Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина І)
The problem of identifying nonlinear systems with periodic external actions is considered in the article. The number of such actions in the system is not limited, and these actions can be either additive or multiplicative. We use a time series of observed system variables to calculate unknown equati...
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2024
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System research and information technologies| _version_ | 1866302987743264768 |
|---|---|
| author | Gorodetskyi, Viktor |
| author_facet | Gorodetskyi, Viktor |
| author_sort | Gorodetskyi, Viktor |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2024-11-16T18:06:34Z |
| description | The problem of identifying nonlinear systems with periodic external actions is considered in the article. The number of such actions in the system is not limited, and these actions can be either additive or multiplicative. We use a time series of observed system variables to calculate unknown equation coefficients. The proven theorem allows us to separate the unknown coefficients of the system into variables and constants. The proposed computational procedure allows us to avoid possible errors caused by the discrete nature of observable time series. Identification of zero coefficients is carried out in two ways, eliminating erroneous zeroing of the terms of the equations. The method is illustrated with a numerical example of identifying a chaotic system with periodic external actions. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2024.3.06 |
| first_indexed | 2025-07-17T10:28:36Z |
| format | Article |
| fulltext |
Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2024
Системні дослідження та інформаційні технології, 2024, № 3 93
UDC 517.925
DOI: 10.20535/SRIT.2308-8893.2024.3.06
IDENTIFICATION OF NONLINEAR SYSTEMS WITH PERIODIC
EXTERNAL ACTIONS (Part I)
V. GORODETSKYI
Abstract. The problem of identifying nonlinear systems with periodic external ac-
tions is considered in the article. The number of such actions in the system is not
limited, and these actions can be either additive or multiplicative. We use a time se-
ries of observed system variables to calculate unknown equation coefficients. The
proven theorem allows us to separate the unknown coefficients of the system into
variables and constants. The proposed computational procedure allows us to avoid
possible errors caused by the discrete nature of observable time series. Identification
of zero coefficients is carried out in two ways, eliminating erroneous zeroing of the
terms of the equations. The method is illustrated with a numerical example of identi-
fying a chaotic system with periodic external actions.
Keywords: identification, ordinary differential equation, external action, periodic
coefficient, constant coefficient.
INTRODUCTION
Nonlinear systems with external actions occur in the study of many real objects
and processes. Such systems are widespread, for example, in biology [1–3], ecol-
ogy [4], epidemiology [5], mechanical engineering [6–8], and electrical engineer-
ing [9].
A significant amount of research is devoted to constructing models of non-
autonomous systems, including those that involve periodic external actions. In
[4], for example, non-autonomous models of the “predator-prey” type are studied
for almost periodic systems that are used in bioinformatics, social networks, and
wireless sensor networks. Study [6] is devoted to the analysis of the influence of
external periodic force on the behavior of the model of single-degree-of-freedom
vibro-impact system. In this work the conditions for the transition of the model
from the chaotic to the regular regime have been studied. In paper [7], a nonlinear
non-autonomous dynamic model of a quarter vehicle with nonlinear spring and
damping was studied. The influences of the damping coefficient, external action
amplitude and frequency on the dynamic responses were analyzed. It was estab-
lished that the system could have chaotic, quasi-periodic or periodic motion. A
study of a bio-reactor model with periodic nutrient forcing is presented in [10].
The paper [11] investigates the global behaviors of the logistic system with peri-
odic impulsive perturbations. The authors formulate condition under which the
system may have periodic solution. A wide class of non-autonomous models is
described and analyzed in [12]. An increasing interest in non-autonomous systems
was admitted in [13]. This survey introduces basic concepts and tools for non-
autonomous dynamical systems and their application to various biological mod-
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 94
els. Investigation [14] is devoted to the analysis of various modes of the oscillator
with periodic external action.
Additionally, extensive research has been conducted to address one of the
specific instances of the inverse problem [15], which is the identification of non-
autonomous systems based on the observed variables. For example, neural net-
works were used in [16; 17] to solve this problem. The study [18] proposes a
method for constructing a nonlinear, non-autonomous model with a hyperbolic
linear part. The articles [19; 20] consider various approaches to identifying sys-
tems of differential equations with an additive external action. A similar problem
for systems of difference equations was solved in [21].
