Paradigm of nonstochastic approach tosystem identification
The paper discusses the construction of linear models, the complexity of which is determined by dimensionality. Within the framework of the non-stochastic approach, a methodological and mathematical foundation for reconstructing models that describe processes in complex systems is developed. Asympto...
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Інститут кібернетики ім. В.М. Глушкова НАН України
2023
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| description | The paper discusses the construction of linear models, the complexity of which is determined by dimensionality. Within the framework of the non-stochastic approach, a methodological and mathematical foundation for reconstructing models that describe processes in complex systems is developed. Asymptotic modeling allows for the formation of classes of models suitable for solving identification problems. An exact description corresponds to an infinite extension, and thus the quality of the model improves with its increasing dimensionality. However, errors in the available data prevent the indefinite increase in dimensionality due to the ill-posedness of the identification problem starting from a certain dimension. The regularization procedure enables the determination of an effective approximate solution to the identification problem, which, in the non-stochastic case, is consistent with the data errors. The properties and features of the proposed approach are illustrated by modeling results.
У статті розглядається побудова лінійних моделей, складність яких визначається розмірністю. У рамках нестохастичного підходу розроблено методологічну та математичну основу реконструкції моделей, що описують процеси у складних системах. Асимптотичне моделювання дозволяє для такої системи формувати класи моделей, які підходять для розв’язання задачі ідентифікації. Точний опис відповідає нескінченному розширенню, тому якість моделі покращується зі збільшенням її розмірності. Однак помилки в наявних даних не дозволяють безмежно збільшувати їхню розмірність через погану обумовленість задачі ідентифікації, починаючи з деякого виміру. Процедура регуляризації дозволяє визначити ефективне наближене рішення задачі ідентифікації, яке для нестохастичного випадку узгоджується з помилками даних. Властивості та особливості пропонованого підходу ілюструються результатами моделювання.
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© V. GUBAREV, S. MELNYCHUK, N. SALNIKOV, 2023
42 ISSN 2786-6491
МЕТОДИ КЕРУВАННЯ ТА ОЦІНЮВАННЯ
В УМОВАХ НЕВИЗНАЧЕНОСТІ
UDС 681.5.015; 519.233
V. Gubarev, S. Melnychuk, N. Salnikov
PARADIGM OF NONSTOCHASTIC
APPROACH TO SYSTEM IDENTIFICATION
Vyacheslav Gubarev
Space Research Institute of NASU and SSAU, Kyiv,
orcid: 0000-0001-6284-1866
v.f.gubarev@gmail.com
Serhii Melnychuk
Space Research Institute of NASU and SSAU, Kyiv,
orcid: 0000-0002-6027-7613
serg.vik@ukr.net
Nikolay Salnikov
Space Research Institute of NASU and SSAU, Kyiv,
orcid: 0000-0001-9810-0963
salnikov.nikolai@gmail.com
The concept of a complex system in this work is understood as a large set
of dynamic interconnected systems, the exact mathematical model of which
is not known or has a very large dimension. In such situation the use of
standard methods for synthesizing feedback becomes difficult or even im-
possible due to the degeneracy of the corresponding mathematical prob-
lems. One way out of this situation is to build an approximation model of
reduced dimension. This can be done using a system of initial equations , if
they are available, or using identification methods based on measurements
of output and input variables acting on the system. In this case, the process of
constructing a mathematical model is reduced to a sequential enumeration
of possible models of increasing complexity. As a criterion for the adequacy of
the model, the norm of deviation of the output of the adjusted model from
the measured value of the output of the system under study is considered.
The article deals with the construction of linear models, the complexity of
which is determined by their dimension. In the framework of nonstochastic
approach it is developed the methodological and mathematical basis for
model reconstruction which describes processes in complex systems. As-
ymptotic modelling allows for such system to form model classes appropri-
ate to solve identification problem. Precise description corresponds to infi-
nite expansion so the model quality is improved when its dimension is in-
S. Melnychuk is grateful to the National Research Fund of Ukraine for support (project 2020.02/0015
«Theoretical and experimental research of global disturbances of natural and man-made origin in the
Earth-Atmosphere-Ionosphere system»).
mailto:v.f.gubarev@gmail.com
mailto:serg.vik@ukr.net
mailto:salnikov.nikolai@gmail.com
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 1 43
creased. However errors in available data do not allow increase their dimen-
sion limitlessly due to ill-conditionality of the identification problem begin-
ning from some dimension. Regularization procedure permits to determine
the effective approximate solution of identification problem which for non-
stochastic case is in agreement with errors in data. Properties and peculiari-
ties of the proposed approach are illustrated by simulation results.
Keywords: system identification, linear regression, regularization, asymptotic
modelling, approximate solution
Introduction
System identification, that is, the construction of a mathematical model of a system
based on data obtained as a result of experiments, is primarily focused on complex dy-
namic processes for which it is impossible to establish the laws of their behaviour in
other ways. This means that even in a general form it is impossible to indicate a class of
mathematical models containing an exact description of the system under study.
