low speeds where the fundamental excitation is low and the observer performance tends to be poor.
The reasons are the observer sensitivity to model parameter variations, unmodeled nonlinearities and
disturbances, limited accuracy of acquisition signals, drifts, and dc offsets. Poor speed estimation at low
speed is attributed to data acquisition errors, voltage distortion due the PWM inverter and stator resistance
drop which degrading the performance of sensorless drive. Moreover, the noises of system and
measurements are considered other main problems. This paper presents a comprehensive study of the
different methods of speed and position estimations for sensorless PMSM drives. A deep insight of the
advantages and disadvantages of each method is investigated. Furthermore, the difficulties faced sensorless
PMSM drives at low speeds as well as the reasons are highly demonstrated.
Keywords: permanent magnet, synchronous motor, sensorless control, speed estimation, position
estimation, parameter adaptation.
1. Introduction
Permanent magnet synchronous motor (PMSM) drives are replacing classic dc and induction motors drives
in a variety of industrial applications, such as industrial robots and machine tools [1-3]. Advantages of
PMSMs include high efficiency, compactness, high torque to inertia ratio, rapid dynamic response, and
simple modeling and control [4]. Because of these advantages, PMSMSs are indeed excellent for use in
high-performance servo drives where a fast and accurate torque response is required [5, 6]. Permanent
magnet machines can be divided in two categories which are based on the assembly of the permanent
magnets. The permanent magnets can be mounted on the surface of the rotor (surface permanent magnet
synchronous motor - SPMSM) or inside of the rotor (interior permanent magnet synchronous motor -
IPMSM). These two configurations have an influence on the shape of the back electromotive force (back-
EMF) and on the inductance variation. In general, there are two main techniques for the instantaneous
torque control of high-performance variable speed drives: field oriented control (FOC) and direct torque
control (DTC) [7, 8]. They have been invented respectively in the 70s and in the 80s. These control
strategies are different on the operation principle but their objectives are the same. They aim both to control
effectively the motor torque and flux in order to force the motor to accurately track the command trajectory
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regardless of the machine and load parameter variation or any extraneous disturbances. The main
advantages of DTC are: the absence of coordinate transformations, the absence of a separate voltage
modulation block and of a voltage decoupling circuit and a reduced number of controllers. However, on the
other hand, this solution requires knowledge of the stator flux, electromagnetic torque, angular speed and
position of the rotor [9]. Both control strategies have been successfully implemented in industrial products.
The main drawback of a PMSM is the position sensor. The use of such direct speed/position sensors
implies additional electronics, extra wiring, extra space, frequent maintenance and careful mounting which
detracts from the inherent robustness and reliability of the drive. For these reasons, the development of
alternative indirect methods becomes an important research topic [10, 11]. PMSM drive research has been
concentrated on the elimination of the mechanical sensors at the motor shaft (encoder, resolver, Hall-effect
sensor, etc.) without deteriorating the dynamic performances of the drive. Many advantages of sensorless
ac drives such as reduced hardware complexity, low cost, reduced size, cable elimination, increased noise
immunity, increased reliability and decreased maintenance. Speed sensorless motor drives are also
preferred in hostile environments, and high speed applications [12, 13].
The main objective of this paper is to present a comparative study of the different speed estimation methods
of sensorless PMSM drives with emphasizing of the advantages and disadvantages of each method.
Furthermore, the problems of sensorless PMSM drives at low speeds are demonstrated.
2. PMSM Model
The PMSM model can be derived by taken the following assumptions into consideration:
- The induced EMF is sinusoidal
- Eddy currents and hysteresis losses are negligible
- There is no cage on the rotor
The voltage and flux equations for a PMSM in the rotor reference (d-q) frame can be expressed as [8]:
ds
ds s ds qs
d
R i
dt

v e = + (1)
qs
qs s qs ds
d
R i
dt

v e = + + (2)
ds d ds r
L i = + (3)
qs q qs
L i =

(4)

The torque equation can be described as:
3
[ ( ) ]
2
e r qs q d ds qs
T P i L L i i = (5)
The equation for the motor dynamic can be expressed as:
1
( )
r
e L r
d
T T F
dt J
e
e = (6)
where the angular frequency is related to the rotor speed as follows:
r
d
P
dt
u
e e = = (7)
where P is the number of pole pairs,
s
R , is the stator winding resistance, e is the angular frequency,
ds
v
,
qs
v
, and
ds
i ,
qs
i are d-q components of the stator winding current and voltage,
ds
and
qs
are d-q
components of the stator flux linkage,
d
L and
q
L are d and q axis inductances, and
r
is the rotor flux
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linkage. F is the friction coefficient relating to the rotor speed; J is the moment of inertia of the rotor; u is
the electrical angular position of the rotor; and
e
T and
L
T are the electrical and load torques of the
PMSM.


3. Speed Estimation Schemes of Sensorless PMSM Drives
Several speed and position estimation algorithms of PMSM drives have been proposed [14]. These methods
can be classified into three main categories. The first category is based on fundamental excitations methods
which are divided into two main groups; non-adaptive or adaptive methods. The second category is based
on saliency and signal injection methods. The third done is based on artificial intelligence methods. These
methods of speed and position estimation can be demonstrated in Figure 1.

