Стратегія забезпечення асимптотичної збіжності процесу нелінійного оцінювання параметрів динамічних об’єктів
The article considers a step-by-step strategy of sequential use and adjustment of a parallel model to an object of identical structure with orthogonal operators, a series-parallel model to an object with the connection of operators of a certain type for orthogonal approximation in order to obtain as...
Gespeichert in:
| Datum: | 2025 |
|---|---|
| Hauptverfasser: | , , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
2025
|
| Schlagworte: | |
| Online Zugang: | https://journal.iasa.kpi.ua/article/view/336060 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | System research and information technologies |
| Завантажити файл: |  |
Institution
System research and information technologies| _version_ | 1867334452338753536 |
|---|---|
| author | Silvestrov, Anton Ostroverkhov, Mykola Spinul, Liudmyla Khalimovskyy, Oleksiy Veshchykov, Heorhii |
| author_facet | Silvestrov, Anton Ostroverkhov, Mykola Spinul, Liudmyla Khalimovskyy, Oleksiy Veshchykov, Heorhii |
| author_institution_txt_mv | [
{
"author": "Anton Silvestrov",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Mykola Ostroverkhov",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Liudmyla Spinul",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Oleksiy Khalimovskyy",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
},
{
"author": "Heorhii Veshchykov",
"institution": "National Technical University of Ukraine \"Igor Sikorsky Kyiv Polytechnic Institute\", Kyiv"
}
] |
| author_sort | Silvestrov, Anton |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2025-07-25T15:56:08Z |
| description | The article considers a step-by-step strategy of sequential use and adjustment of a parallel model to an object of identical structure with orthogonal operators, a series-parallel model to an object with the connection of operators of a certain type for orthogonal approximation in order to obtain asymptotically unbiased estimates of coefficients of a structurally identical to a dynamic object of a mathematical model under conditions of noise of measurements of the initial variable of the identification object and non-convexity of the proximity functional of the initial variables of the object and the model in a space of coefficients of the object’s mathematical model. Structural diagrams of each stage of identification are given using refined parameters and the structure of the model of object. This algorithm was implemented to identify the parameters of the mathematical model of aircraft, provided that the sample of experiment data is limited and there is of the initial a significant range of deviations of state variables from the basic mode. |
| doi_str_mv | 10.20535/SRIT.2308-8893.2025.2.08 |
| first_indexed | 2025-07-27T04:04:09Z |
| format | Article |
| fulltext |
Publisher IASA at the Igor Sikorsky Kyiv Polytechnic Institute, 2025
Системні дослідження та інформаційні технології, 2025, № 2 115
UDC 519.711
DOI: 10.20535/SRIT.2308-8893.2025.2.08
STRATEGY FOR ENSURING ASYMPTOTIC CONVERGENCE
OF THE PROCESS OF NON-LINEAR ESTIMATION
OF DYNAMIC OBJECT PARAMETERS
А. SILVESTROV, M. OSTROVERHOV, L. SPINUL,
O. KHALIMOVSKYY, H. VESHCHYKOV
Abstract. The article considers a step-by-step strategy of sequential use and adjust-
ment of a parallel model to an object of identical structure with orthogonal opera-
tors, a series-parallel model to an object with the connection of operators of a certain
type for orthogonal approximation in order to obtain asymptotically unbiased esti-
mates of coefficients of a structurally identical to a dynamic object of a mathemati-
cal model under conditions of noise of measurements of the initial variable of the
identification object and non-convexity of the proximity functional of the initial
variables of the object and the model in a space of coefficients of the object’s
mathematical model. Structural diagrams of each stage of identification are given
using refined parameters and the structure of the model of object. This algorithm
was implemented to identify the parameters of the mathematical model of aircraft,
provided that the sample of experiment data is limited and there is of the initial
a significant range of deviations of state variables from the basic mode.
Keywords: non-linear estimation, identification, convergence of estimation algo-
rithms, optimization.
