Parameter Estimation of Electrical Vehicle Motor
HAFID OUBOUADDI, FATIMA EZZAHRA EL MANSOURI, ALI BOUKLATA,
RAMZI LARHOUTI, ABDELMALEK OUANNOU, ADIL BROURI
ENSAM, L2MC Laboratory,
Moulay Ismail University,
MOROCCO
Abstract: - Presently, the problem of parameter estimation of motors used in electrical vehicles is discussed. To
make the control more efficient, a model of the electrical motor is developed. Then, a parameter estimation
method is established to determine the parameters of the used electrical motor. The proposed method is easy
and can be easily implemented. Furthermore, a mathematical model of an electrical vehicle’s motor using linear
and nonlinear blocks is obtained. The given model will allow us to perform several experiments without any
extra fees.
Key-Words: - Electrical vehicle motor; Linear and nonlinear systems; Parameter estimation; Polynomial
function; Parallel connection; Excitation signal; Set of frequencies.
Received: January 25, 2023. Revised: September 27, 2023. Accepted: November 11, 2023. Published: December 5, 2023.
1 Introduction
The problem of nonlinear system identification in
industrial systems always holds a special interest,
[1], [2], [3], [4], [5]. The system identification is an
essential step before the control. Several methods
allowing the determination of system parameters
have been proposed in literature, [6], [7], [8].
Many solutions have been evaluated on
practical systems, [9], [10], [11], [12], [13].
The established estimation parameter methods
can be classified into several categories. For
instance, the black-box methods are based on
several approximations without prior knowledge of
the system.
Most available parameter estimation approaches
are focused on the cascading (series) connection of
linear and nonlinear subsystems. Examples of these
nonlinear structures are Wiener systems,
Hammerstein models, Wiener-Hammerstein
systems, and Hammerstein-Wiener structures, [14],
[15], [16], [17], [18], [19].
There are practical systems that cannot be
described by these nonlinear structures. The parallel
connections of linear and nonlinear blocks have
been already proposed, [20], [21], or, [4]. Examples
of industrial systems that can be modeled by this
nonlinear structure are dealt with in, [21]. The
parallel connections of linear and nonlinear
subsystems, [22], or the combinations of parallel
connections and series connections of linear and
nonlinear blocks, [4], can be very efficient and are
more general than the cascade connections.
Presently, the problem of parameter estimation
of real systems is discussed. The considered system
in this study is an electrical machine studied in, [2],
[12], [13]. This motor is used in electric vehicles.
The main issue in the control of electrical vehicle
motors is related to the fact that it is very difficult to
establish an exact model describing the studied
system. Then, even if a model is obtained, how to
practically proceed with the parameter
determination. In this work, a model of an electrical
vehicle motor is given. Then, an estimation method
of the established nonlinear system is developed.
The electrical vehicle motor is modeled by the
parallel connection of linear and nonlinear blocks.
Using this parallel model, a method allowing the
determination of motor parameters is presented. The
estimation parameter solution is based on sine
signals. Specifically, the parallel model of the
electrical vehicle motor by sine signals. Then, using
the input data and the corresponding output, the
parameters of the electrical motor will be obtained.
For convenience, the rest of this paper is organized
as follows. Firstly, the studied problem is presented
in Section 2. Then, the parameter estimation of the
electrical vehicle motor is discussed in Section 3.
Finally, examples of simulation results are
established in Section 4.
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.46
Hafid Oubouaddi, Fatima Ezzahra El Mansouri,
Ali Bouklata, Ramzi Larhouti,
Abdelmalek Ouannou, Adil Brouri
E-ISSN: 2224-2856
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2 Electrical Vehicle Motor and
Problem Statement
The objective presently is to obtain a model
describing the nonlinear behavior of an electrical
vehicle motor. Furthermore, using this model, the
parameters of the electrical vehicle motor can be
determined.
The electrical vehicle studied in this paper is
shown in Figure 1. This machine has several
advantages compared to other machines, [23], [24].
For instance, this machine type is less expensive,
robust, and has fewer losses.
The electrical equation of this proposed machine
is given as follows:
(1)
where is the phase voltage, is the phase
resistor, is the current flowing through the
phase, and is the flux linkage. The latter is
expressed by the following equation:
(2)
where is the phase inductance that depends on
the input signal and the rotor position .
Knowing that the speed of the motor shaft is
related to the rotor position by the expression:
(3)
by replacing the flux linkage with its
expression given by (2) in (1), and using (3) one
immediately gets:
(4)
where the dependence has been removed from the
right-hand side of (2) to alleviate the equation.
