Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models
The article provides a review and generalization of the principles, methods, and problems of designing discrete controllers for controlling complex systems of various nature, whose dynamics are described using difference equations of cognitive maps (CM) impulse processes (Roberts equations). An exte...
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Інститут кібернетики ім. В.М. Глушкова НАН України
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| Cite this: | Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models / V. Romanenko, V. Gubarev, Yu. Miliavskyi // Проблеми керування та інформатики. — 2024. — № 3. — С. 5–21. — Бібліогр.: 12 назв. — англ. |
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| author | Romanenko, V. Gubarev, V. Miliavskyi, Yu. |
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| citation_txt | Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models / V. Romanenko, V. Gubarev, Yu. Miliavskyi // Проблеми керування та інформатики. — 2024. — № 3. — С. 5–21. — Бібліогр.: 12 назв. — англ. |
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| description | The article provides a review and generalization of the principles, methods, and problems of designing discrete controllers for controlling complex systems of various nature, whose dynamics are described using difference equations of cognitive maps (CM) impulse processes (Roberts equations). An external control vector is implemented by discrete controllers based on varying node coordinates or weight coefficients of CM; the controllers are designed based on well-known methods of control theory. The article provides solutions to the following problems: stabilization of unstable impulse processes in CM of complex systems, control by varying weight coefficients, coordinating control of ratios between node coordinates, suppression of external disturbances, modeling impulse processes with multirate sampling of node coordinates, and identification of weight coefficients in CM impulse process models with complete and incomplete data.
У статті наведено огляд та узагальнення принципів, методів і проблем проектування дискретних регуляторів для керування складними системами різної природи, динаміка яких описується рівняннями різниць когнітивних карт (імпульсні процеси, рівняння Робертса). Зовнішній вектор керування реалізується дискретними регуляторами шляхом варіювання координат вузлів або вагових коефіцієнтів когнітивних карт; регулятори проектуються на основі відомих методів теорії керування. Розглянуто задачі стабілізації нестійких імпульсних процесів у когнітивних картах складних систем, керування процесами за допомогою варіювання вагових коефіцієнтів, а також координаційне керування співвідношеннями між координатами вузлів. Запропоновано методи придушення зовнішніх збурень, моделювання імпульсних процесів із багаторівневим відбором координат вузлів та ідентифікації вагових коефіцієнтів у моделях імпульсних процесів когнітивних карт за повними та неповними даними.
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| first_indexed | 2026-03-14T13:59:30Z |
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© V. ROMANENKO, V. GUBAREV, YU. MILIAVSKYI, 2024
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 5
АДАПТИВНЕ КЕРУВАННЯ ТА МЕТОДИ ІДЕНТИФІКАЦІЇ
UDC 62-50
V. Romanenko, V. Gubarev, Yu. Miliavskyi
STAGES AND MAIN PROBLEMS
OF THE CENTURY-LONG CONTROL THEORY
AND SYSTEM IDENTIFICATION DEVELOPMENT.
Part 5. PRINCIPLES AND PROBLEMS IN CONTROL
AND IDENTIFICATION OF COMPLEX SYSTEMS
OF VARIOUS NATURE BASED ON COGNITIVE MAPS
IMPULSE PROCESSES MODELS
Viktor Romanenko
National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»,
Institute for Applied Systems Analysis,
ipsa@kpi.ua; romanenko.viktorroman@gmail.com
Vyacheslav Gubarev
Space Research Institute of NAS of Ukraine and SSA of Ukraine, Kyiv,
https://orcid.org/0000-0001-6284-1866
v.f.gubarev@gmail.com
Yurii Miliavskyi
National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute»
Institute for Applied Systems Analysis,
https://orcid.org/0000-0003-0882-3418
yuriy.milyavsky@gmail.com
The article provides a review and generalization of the principles, methods and
problems of designing discrete controllers for controlling complex systems of
various nature, the dynamics of which are described using difference equations
of cognitive maps (CM) impulse processes (Roberts equations). An external
control vector is implemented by discrete controllers on the basis of varying
nodes coordinates or weight coefficients of CM; the controllers are designed
based on well-known methods of control theory. The article provides a solution
to the following problems of controlling impulse processes in the complex sys-
tems CM: stabilization of unstable impulse processes in the CM of complex sys-
tems based on reference models of closed-loop control systems and on the basis
of the modal control method; control of CM impulse process by varying weight
coefficients; implementation of coordinating control of the ratios between the
CM nodes coordinates in complex systems; suppression of external disturbances
when controlling complex systems based on invariant ellipsoids method; control
of impulse processes in the CM of complex systems with multirate sampling of
nodes coordinates; identification of weight coefficients in CM impulse processes
models with complete and incomplete measurement of nodes coordinates. The
mailto:ipsa@kpi.ua
mailto:romanenko.viktorroman@gmail.com
mailto:v.f.gubarev@gmail.com
mailto:yuriy.milyavsky@gmail.com
6 ISSN 2786-6491
solution to the above problems is based on the described new principles of cont-
rolling CM impulse processes in complex systems using automatic control theo-
ry methods.
Keywords: cognitive maps, impulse processes, reference models, invariant el-
lipsoids, identification, coordinating control.
Introduction
As a means for modeling complex systems of various nature with large dimension,
cognitive maps (CM) are used, which are structural diagrams of cause-and-effect rela-
tionships between the components (coordinates, factors, concepts) of a complex system.
From a mathematical point of view, a CM is a weighted directed graph, the nodes of
which represent the coordinates of complex systems, and the edges describe the rela-
tionships between these coordinates [1, 2]. The construction of the CM is carried out by
experts, which makes it possible to describe qualitatively the relationships between the
components of a complex system and quantitatively display the influence of each CM
coordinate on all others using the edges of directed graphs.
During the operation of a complex system with impulse-type behavior, under the
influence of various disturbances, the coordinates of the CM change over time. In this
case, each node il of the CM takes on a value ( )iy k at discrete time moments
0,1, 2,k = . At the next sampling period, the value ( 1)iy k + is determined by the
value ( )iy k and information about whether other nodes jl adjacent to il at a time in-
stants k have increased or decreased their values. The change in the coordinate of the
node il at a time moment k is called an impulse [2], which is specified by the differ-
ence ( ) ( ) ( 1), 0.j j jP k y k y k k= − − An impulse ( )jP k received at one of the nodes
jl will propagate along the CM chains to the remaining nodes, intensifying or attenuat-
ing. The process of propagation of disturbances along the nodes of the CM is deter-
mined by the difference equation
1
( 1) ( ) ( ), 0,1, 2, , ,
n
i i ij j
j
y k y k a P k i n
=
+ = + = (1)
where ija is the weight coefficient of the directed graph edge that connects the j-th with
i-th one. If there is no edge from node jl to node ,il then the corresponding coefficient
is 0.ija =
The rule for changing the coordinates of the nodes of the CM (1) is usually formu-
lated in the form of a first-order difference equation in increments of variables (Roberts
equations) [2]
1
( 1) ( ),
n
i ij j
j
y k a y k
=
+ = (2)
which describes the impulse process in the CM, and ( ) ( ) ( 1),i i iy k y k y k = − −
0, 1, 2, , .i n=
In vector form, expression (2) is written as follows:
( 1) ( ),y k A y k + = (3)
where A is the weight adjacency matrix, and ( )y k is the vector of increments of the
coordinates of the nodes y of the CM.
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 7
From the point of view of control theory, model (3) describes the dynamics of a
multidimensional system in discrete time in the free motion of the coordinates of the
CM nodes.
1. Principles of controlling impulse processes in cognitive maps of complex systems
In [3], axiomatic definitions of the basic principles of control for complex systems
are introduced, if their mathematical model is presented in the form (2), (3).
The first principle is the formation of an external control vector based on variable
vertices coordinates of the CM nodes of a complex system. To do this, it is necessary to
formulate an equation for the forced motion of the system during an impulse process
1
( 1) ( ) ( ),
n
i ij j i i
j
y k a y k b u k
=
+ = +
where ( ) ( ) ( 1)i i iu k u k u k = − − is the increment of the control action. In vector form
this equation can be written as follows:
( 1) ( ) ( ),y k A y k B u k + = + (4)
where u is the vector of increments of control actions ( ) ( ) ( 1).u k u k u k = − −
Depending on the physical nature of the complex system under study, we indentify
those CM nodes coordinates (values) which can be changed at discrete moments of time
by the decision maker (dm) by changing the available resources. In general, for various
CM of complex systems of various nature, these can be financial, energy, intellectual,
information, economic, technological, administrative, defense, social, scientific, politi-
cal, educational, environmental and other resources that can be changed at each set
sampling period as external controls influencing specific nodes of the CM. In this case,
external controls must have the same physical nature as the nodes on which they influ-
ence. The matrix B in (4) usually contains units and zeros, and the elements of this ma-
trix corresponding to the controls in the vector u are equal to units.