Sensorless Control of PMSM
Fundamental
excitations methods
Saliency & signal
injection methods
Artificial intelligence
methods
Adaptive methods
Non-adaptive
methods

Figure 1. Speed estimation schemes of sensorless PMSM drives.
3.1. Fundamental Excitations Methods
3.1.1 Non-adaptive Methods
Non-adaptive methods use measured currents and voltages as well as fundamental machine equations of the
PMSM. The characteristic of this method is easy to be computed, responded quickly and almost no delay.
But it required high accurate Motor parameters, more suitable for Motor parameters online identification
[2].
A. Estimators using monitored stator voltages, or currents
For estimating the rotor angle u using measured stator voltages and currents different authors follow
different approaches regarding to the used reference frame. This section shows examples for the
perspective in three axis and - coordinates.
[15, 16] propose the approach with the three axis model. Firstly transformation from the d-q coordinates to
the - coordinates have to be done. The transformation matrix as follows:
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cos sin
sin cos
dq
T
o|
u u
u u

(
=
(


(8)
The transformation matrix from the - coordinates to the three axis model is illustrated in equation:
23
1
1
2 2
3 3
2 2
2 2 2
T
1 (

(
(
(
= 0
(
(
1 1 1
(

(

(9)
The final equation for the rotor position angle can be found as:
1
tan ( )
r
A
B
u

= (10)
Where
( ) ( )
3 ( )
bs cs s bs cs d bs cs
r q d as
A R i i L p i i
L L i
v v
e
=

(11)
3( )
( )( )
as s as d as
r q d bs cs
B R i L pi
L L i i
v
e
=
+
(12)
The position of rotor can thus be obtained in terms of machine voltages and currents in the stator frame
provided
r
e in equation can be evaluated in terms of voltages and currents.
[17] Proposes a voltage model and current model based control which works in the - reference frame
with the assumption that
d q
L L L = = . The required speed
r
e can be calculated as follows:
r
C
D
e = (13)
2
2
( )
1
( ( ) ( ))
3
as s as s as
cs s cs s cs bs bs bs
C R i L pi
R i i L p i i
v
v v
=
+

(14)
r
D = (15)
The initial position of the rotor at t=0 can be determined by substitution
r
e =0 in above equation:
1
tan ( )
ro
E
F
u

= (16)
Where:
1
( ( ) ( ))
3
bs cs s bs cs d bs cs
E R i i L p i i v v = (17)
as s as d as
F R i L pi v = (18)
The currents are detected by a current sensor and the voltages are obtained by calculation which using
information on PWM pattern, dc voltage and dead time.
The calculation is direct and easy with a very quick dynamic response, and no complicated observer is
needed. However, the stator current deviation used in above equations will introduce calculation error due
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to measurement noise. And any uncertainty of motor parameters will cause trouble to the motor position
estimation, which is the biggest problem of this method [18, 19].
B. Flux based position estimators
In this method, the flux linkage is estimated from measured voltages and currents and then the position is
predicted by use of polynomial curve fitting [20, 21]. The fundamental idea is to take the voltage equation of the
machine,
d
V Ri
dt

= + (19)
0
( )
t
V Ri dt =
}
(20)
Where, V is the input voltage, i is the current, R is the resistance, and is the flux linkage, respectively. Based
on the initial position, machine parameters, and relationship between the flux linkage and rotor position, the rotor
position can be estimated. At the very beginning of the integration the initial flux linkage has to be known
precisely to estimate the next step flux linkages. This means that the rotor has to be at a known position at the
start [14, 16, and 20]. Last equation (20) written in - coordinates depends on the terminal voltage and the
stator current. Using the - frame the equation for the rotor angle can be written as follows [14, 22]:
1
tan ( )
s s
s s
Li
Li
o o
| |

(21)
where L is the winding inductance.
The actual rotor angle using the d-q frame can be calculated with [14, 22]:
1
tan ( )
ds
qs

= (22)
This method also has an error accumulation problem for integration at low speeds. The method involves lots of
computation and is sensitive to the parameter variation. An expensive floating-point processor would be required
to handle the complex algorithm [20].
Because of the noise, in the last decade a pure investigation of the rotor position has gained less attention.
Solutions with adaptive or observer methods are more common [23, 24].
C. Position estimators based on back-EMF
In PM machines, the movement of magnets relative to the armature winding causes a motional EMF. The
EMF is a function of rotor position relative to winding, information about position is contained in the EMF
waveform [14].
Paper [25] propagates the determination of the back EMF without the aid of voltage probes which reduces
the cost of the system and improves its reliability. Instead of the measured voltages reference voltages are
used. The back EMF is not calculated by the integration of the total flux linkage of the stator phase circuits
because of the integrator drift problem. The estimation of the rotor position is given by the difference of the
arguments of the back EMF in the - reference frame and the arguments of the same one in the rotating d-
q frame:
1 1
tan ( ) tan ( )
s
r
s q qs
e
e L i
|
o

u

= (23)
Previous equation shows, that there is only a quadrature current dependency of the rotor position.
Furthermore the back-EMF in subject to the reference voltage in - coordinates can be described as
following function:
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1
( , )
tan ( )
s s
s s s
s s s
R i
R i
o |
| |
v v
o o
v
v

