Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина ІI)
The article presents the results of the study, which is a continuation of the author’s previous research. This paper considers more complex problems in identifying nonlinear systems with periodic external actions. The article shows that the previously proposed method is applicable when the periods o...
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| author | Gorodetskyi, Viktor |
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| description | The article presents the results of the study, which is a continuation of the author’s previous research. This paper considers more complex problems in identifying nonlinear systems with periodic external actions. The article shows that the previously proposed method is applicable when the periods of external actions in the same differential equation may differ. At the same time, the ratio between the values of the periods can be both integer and fractional. The conditions under which this is possible are formulated. These conditions are based on the theorem proved in the previous work. Part of this study is devoted to the problem of identification of a chaotic system with an external non-sinusoidal action. To create such an external action, a function with three harmonic components was used. A numerical experiment confirmed the effectiveness of the algorithm in this case as well. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.4.05 |
| first_indexed | 2025-07-17T10:28:39Z |
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Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2024
66 ISSN 1681–6048 System Research & Information Technologies, 2024, № 4
UDC 517.925
DOI: 10.20535/SRIT.2308-8893.2024.4.05
IDENTIFICATION OF NONLINEAR SYSTEMS
WITH PERIODIC EXTERNAL ACTIONS (Part II)
V. GORODETSKYI
Abstract. The article presents the results of the study, which is a continuation of the
author’s previous research. This paper considers more complex problems in identi-
fying nonlinear systems with periodic external actions. The article shows that the
previously proposed method is applicable when the periods of external actions in the
same differential equation may differ. At the same time, the ratio between the values
of the periods can be both integer and fractional. The conditions under which this is
possible are formulated. These conditions are based on the theorem proved in the
previous work. Part of this study is devoted to the problem of identification of
a chaotic system with an external non-sinusoidal action. To create such an external
action, a function with three harmonic components was used. A numerical experi-
ment confirmed the effectiveness of the algorithm in this case as well.
Keywords: identification, ordinary differential equation, external action, periodic
coefficient, constant coefficient.
INTRODUCTION
As is known, non-autonomous mathematical models are widely used to describe
various physical processes [1; 2]. The construction of such models can be reduced
to the so-called inverse problem [3]. In this case, the model is built on the basis of
information about the output of the system, that is, the problem of system identifi-
cation is solved. In this case, the usual formulation of the problem assumes the
presence in the system equations of additive periodic actions and information
about the structure of the system [4–6]. The task becomes more complex when
the structure of the system is unknown and external actions can be either additive
or multiplicative. At the same time, the number of such actions may be not lim-
ited. The solution of the mentioned problem was proposed in [7]. This study is a
development of the author’s previous work and demonstrates additional capabili-
ties of the method introduced in [7].
NOTATIONS AND SOME PREVIOUS RESULTS
Using the notations from [7], we consider a system of ordinary differential equa-
tions (ODE) of the form
m
j
jiji ftcx
0
)()( x , (1)
.0],;0[)}.(),...,({;,...,1 1 een ttttxtxni x
Identification of nonlinear systems with periodic external actions (Part II)
Системні дослідження та інформаційні технології, 2024, № 4 67
In equation (1), we consider the time functions )(txi to be known and the
coefficients )(tcij to be unknown. In this case, any of the coefficients can be ei-
ther constant or a continuous time function of a period T .
If in equation (1) all the coefficients ,)( consttcij then to find them we can
apply the well-known relation:
BA=C -1 , (2)
where C is the vector of the required coefficients of equation (1), and B is the
vector of values ,,...,0),( mktx ki A is the matrix of function ))(( kj tf x values,
mj ,...,0 . In [7], a theorem was proven according to which relation (2) can be
used to calculate the coefficients of equation (1) if the moments of time jt are
subject to the relations:
emm tttmtttttt ,0,0;...,,2, 000201 (3)
and wherein T . Based on this theorem, the algorithm described in [7] was
constructed. That is, if conditions (3) are met and T , then applying formula (2)
for any 0t , we will obtain the exact values of all constant coefficients of the identified
equation. Therefore, if T , then for two different 0t : 01t and 02t the relation
min|| 21 ijijj cc , (4)
must be satisfied, where 1
ijc and 2
ijc are the values of the coefficient )(tcij ob-
tained for 01t and 02t , respectively. In order to avoid errors possible for a specific
value of 0t , intervals of 01t and 02t values are used in calculations. The applica-
tion of the algorithm is illustrated in [7] using the example of identifying an equa-
tion with additive and multiplicative periodic actions having the same period.