Here we consider a new concept or paradigm of system identification within a non-
stochastic approach for linear time invariant (LTI) systems. We shell assume the exist-
ence of abstract transfer function matrix ( )G z including nonrational cases, which con-
nects the input and the measured output. According to the concept of asymptotic model-
ling, widely used in computational mathematics, an unknown transfer function can be
written as finite or infinite expansions for some basis functions, which makes it possible
to write a class of models in a form convenient for identification.
At such approach, we are talking about the construction of finite-dimensional approxi-
mating models. At the same time, with an increase in the dimension, the accuracy of the de-
scription of the system increases if certain requirements for the system under study are met.
For example, truncated rational approximation of infinite-dimensional LTI system with fi-
nite-dimensional input and output that induced nuclear type Hankel operator with distinct
singular values convergences to precise model as it dimension tends to infinity [1].
Asymptotically stable system can be written as an infinity expansion
0
( ) ( )k k
k
G z G f z
(1)
where 0,1,2,{ ( )}k kf z is a sequence of orthonormal functions and 0,1,2,{ }k kG is a
sequence of parameters. Then task of identification is to find a finite-dimensional model
1
0
( ) ( ).
n
k k
k
G z G f z
(2)
The accuracy of the model depend on the choice of basis functions and dimen-
sion .n The case of k
kf z corresponds to Infinite Impulse Response (IIR) mode-
ling [2]. In some cases the use of Laguerre or Kautz polynomials gives a better approx-
imation [3, 4]. In general, the advantage in identification and approximation can be
achieved by using such orthonormal basis functions that correspond to dynamics close
to the dynamics of the object under study [5].
A very popular form for rational approximation is the Linear Regression which di-
rectly links input and output variables [2]. Here, for ease of paradigm presentation, we
consider an ARX model with a scalar input ( )u t and output ( )y t
1 1( ) ( 1) ( ) ( 1) ( )
a bn a n by t a y t a y t n b u t b u t n (3)
in discrete time 0, 1,t . Let us denote vectors
T
1[ , , ] ,
aa na a
T
1[ , , ] ,
bb nb b
T( ) [ ( 1), , ( )]a at y t y t n and T( ) [ ( 1), , ( )] ,b bt u t u t n so (3) can be
written as follows
T T( ) ( ) ( ) .a a b by t t t (4)
44 ISSN 2786-6491
The special case 0an gives us Finite Impulse Response (FIR) approximate mo-
del. Any ARX model has an equivalent state-space model representation, so it is easy to pass
from one description to another using linear transformation. It is important because the trun-
cated state-space models are approximating for nuclear type nonrational system [1].
1. Paradigm of stochastic identification
Let us briefly consider identification paradigm in stochastic case according to [6].
Mathematical difficulties in system identification are associated with the presence of
uncertainty in the data. The generally accepted is the stochastic interpretation of uncer-
tainty, which assumes that measurement errors are independent and identically distrib-
uted (i.i.d.) random variables. This paradigm underlies all classical statistical methods in
system identification. Model parameters of a given order are estimated by maximum
likelihood methods (prediction errors), which in most cases are formulated as extremal
problems
2
1
ˆ ˆarg min ( ) ( / ) ,
N
D t
y t y t
(5)
where ( )y t is a measured output, ˆ( / )y t is a prediction by the model, D is a set of
values . There are many publications devoted to the identification problem in such
statement, see, for example [2, 7, 8].
Under concept of asymptotic modeling the key question in identification is a model
order determination. It is known that a higher order model can better approximate the
measured output of the system, i.e. reduce the error called «bias». On the other hand, a
higher-order model is more sensitive to errors in the data, which corresponds to a larger
variance in model parameter estimates. The mismatch between system and model in-
cludes both of these components. Traditional system identification often uses the bias-
variance trade-off to minimize the total mean square error (MSE). Various procedures
have been proposed for this, among which are the Akaikeʼs Information Criteria (AIC)
and Bayesian Information Criteria (BIC) order criteria described in the books men-
tioned above.
2. Nonstochastic approach to EIV-identification
In many cases on practice, errors occur at output and input variables measurement.
Such situation in system identification is called «errors in variables» (EIV) and was
considered by many authors. Results obtained in the framework of stochastic paradigm
are described in [9–13]. Recently EIV identification in frequency domain was consid-
ered in [14]. New results also presented in [15–17].
This article develops an alternative approach based on the non-stochastic para-
digm. We assume that data errors are random variables belonging to known bounded
sets with arbitrary distributions, including worst cases. With this formulation, the max-
imum likelihood method is not appropriate. Early the similar approach was implement-
ed in so-called set membership identification (SMI) in which the main goal was to de-
termine a guaranteed set of models, including the exact one. The description of these
methods is given in [18–21].
The approach developing in this article is an alternative to SMI and aims to find a
single approximate solution of identification problem consistent with errors in availa-
ble data.