INTRODUCTION
An important place in the problems of non-linear programming (identification
[1–8]) is occupied by the form of a proximity functional of initial variables of an
object and a model, whose physical parameters are optimized at the extremum
(usually the minimum) of the variable functional. This functional is simultane-
ously a function of physical parameters of the model being optimized. If the
“input (x) – output (y)” mapping of the object can be represented by the following
differential equation:
)(... 11
1
10 tya
dt
dy
a
dt
yd
a
dt
yd
a nnn
n
n
n
)(... 11
1
10 txb
dt
dx
b
dt
xd
b
dt
xd
b mmm
m
m
m
, (1)
where niai ,0, ; mjb j ,0, are parameters to be determined using entries
,)( kty )( ktx , Nk ,1 , considering that )( kty is measured with uncorrelated
noises )( kt ; then desired coefficients ia of the model (1) enter the equation of
error )( kt between )( kty and output variable )( kM ty non-linearly, since they
are in the denominator of the model operator )( pWM :
А. Silvestrov, M. Ostroverhov, L. Spinul, O. Khalimovskyy, H. Veshchykov
ISSN 1681–6048 System Research & Information Technologies, 2025, № 2 116
)()()()( txpWtyt M ,
where
dt
d
p
pa
pb
pWM ,
)(
)(
)( , ,...)( 1
10 m
mm bpbpbpb npapa 0)(
n
n apa ...1
1 .
Accordingly, parameters ia non-linearly enter mean square 2 of error
)( kt (both for the functional of )( kty , )( kM ty and the functions of parameters
ia , jb ), violating the elliptic form of dependence 2 on deviations ia of esti-
mates ia from their optimal values *
ia :
ji ba
ji ba
,
2** minarg),( . (2)
Violation of ellipticity and strict convexity of function ),(2
ji ba (the “ra-
vine” effect) significantly complicates the recurrent process of convergence of
parameters ia , jb with their optimal values **, ji ba that satisfy the condition (2).
Furthermore, the form of function ),(2
ji ba significantly depends on the
bandwidth of input signal )(tx , because 2 is simultaneously a functional of
)(ty , )(tyM , which, in turn, depend on input stimulus )(tx . For the given )(tx
and the structure of the model (1), the key to successful optimization of the re-
laxation process for convergence of the model (1) parameter estimates with their
optimal values (2) can be a strategy of using a set of different models and a series
of their connection and identification, which ensures convexity and ellipticity
),(2
ji ba .
FORMULATION OF THE PROBLEM
While using an object-parallel:
model with a structure identical to the object (1);
model with a series-parallel connection of operators
)(
)(
pc
pa
and
)(
)(
pc
pb
to
the object (where )( pc is a filter of the degree p exceeding the degree n of poly-
nomial )( pa , which ensures the correctness of differentiation operation )(ty ;
as well as model with orthogonal parameters connected in parallel to the
object; it is necessary to organize the sequence of their connection and adjustment
in such a way as to ensure strict convexity and ellipticity of proximity functions of
the object and the model being optimized, and thus, the convergence of the relaxation
process for identification of the model (1) coefficients under the condition (2).
Strategy for ensuring asymptotic convergence of the process of non-linear…
Системні дослідження та інформаційні технології, 2025, № 2 117
Strategy for implementing the condition of guaranteed convergence of model
(1) parameters with their optimal values (2)
The strategy consists of four steps, where Steps 2, 3, 4 are to be repeated until
condition (2) is met.
Step 1. Identification (Fig. 1) of weight coefficients i of operators )( pWi
of the model (3).
An equation of the model parallel to the object:
n
i
ii
n
i
iiM txtxpWty
11
))(()}()({)( , (3)
where )()()( txpWx ii are linearly independent functions of time t.
Operators )( pWi transform input stimulus )(tx into a system of linearly in-
dependent functions ))(( txi . If )(tx is close to “white noise”, then, using, for
instance, Lager functions as these operators
i
i
i
p
p
pW
)(
)(
)(
, (4)
(where is the operator parameter), we obtain a system of mutually orthogonal
functions ))(( txi . Linear independence, and even more so, orthogonality
))(( txi guarantee strict convexity and ellipticity of mean square 2 of error
)( kt [1], [2]. In the “off-line” mode, the determination of the model (3) coeffi-
cients i can be one-step, if we have a data sample )( ktx , )( kty , Mk ,1 . Then
estimates î of coefficients i are determined by the least squares method (LSM)
under the condition
M
k
k
M 1
2 )(
1
argminˆ ,
where
n
i
ii txkyk
1
))(()()( .
