=Paper=
{{Paper
|id=Vol-3909/Paper_23.pdf
|storemode=property
|title=System-Information Models of Digital Twins with Elements of Controlled Self-Organization
|pdfUrl=https://ceur-ws.org/Vol-3909/Paper_23.pdf
|volume=Vol-3909
|authors=Mykola Korablyov,Sergey Lutskyy,Anatolii Voronin,Ihor Ivanisenko
|dblpUrl=https://dblp.org/rec/conf/iti2/KorablyovLVI24
}}
==System-Information Models of Digital Twins with Elements of Controlled Self-Organization==
https://ceur-ws.org/Vol-3909/Paper_23.pdf
System-information models of digital twins with
elements of controlled self-organization ⋆
Mykola Korablyov1, , Sergey Lutskyy1 , Anatolii Voronin2, , and Ihor Ivanisenko3,4,
1
Kharkiv National University of Radio Electronics, Kharkiv 61166, Ukraine
2
Simon Kuznets Kharkiv National University of Economics, Kharkiv 61166, Ukraine
3
University of Jyväskylä, Jyväskylä 40014, Finland
4
Kharkiv National Automobile and Highway University, Kharkiv 61002, Ukraine
Abstract
The article considers an approach to modeling the processes of controlled self-organization of a digital twin
based on a single system-information space. Information processes are the basic basis for the manifestation
of self-organization of systems. The structure of energy and material connections reflects the properties of
objects of the external world and the internal environment of a self-organizing system. These connections
with significant structural features are information connections. The tasks of controllability of self-
organization of a digital twin are the tasks of dynamics with the definition of the system-information
structure of an object with information stability. A dynamic system based on system-information models
of a digital twin, in addition to the classical one, considers the uncertainty of parameters as an indicator
that affects information stability, optimality, and the degree of self-organization. System-information
criteria for solving problems of the dynamics of a digital twin with elements of controlled self-organization
allow for the choice of structures of systems with information stability that satisfy the conditions of self-
organization. The controllability of self-organization of a digital twin is achieved by selecting the function
of informational coordination of uncertainty of system elements, ensuring the emergence of stable
structures of connections in a dynamic system. The article considers the issues of analysis of elements of
controllability of self-organization, from the position of the system-information approach, models of system
information of digital twin data with elements of controlled self-organization, as well as examples of solving
problems of the dynamics of the process of self-organization of a digital twin.
Keywords
digital twin, controlled self-organization, system information, system-information models1
1. Introduction
Manufacturing technologies in their development have gone through stages of evolution starting
from elementary manual labor, mechanized, automated, automatic, digital, and further to the self-
organization of unmanned production. The development of the concept of a "digital twin" with
elements of self-organization is an urgent need of our time to create unmanned production outside
of human habitation. Self-organization of production is designed to help find non-standard solutions
to problems faster, predict their results more accurately, and ensure the production of high-quality
products. A digital twin with elements of self-organization is a digital copy of a process, system, or
asset that expands the capabilities for transforming new, previously unforeseen solutions to achieve
set goals [1-4].
Information Technology and Implementation (IT&I-2024), November 20-21, 2024, Kyiv, Ukraine
Corresponding author.
These authors contributed equally.
mykola.korablyov@nure.ua (M. Korablyov); lutsk.sv6@gmail.com (S. Lutskyy); voronin61@ukr.net (A. Voronin);
ihor.i.ivanisenko@jvu.fi (I. Ivanisenko)
0000-0002-8931-4350 (M. Korablyov); 0000-0002-5327-6521 (S. Lutskyy); 0000-0003-2570-0508 (A. Voronin); 0000-0002-
2679-959X (I. Ivanisenko)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
285
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
The digital twin as an information system is characterized by classical problems of the general
theory of systems, which require the development of principles for formalizing system problems and
solving them based on system-information models. This is theoretically based on the concept of the
term system information of objects. In production, digital twins with elements of self-organization
can be created for production technologies, specific production lines, the final product, or any other
object within the production process [5].
The term self-organization is not new. There is a special discipline in science "synergetic" [6],
which aims to find, together with other sciences, the principles of self-organization, according to
which the processes of formation of order from chaos are realized in the universe. The main concepts
of synergetic are the bifurcation point, attractor, dissipative processes, and fractals. Hermann Haken
gave the following definition of self-organization within the framework of synergetic [6]: "Self-
organization is a process of ordering (spatial, temporal, or spatio-temporal) in an open system, due
to the coordinated interaction of many elements of its components." However, in recent years, this
concept has increasingly and persistently appeared in a variety of scientific applications. Self-
organization occurs in many physical, chemical, biological, robotic, mathematical, virtual, and
cognitive systems and is an information process. Self-organization is associated with the concept of
emergence [7].
The concept of controlled self-organization began to form in 2008. This approach aims to regulate
self-organization for specific purposes so that a dynamic system can achieve certain attractors or
results [7]. Concerning self-organization of production, the system is characterized by regulated self-
organization of the production and technological process, the operation of equipment and its
software, and has the properties of self-healing, self-adjustment, self-management, self-adaptation
of production systems, etc.