Accordingly, in static experiments the motor speed
boils down to zero and the expression of the
output signal in (4) becomes:
(5)
Without loss of generality, it is readily seen that
the electrical equation of this machine (taking the
signal as input signal and the wave as
output signal) can be modeled by the parallel
connection of linear and nonlinear blocks, [10], and,
[13], for more details. The structure of this nonlinear
system is given by Figure 2, where the function
is a nonlinearity and the linear block is described by
its transfer function .
Note that several industrial systems can be
captured using the parallel connection of linear and
nonlinear subsystems, [20], [25]. Then, the aim
presently is to develop a method to estimate the
parameters of the studied electrical motor. The
details of the parameter determination method will
be discussed in the following section.
Fig. 1: Used electrical motor of type 8/6
Fig. 2: Nonlinear system describing the electrical
vehicle’s motor
3 Determination of Motor Parameters
In this section, we propose a solution to determine
the parameters of an electrical motor. The latter is
described by the parallel connection of linear and
nonlinear blocks as shown in Figure 2.
Firstly, note that when the input signal is a
sine signal, the output signal is thus periodic
but not necessarily a sine signal. Furthermore, it is
commonly known that the system nonlinearity of
this electrical motor can be described by a
polynomial function, [10], [12], and, [13], for more
details. Then, the nonlinearity is a polynomial
function of finite degree ,. Let
denoting the parameter vector of the system
nonlinearity . Specifically, the inner signal
according to the input signal can be
expressed as:
(6)
similarly, the inner signal and the system input
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DOI: 10.37394/23203.2023.18.46
Hafid Oubouaddi, Fatima Ezzahra El Mansouri,
Ali Bouklata, Ramzi Larhouti,
Abdelmalek Ouannou, Adil Brouri
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are related by the following expression:
(7)
where the symbol « » stands for the convolution
product and denotes the impulse response of
linear part .
Finally, it follows from (6)-(7) and Figure 2 that
the output signal can be expressed as:
(8)
where the signal is accounted for the noise
measurement.
In this study, we propose an excitation signal
(i.e., the input signal) of sine type having the
following form:
(9)
where the frequency belongs to a set of given
frequencies . Accordingly, using
(6)-(7) and (9), the inner signals and can
be, respectively, expressed as:
(10)
(11)
where denotes the phase or argument of the
linear block, i.e., . The term
in (11) can be decomposed, for any
integer , as follows, [1], and, [21]:
(12)
Then, it follows from (11)-(12) that the inner signal
can be rewritten as:
(13)
where depend on the maximum amplitude of
the input signal and the coefficients of the
parameter vector . Finally, using (8),
(10), and (13) the output signal can be
expressed as:
(14)
Therefore, it is readily seen that the output signal
can be rewritten as follows:
(15)
where the coefficients are expressed
as:
(16)
and the phases are given as:
(17)
This means that the coefficients
depend on the maximum amplitude of the input
signal and the coefficients of the parameter vector
, while the parameters of the
fundamental signal depend on the
amplitude , the coefficients of the parameter vector
, the gain module , and the
phase .
It is readily shown from (14) that the free-noise
output signal is not necessarily a sine signal,
but periodic of the same period as the
input signal .
This outcome is quite interesting in two ways.
Firstly, using the fact that the output signal is
periodic of known period . Then, a
filtering of the output signal can be performed,
[1], [19], [26], using the following algorithm:
for
(18a)
for (18b)
where N is any large integer. Bearing in mind that
the main aim of this section is to estimate the
parameters of the vehicle motor, i.e., the parameter
vector and parameters of linear block
. This can be achieved by
measuring the amplitude and phase of sine terms
in the output signal .
4 Simulation
This section aims to present some examples of
simulation to show the effectiveness of the obtained
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DOI: 10.37394/23203.2023.18.46
Hafid Oubouaddi, Fatima Ezzahra El Mansouri,
Ali Bouklata, Ramzi Larhouti,
Abdelmalek Ouannou, Adil Brouri
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results. In simulation, the used electrical motor has
the following characteristics:
- Number of rotor teeth Nr=6
- Number of stator teeth Ns=8
- Nominal speed
- Nominal current
- Nominal power
- Resistance R=0,8493 Ω
- Phase inductance minimum
- Phase inductance maximum
Figure 1 shows this motor. Then, to determine the
parameters of the electrical vehicle motor, the
proposed method developed in section 3 will be
used.
Firstly, the used motor of Figure 1 is excited by the
input signal (9) by choosing a set of frequencies
. Then, for the following input signal
:
(19)
where the amplitude is and the frequency
. The plot of the input signal
is given in Figure 3. It is shown in section 3 that
the output signal is not necessarily a sine
signal, but periodic of the same period
as the input signal .
In this respect, the corresponding plot of the
output signal is represented in Figure 4. This
outcome demonstrates that the output signal is
also periodic as the input signal .