Thus, when forming a vector ,u it is necessary to select the coordinates of the
nodes of the CM, which can be influenced by the decision maker by changing the avail-
able resources. For example, for a complex system of socio-economic type, the follow-
ing ones can act as control actions:
— financial costs (capital investments);
— changes in the standard time for performing certain work;
— variations in prices for certain goods;
— development of new types of products;
— increasing the level of scientific research;
— improving the staff qualification through retraining;
— administrative commands of the management of a particular organization.
The second principle is the implementation of a closed-loop control system, in-
cluding a multidimensional discrete controller designed on the basis of methods of au-
tomatic control theory, which forms a vector of controls acting directly on the nodes of
the CM as the output controlled coordinates of a complex system. In [4], the main prop-
erties of models (3) were studied. It is shown that if the CM is stable from the point of
view of the theory of cognitive modeling, then the corresponding model (4) will be as-
ymptotically stable from the point of view of control theory (if we imagine it as a model
in state space). A study of controllability was also carried out and it was shown that if
the system is controllable, then it is possible to form a state control vector in a closed
system, as a result of which the corresponding CM goes into a stable static state.
8 ISSN 2786-6491
The third principle is the use of the possibility of varying the weight coefficients
of the matrix A in (3) to implement control actions ( )iu k in a closed-loop control
system. This principle must be applied in cases where it is impossible or undesirable to
vary the coordinates of the CM nodes (resources) during formation ( )iu k in accor-
dance with the first and second principles. Varying the weight coefficient can take place
when it is possible to change the degree of sensitivity of the influence of one CM value
on another one. The decision maker can implement this principle by changing the
transmission coefficients when forming administrative, financial, political, educational,
information interactions between the coordinates of a complex system, presented in the
form of nodes of the CM. When controlling the impulse CM process by varying the
weight coefficients, we change the degree of influence on the coordinate iy of the remain-
ing coordinates .jy In this case, the magnitude of the control action ( )iu k is formed at the
expense of changing the influence of the remaining coordinates ( )iu k on the node .iy
When implementing all three of the above control principles, it is necessary to
measure (fix) accurately the coordinates of all nodes of the CM. However, many CM
nodes in various complex systems are difficult to formalize and cannot be measured in
real time. These include:
— competitiveness of products;
— level of technology development;
— shadow connections between the economy and business;
— effectiveness of information propaganda;
— level of democratization of the state;
— level of corruption;
— integral level of population health in a given region, etc.
For this case, the work [3] proposed a fourth principle for controlling impulse
processes of the CM, which involves the decomposition of the original CM into two in-
terconnected parts. The first part of the CM is compiled of the measurable coordinates
of the original CM nodes, and the second part includes the unmeasured coordinates of
the nodes. Then, for the first part of the CM, the first model is compiled, which de-
scribes the dynamics of the impulse process of the measured coordinates of the CM:
1 1
( 1) ( ) ( ),
P n
i ij j i
j p
y k a y k a y k
= = +
+ = + (5)
where ,iy 1, 2, ,i P= are the measured coordinates of the CM in real time; ,y
1, ,p n = + are unmeasured coordinates of the nodes of the original CM.
The second model is compiled to describe the dynamics of the impulse process of
unmeasured nodes of the CM:
1 1
( 1) ( ) ( ).
n P
j j j j
j p j
y k a y k a y k
= + =
+ = + . (6)
In this case, unmeasured coordinates are considered as disturbances in model (5).
Expressions (5), (6), respectively, can be written in vector-matrix form:
1 11 1 12 2( 1) ( ) ( ),y k A y k A y k + = + (7)
2 21 1 22 2( 1) ( ) ( ),y k A y k A y k + = + (8)
where the adjacency matrices 11,A 12 ,A 21,A 22A have dimensions ( ),p p
( ( )),p n p − (( ) ),n p p− ( ) ( )n p n p− − respectively.
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 9
In the future, the impulse process model (7) will be used for control.
The fifth control principle involves multirate sampling of the coordinates of the
nodes of the CM for the case when the measurement of all coordinates of the CM can-
not be performed with one sampling period, the choice of which is carried out depend-
ing on the rate of change of the most rapidly changing coordinate of the CM. During the
operation of a complex system, these coordinates can be measured (fixed) at discrete
moments in time with different sampling periods. To determine these periods, it is nec-
essary to have information about the possible rates of change of each coordinate of the
nodes of a complex system. Then the choice of period 0 ,T according to [5], is per-
formed on the basis
0 min
max
,
( )i
i
T
dy t
dt
where is the specified absolute error of the coordinate ,iy which arises due to the in-
accuracy of measurement iy and sampling of the continuous function ( ).iy t The quan-
tity has the physical dimension of coordinates .iy
Let us consider a model of an impulse process of a complex system, which has the
property of functioning on two time scales. In this case, one part of the coordinates is
measured with a sampling period 0 ,T and the other one with a period 0 ,h mT= where
m is an integer greater than one. Then the original model (1) can be written as
0 0
1
( 1)
n
i ij j
j
k k
y h l T a y h lT
m m=
+ + = +
, (9)
where 0,1, , 1;l m= − 1, 2, , ;i n=
k
m
is integer part of k divided by .m
Let us assume that in the CM of a complex system with dimension n there are p
coordinates of nodes iy that are measured with a sampling period 0 ,T and ( )n p− co-
ordinates can be measured with an increased period 0.h mT= Then the model of the
impulse process of CM should be described with multirate sampling of coordinates as
follows:
0 0
1 1
( 1)
P n P
i ij j i p
j
k k k
y h l T a y h lT y h
m m m
−
+
= =
+ + = + +
; (10)
0
1 1
1
P n P
r rj j r p
j
k k k
y h a y h lT y h
m m m
−
+
= =
+ = + +
, (11)
where 1, 2, , ;i p= ( 1), , ;r p n= + 0,1, , ( 1);l m= − p
k
y h
m
+
equal to
the increment of the slowly changing coordinate py + at 0l = and equal to zero at
1, , 1.l m= −
Equations (10), (11) can be written in a generalized vector-matrix form:
1 0 11 1 0 12 2( 1) ;
0,1, , ( 1);
k k k
y h l T W y h lT W y h
m m m
l m
+ + = + +
= −
(12)
10 ISSN 2786-6491
2 21 1 0 22 21 ;
0, 1, , ( 1),
k k k
y h W y h lT W y h
m m m
l m
+ = + +
= −
(13)
where the matrices have dimensions 11( );W p p 12( ( ));W p n p − 21(( ) );W n p p−
22(( ) ( )).W n p n p− − Coordinates 2y that are measured with a sampling period h
will be constant over time 1
k k
h t h
m m
+
.
In work [3], the following statements were formulated and proven.
Statement 1. When calculating 2 1
k
y h
m
+
in model (13), the component
1 0
k
y h lT
m
+
is taken into account according to the formula
1 0 1 0 1 0( 1)
k k k
y h lT y h m T y h T
m m m
+ = + − − −
.
Statement 2. If the first inverse differences in (9) are defined as
0 0 0( 1) ( 1)i i i
k k k
y h l T y h l T y h lT
m m m
+ + = + + − +
,
0 0 0( 1)i i i
k k k
y h lT y h lT y h l T
m m m
+ = + − + −
,
then the transition of these differences to their representation with a large sampling pe-
riod h is carried out on the basis of:
0
1
1
m
i i
k k
y h y h T
m m=
+ = +
;
0
1
1
m
i i
k k
y h y h T
m m=
= − +
.
The formulated principles are intended for the subsequent stabilization of unstable
impulse processes in the CM, control of the ratios between the coordinates of the CM
nodes and for identifying the coefficients of the CM adjacency matrix.
2. Problems of stabilization of unstable impulse processes
in cognitive maps of complex systems
2.1. Stabilization of unstable impulse processes in CM based on reference
models. It is assumed that the original dynamic model of the impulse process of CM (3)
is unstable. In this case, the eigenvalues of the adjacency matrix A will be 1.i To
stabilize an unstable impulse process, the problem of designing a discrete controller to
create a vector of external controls, which, according to a certain law, influences the
nodes iy of the CM in a closed-loop control system, was solved in [6]. For this purpose,
the CM model (4) is presented in increments of variables in the «input–output» form as
1 1( ) ( ) ( ),I Aq y k Bq u k− −− = (14)
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 11
where 1q− is the reverse shift operator by one sampling period. Assume dim dim ,u y =
control actions are applied to all nodes of the CM. The control law of a discrete controller is
formulated as
1 1
0 1( ) ( ) ( ( ) ( )),u k D I F q G k y k− − = + − (15)
where ( )G k is vector of increments of set-point values. In practice in stabilization
systems ( ) 0.G k = Based on (14), (15), the equation of the multidimensional closed-
loop control system is presented over the « G y → » channel in the form
1 1 1 1 1 1 1 1
1 1 0 1( ) ( ){( )( ) } ( ) ( ).y k I F q I Aq I F q q BD Bq I F q G k− − − − − − − − = + − + + + (16)
To ensure the stability of the closed-loop system (16), a reference model of the
characteristic polynomial is used
1 2
1 2 1 1 1
1 0( )( ) .M MI A q A q I Aq I F q q BD− − − − −+ + = − + + (17)
In this case, the roots of the reference model
1 2
1 2det ( ) 0M MI A q A q− −+ + = of the ref-
erence model are selected with a modulus less than one. From equation (17) the matri-
ces of the control law (15) are determined
2
1
1 ;MF A A−= −
2 1
1 1
0 ( ).M MD B A A A A− −= + + (18)
Thus, at each sampling period with measured increments of coordinates ( )y k of
the nodes of the CM, based on the control law (15) and controller parameters (18), the
vector of increments of control actions is determined for ( ) 0G k = as
2 1 2
1 1 1 1 1( ) ( ) ( ) ( ).M M Mu k B A A A A I A A q y k− − − − − = − + + −
The disadvantage of this method is equality dim dim ,u y = so that to imple-
ment control it is necessary to vary the resources of all nodes of the CM. This is pos-
sible only in rare cases. For example, this algorithm was used in [6] to stabilize an un-
stable process in a financial bank (in the event of its bankruptcy).
