Sensorless Adaptive Observers for Linearly Parameterized Class of
Induction Machine
AHMED CHOUYA
Department of Genie Electrical
University of Djilali Bounaama
Khemis-Miliana City, 44 225
ALGERIA
Abstract: In this paper a new adaptive flux, speed, load torque and rotor resistance observer is proposed for induction machine
drive system. The structure of the proposed observer is simple and it is able to give rise to different observers among which
adaptive high gain and adaptive sliding mode. We adopt a mild change of coordinates that allows to easily design of adaptive
observer. Computer simulation results verify the validity of the proposed estimation algorithm.
Key–Words: Adaptive observer, high gain observer, sliding mode observer, induction machine, sensorless speed.
1 Introduction
On line joint estimation of states and parameters in state
space systems is of practical importance for adaptive con-
trol and for fault detection and isolation. The algorithms
designed for this purpose are called adaptive observers.
Some early works on this subject can be found in [1, 2, 3, 4]
.These results are essentially for linear time invariant (LTI)
systems, though some of them have been proposed for
nonlinear systems which can be linearized by coordinate
change and by output injection. Recently, adaptive ob-
servers for multi input multi output (MIMO) linear time
varying (LTV) systems have been developed [5, 6] .Some
results on truly nonlinear systems have also been reported
[7,8,5].
The adaptive observer for MIMO LTV systems pro-
posed in [6] is conceptually simple and computationally
efficient. Its global exponential convergence for joint state
parameter estimation has been proved in the noise free case.
When the considered system is noise corrupted, the conver-
gence in the mean has been established, but not the consis-
tency. In other words, under appropriate assumptions, the
means of the estimation errors tend to zero, but not the er-
rors themselves. Up to our knowledge, this was the first
reported result on the convergence of adaptive observer for
noise
In this paper, we propose an adaptive observers for
a class of uniformly observable nonlinear systems. This
observers, based on techniques of High gain and Sliding
mode, is applied to jointly estimate states (rotor flux, rotat-
ing speed and torque load) and unknown constant parame-
ters (rotor resistance).
Firstly, the convergence of the proposed observer is
guaranteed under a well-defined persistent excitation con-
dition. Secondly, the structure of the proposed observer
is simple and it is able to give rise to different observers
among which adaptive high gain like observers [9, 10, 11]
and adaptive sliding mode like observers [12, 13, 14].
This paper is organized as follows. The next section
introduces the dynamic model of an induction machine. In
Section 3, the observer design is detailed. The equations of
the proposed adaptive observer are given and a full conver-
gence analysis is made. Besides, different expressions of
the observer design function are specified and it is shown
that they give rise to different observers. A simulation of
IM is given in Section 4 in order to illustrate the theory.
2 Description of Induction Machine
Assuming linear magnetic circuits, the dynamics of a bal-
anced non-saturated induction machine in a fixed reference
frame attached to the stator are given in the form follows :
(1)
The states variables accessible to measurement are the
stator currents
but to in no case
rotor flux
and possibly rotat-
ing speed. The source of energy
.
The model parameters are : rotor moment of inertia
, ro-
tor and stator winding’s resistances
,
, mutual induc-
tance
. To simplify notations we use the parameteriza-
tion :
,
,
,
,
Received: March 29, 2022. Revised: February 12, 2023. Accepted: March 7, 2023. Published: April 10, 2023.
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Ahmed Chouya
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Volume 5, 2023
,
,
is the 2- dimensional identity matrix and
is a skew - symmetric matrix.
We need to transformsystem (1) to the triangular form.
One will introduce the change of variable according to:
(2)
Using this transformationand a time derivativeof these
states, we can rewrite from model (1), a following model :
(3)
With
.
One can show that this transformation puts system (1)
under the following form:
(4)
where
and
3 Observers Design
3.1 Observers synthesis
As in the works related to the high gain observers synthesis
[10, 11] and adaptive observers [15], one pose the hypoth-
esis :
:The functions
is globally LIPSCHITZ with re-
spect to
:The functions
is globally LIPSCHITZ with
respect to
uniformly in
.
:The functions
is globally LIPSCHITZ with re-
spect to
Before giving our candidate observers, one introduces
the following notations.
block diagonal matrix and
is his left inverse:
Let
is a block diagonal matrix defined by:
is a real number.