Meaning, )()()(ˆ *T1TT1T YY .
LSM estimation ̂ of the vector of coefficients i , ni ,1 will be unbiased,
since noise )(t is uncorrelated with )(ti , and if )(t is a Gaussian “white
noise”, then estimate ̂ will have minimal variance. If )(tx is a Gaussian “white
Fig. 1. Approximation of the “input-output” mapping (1) of the object in the model (3)
Оbject (1)
Model (4)
( )x t
*( )y t
( )t
( )y t
( )My t
( )t
? Evaluatorβ
А. Silvestrov, M. Ostroverhov, L. Spinul, O. Khalimovskyy, H. Veshchykov
ISSN 1681–6048 System Research & Information Technologies, 2025, № 2 118
noise”, and operators )( pWi are type (4), then matrix T will be diagonal, and
each component î of vector ̂ is determined independently:
M
k
i
M
k
i
i
k
kyk
1
2
1
)(
)()(
ˆ .
In the “on-line” mode, the recurrent process of adjusting coefficients i of
the model (3) is done through the gradient procedure:
)()(
ˆ
tt
dt
d
ii
i
.
If i is limited, the process of approximating estimate )(ˆ ti to the optimal
stationary value is exponential, which achieves an exponentially weighted averag-
ing of the current value )(2 t .
Resulting from operation of the system (Fig. 1) in the first step, with the lim-
ited dimensionality n of the model (3), we have a close approximation of mapping
)(tx onto )(* ty in the model (3) (non-parametric identification). In this case, the
structures of the desired mapping (1) and the model (3) are different, but My no
longer contains noise )(t .
Step 2. Approximation of mapping )(tx onto )(ˆ tyM that uses the series-
parallel model (Fig. 2) of the equation (1).
Total error equation:
)(
)(
)(ˆ
)(
)(
)(ˆ
)( tx
pc
pb
ty
pc
pa
t My , (5)
where )( pc is a filter polynomial with degree exceeding polynomial degree
)( pa ; polynomial structures )(ˆ pa and )(ˆ pb are identical to the polynomial
structures )( pa and )( pb of the object model (1). Then, depending on the “on-
Model (4)
ˆ( )
( )
b p
c p
ˆ( )
( )
a p
c p
( )x t ( )My t
y t
ˆ
y
b
ˆ
y
a
âb̂
Evaluator
ˆˆ,a b
Fig. 2. Approximation of mapping )(tx onto )(ˆ* tyM in the model (1)
Strategy for ensuring asymptotic convergence of the process of non-linear…
Системні дослідження та інформаційні технології, 2025, № 2 119
line” or “off-line” modes of minimizing the mean square of total error )(ty , by
means of the adaptive circuit, the least squares method, or the gradient method,
respectively, the optimal values of coefficients niai ,1,ˆ , mjbj ,1,ˆ are calcu-
lated. Absence of “noise” in signal )(ˆ tyM , and linearity of dependence )(ty on
coefficients iâ , jb̂ being adjusted guarantee that their estimates are obtained.
However, their values are not yet true values ia , jb of the object model (1). This
is due to the proximity of mapping )(* ty in the model (3).
Step 3. Approximation of mapping )(tx onto )(* ty in the model, in the
form of a composition of the series connection of model
)(ˆ
)(ˆ
pa
pb
obtained in the
previous step, and the orthogonal model (3) (Fig. 3).
In this step, an inaccuracy of mapping )(tx onto )(* ty in the model
)(ˆ
)(ˆ
pa
pb
is
compensated by adjusting coefficients i of model
n
i
ii pW
1
)( , which is turned
on in series with model
)(ˆ
)(ˆ
pa
pb
(Fig. 3). The adjustment process i is similar to
the process in the Step 1 (as per LSM in “off-line” mode, or gradient method in
“on-line” mode). Now, however, model
n
i
ii pW
1
)( should map not the mapping
)(tx onto )(* ty , but only mapping )(* tyM onto )(* ty , which is much simpler,
since )(* tyM is already more or less close to )(* ty .