2.2. Stabilization of an unstable impulse process in a CM based on modal control.
When forming control actions to stabilize unstable processes in complex systems based
on models of impulse processes, the CM can be varied only by some of the coordinates
of the CM nodes. In work [7] to control the impulse process (4)
( 1) ( ) ( )y k A y k B u k + = +
controller applied
( ) ( ),pu k K y k = −
which is designed based on the modal control method [8] for the case when
dim ( ) dim ( ).u k y k To do this, the desired spectrum of the closed-loop system
( 1) [ ] ( )py k A BK y k + = − is set
1 0
0 n
=
,
where all 1.j In this case, all eigenvalues j of the matrix ( )pA BK− are cho-
sen to be different, and among j there are no eigenvalues of the matrix A of weight
12 ISSN 2786-6491
coefficients of the CM. It is assumed that the control vector u has dimension ,m where
.m n To design the controller, eigenvectors ,jR 1, ,j n= of the state matrix
pA BK− of the closed-loop system are introduced, for which the equation ( )p jA BK R− =
j jR= holds. This equality can be written in the form
( ) ,j j p j jA I R BK R BP− = = 1, , ,j n= (19)
where the vectors j p jP K R= have dimension m.
An arbitrary matrix ( )P m n is specified so that it has full rank and does not have
zero columns: 1 2( , , , ).nP P P P= Then from expression (19) we can determine
1( ) ,j j jR A I BP−= − 1, ,j n= (20)
and form a matrix 1( , , )nR R R= of dimension ( ),n n which will be non-
degenerate. Since ,j p jP K R= the feedback matrix of the closed-loop control system is
calculated as follows:
1.pK PR−= (21)
By construction pK provides the desired set of roots 1 2, , , n of the charac-
teristic equation of a closed-loop control system det[ ] 0.pIq A BK− + =
Based on the modal control method, work [9] carried out stabilization of an unsta-
ble mode in the impulse process of the CM of a student’s social-educational process,
and work [10] implemented an adaptive system for stabilizing an unstable cryptocurren-
cy rate based on a model of the impulse process of a cognitive map.
3. The problem of controlling the impulse process
of CM by varying the weight coefficients
In [11], a new approach to controlling the CM process is proposed. It is imple-
mented by varying the weight coefficients of the CM edges when forming control ac-
tions ( ).u k Varying the weight coefficients is possible when it is possible to change the
influence of one of the CM nodes on another one. The paper considers the problem of
stabilizing the coordinates of the nodes iy of the CM at given levels iG at each sam-
pling period. In this case, control actions are formed by changing the increments of the
weighting coefficients ( )
l
ia k
on the edges of the CM, which connect the node
( )
i
y k with ( ).iy k For this purpose, the equation of the forced motion of the impulse
process of the CM is written in the full values of the coordinates ( )y k of the CM nodes
in the following form:
1( 1) ( ) ( ) ( ) ( ) ( ),y k I A Aq y k L k a k k−+ = + − + + (22)
where the vector of increments of weighting coefficients ( )a k of dimension ( 1),m
,m n contains only non-zero elements ( ) 0,
l
ia k
where is the number of the
node of the CM, which affects the i-th node iy through the edge of the CM with a coef-
ficient ,ia
and the index 1, 2, ,l m= denotes the sequence number of this coeffi-
cient in the vector ( ).a k If some weight coefficient ia
does not vary, then its incre-
ment ( ) 0
l
ia k
= is not included in the vector ( ).a k
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 13
The matrix ( )L k contains only the measured coordinates of the nodes ( )
i
y k of
the CM, which influence the nodes ( 1),iy k + 1, 2, , ,i n= through the edges of the
CM with variable coefficients ( ).
l
ia k
The matrix ( )L k ( )n m contains at most one
non-zero element
i
y on each i row, which affects the node iy through the increment
of the weight coefficient ( ).
l
ia k
The number of the column in the matrix ( )L k in
which this element is located is equal to the number of the element ( )
l
ia k
in the
vector ( )a k of equation (22).
The design of the optimal control vector ( )a k is implemented based on the min-
imization of the quadratic optimality criterion
T T( 1) {[ ( 1) ] [ ( 1) ] ( ) ( )},J k E y k G y k G a k R a k+ = + − + − + (23)
where G is the set-point vector; R is diagonal positive definite matrix. As a result of
minimization
( )
( 1) min,
a k
J k
+ → the controller control law is found
T 1 T 1( ) [ ( ) ( ) ] ( ){( ) ( ) ( ) }.a k L k L k R L k I A Aq y k k G− − = − + + − + − (24)
Based on equations (22), (23), the equation of the closed-loop control system for
the impulse CM process has the form
T 1 T
1 T 1 T
T 1 T
( 1) { ( )[ ( ) ( ) ] ( )}
( ) ( ) ( )[ ( ) ( ) ] ( )
{ ( )[ ( ) ( ) ] ( )} ( ),
y k I L k L k L k R L k
I A Aq y k L k L k L k R L k G
I L k L k L k R L k k
−
− −
−
+ = − +
+ − + + +
+ − +
(25)
where ( )k is the external disturbance.
The stability of the closed-loop system (25) is determined by the location of the ei-
genvalues i of the matrix
T 1 T{ ( )[ ( ) ( ) ] ( )}.I L k L k L k R L k−− + The following state-
ment was formulated and proven in [11].
Statement 3. Let the equation of the forced motion of the CM impulse process have
the form (22), where ( )a k is the m-dimensional vector of increments of weight coeffi-
cients, and the matrix ( )L k ( )n m is formed at each sampling period based on the co-
ordinates ( ),
l
y k where 1, 2, ,l m= is the number of the corresponding increments
( )
l
ia k
in the vector ( ),a k then the product of the dimensional ( )m m matrices
T ( ) ( )L k L k in the control law ( 24) will be a diagonal matrix with elements 2 ( ).
l
y k
Corollary 1. Since the weight matrix R ( )m m in (24) is diagonal and positive
definite, the inverse matrix
T 1[ ( ) ( ) ]L k L k R −+ in (25) will be diagonal with elements
2
1
,
( )
l
lly k R
+
1, 2, , ,l m= which are always positive.
Corollary 2. The matrix
T 1 T( ) { ( )[ ( ) ( ) ] ( )}M k I L k L k L k R L k−= − + in (25) dimen-
sion ( )n n will be diagonal with elements
2
2 2
( )
1 1
( ) ( )
l
l l
ll
ll ll
y k R
y k R y k R
− =
+ +
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in rows that correspond to non-zero values ( )
l
y k and equal to one in other cases.
Thus, the eigenvalues of the matrix ( )M k will always be 0 1,i 1, 2, , ,i n= and
the closed loop system (25) will be stable.
In work [12], the problem of combined control of impulse processes in a CM was
solved based on varying the weight coefficients and resources of the coordinates of the
CM nodes.
4. The problem of coordinating control of the ratios
between the coordinates of complex system CM in a stochastic environment
The problem of controlling coordinate ratios arises in multidimensional automatic
control systems [13]. A huge number of complex systems represented by CM, must be
described in a stochastic environment due to the following circumstances:
a) many coordinates of socio-economic, political, administrative systems cannot be
measured and they can only be taken into account as stochastic disturbances;
b) some coordinates of the CM are measured with a large error.
