Observers Design For Sensorless PMSMs
AHMED CHOUYA
Department of Genie Electrical
University of Djilali Bounaama
Khemis-Miliana City, 44 225
ALGERIA
Abstract: A state observer is proposed for permanent magnet synchronous motors (PMSMs). The gain of this observer
involves a design function that has to satisfy some mild conditions which are given. Different expressions of such a function
are proposed. Of particular interest, it is shown that high gain observers and sliding mode like observers can be derived by
considering particular expressions of the design function. The simulation is given in order to compare the performance of a
high gain observer and a sliding mode observer obtained through two different choices of the design function. Simulation is
made by the software MATLAB/SIMULINK.
Key–Words: permanent magnet synchronous motors, high gain observer, sliding mode observer
Received: April 9, 2022. Revised: December 26, 2022. Accepted: January 18, 2023. Published: February 28, 2023.
1 Synchronous PMSM Model
Because of mechanical rotor position is practically unavail-
able for measurement devices, the PMSM model is consid-
ered in the
-frame which is more suitable for ob-
server design. According to [1, 2], the PMSM model in the
coordinates is given by:
(1)
Where
,
,
are respectively, the stator currents, the rotor
flu es and the voltages.
and
respectively, denote the
rotor speed and the load torque.
is the
matrix
define as
;
is the motor moment of
inertia;
is the number of pairs of poles. The electrical pa-
rameters
and
are the stator resistor and inductance,
respectively. Notice that the time derivative of the external
load torque is described by an unknown bounded function.
The firs issue one must deal with is, under what conditions
that all the state variables,
,
,
and
can be deter-
mined using only measurements of the electrical variables
i.e. the stator current and supply voltage measurements
and
, respectively.
2 Model Transformation
For clarity our purposes, one introduces the following no-
tations:
with
(2)
In the sequel, the notation
and
will be used to
denote the
identity matrix and the
null
matrix, respectively. The rectangular
null matrix
shall be denoted by
. Model (1) can then be rewritten
under the following condensed form:
(3)
Where
×
×
×
×
Ú
We need to transform system (3) to the triangular form.
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One will introduce the change of variable according to :
(4)
Then
(5)
Where
×
×
×
×
Ú
Ú
and
We will introduce a classical state transformation.
that puts model (1) under a known observable canon-
ical form [3].
The sufficien conditions under which the considered
state transformation is a diffeomorphism. In particular, this
the analysis will emphasize the JACOBIAN matrix (of the
considered state transformation) that is required to be full
rank. Now, let us consider the following change of vari-
ables.
(6)
The map
is one to one. Let
denote its converse.
Before deriving the dynamics of
, let us introduce the fol-
lowing notations :
is the diagonal matrix :
(7)
is left invertible. One shall denote by
its left inverse. Now, one can easily check that :
One can illustrate that the above state transformation
puts system (5) under the following canonical form:
(8)
Where
;
and
, with
;
3 Structure of Observer
For convenience, the system model (3) is given the follow-
ing more compact form :
(9)
where the state
, the
matrix
is the following anti-shift block matrix:
The function
has a triangular structure :
As in the works related to the observers synthesis [5,
6, 7, 8], one pose the hypothesis :
:The function
is globally Lipschitz with re-
spect to
uniformly in
.
Before giving our candidate observers, one introduces
the following notations.
Let
is a block diagonal matrix define by:
(10)
is a real number.
Let
is a definit positive solution of the alge-
braic Lyapunov equation:
(11)
Note that (11) is independent of the system and the
solution can be expressed analytically. For a straight-
forward computation, its stationary solution is given
by:
where
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for
and
; and then we can explicitly de-
terminate the correction gain of (3) as follows:
×
×
(12)
, set
and let
be a vector of smooth functions satisfy-
ing:
(13)
(14)
The system
(15)
is an observer for (9); Where
error in estimation;
satisfie conditions (13) and (14).
3.1 Stability Analysis of the Proposed Ob-
server
Now, we present the stability analysis of the candidate ob-
server (15), for that let use the error consider
, his deriva-
tive :
Where
Notice that
is a lower triangular matrix with ze-
ros on its main diagonal, one can easily deduce that
is bounded.
Now, one can easily check the following identities:
One obtains :
(16)
To prove convergence, let us consider the following
equation of Lyapunov
. By calculating the
derivative of
along the
trajectories, we obtains:
!
By taking account of the (11) and (13) the derivative of
becomes:
(17)
Now, assume that
, then, because of the triangular
structure and the Lipschitz assumption on
, one can show
that :
"
(18)
where
"
is a constant of Lipschitz. Similarly, according
to hypothesis
.