The above characteristics of real production are provided at the virtual level by a digital twin with
elements of self-organization based on a single system information space [8]. System tasks of digital
twins based on system-information models relate to methods of managing the uncertainty of
parameters of processes and systems of real production. The solution of system tasks of a digital
twin in a single information space is implemented based on system-information models taking into
account the presence of uncertainty. The systemic tasks of a digital twin are questions of systems
theory: analysis, synthesis, identification, observability, forecasting, evaluation, solvability, control,
stability, dynamics, optimization, and others.
The single system-information space is based on the methodology of system-information
modeling [9, 10] of processes and systems. System-information modeling complements modern
information theory by introducing a new concept and its mathematical justification system
information. System information is characterized by a quantitative indicator of the communication
ability of an object to exchange information with the environment [11, 12]. It takes into account the
presence of uncertainty in the parameters of the state of objects and their information coordination
in the system based on the formulated system-information laws. In the process of exchanging system
information, objects change their state by an amount multiple of the sensitivity threshold to the
influencing object.
The basic foundation of system-information models are:
1. Formulated information laws for the transformation of system information of physical
quantities (parameters).
2. Modeling of the information measure and norm of system information of a physical quantity
(parameter).
3. Use of Planck units of physical quantities in models, which ensures their high accuracy.
4. Modeling of objects in a single system-information space, which ensures their information
virtual interaction.
286
The mathematical apparatus of system-information modeling adequately reflects the production
processes of both real production and its virtual copy - a digital twin. Analysis of methods for solving
production problems based on system-information models showed additional possibilities for using
intelligent information processing in virtual systems of digital twins. These are based on the
developed algorithms of system information based on the information norm and measure [12].
The use of system-information models in digital twins with elements of self-organization allows
solving the problems of controlled self-organization at the virtual level with the transfer of solutions
to real production. Therefore, developing a methodology for creating digital twins with elements of
controlled self-organization is a pressing task.
2. Analysis of elements of controlled self-organization from the
position of the system-information approach
General principles of self-organization of systems follow from the regularities of hierarchical
systems. These principles are decisive for the functioning of living and non-living organisms, and
the strict evolutionary nature of their development. The concept of controlled self-organization aims
to regulate self-organization for specific purposes so that a dynamic system can achieve certain
attractors or results. Regulation limits the self-organizing process within a complex system, at the
level of local interactions between system components, and does not follow an explicit control
mechanism or a global design plan [13, 14]. Desired results, such as an increase in the resulting
internal structure or functionality, are achieved by combining task-independent global goals that are
task-dependent with constraints on local interactions.
An important ability of a self-organizing system is the ability to change under the influence of
certain factors of the external world that are significant for this self-organizing system [7]. This is
the essence of the manifestation of the sensitivity threshold in the processes of self-organization of
the system. The process of self-organization of the system is dynamic, and the sensitivity threshold
has a stochastic nature. By controlling the value of the sensitivity threshold of the system elements,
the principle of controllability of the self-organization of the system is achieved.
The information system is the basic foundation for the manifestation of the principles of self-
organization of systems. The distinctive property of the information system is that the structure of
energy and material connections is essential for it, which is a reflection of the properties of objects
of the external world and the internal environment of the self-organizing system. These connections
with essential structural features are information connections. The elements of the structure of the
information system are physical quantities of the SI system - parameters.
The basis of controlled self-organization is the system-information principles of reflection and
the laws of transformation of system information [11, 12], which determine the types of algorithms
for managing self-organization. Let's consider the system-information laws.
Axiom. A change in the state of an object occurs as a result of external (internal) influence, starting
from the threshold of the object's sensitivity to this influence.
1. The law of identical reflection of the properties of objects 𝑋𝑖 , 𝑌𝑗 :
𝐼(𝑋𝑖 ) = 𝐼(𝑌𝑗 ) . (1)
2. The law of coordination of the uncertainty of the properties of objects 𝑋𝑖 , 𝑌𝑗 of the reflection
process of the reflection process:
µ µ
𝑙𝑜𝑔2 𝑖 = 𝑙𝑜𝑔2 𝑗 ;
𝑈𝑖 𝑈𝑗
X Y
(2)
𝑙𝑜𝑔2 𝛥𝑥𝑖 = 𝑙𝑜𝑔2 𝛥𝑥𝑖 .
𝑖 𝑗
287
where µ𝑖 , µ𝑗 is the mathematical expectation, 𝑈𝑖 , 𝑈𝑗 is the expanded uncertainty, X 𝑖 , Y𝑖 is the
properties of objects, 𝛥𝑥, 𝛥𝑦 is the sensitivity threshold.
3. The law of systemic properties of stationary, equilibrium reflection space:
𝑛
∑ 𝐼𝑆𝑖 (𝑡) = 𝑐𝑜𝑛𝑠𝑡 , (3)
𝑖=1
where n is the number of system elements, 𝑠𝑖 is the element of system S, 𝐼𝑆𝑖 is the amount of
system information.
4. Additivity of system properties of the reflection space (consequence):
𝑡
𝐼𝑌𝑗 = ∑ 𝐼𝑋𝑖 , 𝑡 = ̅̅̅̅̅
1, 𝑛 ,
(4)
𝑖=1
∑𝑡𝑖=1 𝐼𝑖
µ𝑗 = 𝑈𝑗 ·𝑛 ,
where n is the base of the logarithm, U is the expanded uncertainty, µ is the mathematical
expectation, and 𝐼𝑖 is the system information of the i-th property.
Based on the presented system-information laws, information structures of a digital twin with
controlled self-organization are formed.