In [14], the problem of controlling the ratios between the coordinates of the nodes
of CM in a stochastic environment (coordinating control) is considered. The impulse
process of CM (4) is presented in full coordinates of the nodes by analogy with [15]
1 2 1[ ( ) ] ( ) ( ) ( ),I I A q Aq y k Bq u k k− − −− + + = + (26)
where 1q− is the reverse shift operator; A ( )n n is the adjacency matrix of the CM
with eigenvalues 1,i the control matrix B is usually assumed B I= (it is possible
to vary the resources of all nodes of the CM); ( )k is the vector of stochastic disturb-
ances or errors in measuring the coordinates of nodes in the form of white noise, uncor-
related with the coordinates ( )y k and controls ( ).u k For convenience, it is consi-
dered in further calculations
( 1) ( ) ( ) ( 1),y k I A y k Ay k+ = + − − (27)
which is known part of ( 1)y k + at the time .k
First, a standard criterion for the generalized variance of the discrepancy between the
coordinates ( 1)y k + and the set-point vector G and of control actions ( )u k is introduced:
T T( 1) {[ ( 1) ( )] [ ( 1) ( )] ( ) ( )},GJ k E y k G k y k G k u k R u k+ = + − + − + (28)
where E is the operator of conditional mathematical expectation based on all infor-
mation available at a given time .k The matrix R is chosen to be symmetrical, positive
definite and such that
T( )B B R+ it is not degenerate. To minimize (28) ( 1)y k + is
split into known and unknown parts
( 1) ( 1) ( ) ( 1).y k y k B u k k+ = + + + +
Then criterion (28) will have the form
T( 1) [ ( 1) ( ) ( )] [ ( 1) ( ) ( )]GJ k y k G k B u k y k G k B u k+ = + − + + − + +
T T T( ) ( ) 2 {[ ( 1) ( ) ( )] ( 1)} { ( 1) ( 1)}.u k R u k E y k G k B u k k E k k+ + + − + + + + + (29)
Minimization of this criterion by vector ( )u k leads to the equation of the optimal
controller
T( 1)
2 ( ( ) ( 1) ( )) 2 ( ) 0.
( )
GJ k
B B u k y k G k R u k
u k
+
= + + − + =
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 15
It was shown in [14] that the solution to this equation is a minimum point. Then the
control law is calculated by the formula
T 1 T( ) ( ) { [ ( 1) ( )]}.Gu k B B R B y k G k− = − + + −
The problem of coordinating control is stated based on given M ratios
( ) ,Sy k b= (30)
where b is a given vector with dimension ,M S is a given matrix with dimension
( )M n for .M n Here rank( ) .SB M= The requirement of coordinating control is
that ratios (30) should be satisfied as accurately as possible at each sampling period.
To implement coordinating control, the following optimality criterion is proposed
T( 1) {[ ( 1) ] [ ( 1) ]} min.bJ k E Sy k b Sy k b+ = + − + − → (31)
Let us write this criterion similarly to the previous criterion (29)
T
T T T
( 1) ( 1) ( ) ] [ ( 1) ( ) ][
2 {[ ( 1) ( ) ] ( 1)} { ( 1) ( 1)}.
bJ k y k SB u k b Sy k SB u k bS
E Sy k SB u k b S k E k S S k
+ + + − + + − +=
+ + + − + + + +
The last term does not depend on ( ),u k so it can be ignored. The penultimate
term is equal to zero due to { ( 1)} 0.E k + = The first term is equal to the square of the
Euclidean norm of the vector ( 1) ( ) .Sy k SB u k b+ + − It is known that the squared
norm of a vector is non-negative and equals to zero if and only if the vector is zero.
Therefore, if the equation ( ) ( ( 1) )SB u k Sy k b = − + − has a solution, then this solution
is the global minimum point of this criterion. By condition, a vector ( )u k has dimen-
sion ,n and a vector [ ( 1) ]Sy k b+ − has dimension ,M n and rank( ) .SB M= There-
fore, according to the Kronecker-Capelli theorem, this equation has many solutions, at
which the minimum of criterion (31) is achieved, namely
( ) ( ( 1) ).SB u k Sy k b = − + − (32)
In general, the control for ( )gu k does not satisfy equality (32). Thus, we have a
multicriteria optimization problem ( )u k based on criteria (28) and (31), and criteri-
on (31) by condition has higher priority, but has an ambiguous solution.
In [14], a method of conditional minimization of the discrepancy of ratios and the
generalized variance of CM coordinates is considered. In this case, equation (32) is con-
sidered as a constraint that must be satisfied when minimizing criterion (28), that is, the
problem of unconditional multicriteria optimization is reduced to the problem of condi-
tional single-criteria optimization, which is solved by the Lagrange multiplier meth-
od [16], and then the Frobenius theorem is applied to address block matrix [17]. As a
result, the following theorem was formulated and proven.
Theorem 1. Let the CM impulse process be specified by equation (26) and it is neces-
sary at each time moment to create such a control that will minimize the optimality crite-
ria (31) and (28), and criterion (31) has higher priority, then the optimal control ( )u k will
be determined as follows:
T 1 T T 1 T 1( ) ( ) {[ ( ) ]u k B B R I B S L SB B B R− − − = − + − +
T T T 1[ ( ( 1) ( ))] [ ( 1) ]}B y k G k B S L Sy k b− + − + + − (33)
if the matrices
T( )B B R+ and
T 1 T T( )L SB B B R B S−= + are nonsingular, and
( 1) ( ) ( ) ( 1).y k I A y k Ay k+ = + − −
16 ISSN 2786-6491
In [18], the problem of adaptive coordinating control of the ratios of the coordi-
nates of the nodes of interacting CM in the mode of impulse processes was solved.
5. The problem of suppressing external disturbances of impulse processes
in cognitive maps of complex systems based on invariant ellipsoids
In [19], a study was carried out on the use of the invariant ellipsoid method for
suppressing limited disturbances with partial control of dynamic processes in complex
systems of various nature, represented by mathematical models of impulse processes of
CM. According to the fourth control principle, the original CM is decomposed into two
interconnected parts. The first part of the CM is compiled of the measured coordinates
of the nodes of the original CM (equation (7)), and the second part (8) describes the dy-
namics of the unmeasured coordinates of the nodes. Let us write models (7), (8) in a
more convenient form [19]
( 1) ( ) ( ),x k A x k D y k + = + (34)
( 1) ( ) ( ),y k C y k x k + = + (35)
where x is the vector of measured coordinates of the CM; y is the vector of unmeas-
ured coordinates. In this case, the matrices D and display the interconnections be-
tween the first (34) and second (35) parts of the original CM. In this case, changes in
unmeasured coordinates ( )y k are considered in the first system of equations (34) of
the CM for impulse processes of the CM with measurable parameters as external limited
unmeasured disturbances with unknown probabilistic characteristics.
In [20, 21], a method for suppressing limited external disturbances based on invari-
ant ellipsoids for models of control objects in state space is considered. If we consider
( )y k as a disturbance in model (34), then its norm limitation l will have the form
T 1/2
0
( ) sup[ ( ) ( )] 1.
k
y k y k y k
= (36)
In this case, to describe the characteristics of the influence of disturbances of type
(36) on the trajectory of motion of a dynamic discrete system (34), invariant ellipsoids
along coordinates ( )x k are used in the following form:
T 1{ ( ) : ( ) ( ) 1},n
x x k x k P x k−
= R 0,P (37)
if the condition (0) xx implies the fulfillment of the condition ( ) xx k for
all discrete moments of time 1, 2, 3,k = . The matrix P is called an ellipsoid mat-
rix .x In [20] it is shown that the ellipsoid (37) x will be invariant for the dynamic
system (34) with l-bounded perturbations (36) if and only if the matrix P satisfies
the linear matrix inequality
T
T1
0,
(1 )
DD
APA P− +
−
0 ,P P (0,1). (38)
In [19], a design of static state feedback was carried out, which minimizes the size
of invariant ellipsoids for a controlled dynamic system, the model of which, based
on (34), is supplemented with a control vector ( )u k
( 1) ( ) ( ) ( ).x k A x k B u k D y k + = + + (39)
The control vector ( )u k is formed by the state controller
( ) ( ),pu k K x k = − (40)
which implements static state feedback.
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 17
In the model of a controlled impulse process CM (39), as in the state equation, the
components of the vector ( )x k are completely measured.
The optimality criterion for implementing the controller (40) is minimizing the size
of the invariant ellipsoid (37) based on
*tr ( ) min, 1P a a → (41)
with maximum suppression of disturbances ( )y k that are limited by the maximum
range (36).
Naturally, minimizing the size of an invariant ellipsoid is equivalent to the problem
of minimizing the trace of the matrix (41) under a constraint of the type of linear matrix
inequality (38).
Based on model (39) and control law (40), the equation of the closed-loop control
system for the impulse CM process will take the form
( 1) ( ) ( ) ( ).px k A BK x k D y k + = − + (42)
It is assumed that the pair ( , )A B in model (39) is controllable. Then the linear ma-
trix inequality (38) for the closed system (42) takes the form
T
T1
( ) ( ) 0,
(1 )
p p
DD
A BK P A BK P− − − +
−
(43)
and after multiplying the terms
T
T T T T1
( ) 0.
(1 )
T T
p p p p
DD
APA BK PA APBK B BK PK B P− − + − +
−
(44)
This inequality is nonlinear with respect to P and .pK In [20] replacement
pL K P= was proposed, and the matrix TR R= was introduced, for which an addition-
al constraint is satisfied
T
0.
