J. Software Engineering & Applications, 2010, 3: 221-229
doi:10.4236/jsea.2010.33027 Published Online March 2010 (http://www.SciRP.org/journal/jsea)
Copyright © 2010 SciRes. JSEA
Parameter Identification Based on a Modified PSO
Applied to Suspension System
Alireza Alfi, Mohammad-Mehdi Fateh
Faculty of Electrical and Robotic Engineering, Shahrood University of Technology, Shahrood, Iran.
Email: a_alfi@shahroodut.ac.ir
Received November 25th, 2009; revised January 15th, 2010; accepted January 20th, 2010.
This paper presents a novel modified particle swarm optimization algorithm (MPSO) for both offline and online
parametric identification of dynamic models. The MPSO is applied for identifying a suspension system introduced by a
quarter-car model. A novel mutation mechanism is employed in MPSO to enhance global search ability and increase
convergence speed of basic PSO (BPSO) algorithm. MPSO optimization is used to find the optimum values of
parameters by minimizing the sum of squares error. The performance of the MPSO is compared with other optimization
methods including BPSO and Genetic Algorithm (GA) in offline parameter identification. The simulating results show
that this algorithm not only has advantage of convergence property over BPSO and GA, but also can avoid the
premature convergence problem effectively. The MPSO algorithm is also improved to detect and determine the variation
of parameters. This novel algorithm is successfully applied for online parameter identification of suspension system.
Keywords: Particle Swarm Optimization, Genetic Algorithm, Parameter Identification, Suspension System
1. Introduction
A mathematical model can be provided to describe the
behavior of a system based on obtained data for its inputs
and outputs by system identification. It is necessary to
use an estimated model for describing the relationships
among the system variables for this purpose. The values
of parameters in the estimated model of a system must be
found such that the predicted dynamic response coincides
with that of the real system [1].
The basic idea of parameter identification is to com-
pare the time dependent responses of the system and pa-
rameterized model based on a performance function giv-
ing a measure of how well the model response fits the
system response. It should be mentioned that the model
of system must be regarded fixed through identification
procedure. It means that data are collected from the
process under a determined experimental condition and
after that, the characteristic property of system will stay
the same.
Many traditional techniques for parameter identifica-
tion have been studied such as the recursive least square
[2], recursive prediction error [3], maximum likelihood
[4], and orthogonal least square estimation [5]. Despite
their success in system identification, traditional optimi-
zation techniques have some fundamental problems in-
cluding their dependence on unrealistic assumptions such
as unimodal performance landscapes and differentiability
of the performance function, and trapping in local min-
ima [6].
Evolutionary algorithms (EAs) and swarm intelligence
(SI) techniques seem to be promising alternatives as
compared with traditional techniques. First, they do not
rely on any assumptions such as differentiability, conti-
nuity, or unimodality. Second, they can escape from local
minima. Because of this, they have shown superior per-
formances in numerous real-world applications. Among
them, genetic algorithm (GA) and particle swarm opti-
mization (PSO) are frequently used algorithms in the
area of EAs and SI, respectively. Owing these attractive
features, these algorithms are applied in the area of sys-
tem identification [7–10].
Comparing GA and PSO, both are population based
optimization tools. However, unlike GA, PSO has no
evolution operators such as crossover and mutation. Easy
to implement and the less computational complexity are
advantages of PSO in comparing with GA. The basic
PSO (BPSO) algorithm has good performance when
dealing with some simple benchmark functions. However,
it is difficult for BPSO algorithm to overcome local
minima when handling some complex or multimode
functions. Hence, a modified PSO (MPSO) is proposed
Parameter Identification Based on a Modified PSO Applied to Suspension System
to overcome this shortage. In this paper, a novel mutation
mechanism is introduced to enhance global search of
algorithm. Then, it is demonstrated how to employ the
MPSO method to obtain the optimal parameters of a dy-
namic system.
In order to show the effectiveness of MPSO in system
identification, a quarter-car model of suspension system
is identified as an application. Although a linear model is
proposed for a suspension system for control purposes
[11–14], the MPSO can be applied well to identify the
non-linear systems, as well. It should be noticed that a
suspension system operates under various operating con-
ditions, where parameter variations are unavoidable.
