International Journal of Automation and Computing 12(5), October 2015, 482-489
DOI: 10.1007/s11633-015-0905-y
RIM: A Matlab Software Tool for Recursive
Identification Methods
Aissa Rebai1 Kamel Guesmi2 Boualem Hemici11Process Control Laboratory, National Polytechnic School of Algiers, El Harrach, Algiers 16200, Algeria
2CReSTIC, IUT of Troyes, 09 Rode of Quebec, Troyes 10026, France
Abstract: This paper introduces a recursive identification methods toolbox (called RIM) running under Matlab environment for
dynamic system identification from available data. The RIM includes many methods which are generally used. The RIM helps
users to validate the theoretical results and to carry out comparison between identifications methods without the need of algorithms
programming. Furthermore, the RIM can be used as an education platform to study the identification parameters effect on model
validity and results accuracy. To show its performance and capability, the RIM is evaluated through many application examples.
Keywords: Matlab graphical user interfaces (GUI), system identification, recursive methods, multi-innovation algorithms, auxiliary
model algorithms, model validation.
1 Introduction
In dynamic system theory, the forward problem is the
system behavior description using mathematical models.
These expressions can be established using analytical devel-
opment and physical laws, which is difficult in most cases.
To deal with this problem, the system behavior can be ap-
proximated using the identification theory[1, 2].
In this context, there is several methods of identifica-
tion and the most used are those called recursive meth-
ods. Indeed, they are frequently used because of their high
performance and capability to approximate a large class of
systems[3−6].
In order to simplify the use of these techniques, sev-
eral toolboxes and graphical user interfaces (GUI) were
implemented in Matlab environment and we can men-
tion, frequency domain system identification toolbox[7],
the continuous-time system identification (CONTSID)
toolbox[8], recursive identification algorithms library[9], the
mathworks system identification toolbox[10], the interac-
tive tool for system identification education (ITSIE)[11], the
computer aided program for time-series analysis and iden-
tification of noisy systems (CAPTAIN) toolbox[12], and the
university of newcastle identification toolbox (UNIT)[13].
However, these tools are, in most cases, not easy to use,
commercial and need a minimum of information on Matlab
environment manipulations. Furthermore, these tools are
relatively limited to specific methods and do not allow the
comparison between the main identification techniques.
In this paper, we present RIM which is an identification
platform that includes the main recursive methods. This
platform is developed using Matlab GUI[14], and is non-
Regular paperManuscript received March 15, 2014; accepted October 23, 2014Recommended by Associate Editor Sheng Chenc© Institute of Automation, Chinese Academy of Science and
Springer-Verlag Berlin Heidelberg 2015
commercial and freely available. RIM possesses the advan-
tage to be easy to use by beginners and students to validate
the theoretical results without any need to learn about Mat-
lab programming environment and manipulations.
The paper is organized as follows. In Sections 2 and
3, the main recursive identification methods are presented.
Description of the RIM GUI is given in Section 4. Applica-
tion examples are discussed in Section 5.
2 Recursive identification algorithms
The identification problem can take the form as
y(t) = θTϕ(t) + e(t) (1)
where y(t) is the system output, θ is the parameter vector
to be estimated, ϕ is the observations vector, and e(t) is a
white noise with zero mean value and constant variance. In
this paper, the superscript T denotes the transpose. The
dimensions of ϕ and θ depend on the model choice.
In this section, we give a survey on the most known al-
gorithms used in systems identification.
2.1 Stochastic gradient (SG) algorithm
From (1), we define the criterion J1(t) as
minθ(t)
J1(t) = E{||y(t) − θT(t)ϕ(t)||2}. (2)
Minimizing this criterion and using the stochastic gradient
search principle, we obtain the stochastic gradient identifi-
cation algorithm[15]
θ(t) = θ(t − 1) +ϕ(t)
r(t)
[y(t) − θT(t − 1)ϕ(t)
](3)
r(t) = r(t − 1) + ||ϕ(t)||2, r(0) = 1 (4)
where
A. Rebai et al. / RIM: A Matlab Software Tool for Recursive Identification Methods 483
||X||2 = tr[XXT
].