R L
L P
(45)
Since, according to the Schur formula for 0,P inequality (45) is equivalent to
1 T T ,p pR LP L K PK− = then to satisfy inequality (44) it is sufficient that
T
T T T T T1
( ) 0.
(1 )
DD
APA BLA AL B BRB P− − + − +
−
(46)
Minimization of criterion (41) under restrictions (45), (46) is performed for varia-
bles ,P ,L R using the semidefinite programming method, for example, by using Se-
DuMi Toolbox based on Matlab. Let ̂ ˆ,P ˆ,L R̂ provide the minimum of (41) under
restrictions (45), (46). Then the matrix ˆ
pK of the optimal controller (40) is found as
1ˆ ˆ ˆ .pK LP−= (47)
Thus, changes in unmeasured CM coordinates ( )y k are taken into account as
limited disturbances in model (34) for impulse CM processes with measured coordi-
nates. These disturbances are suppressed by the controller (47).
18 ISSN 2786-6491
In [19], a study of the impulse process control system in the cognitive map of an IT
company was carried out, in which unmeasured coordinates are presented as disturbances
for the controlled part of the CM. Compared to an uncontrolled process, when applying
designed control actions (40), (47), the development time of IT projects was significantly
reduced, the quality of the project was improved, and sales volumes increased.
6. The problem of controlling impulse processes
in the CM of complex systems with multirate sampling of node coordinates
The work [22] describes a method for multirate sampling of the coordinates of the
nodes of CM for the case when measuring all coordinates with one sampling period 0T
is impossible. Based on the fifth principle [3], it is assumed that the first part of the node
coordinates 1y in model (17) is measured at discrete time with a sampling period 0 ,T
and the rest 2y with a period 0 ,h mT= where m is an integer greater than one. To con-
trol the impulse process in model (12), (13), control action vectors 1,u 2u are intro-
duced, which directly affect the nodes 1y and 2.y In this case, the controlled model of
the impulse process (12), (13) will have the form:
1 0 11 1 0
12 2 11 1 0
2 21 1 0 22 2 22 2
( 1)
;
1
k k
y h l T W y h lT
m m
k k
W y h B u h lT
m m
k k k k
y h W y h lT W y h B u h
m m m m
+ + = + +
+ + +
+ = + + +
;
(48)
where 0,1, , ( 1),l m= −
k
m
is the integer part of division k by ,m 11,B 22B are di-
agonal matrices that are selected by the automation system developer. The dynamics of vec-
tors 1( ),y k 2 ( )y k in model (48) are presented in full values of the variables as follows:
1
1 0 11 11 11 1 1 0( 1) ( )
k k
y h l T I W W q y h lT
m m
−
+ + = + − + +
11 1 0 12 2 ;
k k
B u h lT W y h
m m
+ + +
(49)
1
2 22 22 22 2 21 ( )
k k
y h I W W q y h
m m
−
+ = + − +
22 2 21 1 0 ,
k k
B u h W y h lT
m m
+ + +
(50)
where 1,q 2q are the reverse shift operators for the sampling periods 0T and
0 ,h mT= respectively. In equations (49), (50), the terms 12 2
k
W y h
m
and
21 1 0
k
W y h lT
m
+
are perturbations respectively. To create the first control vector
1 0
k
u h lT
m
+
, which is implemented with a sampling period 0 ,T the following op-
timality criterion is formulated in [22]:
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 19
( )
T
1 0 1 0 1( 1) 1
k k
J h l T E y h l T G
m m
+ + = + + −
( ) T
1 0 1 1 0 1 1 01 ,
k k k
y h l T G u h lT R u h lT
m m m
+ + − + + +
(51)
where 1G is the set-point vector for controlling the coordinates of the nodes 1y of the
CM. Based on the minimization of this criterion with respect to vector 1,u taking into
account (49), the equation of the first controller was written.
1 0
T 1
11 11 11 11 1 1 0
1 0
11 1 0 12 2 1 1 1 0
( 1)
2 ( )
2 0,
k
J h l T
m k
B I W W q y h lT
mk
u h lT
m
k k k
B u h lT W y h G R u h lT
m m m
−
+ +
= + − + + +
+ + + − + + =
where the control law of the first controller is created
T 1 T 1
1 0 11 11 1 11 11 11 11 1
1 0 12 2 1
( ) ( )
, 0, 1, , ( 1),
k
u h lT B B R B I W W q
m
k k
y h lT W y h G l m
m m
− −
+ = − + + −
+ + − = −
(52)
which, after expanding the difference 1,u has the form
T 1 T
1 0 1 0 11 11 1 11( 1) ( )
k k
u h lT u h l T B B R B
m m
−
+ = + − − +
1
11 11 11 1 1 0 12 2 1( ) .
k k
I W W q y h lT W y h G
m m
−
+ − + + −
To create the second control vector 2
k
u h
m
, which is implemented with a
period 0 ,h mT= the second optimality criterion is formulated
T
2 2 21 1
k k
J h E y h G
m m
+ = + −
T
2 2 2 2 21 ,
k k k
y h G u h R u h
m m m
+ − +
(53)
where 2G is the set-point vector for controlling the coordinates of the nodes of the CM 2.y
Based on the minimization of criterion (53) with respect to vector 2 ,u taking into ac-
count (50), the equation of the second controller was created
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2
T 1
22 22 22 22 2 2
2
1
2 ( )
k
J h
m k
B I W W q y h
mk
u h
m
−
+
= + − +
22 2 21 1 0 2 2 22 0,
k k k
B u h W y h lT G R u h
m m m
+ + + − + =
where the control law of the second controller is found
T 1 T
2 2 22 22 2 22
1
22 22 22 2 2 21 1 0 2
1 ( )
( ) .
k k
u h u h B B R B
m m
k k
I W W q y h W y h lT G
m m
−
−
= − − +
+ − + + −
In the impulse process model (50), in which the vector 2y is considered at discrete
times with a period ,h frequently measured coordinates 1 0
k
y h lT
m
+
based on
statement 1 are taken into account as follows
1 0 1 0 1 0( 1)
k k k
y h lT y h m T y h T
m m m
+ = + − − −
.
In [23], a new principle for controlling impulse processes in the CM of complex
systems is considered based on varying the coordinates of the nodes and the weight co-
efficients of the CM when forming control actions in closed-loop control systems. In
this case, the coordinates of the nodes of the CM and the weight coefficients of the edg-
es change in discrete time with multirate sampling.
In [24], a cognitive map of cause-and-effect relationships was developed in the
process of the spread of population morbidity due to Covid-19 in a specific region,
models of impulse processes of CM subsystems with multirate sampling were devel-
oped, and combined control subsystems with multirate sampling were designed based
on the formation of control actions by varying node coordinates and CM weight coeffi-
cients. Experimental studies of control systems for impulse processes in the CM of
morbidity for Covid-19 have been carried out.
7. Problems of identifying models of impulse processes
in cognitive maps of complex systems
7.1. Identification in CM in the impulse processes mode with complete infor-
mation. The dynamics of a complex system changes widely during its operation be-
cause of crisis phenomena, emergence of non-standard and conflict situations, human
factor, changes in social and political relationships, etc. If difference equations of im-
pulse processes CM (2) are used as a mathematical model of a complex system, then the
weighting coefficients ija must be periodically assessed based on identification methods.
The identification problem is performed during the impulse CM process, when all coordi-
nates of the CM nodes are in a transition mode or under the influence of external testing
influences. For this purpose, the equation of the impulse process (2) is written in the form
1
( 1) [ ( ) ( )],
n
i ij j j
j
y k a y k q k
=
+ = + (54)
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 21
where ( )iy k is the coordinate value of the i-th node of the CM at time 0 ,t kT=
( ) ( ) ( 1)i i iy k y k y k = − − is the magnitude of the impulse at the i-th node, i =
1, 2, , ;n= ( )jq k is generated known testing influence on the j-th node during
identification. The impulse process (54) for the entire CM is written in vector -
matrix form
( 1) [ ( ) ( )],y k A y k Q k + = + (55)
where A is the weight adjacency matrix of the CM, ( )y k is the vector of increments
of coordinates of the CM nodes, ( )Q k is the vector of increments of testing influences
according to a given program.
In [6, 25], to estimate variable matrix A of coefficients ,ija the recurrent least
squares method (RLSM) was used, which gives unbiased estimates provided that the
disturbances in (54) are discrete white noise. This condition is not always met during
the impulse CM process. When the structure of the CM changes, which is expressed in
the appearance of new weighting coefficients ija that were not previously established
by experts, the RLSM does not evaluate these coefficients at all, since they will be ab-
sent in the RLSM algorithm.
The works [26, 27] consider various options for solving the problem of parametric
identification of the coefficients of the CM adjacency matrix, when all CM nodes coor-
dinates are measured.
Parametric identification of adjacency matrix in the absence of disturbances.