Fig. 3: The input signal
Fig. 4: The output signal
For convenience, this test is repeated for other
periods
, where
.
Then, the input of the system in this case is
given in Figure 5. The corresponding output is given
in Figure 6.
This outcome demonstrates the obtained results
in section 3.
Fig. 5: The input signal of the system
Fig. 6: The output signal
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Hafid Oubouaddi, Fatima Ezzahra El Mansouri,
Ali Bouklata, Ramzi Larhouti,
Abdelmalek Ouannou, Adil Brouri
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Finally, the last experiment is repeated for the
period
, where
.
Then, the input of the system in this case is given
in Figure 7. The corresponding output is given in
Figure 8.
Which validates the obtained results in section 3.
Fig. 7: The input signal of the system
Fig. 8: The output signal
5 Conclusion
The problem of system parameter estimation is
addressed for a more general nonlinear model. In
this respect, the proposed nonlinear system is
composed of the parallel connection of linear and
nonlinear blocks. The aim is to determine the
parameters of an electrical vehicle’s motor using
this nonlinear structure. This solution is easy and
more general.
References:
[1] A. Brouri, Wiener-Hammerstein nonlinear
system identification using spectral analysis,
Int Journal Robust Nonlinear Control, Wiley
Online, Vol. 32, no. 10, pp. 6184-6204, 2022.
Doi: 10.1002/rnc.6135
[2] L. Kadi, A. Brouri, A. Ouannou, K. Lahdachi,
Modeling and Determination of Switched
Reluctance Machine Nonlinearity, 4th IEEE
Conference on Control Technology and
Applications, pp. 898-902, Montréal, Canada,
2020.
[3] A. Brouri, L. Kadi, S. Slassi, Identification of
Nonlinear Systems, European Conference on
Electrical Engineering and Computer Science
(EECS), IEEE, Bern, Switzerland, Nov 17-19,
pp. 286-288, 2017.
http://doi.ieeecomputersociety.org/10.1109/E
ECS.2017.59.
[4] A. Brouri, F. Giri, Identification of series-
parallel systems composed of linear and
nonlinear blocks, International Journal of
Adaptive Control and Signal Processing, Vol:
37, No: 8, pp. 2021-2040, 2023,
DOI: 10.1002/acs3624
[5] Castro, G.R., Agudelo, O.M., Suykens, A.K.,
Impulse response constrained LS-SVM
modelling for MIMO Hammerstein system
identification, International J. of Control,
Vol. 92, no. 4, pp. 908-925, 2019.
[6] A. Brouri, A. Ouannou, F. Giri, H.
Oubouaddi, F.Z. Chaoui, Identification of
Parallel Wiener-Hammerstein Systems, 14th
IFAC – ALCOS, Vol: 55, No: 12, June 29 -
July 1, Casablanca, Morocco, pp. 25-30, 2022.
[7] M. Pawlak and Z. Hasiewicz, Nonlinear
system identification by the Haar
multiresolution analysis, IEEE Transactions
on Circuits and Systems I: Fundamental
Theory and Applications, Vol. 45 (9), pp. 945-
961, 1998.
[8] F. Giri, Y. Rochdi, A. Brouri, A. Radouane,
F.Z. Chaoui, Frequency identification of
nonparametric Wiener systems containing
backlash nonlinearities, Automatica, Vol. 49,
pp. 124-137, 2013.
[9] L. Kadi, A. Brouri, A. Ouannou, Frequency-
Geometric Identification of Magnetization
Characteristics of Switched Reluctance
Machine, Advances in Systems Science and
Applications, Vol: 20, No: 4, pp. 11-26, 2020.
[10] A. Brouri, L. Kadi, A. Tounzi, A. Ouannou &
J. Bouchnaif, Modelling and identification of
switched reluctance machine inductance,
AJEEE Taylor & Francis, Vol: 18, No: 1, pp.
8-20, 2021.
https://doi.org/10.1080/1448837X.2020.18662
69.
[11] I. W. Hunter and M. J. Korenberg, The
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
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Abdelmalek Ouannou, Adil Brouri
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Volume 18, 2023
identification of nonlinear biological systems:
Wiener and Hammerstein cascade models,
Biological Cybernetics, Vol. 55 (2-3), pp.
135-144, 1986.
[12] F. Z. El Mansouri, L. Kadi, A. Brouri, F. Giri,
F.Z. Chaoui, Identification of Magnetization
Characteristics of Switched Reluctance
Machine, 14th IFAC – ALCOS, Vol: 55, No:
12, June 29 - July 1, Casablanca, Morocco,
pp. 126-131, 2022.
[13] A. Ouannou, A. Brouri, F. Giri, L. Kadi, H.
Oubouaddi, S.A. Hussien, N. Alwadai, M.I.
