Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина III)
The article considers the problem of identifying a mathematical model in the form of a system of ordinary differential equations. The identified system can have constant and periodic coefficients. The source of information for solving the problem is time series of observed variables. The article stu...
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| author | Gorodetskyi, Viktor |
| author_facet | Gorodetskyi, Viktor |
| author_institution_txt_mv | [
{
"author": "Viktor Gorodetskyi",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
}
] |
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| description | The article considers the problem of identifying a mathematical model in the form of a system of ordinary differential equations. The identified system can have constant and periodic coefficients. The source of information for solving the problem is time series of observed variables. The article studies the effect of uniformly distributed noise on the identification result. To solve the problem, the algorithm proposed by the author in previous works was used. It is shown that the method has different sensitivity to noise depending on which of the observed variables is contaminated with noise. The implementation of the method is illustrated by numerical examples of identifying nonlinear differential equations with polynomial right-hand sides. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2025.1.04 |
| first_indexed | 2025-07-17T10:28:43Z |
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Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2025
44 ISSN 1681–6048 System Research & Information Technologies, 2025, № 1
UDC 517.925
DOI: 10.20535/SRIT.2308-8893.2025.1.04
IDENTIFICATION OF NONLINEAR SYSTEMS WITH PERIODIC
EXTERNAL ACTIONS (PART III)
V. GORODETSKYI
Abstract. The article considers the problem of identifying a mathematical model in
the form of a system of ordinary differential equations. The identified system can
have constant and periodic coefficients. The source of information for solving the
problem is time series of observed variables. The article studies the effect of uni-
formly distributed noise on the identification result. To solve the problem, the algo-
rithm proposed by the author in previous works was used. It is shown that the meth-
od has different sensitivity to noise depending on which of the observed variables is
contaminated with noise. The implementation of the method is illustrated by nu-
merical examples of identifying nonlinear differential equations with polynomial
right-hand sides.
Keywords: identification, ordinary differential equation, periodic coefficient, con-
stant coefficient, uniformly distributed noise.
INTRODUCTION
When developing mathematical methods for studying various physical systems, it
is necessary to evaluate reliability of their results if applied to systems in real
world. One of the common tasks in applied mathematics is the problem of identi-
fying a mathematical model of a certain process. The initial data for solving this
problem can be the observed variables of the process. If we study real systems,
the results of measurements of the observed variables may contain noise [1–3].
This circumstance can complicate the identification of the model.
BACKGROUND AND TASK FOR RESEARCH
We follow the results obtained in [4; 5]. There, the problem of identifying a
system of n ordinary differential equations with constant and periodic coeffi-
cients )(tcij ( ;,...,1 ni mj ,...,1 ) was considered. The initial data for identifi-
cation was time series of observed variables ,)(txi .0],;0[ ee ttt To solve the
problem, the theorem proved in [4] was used. According to this theorem, simple
relationships that are used to identify equations with constant coefficients can be
used to identify differential equations with periodic coefficients. For this purpose,
the calculations must use the values of the functions )(txi at moments of time jt ,
separated from each other by the value ,...3,2,1, qqT , where T is the period of
the periodic coefficients. In other words, this time moments obey the relations
emm tttmtttttt ,0,0;,...,2, 000201 , (1)
where qT .
Identification of nonlinear systems with periodic external actions (Part III)
Системні дослідження та інформаційні технології, 2025, № 1 45
x1(t)
t
Fig. 1. Time series of )(1 tx contaminated
with noise
Examples of the method application were demonstrated in [4; 5]. In this
study, we try to apply the proposed algorithm to identify equations by observed
variables with noise. As in the mentioned articles, we use as an example a system
of the form
,)()()(
,
,
3136333303
212
321
xxtcxtctcx
dxxx
xxx
(2)
obtained on the basis of the Rössler system [6]. The parameters of the third equa-
tion of the system (2) are as following:
.2,5)(,20)(,
2
sin4.05.0)( 03633
0
30 sTtctc
T
t
tc
The generalized structure for the purpose of identifying the desired equation
has the form
2135
2
134333232131303 )()()()()()( xxtcxtcxtcxtcxtctcx
.)()()()( 2
3393238
2
2373136 xtcxxtcxtcxxtc (3)
Next, we will consider how adding noise to different observables affects the
identification result.
IDENTIFICATION OF AN EQUATION WITH A VARIABLE )(1 tx AFFECTED
BY NOISE
For the study, we add noise with a uniform distribution to the observable )(1 tx .