GENERALIZATION OF THE PROPOSED METHOD
It is easy to show that the method can be effective for solving more complex
problems. Let an equation of the form (1) have two external periodic actions with
periods 1T and 2T , respectively. Let also there exist 1q and 2q , ,...3,2,1, 21 qq
and etT , such that
TTqTq 2211 , (5)
where T is the least common multiple of 1T and 2T . That is, we have period T ,
which is common for both external actions. Therefore, the condition of the theo-
rem from [7] is met.
Identification of equations with an integer ratio of periods of external actions
Let us consider the case when relation (4) is satisfied, and at the same time
,...3,221 TT or ,...3,212 TT . As an example of using the method, consider
identification of a system
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 68
,)()()(
,
,
3136333303
212
321
xxtcxtctcx
dxxx
xxx
(6)
obtained on the basis of the well-known Rössler system [8]. The system has the
following coefficients:
,
2
sin4.05.0)(,15.0
0
30
T
t
tcd
.4,2,
2
2
sin5.25)(,20)( 60
6
3633 sTsT
T
t
tctc
It is obvious that for the external actions of the system (6), we have
206 TT . That is, condition (5) is satisfied with TT 6 . Fig. 1 shows the
time series of the variables of the system (6) and Fig. 2 shows its phase tra-
jectories.
To identify the third equation of system (6), its general structure was chosen
in the form of a polynomial of the second degree:
2135
2
134333232131303 )()()()()()( xxtcxtcxtcxtcxtctcx
.)()()()( 2
3393238
2
2373136 xtcxxtcxtcxxtc (7)
x1 x2
x3
t t
t
Fig. 1. Time series of system (6) variables
Identification of nonlinear systems with periodic external actions (Part II)
Системні дослідження та інформаційні технології, 2024, № 4 69
The results of applying the algorithm [7] are presented in Table 1.
T a b l e 1 . The values for which the j value is closest to zero. The first row
shows the values at which j takes on the least values. As the row number
increases, the j value also increases
The values calculated for the coefficients of equation (7) at min
№
)(30 tc )(31 tc )(32 tc )(33 tc )(34 tc )(35 tc )(36 tc )(37 tc )(38 tc
)(39 tc
1 2.18 4.75 8.00 9.10 4.00 8.00 6.18 4.00 8.00 2.71
2 4.31 4.00 4.00 8.00 8.00 4.00 0.58 8.00 4.00 8.00
3 1.88 8.00 1.81 5.30 3.91 6.25 6.16 10.33 4.44 8.33
4 1.26 3.41 7.98 4.89 9.76 3.95 4.09 7.99 4.01 5.39
5 7.99 1.06 3.82 3.20 5.78 6.07 6.19 3.98 7.98 6.48
In Table 1, the values that are repeated or multiples are highlighted in bold.
The theorem in article [7] suggests that they can correspond to the real value of
the period. It also follows from the theorem that the presence of such values in a
certain column of the table indicates that this coefficient is constant.
Note that when T , relation (4) must be satisfied. That is, the values
highlighted in bold (which correspond to the real value of the period) must be lo-
cated in the top row of the table. However, this is not observed for some coeffi-
cients, which is explained by the possible presence of computational errors [7].
As can be seen from Table 1, the least period obtained as a result of the calcu-
lation is sT 4 . The value of s8 from the table obviously corresponds to T2 .
Now that the period T of the external actions is known, it is possible to deter-
x2
x3
x1
x3
x2
x1
Fig. 2. Phase trajectories of system (6)
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 70
mine the form of all functions )(3 tc j using the final part of the algorithm [7]. To
do this, we form matrix A and vector B for system (2) taking into account rela-
tions (3) with T and solve system (2) for 0t values from a certain range.
Thus, we obtain the values of the functions )(3 tc j at all points in this range.