The equation of the impulse process (55) for moments in time 1, 2, , 1k N= − can be
represented as a sequence of systems of equations
(1) [ (0) (0)],
(2) [ (1) (1)],
( ) [ ( 1) ( 1)].
y A y Q
y A y Q
y N A y N Q N
= +
= +
= − + −
(56)
In the deterministic case, based on (56), the problem of determining the matrix A
from measurement data is solved when there is no noise, i.e. sequences of vectors
{ ( )}y k are specified exactly. In this case, the impacts { ( )}Q k are formed according
to the program.
Then system (56) can be written in a unified form
[ ] ,i iy Y Q a = + 1, 2, , ,i n= (57)
where iy is the vector representing the dynamics of the i-th node of the CM from the
1st to the n-th sampling period
T [ (1) (2) ( )],i i i iy y y y n = (58)
and
1 2
T ( )
ni i i ia a a a= is a vector containing the elements of the i-th row of the ma-
trix .A The matrix Y is composed of discrete measurements of the coordinates of the
nodes of the CM from the zero to the ( 1)n− -th sampling periods as follows:
22 ISSN 2786-6491
1 2
1 2
1 2
(0) (0) (0)
(1) (1) (1)
( 1) ( 1) ( 1)
n
n
n
y y y
y y y
Y
y n y n y n
=
− − −
. (59)
The matrix of known testing influences is formed according to
1 2
1 2
1 2
(0) (0) (0)
(1) (1) (1)
( 1) ( 1) ( 1)
n
n
n
q q q
q q q
Q
q n q n q n
=
− − −
. (60)
Thus, according to (57), the weighting coefficients of the adjacency matrix A are
determined based on
1( ) ,i ia Y Q y−= + 1, 2, , .i n= (61)
Solvability of (61) is ensured when det ( ) 0.Y Q + When implementing (61),
it is assumed that the matrix A elements remain constant over the time interval
1, 2, , .k n=
When forming sequences of testing influences ,iq 1, 2, ,i n= it is necessary to
take into account that not all nodes of the CM can be varied according to the program
established by the decision maker. In this case, some iq in equation (57) will be equal
to zero and will not be taken into account when estimating coefficients ia based
on (61). So during identification the corresponding coordinates iy will change only
when influenced through weighting coefficients ija when varying the coordinates jy to
which testing influences jq will be applied.
Identification when there is noise in the measured data. In this case, instead of
the exact values of the matrix Y and vector ,iy there is information about the ap-
proximate values of the matrix Y and vector .iy Therefore, when solving (61), an
approximate estimate of the coefficients ˆ
ia is obtained. When carrying out identifica-
tion, it is very important to know how approximate information differs from the exact
values. It should be noted that the estimate ˆ
ia significantly depends on the condition
number of the summary matrix ( ).Y Q + If this matrix is ill-conditioned or close to
degenerate, then even small disturbances or inaccuracies in measuring the coordinates
of the nodes of the CM can lead to the identification problem becoming incorrectly
posed. Poor conditioning depends on two factors: the initial state (0),y namely the
signal level of each of the modes of the system, and the parameters of the system gener-
ating the data. In this case, the condition number, even with equal excitation of all
modes of the CM, grows rapidly with increasing dimension of the CM. Therefore, for
large ,n , it is necessary to evaluate the deviation of the resulting approximate solution
from the exact one.
Solving the problem using the combinatorial method. Measuring the coordi-
nates of the CM nodes in the presence of noise is specified in the form
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 23
,i i iy y = + ,i (62)
where iy is the exact increment of the coordinate vector (58), i is the vector of er-
rors in measuring the i-th coordinate at discrete moments in time over the observation
interval 1, 2, , ,k n= and is a fairly small value.
Consider a system of linear quadratic equations
1
ˆ ,i iy Y a = + (63)
where 1 .Y Y Q = + In system (63), the matrix Y and vector iy are specified
with an error, and
1( )i i iy Y a − = + (64)
corresponds to an exact equation in which is an additive matrix with elements that
are the measurement errors of the coordinates of the nodes of the CM, taken with a mi-
nus sign. Then, to obtain a component-wise estimate of the membership intervals of ex-
act values, the following theorem is applied [28].
Theorem 2. Let an approximate non-degenerate system (63) and an exact sys-
tem (64) be given such that
1 ,Y
i iy
. (65)
If the condition number 1( )Y satisfies the condition
1( ) 1,Y r = (66)
then 1Y + it is also non-degenerate and
1
1 1
2ˆ ˆ .
1
i i ia a Y Y a
r
−
−
−
(67)
The value 1
1 1 1( )Y Y Y−
= in (66) is the condition number of the
matrix 1Y according to the « » norm, and 1
1 1Y Y−
in (66) it is also
called the Shkeel condition number. In this theorem, specific realizations and i are
unknown, and, therefore, it is not possible to determine and r from (65), (66). How-
ever, if we focus on the most unfavorable realization, the probability of which is small,
then, based on (65), we can obtain a guaranteed membership interval of the exact value
for any realizations that satisfy (62). With this realization we have
1,= ,i e = (68)
where 1 is a matrix in which all elements are equal to one, and e is a vector with all
unit elements. Then
,n
= .
i
= (69)
The obtained result allows us to apply a combinatorial method for solving the iden-
tification problem using approximate data, which is implemented when condition (66)
of the above theorem is satisfied. Therefore, its feasibility is first checked. To do this,
find the smallest value at which the loose inequalities are satisfied
,
Y
n
.iy
(70)
24 ISSN 2786-6491
After this, the feasibility of strict inequality (66) is checked. In this case, it
should be expected that if the matrix Y is poorly conditioned, the value r at the
smallest will not be less than one. This can actually mean a large spread of solu-
tions obtained based on quadratic equations formed from (56), that is, the solution
method based on a guaranteed result cannot be implemented. In this case, when (66)
is satisfied for the set of quadratic systems formed from (56), the combinatorial
method will be effective.
Its description is given in [29]. The advantage of the method is the successful com-
bination of the properties of statistics with guaranteed interval estimation. This makes it
possible, based on multiple estimates and with certain properties of statistics, to signifi-
cantly reduce the guaranteed membership intervals of the exact values of the vector
components .ia
The solution algorithm using the combinatorial method is as follows:
— from the overdetermined system (56) we create a set of square equations, dis-
carding unnecessary equations in an arbitrary manner. There will be a total n
NC of such
combinations;
— we leave only those systems for which condition (66) is satisfied, that is, non-
degenerate and well conditioned systems. Let such systems remain ;S
— we solve the remaining quadratic systems and find approximate estimates ˆ
ija of
the parameters .ija
Each s-th square system 1, 2, ,s S= according to (67) is characterized by a
guaranteed component-wise error
1
1 1
2 ˆ .
1
is iY Y a
r
−
=
−
Inequality (67) in this case allows us to write an element-wise estimate for ija in
the form ˆ ,s s
ij ij isa a− or
ˆ ˆ ,s s
ij is ij ij isa a a− + (71)
For different s and fixed ,i j (71) gives a system of guaranteed estimates, from
which it follows that the exact value must belong to the truncated smallest interval
ˆ ˆmax ( ) min ( ).s s
ij is ij ij is
ss
a a a− + (72)
From (72) it follows that with increasing S and spread of realizations ˆ ,s
ija the ac-
curacy of estimation should improve. It is advisable to take the middle of the interval
ˆ ˆ[max( ); min( )]s s
ij is ij isss
a a− + as an estimate ija integrated over ,S and half of its
width will characterize the accuracy of the estimate.
Solving the problem using least squares method. The combinatorial method is
not applicable when condition (66) is not satisfied, since estimate (67) will not be fair.
In this case, it is proposed to use ordinary or weighted OLS. The sequence of equa-
tions (56) allows us to construct an overdetermined system of equations (63) with the
same desired vector ˆ ,ia in which
T[ (1) (2) ( )] ,i i i iy y y y N =
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 25
1 2
1 2
1 2
(0) (0) (0)
(1) (1) (1)
( 1) ( 1) ( 1)
n
n
n
y y y
y y y
Y
y N y N y N
=
− − −
,
1 2
1 2
1 2
(0) (0) (0)
(1) (1) (1)
( 1) ( 1) ( 1)
n
n
n
q q q
q q q
Q
q N q N q N
=
− − −
.
Here it is assumed that the input impulses ( )iq k are known exactly. However, all
the results that will be obtained below remain valid even if instead Q we have an ap-
proximate matrix .Q
If we use the least squares method to solve an overdetermined system of linear al-
gebraic equations with an approximate right-hand side iy and matrices Y and ,Q
then the vector estimate ˆ
ia is determined as
T 1 Tˆ [( ) ( )] ( )i ia Y Q Y Q Y Q y−= + + + . (73)
When using a weighted least squares method, in which the weighting coefficients
are specified by the elements of the matrix D (diagonal, dimension ),K formula (73)
takes the form
T 1 Tˆ [( ) ( )] ( ) .i ia Y Q D Y Q Y Q D y−= + + + (74)
Both these methods are most effective when the matrix Q is formed from per-
manently exciting input impulses [30]. Regarding the problem under consideration, they
are defined as follows. Let us have a sequence of input impulses ,Q presented in the
form
T
1 1 1[ (0) (0) (1) (1) ( 1) ( 1)].n n nQ q q q q q N q N = − −
A sequence Q is considered permanently exciting if
T( ) .rank Q Q nN = (75)
When the signal Q is stationary zero-mean white noise, the sequence ( )Q j has
the statistical property
T T T 2
( ) 0 0
0 0( 1)
[ ( ) ( 1) ( 1)]
0 0( 1)
n
n
n
n
Q j I
IQ j
E Q j Q j Q j k
IQ j k
+
+ + − =
+ −
,
where
T
1( ) [ ( ) ( )] ,nQ j q j q j = E is the mathematical expectation of a stationary
random sequence, and
2
n is the variance.