The noise value is 11 01.0 xu , min1max11 xxx , max1x and min1x are,
respectively, the maximum and the minimum of the observable )(1 tx over the
studied interval of 100 s. Fig. 1 shows a fragment of the time series )(1 tx with
added noise.
The first stage of model identifica-
tion with this algorithm is to find those
values of (see (1)) that can be equal to
or multiples of the expected .T The val-
ues obtained by applying the algorithm
are presented in Table 1. The table shows
in bold the values that are repeated or
multiples of other values. This means that
they may be the sought-for values of T or
multiples of this value. For example, the
values 1.04, 3.01, 4.88, 5.30, 5.67, 10.96
are repeated. The set: 2.65, 5.30, 10.60 is
also highlighted in bold because the second and the third values of it are multiples
of the first one. Similarly, the value 6.00 is a multiple of 3.00. The values 2.20
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 46
and 7.70 are multiples of 1.10, which is not in the table, but which may poten-
tially be the sought-for period. For the same reason, 6.00, 8.00, 10.00 are high-
lighted, which are multiples of 2.00, which is not in the table.
T a b l e 1 . The result of applying the algorithm for observable )(1 tx with noise
The values calculated for the coefficients of the third equation of system (2)
№
)(30 tc )(31 tc )(32 tc )(33 tc )(34 tc )(35 tc )(36 tc )(37 tc )(38 tc
)(39 tc
1 9.25 5.24 5.18 4.08 5.95 7.26 1.84 6.00 1.72 2.63
2 0.88 1.27 3.34 1.29 5.31 5.86 2.65 6.03 3.66 0.86
3 7.70 10.96 3.00 8.18 8.98 6.00 5.18 6.64 3.01 5.67
4 1.04 3.31 9.91 2.98 5.91 3.20 5.36 0.81 5.85 2.58
5 8.23 2.30 10.94 10.95 3.33 7.72 5.67 7.07 8.23 7.45
6 3.01 4.36 5.86 3.01 4.74 5.30 5.39 7.79 10.60 1.91
7 9.17 4.88 5.91 5.32 10.96 4.97 9.07 8.92 9.06 1.08
8 4.81 10.00 1.85 3.49 9.80 5.44 3.01 7.99 6.00 7.44
9 4.88 2.20 4.79 7.58 8.00 5.98 5.91 1.53 10.66 9.24
10 2.98 4.43 7.99 5.30 1.48 7.59 1.47 9.50 4.02 1.04
We have to, based on the data in Table 1, reject the excess coefficients of the
desired equation and estimate the type and values of the remaining coefficients.
For this, we use the second part of the algorithm. Namely, for each value selected
in Table 1, we solve a system of the form
BA=C -1 , (4)
where C is the vector of the required coefficients of the third equation of sys-
tem (2), and B is the vector of values ,,...,0),( mktx ki A is the matrix of
function ))(( kj tf x values, mj ,...,0 , },...,{ 1 nxxx . In this study we consider
as functions ))(( kj tf x the products of ix in each monomial of the right-hand side
of (3). We set some interval of change 0t from (1) and obtain the calculated val-
ues of the coefficient on this interval. The resulting time series of coefficients al-
low us to estimate the type of coefficient and the possible value of .T
To begin with, let us try to identify zero coefficients in the analyzed equa-
tion, provided that the selected value of
from Table 1 can be the desired period .T
For example, Fig. 2 shows the time series
of the calculated coefficient )(34 tcc for
s04.1 on an interval of 4 s.
As can be seen, this coefficient has
values close to zero on this segment. At the
same time, its greatest deviation from zero,
including at singular points, is .1|)(| 34 tcc
We will assume that if these conditions are
met, this coefficient is a candidate for
zeroing. Such a criterion was applied to all
Fig. 2. Time series of calculated coef-
ficient )(34 tcc
t
)(34 tcc
Identification of nonlinear systems with periodic external actions (Part III)
Системні дослідження та інформаційні технології, 2025, № 1 47
coefficients for all selected from Table 1. The results of the analysis are
presented in Table 2, where the coefficients that may be zero in the desired
equation are marked with a “+” sign. Note that this table has been supplemented
with the values s10.1 and s00.2 , which, as explained above, may also
be values of the period T .