Fig. 3 shows the time series )(30 tcс , )(31 tcс , )(33 tcс , )(36 tcс , obtained as a result of
the calculation at the interval st 20,...,00 . The figure shows time series of coef-
ficients of different types: variables )(30 tcс and )(36 tcс , constant zero )(31 tcс , and
constant non-zero )(33 tcс . The numerical values of the constant coefficients can
be estimated from the form of the obtained time series )(3 tc j . More accurate val-
ues can be obtained using their values obtained by solving system (2) for 01t and
02t , for which T or is a multiple of T , see Table 2 in [7].
Based on the graphs and estimated values of the constant coefficients, the
zero coefficients can be eliminated. As a result, the structure of the general
equation (7) is reduced to the structure of the third equation of system (6). Since
we know T value and have a simplified equation structure, we can re-identify the
equation. Time series of the obtained coefficients are presented in Fig. 4.
Note that according to the calculations (Table 1), condition (3) was satisfied
for s4 . This value obviously corresponds to the period of external action 6T .
At the same time, as expected, the value of 0T was not determined as a result of
the calculation. However, after comparing the form of the external action )(30 tcс
with )(36 tcс in Fig. 3 or 4, it can be argued that sTT 2260 .
t
)(30 tcc )(31 tcc
)(33 tcc
)(36 tcc
t t
t
Fig. 3. Time series of calculated coefficients of the third equation of system (6). Calcula-
tion was performed using structure (7) of equation
Identification of nonlinear systems with periodic external actions (Part II)
Системні дослідження та інформаційні технології, 2024, № 4 71
Identification of equations with a fractional ratio of periods of external actions
Let’s consider a more general case when the periods of actions 1T and 2T in the
equation are subject to condition (5) and in this case ,...3,2, 1221 TTTT . Let the
external actions in the third equation of the system (6) have the form:
.2,3,
4
2
sin5.25)(,
2
sin4.05.0)( 60
6
36
0
30 sTsT
T
t
tc
T
t
tc
(8)
Time series and phase trajectories of system (6) under input actions (8) are
presented in Fig. 5 and 6, respectively.
Obviously, with 21 q and 32 q we get sTTTqTq 6,6201 . That is,
relation (5) is satisfied. Then, considering T to be the only period of external
actions in the equation, we can apply the theorem and algorithm from [7]. System
(6) with external actions (8) was solved on an interval of s100 with a step of
st 01.0 . According to the algorithm, the initial times stst 4.0,15.0 0201
were selected. The value was chosen from the interval ];[ eb , sb 1 ,
)(36 tcc
t
)(33 tcc
tt
)(30 tcc
Fig. 4. Time series of calculated coefficients of the third equation of system (6). Calculation
was performed using simplified structure of the equation
x1 x2 x3
t t t
Fig. 5. Time series of variables of system (6) with external actions (8)
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 72
se 11 . Table 2 presents the values for which relation (4) is satisfied. As can
be seen from the table, in most of its columns the value s6 is found (shown in
bold in the table).
T a b l e 2 . The same as in Table 1, for system (6) with external actions (8)
The values calculated for the coefficients of equation (7) at min
№
)(30 tc )(31 tc )(32 tc )(33 tc )(34 tc )(35 tc )(36 tc )(37 tc )(38 tc )(39 tc
1 9.81 2.91 2.00 10.02 6.00 6.00 6.32 6.00 6.36 2.90
2 2.69 6.00 8.03 7.29 1.36 2.70 5.94 9.98 9.99 1.86
3 9.48 4.89 7.62 8.18 8.05 1.43 7.99 2.59 6.00 4.49
4 9.68 9.12 8.00 1.25 2.55 6.91 6.55 1.47 2.61 6.00
5 8.91 2.90 6.00 1.27 10.11 7.19 1.96 6.01 5.94 3.00
After an analysis similar to that carried out in the previous section and elimi-
nation of zero coefficients, we obtain time series of constant and variable coeffi-
cients presented in Fig. 7.
x1 x2 x3
t t t
Fig. 6. Phase trajectories of system (6) with external actions (8)
)(30 tcc )(33 tcc
)(36 tcc
t t
t
Fig. 7. Time series of calculated coefficients of the third equation of system (6) with ex-
ternal actions (8). Calculation was performed using simplified structure of the equation
Identification of nonlinear systems with periodic external actions (Part II)
Системні дослідження та інформаційні технології, 2024, № 4 73
If the data in Table 2 are not informative enough to confidently determine
periods of external actions, one can repeat the numerical experiment on a larger
time interval or/and use the results for more than five least values when creat-
ing the table. Such a numerical experiment was carried out for system (6) with
external actions (8), which was solved over an interval of s200 . As a result of
applying the algorithm, the values given in Table 3 were obtained.