Accurate knowledge of these parameters is important to
form the control laws. Therefore, it is of our interest to
investigate an efficient model parameter tracking ap-
proach to achieve precise modeling results under differ-
ent conditions without using complicated model structures.
In this paper, the MPSO is compared to GA and BPSO
in offline parameter identification of suspension system.
It can be shown that the MPSO has a better performance
than the aforementioned algorithms in solving the pa-
rameter estimation of suspension system. Because of the
superiority of MPSO in offline identification, it can be
used for online parameter identification of suspension
system, as well. In the propose method, the estimated
parameters will not be updated unless any changes in
system parameters is detected by algorithm. A sentry
particle is introduced to detect any change in system pa-
rameters. If a change is detected, the algorithm scatters the
particles around the global best position and forces the algo-
rithm to forget its global memory, then runs the MPSO to
find the new values for parameters. Therefore, MPSO runs
further iterations if any changes in parameters are detected.
The rest of the paper is organized as follow: Next sec-
tion describes problem description. Section 3 introduces
optimization algorithms. The proposed algorithms in
both offline and online parametric identification are pre-
sented in Section 4. Simulation results are shown in Sec-
tion 5. Finally, conclusion and future works are presented
in Section 6.
2. Problem Description
This section presents a quarter-car model of suspension
system and a proper fitness function for optimization
2.1 Suspension System Dynamics
Modeling of vehicle suspension system has been stud-
iedfor many years. In order to simplify the model, a quar-
ter-car model was introduced in response the vertical force
for the suspension system [15] as shown in Figure 1. In
this figure, b is damping coefficient, and are un-
sprung and sprung mass, respectively, and are tire
Figure 1. Schematic diagram of the quarter-car model
and suspension stiffness, respectively, u is the road dis-
placement and y is the vertical displacement of sprung
mass. The linearized dynamic equations at equilibrium
point with an assumption that the tire is in contact with the
road are given as:
)()()( 12
2.2 Problem Statement
When the model of system is fixed through identification
procedure, the parameter identification problem can be
treated as an optimization problem. The basic idea of
parameter estimation is to compare the system responses
with the parameterized model based on a performance
function giving a measure of how well the model re-
sponse fits the system response. Moreover, a common
rule in identification is to use excitation signals that cor-
respond to a realistic excitation of the system such that
the identified linear model is a good approximation of the
system for that type of excitation. Consequently, in order
to estimate the system parameters, excitation signal is
chosen Gaussian band-limited white noise. The band-
width is set to 50 Hz, which is sufficiently higher than
the desired closed-loop bandwidth [14].
Considering Figure 2 the excitation input is given to
both the real system and the estimated model. Then, the
outputs from the real system and its estimated model are
input to the fitness evaluator, where the fitness will be
calculated. The sum of squares error between real and
estimated responses for a number of given samples is
considered as fitness of estimated model. So, the fitness
function is defined as follow:
SSE (()())
 
where N is the number of given sampling steps,
)( s
Copyright © 2010 SciRes. JSEA
Parameter Identification Based on a Modified PSO Applied to Suspension System 223
Real System
Fitness Evaluator
Identifier Algorithms
Estimated Model
Figure 2. The estimation process
and are real and estimated values in each sample
time, respectively. The calculated fitness is then input to
the identifier algorithms, i.e. GA- BPSO and MPSO, to
identify the best parameters for estimated system in fit-
ting procedure by minimizing the sum of square of re-
sidual errors in response to excitation input.
3. Optimization Algorithms
As mentioned before, the parameter identification prob-
lem can be treated as an optimization problem. The pro-
posed MPSO optimization algorithm is compared with
frequently used algorithms in optimization problems,
namely GA and BPSO in the optimization problem in
hand. These algorithms are taken from two main optimi-
zation groups namely evolutionary algorithms (EAs) and
swarm intelligence (SI). These algorithms are currently
used for numerical optimization problems of stochastic
search algorithms.
3.1 Evolutionary Algorithms (EAs)
EAs algorithms are population based, instead of using a
single solution. EAs mimic the metaphor of natural bio-
logical evolution. EAs operate on a population of poten-
tial solutions applying the principle of survival of the
fittest to produce better approximations to a solution. At
each generation, a new set of approximations is created
by two processes. First, selecting individuals according
to their level of fitness in the problem domain. Second,
breeding them together using operators borrowed from
natural genetics.