T a b l e 2 . Results of the analysis of time series obtained for the selected values
of from Table 1
Possible values of T or its multiples
№
1.04 1.10 2.20 7.70 2.00 6.00 8.00 10.00 3.00 3.01 2.65 5.30 10.60 4.88 5.67 10.96
34c + +
35c + + + + +
37c + + + + + + + +
According to Table 2, the most likely candidates for zeroing are coefficients
35c and 37c . Taking 03735 cc and performing an analysis similar to the pre-
vious one, we obtain Table 3.
T a b l e 3 . The same as in Table 2 with 03735 cc
Possible values of T or its multiples
№
1.04 2.00 4.00 6.00 8.00 10.00 3.00
10.60 4.88 10.96
31c +
32c + + +
34c + + + + + +
Note that Table 3 does not include the values from Table 2, for which (ac-
cording to Table 2) it is not possible to determine the coefficients that may be
subject to zeroing. It should be noted that the value s00.4 is added to the ta-
ble because it is a multiple of s00.2 . One can also pay attention to the graph
)(30 tc obtained, for example, at s00.4 (see Fig. 3). We can already assume
from it that sT 00.2 .
Based on Table 3, the next step
should be to zero out the coefficients in-
cluded in it. As a result, time series of
the remaining coefficients )(30 tcc , )(33 tcc ,
)(36 tcc , )(38 tcc , )(39 tcc were obtained.
For further analysis, let us consider
graphs )(38 tcc and )(33 tcc shown in Fig. 4.
Despite the large number of points in
which the calculated value of the coeffi-
cient )(38 tcc deviates significantly from
zero, we can assume that 038 c . Graph
)(39 tcc looks similar, which also allows us
Fig. 3. Time series of calculated coeffi-
cient )(30 tcc
t
)(30 tcc
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 48
to exclude it from the desired equation. On the contrary, most points of graph
)(33 tcc clearly have values different from zero. Graph )(36 tcc looks similar to
)(33 tcc . Therefore, at the next step, it is advisable to zero out the coefficients 38c
and 39c . As a result, this method gives us the structure of equation analogous to
the apriori equation (2). Time series of calculated coefficients are shown in Fig. 5.
IDENTIFICATION OF AN EQUATION WITH A VARIABLE )(2 tx AFFECTED
BY NOISE
A study similar to that performed in the previous section was also performed for
a variable with the same noise level 22 01.0 xu . The result is shown in Table 4.
Fig. 4. Time series of calculated coefficients )(38 tcc and )(33 tcc
)(38 tcc )(33 tcc
t t
a b
Fig. 5. Time series of calculated coefficients of the identified equation with the desired
structures
t
)(36 tcc
)(33 tcc
t
)(30 tcc
t
Identification of nonlinear systems with periodic external actions (Part III)
Системні дослідження та інформаційні технології, 2025, № 1 49
T a b l e 4 . The result of applying the algorithm for observable )(2 tx with noise 1%
The values calculated for the coefficients of the third equation of system (2)
№
)(30 tc )(31 tc )(32 tc )(33 tc )(34 tc )(35 tc )(36 tc )(37 tc )(38 tc )(39 tc
1 2.98 4.00 10.00 4.00 4.00 8.00 4.00 10.00 2.00 4.00
2 2.76 8.00 2.00 8.00 8.00 10.00 8.00 2.00 10.00 10.00
3 1.88 10.00 8.00 10.00 10.00 2.00 10.00 4.00 8.00 8.00
4 3.34 2.00 7.63 2.00 2.00 4.00 2.00 6.00 4.00 0.86
5 8.71 1.27 4.00 10.95 9.80 5.91 1.88 8.00 10.67 9.24
Based on the corollary of Theorem
[4], we can conclude that the equation has a
single variable coefficient )(30 tc , and the
other coefficients are constant. That is, in
this case, the identification occurs in the
same way as for equations without noise,
see [4, 5]. Moreover, a similar result was
obtained by increasing the noise level
added to the observable .)(2 tx Fig. 6
shows this variable with noise
22 2.0 xu , and Table 5 demonstrates
the result of applying the algorithm.
Based on the comparison of these two tables, it can be concluded that the al-
gorithm has low sensitivity to noise in this case. This can also be illustrated by
Table 6, which presents the calculated values of the constant coefficients at 20%
noise for two different 0t . It is clear from the table that in the structure (3) all
terms that include the variable 2x are subject to zeroing. Therefore, in the subse-
quent steps of identification, only the observables 1x and 3x will be used, which
in this case are noise-free. This simplifies the task.