We have the following measurements of true values ( )u t and ( )y t
( ) ( ) ( ),uu t u t t ( ) ( ) ( ),yy t y t t (6)
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 1 45
where ( )u t and ( )y t are errors. According to the mentioned assumption ( )u t and
( )y t are unknown arbitrary random sequences satisfying conditions
( ) ,u ut ( ) .y yt (7)
An important issue in identification is the informative input that excites all modes
of the system. In stochastic methods, a persistent excitation signal of a given order is
usually used for this. In the case of non-stochastic identification, we propose an active
experiment in which each mode of the system at a certain moment of observation makes
the maximum possible contribution to the output. For an asymptotically stable system,
this can be achieved by two ways. The first is to collect data from separate experiments
consisting of the intervals of excitation and following relaxation. An alternative is a sin-
gle continuous experiment where excitation intervals alternate with relaxations. In both
cases, the duration of relaxation should exceed the transient time.
Consider an observable and controllable discrete LTI system. Such a system can be
represented by linear regression or its state-space equivalent
( 1) ( ) ( ),x t Ax t bu t T( ) ( ) ( )y t c x t du t (8)
where t is a discrete time and «T» denotes a transposition. Assume that this system
does not have multiple eigenvalues. Then we can choose the Jordan block realization:
diag ( ),pA A col( ),pb b col( ),pc c where real eigenvalues p p corre-
spond to blocks
,p pA ,c
p pb b ,c
p pc c (9)
and complex eigenvalues p p pi correspond to blocks
p p
p
p p
A
,
c
p
p
s
p
b
b
b
,
c
p
p
s
p
c
c
c
. (10)
In case of finite-dimensional system with ( ) 0y t at 0,t ( ) 0u t at 0t and
( ) 0u t at 0t input-output ration (8)–(10) is equivalent to
1
0
0 1
( ) ( ) ( ),
t P
p
j p
y t h t j u j
(11)
where 0 ( ) [ cos sin ],k c s
p p p p p ph k f k f k p p , arg ,p p c
pf
,c c s s
p p p pc b c b .s c s s c
p p p p pf c b c b Parameters ,p ,p ,c
pf
s
pf completely deter-
mine the system dynamics. In case of real eigenvalues we have ,p p 0,p
0.s s
p pc b
In EIV identification the informativeness of the input is defined by the signal-to-
noise ratio (SNR). The larger SNR of the output for each mode (10), the more informa-
tive the input is. Therefore, we will shape the input so that at the end of the excitation
interval, a certain mode has the largest output signal. Different ( )u t across experiments
should effectively excite each of the modes. Two types of input signal will be used for
this. The first is a rectangular pulse
46 ISSN 2786-6491
1, 1,
( )
0, else
it
u t
(12)
where i is of a varied duration across experiments: ,i i max1, 2, , .i i Short rec-
tangular pulses provide stronger excitation of fast modes. Increasing the duration will
add a contribution from slow modes. Therefore a set of rectangular pulses of different
duration should provide separate excitation of individual modes corresponding to real
eigenvalues (9).
The second type of input is a single harmonic
sin , 1,
( )
0, else
jt t
u t
(13)
where frequency j change across experiments ,j j max1, 2, ,j j and cover
the range (0, π / 2). For modes corresponding to complex eigenvalues (10), we aim to
catch resonant excitation when j happens to be close to the natural frequency .p
When j are varied with a small step, it is possible to provide resonant excitation of all
oscillating modes of the system. In this case the duration is chosen to be sufficient to
establish steady-state forced oscillations. With an appropriate choice of parameters
max ,i max ,j and it is possible to collect informative data with acceptable SNR
with a fairly general knowledge of the system under study. More detailed information
on this can be found in [22].
3. Method of identification
Here will be developed the identification method in the framework of nonstohastic
approach. The goal is to obtaine a regularized solution that gives an approximation with
an accuracy corresponding to the noise in data.
As mentioned, the asymptotic class of linear autoregressive models is considered to
identify complex systems that in this class have a large or infinite dimension. Conception of
approximate regularized solution means that the model dimension may be less then the order
of the true system. Therefore, identification must include the choice of model order.
3.1. Model order selection. The type of control signals proposed above makes it
possible to divide the identification process into independent tasks. Measurements on
relaxation intervals are used to determine the vector a and its dimension. For this we
will use the equation
1 2( ) ( 1) ( 2) ( ),ny t a y t a y t a y t n (14)
which describe free motion on relaxation intervals. Applying controls (12) and (13), we
collect the corresponding outputs into the following matrix:
pulse
relax
harm
relax
pulse pulse pulse
1relax
harm harm harm
relax 2
,
col ([ ( 1) ( ), 1, ,
col ([ ( 1) ( )]), 1, ,
i ii i
j j
Y
Y
Y
Y y y l i k
Y y y l j k
(15)
where
pulse
iy is a measured response on the i -th rectangular pulse (12), 1k is a total
number of rectangular pulses,
harm
jy is a measured response on the j-th harmonic in-
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 1 47
put (13), 2k is a total number of input harmonics and l is the relaxation duration, the same
for all experiments. We denote the total number of rows of Y by 1 2.k k k Value l
should be more then transient time and both k and l must exceed supposed model order.
Let us produce singular value decomposition (SVD) of the matrix Y
T,Y U V (16)
where ,U , V are matrices of dimensions ,k k k l and l l respectively.