This process leads to the evolution of populations of
individuals that are better suited to their environment
than they were created from, just as in natural adaptation.
Evolutionary algorithms model natural processes, such as
selection, recombination, mutation, migration, locality
and neighborhood. The majority of the present imple-
mentations of EA come from any of these three basic
types, which are strongly related although independently
developed: Genetic Algorithms (GA), Evolutionary Pro-
gramming (EP) and Evolutionary Strategies (ES). Hence,
in this paper, the proposed method is compared to GA.
3.2 Swarm Intelligence (SI)
SI is the artificial intelligence based on the collective be-
havior of decentralized and self-organized systems. SI
systems are typically made up of a population of simple
agents interacting locally with one another and with their
environment. The agents follow very simple rules. Al-
though there is no centralized control structure dictating
how individual agents should behave, local interactions
between such agents lead to the emergence of complex
global behavior. Natural examples of SI include ant colo-
nies, bird flocking, animal herding, bacterial growth, and
fish schooling. Among them, PSO is a new and frequently
used SI technique.
3.2.1 Basic PSO
PSO is used to search for the best solution by simulating
the movement and flocking of birds [16]. The algorithm
works by initializing a flock of birds randomly over the
searching space, where every bird is called as a “particle”.
These “particles” fly with a certain velocity and find the
global best position after some iteration. At each iteration,
each particle can adjust its velocity vector based on its
momentum and the influence of its best position as well as
the best position of the best individual. Then, the particle
flies to a new computed position. Suppose that the search
space is n-dimensional, and then the position and velocity of
particle are represented by
and , respectively. The fitness of
each particle can be evaluated according to the objective
function of optimization problem. The best previously
visited position of the particle is noted as its personal
best position denoted by . The
position of the best individual of the swarm is noted as
the global optimum position . At
each step, the velocity of particle and its new position
will be assigned as follows:
Xxx x
iii in
Pp pp
,, ... ,T
Gg gg
i in
v v
)()()()1( 2211iiiii XGrcXPrctVtV 
)1()()1( 
tVtXtXiii (5)
is called the inertia weight that controls the
impact of previous velocity of particle on its current one.
and are independently uniformly distributed ran-
dom variables in a range of [0,1]. and are posi-
tive constant parameters called acceleration coefficients
which control the maximum step size. In the references
[17,18], several strategies of inertial weight
given. Generally, the inertial weight
should be re-
duced rapidly in the beginning stages of algorithm but it
should be reduced slowly around optimum. If the veloc-
ity exceeds the predefined limit, another restriction called
is used.
In BPSO, (4) is used to calculate the new velocity ac-
Copyright © 2010 SciRes. JSEA
Parameter Identification Based on a Modified PSO Applied to Suspension System
cording to its previous velocity, the distance of its current
position from both its own personal best position and the
global best position of the entire population. Then the
particle flies toward a new position according (5). This
process is repeated until a stopping criterion is reached.
3.2.2 The Proposed Modified PSO
As mentioned before, possible trapping in local minima
when handling some complex or multimode functions is
a shortage of BPSO [19–21]. Hence, the motivation of
the proposed method is to overcome this drawback. In
BPSO, as time goes on, some particles become quickly
inactive because their states are similar to the global op-
timum. As a result, they lose their velocities. In the sub-
sequent generations, they will have less contribution to
the search task due to their very low global search activ-
ity. In turn, this will induce the emergence of a state of
premature convergence.
To deal with the problem of premature convergence,
several investigations have been undertaken to avoid the
premature convergence. Among them, many approaches
and strategies are attempted to improve the performance
of PSO by variable parameters. A linearly decreasing
weight into PSO was introduced to balance the global
exploration and the local exploitation in [17]. PSO was
further developed with time-varying acceleration coeffi-
cients to modify the local and the global search ability
[18]. A modified particle swarm optimizer with dynamic
adaptation of inertia weight was presented [19]. More-
over, some approaches aim to divert particles in swarm
among the iterations in algorithm. Mutation PSO em-
ployed to improve performance of PSO [20]. The con-
cepts of ‘‘subpopulation” and ‘‘breeding” adopted to
increase the diversity [21]. An attractive and repulsive
PSO developed to increase the diversity [22].