FORMULATION OF THE PROBLEM
In the works mentioned above, the problem of identifying a system with a known
model structure and additive external action is typically addressed. We are at-
tempting to solve a more general problem. Namely, we propose a method for
finding external actions, both additive and multiplicative, without limiting their
quantity. This task is complicated by the lack of information regarding which co-
efficients of the differential equation are constant and which are periodic (repre-
senting external actions). Also, the task can become more difficult when we study
systems with deterministic chaos. As is known [22], the behavior of such systems
essentially depends on the initial conditions.
Consider a system consisting of ordinary differential equations (ODEs) of
the form
m
j
jiji ftcx
0
)()( x , (1)
where )}.(),...,({;,...,1 1 txtxni n x We assume that the right-hand sides of
equation (1) satisfy the conditions for the existence and uniqueness of the solution
on a certain time interval when 0],;0[ ee ttt .
The coefficients in equation (1) can be of three types: const)( tcij ,
0)( tcij , and ),()( tptc ijij where )(tpij is a continuous periodic function with
a period T . This function represents an external action on the system. Each equa-
tion of the system can have many periodic coefficients that correspond to external
actions. Moreover, all periodic coefficients in each equation have the same pe-
riod. Generally speaking, the external actions of equation (1) are multiplicative.
If, for example, 1)(0 xf , then the external action )(0 tci becomes additive.
A method is proposed for solving the following problem. We assume that the
functions )(txi , )(txi , and )(xijf in equation (1) are known. It is necessary to
define the following:
1. Determine which coefficients of equation (1) are periodic, which are con-
stant, and which are zero. We assume that equation (1) has at least one coefficient
of each type. At the same time, the number of coefficients of each type is limited
only by the total number of coefficients in the equation, which is 1m .
2. Find the period of the functions )(tpij .
Identification of nonlinear systems with periodic external actions (Part I)
Системні дослідження та інформаційні технології, 2024, № 3 95
3. Find the values of the constant coefficients.
4. Find the form of the functions )(tpij .
METHOD
To solve this problem, we will use the following approach. Consider the case
when, in equation (1), const)( tcij mj ,...,0 . Then, to find these coefficients,
we can use a system of 1m linear algebraic equations (SLAE) compiled for
1m time points mtt ,...,0 :
)).((...))(())(()(
....................................................................................
)),((...))(())(()(
)),((...))(())(()(
1100
11111001
00110000
mmimmimimi
mimiii
mimiii
tfctfctfctx
tfctfctfctx
tfctfctfctx
xxx
xxx
xxx
(2)
The SLAE (2) coefficients can be calculated using the well-known relation [23]:
BA=C -1 , (3)
where
,
...
1
0
im
i
i
c
c
c
C
))((...))(())((
...
))((...))(())((
))((...))(())((
10
11110
00100
mmmm
m
m
tftftf
tftftf
tftftf
xxx
xxx
xxx
A ,
)(
...
)(
)(
1
0
mi
i
i
tx
tx
tx
B .
To calculate constant coefficients using formula (3), it is sufficiently to do
the following:
1. Select arbitrary moments of time mtt ,...,0 .
2. Form matrix A and vector B for these moments. We assume that matrix
A is not singular.
3. Use formula (3) to obtain vector C .
Note that, since in this case all the imi cc ,...,0 coefficients are constant, it is
sufficiently to perform all the listed above operations for one set of mtt ,...,0 .
If at least one of the required coefficients is a function of time var)( tcij , it
is necessary to find its values at each point in the time interval ];0[ ett . Using
the procedure described above for some arbitrary set of mtt ,...,0 , we will obtain
some value of coefficient )(tcij . At the same time, it is not known to which point
in time from the interval ];0[ et this value corresponds. Thus, the function )(tcij
cannot be constructed. As will be shown below, to eliminate this uncertainty
when calculating periodic coefficients using formula (3), it is sufficiently to im-
pose some conditions on the moments of time for which matrix A and vector B
are formed.
Let the time moments for which the SLAE (2) is formed obey the relations:
emm tttmtttttt ,0,0;,...,2, 000201 . (4)
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 96
Theorem. Let equation (1) has constant coefficients and periodic coeffi-
cients with a period of T . Additionally, let the SLAE be formed in the form of
equation (2) for the time moments, subject to the conditions (4) and when T .