Step 4 repeats Step 2 (Fig. 2), but for mapping )(tx onto )(* ty clarified in
the model
n
i
ii pW
1
)( . Theoretically, if in Step 4 n of the model
n
i
ii pW
1
)( is
unlimited, we will already get estimates iâ , jb̂ of parameters ia , jb of the iden-
tification object’s accurate model (1), which will be unbiased by noise )(t .
( )x t *( )y t
( )t
( )y t
( )t
ˆ( )
ˆ( )
b t
a t
ˆ( )y t* ( )My t
1
( )
n
i i
i
W t
Object (1)
? Evaluatorβ
Fig. 3. Structure of the identification system in Step 3
А. Silvestrov, M. Ostroverhov, L. Spinul, O. Khalimovskyy, H. Veshchykov
ISSN 1681–6048 System Research & Information Technologies, 2025, № 2 120
In practice, if n is limited, Steps 2–4 are repeated, and )(ˆ ty gradually ap-
proaches )(* ty of the object, coefficients i of the model (3) approach zero, ex-
cept for 0 , which approaches one with the unit operator 1)(0 pW , and coeffi-
cient estimates iâ , jb̂ approach true values ia , jb of the model (1).
A flow chart of the strategy of using three types of models is presented in Fig. 4.
We will consider the feasibility
of using the proposed strategy on the
example of identifying coefficients of
the transfer function of an aircraft in
longitudinal short-periodic movement
[4]. The problem of identification lies
in a concept of the transfer function
being valid only under conditions of
linearity and stationarity of the map-
ping of the control stimulus (deviation
)(th of the altitude control) in the
deviation of attack angle )(t of the
angle between the longitudinal axis of
the aircraft and the direction of the
oncoming air flow.
Short-periodic movement means
movement in a short time interval dur-
ing which factors not taken into ac-
count in model (1) hardly change.
These include non-linearity, non-
stationarity, speed changes, height,
weight, and dimensions of the aircraft,
etc. Therefore, the mathematical mod-
el of the aircraft movement will be
more accurate, the smaller the time
and deviation of the variables from a
certain basic balancing mode are. If
the time and range of deviations of the
aircraft state variables from the basic
mode tend to zero, the model (1) tends
to being perfectly accurate. But in re-
ality, the measurements of the aircraft
state variables give not only accurate,
but a random component of noise.
Then, when the range of the variable
deviations decreases, the “noise-to-
signal” ratio increases, and, in a lim-
ited time interval, the variance of es-
timates of aircraft parameters will in-
crease unacceptably. Desire to reduce impact of non-linearity on a linearized air-
craft model leads to an increase in the “noise-to-signal” ratio and, as a result, an
increase in the random component of the error in estimating the parameters of the
mathematical model (MM) of the aircraft. This range increase leads to a shift in
the linear MM parameters’ estimates due to the influence of non-linearity of the
Start
Input
( ), ( )x k y k
i
ˆˆ ,i ja b
i
Calculation
ˆˆ ,i ja b
in the block 4
Estimating the
difference
1ˆ ˆk ka a
1
ˆ ˆ
k kb b
End
Calculation
in the block 3
Calculation
in the block 2
Calculation
in the block 1
Less given error
Fig. 4. Flow chart of the identification algo-
rithm for model (1) parameters
Strategy for ensuring asymptotic convergence of the process of non-linear…
Системні дослідження та інформаційні технології, 2025, № 2 121
aircraft characteristics. If it is impossible to fulfill other conditions (a large sample
of experimental data and deviation range of the state variables from the basic
mode), the condition of ellipticity and strict quadraticity of the error functional
between )(ty and )(tyM , as a function of the optimized parameters of the aircraft
MM, allow to successfully solve the problem of identifying parameters iâ and jb̂
of the MM on the level of their proximity to real physical values ia and jb of the
accurate MM (1).