In this paper, a modified particle swarm optimization
(MPSO) algorithm is proposed to avoid premature con-
vergence and increase the convergence speed of algo-
rithm. In our proposed method, after some iteration, the
algorithm measures the search ability of all particles and
mutates a percentage of particles which their search abil-
ity is lower than the others. Our motivation is that parti-
cles with low search ability become inactive as their fit-
ness do not grow and need to mutate for getting a chance
to search new areas in solution space, which may not
been meet already. Also the mutation rate is not constant
and if the global best doesn’t grow, the rate of mutation
is increased. If the fitness of global optimum does not
grow, the algorithm can get stuck in local minima forever
or at least for some iteration, which lead to a slow con-
vergence speed. In the other words, if the global best of
the present population is equal to that of the previous
population (solution converges), mutation rate is set to a
higher value , such that the diversity of particles is
increased so to avoid premature convergence. Otherwise
(solution diverges), Pm is set to a lower value Pml, since
the population already has enough diversity. The adap-
tive mutation rate scheme is described as
;if G(t)=G(t-1)
;if G(t)>G(t-1)
where and are 0.2 and 0.1, respectively. Con-
sequently the mutation rate will be increased by 0.1, if
the global optimum doesn’t grow, until a growth on the
fitness of global optimum occurs. In order to measure the
search ability for particle iat each iteration, we take
advantage of the fitness increment of the local optima in
a designated interval
, from iteration Tt
to t.
consequently; the parameter is used to measure the
search capability of particle.
))(())((TtPFtPFC iii  (7)
where is the fitness of the best position for i-th
particle in t-th iteration, and is a designated
interval. Particles with low search ability (small) have
low increment in the best local value for their low global
search activity and so in our algorithm, they have a big-
ger change to mutate to get a chance to search a new area
in the search space.
))((tPF i
Generally, In MPSO algorithm, after iterationT
, all
particles are sorted according to their parameter. Then
the swarm is divided into two parts: The active part includ-
ing the top
with higher Cand the inactive
part consisting of the rest particles with smaller
whereas is size of the swarm. Particles in the first
part update their velocities and position the same as BPSO
algorithm. Finally, in order to increase the diversity of the
swarm, the inactive particles are chosen to mutate by
adding a Gauss random disturbance to them as follows:
ijijij xx
where is the j-th component of the i-th inactive parti-
is a random variable, which follows a Gaussian
distribution with a mean value of zero and a variance
value of 1, namely(0,1)
ij N
. This strategy prevents
all particles to divert from the local convergence. Instead,
only inactive particles are mutated.
4. Implementation of the MPSO
In this section, the procedure of MPSO in online and
offline system parameter identification is described.
Copyright © 2010 SciRes. JSEA
Parameter Identification Based on a Modified PSO Applied to Suspension System 225
4.1 Offline Identification
MPSO algorithm is applied to find the best system pa-
rameter, which simulates the behavior of dynamic system.
Each particle represents all parameters of estimated model.
The procedure for this algorithm can be summarized as
Step 1: Initialize positions and velocities of a group of
particles in an M-dimensional space as random points
where M denotes the number of system parameters;
Step 2: Evaluate each initialized particle’s fitness
value using (3);
Step 3: Set as the positions of current particles
while is the global best position of initialized particles.
The best particle of current particles is stored;
Step 4: The positions and velocities of all particles are
updated according to (4) and (5), and then a group of
new particles are generated;
Step 5: Evaluate each new particle’s fitness value. If the
new position of i-th particle is better than, set as the
new position of the i-th particle. If the fitness of best posi-
tion of all new particles is better than fitness of , then
is updated and stored;
Step 6: If iteration, calculate the mutation rate
() and search ability of each particle using (6) and (7),
respectively. Then mutate number of particle
with lower search ability;
Step 7: Update the velocity and location of each parti-
cle according to the (4) and (5). If a new velocity is be-
yond the boundary , the new velocity will be
set as or ;
],[ maxmin VV
Step 8: Output the global optimum if a stopping crite-
ria is achieved, else go to Step 5.
When a stopping criterion is occurred, the global op-
timum is the best answer for the problem in hand (the
best estimated system parameters).