Using inequalities (14) inequality (17) becomes:
#
"
$%
&
where
&
%
"
$%
with
#
and
#
being respectively the smallest and the largest
eigenvalues of
and
%
.
Now taking
&
and using the fact that
for
,
, one can show that for
, one has :
%
&
It is easy to see that
#
and
'
needed by the result 1
are:
#
%
and
'
. This completes the proof.
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3.2 Observers Equations in the Original Co-
ordinates
Proceeding as in [5], one can show that observer (15) can
be written in the original coordinates
as follows:
(19)
Some expressions of
that satis-
fying conditions (13) and (14) shall be given in this section
and the so-obtained observers are discussed. These expres-
sions will be given in the new coordinates
in order to
easily check conditions (13) and (14) as well as in the orig-
inal coordinates
in order to easily recognize the structure
of the resulting observers.
3.3 High Gain Observer
Consider the following expression of
:
(20)
One can easily check that expression (20) satisfie condi-
tions (13) and (14). Replacing
by expression (20) in
(15) gives rise to a high gain observer (see e.g. [5, 7, 10]):
(21)
Or
×
×
×
×
Ú
Ú
(22)
Referring to (4), the rotor flu is governed by the fol-
lowing equations:
(23)
3.4 Sliding Mode Observers
At firs glance, the following vector seems to be a potential
candidate for the expression of
:
sign
sign
sign
(24)
where sign is the usual signe function with sign
sign
sign
; then:
sign
(25)
Or
×
×
×
×
sign
Ú
sign
Ú
sign
(26)
Indeed, condition (13) is trivially satisfie by (24).
Similarly, for bounded input bounded output systems.
However, expression (24) cannot be used due the discon-
tinuity of sign function. Indeed, such discontinuity makes
the stability problem not well posed since the LYAPUNOV
method used throughout the proof is not valid. In order
to overcome these difficulties one shall use continuous
functions which have similar properties that those of the
signfunction. This approach is widely used when imple-
menting sliding mode observers. Indeed, consider the fol-
lowing function:
3.4.1
!
Function
!
!
(27)
where
!
denotes the hyperbolic tangent function; then:
!
(28)
Or
×
×
×
×
!
Ú
!
Ú
!
(29)
3.4.2
"#
Function
"#
"#
(30)
Similarly to the hyperbolic tangent function, one can
easily check that the inverse tangent function:
"#
(31)
Or
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×
×
×
×
"#
Ú
"#
Ú
"#
(32)
4 Comparison of Sensorless Ob-
servers
To examine practical usefulness, the proposed observer has
been simulated for a PMSM (see [11, 12]), whose parame-
ters are depicted in Table 1.
Table 1: PMSM parameters used in simulations.
Parameters Notation Value Unit
Pairs number 2
of poles
Frequency 50
(
Inductance 0.02682
)
$
Flux linkage 0.1717
(
established
Stator phase
%
&
resistance
Friction factor
'(
* (
Inertia
)
+
Torque
*
*
In order to evaluate the observer behaviour in the re-
alistic situation, the measurements of
issued from the
model simulation have been corrupted by noise measure-
ments with a zero mean value. The torque lead takes the
step value.
4.1 High Gain Observer
The adjustment parameter of the observer (22) is to cho-
sen
*
. The dynamic behaviour of the error of rotor
flu is depicted in Figure Fig.1 graph (a); when graph (b)
shows the gaussian errors density and empirical errors his-
togram of rotor flu error. The means of error flu equal
with very small variance
&%%
this is almost surety. The pace of speed error is given by
the figur Fig.2 graph (a) and the gaussian errors density
and empirical errors histogram of rotor speed error are pre-
sented in graph (b) where means of error rotating speed
equal
*&
and variance equal
&
(*(
; the curve
of load torque is illustrated on figur Fig.3 graph (a). In
graph (b) appear gaussian errors density and empirical er-
rors histogram of load torque error where means of error
load torque equal
**(
and variance equal
.
0 20 40 60 80
0
0.02
0.04
0.06
0.08
0.1
0.12
Time[s]
Flux error [Wb]
−0.018−0.016−0.014−0.012 −0.01
0.5
1
1.5
2
2.5
x 10
4
(b)
Histogram
Gaussian
Figure 1: (a) Flux error. (b) Gaussian and histogram of
error flux
0 0.05 0.1
−200
−150
−100
−50
0
50
100
Time [s]
Speed error [rad/s]
−5 0 5
0
0.5
1
1.5
2
x 10
5
(b)
Figure 2: (a) Speed error. (b) Gaussian and histogram of
error speed.