When solving the SLAE using relation (3), we obtain a vector C , which consists
of the values of the periodic coefficients of equation (1) at time 0t as well as the
values of the constant coefficients of this equation.
Proof. Let us first assume that in equation (1), one of the coefficients, for
example, )(0 tci , is periodic with a period T , while the remaining coefficients are
constant. Then the SLAE (2) will take the following form:
)).((...))(())(()()(
..........................................
)),((...))(())(()()(
)),((...))(())(()()(
1100
111110101
001100000
mmimmimmimi
mimiii
mimiii
tfctfctftctx
tfctfctftctx
tfctfctftctx
xxx
xxx
xxx
(5)
Let us form a matrix A and vector B to solve system (5). To do this, we
choose the moments of time according to the condition stated in the theorem,
which is condition (4). If at the same time T , then, since the function is peri-
odic with a period of T , we get
)(...)()( 01000 miii tctctc , (6)
where
.,...,2, 00201 mTttTttTtt m
That is, in the system (5), the periodic coefficient )(0 tci becomes constant.
Therefore, applying formula (3) to solve SLAE (5) is correct. The resulting vector
of coefficients of the system (5) will include the values of the constant coefficients
of equation (1) and the value of the periodic coefficient at time moments (6).
According to the conditions of the problem, all periodic coefficients of equa-
tion (1) have the same period. Therefore, the above reasoning is valid if equation
(1) has more than one periodic coefficient. That is, when forming the SLAE of the
form (2) and considering relations (4) at T , all periodic coefficients of equa-
tion (1) will have constant values that correspond to the time moment 0t . Then,
by solving the SLAE (2), we can determine these constant values of the periodic
coefficients of equation (1) as well as the values of the constant coefficients of
equation (1). This completes the proof.
Corollary. To separate the desired coefficients into constant and periodic
ones, it is sufficient to form two systems of the form (2) for two values of 0t : 01t
and 02t while considering conditions (4). If T , 0201 tt , and kTtt 0201
,...)2,1( k , then when solving these two SLAEs, we obtain the same values for
the constant coefficients in equation (1).
This Corollary was used to construct the identification algorithm described
in the next section. It is also necessary to note an important special case that may
arise with an arbitrary choice of 01t and 02t values in the proposed algorithm and
which may lead to incorrectness of results. This situation will be considered in the
Special case section.
Identification of nonlinear systems with periodic external actions (Part I)
Системні дослідження та інформаційні технології, 2024, № 3 97
NUMERICAL RESULTS
On the basis of proven Theorem and its Corollary, an algorithm was developed
that can be divided into two stages. At the beginning, we can address the first
three points of the formulated problem. In the second stage, the form of the func-
tions )(tpij is determined.
To illustrate the method, we used system (7), which was built based on the
well-known Rössler system [24]:
,)()()(
,
,
3136333303
212
321
xxtcxtctcx
dxxx
xxx
(7)
where
.
2
2
sin5.25)(,20)(,
2
sin4.05.0)(,15.0 363330
T
t
tctc
T
t
tcd
As we can see, the third equation of the system has two external actions: an
additive one )(30 tс and a multiplicative one )(36 tс . The period of external
actions was taken to be sT 11.2 . The system was solved over an interval of
100 s with a step size of .01.0 st Fig. 1 shows the time series of the variables
in the system (7), and Fig. 2 displays its phase trajectories.
The object of study is the third equation of the system (7). We will identify it
based on the general structure of the form (8), which includes a second-degree
polynomial on the right-hand side:
2135
2
134333232131303 )()()()()()( xxtcxtcxtcxtcxtctcx
.)()()()( 2
3393238
2
2373136 xtcxxtcxtcxxtc (8)
x1
x2
t t
t
x3
Fig. 1. Time series of system (7) variables
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 98
The solution to the formulated problem can be significantly simplified if it is
possible to first estimate the period of external actions on the system. This possi-
bility exists, for example, with resonance [25]. In this case, the period of external
action can be estimated based on the period of oscillation of the observed vari-
ables. It is easy to show that for a chaotic system (7) such an approach will not
lead to proper results. Considering that the external actions of this system are in-
cluded in its third equation, then, first of all, these actions can affect the shape of
the function )(3 tx . As a result, this function can become periodic. But, as follows
from Fig. 1, this function has no periodicity. Also, the period of external actions
cannot be estimated based on an analysis of the shape of the functions )(1 tx and
)(2 tx . For example, Fig. 3 shows the time
series )(2 tx and )(36 tv , where the latter is the
variable component of the external action
)(36 tс , ).22(sin5.236 Ttv It is obvious
that the quasi-period of a function )(2 tx does
not coincide with the period of )(36 tv and is
not a multiple. Therefore, we cannot
preliminarily estimate the period of external
actions and thus simplify the solution of the
problem. Moreover, due to the lack of
information about the existence of external
periodic action, we may erroneously assume
that the model has only constant coefficients. Such an initial assumption may lead
to the construction of an inadequate model.