2.2 Recursive least squares (RLS) algo-rithm
For this algorithm, the criterion J2(t) is used:
minθ(t)
J2(t) =1
2
t∑i=1
[y(i) − θTϕ(i)]2. (5)
Minimizing this criterion, we obtain
θ(t) = θ(t − 1) + F (t)ϕ(t)ε(t) (6)
with
ε(t) = y(t) − θT(t − 1)ϕ(t)
and
F (t) = F (t − 1) − F (t − 1)ϕ(t)ϕT(t)F (t − 1)
1 + ϕT(t)F (t − 1)ϕ(t)(7)
where F (t) is the adaptation gain and ε(t) is the prediction
error.
The formula of the adaptation gain is generalized by in-
troducing factors λ1(t), λ2(t)[16] and it is given by
F (t) =1
λ1(t)
(F (t − 1) − F (t − 1)ϕ(t)ϕT(t)F (t − 1)
λ1(t)λ2(t)
+ ϕT(t)F (t − 1)ϕ(t)
)
(8)
with
0 ≤ λ1(t) ≤ 1 and 0 ≤ λ2(t) ≤ 2.
The effects of λ1(t) and λ2(t) on the adaptation gain are
summarized in Table 1.
Table 1 Effects of λ1(t) and λ2(t)
Adaptation gain λ1(t) and λ2(t) values
Decreasing gain λ1(t) = 1
λ2(t) = 1
Constant forgetting factor λ1(t) = λ1,
0.95 ≤ λ1 ≤ 0.99
λ2(t) = 1
Variable forgetting factor λ1(t) = λ0λ1(t − 1) + 1 − λ0,
0.95 ≤ λ0 ≤ 0.99
λ2(t) = 1
Constant gain λ1(t) = 1
λ2(t) = 0
2.3 Multi-innovation stochastic gradient(MISG) algorithm
The multi-innovation stochastic gradient (MISG) was
proposed in [17], in order to achieve fast convergence rates
and improve the parameter estimation accuracy[18, 19]. The
basic idea of this algorithm is to use not only the current
data {y(t), ϕ(t)} but also the past data {y(t−i), ϕ(t−i), i =
1, 2, · · · , p − 1}, where p is a positive integer denoting the
innovation length.
Define the stacked output vector Y (p, t) and stacked in-
formation matrix φ(p, t) as
Y (p, t) =
⎡⎢⎢⎢⎢⎣
y(t)
y(t − 1)...
y(t − p + 1)
⎤⎥⎥⎥⎥⎦
, φT(p, t) =
⎡⎢⎢⎢⎢⎣
ϕT(t)
ϕT(t − 1)...
ϕT(t − p + 1)
⎤⎥⎥⎥⎥⎦
.
Minimize the quadratic criterion defined by
minθ(t)
J3(t) = ||Y (p, t) − φT(p, t)θ(t)||2. (9)
Finally, the multi-innovation stochastic gradient is obtained
as
θ(t) = θ(t − 1) +φ(p, t)
r(t)
[Y (p, t) − φ(p, t)θ(t − 1)
](10)
r(t) = r(t − 1) + ||ϕ(t)||2, r(0) = 1 (11)
φ(p, t) = [ϕ(t), ϕ(t − 1), · · · , ϕ(t − p + 1)] (12)
Y (p, t) = [y(t), y(t − 1), · · · , y(t − p + 1)]T . (13)
When p = 1, the MISG algorithm reduces to the SG algo-
rithm.
2.4 Multi-innovation least squares (MILS)algorithm
The minimized criterion for this algorithm is given by
minθ(t)
J4(t) =1
2
t∑i=1
[y(i) − φ(p, i)Tθ]2. (14)
Then, the multi-innovation recursive identification algo-
rithm is obtained[19, 20] as
θ(t) = θ(t − 1) + L(t)[Y (p, t) − φT(p, t)θ(t − 1)
](15)
L(t) = F (t − 1)φ(p, t)[Ip + φT(p, t)F (t − 1)φ(p, t)
]−1
(16)
F (t) = F (t − 1) − L(t)φT(p, t)F (t − 1) (17)
φ(p, t) = [ϕ(t), ϕ(t − 1), · · · , ϕ(t − p + 1)] (18)
Y (p, t) = [y(t), y(t − 1), · · · , y(t − p + 1)]T . (19)
When p = 1, the MILS algorithm reduces to the RLS algo-
rithm.
3 Recursive identification techniques
The recursive identification techniques are divided into
two categories. The first one is based on the whitening of
the prediction error. The second is based on the uncorre-
lation of observation vector and prediction error[21, 22]. In
this section, we give a survey on the most known techniques
of these two categories.
For the first category, the four well known methods are
the RLS, extended least squares (ELS), generalized least
squares (GLS) and output error with extended prediction
model (OEEPM).