T a b l e 6 . The calculated values of the constant coefficients at 20% noise for two
different 0t
Coefficients
0t
31c 32c 33c 34c 35c 36c 37c 38c 39c
st 15.001 -1.307 10-5 -7.991 10-6 -20.008 -1.288 10-6 2.425 10-6 5.002 1.367 10-7 3.321 10-4 4.862 10-4
st 4.002 -3.51 10-6 2.514 10-6 -20.026 8.848 10-7 -3.002 10-7 5.006 3.638 10-7 6.584 10-4 1.585 10-3
T a b l e 5 . The result of applying the algorithm for observable )(2 tx with noise 20%
The values calculated for the coefficients of the third equation of system (2)
№
)(30 tc )(31 tc )(32 tc )(33 tc )(34 tc )(35 tc )(36 tc )(37 tc )(38 tc )(39 tc
1 10.75 8.00 6.00 8.00 8.00 4.00 10.35 6.00 6.00 8.00
2 1.20 10.00 4.00 1.46 10.00 6.00 8.00 10.00 10.00 10.00
3 1.05 2.00 10.00 10.00 2.00 10.00 10.00 8.00 4.00 6.73
4 6.89 5.29 8.00 0.72 4.00 8.00 0.95 2.00 8.00 2.00
5 2.60 1.86 2.00 8.75 6.00 2.00 4.00 4.00 2.00 4.00
t
x2(t)
Fig. 6. Time series of )(2 tx con-
taminated with noise 20%
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 50
IDENTIFICATION OF AN EQUATION WITH A VARIABLE )(3 tx AFFECTED
BY NOISE
Unlike the previous case, the variable 3x is present in the desired equation, both
in the right-hand side and in the left-hand side in the form of a time derivative.
A fragment of the noisy series )(3 tx with 33 01.0 xu is shown in Fig. 7, a.
In order to reduce noise, smoothing was performed using the moving aver-
age method according to the formula
7
3323133132333
3
xxxxxxx
xs , (5)
where sx 3 is the value of the function )(3 tx at the point with the number
after smoothing by formula (5). A fragment of the function )(3 txs is shown in
Fig. 7, b. To form the vector B of the left sides in the system (4), it is necessary to
perform numerical differentiation of the function )(3 txs . For this, the formula
t
xx
x
ss
s
2
1313
3
was used, where t is a step of time series )(3 tx representation. The time series
)(3 txs
is shown in Fig. 7, c, which demonstrates that noise is significantly ampli-
fied when numerical differentiation is computed.
Fig. 7. Time series of )(3 tx : a — fragment of time series )(3 tx contaminated with noise
1%; b — the same fragment after smoothing; c — time derivative of graph (b)
)(3 tx sx3(t)
t t
a b)(3 tx s
t
c
Identification of nonlinear systems with periodic external actions (Part III)
Системні дослідження та інформаційні технології, 2025, № 1 51
Unfortunately, the application of the algorithm did not allow us to obtain an ade-
quate result. Obviously, the reason for this is insufficient smoothing of the time
series noise and the appearance of significant computational noise during numeri-
cal differentiation.
DISCUSSION AND CONCLUSIONS
From the previous sections it is clear that the most difficult identification is when
the variables )(1 tx and )(3 tx , which are included in the desired equation, are
contaminated with noise. The results obtained can be explained using Cramer's
rule. When determining the coefficients of equation (3), it will have the form:
mjс j
j ,...,0,
)(det
)(det
3
A
A
,
where )(det A is the main determinant of the system of linear algebraic equations
(4), formed taking into account conditions (1), )(det jA is the determinant obtained
by replacing the j-th column of the determinant )(det A with the vector B from (4).
Let the variable with noise be )(1 tx . Then, to find, for example, the coeffi-
cient 31с , we create matrix 1A by replacing column 1 with B in matrix A ,
which in this case consists of the time derivatives of the values of )(3 tx . That is,
we replace the column of values of noisy )(1 tx with the column B without noise.
Here we assume that the differentiation of the variable )(3 tx without noise is per-
formed correctly, without significant errors. Therefore, such a replacement, at
a minimum, should not increase the error in calculating 31с .
On the contrary, if the observed variable with noise is )(3 tx , then when dif-
ferentiating it numerically, computational noise will appear, see Fig. 7, c. Then,
when calculating, for example, the coefficient 33с , we replace the column num-
ber 3 in A with vector B . The column number 3 of A initially contains the vari-
able )(3 tx , which contains noise. However, comparing Fig. 7, b and 7, c, we see
that the noise in the column B is much greater and can of course be a source of
significant errors.