(24)
The relation between the actual and reference voltages may be written in the form in which the variations
s o
ov and
s |
ov are due to the phase difference.
*
s s s o o o
v v ov = + (25)
*
s s s | | |
v v ov = + (26)
Substituting from previous equations into equation (24), one gets:
* *
( , )
( , )
*
1
*
tan ( )
s s
s s
s s
s s
s s s
r
s s s
R i
V
T
E
R i
o |
o |
v v o |
v v
o |
| |
o o
ov ov
v v
v
e
v

c c
= + +
c c

=
(27)
The second term in this equation is the dependent of the back EMF on the rotor speed. V, E and T are the
rms values of the stator voltages, the back EMF and the lag time introduced by the inverter respectively.
Thus, we get for the estimated position
*
u :
*
*
1 1
*
tan ( ) tan ( )
s s s
r
r
q qs
s s s
R i
V
T
E L i
R i
| |
o o
v

u e
v

(28)
The proposed algorithm in [25] appears to be robust against parameter variation. Furthermore the electrical
drive has a good dynamic performance.
Many control methods suitable for SPMSM cannot be used directly to IPM. In the mathematical model of
the IPM, position information is included not only in the flux or EMF term but also in the changing
inductance because of its saliency. The model of SPMSM is a special symmetrical case of IPM, which is
relatively easy for mathematical procession. In order to apply the method suitable for SPM to a wide class
of motors, i.e. the IPM, in [26-28] a novel IPM models are suggested with an extended EMF (EEMF).
By rewriting motor voltage equations into a matrix form:
0
s d r q ds ds
qs qs r d s q
r r
R pL L i
i L R pL
e v
v e
e
+ ( ( (
( ( (
+
(

(
+
(

-
=



(29)
There are two trigonometric functions of 2, which result from changing stator inductance. A reason why
2 terms appear can be concluded as that the impedance matrix is asymmetrical. If the impedance matrix is
rewritten symmetrically as:
0
( )( )
s d r q ds ds
qs qs r q s d
d q r ds qs r r
R pL L i
i L R pL
L L i pi
e v
v e
e e
+ ( ( (
( ( (
+
(

(
+
(
+

-
=



(30)
The circuit equation on - coordinate can be derived as follows, in which there is no 2 term.
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( )
( )
sin
{( )( ) }
cos
s d r d q s s
s s r d q s d
d q r ds qs r r
R pL L L i
i L L R pL
L L i pi
o o
| |
e v
v e
u
e e
u
+
+

+ +
( ( (
( ( (

(
(


=
-

(31)
The second term on the right side of (31) is defined as the extended EMF (EEMF). In this term, besides the
traditionally defined EMF generated by permanent magnet, there is a kind of voltage related to saliency of
IPMSM. It includes position information from both the EMF and the stator inductance. If the EEMF can be
estimated, the position of magnet can be obtained from its phase just like EMF in SPMSMs. Generally the
position estimation calculated from the EEMF [14, 18]:
sin
{( )( ) }
cos
s
s
d q r ds qs r r
e
e
e
L L i pi
o
|
u
e e
u
(
=
(

(
= +
(

(32)
1
tan ( )
( ) ( )
( )
( ) ( )
s
s
s s d s r d q s
s s d s r d q s
e
e
R pL i L L i
R pL i L L i
|
o
o o |
| | o
u
v e
v e

=
+
=
+ +

(33)
The problem is that, the EEMF is influenced by stator current ids and iqs, which vary during motor transient
state. This will cause troubles to the speed estimation. In the low speed range, the signal to noise ratio of
the EEMF is relatively small and the speed estimation result is still not so good. To overcome this difficulty
several authors uses observer and adaptive methods [14].
3.1.2 Adaptive Methods
In this category, various types of observers are used to estimate rotor position. The fundamental idea is that
a mathematical model of the machine is utilized and it takes measured inputs of the actual system and
produces estimated outputs. The error between the estimated outputs and measured quantities is then fed
back into the system model to correct the estimated values adaptation mechanism. The biggest advantage of
using observers is that all of the states in the system model can be estimated including states that are hard to
obtain by measurements. Also, in the observer based methods, the error accumulation problems in the flux
calculation methods do not exist [20], but the weakness is poor speed adjustable at low speed, complicated
algorithm and huge calculation [2]. Observers have been implemented in sensorless PM motor drive
systems. The adaption mechanism base on the following three methods criteria of super stability theory
(Popov), kalman filter, and method of least error square [29]. Methods using the Popov are criteria model
reference adaptive system and luenberger observer.
A. Estimator Based on Model Reference Adaptive System
A model reference adaptive system (MRAS) can be represented by an equivalent feedback system as shown
in Figure 2.
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Reference
Model
Adaptive
Model
Adaptation
mechanism
+
-
s
u
s
i
x c
y

Figure 2. Rotor speed estimation structure using MRAS.
Where c represents the error between the reference model and the adaptive model. The difference between
real and estimated value can be expressed with the dynamic error equation [29]:

? ( ) ( )

( - ) ( - )
d d
x x Ax Ax K Cx Cx
dt dt
A KC A A x
c
c
= =
= +
(34)
Where x is the state vector, u the system input vector, y the output vector, the matrices A , B and C the
parameter of the PMSM and the matrix K a gain coefficient respectively. All elements with ^ are the
estimated vectors and matrices.
It should be noted that, speed estimation methods using MRAS can classified into various types according
to the state variables. The most commonly used are the rotor flux based MRAS, back-emf based MRAS,
and stator current based MRAS. For all mentioned states can be applied one adaption model [29]:
0
? ( ) ( )
T
p q d q d i q d q d
y k x x x x k x x x x dt = +
}
(35)
Stability and speed of the calculation of y depends on the proportional and integral part of (35).
q
x and
d
x represent the states of the PMSM in quadrature and direct coordinates.
As the rotor speede is included in current equations, we can choose the current model of the PMSM as the
adaptive model, and the motor itself as the reference model. These two models both have the output
ds
i
and
qs
i , According to the difference between the outputs of the two models, through a certain adaptive
mechanism, we can get the estimated value of the rotor speed. Then the position can be obtained by
integrating the speed [30].
The equations (1) to (4) can be written as below form:
-
ds
d s ds q qs ds
di
L R i L i
dt
e v = + + (36)
- - -
qs
q s qs d ds r qs
di
L R i L i
dt
e e v = + (37)
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The d-q axis currents
ds
i ,

qs
i are the state variables of the current model of the PMSM, which is described
by (36) and (37).
By rewriting equations (36) and (37) into a matrix form as below:
-
q
s
r r
ds ds
d d
d d
d s
qs qs
q q
ds s r
d d
qs
q
L
R
i i
L L
d
L L
L R dt
i i
L L
R
L L
L
e
e
v
v
(

( (
(
+ +
( (
(
=
( (
(
( (
(

(

(
+
(
(
+
(
(
(





(38)
For the convenience of stability analysis, the speede has been confined to the system matrix:
-
q
s
d d
d s
q q
L
R
L L
A
L R
L L
e
e
(

(
(
=
(

(
(



(39)
To be simplified, define:
1
2
r
ds
d
qs
i
x
L
x
x
i
(
+
(
(
= =
(
(

(


(40)

1
2
ds s r
d d
qs
q
R
L L
u
u
u
L
v
v
(
+
(
(
(
= =
(
(

(
(


(41)
Then the reference model can be rewritten as:
d
x Ax u
dt
= + (42)
The adaptation mechanism uses the rotor speed as corrective information to obtain the adjustable parameter
current error between two models in order to drive the current error to zero, when we can take the
estimation value as a correct speed. The process of speed estimation can be described as follows:
1 1 1
2 2 2

?
?
-
q
s
d d
d s
q q
L
R
L L x x u
d
x L R x u dt
L L
e
e
(

(
( ( (
(
= +
( ( (
(

(
(



(43)
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Where e is to be estimated, (43) can be simplified as below:

?
d
x Ax u
dt
= + (44)
The error of the state variables is:
e x x = (45)
According to equation (42) and (44), estimation equation can be written as:
d
e Ae Iw
dt
= (46)
De v = (47)
Where

( ) w A A x = , choose D I = , then
Ie e v = = (48)
According to Popov super stability theory, if
(1)
1
( ) ( - ) H s D SI A

= is a strictly positive matrix,
(2)
0
0
0
(0, )
t
T
t wdt q v =
}
,
0
0 t > , where
2
0
is a limited positive number, then lim ( ) 0
t
e t

= .
The MARS system will be stable.
Finally, the equation of e can be achieved as:
1 1 2 2 1
0
2 1 2 2 1 (0)
( )
( )
t
k x x x x dt
k x x x x
e
e
=
+ +
}


(49)
Where
1
k ,
2
k 0
Replacing x with i :
1
0
2 (0)
?
( ( ))
?
( ( ))
t
r
ds qs qs ds qs qs
d
r
ds qs qs ds qs qs
d
k i i i i i i dt
L
k i i i i i i
L

e
=
+ +
}

(50)
In the equation (50),

ds
i ,

qs
i can be calculated through the adjustable model,

ds
i ,
qs
i can be obtained by
the transformation of the measured stator currents.
The rotor position can be obtained by integrating the estimated speed:
0

t
dt u e =
}
(51)
The MRAS scheme is illustrated in Figure 3.
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PMSM
Coordinate
Transform
Coordinate
Transform
1
0
2 (0)

( ( ))