3. Models of system information of a digital twin with controlled self-
organization
System-information modeling of a single information space of a digital twin is based on the principles
of superposition. The sum of the system's reactions to individual disturbances from identical initial
states is equal to the reaction of this system to the total impact from the same initial state. The
elementary reaction of the system to an external disturbance is the threshold of the system's
sensitivity to the impact.
One of the solutions to information system problems is to assess the uncertainty value of the
parameters of a technical (closed) system based on a system-information model of the dynamic
processes of a digital twin, which characterizes the stability of the system and the optimization of its
state.
The object of this study is a stochastic system a set of elements X, which are in information
links with each other and form a certain integrity and unity. System information, possessing
elements of set X, is characterized by the interval of the upper X 𝑚𝑎𝑥 and lower X 𝑚𝑖𝑛 limits of its
manifestation, as well as the uncertainty 2𝑈𝑥 = µ𝑥 − X 𝑚𝑖𝑛 ⁄𝑛, where µ𝑥 = 𝑛𝑈 + X 𝑚𝑖𝑛 discrete
variable value on the interval X 𝑚𝑎𝑥 − X 𝑚𝑖𝑛 , 2𝑈𝑥 uncertainty/sensitivity threshold.
The structure of a single system-information space of a digital twin with elements of controlled
self-organization consists of three main blocks [4]. The central block is a fragment of the reality of a
set of properties of an object, the so-called intensity of properties. The intensity of properties is
characterized by the maximum number of perceived properties and their values max and min. The
modeled factor is the scale of the intensity of properties, which is characterized by the sensitivity
threshold.
The other two blocks are the duration block (time) and the extension block (length). These blocks
are characterized by the extreme boundaries of duration and extension, the sensitivity thresholds of
duration and extension, the scales of duration variability, and extension heterogeneity. The
combination of intensity, duration, and extension block elements in various configurations forms
layers of reality fragments.
The source of the content of system information of the probable physical quantity xi is the
mathematical expectation of the manifestation of the properties of processes and systems
288
𝑛
𝜇 = ∑ 𝑥𝑖 𝑝𝑖 . (5)
𝑖=1
The expression for calculating the amount of system information of the assumed physical
quantity (PQ) is determined based on the logarithmic indicator of communication capacity:
𝜇 ∑𝑛𝑖=1 𝑥𝑖 𝑝𝑖 .
𝐼 = 𝑙𝑜𝑔2 = 𝑙𝑜𝑔2 , (6)
U K√𝐷
where D is the variance, U is the expanded uncertainty, K is the coverage factor, µ is the
mathematical expectation, p is the probability of the event.
In the algorithms of dynamic processes of digital twins with controlled self-organization, models
of objects with an information measure and norm are used [15, 16].
The information measure characterizes the system information of an object |I(X)|, which is a
function of the absolute value of the qualitative or quantitative proportion of the relationship. The
information measure |(X)| is a function of the ratio of the value of the interval of the object's feature
to its sensitivity threshold:
𝑋𝑚𝑎𝑥 − 𝑋𝑚𝑖𝑛
|𝐼(𝑋)| = 𝑓 ( ). (7)
2𝑈𝑥
The amount of system information of the first kind will be calculated based on the information
measure:
𝑋𝑚𝑎𝑥 − 𝑋𝑚𝑖𝑛
𝑙𝑜𝑔2 |𝐼(𝑋)| = 𝑙𝑜𝑔2 ( ), (8)
2𝑈𝑥
where 𝑋𝑚𝑎𝑥 , 𝑋𝑚𝑖𝑛 are the upper and lower boundaries of the interval of values of the object's
properties, 2𝑛𝑈𝑥 – uncertainty/sensitivity threshold.
The information norm ‖(𝑋)‖ is a function of the ratio of the interval value of an object's feature
to its particular variable value:
𝑋𝑚𝑎𝑥 − 𝑋𝑚𝑖𝑛 𝑋𝑚𝑎𝑥 − 𝑋𝑚𝑖𝑛
‖(𝑋)‖ = 𝑓 ( )=( ). (9)
µ𝑥 2𝑛𝑈𝑥 + 𝑋𝑚𝑖𝑛
The amount of system information of the second kind will be calculated based on the information
norm:
𝑋𝑚𝑎𝑥 − 𝑋𝑚𝑖𝑛 𝑋𝑚𝑎𝑥 − 𝑋𝑚𝑖𝑛
𝑙𝑜𝑔2 ‖(𝑋)‖ = 𝑙𝑜𝑔2 ( ) = 𝑙𝑜𝑔2 ( ), (10)
µ𝑥 2𝑛𝑈𝑥 + 𝑋𝑚𝑖𝑛
where µ𝑥 = 𝑛𝑈 + X 𝑚𝑖𝑛 .
The presented stochastic models of system information of a digital twin in combination with
system-information methods are used in solving problems of controlled self-organization.
4. General approach to solving the problems of the dynamics of the
self-organization process of a digital twin
The solution of systemic problems of a digital twin based on system-information models refers to
methods of managing the self-organization of parameters of processes and systems of real production
[14]. The tasks of managing the self-organization of a digital twin are problems of dynamics with
the definition of the system-information structure of an object with information stability. A dynamic
system based on system-information models of parameters of processes and systems of a digital twin,
in addition to the classical one, considers the uncertainty of parameters as an indicator that affects
the information stability, optimality, and self-organization of the system. System-information
criteria for solving problems of the dynamics of a digital twin with self-organization management
289
allow selection structures with information stability that satisfy the conditions of the problem.