Let’s suppose for small deviations from the basic longitudinal horizontal
movement the aircraft MM looks as follows:
)()(*
0
*
12
*2
tbxtya
dt
dy
a
dt
yd
,
where )(* ty is the deviation of attack angle )(t , and )(tx is the altitude control
step deviation h . Desired aerodynamic coefficients are ,11 ca ,30 a 5,0b .
Reaction )(* ty to a single step stimulus )(tx is shown in Fig. 5.
The test corresponds to real conditions of aircraft identification: observation
time 5T s, discretion step t in time was 0.02 s. Meaning, number M of
discrete values kt was 100. Measurements of attack angle )(ty in 100 discrete-
times kt consist of the exact value )(*
kty (Fig. 5), and adaptive noise in the form
of Gaussian “white noise”. Input stimulus )(tx is measured accurately.
The computer modeling was carried out for various noise )(ty to signal
)(* ty ratios, therefore, due to a limited data sample and the presence of random
noise, estimates baa ˆ,ˆ,ˆ 10 are random (Table). The estimates were obtained using
the series-parallel model (Fig. 2, Step 2) and the three models strategy (Figs. 1–3,
Steps 1–4). The model (3) uses three type (4) operators, while the model (5), as a
filter )( pC , uses operator
123 )765()( ppppWf ,
which significantly smooths noise components )(t .
Table shows estimates baa ˆ,ˆ,ˆ 10 obtained through a one-step algorithm that
uses only the series-parallel model (Step 2), and a multi-step (Steps 1–4) algo-
rithm that uses the proposed strategy (Fig. 4).
1
1.5
x, y*
y*(t)
x(t)
t, c
1 2 3 4 5
Fig. 5. Reaction )(* ty to step impact and )(tx
А. Silvestrov, M. Ostroverhov, L. Spinul, O. Khalimovskyy, H. Veshchykov
ISSN 1681–6048 System Research & Information Technologies, 2025, № 2 122
The estimates, obtained by the one-step and multi-step algorithms
0â 1â b̂ Noise/signal
Step 2 Steps 1–4 Step 2 Steps 1–4 Step 2 Steps 1–4
0 3 3 1 1 0.5 0.5
0.5 3.15 3.1 1.01 1.1 0.517 0.516
1 2.69 3.15 0.71 1.14 0.6 0.52
2 1.56 2.82 0.16 0.96 0.49 0.48
3 1.39 3.22 0.02 0.95 0.53 0.58
Therefore, despite a random component of estimates baa ˆ,ˆ,ˆ 10 , associated
with a limited data sample and a significant noise )( kt to signal )(*
kty ratio, in
the dependences 1â and 0â on the noise-to-signal ratio, we can observe a pattern
that has a statistically significant value, i.e. a significant decrease in estimates 1â
and 0â with an increase in the noise level )( kt (Fig. 6).
To explain the effect of lower estimates iâ and 0â , we will consider a
structural diagram with a series-parallel model (Step 2), where an object is
connected instead of a model (Fig. 7).
The LSM evaluator determines estimates baa ˆ,ˆ,ˆ 10 of parameters baa ,, 10
under the condition of the minimum mean square 2
y of error )( ky t [5; 6], i.e.,
under the following conditions:
0
ˆ
)(
)(
1
1 0
N
k
ky
ky a
t
t
N
, (6)
0
ˆ
)(
)(
1
1 1
N
k
ky
ky a
t
t
N
, (7)
0
ˆ
)(
)(
1
1
N
k
ky
ky
b
t
t
N
, (8)
Total error:
)(ˆˆ)(ˆ)(ˆ
ˆ)(ˆˆ)(
2
2
10 k
kk
kky txb
dt
tyd
dt
tyd
atyat . (9)
0
0
â
a
1
0.5
1 2 3
1
1
â
a
noise-to-signal
Fig. 6. Dependence
1
1ˆ
a
a and
0
0ˆ
a
a on the noise level )(t in signal )(ty
Strategy for ensuring asymptotic convergence of the process of non-linear…
Системні дослідження та інформаційні технології, 2025, № 2 123
Error sensitivity functions for the relevant parameters:
)(ˆ
ˆ
)(
k
ky tx
b
t
,
)(ˆ)(ˆ)(ˆ
ˆ
)( *
0
kkk
ky ttyty
a
t
,
dt
td
dt
tyd
dt
tyd
a
t kkkky )(ˆ)(ˆ)(ˆ
ˆ
)( *
1
.