Mosaad, Parameter Identification of Switched
Reluctance Motor SRM Using Exponential
Swept-Sine Signal, Machines, Vol. 11, 625,
2023, DOI: 10.3390/machines11060625
[14] A. Brouri, F. Giri, F. Ikhouane, F.Z. Chaoui,
and O. Amdouri, Identification of
Hammerstein-Wiener systems with Backlash
input nonlinearity bordered by straight lines,
In 19th IFAC, Cape Town, pp. 475-480, 2014.
[15] A. Brouri, L. Kadi, Identification of Nonlinear
Systems, SIAM CT'19, Chengdu, China, Jun
19-21, pp. 22-24, 2019.
https://doi.org/10.1137/1.9781611975758.4.
[16] A. Brouri, L. Kadi, and S. Slassi. Frequency
Identification of Hammerstein-Wiener
Systems with Backlash Input Nonlinearity.
International Journal of Control, Automation
and Systems, Springer, Vol. 15, No. 5, pp.
2222-2232, 2017.
[17] F. Giri, A. Radouane, A. Brouri, and F.Z.
Chaoui, Combined frequency-prediction error
identification approach for Wiener systems
with backlash and backlash-inverse operators,
Automatica, Vol. 50, pp. 768-783, 2014.
[18] A. Brouri, F.Z. Chaoui, O. Amdouri, F. Giri,
Frequency Identification of Hammerstein-
Wiener Systems with Piecewise Affine Input
Nonlinearity, In 19th IFAC World Congress,
Cape Town, South Africa, August 24-29, pp.
10030-10035, 2014.
[19] A. Brouri, F.Z. Chaoui, F. Giri, Identification
of Hammerstein-Wiener Models with
Hysteresis Front Nonlinearities, International
Journal of Control, Taylor Francis, pp. 1-15,
2021, DOI: 10.1080/00207179.2021.1972160.
[20] A. Brouri, L. Kadi, K. Lahdachi,
Identification of nonlinear system composed
of parallel coupling of Wiener and
Hammerstein models, Asian Journal of
Control, Vol. 24, no. 3, pp. 1152-1164, 2022,
DOI: 10.1002/asjc.2533
[21] A. Ouannou, F. Giri, A. Brouri, H.
Oubouaddi, C. Abdelaali, Parameter
Identification of Switched Reluctance Motor
Using Exponential Swept-Sine Signal, 14th
IFAC – ALCOS, Vol: 55, No: 12, June 29 -
July 1, Casablanca, Morocco, pp. 132-137,
2022.
[22] Oubouaddi, Hafid, El Mansouri, Fatima
Ezzahra, et BROURI, Adil. Parameter
Estimation of Linear and Nonlinear Systems
Connected in Parallel. WSEAS Transactions
on Mathematics, 2023, vol. 22, p. 373-377,
https://doi.org/10.37394/23206.2023.22.44.
[23] T. Rabyi, A. Brouri, Ant colony optimization
algorithm and fuzzy logic for switched
reluctance generator control, AIMS Energy,
Vol: 10, No: 5, pp. 987-1004, 2022,
doi: 10.3934/energy.2022045.
[24] L. Kadi, A. Brouri, A. Ouannou, K. Lahdachi,
Modeling and Determination of Switched
Reluctance Machine Nonlinearity, IEEE Conf.
on Control Tech. and App., pp. 898-902,
Montréal, Canada, 2020.
[25] D. Westwick, M. Ishteva, P. Dreesen, and J.
Schoukens. Tensor Factorization based
Estimates of Parallel Wiener Hammerstein
Models. IFAC Papers Online, Vol. 50 (1), pp.
9468-9473, 2017.
[26] M. Benyassi, A. Brouri, T. Rabyi and A.
Ouannou "Identification of nonparametric
linear systems", International Journal of
Mechanics, Vol: 13, pp. 60-63, 2019.
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DOI: 10.37394/23203.2023.18.46
Hafid Oubouaddi, Fatima Ezzahra El Mansouri,
Ali Bouklata, Ramzi Larhouti,
Abdelmalek Ouannou, Adil Brouri
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Hafid Oubouaddi: Methodology,
conceptualization, software, validation.
- Fatima Ezzahra El Mansouri: Methodology,
software, validation.
- Ali Bouklata: Methodology, software, validation.
- Ramzi Larhouti: Methodology, simulation.
- Abdelmalek Ouannou: Project administration,
supervision.
- Adil Brouri: Conceptualization, formal analysis,
investigation, project administration, supervision.
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 authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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DOI: 10.37394/23203.2023.18.46
Hafid Oubouaddi, Fatima Ezzahra El Mansouri,
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Abdelmalek Ouannou, Adil Brouri
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