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Parameter Estimation of Linear and Nonlinear Systems Connected in
Parallel
HAFID OUBOUADDI, FATIMA EZZAHRA EL MANSOURI, ADIL BROURI
ENSAM, L2MC Laboratory,
Moulay Ismail University,
MOROCCO
Abstract: - Presently, the problem of 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 interconnected structure of linear and nonlinear blocks systems leads to a highly nonlinear problem
involving several unknown parameters and inaccessible internal signals. In this parameter estimation method,
the linear and nonlinear block parameters are estimated in one stage by using simple a sine signal or multi-
cosine wave.
Key-Words: - Linear and nonlinear systems; Parameter estimation; Sine signals.
Received: September 28, 2022. Revised: April 15, 2023. Accepted: May 8, 2023. Published: May 22, 2023.
1 Introduction
The problem of nonlinear system parameter
estimation has been given a great deal of interest,
[1], [2], [3], [4], [5], [6]. Most available parameter
estimation approaches are focused on the cascading
link of linear and nonlinear subsystems (Wiener and
Hammerstein systems), [7], [8], [9], [10], [11], [12].
The latter are obtained using the series connection
of linear and nonlinear subsystems. The Wiener and
Hammerstein nonlinear systems are very popular
models and can model several practical systems,
[13], [14], [15], [16], [17].
There exist many parameter estimation methods in
the literature to deal with the problem of nonlinear
systems composed by the series connections of
linear and nonlinear blocks, [18], [19], [20], [21],
[22], [23].
To increase the modelling capacity, the parallel
connection of linear and nonlinear subsystems can
be very efficient, [24], [25]. The parallel connection
of linear and nonlinear subsystems is more general
than the series connection [26], [27].
Presently, the parameter estimation of a
nonlinear system composed of the parallel
connection of linear and nonlinear subsystems is
presented. This nonlinear system is shown in Fig. 1.
Note that several real systems can be captured
using the parallel connection of linear and nonlinear
subsystems, [14], [28]. We propose a solution to
estimate the parameters of linear subsystem
and those of nonlinear subsystem
.. This approach is based on sine signals. In this
respect, note that when linear and nonlinear systems
are excited by sine signal, their output is thus
periodic of the same frequency as the input, [29],
[30], [31].
This means that the output is also periodic but not
necessarily a sine signal.
Then, it is quite interesting to note that very few
works have treated this problem. The most of
existing parameter estimation problems concern the
classical Wiener and Hammerstein systems.
At this stage, it is worth emphasizing that this
nonlinear system can describe several nonlinear
structures.
Fig. 1: Nonlinear system composed by the parallel
connection of linear and nonlinear subsystems
For convenience, the rest of this paper is
organized as follows. Firstly, the problem statement
in this study is introduced in Section 2. Then, the
parameter estimation method is described in Section
3. Finally, the performances of the parameter
estimation method are presented in Section 4.
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2 Problem Statement
This paper aims to estimate the parameters of the
considered nonlinear system. The latter is obtained
by the parallel link of linear and nonlinear
subsystems (Fig. 1).
Firstly, the nonlinear part can be described as
follows:
∑
(1)
where and ,..., are the degree and
coefficient vector of the nonlinearity ..
Then, the linear part is analytically described as
follows:
∗
(2)
The Equation (2) can be rewritten in the Laplace
domain as:
(3)
where and are the Laplace transforms of
the input an output of the linear
subsystem, respectively.
Then, one immediately gets using Fig. 1 that:
(4)
Finally, it follows from (1)-(2) and (4) that the
output of the nonlinear system can be expressed as:
∗∑
(5)
For convenience, let us consider the signal
defined as:
(6)
Then, the signal can be rewritten as follows:
∗
∑
(7)
In this study, we propose an input of the form:
(8)
where ∈, .., is any arbitrarily chosen
set. It follows (8) that the signal is of period
2/
. Then, it is shown that the output of a
linear subsystem is of period 2/
(e.g., see,
[1]).
In this respect, it is also shown that the output of a
nonlinear subsystem is not necessarily a sine signal,
but of period 2/
(e.g., see, [1], [24]).
Specifically, using (8) the signal in Fig. 1 can
be expressed as follows:
∑
(9)
3 Determination of Linear and
Nonlinear Parameters
This section aims to present an estimation solution
to give ||, and ,...,.
Firstly, it is readily seen that the signal for
can be expressed as follows:
||
(10)
where the set ||, are the
parameters of the linear subsystem to be estimated.
Then, using (9) one immediately gets that the signal
can be expressed as (for more details see, [1],
[24]):
∑,,...,
(11)
where , for 0..., is known real constant,
and the expression . according to the variables
... is already determined, but . is not
known. Finally, the output can be expressed as
follows:
||
∑,,...,
(12)
Accordingly, it is seen that the signal
defined by (6) is related to the output by the
expression:
(13)
Furthermore, knowing that the signals and
are of period 2/
, using (6) we
conclude that the signal is of period
2/. This result is very interesting, it allows to
filter the output . Specifically, the filtered
output noted , can be given, for any large
integer N, by:
∑
for 0
(14a)
otherwise (14b)
It is readily seen that the filtered output
converges to the signal , which is given by:
||
∑,,...,
(15)
It is readily seen that the signal is also of the
period 2/
. Furthermore, using the filtered
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DOI: 10.37394/23206.2023.22.44
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signal , the parameters of linear subsystem
||, and those of nonlinear
subsystem ,..., can be obtained.
4 Simulation
This section aims to present some examples of
simulation to show the effectiveness of the obtained
results. The studied system (Fig. 1) is described by
the following linear subsystem:
.. (16)
and by the following nonlinear subsystem:
0.1
0.50.3 (17)
The nonlinearity is plotted in Fig. 2.
Fig. 2: The nonlinearity .
Then, using the proposed method established in
section 3, all variables can be estimated. Firstly, the
system of Fig. 1 is excited by the signal (8) by
choosing 0.02/, where ∈
, .., is the chosen set of applied
frequencies. It is shown that the signal is of
period 2/
314.16. Then, the output of
this system is given in Fig. 3. This result shows that
the signal is of period 2/
314.16,
but affected by disturbances. Using the filtered
signal , given by (14a-b), one gets the signal
plotted in Fig. 4. This figure confirms that the signal
(or the estimate of ) is also of period
2/ 314.16.
Fig. 3: The output of the system
Fig. 4: The filtered signal
This test is repeated for other periods
2/
, where 0.1/. Then, use the signal (8)
in the input of the system. Then, the output of
this system is given in Fig. 5. This result shows that
the signal (but affected by disturbances) is of
period 2/
62.83. Then, the filtered
signal , obtained using (14a-b), is given in Fig.
6.
Fig. 5: The output of the system
0 20 40 60 80 100 120 140 160 180
Time
(
s
)
-15
-10
-5
0
5
10
15
20
25
30 Output of the system
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Fig. 6: The filtered signal
Finally, using the filtered signals , obtained
using (14a-b) and represented by Fig. 4 and Fig. 6,
the parameters of linear subsystem
||, and those of nonlinear
subsystem ,..., can be estimated (using
(15)).
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. This solution is easy and more
general.
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DOI: 10.37394/23206.2023.22.44
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-Adil Brouri: Conceptualization, formal analysis,
investigation, project administration, supervision.
-Hafid Oubouaddi: Methodology, software,
validation.
-Fatima Ezzahra El Mansouri: Methodology,
software, validation
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: 10.37394/23206.2023.22.44
Hafid Oubouaddi, Fatima Ezzahra El Mansouri, Adil Brouri
E-ISSN: 2224-2880
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Volume 22, 2023