Taking into account the above, for successful identification of systems from
time series of observations with noise, sometimes it is necessary to use various
noise filtering methods [7–12] that are more effective than (5). The use of more
effective, although more complex, methods of numerical differentiation [13–15]
is also justified.
REFERENСES
1. P.R. Deboeck, S.M. Boker, “Modeling noisy data with differential equations using
observed and expected matrices,” Psychometrika, vol. 75, no. 3, pp. 420–437, Sept.
2010. doi: 10.1007/S11336-010-9168-2
2. H. Liang, H. Miao, and H. Wu, “Estimation of constant and time-varying dynamic
parameters of HIV infection in a nonlinear differential equation model,” Ann. Appl.
Stat., no. 4(1), pp. 460–483, Mar. 1, 2010. doi: 10.1214/09-AOAS290
3. J.O. Ramsay, G. Hooker, D. Campbell, and J. Cao, “Parameter estimation for differ-
ential equations: a generalized smoothing approach,” J. Roy. Statistic. Soc., Ser. B,
vol. 69, no. 5, pp. 741–796, Nov. 2007. doi: 10.1111/j.1467-9868.2007.00610.x
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2025, № 1 52
4. V. Gorodetskyi, “Identification of nonlinear systems with periodic external actions
(Part 1),” System Research & Information Technologies, no. 3, pp. 93–106, 2024.
doi: 10.20535/SRIT.2308-8893.2024.3.06
5. V. Gorodetskyi, “Identification of nonlinear systems with periodic external actions
(Part 2),” System Research & Information Technologies, no. 4, pp. 66–76, 2024. doi:
10.20535/SRIT.2308-8893.2024.4.05
6. O.E. Rössler, “An equation for continuous chaos,” Phys. Lett. A., vol. 57, no. 5,
pp. 397–398, 1976. doi: 10.1016/0375-9601(76)90101-8
7. M. Mafi, H. Martin, M. Cabrerizo, J. Andrian, A. Barreto, and M. Adjouadi,
“A comprehensive survey on impulse and Gaussian denoising filters for digital images,”
Sign. Proc., vol. 157, pp. 236–260, Apr. 2019. doi: 10.1016/j.sigpro.2018.12.006
8. L. Tan, J. Jiang, “Finite Impulse Response Filter Design,” in Digital Signal
Processing (Third Edition): Fundamentals and Applications, 2019, pp. 229–313.
9. J.P. do Vale Madeiro, P.C. Cortez, J.M. da Silva Monteiro Filho, and P.R.F. Rodri-
gues, “Techniques for Noise Suppression for ECG Signal Processing,” in Develop-
ments and Applications for ECG Signal Processing. Modeling: Segmentation, and
Pattern Recognition, pp. 53–87, 2019. doi: 10.1016/B978-0-12-814035-2.00009-8
10. S.V. Vaseghi, Advanced Digital Signal Processing and Noise Reduction. (Fourth
Edition). London, UK: Wiley and Sons, 2008.
11. Y. Chen, S. Fomel, “Random noise attenuation using local signal-and-noise orthogo-
nalization,” Geophys., vol. 80, no. 6, pp. WD1–WD9, Nov.-Dec., 2015. doi:
10.1190/GEO2014-0227.1
12. T. Kelemenová, O. Benedik, and I. Koláriková, “Signal noise reduction and filtering,”
Acta Mechatronica, vol. 5, no. 2, pp. 29–34, Jun. 2020. doi: 10.22306/am.v5i2.65
13. I. Knowles, R.J. Renka, “Methods for numerical differentiation of noisy data,”
Electr. J. Dif. Eq., Conference 21, pp. 235–246, 2014, presented at Variational and
Topological Methods: Theory, Applications, Numerical Simulations, and Open
Problems, 2012.