Information stability of information links of system elements is characterized by three states [16]:
• Δx⁄𝑈(𝑥) = 1 sufficient stability;
• Δx⁄𝑈(𝑥) < 1 insufficient stability;
• Δx⁄𝑈(𝑥) > 1 excess stability.
The safety factor of the information stability of the parameter is determined by the expression:
𝐾𝑖𝑠 = Δx⁄𝑈(𝑥) − 1,
where Δx is the parameter sensitivity threshold, 𝑈(𝑥) is the expanded uncertainty of the
parameter.
Example 1.
The conditions of the system-information process have the form:
𝑥𝑖 (𝑡), 𝑦(𝑡), 𝐼𝑌 = ∑𝑁𝑖=1 𝐼𝑋𝑖 , ∆𝑥𝑖 = 𝑈𝑋𝑖 , ∆𝑦 = 𝑈𝑦 ,
at the same time
𝑥1 𝑥 𝑈1 𝑈
= 2, 𝑥1 = , d𝑥1 = 𝑑 ( 1 ) ,
𝑈1 𝑈2 𝑈2 𝑈2
where: 𝑥1, , 𝑥2 – elements of the system, 𝑈1, , 𝑈2, – expanded uncertainty.
The increment of system information I(d𝑥1, )=I(d𝑥2, ) determines the characteristics of the
dynamics of the information connection of the elements 𝑥1 (𝑡) and 𝑥2 (𝑡) depending on the values
𝑈1 , 𝑈2 , which can be a function of time 𝑈𝒊 (𝑥𝑖 ).
The task is set: to formalize the dynamics of the information system [12] based on the uncertainty-
matching function of the elements:
𝑁
𝐼𝑌 (𝑡) = ∑ 𝐼𝑋𝑖 (𝑡) ,
𝑖=1
𝑁 (11)
𝑦(𝑡) 𝑥𝑖 (𝑡)
𝑙𝑜𝑔2 = ∑ 𝑙𝑜𝑔2 .
∆𝑦 𝑖=1 ∆𝑥𝑖
The problem of system information dynamics is solved in several stages.
1. A matrix of information links of elements xi (t) is built (Table 1).
The time variable 𝑥𝑖 (𝑡) can be an argument of the function (𝑥1𝑡 ), then you need to solve the
equation and take positive roots as a separate element 𝑥𝑖 (𝑡).
Table 1
A matrix of Information Links of Elements
N/N 𝑥1 𝑥2 𝑥𝑁
𝑥1 𝐾11 = 𝑥1 ⁄𝑥1 𝐾12 = 𝑥1 ⁄𝑥2 𝐾12 = 𝑥1 ⁄𝑥𝑁
𝑥2 𝐾21 = 𝑥2 ⁄𝑥1 𝐾22 = 𝑥2 ⁄𝑥2 𝐾22 = 𝑥2 ⁄𝑥𝑁
𝑥𝑁 𝐾𝑁1 = 𝑥𝑁 ⁄𝑥1 𝐾𝑁2 = 𝑥𝑁 ⁄𝑥2 𝐾𝑁2 = 𝑥𝑁 ⁄𝑥𝑁
2. We compose a system of equations of dynamics for each pair of elements, such pairs will be
equal to N × N:
∆𝑥𝑖 (𝑘) = 𝑥𝑖 (𝑘 + 1) − 𝑥𝑖 (𝑘) , 𝑖 = 1, … , 𝑁 ,
where: k discrete time, ∆𝑥𝑖 sensitivity threshold;
∆𝑥 ∆𝑥 ∆𝑥 ∆𝑥
𝑎12 = ∆𝑥1 , . . . , 𝑎1𝑁 = ∆𝑥 1 , … , 𝑎𝑁1 = ∆𝑥𝑁 𝑎𝑁𝑁 = ∆𝑥𝑁 ,
2 𝑁 1 𝑁
290
∆𝑥1 𝑥 𝑥1
𝑥1 (0), … , 𝑥𝑁 (0), = 1, ∆𝑥1 = ∆𝑥𝑁 ,
∆𝑥𝑁 𝑥𝑁 𝑥𝑁
𝑥1 (𝑘 + 1) − 𝑥1 (𝑘) = 𝑎11 𝑥1 (𝑘) + 𝑎12 𝑥2 (𝑘) + ⋯ + 𝑎1𝑁 𝑥𝑁 (𝑘),
𝑥 (𝑘 + 1) − 𝑥2 (𝑘) = 𝑎21 𝑥1 (𝑘) + 𝑎22 𝑥2 (𝑘) + ⋯ + 𝑎2𝑁 𝑥𝑁 (𝑘),
{ 2 },
…
𝑥𝑁 (𝑘 + 1) − 𝑥𝑁 (𝑘) = 𝑎𝑁1 𝑥1 (𝑘) + 𝑎𝑁2 𝑥2 (𝑘) + ⋯ + 𝑎𝑁𝑁 𝑥𝑁 (𝑘),
or
𝑥1 (𝑘 + 1) = (1 + 𝑎11 )𝑥1 (𝑘) + 𝑎12 𝑥2 (𝑘) + ⋯ + 𝑎1𝑁 𝑥𝑁 (𝑘),
𝑥2 (𝑘 + 1) = 𝑎21 𝑥1 (𝑘) + (1 + 𝑎22 )𝑥2 (𝑘) + ⋯ + 𝑎2𝑁 𝑥𝑁 (𝑘),
{ },
…
𝑥𝑁 (𝑘 + 1) = 𝑎𝑁1 𝑥1 (𝑘) + 𝑎𝑁2 𝑥2 (𝑘) + ⋯ + (1 + 𝑎𝑁𝑁 )𝑥𝑁 (𝑘),
or finally
𝑥1 (𝑘 + 1) (1 + 𝑎11 ) 𝑎12 … 𝑎1𝑁 𝑥1 (𝑘)
𝑥2 (𝑘 + 1) 𝑎21 (1 + 𝑎22 ) … 𝑎2𝑁 ] [ 𝑥2 (𝑘)] . (12)
[ ]=[ …
… …
𝑥𝑁 (𝑘 + 1) 𝑎𝑁1 𝑎𝑁2 … (1 + 𝑎𝑁𝑁 ) 𝑥𝑁 (𝑘)
The solution of the given equations of the dynamics of the information system (12) relative to the
information stability of the connections of each pair of elements of the system allows us to determine
a set of stable structures of connections of elements of the information system. The choice of the
structure of the information system that satisfies the conditions of the problem is the process of
managing the self-organization of the digital twin.