As one can see, the data in Table 3 confirm the correctness of the relation-
ship sT 6 obtained as a result of the analysis of the data in Table 2. The
values of s12 and s18 in Table 3 are additional arguments for such a con-
clusion, since these values are obviously the multiples of s6 .
T a b l e 3 . The same as in Table 2, for system (6) with external actions (8). We
used a time interval of s200 and ten values for which min
The values calculated for the coefficients of equation (7) at min
№
)(30 tc )(31 tc )(32 tc )(33 tc )(34 tc )(35 tc )(36 tc )(37 tc )(38 tc
)(39 tc
1 9.81 2.91 2.00 16.51 6.00 18.00 6.32 6.00 14.13 2.90
2 2.69 6.00 8.03 10.02 18.00 6.00 5.94 18.00 6.36 19.48
3 9.48 18.00 7.62 7.29 1.36 2.70 7.99 12.00 16.95 17.79
4 16.82 4.89 8.00 8.18 8.05 17.06 6.55 9.98 12.00 1.86
5 9.68 17.29 18.74 11.40 12.31 12.00 12.90 14.05 9.99 14.91
6 8.91 9.12 18.00 19.25 2.55 14.14 12.89 2.59 17.45 4.49
7 1.96 2.90 6.00 15.50 10.11 1.43 10.88 1.47 14.97 15.00
8 11.29 15.96 12.00 13.56 1.02 14.18 18.50 12.16 6.00 16.69
9 5.20 1.37 7.33 15.70 19.09 18.68 19.65 16.21 18.00 6.00
10 2.87 9.72 6.72 12.50 19.39 17.94 12.32 6.01 17.98 3.00
IDENTIFICATION OF THE EQUATION UNDER NON-SINUSOIDAL
PERIODIC EXTERNAL ACTION
In this section we investigate system (6) with such coefficients:
,20)(,5.0)(,15.0 3330 tctcd
.2,
2
8
sin8.0
4
sin25.1
2
2
sin25)(36 sT
T
t
T
t
T
t
tc
(9)
System (6) with coefficients (9) was solved on the interval of s100 with
a step .01.0 st Fig. 8 shows the time series of system under study and its ex-
ternal action )(36 tc . Fig. 9 shows the phase trajectories of this system.
Our goal was to identify the third equation of system (6) with coeffi-
cients (9). For this purpose, the general structure of an equation of the form (7)
was used. As a result of applying the algorithm, the data presented in Table 4
were obtained. As can be seen from this table, the smallest possible value of the
period is sT 2 .
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 74
Fig. 10 shows the calculated time series of coefficients of different types:
constant non-zero )(30 tсс , )(33 tсс , variable )(36 tсс and constant zero )(31 tсс . The
rest calculated coefficients of this equation have time series similar to )(31 tсс ,
that is, they are zero.
T a b l e 4 . The same as in Table 3, for system (6) with coefficients (9)
The values calculated for the coefficients of equation (7) at min
№
)(30 tc )(31 tc )(32 tc )(33 tc )(34 tc )(35 tc )(36 tc )(37 tc )(38 tc
)(39 tc
1 8.00 4.00 1.49 8.00 10.00 10.00 2.11 10.00 8.00 10.57
2 9.23 2.00 10.00 10.00 8.00 8.00 2.75 8.00 10.00 7.56
3 10.00 9.24 8.00 6.62 4.00 4.54 2.77 2.00 2.00 5.78
4 10.31 10.00 10.67 5.67 2.00 2.00 8.71 5.60 4.00 4.50
5 4.00 8.00 2.00 4.00 10.97 4.00 1.63 4.00 7.60 2.00
x1 x2
x3
t t
t t
x36(t)
Fig. 8. Time series of system (6) with coefficients (9) and its external action 36 ( )c t
x1 x2 x3
t t t
Fig. 9. Phase trajectories of system (6) with coefficients (9)
Identification of nonlinear systems with periodic external actions (Part II)
Системні дослідження та інформаційні технології, 2024, № 4 75
After simplifying the structure, re-identification was carried out and the time
series presented in Fig. 11 were obtained. Fig. 11, d shows the calculated time series
of external action )(36 tсс (line 1) and the original one )(36 tc (line 2). As we can see,
these time series practically coincide, with the exception of points with singularity.