In general, the main steps of the first stage of the proposed algorithm in rela-
tion to equation (8) are as follows:
1. We set 01t , form the SLAE (2), and solve it by setting the values of
within a certain range of );( eb which presumably includes the desired value of
T . Thus, we obtain the values of all coefficients )(3 tc j .
x2
x1
x3
x1
x3
x2
Fig. 2. Phase trajectories of system (7)
t
x2
ν36
x 2
, ν
36
Fig. 3. Comparison of oscillation
periods of functions )(36 tv and )(2 tx
Identification of nonlinear systems with periodic external actions (Part I)
Системні дослідження та інформаційні технології, 2024, № 3 99
2. Repeat step 1 for 0102 tt .
3. For each coefficient )(3 tc j , we find a value at which
min)( 22
3
1
3 jjj cc , (9)
where 1
3 jc and 2
3 jc represent the values of the coefficient )(3 tc j obtained for 01t
and 02t , respectively.
The calculations were carried out for ,15.001 st ,4.002 st ,0.1 sb
.5.11 se The calculation results are shown in Table 1. The first line of Table 1
shows the values for which relation (9) is satisfied. Since the calculations in-
volve discrete time series instead of continuous functions )(txi , there is a possi-
bility of errors when calculating (9). Therefore, in order to obtain more complete
information for analyzing the results, lines 2–5 of Table 1 show the values for
which the j value is closest to zero. As the line number increases, the j value
also increases.
T a b l e 1 . The first line shows the values at which j takes on the least values
The values calculated for the coefficients of equation (8) at min j
№
)(30 tc )(31 tc )(32 tc )(33 tc )(34 tc )(35 tc )(36 tc )(37 tc )(38 tc )(39 tc
1 2.08 8.44 2.11 8.44 8.44 2.11 8.78 2.11 8.44 2.11
2 10.10 2.11 8.44 10.05 2.11 8.44 4.16 8.44 2.11 8.44
3 5.26 4.22 10.55 10.55 4.22 10.55 6.67 4.22 10.55 1.31
4 10.03 10.55 4.22 2.11 10.55 2.05 4.06 10.55 6.33 4.22
5 3.34 9.38 4.71 4.22 8.10 4.22 5.44 8.37 4.22 5.42
Based on the data presented in Table 1, the following conclusions can be drawn.
1. In the columns corresponding to the coefficients ),(),( 3231 tctc ),(33 tc
),(34 tc ),(),( 3735 tctc )(),( 3938 tctc , the predominant values are s11.2 or
multiples: s22.4 , s33.6 , s44.8 and s55.10 (these values are highlighted in bold
in the table). Based on the corollary of the theorem, it can be argued that these
coefficients are constant. Such regularity is not observed for the coefficients
)(30 tc and )(36 tc , suggesting that these coefficients are variable.
2. In addition, we can infer from the table that the search value for the period
of external action is sT 11.2 , and consequently, ,222.4 Ts ,333.6 Ts
,444.8 Ts .555.10 Ts
The values of the coefficients 1
3 jc and 2
3 jc , which satisfy relation (9) and
were used to fill the first line of Table 1, are indicated in Table 2. An examination
of these values suggests that equation (8) has only one non-zero constant coeffi-
cient, 33c . The final conclusion can be reached after further analysis.