Example 2.
1 2 of the information system based
on the uncertainty matching function of the elements:
𝑥1 (𝑡) 𝑥2 (𝑡)
𝑑𝑡
= ℓ𝑥2 , 𝑑𝑡
= 𝜑𝑥1 ,
∆𝑥 𝑈 ∆𝑥 𝑈
ℓ = 𝐾12 = ∆𝑥1 = 𝑈1 , 𝜑 = 𝐾12 = ∆𝑥2 = 𝑈2 , (13)
2 2 1 1
∆𝑥1 , ∆𝑥2 = 𝑐𝑜𝑛𝑠𝑡, 𝑚𝑖𝑛 ≤ 𝑥𝑖 ≤ 𝑚𝑎𝑥 ,
where: 𝑥𝑖 variables, ∆𝑥𝑖 sensitivity threshold, 𝑈𝑖 expanded uncertainty, 𝐾𝑖𝑗 information
link coefficient. Then we have:
𝑥1 (𝑛 + 1) = 𝑥1 (𝑛) + ℓ𝑥2 (𝑛),
(14)
𝑥2 (𝑛 + 1) = 𝑥2 (𝑛) + 𝜑𝑥2 (𝑛), 𝑥1 (0), 𝑥2 (0).
We apply the discrete Z-Laplace transform:
∞
𝑥(𝑧) = ∑ 𝑥(𝑛)𝑧 −𝑛 . (15)
𝑛=0
or
𝑥 (𝑧) 𝑧 𝑧−1 ℓ 𝑥 (0)
[ 1 ]= 2 ∙[ ]∙[ 1 ]. (16)
𝑥2 (𝑧) 𝑧 − 2𝑧 + 1 − ℓ𝜑 𝜑 𝑧 − 1 𝑥2 (0)
Let's put 𝜎 2 = 𝑙 ∙ 𝜑 ≥ 0, then we can determine the roots of the equation:
𝑧 2 − 2𝑧 + 1 − 𝜎 2 = 0,
(𝑧 − (1 − 𝜎)) ∙ (𝑧 − (1 + 𝜎)) = 0, (17)
𝑧1 = 1 − 𝜎, 𝑧2 = 1 + 𝜎.
Solution (17) takes the form:
𝑧2 − 𝑧 ℓ𝜑
𝑥1 (𝑧) = 2 ∙ 𝑥1 (0) + 2 ∙ 𝑥 (0) ,
𝑧 − 2𝑧 + 1 − 𝜎 2 𝑧 − 2𝑧 + 1 − 𝜎 2 2 (18)
𝜑𝑧 𝑧2 − 𝑧
𝑥2 (𝑧) = 2 ∙ 𝑥 (0) + ∙ 𝑥 (0) .
𝑧 − 2𝑧 + 1 − 𝜎 2 1 𝑧 2 − 2𝑧 + 1 − 𝜎 2 2
Find the inverse transformation to the temporary variable n:
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𝑧2 − 𝑧 1 𝑧 𝑧
2 2
= ∙( + )⇒
𝑧 − 2𝑧 + 1 − 𝜎 2 𝑧 − (1 − ℓ) 𝑧 − (1 + ℓ)
1 1
⇒ 2 ∙ (1 − 𝜎)𝑛 + 2 ∙ (1 + 𝜎)𝑛 ,
𝑧 𝜎−1 1 𝜎+1 1 (19)
2 2
= + ⇒
𝑧 − 2𝑧 + 1 − 𝜎 2𝜎 𝑧 − (1 − 𝜎) 2𝜎 𝑧 − (1 + 𝜎)
1 1
⇒ ∙ (1 + 𝜎)𝑛 + ∙ (1 − 𝜎)𝑛 .