0 20 40 60 80
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Time [s]
Turque error [Nm]
(a)
0.25 0.3 0.35 0.4 0.45
0
500
1000
1500
2000
2500
3000
3500
4000
4500
(b)
Histogram
Gaussain
Figure 3: (a) Load torque error. (b) Gaussian and histogram
of error load torque.
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4.2 Sliding Mode Observer With
Func-
tion
Estimation results of the proposed algorithm (29) with
'
is reported in Figure Fig.4, Fig.5 and Fig.6. The be-
haviour of the error of rotor flu is depicted in figur Fig.4
graph (a); when graph (b) shows the gaussian errors den-
sity and empirical errors histogram of rotor flu error. The
means of error flu equal
'&
with very small vari-
ance
(
this is almost surety. The pace of speed
error is given by the figur Fig.5 graph (a) and the gaussian
errors density and empirical errors histogram of rotor speed
error are presented in graph (b)where means of error rotat-
ing speed equal
)
*&
and variance equal
(
(*%
;
the curve of load torque is illustrated on figur Fig.6 graph
(a).In graph (b) appear gaussian errors density and empir-
ical errors histogram of load torque error where means of
error load torque equal
)*(*
and variance equal
(
0 20 40 60 80
0.1
0.15
0.2
0.25
0.3
0.35
Time [s]
Flux error [Wb]
(a)
0.1 0.2 0.3 0.4 0.5
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
(b)
Histogram
Gaussian
Figure 4: (a) Flux error. (b) Gaussian and histogram of
error flux
0 0.5 1 1.5 2 2.5
−200
−100
0
100
200
300
400
500
600
700
Time [s]
Speed error [rad/s]
(a)
−100 0 100
0
2000
4000
6000
8000
10000
(b)
Histogram
Gaussian
Figure 5: (a) Speed error. (b) Gaussian and histogram of
error speed.
0 5 10 15
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
Time [s]
Turque load [N.m]
(a)
−0.9 −0.8 −0.7 −0.6 −0.5 −0.4
0
200
400
600
800
1000
1200
1400
1600
1800
(b)
Histogram
Gaussian
Figure 6: (a) Load torque error. (b) Gaussian and histogram
of error load torque.
4.3 Sliding Mode Observer
Function
Under the same conditions with the function
!
. One
simulates for the function
"#
. The figur Fig.7, Fig.8
and Fig.9 illustrates the pace of error flux error speed and
error load torque in respectively. The behaviour of the er-
ror of rotor flu is depicted in Figure Fig.7 graph (a); when
graph (b) shows the gaussian errors density and empirical
errors histogram of rotor flu error. The means of error flu
equal
)
%
with very small variance
*
)
this is almost surety. The pace of speed error is given
by the figur Fig.8 graph (a) and the gaussian errors den-
sity and empirical errors histogram of rotor speed error are
presented in graph (b)where means of error rotating speed
equal
%
(*(
and variance equal
*
)
; the curve
of load torque is illustrated on figur Fig.9 graph (a).In
graph (b) appear gaussian errors density and empirical er-
rors histogram of load torque error where means of error
load torque equal
&*
and variance equal
&
0 5 10
0.1
0.12
0.14
0.16
0.18
0.2
0.22
Time [s]
Flux error [Wb]
(a)
0.05 0.1 0.15 0.2 0.25 0.3
0
50
100
150
200
250
(b)
Histogram
Gaussian
Figure 7: (a) Flux error. (b) Gaussian and histogram of
error flux
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0 0.5 1 1.5 2 2.5
−200
−100
0
100
200
300
400
500
600
700
Time [s]
Speed error [rad/s]
(a)
−200 −100 0 100 200
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
(b)
Histogram
Gaussian
Figure 8: (a) Speed error. (b) Gaussian and histogram of
error speed.
0 5 10
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
Time [s]
Turque load [Nm]
(a)
−0.8 −0.7 −0.6 −0.5
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
(b)
Histogram
Gaussian
Figure 9: (a) Load torque error. (b) Gaussian and histogram
of error load torque.
5 Conclusions
In this paper, high gain and alternative form for a sliding
mode observers are presented. they is observer makes pos-
sible to observe, rotor flux rotor speed and load torque.
An observer with high gain and three others with sliding
mode which the functions sign,
!
and
"#
. Observer
whose sign gives chattering. High gain observer is good
for the observation of rotor flux rotating speed and load
torque.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
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