Table 2. Calculated values of the constant coefficients of equation (8)
Coefficients
0t
31c 32c 33c 34c 35c 37c 38c 39c
st 15.001 3.417*10-5 5.070*10-5 -20.019 2.399*10-6 6.222*10-6 3.682*10-7 -9.514*10-4 9.320*10-2
st 4.002 -2.017*10-5 -5.214*10-7 -20.022 -1.450*10-6 2.168 *10-7 2.094*10-7 3.027*10-4 4.599*102
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 100
Moreover, it will be possible to answer the fourth point of the formulated
problem, which is to determine the form of the functions )(30 tc and )(36 tc . The
second part of the algorithm is dedicated to solving this problem. With the already
known value of T , the SLAE (2) is formed for 0t varying within a certain range.
By solving the SLAE at all points within this range, we can evaluate the values of
all coefficients (both constant and variable) at these specific time intervals. The
time series of certain coefficients, obtained from the calculation for
sts 3010 0 , are shown in Fig. 4. In the graphs, the calculated values of the
coefficients are indicated by ...,, 3130
cc сс The graphs in this figure have singulari-
ties, i.e., some points where the values of ijc differ significantly from neighboring
points. This occurs because the matrix A formed for calculating the coefficients
of ijc , in this case, has a determinant 0det A .
We also note that, for example, in Fig. 4, b, 031
cс at points where the sin-
gularity does not occur. Here, graphs for some of the coefficients from Table 2
are not shown since all of them, except for )(33 tc , have a similar form to Fig. 4,
b. That is, the values of these coefficients are close to zero. This fact confirms the
preliminary assessment based on the data in Table 2, namely:
039383735343231 ccccccc .
Re-identification of equation (8) using non-zero coefficients ),(30 tc ,)(33 tc
)(36 tc allowed us to obtain the time series, as shown in Fig. 5, a, b, c. Fig. 5, d
Fig. 4. Time series of )(),(),(),( 36333130 tсtсtсtс cccc obtained as a result of the calculation
b
)(31 tсc
t
d
)(36 tсc
t
)(30 tс c
a
t
c
)(33 tсc
t
Identification of nonlinear systems with periodic external actions (Part I)
Системні дослідження та інформаційні технології, 2024, № 3 101
shows the original and identified time series of the )(30 tc coefficient on the inter-
val .2.222.21 sts This figure illustrates the proximity of these series, except
for points with a singularity.
Fig. 6 shows the time series of errors: ,303030 ccc с ,333333 ccc с
363636 ccc c , which allows us to visually estimate the accuracy of identifica-
tion.
SPECIAL CASE
As noted in the Method section, for a certain set of calculation parameters the re-
sult may be incorrect. This situation is possible when a periodic function )(tcij
has the period T and the following conditions are met
)()( 0201 tctc ijij , ,...)3,2(,0201 a
a
T
tt . (10)
Let us illustrate the features of the algorithm application in this case with an
example. Let the system (7) have the following parameters:
.2,5)(,20)(,
2
sin1)(,15.0 363330 sTtctc
T
t
tcd
(11)
Fig. 5. Time series of calculated coefficients: a — 30( )сc t ; b — 33( )сc t ; c — 36( )сc t ; d — initial
30( )c t and calculated 30( )сc t time series on the interval sts 2.222.21
t
c
)(36 tсc
a
t
)(30 tс c
)(30 tс
)(30 tс c
d
)(30 tс
t
b
)(33 tсc
t
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 102
The system was solved over an interval of 100 s with a step size of
.01.0 st Identification was carried out according to the algorithm described
above for stst 2,1 0201 . The input action graph is shown in Fig. 7. As fol-
lows from (11) and illustrated by the graph, the following relationships hold:
)()( 02300130 tctc , s
T
tt 1
20201 .
It can be noted that these
relationships correspond to conditions
(10). As a result of applying the
algorithm, the data presented in Table 3
were obtained.
An analysis of these data similar to
the analysis of Table 1 may lead us to
incorrect conclusions.
1. Since all columns of Table 3
contain values ...,2,1 , all the coeffi-
cients are constant.
2. We can also mistakenly assume
that sT 1 , and the values sss 10...,,3,2
are multiples.