2𝜎 2𝜎
It follows that
1 ℓ 1 ℓ
𝑥1 (𝑛) = (𝑥1 (0) − 𝑥2 (0)) (1 − 𝜎)𝑛 + (𝑥1 (0) + 𝑥2 (0)) (1 + 𝜎)𝑛 ,
2 𝜎 2 𝜎 (20)
1 𝜑 𝑛
1 𝜑 𝑛
𝑥2 (𝑛) = (𝑥2 (0) − 𝑥1 (0))(1 − 𝜎) + (𝑥2 (0) + 𝑥1 (0)) (1 + 𝜎) .
2 𝜎 2 𝜎
Thus, in the closed information system of the digital twin, the stability of the dynamic process is
ensured by a limitation:
𝑚𝑖𝑛 ≤ 𝑥𝑖 ≤ 𝑚𝑎𝑥 , (21)
and the condition
𝑁 𝑥𝑖
∑ = 𝑐𝑜𝑛𝑠𝑡 . (22)
𝑖=1 ∆𝑥𝑖
The analysis of the dynamics of the self-organization process of the digital twin showed that the
dynamics of the systemic connections of the elements {𝑥2 𝑥1 , 𝑥1 𝑥2 } of the system is self-
developing and represents an evolutionary spiral, and the dynamics of the systemic connections of
the elements {1/𝑥2 𝑥1 , 1/𝑥2 𝑥1 } of the system is damped. In a stable information system of real
production, there are connections between the elements of both dynamics.
5. Examples of system-information analysis of controlled self-
organization processes
Example 3. Let us consider an example of constructing a system-information model of a dynamically
stable process of a digital twin with the definition of the coefficients of the transfer matrix 𝐴𝑁𝑁 and
the function of the sensitivity thresholds ∆𝑥𝑖 = 𝑓(𝑥𝑖 ), ∆𝑥𝑗 = 𝑓(𝑥𝑗 ), ensuring the dynamic stability
of the digital twin.
Given:
x x 𝑥𝑖
𝑙𝑜𝑔2 𝑖 = 𝑙𝑜𝑔2 𝑗 , 𝑖, 𝑗 = ̅̅̅̅̅
𝛥𝑥𝑖 𝛥𝑥𝑗
1, 𝑁 , ∑𝑁
𝑖=1 = 𝑐𝑜𝑛𝑠𝑡 ,
∆𝑥𝑖
x𝑖 x𝑗 𝛥x𝑖 𝛥x
= , x𝑖 = x𝑗 , k 𝑖𝑗 = 𝑖 ,
𝛥𝑥𝑖 𝛥𝑥𝑗 𝛥𝑥𝑗 𝛥𝑥𝑗
x1 (0), x2 (0), … , x𝑛 (0), x1 (𝑡), x2 (𝑡), … , x𝑛 (𝑡) ,
𝛥x𝑖
k 𝑖𝑗 = 𝛥𝑥 , 𝑘𝑖𝑗 =1/𝑘𝑗𝑖 , k 𝑖𝑗 (𝑡) = 𝜑𝑖𝑗 (𝑡 − 𝑡0 ) .
𝑗
Find: 𝜑𝑖𝑗 (𝑡 − 𝑡0 ), 𝛥𝑥𝑖 = 𝑓(𝑥𝑖 ), 𝛥𝑥𝑗 = 𝑓(𝑥𝑗 ).
It is necessary to solve the system of equations
x1 (𝑡) 𝜑11 (𝑡 − 𝑡0 ) 𝜑12 (𝑡 − 𝑡0 ) … 𝜑1𝑁 (𝑡 − 𝑡0 ) x1 (𝑡0 )
x (𝑡) 𝜑 (𝑡 − 𝑡0 ) 𝜑22 (𝑡 − 𝑡0 ) … 𝜑2𝑁 (𝑡 − 𝑡0 ) x2 (𝑡0 )
[ 2 ] = [ 21 ][ ] (23)
… … …
x𝑁 (𝑡) 𝜑11 (𝑡 − 𝑡0 ) 𝜑12 (𝑡 − 𝑡0 ) … 𝜑1𝑁 (𝑡 − 𝑡0 ) x𝑁 (𝑡0 )
to determine the roots 𝜆𝑖 𝑖 = ̅̅̅̅̅
1, 𝑁.
When the conditions are met 𝜆𝑖 < 0, 𝑖 = ̅̅̅̅̅1, 𝑁 the system is dynamically stable.
̅̅̅̅̅
When the conditions are met 𝜆𝑖 ≥ 0, 𝑖 = 1, 𝑁 the system is dynamically unstable.
Let's consider a special case when 𝑁 = 2. Then we have:
292
k k 1
𝐾2⨯2 = [ 11 12 ] , k11 = k 22 = 1, k12 = 𝜑, k 22 = .
k 21 k 22 𝜑
Then
𝑑𝑥1
1 𝜑 1 𝜑 𝑥
𝑑𝑡 1
𝐾2⨯2 = [ 1 ] , [𝑑𝑥 ] = [1 ] · [𝑥 ] ,
𝜑
1 2
𝜑
1 2
𝑑𝑡
where
1−𝜆 𝜑
𝑑𝑒𝑡|𝐴𝑥1 − 𝜆 𝐸𝑥2 | = 0, [ 1 ] = 0,
𝜑
1−𝜆
1
(1 − 𝜆) 2 − 𝜑 𝜑 = 0, (𝜆2 − 2𝜆) = 0.
From here we find the roots of the system: 𝜆1 = 0, 𝜆2 = 2.
By analyzing the obtained roots of system 𝑁2⨯2 , we can conclude that this system is unstable.