14. R. Chartrand, “Numerical differentiation of noisy, nonsmooth data,” Intern. Schol.
Res. Netw. Appl. Math., vol. 2011, Art. no. 164564. doi: 10.5402/2011/164564
15. K. Ahnert, M. Abel, “Numerical differentiation of experimental data: local versus
global methods,” Comput. Phys. Commun. vol. 177, no. 10, pp. 764–774, Nov. 2007.
doi: 10.1016/j.cpc.2007.03.009
Received 14.01.2025
INFORMATION ON THE ARTICLE
Viktor G. Gorodetskyi, ORCID: 0000-0003-4642-3060, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: v.gorodetskyi@ukr.net
ІДЕНТИФІКАЦІЯ НЕЛІНІЙНИХ СИСТЕМ З ПЕРІОДИЧНИМИ ЗОВНІШНІМИ
ДІЯМИ (Частина ІIІ) / В.Г. Городецький
Анотація. Розглянуто проблему ідентифікації математичної моделі у вигляді
системи звичайних диференціальних рівнянь. Ідентифікована система може
мати сталі та періодичні коефіцієнти. Джерелом інформації для розв’язання
поставленої задачі є часові ряди спостережуваних змінних. Досліджено вплив
шуму з рівномірним розподілом на результат ідентифікації. У ході досліджен-
ня використовувався алгоритм, запропонований автором у попередніх працях.
Розглянуто особливості застосування цього алгоритму для даної задачі. Пока-
зано, що метод має різну чутливість до шуму залежно від того, яка із спосте-
режуваних змінних забруднена шумом. Реалізацію методу проілюстровано чи-
сельними прикладами ідентифікації нелінійних диференціальних рівнянь із
поліноміальними правими частинами.
Ключові слова: ідентифікація, звичайне диференціальне рівняння, періодич-
ний коефіцієнт, сталий коефіцієнт, шум із рівномірним розподілом.
|
| id | journaliasakpiua-article-329343 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-09-17T09:26:02Z |
| publishDate | 2025 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/9c/5e9a044104f0a337d8755681b9c2319c.pdf |
| spelling | journaliasakpiua-article-3293432025-05-20T17:56:07Z Identification of nonlinear systems with periodic external actions (Part III) Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина III) Gorodetskyi, Viktor ідентифікація звичайне диференціальне рівняння періодичний коефіцієнт сталий коефіцієнт шум із рівномірним розподілом identification ordinary differential equation periodic coefficient constant coefficient uniformly distributed noise The article considers the problem of identifying a mathematical model in the form of a system of ordinary differential equations. The identified system can have constant and periodic coefficients. The source of information for solving the problem is time series of observed variables. The article studies the effect of uniformly distributed noise on the identification result. To solve the problem, the algorithm proposed by the author in previous works was used. It is shown that the method has different sensitivity to noise depending on which of the observed variables is contaminated with noise. The implementation of the method is illustrated by numerical examples of identifying nonlinear differential equations with polynomial right-hand sides. Розглянуто проблему ідентифікації математичної моделі у вигляді системи звичайних диференціальних рівнянь. Ідентифікована система може мати сталі та періодичні коефіцієнти. Джерелом інформації для розв’язання поставленої задачі є часові ряди спостережуваних змінних. Досліджено вплив шуму з рівномірним розподілом на результат ідентифікації. У ході дослідження використовувався алгоритм, запропонований автором у попередніх працях. Розглянуто особливості застосування цього алгоритму для даної задачі. Показано, що метод має різну чутливість до шуму залежно від того, яка із спостережуваних змінних забруднена шумом. Реалізацію методу проілюстровано чисельними прикладами ідентифікації нелінійних диференціальних рівнянь із поліноміальними правими частинами. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2025-03-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/329343 10.20535/SRIT.2308-8893.2025.1.04 System research and information technologies; No. 1 (2025); 44-52 Системные исследования и информационные технологии; № 1 (2025); 44-52 Системні дослідження та інформаційні технології; № 1 (2025); 44-52 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/329343/318902 |
| spellingShingle | ідентифікація звичайне диференціальне рівняння періодичний коефіцієнт сталий коефіцієнт шум із рівномірним розподілом Gorodetskyi, Viktor Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина III) |
| title | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина III) |
| title_alt | Identification of nonlinear systems with periodic external actions (Part III) |
| title_full | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина III) |
| title_fullStr | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина III) |
| title_full_unstemmed | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина III) |
| title_short | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина III) |
| title_sort | ідентифікація нелінійних систем з періодичними зовнішніми діями (частина iii) |
| topic | ідентифікація звичайне диференціальне рівняння періодичний коефіцієнт сталий коефіцієнт шум із рівномірним розподілом |
| topic_facet | ідентифікація звичайне диференціальне рівняння періодичний коефіцієнт сталий коефіцієнт шум із рівномірним розподілом identification ordinary differential equation periodic coefficient constant coefficient uniformly distributed noise |
| url | https://journal.iasa.kpi.ua/article/view/329343 |
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