Obviously, both of these conclusions are incorrect. But this result can be eas-
ily corrected by changing the calculation parameters of the algorithm to violate
the conditions (10). In this case, the value st 01.202 was used instead of
st 202 . That is, the moment of time 02t was shifted by 1 step compared to the
previous case. The calculation results are given in Table 4.
c
30
t
c
33
t
c
36
t
Fig. 6. Time series of errors
Fig. 7. Time series of external action in (11)
c 3
0
t
Identification of nonlinear systems with periodic external actions (Part I)
Системні дослідження та інформаційні технології, 2024, № 3 103
T a b l e 3 . The same as in Table 1 for system (7) with parameters (11)
The values calculated for the coefficients of equation (8) at min j
№
)(30 tc )(31 tc )(32 tc )(33 tc )(34 tc )(35 tc )(36 tc )(37 tc )(38 tc
)(39 tc
1 1.00 9.00 4.00 7.00 3.00 10.00 6.07 3.00 4.00 6.00
2 7.00 2.00 9.00 1.00 2.00 4.00 6.00 8.00 2.00 5.72
3 9.00 10.00 8.00 6.00 10.00 8.00 2.00 4.00 9.00 5.64
4 2.00 3.00 10.00 9.00 1.00 5.00 5.00 2.00 8.00 5.99
5 3.00 1.00 2.00 5.00 5.00 9.00 9.00 7.00 3.42 10.00
T a b l e 4 . The same as in Table 3 with st 01.202
The values calculated for the coefficients of equation (8) at min j
№
)(30 tc )(31 tc )(32 tc )(33 tc )(34 tc )(35 tc )(36 tc )(37 tc )(38 tc
)(39 tc
1 9.07 10.00 4.00 6.00 2.00 10.00 2.00 8.00 4.00 6.00
2 2.49 2.00 10.00 2.00 10.00 4.00 6.00 4.00 7.65 6.10
3 9.47 4.51 8.00 8.40 7.50 8.00 8.00 2.00 2.00 8.53
4 5.72 4.84 2.00 8.97 8.00 2.00 4.00 10.00 8.00 5.67
5 0.58 8.00 6.00 8.00 4.00 2.28 10.00 6.00 10.00 10.00
The data in Table 4 already allow us to draw the correct conclusions.
1. In the identified equation, the coefficient 30с is periodic, the other coeffi-
cients are constant.
2. The function )(30 tc has a period sT 2 and the values ssss 10,8,6,4
are multiples.
It should be noted that in a real study, repeating the identification with
changed 02t is not mandatory to obtain the correct result. It is sufficient to apply
the last stage of the algorithm based on the data in Table 3, namely, to try to ob-
tain the form of the input action with already known possible values of T . For
this purpose, the SLAE (2) is formed taking into account the conditions (6) for 0t
varying within a certain range. By solving the SLAE at all points within this
range, we can evaluate the values of ijc coefficients for all these points. Thus, we
obtain the function )(tcij .
On the contrary, when conditions (10) are met, the condition
)(...)()( 10 mijijij tctctc is met only for specific 0t . If we choose sT 1 ,
then, for example, at st 00 we will get according to (6) st 11 . As it is seen
from Fig. 7, )1()0( 3030 scsc . However, at st 5.00 we will get st 5.11 and
)5.1()5.0( 3030 scsc . Thus, we cannot construct a function )(tcij for all 0t
within a given time interval.
This feature becomes apparent when we perform the second stage of the al-
gorithm. Time series of all coefficients at sT 1 were calculated. Two of them
are shown in Fig. 8, a and 8, b. For comparison, Fig. 8, c and 8, d show the graphs
obtained for the same functions, but at sT 2 . The choice of the correct T value
when considering these four graphs is obvious.
V. Gorodetskyi
ISSN 1681–6048 System Research & Information Technologies, 2024, № 3 104
CONCLUSIONS
The proven theorem and its corollary make it possible to solve the inverse prob-
lem with many unknowns. Such unknowns can be the number and values of con-
stant ODE coefficients, the number, period and forms of external actions. The
latter can be both additive and multiplicative. The number of external actions in
each equation of the system is unlimited. The only restriction is that all the exter-
nal actions in each equation must have the same period. The method allows us to
detect unknown periodic actions that cannot be identified based on the form of
observed variables.
To solve the formulated problem, it is not necessary to know in advance
which coefficients in the ODE system’s equations are variables, constants, or ze-
ros. To use proposed method for solving formulated problem it is sufficiently to
have time series of observed variables.