In condition (7), considering expressions (8) and (9), there is no square of
noise )(t . If (9) is weakly correlated with the inaccuracy of determining coeffi-
cients 0â and 1â in (9), estimate b̂ of parameter b is almost unbiased (see Table,
Step 2).
It is different for estimates 0â and 1â . In equations (6), (7), there is a square
of noise )(t or its derivative. This leads to a bias in estimates 0â and 1â propor-
tional to the noise level )(t . This is the main drawback of the series-parallel
model (Fig. 7), which is eliminated by the proposed strategy (see Table, Steps 2–4).
CONCLUSION
In order to guarantee unbiased physical parameter estimates for mathematical
models of dynamic objects in real conditions of limited data samples and dynamic
ranges of state variables of an object, an effective strategy consists of a step-by-
step use of a parallel model with orthogonal operators, a series-parallel model for
approximating the orthogonal model, and a subsequent use of these models for a
more accurate “input-output” mapping of the object and, consequently, an unbi-
ased estimation of desired dynamic object parameters.
Fig. 7. Evaluation of object parameters using the series-parallel model
0â
1â
p2
p
1
( )c p
1
( )c p
b̂
2
1 0
b
p a p a
1â0âb̂
МНК-
оцінювач
( )x t
*( )y t
( )t
( )y t
ˆ( )x t ˆ( )y t
( )y t
( )y kt( )
ˆ
y kt
b
1
( )
ˆ
y kt
a
0
( )
ˆ
y kt
a
LSM
Evaluator
А. Silvestrov, M. Ostroverhov, L. Spinul, O. Khalimovskyy, H. Veshchykov
ISSN 1681–6048 System Research & Information Technologies, 2025, № 2 124
REFERENCES
1. A.A. Pavlov, V.V.Kalashnik, “Recommendations for choosing the zone of an active
experiment for one-dimensional polynomial regression analysis,” Visnyk NTUU
“KPI”. Informatics, Operation and Computer Science, no. 60, pp. 41–45, 2014.
2. K.Kh. Zelenskyi, V.M. Yhnatenko, and A.P. Kots, Computer methods of applied
mathematics. Kyiv: Ukrainskyi Litopys, 2000, 207 p.
3. A.M. Silvestrov, V.M. Sineglazov, Theory of identification: textbook. Kyiv: NAU,
2015, 477 p. Available: https://ela.kpi.ua/handle/123456789/30302
4. V.M. Sineglazov, M.K. Filiashkin, Automated aircraft control systems: textbook.
Kyiv: NAU, 2003, 502 p. Available: https://er.nau.edu.ua/handle/NAU/ 52668\
5. V.V.Samsonov, A.M. Silvestrov, Essays on the theory of identification. Kyiv:
NUFT, 2021, 166 p. Available: https://ela.kpi.ua/handle/123456789/30522
6. V.V. Samsonov, А.M. Silvestrov, and L.Yu. Spinul, Multiple adaptive system of
identification. Kyiv: NUFT, 2018, 228 p. Available: https://ela.kpi.ua/ han-
dle/123456789/30306
7. I. Levchuk, G. Manko, V. Tryshkin, and V. Korsun, Theory and practice of control
objects identification: monograph. Dnipro: Ukrainian State University of Chemical
Technology , 2019, 203 p.