26 ISSN 2786-6491
Regularized solution. Under certain conditions, the identification problem may be sig-
nificantly incorrectly formulated. This happens when n is large and the information matrix
Y Q + is ill-conditioned. This leads to the solution becoming sensitive to errors in the in-
put data. In such cases, it is advisable to use additional information about the desired solution
and, on its basis, introduce a regularization procedure into the solution algorithm, which
makes it possible to find stable solutions with respect to error variations. Additional infor-
mation about the values of the adjacency matrix, that is, about the connections between the
nodes of the CM, can be very effective. The absence of certain connections means that in the
matrix A coefficients ija are equal to zero. This leads to a decrease in the dimension of the
vector ia and the dimension of the information matrices in (73). If this is not enough to obtain
a stable solution and in the absence of other additional information, one can use Tikhonov’s
regularization method [31] as applied to solving ill-conditioned systems of linear algebraic
equations with inaccurately specified right-hand side and the main information matrix.
A stabilizer is introduced:
2
2
( ) ,i ia a = (76)
in which is the Euclidean norm. In accordance with [31], the smoothing functional
is considered
22
2 2
( , , ) ( ) ) .a
i i i i iM a Y Y Q Y Q a Y a + = + − + (77)
A decreasing sequence { }j is suggested, for example, geometrically decreasing.
After this, a sequence of problems is solved for these .j The regularized solution will
be at the smallest value ,j which is found from the residual principle. According to
this principle, the minimum value j is determined from the inequality
12 2 2
( ) ( ) )i iY Q a Y e + − + (78)
if for 1j− expression (78) is not satisfied. The element ˆ
ia that provides the mini-
mum (77) will be the regularized value of the parametric identification problem, that is,
a solution robust to errors in the data.
7.2. Identification in CM in the impulse processes mode with incomplete meas-
urement of node coordinates [32]. It is assumed that some of the measured nodes of the
CM can be directly influenced by the decision maker by applying external impulses (test
signals or controls) to them. Then the equation of forced motion (4) is written in the form
( 1) ( ) ( ).y k A y k Bu k + = + (79)
Let the vector u have dimension ,r then the matrix ( )B n r is composed of ze-
ros and ones:
0
rI
B
=
, where rI is the unit matrix of dimension .r
It is assumed that some of the coordinates iy of the nodes are measurable. Then
the measured subvector z (dimension )p can be described in the form of a measure-
ment equation
( ) ( ) ( ),z k C y k k = + (80)
where the matrix ( )C p n is obviously composed of zeros and ones [ 0],pC I= where
pI is the unit matrix of size .p
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 27
Thus, equations (79), (80) is a system of equations for a discrete dynamic system in
state space.
Identification of CM using the 4SID method. To identify systems in the sub-
space state space, the so-called 4SID methods [30] are most often used — subspace
state space system identification methods. Let us adapt these methods to the problem at
hand.
For identification the inputs and outputs of the system are observed over a time
interval 1 2N N+ of sampling periods. The values 1,N 2N are chosen so that
2 1( ).N N r p + Then, Hankel matrices are composed of input and output values on the
considered interval:
1
2
2
1,
1 1 2 1
(0) (1) ( 1)
(1) (2) ( )
( 1) ( ) ( 2)
N
u u u N
u u u N
U
u N u N u N N
−
=
− + −
, (81)
1
2
2
1,
1 1 2 1
(0) (1) ( 1)
(1) (2) ( )
( 1) ( ) ( 2)
N
z z z N
z z z N
Y
z N z N z N N
−
=
− + −
. (82)
Test signals u are selected so that the matrix
11,NU is full-rank and well-
conditioned. It is also advisable to choose 2N so that the composite matrix of inputs
and outputs
1
1
1,
1,
N
N
U
Y
does not differ much from the square one.
Quite important is the question of the nature of the input (test) influences. It is
known that informative signals of infinite order are signals such as white noise
or -functions. Therefore, white noise is used as input testing signals.
An RQ-decomposition of a composite matrix
1
1
1,
1,
N
N
U
Y
is carried out with a block
representation of the decomposition in the form
1
1
1, 111
21 221, 2
0N
N
U QR
R RY Q
=
.
For the block 22 ,R its SVD (Singular Value Decomposition) is calculated, that is
T
22 ,R U V= (83)
where is a square diagonal matrix consisting of singular numbers 22R on the main
diagonal.
The decomposition matrix (83) is represented in the following block form
28 ISSN 2786-6491
T
1 1
22 1 2 T
2 2
0
[ ]
0
V
R U U
V
=
where the matrix 1U has dimension 1 ,N p n and matrix 2U — 1 1( ),N p N p n − mat-
rix 1 — ( ),n n 2 — 1 1( ) ( ).N p n N p n− −
According to the realizations theory, some realization ( , , )A B C is found for
system (79), (80), without immediately trying to find the original system
( , , ).A B C In this case, the matrix A is found from an overdetermined system of
matrix equations
(1) (2)
1 1 ,U A U = (84)
where matrix
(1)
1U is a submatrix of the matrix 1U in which the last p rows are crossed
out, and matrix
(2)
1U is a submatrix of the matrix nU in which the first p rows are
crossed out. When obtaining (84), the property of shift invariance is used. In this case,
the matrix 1U is considered as an observability matrix for some realization, which fol-
lows from the realizations theory. Then the matrix C for the same realization will be a
matrix consisting of the first p rows of the matrix 1.U
To find ,B an auxiliary matrix 1
2 21 11
TU R R− = is introduced (test signals must be
selected so that 11R is invertible) and the following equation is solved
( )
( )
( )
T T T
2 2 2 2 2
T T
2 2
T
2 1 1
(1)
1
1 1
(1: ) ( 1: 2 ) ( 1) 1:
( 1: 2 ) (2 1: 3) 0
( 1) 1: 0 0
(1: )
( 1: 2 )0 0
,
0
( 1) 1:
p
U p U p p U p N pN
U p p U p
U p N pN
r
r rI
U B
r N rN
+ − +
+ +
− +
+
=
− +
where for all matrices ( : )D a b denote the submatrix of the matrix D from the a -th to
the b -th columns. Thus, the system has been identified. However, for the practical use
of the results, it is necessary to go to the original realization, that is, the one in which the
matrices ,B C have the known form
0
rI
B
=
, [ 0].pC I=
To do this, equations from the realizations theory are used, which relate different
realizations to each other using a non-singular transformation matrix :T
1A T A T− = , (85)
,B TB = .C C T= (86)
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 29
To find the matrix T connecting the original realization ( , , )A B C and the result-
ing realization ( , , ),A B C it is necessary to solve the system of equations (86), in
which the matrix T (dimension ( )n n ) is unknown. The system will be underdeter-
mined if the number of unknowns is greater than the number of equations, that is if
2 ,n n p + which is almost always true. Thus, it is enough to find any of the set of its
solutions to obtain ,T and then find A from (85) in the original realization.
To assess the quality of identification of coefficients of the adjacency matrix CM,
comparison of the original and resulting matrices A is not always sensible, since the
4SID method ensures equivalence of the original and identified systems by output, that
is, to any identical input influence both models provide the same response; but the
method does not guarantee the identity of the realizations of the state matrices. There-
fore, it is advisable to compare the original and identified systems on the basis of some
invariants that do not depend on the specific realization.
From (85) it is obvious that the most natural invariants are the eigenvalues, which
are identical for the pair ,A .A
The works [33, 34] consider the problem of structural-parametric identification of a
complex multidimensional discrete system in the class of state-space models. It is as-
sumed that only the input and output coordinates of the system over a certain time inter-
val and the measurement error range are known. The selected subspace method is used
as a basis, which assumes that the dimension of the system (state vector) is known.
However, this is not always true in practice, especially when developing cognitive maps
of complex systems. In addition, due to the dependence on the noise level, it is impossi-
ble to identify correctly a high-dimensional system. Therefore, in [33] it was proposed
to consider dimension as a regularizing parameter. Three methods have been developed
for selecting the approximate dimension of the model depending on the length of the
observation interval and the possibility of an active experiment. The proposed methods
are developed using the example of problems of identifying a cognitive map of a com-
mercial bank in an impulse process.