In the deterministic case for a finite dimensional system the rank of Y is equal to
true dimension n and only the first n singular numbers 1, , n are non-zero. Then
the dimension of the system can be precisely determined by the number of non-zero di-
agonal elements of .
In the non-deterministic case the matrix Y almost always will be of full rank and
all singular numbers will be positive. Even then the true dimension n can sometimes be
determined by singular values, if there is a gap between the values n and 1.n How-
ever there is often no clear gap. Result depends on the SNR of individual modes. Modes
of the system with a low SNR becomes indistinguishable from background noise, mak-
ing impossible in the non-stochastic case to establish the true dimension. In such cases,
we can find a model of reduced dimension applying the known in regularization princi-
ple that approximate solution should be consistent in accuracy with the data errors.
For any chosen value n̂ we can divide the measured output into «signal» and
«noise» part using a low-rank approximation in Frobenius norm. For this we split (16)
T T0
[ ] [ ] ,
0
S
S N S N
N
U V U U V V
(17)
where S contains n̂ largest singular values. Indices «S» and «N» denote «signal» and
«noise» part. As a result we get decomposition
ˆ ˆ
,S N
n n
Y Y Y (18)
where T
ˆ
( )S S S S
n
Y U V and T
ˆ
( ) .N N N N
n
Y U V Expansion (18) allows one to find
the matrix of the given rank closest to full-rank matrix Y in Frobenius norm using -rank
property [23]. So we have
ˆ
ˆrank( )S
n
Y n and
ˆ 1 ˆ
,N
n n F
Y (19)
where subscript «F» means Frobenius norm, defined as 2
, iji jF
A a for any ma-
trix ( ).ijA a Matrix
ˆ
S
n
Y will be considered as the output of the desired model.
Again we represent the matrix Y in the form
,Y Y Y (20)
where Y corresponds to noise-free output generated by true system, and Y corre-
sponds to the noise, which, according to condition (7), is bounded in the infinity norm
.yY
(21)
Since all available information about Y is limited only by condition (21), then, ac-
cording to the guaranteed approach, any Y that satisfies the inequality Y Y
48 ISSN 2786-6491
can be considered as corresponding to a noise-free output of the exact model. Among all
exact models, we are looking for a model of minimal order, which corresponds to the min-
imal rank of the matrix .Y Using decomposition (18) we determine the model by its out-
put as
ˆ
,S
n
Y increasing the dimension n̂ until condition
ˆ
S
yn
Y Y
is satisfied. From
here comes the criterion
ˆ ˆ 1
.N N
yn n
Y Y
(22)
Using equivalence of norms ,
F
A A A k l
which is valid for an arbi-
trary matrix ,k lA R from (22) and (19) we derive inequality
ˆ 1
ˆ ,n
y n
kl
or
ˆ
ˆ 1
,
.
n y
n y kl
(23)
The condition (23) can be used to determine the dimension of the model that is
consistent in accuracy with the errors.
3.2. Parametric identification. After determining the order ˆan n of vector ,a it
can be found from the basic equations (3), (4), where ( ) 0.b t u For this we construct
an overdetermined linear system
,A a AW w (24)
where the matrix AW and vector Aw are formed from the matrix
a
S
n
Y as follows:
1
2
( )a
A
A
A
l n
A
W
W
W
W
,
(1)
(2)
( )a
A
A
A
l n
A
w
w
w
w
.
Here,
(1)
AW contains the first an columns of the matrix ,
a
S
n
Y the matrix
(2)
AW also
contains an columns, starting from the second, and so on. The vector
(1)
Aw is the
( 1)an –th column of ,
a
S
n
Y
(2)
Aw is the ( 2)an –th column, etc. The last one vector
( )al n
Aw
is the first column of the matrix .
a
N
n
Y
System (24) is a strongly overdetermined for large r and .l Then we can discard non-
informative equations with small SNR, i.e. having max / yy less than some threshold,
where maxy is the maximum modulus of regressors. As a result, we obtain a truncated but
still overdetermined system
.A a AW w (25)
The solution of (25) can be found by ordinary least squares (OLS) or by total least
squares (TLS) [23]. However, in practice, very often problem (25) turns out to be ill-
posed, having an ill-conditioned matrix .AW Therefore, it is proposed to implement the
LS based on the SVD decomposition of the matrix
anY
T
1 1 1 .AW U V (26)
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 1 49
Substituting (26) into (25), we obtain
1 1 ,Aw (27)
where T
1 1 ,aV T
1 .A Aw U w The ratio of the first singular value of the matrix 1
to the last one determines the conditionality of the problem. Under good conditionality,
the least squares solution of (26) is
1 T
1 1 1 .a AV U w (28)
When the matrix AW is ill-conditioned, one should find a regularized solution, the
construction of which is given below.