8. Kamil Aida-zade, Yegana Ashrafova, “Determination of the parameters of dynamic
objects, arbitrarily related by conditions,” Journal of Industrial and Management
Optimization, vol. 20, issue 7, pp. 2477–2499, 2024. doi: 10.3934/jimo.2024010
Received 13.03.2024
INFORMATION ON THE ARTICLE
Anton M. Silvestrov, ORCID: 0000-0002-2511-5029, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: silvestrovan-
ton@gmail.com
Mykola Ya. Ostroverkhov, ORCID: 0000-0002-7322-8052, National Technical Univer-
sity of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail:
n.ostroverkhov@hotmail.com
Liudmyla Yu. Spinul, ORCID: 0000-0002-4234-6072, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: spinul20@gmail.com
Oleksiy M. Khalimovskyy, ORCID: 0000-0003-3672-8530, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: o.khalimovskyy@ukr.net
Heorhii V. Veshchykov, ORCID: 0009-0002-0606-9765, National Technical University of
Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Ukraine, e-mail: veschikov98@gmail.com
СТРАТЕГІЯ ЗАБЕЗПЕЧЕННЯ АСИМПТОТИЧНОЇ ЗБІЖНОСТІ ПРОЦЕСУ
НЕЛІНІЙНОГО ОЦІНЮВАННЯ ПАРАМЕТРІВ ДИНАМІЧНИХ ОБ’ЄКТІВ/
А.М. Сільвестров, М.Я. Островерхов, Л.Ю. Спінул, О.М. Халімовський, Г.В. Вещиков
Анотація. Розглянуто покрокову стратегію послідовного використання і на-
лаштування паралельної до об’єкта моделі ідентичної структури з ортогональ-
ними операторами, послідовно-паралельної моделі до об’єкта з підключенням
операторів певного типу для ортогональної апроксимації з метою отримання
асимптотично незміщених оцінок коефіцієнтів структурно ідентичної до ди-
намічного об’єкта математичної моделі в умовах зашумленості вимірів вихід-
ної змінної об’єкта ідентифікації і невипуклості функціонала близькості вихід-
них змінних об’єкта і моделі в просторі коефіцієнтів математичної моделі
об’єкта. Наведено структурні схеми кожного етапу ідентифікації з викорис-
танням уточнених параметрів і структури моделі об’єкта. Алгоритм реалізова-
но для ідентифікації параметрів математичної моделі літальних апаратів за
умови обмеженості вибірки даних експерименту і значного діапазону відхи-
лення змінних стану від базового режиму.
Ключові слова: нелінійне оцінювання, ідентифікація, збіжність алгоритмів
оцінювання, оптимізація.
|
| id | journaliasakpiua-article-336060 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-09-17T09:26:03Z |
| publishDate | 2025 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/f9/14e2ee8a7cc40c10e37600c2174d2bf9.pdf |
| spelling | journaliasakpiua-article-3360602025-07-25T15:56:08Z Strategy for ensuring asymptotic convergence of the process of non-linear estimation of dynamic object parameters Стратегія забезпечення асимптотичної збіжності процесу нелінійного оцінювання параметрів динамічних об’єктів Silvestrov, Anton Ostroverkhov, Mykola Spinul, Liudmyla Khalimovskyy, Oleksiy Veshchykov, Heorhii нелінійне оцінювання ідентифікація збіжність алгоритмів оцінювання оптимізація non-linear estimation identification convergence of estimation algorithms optimization The article considers a step-by-step strategy of sequential use and adjustment of a parallel model to an object of identical structure with orthogonal operators, a series-parallel model to an object with the connection of operators of a certain type for orthogonal approximation in order to obtain asymptotically unbiased estimates of coefficients of a structurally identical to a dynamic object of a mathematical model under conditions of noise of measurements of the initial variable of the identification object and non-convexity of the proximity functional of the initial variables of the object and the model in a space of coefficients of the object’s mathematical model. Structural diagrams of each stage of identification are given using refined parameters and the structure of the model of object. This algorithm was implemented to identify the parameters of the mathematical model of aircraft, provided that the sample of experiment data is limited and there is of the initial a significant range of deviations of state variables from the basic mode. Розглянуто