Easy computations allow us to check the following
identities:
where
is matrix:
Let
is a definite positive solution of the AL-
GEBRAIC LYAPUNOV EQUATION:
(5)
Note that (5) is independent of the system and the so-
lution can be expressed analytically. For a straightfor-
ward computation, its stationary solution is given by:
where
for
and
; and then we can explicitly determi-
nate the correction gain of (3) as follows:
(6)
, set
and let
be a vector of smooth functions satisfy-
ing:
(7)
(8)
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The system
!
!
!
!
!
(9)
With
and
!
!
means that the ini-
tial condition of the Ordinary Differential Equation (ODE)
governing
!
is chosen Symmetric Positive Definite (SPD).
is an adaptive observerfor system (4) with an exponen-
tial error convergence for relatively high values of
.
The proof of the stability is given in the next subsec-
tion. However, before detailing this proof, one shall give
some comments and facts which will be used throughout
the proof.
Please notice that the time derivative of
given in (9)
can be written as follows:
(10)
The equation of
consists in a copy of the model (4)
with a correcting term. The correcting term is composed
by two terms. The first one,
is rather
classical and is met in classical high gain state observers
[16]. The second term,
is similar to the expression
used for updating the unknown parameters, i.e. the term
used in the expression of
.
3.2 Convergence analysis
Set
and
From (9) and (10), one has
(11)
Set
. Using the notations 2 from subsection
.3.1, one obtains:
(12)
Now, as in [17], set:
"
. For writing conve-
nience and as long as there is no ambiguity, the time vari-
able
#
shall be omitted in the sequel. Using the fact that
is governed by the ODE given in (9), one can show that:
"
"
(13)
Set
$
"
" $
!
where
!
is given
in (9) and let
$
$
$
be a LYAPUNOV CANDIDATE
FUNCTION. Using (5), one gets:
$
"
"
"
"
"
"
"
"
(14)
It is obvious that
"
(15)
By the Mean Value Theorem, one gets:
%
%
%
%
(16)
As a result, for
and from (16), one obtains
%
%
%
%
&
(17)
where
&
is a constant which does not depend on
for
. Using (17) and (15) , one obtains:
&
"
&
(18)
where
&
&
. Therefore, one has:
"
"
&
"
&
"
&
'
$
&
'
'
!
$
$
&
$
&
$
$
(19)
where
&
&
,
&
&
,
&
and
&
are positive constants which do
not depend on
,
'
and
'
!
denoting the
smallest eigenvalue of the matrix
and
!
.
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In ways similar, we will have:
&
"
&
(20)
where
&
&
. Therefore, one has:
"
"
&
"
&
"
&
'
$
&
'
'
!
$
$
&
$
&
$
$
(21)
where
&
&
,
&
&
,
&
and
&
are positive constants which do not
depend on
.
Since each column of the matrix
assumes a tri-
angular structure and since
is bounded, the argu-
ments developed above are still be valid for bounding
"
and indeed by proceeding in
a similar way as above, one obtains:
"
&
$
&
$
$
(22)
where
&
and
&
are positive constants (depending on
the bounds of
which do not depend on
.
And
"
&
$
&
$
$
(23)
Using (19), (21), (22) and (23), inequality (14) can be
written as follows:
$
$
"
"
"
"
(
$
(
$
$
(24)
with
(
&
&
&
&
and
(
&
&
&
&
.Let us now derive the time derivative of
$
. One has:
$
!
!
!!
$
$
(25)
Hence, using (24) and (25), one obtains
$
$
$
(
$
$
(
$
$
"
"
"
"
(
$
$
(
$
$
(
$
(
$
$
$
(26)
The last inequality is obtained according to the in-
equality (7). Now, set
$
(
$
,
$
$
and
$
$
$
.Please notice that
$
(
$
.
Inequality (26) yields to
$
$
(
(
$
(
(
(
$
(27)
Now, it suffices to choose
such that
. This ends the proof.
3.3 Adaptive high gain observers
Consider the following expression of
:
(28)
Replacing
by expression (28) in (10) gives rise to
a high gain observer :
(29)
!
!
!
!
!
Referring to (2), the rotor flux is governed by the fol-
lowing equations:
(30)
3.4 Adaptive sliding mode like observers
At first glance, the following vector seems to be a potential
candidate for the expression of
)
:
sign
(31)
sign
sign
(32)
where sign is the usual signe function with sign
sign
sign
; then:
sign
(33)
!
sign
!