At the last stage of identification, the dimension bn and coefficients of the
vector b are determined. It is preceded by extracting data corresponding to a purely
forced part of model. With an estimated vector ,a such a signal ( )y t for the moment t
is determined by the relation
1 2( ) ( ) ( 1) ( 2) ( ).n ay t y t a y t a y t a y t n (29)
Then the basic equation for finding the vector b is
1( ) ( 1) ( ),m by t b u t b u t n (30)
where dimension bn of the vector b in the main not exceed dimension of vector .a In
principle, case b an n is possible if there is a static connection between input and output,
caused by very fast modes having a transient process shorter than the sampling step.
After obtaining ( )y t from (29), we select samples corresponding to excitation in-
tervals defined by (12) and (13) and form an overdetermined system
,B b BW w (31)
similar to the construction of AW and using .b an n Elements of the matrix BW are
measured control signals ( ).u t The chosen method of excitation (12), (13) provides
good conditionality of the matrix ,BW so system (31) can be solved by TLS.
This completes the solution of the identification problem. As a result we obtain an
approximate mathematical model, consistent in accuracy with the errors in the data de-
termined by conditions (6), (7).
4. Ill-conditioning
The solvability of the identification problem in the formulation under consideration is
determined by the properties of the matrices AW in (24) and BW in (31), and, first of all, by
their condition number. It was proposed to solve (24) and (31) using instead of OLS, a com-
pletely equivalent method based on the SVD decomposition of these matrices. Then the con-
dition number of these matrices with respect to the norm
2
can be found from the relation
1 ,
n
(32)
where ,a bn n n correspond to the dimensions of the allocated blocks when choosing
the order of the model. Due to the properties of the SVD decomposition, as the
50 ISSN 2786-6491
number n increases, the number can only grow. The growth nature of ( )n of
matrices AW and BW is usually different, and each of them depends on its own factors.
Since the matrix BW is formed from input signals, it is possible to control and influence
behavior of ( )bn in active experiments. The method of excitation with alternating in-
tervals of excitation and relaxation makes it possible to provide a rather weak change of
( )bn with increasing .bn Moreover, there is every reason to believe, that with values
of ( )bn remaining close to unity, we will have the most informative signal.
The behavior of condition number of the matrix AW depends on the following fac-
tors. What remains is its dependence on the informative input action, i.e. its ability to
excite all modes of the system with an acceptable SNR. The SNR indicator, which is a
relative value, naturally includes all errors acting on the system. This equally applies to
both matrices AW and .BW The value of ( )an significantly depends on the dynamic
features of the system, determined by the values of the invariants p and ,p as well
as
c
pf and .s
pf There is a rapid increase in the condition number with increase of n
when the eigenvalues of the system are clustered or when there are very fast modes in
the system. Small values of
c
pf and
s
pf lead to small output signals of the correspond-
ing modes, so that for them the SNR becomes of the order of unity. Nevertheless, even
for systems with the most favorable values of these invariants and with an informative
input, the condition number of matrix AW grows rather quickly as an increases, which
is clearly seen from the Table.
Table
an 10 12 14
( )an 310 510 710
Table shows only the order of the condition number and it is calculated here only
with exact data. With more unfavourable dynamic properties of the system, ( )an
grows only faster, which makes the problem of parametric identification with increasing
an more sensitive to errors in the data, i.e. we have a manifestation of bad conditionali-
ty of .AW This property is fundamental and does not essentially depend on the dynamic
properties of systems and methods that solve the problem of parametric identification.
Therefore, at larger dimensions, when the problem becomes ill-posed, it is necessary to
use regularization procedures, which can be used to ensure the stability of the obtained
solutions and improve the quality of the models. Poor conditionality is also inherent to
stochastic identification problems, which led to a shift in a paradigm for system identi-
fication [6].
The main attention in stochastic case was paid to the choice of various stabilizers
that make it possible to ensure the consistency of the estimation. The most widely used
is the Kernel structure for stabilizator, which contains parameters that are subject to tun-
ing in order to provide optimal regularization. Various types of Kernel structure can be
found in works [24–32]. Here in the framework of nonstohastic approach will be con-
sidered regularization more familiar to classical one [33].
5. Regularization
Since an ill-posed problem arises when solving (24), it is proposed to find a solu-
tion from (27) using the stabilizer stab in the following form:
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 1 51
1
2
stab
1
/ 0 0 0
0 / 0 0
0 0 / 0
0 0 0 1
n
n
n n
,
where 1, 2 , …,
an are singular values of 1. Instead of (27) we get the system
1
1 stab 1( ) ,y (33)
where — is a regularization parameter, which is chosen from the residual principle
by [33]. Parameter changes in a geometric progression
0{ } { }, 0 1, 0,1, 2, ,k
k k
where 0 is chosen to be of order 10-2. With 0k we have an regularized solu-
tion (0)
a
(0) 1 T
1 1 0 stab 1( ) .a V U y (34)
Without regularization, the vector
(0)
b
is found and the impulse response function
of the model is calculated. It is compared with the impulse response function of the sys-
tem, which can be taken as the free motion on the relaxation interval after excitation by
a long rectangular pulse. Let 0ˆ ( )h t be the impulse response function of the model, and
( )h t — impulse response function of the systems. The accuracy of the solution is de-
fined as
0 0ˆ ˆ( ) ( ) max ( ) ( ) .
t
h t h t h t h t
Next, we check the fulfillment of the condition
0ˆ( ) ( ) 2 .yh t h t
(35)
When the condition (35) is met, we obtain a regularized solution that is consistent
in accuracy with the data error. Otherwise, we take 2 and repeat the above steps.