484 International Journal of Automation and Computing 12(5), October 2015
3.1 Recursive least squares
The selected model of this technique is autoregressive
with external input (ARX) given by
A(q−1)y(t) = q−dB(q−1)u(t) + e(t) (20)
with
A(q−1) = 1 +
nA∑i=1
aiq−i
B(q−1) =
nB∑i=1
biq−i.
Hence, the model of (20) can be written as
y(t) = θTϕ(t) + e(t) (21)
where
θT = [a1, · · · , anA , b1, · · · , bnB ]
ϕT = [−y(t − 1), · · · ,−y(t − nA), u(t − 1 − d), · · · ,
u(t − d − nB)].
3.2 Extended least squares
In this method, the model is autoregressive moving aver-
age with external input (ARMAX) given by
A(q−1)y(t) = q−dB(q−1)u(t) + C(q−1)e(t) (22)
where
C(q−1) = 1 +
nC∑i=1
ciq−i.
From (22), we have
θT = [a1, · · · , anA , b1, · · · , bnB , c1, · · · , cnC ]
ϕT = [−y(t − 1), · · · ,−y(t − nA), u(t − 1 − d), · · · ,
u(t − d − nB), e(t − 1), · · · , e(t − nC)].
We replace e(t) by ε(t) in the observation vector because
e(t) is not observable. Finally, we obtain
θT = [a1, · · · , anA , b1, · · · , bnB , c1, · · · , cnC ]
ϕT = [−y(t − 1), · · · ,−y(t − nA), u(t − 1 − d), · · · ,
u(t − d − nB), ε(t − 1), · · · , ε(t − nC)].
3.3 Generalized least squares
This method is adapted to ARARX models as
A(q−1)y(t) = q−dB(q−1)u(t) +e(t)
C(q−1). (23)
Then, we have
θT = [a1, · · · , anA , b1, · · · , bnB , c1, · · · , cnC ]
ϕT = [−y(t − 1), · · · ,−y(t − nA), u(t − 1 − d), · · · ,
u(t − d − nB),−α(t − 1), · · · ,−α(t − nC)].
The estimated value of α(t) is given by
α(t) = A(q−1)y(t) − q−dB(q−1)u(t). (24)
3.4 Output error with extended predic-tion model
The algorithm of this one is the same as the ELS with
θT = [a1, · · · , anA , b1, · · · , bnB , h1, · · · , hnC ]
ϕT = [−y(t − 1), · · · ,−y(t − nA), u(t − 1 − d), · · · ,
u(t − d − nB), ε(t − 1), · · · , ε(t − nC)]
ci = hi + ai.
For the second category, the 4 main methods are the instru-
mental variable with auxiliary to model (IVAM), output
error with fixed compensator (OEFC), output error with
filtered observations (OEFO) and output error with adap-
tive filtered observations (OEAFO).
3.5 Instrumental variable with auxiliarymodel
For this technique, we have
θT = [a1, · · · , anA , b1, · · · , bnB ]
ϕT = [−yIV (t − 1), · · · ,−yIV (t − nA), u(t − 1 − d), · · · ,
u(t − d − nB)]
where the prediction auxiliary model is defined by
yIV (t) = −nA∑i=1
aiyIV (t − i) +
nB∑i=1
biu(t − d − i). (25)
3.6 Output error with fixed compensator
The disturbed output y(t) is replaced by the prediction
y(t) in the RLS predictor and we obtain
θT = [a1, · · · , anA , b1, · · · , bnB ]
ϕT = [−y(t − 1), · · · ,−y(t − nA), u(t − 1 − d), · · · , u(t−d − nB)].
3.7 Output error with filtered observa-tions
This method is based on the same principle of the pre-
vious one with a filtered observation vector obtained by a
stable filter A0(q−1). Thus,
ε(t) = y(t) − θTϕ(t)
F (t) = F (t − 1) − F (t − 1)ϕf (t)ϕf (t)TF (t − 1)
1 + ϕf (t)TF (t − 1)ϕf (t)
θ(t) = θ(t − 1) + F (t)ϕf (t)ε(t)
with
θT = [a1, · · · , anA , b1, · · · , bnB ]
ϕT = [−y(t − 1), · · · ,−y(t − nA), u(t − 1 − d), · · · ,
u(t − d − nB)]
ϕTf =
1
A0(q−1)ϕT.