Conclusion
The problem of controlling the dynamics of complex systems of various nature
based on methods of automatic control theory developed for technical objects is a rele-
vant problem. There are works where cognitive maps are used as models of complex
systems, and control is implemented using fuzzy controllers.
This article provides a review and generalization of the principles and main prob-
lems of a new direction in the control of complex systems based on modified mathemat-
ical models of impulse processes in cognitive maps (Roberts difference equations),
which were developed to describe the dynamics of the free motion of complex systems
in discrete time.
The novelty of this direction lies in the modernization of models of CM impulse
processes by introducing a control vector based on the external control actions, which
are generated by a discrete controller and implemented by varying the coordinate re-
sources of the nodes or changing the weighting coefficients of the CM in closed-loop
control systems. The article discusses the main problems of designing discrete control-
lers for control of complex systems: stabilization of unstable processes, coordinating
control of the ratios of the CM nodes coordinates, suppression of external disturbances
with unknown probabilistic characteristics affecting the coordinates of a complex sys-
tem, control of complex systems with multirate sampling of various output controlled
variables, identification CM weighting coefficients. In this case, these problems are
solved for two options, namely with fully measured coordinates of the CM nodes and
30 ISSN 2786-6491
with incomplete measurement. To design discrete controllers, well-known methods of
control theory are used: modal control, linear-quadratic control, linear matrix inequali-
ties technique, etc.
The main difficulty in implementing control systems for complex systems based on
models of CM impulse processes is the organization of the execution of control actions
associated with active intervention of the decision maker into different processes of a
complex system, by means of varying the resources of the CM nodes coordinates or by
changing the degree of effect of some CM nodes on others at discrete time moments.
В.Д. Романенко, В.Ф. Губарев, Ю.Л. Мілявський
ЕТАПИ ТА ОСНОВНІ ЗАДАЧІ
СТОЛІТНЬОГО РОЗВИТКУ ТЕОРІЇ СИСТЕМ
КЕРУВАННЯ ТА ІДЕНТИФІКАЦІЇ.
Частина 5. ПРИНЦИПИ І ЗАДАЧІ КЕРУВАННЯ
ТА ІДЕНТИФІКАЦІЇ В СКЛАДНИХ СИСТЕМАХ
РІЗНОЇ ПРИРОДИ НА ОСНОВІ МОДЕЛЕЙ
ІМПУЛЬСНИХ ПРОЦЕСІВ КОГНІТИВНИХ КАРТ
Романенко Віктор Демидович
Національний технічний університет України «Київський політехнічний інститут
імені Ігоря Сікорського», Навчально-науковий інститут прикладного системного
аналізу, м. Київ,
ipsa@kpi.ua; romanenko.viktorroman@gmail.com
Губарев Вячеслав Федорович
Інститут космічних досліджень НАН України та ДКА України, м. Київ,
v.f.gubarev@gmail.com
Мілявський Юрій Леонідович
Національний технічний університет України «Київський політехнічний інститут
імені Ігоря Сікорського», Навчально-науковий інститут прикладного системного
аналізу, м. Київ,
yuriy.milyavsky@gmail.com
Виконано огляд і узагальнення принципів, методів та задач проєкту-
вання дискретних регуляторів для керування складними системами різ-
ної природи, динаміка яких описана за допомогою різницевих рівнянь
імпульсних процесів когнітивних карт (рівнянь Робертса). В ці рівнян-
ня введено вектор керування, який реалізується на основі варіювання
координат вершин або вагових коефіцієнтів когнітивних карт (КК),
сформованого дискретними регуляторами, які проєктуються на основі
відомих методів теорії керування. В статті наведено розв’язок наступ-
них задач керування імпульсними процесами в КК складних систем:
стабілізація нестійких імпульсних процесів в КК складних систем на
основі еталонних моделей замкнених систем керування та на основі ме-
тоду модального керування; керування імпульсним процесом КК на ос-
нові варіювання ваговими коефіцієнтами; реалізація координувального
управління співвідношеннями координат вершин КК складних систем;
приглушення зовнішніх збурень при керуванні складними системами
на основі методу інваріантних еліпсоїдів; керування імпульсними про-
цесами в КК складних систем з різнотемповою дискретизацією коорди-
нат вершин КК; ідентифікація вагових коефіцієнтів матриці суміжності
mailto:ipsa@kpi.ua
mailto:romanenko.viktorroman@gmail.com
mailto:v.f.gubarev@gmail.com
mailto:Yuriy.milyavsky@gmail.com
Міжнародний науково-технічний журнал
Проблеми керування та інформатики, 2024, № 3 31
в моделі імпульсних процесів КК при повному і неповному вимірюван-
ні координат вершин КК. Розв’язування наведених задач виконується
на основі описаних нових принципів керування імпульсними процесами в
КК складних систем на основі методів теорії автоматичного керування.
Ключові слова: когнітивні карти, імпульсні процеси, еталонні моделі, ін-
варіантні еліпсоїди, ідентифікація, координуюче керування, різнотемпова
дискретизація.
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Submitted 07.05.2024
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https://www.dl.begellhouse.com/ru/journals/2b6239406278e43e.html
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https://doi.org/10.20535/SRIT.2308-8893.2022.3.03
http://dx.doi.org/10.1007/978-3-319-40673-2_19
https://doi.org/10.1080/00207179208934363
https://doi.org/10.1007/s10559-019-00198-5
|
| id | nasplib_isofts_kiev_ua-123456789-211155 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0572-2691 |
| language | English |
| last_indexed | 2026-03-14T13:59:30Z |
| publishDate | 2024 |
| publisher | Інститут кібернетики ім. В.М. Глушкова НАН України |
| record_format | dspace |
| spelling | Romanenko, V. Gubarev, V. Miliavskyi, Yu. 2025-12-25T08:52:45Z 2024 Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models / V. Romanenko, V. Gubarev, Yu. Miliavskyi // Проблеми керування та інформатики. — 2024. — № 3. — С. 5–21. — Бібліогр.: 12 назв. — англ. 0572-2691 https://nasplib.isofts.kiev.ua/handle/123456789/211155 62-50 10.34229/1028-0979-2024-3-1 The article provides a review and generalization of the principles, methods, and problems of designing discrete controllers for controlling complex systems of various nature, whose dynamics are described using difference equations of cognitive maps (CM) impulse processes (Roberts equations). An external control vector is implemented by discrete controllers based on varying node coordinates or weight coefficients of CM; the controllers are designed based on well-known methods of control theory. The article provides solutions to the following problems: stabilization of unstable impulse processes in CM of complex systems, control by varying weight coefficients, coordinating control of ratios between node coordinates, suppression of external disturbances, modeling impulse processes with multirate sampling of node coordinates, and identification of weight coefficients in CM impulse process models with complete and incomplete data. У статті наведено огляд та узагальнення принципів, методів і проблем проектування дискретних регуляторів для керування складними системами різної природи, динаміка яких описується рівняннями різниць когнітивних карт (імпульсні процеси, рівняння Робертса). Зовнішній вектор керування реалізується дискретними регуляторами шляхом варіювання координат вузлів або вагових коефіцієнтів когнітивних карт; регулятори проектуються на основі відомих методів теорії керування. Розглянуто задачі стабілізації нестійких імпульсних процесів у когнітивних картах складних систем, керування процесами за допомогою варіювання вагових коефіцієнтів, а також координаційне керування співвідношеннями між координатами вузлів. Запропоновано методи придушення зовнішніх збурень, моделювання імпульсних процесів із багаторівневим відбором координат вузлів та ідентифікації вагових коефіцієнтів у моделях імпульсних процесів когнітивних карт за повними та неповними даними. en Інститут кібернетики ім. В.М. Глушкова НАН України Проблеми керування та інформатики Адаптивне керування та методи ідентифікації Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models Етапи та основні проблеми столітнього розвитку теорії керування та ідентифікації систем. Частина 5. Принципи та проблеми керування і ідентифікації складних систем різної природи на основі моделей імпульсних процесів когнітивних карт Article published earlier |
| spellingShingle | Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models Romanenko, V. Gubarev, V. Miliavskyi, Yu. Адаптивне керування та методи ідентифікації |
| title | Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models |
| title_alt | Етапи та основні проблеми столітнього розвитку теорії керування та ідентифікації систем. Частина 5. Принципи та проблеми керування і ідентифікації складних систем різної природи на основі моделей імпульсних процесів когнітивних карт |
| title_full | Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models |
| title_fullStr | Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models |
| title_full_unstemmed | Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models |
| title_short | Stages and Main Problems of the Century-Long Control Theory and System Identification Development. Part 5. Principles and Problems in Control and Identification of Complex Systems of Various Nature Based on Cognitive Maps Impulse Processes Models |
| title_sort | stages and main problems of the century-long control theory and system identification development. part 5. principles and problems in control and identification of complex systems of various nature based on cognitive maps impulse processes models |
| topic | Адаптивне керування та методи ідентифікації |
| topic_facet | Адаптивне керування та методи ідентифікації |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211155 |
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