Note that this system-information model of the digital twin of order 𝑁2⨯2 represents the
evolutionary spiral of development of a dynamic system. The dynamically stable state of the system
is achieved due to two processes when 𝑥𝑖 𝑥𝑗 and when the stochastic function of the dependence
of the sensitivity threshold on the parameter ∆𝑥𝑁 = 𝑓(𝑥𝑁 ) of the digital twin is determined.
Example 4. Let us now consider an example of determining the information stability of digital
twin processes. The systemic information approach to the analysis of the information stability of
digital twin processes allows the establishment of quantitative links between the assessment of the
accuracy tolerance for the size and the uncertainty of the parameter. The criterion of information
stability is the condition when the expanded uncertainty is equal to or less than the accuracy
tolerance of the parameter
U (𝑥𝑖 ) ≤ IT (𝑥𝑖 ).
y x x x
𝑙𝑜𝑔2 𝛥𝑦 = 𝑙𝑜𝑔2 U(𝑥1 ) + 𝑙𝑜𝑔2 U(𝑥2 ) + … + 𝑙𝑜𝑔2 U(𝑥𝑛 ) ,
U𝑦 1 2 𝑛
y x x x
𝑙𝑜𝑔2 𝛥𝑦 = 𝑙𝑜𝑔2 IT(𝑥1 ) + 𝑙𝑜𝑔2 IT(𝑥2 ) + … + 𝑙𝑜𝑔2 IT(𝑥𝑛 ) , (24)
IT𝑦 1 2 𝑛
y x x x
𝑙𝑜𝑔2 = 𝑙𝑜𝑔2 1 + 𝑙𝑜𝑔2 2 + … + 𝑙𝑜𝑔2 𝑛 ,
𝛥𝑦IT𝑦 IT(𝑥1 ) IT(𝑥2 ) IT(𝑥𝑛 )
y x
= ∏𝑛𝑖=1 𝑖 , 𝛥𝑦IT𝑦 = ∏𝑛𝑖=1 𝐼𝑇(𝑥𝑖 ) , 𝛥𝑦U𝑦 = ∑𝑛𝑖=1 𝑈x𝑖 ,
𝛥𝑦IT𝑦 IT(𝑥𝑖 )
∑𝑛𝑖=1 𝑈x𝑖
𝑘𝑦 = 𝑛 .
∏𝑖=1 IT(𝑥𝑖 )
When the conditions are met 𝑘𝑦 ≤ 1, the system is information stable
When the conditions are met 𝑘𝑦 > 1, the system is information unstable.
The conducted system-information analysis of the controlled self-organization of the digital twin
examines the information stability, which provides the self-organizational structure with the
management of the quality of the parameters of the digital twin. It creates the basis for a logical and
consistent approach to the problem of decision-making from the position of the dynamic and
information stability of the digital twin. The peculiarity of the system-information approach to the
process of controlled self-organization of the digital twin is that the categories of processes of
dynamic and information systems are considered. The effectiveness of solving problems using
system-information analysis is determined by constructing the self-organization structure of the
solved problems of the digital twin. The result of system-information research is the definition of the
structure of the parameters of the digital twin.
6. Conclusions
The concept of modeling digital twins with elements of controlled self-organization based on a single
system-information space is presented. It aims to regulate self-organization for specific purposes,
293
providing a dynamic system with the ability to achieve certain attractors or required results. System-
information regulation limits the self-organizing process inside a complex system at the level of
information functions for coordinating the uncertainty of local interactions between components of
a dynamic system. Desired results, such as increased internal structure or functionality, are achieved
by combining tasks independent of global goals with tasks that restrict local interactions.
The tasks of controllability of self-organization of a digital twin are the tasks of dynamics with
the definition of the system-information structure of an object with information stability. System-
information criteria for solving the tasks of dynamics of a digital twin with elements of controlled
self-organization allow for the choice of structures of systems with information stability that satisfy
the conditions of self-organization.
The results of solving the problems show the dependence of the dynamic system reaching the
attractor point on the information function of coordinating the uncertainty of the system elements.
By choosing the information function of coordinating uncertainties, the self-organization of the
digital twin is controlled to achieve the required goals. In general, the methodology of system-
dynamics problems and allows developing methods for managing the information function of
coordination when solving problems of self-organization of a digital twin, due to parameter
uncertainty in the model.
An example of constructing a system-information model of a dynamically stable process of a
digital twin is given, and the conditions of its dynamic stability are defined. An example of
determining the information stability of processes of a digital twin is also given, and the conditions
of its information stability are defined.
Declaration on Generative AI
The authors have not employed any Generative AI tools
References
[1] A. Sharma, E. Kosasih, J. Zhang, A. Brintrup, A, Calinescu. Digital Twins: State of the Art Theory
and Practice, Challenges, and Open Research Questions. Journal of Industrial Information
Integration. Vol. 30 (2022) 100383. DOI: 10.1016/j.jii.2022.100383.
[2] J. Leng, D. Wang, W. Shen, X. Li, Q. Liu, X. X. Chen. Digital twins-based smart manufacturing
system design in Industry 4.0: A review. Journal of Manufacturing System. Vol. 60 (9) (2021)
119-137. DOI:10.1016/j.jmsy.2021.05.011.