( ( ))
t
r
ds qs qs ds qs qs
d
r
ds qs qs ds qs qs
d
k i i i i i i dt
L
k i i i i i i
L

e
=
+ +
}

}
Adjustable
Model
u
s
u
s
i
s
i
s
i
ds
i
qs
u
ds
u
qs

u
e

ds
i

qs
i

Figure 3. Control block scheme of MRAS

B. Observer-Based Estimators
Observer methods use instead of the reference model the real motor. The observer is the adaptive model
with a constantly updated gain matrix K which is selected by choosing the eigenvalues in that way, that the
system will be stable and that the transient of the system will be dynamically faster than the PM machine
[29].
1) Luenberger Observer
A full order state observer with measureable estimated state variables, generally stator current, and not
measurable variables like rotor flux linkages, back-EMF and rotor speed can be described in the form of
state space equations for control of time invariant systems [29]:
x Ax Bu
y Cx
= +
=
(52)
where A is the state matrix of the observer a function of the estimated rotor speed, B the input matrix and C
output matrix.
In the following example the estimation algorithm observer based on back-EMF in - frame is considered.
With the assumption that the back-EMF vector has the following form:
sin
{( )( ) }
cos
s
s
d q r ds qs r r
e
e
e
L L i pi
o
|
u
e e
u
(
=
(

(
= +
(

(53)
and that electrical systems time constant is much smaller than the mechanical one.
r
e is regarded as a
constant parameter. The linear state equation can be described as follows [26, 28, and 29]:

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s s
s
s s
i i
d
A B u W
e e dt
o| o|
o|
o| o|
( (
= + +
( (
( (

(54)

.
s
s
s
i
i C
e
o|
o|
o|
(
=
(
(

(55)
Where,
[ ]
T
s s s
i i i
o| o |
= , [ ]
T
s s s
e e e
o| o |
=

- ( ) -1 0
( ) 0 -1
1
0 0 0 -
0 0 0
r d q
r d q
d
r
r
s
s
R L L
L L R
A
L
e
e
e
e


=
(
(
(
(
(
(







1 0
0 1
1
0 0
0 0
d
B
L
(
(
(
=
(
(





,
1 0 0 0
0 1 0 0
C
(
=
(





?
sin
( )( )
cos
d q r ds qs
W L L i i
u
e
u
(
=
(

The term W is the unknown linearization error and appears only, when
ds
i or
qs
i is changing.
The state equation of the Luenberger observer can be written as:

( )
s
s s
s
s s
s s
u
i i
d
A B
u dt
e e
K i i
o|
o| o|
o|
o| o|
o| o|
( (
(
= + ( (
(
( ( (


+

(56)

s
s
s
i
i C
e
o|
o|
o|
(
= (
(

(57)
Where ^ denotes estimated values.

A is a function of the rotor speed. Therefore the speed

s s
i i
o o o
c = and

s s
i i
| | |
c = are the - axis current errors respectively. For obtaining error dynamic (34) can be used. It
can be seen, that the dynamics are described by the eigenvalues of - A KC . To determine the stability of
the error dynamics of the observer it can be used Popovs super stability theorem or Lyapunovs stability
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theorem which gives a sufficient condition for the uniform asymptotic stability of non-linear system by
using the Lyapunov function V. A sufficient condition for the uniform asymptotic stability is that the
derivate of V is negative definite. If the observer gain K is chosen that ( - )
T
A KC is negative semi-
definite, then the speed observer will be stable [29].
Further literature in flux-based observer can be found in [31, 32].
Systems with a Luenberger observer generally have a better performance than MRAS based Systems.
MRAS based systems have a higher error in the estimated values. Furthermore, the Luenberger approach
has a less tendency to oscillate in the range of low speed and need in comparison with the kalman Filter
method less computation time and memory requirement [29, 33].
2) Reduced Order Observer
Paper [34] is included on the idea of the reduced order observer. The design method considers a general
dynamic system in the form of equation (52).
Where the pair (C, A) is observable. If the output y can be written as a combination of the state vector as:
1 1 2 2
y C x C x = + ;
2
det ( ) 0 C = (59)
Then, it is sufficient to design an observer for the partial state
1
x . If
1
x is the estimate of
1
x , the partition
2
x of the state vector can be calculated as:
1
2 2 1 1
( ) x C y C x

= (60)
The reduced order observer allows for order reduction, is simpler to implement and the state partition
2
x is
found using the algebraic equation (60).
The design methodology of the reduced order observer requires transformation of the original system to the
form:
1 11 1 12 1
x A x A y B u = + + (61)
21 1 22 2
y A x A y B u = + + (62)
A new variable x ' is introduced:
1 1
x x L y ' = + (63)
where
1
L is a nonsingular gain matrix. After the differentiation and algebraic manipulation of (63), the
following form is obtained:
11 1 21 12 1 22 11 1 1 21 1
1 1 2
( ) ( )
( )
A L A x A L A A L L A L y
B L B u
x
'
= + + +
+ +
'

(64)
An observer for x ' is designed in the form:
11 1 21 12 1 22 11 1 1 21 1
1 1 2

( ) ( )
( )
x A L A x A L A A L L A L y
B L B u
' '
= + + +
+ +
(65)
After subtraction, the dynamics of the mismatch is:
11 1 21
( ) x A L A x ' = + (66)
With the known matrices A
11
and A
21
, the gains in L
1
can be selected to obtain desired eigenvalues.
Therefore, the mismatch tends to zero with the desired rate of convergence. Once ' x has been estimated,
the state partition
1
x follows from (63) and