The decrease of continues until (35) is fulfilled.
If, starting from some ,k the monotonic character of the residual in (28) is violat-
ed, we increase an and bn by one and continue the search for a regularized solution.
The increase in dimensions is interrupted if there is no improvement in the quality of the
model. In this case, the model of the smallest dimension is taken, among those that give
a suitable approximation.
6. Simulation results
This section presents experimental studies of the fundamental features of identifi-
cation for systems with different dynamic properties and structures using numerical
simulation. They are mainly determined by the system dimension and invariants, includ-
ing eigenvalues and identifiability parameters characterizing the contribution of each
individual mode to the output.
52 ISSN 2786-6491
To do this, we use the model representation (10), (11), in which the specified in-
variants are represented by parameters ,p p and , .c s
p pf f The complexity of the sys-
tem is characterized by dimension .n Errors in measured output and input were as-
sumed to be uniformly distributed over the membership intervals (7).
For a given model dimension, the main characteristics that determine the quality of
parameter estimation are the conditionality of the matrix AW in the overdetermined sys-
tem (25) and the matrix BW in a similar system (31). The proposed excitation of the
system by a large number of signals of types (12), (13) provides good conditionality of
the matrices BW in a wide range of values for any system under study.
The conditionality of the matrix AW essentially depends on the dynamic properties
of the system, determined by its dimension and the values of the above invariants. As
the dimension of the system and the approximating model increase, the conditionality
grows rapidly. This is clearly seen from Fig. 1 and Fig. 2, where dependence of condi-
tionality of the matrix AW on model order is shown for the systems of order 6 (Fig. 1) and
of order 15 (Fig. 2). The dashed line shows the case with an uncertainty level of 610 .
On Fig. 1, the data generating system has dimension 6,n with two real and two
complex conjugate eigenvalues. The conditionality of the system with 15,n which
has three real and six complex conjugate eigenvalues, is shown in Fig. 2.
Fig. 1
Fig. 2
6 2 10 14 18
100
104
108
1012
1016
6
2
10
14
18
100
10
4
108
1012
1016
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 1 53
The conditionality of the matrix AW for approximate models of different dimen-
sions is shown to the left of .an n The solid line shows the deterministic case, when
there is no uncertainty in the data. The dashed line shows the case with an uncertainty
level of 610 .
As can be seen from the figures, starting from a certain dimension, the condition
number reached a constant level. In the deterministic case, the growth of conditionality
stops when 1,an n and is set at the level of the computational error 16(10 ). In the
non-deterministic case, the increase in conditionality may stop earlier, as soon as it
reaches a level corresponding to the uncertainty in the data 16(10 ). Then the dimen-
sion is determined to be smaller than the true one, since some of the system modes have
a signal with too low SNR.
The results presented in Fig. 1, 2 are obtained for a well-identified system, when
p is uniformly distributed over the interval 0,
2
, all p are close to unity, and all
values of
s
pf and
c
pf are the same. If the eigenvalues are clustered or there are modes
with relatively small coefficients s
pf and ,c
pf then the growth of the conditionality AW
with increasing dimension accelerates even more.
Fig. 3
Fig. 3 shows the conditionality of AW for a system with 15,n whose eigenval-
ues form clusters near the stability boundary. It can be seen that for a well-identified
system, the condition number for model of order 10an has a value of about 410 , and
for a system with clustering of eigenvalues, it has a value of about 610 .
The property of asymptotic convergence for the described identification method
was investigated. The same system that was used to build Fig. 2 was chosen. The re-
sponses of the system and asymptotic models for different an to a rectangular pulse are
shown in Fig. 4–6. On Fig. 4 the model has a dimension of 10,an in Fig. 5 —
13,an in Fig. 6 — 15,an i.e. coincided with the dimension of the original system.
The plots to the left at the figures corresponds to the deterministic case and the right
plots correspond to the non-deterministic case with input and output uncertainty of
value 610 . True system response is shown as solid line, and model response is
shown as dashed line.
6
2
10
14
18
100
104
108
1012
1016
54 ISSN 2786-6491
Fig. 4
Fig. 5
Fig. 6
Since the conditioning is bad in all cases, the regularization procedure (34), (35)
was used. When approaching the exact dimension, the quality of the model improved.
When an exceed ,n in the deterministic case, the coincidence of impulse response
functions was obtained, which indicates the construction of a non-minimal model. For
such models, the rank of the product of the controllability and observability matrices
remained equal to .n In the non-deterministic case, the rank of the identifiability matrix
increased with a probability of almost one, but the quality of the model did not improve
significantly.