CONCLUSION
This study examines a number of special cases of using the proposed method for
identifying nonlinear oscillatory systems with external periodic actions. The most
complex part of the identification problem in this case is finding the periods of
external actions. The first part of the algorithm is devoted to solving this problem.
It can be noted that this method makes it relatively easy to find periods of external
actions when identifying systems with an integer value of the ratio 21 TT or
Fig. 11. Time series of calculated coefficients of the third equation of system (6) with
coefficients (9). Calculation was performed using simplified structure of the equation
)(33 tcc
tt
)(30 tcc
t
)(36 tcc
1
t
2
1– )(36 tcc
2– )(36 tc
1
)(36 tc
a b
c d
Fig. 10. Time series of calculated coefficients of the third equation of system (6) with
coefficients (9)
t
)(36 tcc )(31 tcc
t
t
)(33 tcc
t
)(30 tcc
a b
c d
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 4 76
12 TT . If one of these conditions is met, estimating the values of 1T and 2T using
this algorithm does not differ from the case 21 TT considered in [7].
With a fractional ratio 21 TT or 12 TT , much longer observations of sys-
tem’s functioning may be required in order to make an estimation.
The study of the system with non-sinusoidal periodic external action demon-
strates that the proposed method is as effective as in the case of sinusoidal action.
A possible prospect for further development of the method could be, for ex-
ample, a study of the dependence of the magnitude of the algorithm error on vari-
ous parameters of the identified equations. It is also of interest to assess the influ-
ence of noise on the result of applying the algorithm.
REFERENСES
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and Monographs, vol. 176. Providence, Rhode Island: American Mathematical Society, 2011.
doi: 10.1090/surv/176
2. P.E. Kloeden, C. Potzsche, “Nonautonomous dynamical systems in the life sciences,”
Lecture Notes in Mathematics, vol. 2102, pp. 3–39, 2013. doi: 10.1007/978-3-319-03080-7_1
3. A. Tarantola, Inverse problem theory and methods for model parameter estimation.
Philadelphia: Society for Industrial and Applied Mathematics, 2005. doi:
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4. B.P. Bezruchko, D.A. Smirnov, “Constructing nonautonomous differential equations
from a time series,” Phys. Rev. E., vol. 63, art. no. 016207, 2001. doi:
10.1103/PhysRevE.63.016207
5. T. Sauer, “Observing periodically forced systems of difference equations,” J. Difference Equ.
Appl., vol. 16, no. 2-3, pp. 269–273, 2010. doi: 10.1080/10236190902870439
6. V.G. Gorodetskyi, “Identification of nonlinear systems with additive external ac-
tion,” J. Aut. & Inf. Sci., vol. 50, no. 4, pp. 13–24, 2018. doi:
10.1615/JAutomatInfScien.v50.i4.20
7. V.G. Gorodetskyi, “Identification of nonlinear systems with periodic external actions
(Part 1),” System Research and Information Technologies, no. 3, pp. 93–106, 2024. doi:
10.20535/SRIT.2308-8893.2024.3.06
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Received 12.09.2024
INFORMATION ON THE ARTICLE
Viktor G. Gorodetskyi, ORCID: 0000-0003-4642-3060, National Technical Universityof
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: v.gorodetskyi@ukr.net
ІДЕНТИФІКАЦІЯ НЕЛІНІЙНИХ СИСТЕМ З ПЕРІОДИЧНИМИ ЗОВНІШНІМИ
ДІЯМИ (Частина ІI) / В.Г. Городецький
Анотація. Подано результати дослідження, яке є продовженням попередніх
досліджень автора. Розглянуто більш складні задачі ідентифікації нелінійних
систем з періодичними зовнішніми впливами. Показано, що запропонований
раніше метод також застосовний, коли періоди зовнішніх дій в одному дифе-
ренціальному рівнянні можуть відрізнятися. При цьому співвідношення між
значеннями періодів може бути як цілим, так і дробовим. Сформульовано умо-
ви, за яких це можливо, і які базуються на теоремі, доведеній у попередній ро-
боті. Частину цього дослідження присвячено проблемі ідентифікації хаотичної
системи з вхідною несинусоїдальною дією. Для створення такої зовнішньої дії
використано функцію з трьома гармонічними складовими. Чисельний експе-
римент підтвердив ефективність алгоритму і в цьому випадку.