2
x is calculated using (60).
Further literature in reduced order observer can be found in [35-37].
3) Sliding Mode Observer
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In [38, 39] present a sliding mode observer (SMO) for the estimation of the EMFs and the rotor position of
the PMSM. The observer is constructed based on the full PMSM model in the stationary reference frame.
The proposed sliding mode observer is developed based on the equations of the PMSM with respect to the
currents and EMFs:
s s
pe e
o |
e = (67)
s s
pe e
| o
e = (68)
1 1
s
s s s s
R
pi i e
L L L
o o o o
v = + (69)
1 1
s
s s s s
R
pi i e
L L L
| | | |
v = + (70)
The voltages
s o
v ,
s |
v and currents
s
i
o
,
s
i
|
are measured and considered known. In the observer, a speed
estimate is used according to (71); the speed estimate is considered different than the real speed (note
that e A is unknown).
e e e = +A (71)
The observer equations are:
11
( )
s s s
pe e l u
o | o
e e = +A + (72)
22
( )
s s s
pe e l u
| o |
e e = +A + (73)
1 1 1

s
s s s s s
R
pi i e u
L L L L
o o o o o
v = + (74)
1 1 1

s
s s s s s
R
pi i e u
L L L L
| | | | |
v = + (75)
where the switch (sliding mode) controls
s
u
o
,
s
u
|
are:
. ( )
. ( )
s
s
u M sign s
u M sign s
o o
| |
=
=

s s
s s
s i i
s i i
o o o
| | |
=
=
(76)
Note that M is a design gain, M> 0; l11 and l22 are design parameters. After the original equations (67) to
(70) are subtracted from (72) to (75), system is obtained by following equations:
11

s s s s
pe e e l u
o | | o
e e = A + (77)
22

s s s s
pe e e l u
| o o |
e e = A + (78)
1 1
s
s s
R
s s e u
L L L
o o o o
= + (79)
1 1
s
s s
R
s s e u
L L L
| | | |
= + (80)
In the last two equations (79) and (80), note that if the sliding mode gain M is high enough, the
manifolds s
o
, s
|
and their time derivatives have opposite signs. As a result, the manifolds tend to zero and
sliding mode occurs; i.e. 0 s
o
and 0 s
|
. Once sliding mode starts, s
o
, s
|
and their derivatives are
identically equal to zero. The equivalent controls are:
,
,
s eq s
s eq s
u e
u e
o o
| |
=
=

(81)

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In order to study the behavior of the mismatches
s
e
o
,
s
e
|
, the terms
s
u
o
and
s
u
|
are replaced with the
equivalent controls in the first two equations of (77) and (78). The resulting dynamics of the EMF
mismatches is:

11

s s s s
pe e e l e
o | | o
e e = A + (82)
22

s s s s
pe e e l e
| o o |
e e = A + (83)
Next, select the candidate Lyapunov function which is positive definite, V> 0.
2 2
1
( )
2
s s
V e e
o |
= + (84)
After differentiation, the expression of V is:
s s s s
V e e e e
o o | |
= + (85)
After replacing the derivatives from (82) and (83), this becomes:
2 2
11 22

s s s s s s
V l e l e e e e e
o | | o o |
e e = + A +A (86)
If
11 22
l l k = = where 0 k > , the derivative of V is:
2 2
( ) ( )
s s s s s s
V k e e e e e e
o | | o o |
e = + +A (87)
Equation (87) will be used to study the convergence of the observer. Note that
0 V <
If 0 e A = and the
observer is asymptotically stable. There are two terms in (87): the first one is always negative (and can be
increased using the design parameter k) while the second one has unknown sign. As long as the mismatches
s
e
o
and
s
e
|
are significant, the first term overcomes the second and
V
is negative (as a result, function
V decays). The Lyapunov function stops decaying when
0 V =
and this is equivalent to:
2 2
( ) ( )
s s s s s s
k e e e e e e
o | | o o |
e + = A (88)
Since the mismatches
s
e
o
,
s
e
|
should be of the same order of magnitude, using the notation
s s
me e
o |
=
(where m is unknown), equation (88) is manipulated to give the value of the mismatch
s
e
o
at which the
function V stops decaying:
2

(1 )
s s
s
m e e
e
k m
| o
o
e

= A
+
(89)
The Lyapunov function settles to the vicinity given by the mismatch in (89), and the size of this vicinity
(and the mismatch) can be reduced by increasing k (which is a design parameter). The analysis shows that
the influence of the speed mismatch e A (caused by the speed observer) can be made irrelevant by proper
design of the SM observer gains; the mismatch between the real and estimated EMFs can be made as small
as desired according to (89). Once the EMFs have been found, the rotor position is computed directly with:
1

tan ( )