!
!
!
Indeed, condition (7) is trivially satisfied by (31). Sim-
ilarly, for bounded input bounded output systems. How-
ever, expression (31) cannot be used due the discontinu-
ity of sign function(see.[18]). Indeed, such discontinuity
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makes the stability problem not well posed since the LYA-
PUNOV method used throughout the proof is not valid. In
order to overcome these difficulties, one shall use contin-
uous functions which have similar properties that those of
the signfunction. This approach is widely used when im-
plementing sliding mode observers. Indeed, consider the
following function:
Tanh function:
)
(34)
where
denotes the hyperbolic tangent function; then:
(35)
!
!
!
!
!
Arctan function:
)
(36)
Similarly to the hyperbolic tangent function, one can easily
check that the inverse tangent function:
(37)
!
!
!
!
!
4 Simulation of Sensorless Ob-
servers
To examine practical usefulness, the proposed observer
has been simulated for a three-phase 1.5kw induction ma-
chine(see [19]), whose parameters are depicted in Table1.
Table 1: Induction machine parameters used in simulations.
Notations value unit
p 2
*
50
0.464
0.464
0.4417
5.717
3
0.0049
+
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.
High gain observer :
The adjustment parameter of the observer (29) is to
chosen
. The dynamic behaviour of the er-
ror of rotor flux is depicted in Figure 1 graph (a);
when graph (b) shows the gaussian errors density and
empirical errors histogram of rotor flux error. The
means of error flux equal
with small variance
!
!!
. The pace of speed error is given by the
figure 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 figure 3 graph (a).In
graph (b) appear gaussian errors density and empiri-
cal errors histogram of load torque error where means
of error load torque equal
!
and variance
equal
#
; the curve of stator resistance
is illustrated on figure 4 graph (a).In graph (b) appear
gaussian errors density and empirical errors histogram
of stator resistance error where means equal
and variance equal
.
0 0.1 0.2 0.3 0.4 0.5
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Time[s]
Flux error [Wb]
(a)
−0.05 0 0.05 0.1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
Gaussian
Histogram
Figure 1: (a) Flux error. (b) Gaussian and histogram of
error flux.
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0 0.5 1 1.5
0
20
40
60
80
100
Time[s]
Speed error[rad/s]
(a)
−10 0 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
Gaussian
Histogram
Figure 2: (a) Speed error. (b) Gaussian and histogram of
error speed.
0.1 0.2 0.3 0.4 0.5
−2
0
2
4
6
8
10
12
14
x 10
−5
Time[s]
Load torque [N.m]
(a)
1.2 1.4 1.6 1.
8
x 10
−4
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
(b)
Gaussian
Histogram
Figure 3: (a) Load torque error. (b) Gaussian and histogram
of error load torque.
0 0.1 0.2 0.3 0.4 0.5
−8
−6
−4
−2
0
2
4
6
x 10
−3
Time[s]
Rotor resistance error [Ω]
(a)
−4 −3.5 −3 −2.5
x 10
−3
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Gaussian
Histogram
Figure 4: (a) stator resistance error. (b) Gaussian and his-
togram of error rotor resistance.
Sliding mode observer with
:
Estimation results of the proposed algorithm(35) with
is reported in figure 5, 6 and 7. The be-
haviour of the error of rotor flux is depicted in fig-
ure 5 graph (a); when graph (b) shows the gaussian
errors density and empirical errors histogram of ro-
tor flux error. The means of error flux equal
#
with very small variance
!
!
this is almost
surety. The pace of speed error is given by the figure
6 graph (a) and the gaussian errors density and em-
pirical 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 figure 7 graph (a).In
graph (b) appear gaussian errors density and empiri-
cal errors histogram of load torque error where means
of error load torque equal
and variance
equal
##
; the curve of stator resistance
is illustrated on figure 8 graph (a).In graph (b) ap-
pear gaussian errors density and empirical errors his-
togram of stator resistance error where means equal
#
and variance equal
!
.
0 0.2 0.4 0.6
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Time[s]
Flux error [Wb]
(a)
−0.05 0 0.05 0.1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
Gaussian
Histogram
Figure 5: (a) Flux error. (b) Gaussian and histogram of
error flux.