At a higher level of uncertainty, this effect appears for asymptotic models starting
from a certain dimension an less than ,n so further increase in dimension did not sig-
nificantly improve the quality of the model. This value can be considered as the most
appropriate dimension of the asymptotic model. It corresponds as a rule, to the stop of
the growth of the conditionality of the matrix ,
anY according to Fig. 3. Thus, there is
reason to believe that the dimension ,an at which the condition number reaches satura-
tion, is the most appropriate for the approximating model.
The dependence of the singular values of the matrices Y (15) and AW (25) on dy-
namic invariants and the magnitude of the uncertainty was studied. It has been found
that the singular values can be used to judge the dynamic properties of the system, the
number of essentially excited modes, and the level of noise.
20
0
40
60
80
14
100
10
– 2
2
6
20
0
40
60
80
14
100
10
– 2
2
6
20
0
40
60
80
14
100
10
– 2
2
6
20
0
40
60
80
14
100
10
– 2
2
6
20
0
40
60
80
14
100
10
– 2
2
6
20
0
40
60
80
14
100
10
– 2
2
6
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 1 55
For example, Fig. 7 shows the singular values of the matrix Y (15) with different
noise levels. Generating system has dimension 15 with 6 complex conjugate pairs and 3
real eigenvalues. The values of the invariants c
pf and s
pf for different modes differ up
to 100 times. Solid line shows singular values in the deterministic case, dashed and
dashed-dotted lines correspond to the noise 610u y
and 310 ,u y
re-
spectively.
Fig. 7
Dealing specifically with this system in the deterministic case, one can hope to
build a full-size model. In the case
610 ,u y
the successful construction of a full-
size model is unlikely. In the case 310 ,u y
a model with 10an is non-
minimal, because all further modes will have an SNR below one.
Conclusion
The non-stochastic approach to the identification of complex systems in the
framework of asymptotic modeling is more realistic for practical use, since it uses less
stringent restrictions on the uncertainty in the data, including finite-time observation. To
ensure the effectiveness of this method, two main requirements must be met.
The first, it is necessary that the input signal be universally informative, i.e. for a
system with arbitrary dynamic properties, the output of each mode of the system had an
acceptable SNR. Otherwise, low SNR modes are poorly identified. The proposed design
of active experiment with alternating intervals of excitation and relaxation partially
solves this problem, due to the use of resonance effects. If the frequencies of the input
signals vary with a small step, then the informativeness of the input becomes more uni-
versal. In principle, the most versatile informative input signals are white noise signals.
However, their implementation in practice can cause certain problems. The proposed
method of excitation allows you to control its spectrum, which allows you to use, for
example, a priori knowledge about the dynamic properties of the system.
The second important aspect of non-stochastic identification is related to the choice
of model dimension, which is the best in each particular case. To do this, within the
framework of the described method, it is proposed to use the principle of choosing a
model that is consistent in accuracy with the errors in the available data. In the first ap-
proximation, it is used in condition (20). Further, for the dimension established by (23),
the problem of parametric identification is solved, after which the consistency of the
6
2
10
14
18
10–15
10–9
10–3
100
56 ISSN 2786-6491
model and the system is checked by the difference in responses. If the principle is ful-
filled, then the taken dimension of the model is final. If the model error is greater, we
increase its dimension and perform parametric identification again. The principle of
consistency is checked and a decision is made to stop or to continue increasing the di-
mension of the model. In this case, if the problem is ill-posed, regularization should be
used. However, the case is not ruled out when the quality of the model reaches satura-
tion and does not improve. Then, as the desired dimension, we take the smallest one
corresponding to saturation.
In conclusion, we note that the process of identifying complex systems cannot be
reduced to a formal solution of a strictly formulated mathematical problem. It is rather a
research process of establishing patterns of system behavior based on the data of one or
more experiments with the analysis of their results and the use of mathematical tools,
the basics of which are described in this paper.
В.Ф. Губарев, С.В. Мельничук, М.М. Сальніков
ПАРАДИГМА НЕСТОХАСТИЧНОГО ПІДХОДУ
ДО ІДЕНТИФІКАЦІЇ СИСТЕМ
Губарев Вʼячеслав Федорович
Інститут космічних досліджень НАН України та ДКА України, м. Київ,
v.f.gubarev@gmail.com
Мельничук Сергій Вікторович
Інститут космічних досліджень НАН України та ДКА України, м. Київ,
serg.vik@ukr.net
Сальніков Микола Миколайович
Інститут космічних досліджень НАН України та ДКА України, м. Київ,
salnikov.nikolai@gmail.com
Під поняттям складна система у цій роботі розуміється велика сукуп-
ність динамічних взаємодіючих систем, точна математична модель
яких невідома чи має дуже велику розмірність. Застосування стандарт-
них методів синтезу зворотних звʼязків для таких систем стає складним
і навіть неможливим із-за виродженості відповідних математичних за-
дач. Один із виходів із такої ситуації полягає у побудові апроксимацій-
ної моделі зниженої розмірності. Це може бути зроблено з використан-
ням системи вихідних рівнянь, якщо вони є, або методів ідентифікації
на основі вимірювань вихідних і вхідних змінних, що діють на систему.
У цьому випадку процес побудови математичної моделі зводиться до
послідовного перебору можливих моделей зі зростаючою складністю.