A. Rebai et al. / RIM: A Matlab Software Tool for Recursive Identification Methods 485
3.8 Output error with adaptive filteredobservations
In this method, the observation vector is filtered by an
estimated filter A(q−1). Hence,
ε(t) = y(t) − θTϕ(t)
F (t) = F (t − 1) − F (t − 1)ϕf (t)ϕf (t)TF (t − 1)
1 + ϕf (t)TF (t − 1)ϕf (t)
θ(t) = θ(t − 1) + F (t)ϕf (t)ε(t)
where
θT = [a1, · · · , anA , b1, · · · , bnB ]
ϕT = [−y(t − 1), · · · ,−y(t − nA), u(t − 1 − d), · · · ,
u(t − d − nB)]
ϕTf = 1
A(q−1)ϕT.
3.9 Auxiliary model identification algo-rithm
This technique was proposed in [20, 23] for output error
moving average (OEMA) system given by
y(t) =B(q−1)
A(q−1)u(t) + D(q−1)e(t). (26)
From (26), we have
y(t) = x(t) + D(q−1)e(t) (27)
with
D(q−1) = 1 +
nD∑i=1
diq−i.
Define the parameter vector
θ =
[θs
θn
]
θTs = [a1, · · · , anA , b1, · · · , bnB ]
θTn = [d1, d2, · · · , anD ]
and the observations vector
ϕ =
[ϕs
ϕn
]
ϕTs = [−x(t − 1), · · · ,−x(t − nA), u(t − 1), · · · , u(t − nB)]
ϕTn = [e(t − 1), · · · , e(t − nD)].
We choose Ba(q−1)/Aa(q−1) as the transfer function of the
auxiliary model and xa(t) as its output. The unknown inner
variables x(t − i) are replaced by xa(t − i), and unmeasur-
able noise terms e(t − i) are replaced by their estimated
residuals ε(t − i).
Finally, we obtain
θT = [a1, · · · , anA , b1, · · · , bnB , d1, · · · , dnD ]
ϕT = [−xa(t − 1), · · · ,−xa(t − nA), u(t − 1), · · · ,
u(t − nB), ε(t − 1), · · · , ε(t − nD)].
For each identification method, one of the previous algo-
rithms described in Section (2) can be used to estimate the
parameter vector.
For validation of the various identified models, the esti-
mations of the normalized autocorrelations RN(i) are cal-
culated. RN(i) is defined for the 1st category of identifica-
tion methods and the whiteness test is
RN(i) =R(i)
R(0), i = 1, 2, · · · , max(nA, nB + d) (28)
with
R(i) =1
N
N∑t=1
ε(t)ε(t − i), R(0) =1
N
N∑t=1
ε2(t). (29)
For the 2nd category, the uncorrelation test is applied and
we have
RN(i) =R(i)√√√√
(1
N
N∑t=1
y2(t)
)(1
N
N∑t=1
ε2(t)
) (30)
with
R(i) =1
N
N∑t=1
ε(t)y(t − i) (31)
where ε(t) is output prediction error, y(t) is system output,
y(t) is model output, and N is data number.
The identified model is valid if |RN(i)| ≤ 2.17√N
. In prac-
tice, the used criterion is |RN(i)| ≤ 0.15[22] .
4 Description of the RIM toolbox
The proposed RIM GUI is developed using tools available
in Matlab/GUI and all algorithms described in Sections 2
and 3 are programmed using Matlab environment in such
way that the user does not need Matlab or any other pro-
gramming language experience.
To start the RIM platform, you have to enter into the
Matlab command line: RIM.
The RIM window appears on the screen as shown in
Fig. 1.
Fig. 1 RIM window
486 International Journal of Automation and Computing 12(5), October 2015
In this platform, the user feels free to use classical menu
or buttons. Indeed, in the menu, we set the links of differ-
ent buttons. Furthermore, RIM allows user to save or print
the identification process results in several formats such as
*.m, *.txt and *.doc using the option “Save as · · · ” and
“Print” in the menu. RIM also allows plotting the vari-
ables (identification error, error autocorrelation, etc.) in
external figures. Furthermore, the RIM shows a message
indicating if the model is valid or not based on the values
of the error autocorrelation.
The user can also jump from the identification category
to the other by pressing the button “Next” or “Previous”.
For each button, we set a shortcut such as CTRL+S for the
option “Save as · · · ” , CTRL+P for “Print”, etc.