0 0.2 0.4 0.6
0
20
40
60
80
100
120
Time[s]
Speed error[rad/s]
(a)
−10 0 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
Gaussian
Historam
Figure 6: (a) Speed error. (b) Gaussian and histogram of
error speed.
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0 1 2 3
−1.5
−1
−0.5
0
0.5
1
1.5
x 10
−5
Time[s]
Load torque[Nm]
(a)
0.5 1 1.5
x 10
−5
100
200
300
400
500
600
700
800
900
1000
(b)
Gaussian
Histogram
Figure 7: (a) Load torque error. (b) Gaussian and histogram
of error load torque.
0 0.1 0.2 0.3 0.4 0.5
−0.5
0
0.5
1
1.5
2
2.5
3
x 10
−3
Time[s]
Rotor resistance error [Ω]
(a)
−6 −4 −2 0
x 10
−4
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Gaussian
Histogram
Figure 8: (a) stator resistance error. (b) Gaussian and his-
togram of error rotor resistance.
Sliding mode observer with
:
Under the same conditions with the function
.
One simulates for the function
. The figure 9,
10 and 11 illustrates the pace of error flux, error speed
and error load torque in respectively. The behaviour
of the error of rotor flux is depicted in figure 9 graph
(a); when graph (b) shows the gaussian errors density
and empirical errors histogram of rotor flux error. The
means of error flux equal
#
with small variance
this is almost surety. The pace of speed error
is given by the figure 10 graph (a) and the gaussian
errors density and empirical errors histogram of ro-
tor 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 figure 11 graph (a).In graph (b) appear gaussian
errors density and empirical errors histogram of load
torque error where means of error load torque equal
"#
and variance equal
!
;
the curve of stator resistance is illustrated on figure 12
graph (a).In graph (b) appear gaussian errors density
and empirical errors histogram of stator resistance er-
ror where means equal
"
and variance
equal
!"
.
0 0.2 0.4 0.6 0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
Time[s]
Flux error [Wb]
(a)
−0.05 0 0.05
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
Gaussian
Histogram
Figure 9: (a) Flux error. (b) Gaussian and histogram of
error flux.
0 0.2 0.4 0.6
0
20
40
60
80
100
120
Time[s]
Speed error [rad/s]
(a)
−10 0 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
Gaussian
Histogram
Figure 10: (a) Speed error. (b) Gaussian and histogram of
error speed.
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0 0.2 0.4 0.6
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
x 10
−5
Time[s]
Speed error[rad/s]
(a)
2 2.5 3
x 10
−5
200
400
600
800
1000
1200
(b)
Gaussian
Histogram
Figure 11: (a) Load torque error. (b) Gaussian and his-
togram of error load torque.
0 0.2 0.4 0.6 0.8
−1
0
1
2
3
4x 10
−3
Time[s]
Rotor resistance error [Ω]
(a)
−10 −8 −6 −4
x 10
−4
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
Gaussian
Histogram
Figure 12: (a) stator resistance error. (b) Gaussian and his-
togram of error rotor resistance.
Table 2, 3, 4 and 5 are collection of the preceding re-
sults, we notice the errors mean of observation with the
high gain observer being very near to
with the very small
variance (almost surely), then the sliding mode observer
with the function
is the good in our case.
Table 2: Means and variances of Flux error.
Flux error
H.G Mean
Var
!
!!
Mean
#
Var
!
!
Mean
#
Var
Table 3: Means and variances of speed error.
Speed error
H.G Mean
" #
Var
"
Mean
"#
Var
!
#
Mean
Var
#"
Table 4: Means and variances of load torque error.
Load Torque error
H.G Mean
!
Var
#
Mean
Var
##
Mean
"#
Var
!
Table 5: Means and variances of rotor resistance error.
Rotor resistance error
H.G Mean
Var
Mean
#
Var
!
Mean
"
Var
!"
5 Conclusions
In this paper, adaptive high gain and alternative form for a
adaptive sliding mode observers are presented. they is ob-
server makes possible to observe, rotor flux, rotor speed,
load torque and rotor resistance. An observer with adap-
tive high gain and three others with adaptive sliding mode
which the functions sign,
and
. Observer
whose sign gives chattering. Adaptive sliding mode ob-
server with function
is good for the observation of
rotor flux, rotating speed, load torque and rotor resistance.
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Engineering World
DOI:10.37394/232025.2023.5.1
Ahmed Chouya
E-ISSN: 2692-5079
9
Volume 5, 2023