4.2 Online Identification
The proposed algorithm sequentially gives a data set by
sampling periodically. The optimized values of parameters
for the first data set are determined by using a procedure
described in Subsection 4.1. The estimated parameters will
not be updated unless a change in the system parameters is
detected. In order to detect any change in system pa-
rameters, the global optimum in the later period is no-
ticed as a sentry particle. In each period, the sentry parti-
cle is evaluated at first and if the fitness of the sentry
particle in the current period is bigger than the previous
one, the changes in parameters are confirmed. If no
changes are detected, the algorithm leaves this period
without changing the positions of particles. When any
changes in parameters occur, the algorithm runs further
to find the new optimum values. For this purpose, a new
coefficient (
) is introduced as follows:
(S ( ))(S (1))
(S ( ))
fitness i
 (0 1) (9)
where is the sentry particle in the i-th period.
will be bigger than zero if the fitness of the sentry parti-
cle at the current period is bigger than the previous one.
Thus, changes in model parameters are detected by in-
at each period. In this case, the particles in
population must forget their current global and personal
memories in order to find the new global optimum. The
fitness of global optimum particle and personal bests of
all particles are then evaporated at the rate of a big
evaporation constant. As a result, other particles have a
chance of fitness bigger than the previous global opti-
mum. Moreover, the velocities of particles are increased
to search in a bigger solution space for new optimal solu-
tion. When a change in system parameters is detected,
the following changes are considered.
SiTPfitnessPfitness ii,...,1)()( 
GG() ()
new old
itnessfitness T
maxnew old
 (12)
T is an evaporation constant. Also (12) shows that the
velocity of particles increase by only in one
iteration. Notice that a bigger , i.e. greater changes in
parameters, causes a bigger velocity. This means that if
significant changes in system parameters occur, the par-
ticles must search a bigger space and if a little change
occurs, particle search around the previous position to
find the new position. This strategy accelerates conver-
gence speed of the algorithm, which is an important issue
in online identification.
5. Simulation Results
In this section the proposed MPSO algorithm is applied
to identify parameters of a suspension system, which its
nominal parameters are summarized in Table 1 [15]. In
order to show the performance of the proposed MPSO in
Table 1. Suspension parameters [15]
ParametersNominal value
m 26 kg
m 253 kg
k 90000 N/m
k 12000 N/m
b 1500 N·sec/m
Copyright © 2010 SciRes. JSEA
Parameter Identification Based on a Modified PSO Applied to Suspension System
the problem in hand, it is compared to two frequently
used optimization algorithms, including GA and BPSO.
Simulation results have been carried out in two parts.
In the first part, in order to show the effectiveness of
the proposed MPSO in offline identification, it has been
compared with GA and BPSO. In the second part, the
proposed MPSO is applied to online parameter identifi-
cation for suspension system. In both BPSO and MPSO
algorithms, , and the inertia weight is set to 0.8.
Also, the simulation results are compared with GA,
where the crossover probability and the mutation
probability are set to 0.8 and 0.1, respectively.
5.1 Offline Parameter Identification
Owing to the randomness of the heuristic algorithms,
their performance cannot be judged by the result of a
single run. Many trials with different initializations
should be made to acquire a useful conclusion about the
performance of algorithms. An algorithm is robust if it
gives consistent result during all the trials. The searching
ranges are set as follows:
20 30m
10000 1k
, ,,
, .
200 300m
1200 1700b
85000 90000k
In order to run BPSO, MPSO and GA algorithms, a
population with a size of 10 for 100 iterations is used.
Comparison of results on sum of squares error resulted
from 20 independent trials with N = 1000, 2000 and 2500
are shown in Tables 2–4, respectively. This comparison
shows that the MPSO is superior to GA and BPSO.
Moreover, MPSO is significantly more robust than other
algorithms because the best and the mean values ob-
tained by MPSO are very close to the worst value. In
addition, the convergence speed of GA, BPSO and
MPSO are compared. Figure 3 shows the convergence
speed of these algorithms during 100 iterations which
proves that the convergence speed of the proposed
MPSO is faster than GA and BSO which can be conclude
that MPSO is more proper than aforementioned algo-
rithms. Figure 4 confirms the success of optimization
process by using MPSO algorithm. The identified pa-
rameters are, , , and b, respectively. In
this figure, the data set is formed by 1000 samples. In
addition, to compare computational time of these algorithms,
a threshold of is considered as stopping condition, in
contrast to a predefined number of generation. Then each
algorithm runs 20 times and the average of elapsed time is
considered as a criteria for computational time. Table 5
illustrates the results obtained by GA, BPSO, and MPSO. It
is clearly obvious that, the proposed algorithm spends ex-
tremely fewer iteration and less computational time to reach
a predefined threshold as compared with other algorithms.