The use of RIM can be summarized on 3 steps and they
are the main steps of any identification process as shown in
Fig. 2. The first step is the generation of I/O file or data
file. The second is the choice of the techniques to be used
for system identification. And the last one is the selection
of the identification parameters such as the polynomials or-
ders nA, nB and nC , the initial adaptation gain F (0) and
the forgetting factor λ1(t).
Fig. 2 Identification steps
5 Application examples
5.1 Example 1: Identification of a piezo-electric actuator
In order to illustrate the application and the performance
of the RIM GUI, this section discusses the identification of
a piezoelectric actuator (PEA) which is a nonlinear system.
The experimental data used for the identification cor-
responds to the piezoelectric actuator APA-120ML. The
excitation signal is a sinusoidal signal of frequency 50 Hz:
u(t) = 68.5 sin(2π × 50t + 0.44) + 61.5(V).
5.1.1 Identification of ARX model
The piezoelectric actuators are generally modeled by a
second-order system. Therefore, the polynomials orders of
the model are selected as follows: nA = 2, nB = 2, nC = 1
and d = 0. The identification parameters used in this paper
are summarized in Table 2.
Table 2 Identification parameters
Parameter Value
nA 2
nB 2
nC 1
d 0
F (0) 1 × 105
A0 [1 0.3]
λ0 0.97
λ1 0.97
N 2 × 104
The identification results using the traditional RLS
method are shown in Fig. 3. From Fig. 3, the values of
|RN(i)| are all less than 0.15, which means that this model
is valid and the system can be represented by the ARX
model as
y(t) = − a1y(t − 1) − a2y(t − 2) + b1u(t − 1)+
b2u(t − 2) + e(t)
with a1 = −1.0577, a2 = 0.0575, b1 = −17.49 × 10−5 and
b2 = 17.43 × 10−5.
Fig. 3 Identification results using the RLS method
Based on RIM, more information can be illustrated (see
Figs. 4–6) to conclude about the model validity. Fig. 4 rep-
resents a comparison between the physical system response
and the model. It can be seen that the model describes
accurately the real system behavior.
From Fig. 5, the input-output feature indicates that the
hysteresis phenomenon, which is the main property of piezo-
electric actuators and can be described accurately by the
proposed ARX model. From Fig. 6, the maximum relative
error is 5%, which is in the accepted rang.
A. Rebai et al. / RIM: A Matlab Software Tool for Recursive Identification Methods 487
Fig. 4 Outputs curves
Fig. 5 Input-output feature
Fig. 6 Identification error
5.1.2 Comparative study
As mentioned before, one of the advantages of RIM is its
capability to carry out a comparison between the various
identification methods based on the normalized autocorre-
lation values. The quantification of results are shown in
Table 3.
The comparison can be carried out by two criteria,
the first one is the autocorrelation boundaries (|RN |max,
|RN |min) for each category. The second one is based on the
choice of the model structure. We compare the values of
|RN |max and |RN |min for both models ARX and ARMAX.
From Table 3, it can be seen that the ELS and the IVAM
methods are more valid than the others and the PEA can
be identified by these two techniques.
Table 3 Comparison between different identification methods
Method |RN |max |RN |min
RLS 3.8665 × 10−4 1.7747 × 10−6
ELS 3.8299 × 10−4 4.5152 × 10−7
GLS 3.8386 × 10−4 4.3394 × 10−6
OEEPM 3.8793 × 10−4 2.0307 × 10−6
IVAM 4.8068 × 10−5 4.7702 × 10−6
OEFC 6.0198 × 10−5 5.1257 × 10−4
OEFO 5.4387 × 10−4 5.1106 × 10−4
OEAFO 0.1160 0.1153
For the second criterion, the |RN |max and |RN |min of the
ELS method are the smallest for the methods based on AR-
MAX structure. For the methods based on ARX structure,
the IVAM has the smallest values of |RN |max and |RN |min.
5.1.3 Effect of identification parameters
RIM also allows user to study the influence of identifi-
cation parameters on the result. For example, in the case
of decreasing gain, the model is valid. But with a constant
gain, the model is not valid. A summary result between the
different cases, presented in Table 1 using the parameters
identification of Table 2, is illustrated in Table 4.
Table 4 Influence of adaptation gain on model validity
Adaptation gain Model validity
Decreasing gain Valid
Constant forgetting factor Not valid
Variable forgetting factor Valid
Constant gain Not valid
5.2 Example 2
Let′s consider the following system
A(q−1)y(t) = B(q−1)u(t) + e(t)
A(q−1) = 1 + a1q−1 + a2q
−2 = 1 − 1.70q−1 + 0.85q−2
B(q−1) = b1q−1 + b2q
−2 = 0.32q−1 + 0.60q−2.