Ключові слова: ідентифікація, звичайне диференціальне рівняння, зовнішня
дія, періодичний коефіцієнт, сталий коефіцієнт.
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| id | journaliasakpiua-article-322499 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:39Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
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| resource_txt_mv | journaliasakpiua/8c/8fe347ebad371d01abad32cf06e8b18c.pdf |
| spelling | journaliasakpiua-article-3224992025-02-09T21:55:38Z Identification of nonlinear systems with periodic external actions (Part II) Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина ІI) Gorodetskyi, Viktor ідентифікація звичайне диференціальне рівняння зовнішня дія періодичний коефіцієнт сталий коефіцієнт identification ordinary differential equation external action periodic coefficient constant coefficient The article presents the results of the study, which is a continuation of the author’s previous research. This paper considers more complex problems in identifying nonlinear systems with periodic external actions. The article shows that the previously proposed method is applicable when the periods of external actions in the same differential equation may differ. At the same time, the ratio between the values of the periods can be both integer and fractional. The conditions under which this is possible are formulated. These conditions are based on the theorem proved in the previous work. Part of this study is devoted to the problem of identification of a chaotic system with an external non-sinusoidal action. To create such an external action, a function with three harmonic components was used. A numerical experiment confirmed the effectiveness of the algorithm in this case as well. Подано результати дослідження, яке є продовженням попередніх досліджень автора. Розглянуто більш складні задачі ідентифікації нелінійних систем з періодичними зовнішніми впливами. Показано, що запропонований раніше метод також застосовний, коли періоди зовнішніх дій в одному диференціальному рівнянні можуть відрізнятися. При цьому співвідношення між значеннями періодів може бути як цілим, так і дробовим. Сформульовано умови, за яких це можливо, і які базуються на теоремі, доведеній у попередній роботі. Частину цього дослідження присвячено проблемі ідентифікації хаотичної системи з вхідною несинусоїдальною дією. Для створення такої зовнішньої дії використано функцію з трьома гармонічними складовими. Чисельний експеримент підтвердив ефективність алгоритму і в цьому випадку. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-12-25 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/322499 10.20535/SRIT.2308-8893.2024.4.05 System research and information technologies; No. 4 (2024); 66-76 Системные исследования и информационные технологии; № 4 (2024); 66-76 Системні дослідження та інформаційні технології; № 4 (2024); 66-76 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/322499/312902 |
| spellingShingle | ідентифікація звичайне диференціальне рівняння зовнішня дія періодичний коефіцієнт сталий коефіцієнт Gorodetskyi, Viktor Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина ІI) |
| title | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина ІI) |
| title_alt | Identification of nonlinear systems with periodic external actions (Part II) |
| title_full | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина ІI) |
| title_fullStr | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина ІI) |
| title_full_unstemmed | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина ІI) |
| title_short | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина ІI) |
| title_sort | ідентифікація нелінійних систем з періодичними зовнішніми діями (частина іi) |
| topic | ідентифікація звичайне диференціальне рівняння зовнішня дія періодичний коефіцієнт сталий коефіцієнт |
| topic_facet | ідентифікація звичайне диференціальне рівняння зовнішня дія періодичний коефіцієнт сталий коефіцієнт identification ordinary differential equation external action periodic coefficient constant coefficient |
| url | https://journal.iasa.kpi.ua/article/view/322499 |
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