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Received 15.01.2024
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 І) / В.Г. Городецький
Анотація. Розглянуто проблему ідентифікації нелінійних систем з періодич-
ними зовнішніми діями. Кількість таких дій у системі не обмежена, і ці дії мо-
жуть бути як адитивними, так і мультиплікативними. Для обчислення невідо-
мих коефіцієнтів рівнянь використано часові ряди спостережуваних змінних
системи. Доведена теорема дозволяє розділити невідомі коефіцієнти системи
на змінні та сталі. Запропонована обчислювальна процедура дозволяє уникну-
ти можливих помилок, спричинених дискретністю спостережуваних часових
рядів. Ідентифікацію нульових коефіцієнтів виконано двома способами, що
виключає помилкове обнулення членів рівнянь. Метод ілюстровано числовим
прикладом ідентифікації хаотичної системи з періодичними зовнішніми діями.
Ключові слова: ідентифікація, звичайне диференціальне рівняння, зовнішня
дія, періодичний коефіцієнт, сталий коефіцієнт.
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| id | journaliasakpiua-article-315266 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:28:36Z |
| publishDate | 2024 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/4a/a4b4bef85ea4b55f57a42482a8292a4a.pdf |
| spelling | journaliasakpiua-article-3152662024-11-16T18:06:34Z Identification of nonlinear systems with periodic external actions (Part I) Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина І) Gorodetskyi, Viktor identification ordinary differential equation external action periodic coefficient constant coefficient ідентифікація звичайне диференціальне рівняння зовнішня дія періодичний коефіцієнт сталий коефіцієнт The problem of identifying nonlinear systems with periodic external actions is considered in the article. The number of such actions in the system is not limited, and these actions can be either additive or multiplicative. We use a time series of observed system variables to calculate unknown equation coefficients. The proven theorem allows us to separate the unknown coefficients of the system into variables and constants. The proposed computational procedure allows us to avoid possible errors caused by the discrete nature of observable time series. Identification of zero coefficients is carried out in two ways, eliminating erroneous zeroing of the terms of the equations. The method is illustrated with a numerical example of identifying a chaotic system with periodic external actions. Розглянуто проблему ідентифікації нелінійних систем з періодичними зовнішніми діями. Кількість таких дій у системі не обмежена, і ці дії можуть бути як адитивними, так і мультиплікативними. Для обчислення невідомих коефіцієнтів рівнянь використано часові ряди спостережуваних змінних системи. Доведена теорема дозволяє розділити невідомі коефіцієнти системи на змінні та сталі. Запропонована обчислювальна процедура дозволяє уникнути можливих помилок, спричинених дискретністю спостережуваних часових рядів. Ідентифікацію нульових коефіцієнтів виконано двома способами, що виключає помилкове обнулення членів рівнянь. Метод ілюстровано числовим прикладом ідентифікації хаотичної системи з періодичними зовнішніми діями. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2024-09-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/315266 10.20535/SRIT.2308-8893.2024.3.06 System research and information technologies; No. 3 (2024); 93-106 Системные исследования и информационные технологии; № 3 (2024); 93-106 Системні дослідження та інформаційні технології; № 3 (2024); 93-106 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/315266/306080 |
| spellingShingle | ідентифікація звичайне диференціальне рівняння зовнішня дія періодичний коефіцієнт сталий коефіцієнт Gorodetskyi, Viktor Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина І) |
| title | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина І) |
| title_alt | Identification of nonlinear systems with periodic external actions (Part I) |
| title_full | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина І) |
| title_fullStr | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина І) |
| title_full_unstemmed | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина І) |
| title_short | Ідентифікація нелінійних систем з періодичними зовнішніми діями (Частина І) |
| title_sort | ідентифікація нелінійних систем з періодичними зовнішніми діями (частина і) |
| topic | ідентифікація звичайне диференціальне рівняння зовнішня дія періодичний коефіцієнт сталий коефіцієнт |
| topic_facet | identification ordinary differential equation external action periodic coefficient constant coefficient ідентифікація звичайне диференціальне рівняння зовнішня дія періодичний коефіцієнт сталий коефіцієнт |
| url | https://journal.iasa.kpi.ua/article/view/315266 |
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