This example can be rewritten as
y(t) = θTϕ(t) + e(t).
The input u(t) is an uncorrelated persistent excitation sig-
nal sequence with zero mean and unit variance, and e(t)
is a white noise sequence with zero mean and variance 0.5.
The data number is chosen as N = 500. Using RIM, the
model parameters are estimated applying traditional RLS
algorithm, MILS algorithm and MISG algorithm. The iden-
tification results for the MILS algorithm with p = 3 are
shown in Fig. 7. The complete results are summarized in
Table 5.
488 International Journal of Automation and Computing 12(5), October 2015
Fig. 7 Identification results using the MILS method (p = 3)
Table 5 Identification results of an ARX system
Method a1 a2 b1 b2
RLS (MILS p = 1) −1.6992 0.8491 0.3214 0.6019
MILS p = 3 −1.7008 0.8507 0.3201 0.6002
MISG p = 8 −1.7125 0.8312 0.3291 0.5715
From Table 5, It can be seen that the MILS algorithm
presents better results than the traditional RLS and MISG
algorithms.
5.3 Example 3
This example considers an OEMA system to evaluate the
multi-innovation auxiliary model algorithm performance in
parameters estimation.
The system is given by
y(t) =B(q−1)
A(q−1)u(t) + D(q−1)e(t)
A(q−1) = 1 + a1q−1 + a2q
−2 = 1 − 1.60q−1 + 0.80q−2
B(q−1) = b1q−1 + b2q
−2 = 0.23q−1 + 0.70q−2
D(q−1) = 1 + d1q−1 = 1 − 0.40q−1.
Simulation parameters are the same as those used in
the previous example. Using RIM, the estimated param-
eters with auxiliary model extended least squares (AM-
ELS) and multi-innovation auxiliary model extended least
squares (MI-AM-ELS) (p = 3, p = 5) algorithms are given
in Table 6.
From Table 6, the MI-AM-ELS algorithm presents bet-
ter computational efficiency compared with the AM-ELS
algorithm. These results can be improved by increasing the
innovation length p.
6 Conclusions
In this paper, we presented a Matlab GUI platform for
systems identification. The proposed toolbox deals with the
recursive identification methods and contains the necessary
tools to validate many theoretical aspects in the domain of
system identification.
The RIM toolbox allows users to make simulations with-
out the need to program the identification techniques or to
have programming experience in Matlab environment. RIM
gives to users the possibility to study the effects of different
identification parameters and makes users more insight into
the system identification theory.
Table 6 Identification results of an OEMA system
Method a1 a2 b1 b2 d1
AM-ELS (MI-AM-ELS p = 1) −1.6128 0.8126 0.2365 0.6922 0.3456
MI-AM-ELS p = 3 −1.5987 0.8054 0.2312 0.6970 0.3825
MI-AM-ELS p = 5 −1.6010 0.8026 0.2311 0.7012 0.3957
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Aissa Rebai received the B.Eng. andM. Sc. degrees in control engineering fromNational Polytechnic School of Algiers, Al-geria in 2012. He is currently a Ph. D. de-gree candidate at the Department of Con-trol, National Polytechnic School of Algiers,Algeria.
His research interests include systemidentification, intelligent control of mecha-
tronic systems and optimization algorithms.E-mail: [email protected] iD: 0000-0001-5935-731X
Kamel Guesmi received the B.Eng. de-gree in electrical engineering from the Uni-versity of Djelfa, Algeria in 2000, the M. Sc.and Ph.D. degrees in electrical engineeringfrom Reims University, France in 2003 and2006, respectively. Currently, he is an asso-ciate professor in the Department of Elec-trical Engineering, Djelfa University, Alge-ria.
His research interests include intelligent control, robust con-trol, power electronics and nonlinear dynamics.
E-mail: [email protected] (Corresponding author)ORCID iD: 0000-0002-3492-8339
Boualem Hemici graduated from theNational Polytechnic School (ENP) of Al-giers, Algeria in 1983. He received thePh. D. degree from National PolytechnicSchool of Algiers, Algeria in 2007. Cur-rently, he is an associate professor in theDepartment of Control, National Polytech-nic School of Algiers, Algeria.
His research interests include power elec-tronics, electrical machinery drives and process control.
E-mail: [email protected]
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