Hence, it can be concluded that IPSO is more proper than
Table 2. Comparison of GA, BPSO and MPSO in offline
identification for N=1000
Best Mean Worst
GA 3.116
BPSO 6.45 8
2.18 5
MPSO 3.12 10
8.46 9
Table 3. Comparison of GA, BPSO and MPSO in offline
identification for N = 2000
Best Mean Worst
GA 6.986
4.65 3
BPSO 7.75 8
5.13 5
MPSO 2.32 9
6.98 8
Table 4. Comparison of GA, BPSO and MPSO in offline
identification for N = 2500
Best Mean Worst
GA 7.246
1.12 2
BPSO 8.12 7
4.65 5
MPSO 5.43 9
7.63 8
Table 5. Iterations and time required
Algorithm GA BPSO MPSO
Iterations 182 121 49
Elapse Time (sec) 28.21 10.34 4.69
Figure 3. Comparison of convergence speed for GA, BPSO
and MPSO
Copyright © 2010 SciRes. JSEA
Parameter Identification Based on a Modified PSO Applied to Suspension System 227
(a) Unsprung mass m1
(b) Sprung mass m2
(c) Tire stiffness k1
(d) Suspension stiffness k2
(e) Damping coefficient b
Figure 4. MPSO process for identification of suspension
aforementioned algorithms in terms of accuracy and con-
vergence speed.
5.2 Online Parameter Identification
Based on the pervious section, the MPSO has more ac-
curacy and faster convergence speed than GA and BPSO
in off-line identification. Because of this, the proposed
method is applied for online identification of suspension
system parameters. During online simulation, the sam-
pling frequency is set to 100 kHz such that 1000 pairs of
data are sampled within 0.01 msec in each period to form
a data set. If a change in the model parameter is detected
by sentry particle in a period, the MPSO continues to run.
When the fitness of global best becomes lower than a
threshold, the simulation for this period is then stopped.
There will be no MPSO iteration unless another change
in system parameter detect.
Figure 5 shows the fitness evaluation of the proposed
method when some changes in system parameters are
occurred. First nominal values of parameters are used
and MPSO detects these parameters after 37 iterations
for a threshold 10-5. If changes in parameters occur the
MPSO algorithm runs further. To show the performance
of the proposed method in tracking time-varying pa-
rameters, two sudden changes are applied to suspension
parameters. At the first stage, damping coefficient is
changed from 1500 to 1550. At the second stage, tire
stiffness is varied from 90000 to 95000. It can be seen
that after the first change the algorithm detects new op-
timal parameters after only 19 iterations. And, after the
second change the algorithm finds the new optimal pa-
rameters after only 27 iterations. It can see that the pro-
posed method can track any change in parameters. Also
since the particles are scattered around the previous
global optimum depending on the values of changes in
parameters, the new global optimum is found fast. Fig-
ures 6 and 7 show the online identification results of the
proposed algorithm when k1 and b are considered as time
varying parameters. It can be seen that the proposed ap-
proach can identify time-varying parameters successfully.
The dashed lines in Figures 5–7 signify the moment that
the sentry particle has detected some change in system
1500 1550
90000 95000
Figure 5. MPSO process in online identification of suspen-
sion system
Copyright © 2010 SciRes. JSEA
Parameter Identification Based on a Modified PSO Applied to Suspension System
Figure 6. Identifying a time-varying damping coefficient
parameter by MPSO
90000 95000
1500 1550
Figure 7. Identifying a time-varying tire stiffness parameter
6. Conclusions
A quarter-car model of suspension system was used to
show the effectiveness of MPSO in system identification.
It has been shown that MPSO is superior to GA and
BPSO in offline identification. Owing these attractive
features, MPSO is applied to online identification. The
estimated parameter will be updated only if a change in
system parameters is detected. Thus, the proposed algo-
rithm is a promising particle swarm optimization algorithm
for system identification. Future works in this area willin-
clude considering variable parameters in nonlinear sus-
pension model.
7. Acknowledgments
Our Sincere thanks to Mr. Hamidreza Modarress for his
helpful comments and his advice to improve this research
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