[3] B. Tekinerdogan. On the Notion of Digital Twins: A Modeling Perspective. Systems 11 (2023)
15. DOI: 10.3390/systems11010015.
[4] L. Li, B. Lei, .C Mao. Digital twin in smart manufacturing. Journal of Industrial Information
Integration. Vol. 26 (2022). DOI: 10.1016/j.jii.2021.100289.
[5] S. Lutskyy. System-information approach to uncertainty of process and system parameters.
Innovative technologies and scientific solutions for industries. No.3(17) (2021) 91-106. DOI:
10.30837/ ITSSI.2021.17.091.
[6] H. Haken. Synergetics: Introduction and Advanced Topics. Printing Springer, Berlin, Vol. 1, 3rd
ed. (2004). DOI: 10.1007/978-3-662-10184-1_2.
[7] B. Feltz, M. Crommelinck, and P. Goujon. Self-Organization and Emergence in Life Sciences.
Springer Library, Vol. 331 (2006). DOI: 10.1162/artl.2008.14.2.239.
[8] M. Korablyov, S. Lutskyy. System Models of Single Information Space of Digital Twins. ICICT
2023, Springer, Vol. 3 (2023) 361-371, DOI: 10.1007/978-981-99-3091-3_29.
[9] P.E.D. Love; J. Zhou; J. Matthews; C.-P. Sing, D.J. Edwards. System information modeling in
practice: Analysis of tender documentation quality in a mining mega-project. Automation in
Construction, 84 (2017) 176 183. DOI: 10.1016/j.autcon.2017.08.034.
294
[10] G. Muller. System Modeling and Analysis: Practical Approach. University of South-Eastern
Norway-NISE, 120 (2023) 128.
[11] M. Korablyov, S. Lutskyy, I. Ivanisenko, O. Fomichov. System-Information Models of Digital
Twins. Intelligent Sustainable Systems. WorldS4 2023. Lecture Notes in Networks and Systems,
Springer, Vol. 812 (2024) 101-109. DOI: 10.1007/978-981-99-8031-4_10.
[12] M. Korablyov, S. Lutskyy, A. Voronin, I. Ivanisenko. System Tasks of Digital Twin in Single
Information Space. Advances in Information and Communication. FICC-2024, Springer, Vol. 921
(2024) 47-56. DOI: 10.1007/978-3-031-54053-0_4.
[13] B. Molnár, A. Benczúr, A. Béleczki. Formal approach to modeling of modern information
systems. International Journal of Information Systems and Project Management, Vol. 4, No. 4,
(2016) 69-89. DOI: 10.12821/ijispm040404.
[14] A. Jeyaraj. Models of Information Technology Use: Meta-Review and Research Directions.
Journal of Computer Information Systems, Vol. 63, Iss. 4 (2023) 809-824. DOI:
10.1080/08874417.2022.2106462.
[15] A. Daneshkhah, T. Bedford, K.J. Wilson. Assessing parameter uncertainty on coupled models
using minimum information methods. Reliability Engineering & System Safety, 125(6) (2013).
DOI: 10.1016/j.ress.2013.05.011.
[16] N. Prakash, D. Prakash. Novel Approaches to Information Systems Design. Publisher:
Engineering Science Reference (2020) 320. ISBN-13: 978-1799829768.
[17] L. Gao, M. Jia, D. Liu. Process Digital Twin and Petrochemical Industry. Journal of Software
Engineering and Applications, Vol.15 No.8 (2022). DOI: 10.4236/jsea.2022.158018.
[18] F. F. Hasan. A Review Study of Information Systems. International Journal of Computer
Applications, 17(18) (2018) 15-19. DOI: 10.5120/ijca2018916307.
[19] M. Pant, L. Nagdeve, G.C. Moona, Harish Kumar. Estimation of Measurement Uncertainty of
Additive Manufacturing Parts to Investigate the Influence of Process Variables. Journal of
Metrology Society of India, 37(4) (2022). DOI: 10.1007/s12647-022-00592-z.
[20] P. Majda, J. Measurement uncertainty of generalized stiffness of machine tools.
Measurement, Vol. 170 (2021) 10869. DOI: 10.1016/j.measurement.2020.108692.
[21] V.V. Bannikov, T.V. Savitskaya. The development of a decision-support information-modeling
system for safety in the chemical industry. Fine Chemical Technologies 14(4) (2019) 59-68. DOI:
10.32362/2410-6593-2019-14-4-59-68.
[22] M. Korablyov, S. Lutskyy, N. Axak, I. Ivanisenko, I. Kobzev. Features of System-Information
Models of the Mechanical Process Based on the Platform (USIS + PLSI) of Digital Twins. Selected
Papers of the X International Scientific Conference "Information Technology and
Implementation" (IT&I). Conference Proceedings, Kyiv, Ukraine, (2023) 36-47. https://ceur-
ws.org/Vol-3624/
[23] M. Korablyov, S. Lutskyy, I. Ivanisenko, O. Fomichov. System-information rationing of digital
twins accuracy. Proceedings of IEEE 13th Annual Computing and
Las Vegas, USA, (2023) 653-660. DOI:
10.1109/CCWC57344.2023.10099319.
[24] Y. Wang, S. H. Zhang, Optimal production and inventory rationing policies with selective-
information sharing and two demand classes. European Journal of Operational Research, Vol.
288 (2021) 394-407. DOI: 10.1016/j.ejor.2020.05.051.
295