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 selforganization 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 selforganization. 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.
Manufacturing technologies in their development have gone through stages of evolution starting from elementary
manual labor, mechanized, automated, automatic, digital, and further to the
selforganization 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 [
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 [
The term self-organization is not new. There is a special discipline in science "synergetic" [
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 [
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 [
The single system-information space is based on the methodology of system-information
modeling [
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.
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 [
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 [
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 [
The information system is the basic foundation for the manifestation of the principles of selforganization 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 [
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 , : 2. The law of coordination of the uncertainty of the properties of objects , of the reflection process of the reflection process:
( ) = ( ) . 2 µ µ ;
= 2 2 X = 2 Y . (1) (2) where µ , µ is the mathematical expectation, , is the expanded uncertainty, X , Y is the properties of objects, ,
is the sensitivity threshold.
3. The law of systemic properties of stationary, equilibrium reflection space: system information.
where n is the number of system elements, is the element of system S, is the amount of 4. Additivity of system properties of the reflection space (consequence): ∑ =1 ( ) =
, =1 = ∑ , = ̅1̅,̅̅̅,
∑ µ = · =1 ,
where n is the base of the logarithm, U is the expanded uncertainty, µ is the mathematical expectation, and is the system information of the i-th property. controlled self-organization are formed.
Based on the presented system-information laws, information structures of a digital twin with 3. Models of system information of a digital twin with controlled selforganization 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 and lower X manifestation, as well as the uncertainty 2 = µ − X ⁄ , where µ = (4)
) . 2 | ( )| = 2 (
− 2 ) ,
The amount of system information of the first kind will be calculated based on the information measure: norm: properties, 2 – uncertainty/sensitivity threshold. to its particular variable value: where ,
are the upper and lower boundaries of the interval of values of the object's The information norm ‖( )‖ is a function of the ratio of the interval value of an object's feature The amount of system information of the second kind will be calculated based on the information 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 [
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:
(5)
(6)
(7)
(8)
(9)
(10)
where µ =
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
[
) ,
) .
−
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 [
excess stability.
The safety factor of the information stability of the parameter is determined by the expression: = Δx⁄ ( ) − 1, where Δx is the parameter sensitivity threshold, ( ) is the expanded uncertainty of the parameter.
Example 1. at the same time The conditions of the system-information process have the form:
( ), ( ), = ∑ =1 , ∆ = , ∆ = , 1 = where: 1,, 2 – elements of the system, 1,, 2, – expanded uncertainty.
The increment of system information I(d 1,)=I(d 2,) determines the characteristics of the dynamics of the information connection of the elements 1 ( ) and 2 ( ) depending on the values 1, 2, which can be a function of time ( ).
The task is set: to formalize the dynamics of the information system [
( ) ,
The time variable ( ) can be an argument of the function ( 1), then you need to solve the equation and take positive roots as a separate element ( ). (11)
2
1 11 = 1⁄ 1 21 = 2⁄ 1 1 = ⁄ 1
We compose a system of equations of dynamics for each pair of elements, such pairs will be equal to N × N: where: k discrete time, ∆
sensitivity threshold; ∆ ( ) = ( + 1) − ( ) ,
= 1, … , , 12 = ∆ 1 , . . . , 1 = ∆∆ 1 , … , 1 = ∆ = ∆∆ , or or finally 1(0),… , (0), ∆ 1 = 1, ∆ ∆ 1 = ∆ ] [ 2…( )] .
1( + 1) = 1( )+ ℓ 2( ), 2( + 1) = 2( )+ 2( ), 1(0), 2(0).
( ) = ∑
( ) − . ∞ =0
[ 12(( ))] = 2 − 2 + 1 − ℓ ∙ [ − 1 −ℓ1] ∙ [ 12((00))] .
Let's put 2 = ∙ ≥ 0, then we can determine the roots of the equation: 2 − 2 + 1 − 2 = 0, ( − (1 − )) ∙ ( − (1 + )) = 0,
1 = 1 − , 2 = 1 + . 2 −
1( ) = 2 − 2 + 1 − 2 ∙ 1(0)+ 2 − 2 + 1 − 2 ∙ 2(0),
2( ) = 2 − 2 + 1 − 2 ∙ 1(0)+ 2 − 2 + 1 − 2 ∙ 2(0).
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. on the uncertainty matching function of the elements: 1 2 of the information system based 1( ) = ℓ 2, 2( )= 1, ∆ 2
2 ℓ = 12 = ∆ 1 = 1 , = 12 = ∆ 2 =
∆ 1 ∆ 1,∆ 2 , ≤ ≤ 21 , , where: variables, ∆ sensitivity threshold, expanded uncertainty, information link coefficient. Then we have: (12) (13) (14) (15) (16) (17) (18) 1 2 1 2
It follows that ensured by a limitation: and the condition 1
− (1− ℓ)+ − (1 + ℓ)) ⇒ 2 − 2 + 1− 2 = 2 − (1− )+ 2 − (1+ ) ⇒ ⇒ 21 ∙ (1+ ) + 21 ∙ (1− ) . Thus, in the closed information system of the digital twin, the stability of the dynamic process is ≤ ≤ , ∑ =1∆ dynamics of the systemic connections of the elements { 2 2} of the system is selfdeveloping and represents an evolutionary spiral, and the dynamics of the systemic connections of the elements {1/ 2 1, 1/ 2
1} 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 selforganization 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 and the function of the sensitivity thresholds ∆ = ( ), ∆ = ( ), ensuring the dynamic stability of the digital twin.
Given:
It is necessary to solve the system of equations x1( ) x ( )
11( − 0) 12( − 0) … 1 ( − 0) x1( 0) [x2…( )] = [ 21( − 0) 22( − 0) … 2 ( − 0)] [x2…( 0)]
… 11( − 0) 12( − 0) … 1 ( − 0) x ( 0) to determine the roots = ̅1̅,̅̅̅.
When the conditions are met < 0, = ̅1̅,̅̅̅ the system is dynamically stable. When the conditions are met ≥ 0, = ̅1̅,̅̅̅ Let's consider a special case when = 2. Then we have: the system is dynamically unstable.
(19)
(20) (21) (22) (23) Then where 1 1] , [ 2] = [ 1 1
1] · [ 12] , 1 1 −
1 − (24) From here we find the roots of the system: 1 = 0, 2 = 2.
By analyzing the obtained roots of system 2⨯2, we can conclude that this system is unstable.
Note that this system-information model of the digital twin of order 2⨯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
and when the stochastic function of the dependence of the sensitivity threshold on the parameter ∆ = ( ) 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
, x x x2
x2 x =1 IT( )
, IT = ∏ =1 ( ) ,
U = ∑ =1 x , = ∑ =1 x .
∏ =1 IT( )
the system is information stable 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.
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, providing a dynamic system with the ability to achieve certain attractors or required results. Systeminformation 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. Systeminformation 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 systemdynamics 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.