s
s
e
e
o
|
u

= (90)
Further literature in sliding mode observer (SMO) can be found in [40-43].
The main difference between the Luenberger and the Sliding mode observer (SMO) lies in the observer
structure. The SMO uses a sign-function of the estimation error instead of the linear value as correction
feedback [44].
4) Kalman Filter
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The kalman filter is in principle a state observer that establishes the approximation for the state variables of
a system, by minimization of the square error, subjected at both its input and output to random
disturbances. If the dynamic system of which the state is being observed is non-linear, then the kalman
filter is called an extended kalman filter (EKF). The EKF is basically a full-order stochastic observer for the
recursive optimum state estimation of a nonlinear dynamical system in real time by using signals that are in
noisy environments. The EKF can also be used for unknown parameter estimation or joint state [45]. The
linear stochastic systems are described by relations [7]:
( ) ( ) ( ) ( ) x t Ax t Bu t w t = + + ; ( )
o o
x t x = (91)
( ) ( ) y t Cx t = (92)
( ) ( ) ( ) z t y t t v = + (93)
Where:
x , y , u , A , B ,C have the significance known from deterministic system; ( ) w t represents the vector
of disturbances applied at the system input; ( ) z t is the vector of the measurable outputs, affected by the
random noise ( ) t v .
It can be considered that, besides the input disturbances, vector ( ) w t includes some uncertainties referring
to the process model. It will be assumed that the vector functions ( ) w t and ( ) t v are not correlated and
zero-mean stochastic processes. From statistic point of view, the stochastic processes ( ) w t and ( ) t v are
characterized by the covariance matrices Q and R, respectively. It is further assumed that the initial state
o
x is a vector of random variables, of mean
o
x and covariance
0
P , not correlated with the stochastic
processes ( ) w t and ( ) t v over the entire interval of estimation.
The covariance matrices Q, R,
0
P characterizing the noise sources of system (94)-(96) are, by definition,
symmetrical and positively semi-definite, of dimensions (n x n), (m x m) and (n x n) respectively, where n
and m represent the number of state and output variables, respectively.
For linear time invariant systems, the following relations of recurrent computation describe the general
form of the kalman filter implementation algorithm:
1
1 1
( )
T T
k k k k k
K P C CP C R


= + (94)
1 1
? ( )
k k k k k k k k
x x K y Cx

= + (95)
1
( - )
n k k k k k
P I K C P

= (96)
1

d s k k k k k
x A x T Bu
+
= + (97)
1
T
d d k k k k
P A P A Q
+
= + (98)
Where T
s
represents the sampling period and A
d
is the matrix of the discrete linearized system:
d n s
A I T A = + (99)
In these relationships, the (n x m) matrix K represents the kalman gain; P represents the covariance state
matrix and I
n
is the (n x n) unit matrix. In the recurrent computation relationships, the subscript index
notations of type k/k-1 show that the respective quantities (state vectors or their covariance matrices) are
computed for sample k, using the values of similar quantities from the previous sample.
For non-linear stochastic systems, the dynamic state model is described by the following expressions:
( ) ( ( ), ( ), ) ( ) x t f x t u t t w t = + (100)
( ) ( ( ), ) y t h x t t = (101)
( ) ( ( ), ) ( ) z t h x t t t v = + (102)
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where f and h are (n x 1) and (m x 1) function vectors, respectively. The A and C matrices of the EKF
structure are dependent now upon the state of the system and are determined by:
( ) ( )
( ( ), ( ), )
( ( ), )
( )
x t x t T
f x t u t t
A x t t
x t
=
c
=
c
(103)
( ) ( )
( ( ), )
( ( ), )
( )
x t x t T
h x t t
C x t t
x t
=
c
=
c
(104)
Additionally to the fact that matrices A and c have now become dependent on the state of the system, the
algorithm suffers some further changes, which affect relationships (95) and (97), now expressed as:
1 1
? [ ( )]
k k k k k k k k
x x K y h x

= + (105)
1
? ( ( ), ( ))
s k k k k
x x T f x t u t
+
= + (106)
The kalman filter algorithm is initiated by adopting adequate values for the covariance matrix of the initial
state
0 0 1
P P

= , as well as for the weighting matrices Q and R, the latter two being constant during the
estimation.
EKF is known for its high convergence rate, which significantly improves transient performance. However,
the long computation time is a main drawback of the EKF [45]. Although last-generation floating-point
digital signal processors can easily overcome the EKF real-time calculations [46], this is not suitable for
low-cost PMSM applications. Moreover, long computation requirements disturb other program service
routines such as fault diagnosis or custom programs installed in products. [47] presents the optimal two-
stage kalman estimator (OTSKE) which is composed of two parallel filters: a full order filter and another
one for the augmented state. This estimator has the advantage of reducing the computational complexity
compared to the classical EKF.
The application of the EKF in the sensorless PMSM drive system will be increased a lot [29, 48]. The main
steps for a speed and position sensorless PMSM drive implementation using a discredited EKF algorithm
can be found in [45, 48-51].
3.2 Saliency and Signal Injection Methods
In signal injection methods, the feature of salient-pole PMSMs such that the inductance varies with the
rotor position is used. High frequency voltage or current signal is injected on the top of the fundamental,
and signal processing (vector filters with inevitable time delay) is employed to extract currents harmonics
that contain rotor position. Thus, the position can be estimated even at standstill and low speeds, robustly to
parameter variations [52]. For wide speed range, hybrid methods are used [53]. High frequency signal
injection techniques, Figure 4, are considered to be superior to other sensorless control schemes of ac
machines for low and zero speed operation.
Fundamental
excitation
High
frequency
signal
PWM
Inverter
AC
Machine
Band pass
filter
Heterodyning
demodulation
Speed
estimation
Position
estimation
e