покрокову стратегію послідовного використання і налаштування паралельної до об’єкта моделі ідентичної структури з ортогональними операторами, послідовно-паралельної моделі до об’єкта з підключенням операторів певного типу для ортогональної апроксимації з метою отримання асимптотично незміщених оцінок коефіцієнтів структурно ідентичної до динамічного об’єкта математичної моделі в умовах зашумленості вимірів вихідної змінної об’єкта ідентифікації і невипуклості функціонала близькості вихідних змінних об’єкта і моделі в просторі коефіцієнтів математичної моделі об’єкта. Наведено структурні схеми кожного етапу ідентифікації з використанням уточнених параметрів і структури моделі об’єкта. Алгоритм реалізовано для ідентифікації параметрів математичної моделі літальних апаратів за умови обмеженості вибірки даних експерименту і значного діапазону відхилення змінних стану від базового режиму. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2025-06-28 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/336060 10.20535/SRIT.2308-8893.2025.2.08 System research and information technologies; No. 2 (2025); 115-124 Системные исследования и информационные технологии; № 2 (2025); 115-124 Системні дослідження та інформаційні технології; № 2 (2025); 115-124 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/336060/324828 |
| spellingShingle | нелінійне оцінювання ідентифікація збіжність алгоритмів оцінювання оптимізація Silvestrov, Anton Ostroverkhov, Mykola Spinul, Liudmyla Khalimovskyy, Oleksiy Veshchykov, Heorhii Стратегія забезпечення асимптотичної збіжності процесу нелінійного оцінювання параметрів динамічних об’єктів |
| title | Стратегія забезпечення асимптотичної збіжності процесу нелінійного оцінювання параметрів динамічних об’єктів |
| title_alt | Strategy for ensuring asymptotic convergence of the process of non-linear estimation of dynamic object parameters |
| title_full | Стратегія забезпечення асимптотичної збіжності процесу нелінійного оцінювання параметрів динамічних об’єктів |
| title_fullStr | Стратегія забезпечення асимптотичної збіжності процесу нелінійного оцінювання параметрів динамічних об’єктів |
| title_full_unstemmed | Стратегія забезпечення асимптотичної збіжності процесу нелінійного оцінювання параметрів динамічних об’єктів |
| title_short | Стратегія забезпечення асимптотичної збіжності процесу нелінійного оцінювання параметрів динамічних об’єктів |
| title_sort | стратегія забезпечення асимптотичної збіжності процесу нелінійного оцінювання параметрів динамічних об’єктів |
| topic | нелінійне оцінювання ідентифікація збіжність алгоритмів оцінювання оптимізація |
| topic_facet | нелінійне оцінювання ідентифікація збіжність алгоритмів оцінювання оптимізація non-linear estimation identification convergence of estimation algorithms optimization |
| url | https://journal.iasa.kpi.ua/article/view/336060 |
| work_keys_str_mv | AT silvestrovanton strategyforensuringasymptoticconvergenceoftheprocessofnonlinearestimationofdynamicobjectparameters AT ostroverkhovmykola strategyforensuringasymptoticconvergenceoftheprocessofnonlinearestimationofdynamicobjectparameters AT spinulliudmyla strategyforensuringasymptoticconvergenceoftheprocessofnonlinearestimationofdynamicobjectparameters AT khalimovskyyoleksiy strategyforensuringasymptoticconvergenceoftheprocessofnonlinearestimationofdynamicobjectparameters AT veshchykovheorhii strategyforensuringasymptoticconvergenceoftheprocessofnonlinearestimationofdynamicobjectparameters AT silvestrovanton strategíâzabezpečennâasimptotičnoízbížnostíprocesunelíníjnogoocínûvannâparametrívdinamíčnihobêktív AT ostroverkhovmykola strategíâzabezpečennâasimptotičnoízbížnostíprocesunelíníjnogoocínûvannâparametrívdinamíčnihobêktív AT spinulliudmyla strategíâzabezpečennâasimptotičnoízbížnostíprocesunelíníjnogoocínûvannâparametrívdinamíčnihobêktív AT khalimovskyyoleksiy strategíâzabezpečennâasimptotičnoízbížnostíprocesunelíníjnogoocínûvannâparametrívdinamíčnihobêktív AT veshchykovheorhii strategíâzabezpečennâasimptotičnoízbížnostíprocesunelíníjnogoocínûvannâparametrívdinamíčnihobêktív |