Як критерій адекватності моделі розглядається норма відхилення вихо-
ду моделі від виміряного значення виходу досліджуваної системи.
У статті розглядається побудова лінійних моделей, складність яких ви-
значається розмірністю. У рамках нестохастичного підходу розроблено
методологічну та математичну основу реконструкції моделей, що опи-
сують процеси у складних системах. Асимптотичне моделювання доз-
воляє для такої системи формувати класи моделей, які підходять для
розв’язання задачі ідентифікації. Точний опис відповідає нескінченно-
му розширенню, тому якість моделі покращується зі збільшенням її роз-
мірності. Однак помилки в наявних даних не дозволяють безмежно
mailto:v.f.gubarev@gmail.com
mailto:serg.vik@ukr.net
mailto:salnikov.nikolai@gmail.com
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2023, № 1 57
збільшувати їхню розмірність через погану обумовленість задачі іден-
тифікації, починаючи з деякого виміру. Процедура регуляризації до-
зволяє визначити ефективне наближене рішення задачі ідентифікації,
яке для нестохастичного випадку узгоджується з помилками даних.
Властивості та особливості пропонованого підходу ілюструються ре-
зультатами моделювання.
Kлючові слова: ідентифікація системи, лінійна регресія, регуляризація,
асимптотичне моделювання, наближений розв’язок.
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| id | nasplib_isofts_kiev_ua-123456789-210935 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | Ukrainian |
| last_indexed | 2026-03-13T00:52:50Z |
| publishDate | 2023 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Gubarev, V. Melnychuk, S. Salnikov, N. 2025-12-21T10:53:01Z 2023 Paradigm of nonstochastic approach tosystem identification / V. Gubarev, S. Melnychuk, N. Salnikov // Проблеми керування та інформатики. — 2023. — № 1. — С. 42–58. — Бібліогр.: 32 назв. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/210935 681.5.015; 519.233 10.34229/1028-0979-2023-1-4 The paper discusses the construction of linear models, the complexity of which is determined by dimensionality. Within the framework of the non-stochastic approach, a methodological and mathematical foundation for reconstructing models that describe processes in complex systems is developed. Asymptotic modeling allows for the formation of classes of models suitable for solving identification problems. An exact description corresponds to an infinite extension, and thus the quality of the model improves with its increasing dimensionality. However, errors in the available data prevent the indefinite increase in dimensionality due to the ill-posedness of the identification problem starting from a certain dimension. The regularization procedure enables the determination of an effective approximate solution to the identification problem, which, in the non-stochastic case, is consistent with the data errors. The properties and features of the proposed approach are illustrated by modeling results. У статті розглядається побудова лінійних моделей, складність яких визначається розмірністю. У рамках нестохастичного підходу розроблено методологічну та математичну основу реконструкції моделей, що описують процеси у складних системах. Асимптотичне моделювання дозволяє для такої системи формувати класи моделей, які підходять для розв’язання задачі ідентифікації. Точний опис відповідає нескінченному розширенню, тому якість моделі покращується зі збільшенням її розмірності. Однак помилки в наявних даних не дозволяють безмежно збільшувати їхню розмірність через погану обумовленість задачі ідентифікації, починаючи з деякого виміру. Процедура регуляризації дозволяє визначити ефективне наближене рішення задачі ідентифікації, яке для нестохастичного випадку узгоджується з помилками даних. Властивості та особливості пропонованого підходу ілюструються результатами моделювання. uk Інститут кібернетики ім. В.М. Глушкова НАН України Проблеми керування та інформатики Методи керування та оцінювання в умовах невизначеності Paradigm of nonstochastic approach tosystem identification Парадигма нестохастичного підходу до ідентифікації систем Article published earlier |
| spellingShingle | Paradigm of nonstochastic approach tosystem identification Gubarev, V. Melnychuk, S. Salnikov, N. Методи керування та оцінювання в умовах невизначеності |
| title | Paradigm of nonstochastic approach tosystem identification |
| title_alt | Парадигма нестохастичного підходу до ідентифікації систем |
| title_full | Paradigm of nonstochastic approach tosystem identification |
| title_fullStr | Paradigm of nonstochastic approach tosystem identification |
| title_full_unstemmed | Paradigm of nonstochastic approach tosystem identification |
| title_short | Paradigm of nonstochastic approach tosystem identification |
| title_sort | paradigm of nonstochastic approach tosystem identification |
| topic | Методи керування та оцінювання в умовах невизначеності |
| topic_facet | Методи керування та оцінювання в умовах невизначеності |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210935 |
| work_keys_str_mv | AT gubarevv paradigmofnonstochasticapproachtosystemidentification AT melnychuks paradigmofnonstochasticapproachtosystemidentification AT salnikovn paradigmofnonstochasticapproachtosystemidentification AT gubarevv paradigmanestohastičnogopídhodudoídentifíkacíísistem AT melnychuks paradigmanestohastičnogopídhodudoídentifíkacíísistem AT salnikovn paradigmanestohastičnogopídhodudoídentifíkacíísistem |