u

Figure 4. High frequency signal injection block diagram

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Based on the injection direction of the excitation signal, the high frequency injection schemes could be
classified as rotating injection method, in which the carrier signal is a rotating sinusoidal signal in the
stationary reference frame; pulsating injection method, in which an ac voltage signal is injected on the
estimated rotor d-axis, rotating with the rotor and the rotor position offset could be observed from the high
frequency component of the estimated q-axis current. However, the drive efficiency of the schemes is hard
to be accessed and the development of the sensorless strategies suffers the unknown nonlinearity of the
model [54].
Other methods are based on the signal processing of PWM excitation without signal injection. The SV-
PWM waveforms provide sufficient excitation to extract the position signal from the stator current. These
methods are: Indirect Flux detection by On-line Reactance Measurement (INFORM) method, and methods
based on the measurements of di/dt of the stator currents induced by SV-PWM. Typically, the injected
signal is a sinusoidal type. To eliminate the time delay introduced by filters, a new square wave signal
injection method is proposed, based only on the measurement of the corresponding induced stator current
variations, which leads to high dynamics and robust position estimation [52].
Magnetic saliency methods are relatively complicated for real time implementation and are less portable
from one machine to another; however, they work well at low speed. The rotor position of the PMSM can
also be estimated at standstill; as a result, the motor can be started with the correct rotor position from the
beginning of the motion [38].
Further literatures which use the magnetic saliency methods can be found in [55-58].
3.3 Artificial Intelligence Methods
Artificial intelligence describe neural network (NN), fuzzy logic based systems (FLS) and fuzzy neural
networks (FNN). The use of artificial intelligence (AI) to identify and control nonlinear dynamic systems
has been proposed because they can approximate a wide range of nonlinear functions to any desired degree
of accuracy [59]. Moreover, they have been the advantages of immunity from input harmonic ripples and
robustness to parameter variations. However, ANN controller synthesis requires design of the control
structure which includes selecting the neural network structure, weight coefficients and activation function.
The selection of neural structure as the initial step is done by trial and error method since there is no proper
procedure for this [60]. The complex of the selected neural network structure is a compromise between the
high quality of control robustness and the possibility of control algorithm calculation in real time. This
gives rise to inaccuracies [60]. Recently, there have been some investigations into the application of AI to
power electronics and ac drives, including speed estimation [60-63].
Paper [64] presents a robust control strategy with NN flux estimator for a position sensorless SPMSM
drive. In the proposed algorithm stator flux is estimated from stationary - axis stator currents and speed
error, E

. An equivalent Recurrent Neural Network (RNN) is then proposed which results in the following
matrix equation:
13 11
22 23
15 14
24 25
( 1) ( ) 0
( )
( 1) ( ) 0
( ) ( )
s s
s
s s
s w
k k W W
i k
k k W W
W W
i k E k
W W
o o
o
| |
|


+
= +
+
+ +
( ( ( (
( ( ( (

( (
( (

(107)

where, W11 , W13 ,W22 , W23 , W14 etc. are the weights of the RNN.
The estimated stator flux using RNN is used to find out the rotor position as following:
u o = + (108)
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Where, is the flux angle, and o is the torque angle. Where
1
tan ( )
s
s
|
o

=

4. Parameter Adaptation
Motion-sensorless PMSM drives may have an unstable operating region at low speeds. Since the back
electromotive force (EMF) is proportional to the rotational speed of the motor, parameter errors have a
relatively high effect on the accuracy of the estimated back EMF at low speeds [65]. Improper observer
gain selections may cause unstable operation of the drive even if the parameters are accurately known [66].
In practice, the stator resistance varies with the winding temperature during the operation of the motor, so
there is often a mismatch between the actual winding resistance and its corresponding value in the model
used for speed estimation. This may lead not only to a substantial speed estimation error but to instability as
well.
Parameter variation not only degrades the control performance but also causes an error in the estimated
position [67]. As consequence, numerous online schemes for parameter identification have been proposed,
recently [65-69].
5. Conclusion
Recently, permanent magnet synchronous motor (PMSM) drives are replacing classic dc and induction
machine drives in a variety of industrial applications. PMSM drive research has been concentrated on the
elimination of the mechanical sensors at the motor shaft without deteriorating the dynamic performances of
the drive. Many advantages of sensorless ac drives such as reduced hardware complexity, low cost, reduced
size, cable elimination, increased noise immunity, increased reliability and decreased maintenance.in this
paper, a review of different speed and rotor position estimation schemes of PMSM drives has been
introduced. Each method has its advantages and disadvantages. Although numerous schemes have been
proposed for speed and rotor position estimation, many factors remain important to evaluate their
effectiveness. Among them are steady state error, dynamic behavior, noise sensitivity, low speed operation,
parameter sensitivity, complexity, and computation time. In particular, zero-speed operation with
robustness against parameter variations yet remains an area of research for speed sensorless control.
As a final comment, each speed estimation method of sensorless application requires a specific design,
which takes into consideration the required performance, the available hardware and the designer skills.
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[7] V. Comnac, M. N. Cirstea, F. Moldoveanu, D. N. Ilea, and R. M. Cernat, "Sensorless Speed and Direct
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