The article considers the mathematical and systematic apparatus for describing structures with subsystems and systems with decomposition with the corresponding functional links in the form of operators. In the analysis of literature sources, it is substantiated that the problem of managing complex systems is not fully solved under the influence of structural, informational, psychological threats, and the problem of structuring the system as a basis for the formation of targeted decisions by operational personnel under active threats is relevant in the future. The mathematical and systematic apparatus for describing structures with a set of relevant interconnected components is presented. The functional blocks of the system with the corresponding functions and characteristics such as functional transformations, cascade connection in the technological structure, models of adaptive and multiplicative interaction, systems with feedback and hybrid connections and mathematical operations are presented. A method for risk assessment in management decision-making in hierarchical systems under extreme conditions, taking into account the cognitive components of operators, has been developed.
Systems analysis arose as a result of attempts to apply the methods and tools of systems theory
to solve problems of managing complex hierarchical systems in normal and emergency modes.
With the development of man-made structures, information barriers arose, which formed
complex management tasks [
This led to the creation of information systems (as a tool for improving the validity and efficiency of decision-making management) and information technologies for synthesizing strategies for achieving goals, tactics, and system planning of actions at the facility.
In the process of solving complex problems, there are the following levels of hierarchies [
Books [
Work [
Books [
Works [17, 18, 22] consider the problem of improving the quality of control processes in complex systems with a hierarchical structure, intellectualization of situation control, knowledge features of decision-making, synthesis of robust control strategies, cognitive features of the ACS operator's thinking process, complex models of energy-active facilities management in the face of threats and information attacks.
Monograph and work [
Works [8, 9, 13] consider the problem of cybersecurity based on logical and cognitive methods, information and intelligent technologies for processing data flows and event scenarios in infrastructure, taking into account the level of risks using categorical models for representing the organization of complex man-made industries.
In [10] substantiate information technologies for developing methods to ensure the cognitive stability of operational management processes for personnel at all levels of the hierarchy of complex systems.
Works [11, 16] considers the categorical models of representation of the complex systems structure and their effectiveness in forming strategies in infrastructure in case of internal and external conflicts.
In [12], information technologies for identifying the structure of systems, cognitive methods for assessing situations when factors affect the management process and increasing sustainability are developed.
Works [15, 19, 20] substantiate the influence of the cognitive characteristics of operational personnel in the implementation of management in extreme situations, assessing the level of threats at the terminal intervals of formation and decision-making for targeted management, and risk assessment in case of errors in strategic decisions.
Important in the procedure for finding a way to solve problems is the acquisition of data and a
model for identifying knowledge in an active diagnostic and expert mode, identifying their
logical and cognitive structure [
The next step is to isolate the control object from the environment of the technogenic
system, and define its boundaries, functional and information structure [
The responsible procedure is the object aggregation scheme and the construction of a
mathematical model of the hierarchy of the resource and information components of the system
in the form of a scheme for structuring relations (linkage matrix, graphs, Petri nets) [
At the same time, there are hierarchies of the forward and reverse process [
The next stage is the hierarchy of procedures for streamlining the stages of action planning
and structuring the management system in accordance with the global goal [
The following stages are carried out in accordance with the global goal [
In accordance with the current situation, let’s describe hierarchical structures as an organization for the implementation of targeted tasks, taking into account active attacks and threats and the cognitive component of the operator.
Definition. A subsystem S′ of a system S will be any subset of S′ ⊂ X × Y, and an element of systems will be a set of appropriately connected components by which the systemS S = (S1 … Sn) can be restored.
Definition. Definition. A decomposition of a system S is a set of (S1, S2, … Sn), for which S = (S1 + S2+. . . +Sn) and X = (X1 × X2 ×. . .× Xn), Y = (Y1 × Y2 ×. . .× Yn), are components of the system.
Functional relationships in the system are described in the form of operators.
For two given systems S1 ⊂ X1 × Y1 and S2 ⊂ X2 × Y2, a design operator is introduced defining the structure similarity class, whose representation is as follows: Π 1: ( 1 × 2) × ( 1 × 2) → ( 1 × 1), Π 2: ( 1 × 2) × ( 1 × 2) → ( 2 × 2) (1)
Accordingly then it is possible to carry out an independent decomposition into two subsystems of unrelated type: ⊂ ( 1 × 2) × ( 1 × 2), 1 = Π 1( ) 2 = Π 2( ), Π 1: ( 1, 2, 1, 2) = ( 1 × 1), Π 2: ( 1, 2, 1, 2) → ( 2 × 2). (2) with state spaces, control objects ( ), control systems ( ), information structures ( ), and mode parameters ( ).
The decomposition procedure is the basis for the allocation of functional blocks with appropriate characteristics and functions.
П1 – functional transformations ∶ → that define resource, technological, and measurement transformations, executive team actions, and one-step operations. П2 – cascade connection in the technological unit structure, information and measurement → Y1 →
S1 : X1 → S2 : (У1* ⊕ У*2 ) }SitПi operations
П3 – model of additive and multiplicative interaction in the course of technological and information operations:
Transformation of material and energy resources (units, power units) into a product (
→ ( ) → ); Transformation of data flows into an information resource based on data processing operations { } → ({ / }) → ( ); Transformation of information resource and data flows into a cognitive knowledge on goal-oriented strategies Sens [Sitou ( , )] StratU ⟶
( ) → ( / ) Х1
X2
Z1 (S1 / S2) X
Y1 Y2 (4) (5) (6)
According to the components of the structure, dynamics models are formed.
( , ′) ∈
( , ′) ∈
⇔ ( ) = ( ′)
⇔ ( ) = ( ′)
control operations:
control actions:
with canonical representations of transformations in the form of a chain of information and
→ →
→ →
accordingly, the transformation coordinates of state parameters will be set according to the
: → /
: → /
([ ], ) ∈ −1 ⇔ [ ] = ( ) - determine the transformation of the parameters of the state
spaces of the system S when management actions affect the object and then : ( ) →
+ , ∈ ( + ) - output state.
3.1.3. Autonomy of functional systems that are part of a hierarchical organization
In the informational sense, decision-making in the autonomous functioning of the system is
achieved by introducing feedback, which provides a logical structure for the decision-making
process. The decision-making logic is based on [
Let’s consider some aspects of the systems' functionality.
functionality; ) ( ′, , ) ∈
Consider a system ⊂ ( × 3) × ( 1 × 4) whose feedback loop link includes the element ⊂ × . Accordingly, the condition fulfilled for the system, and the system is defined in ( , , , ∈
⇒ = , ∈ ⊂ × ) × space.
The structure of the feedback control system is represented as follows (Fig. 4).
Let’s define additional properties of feedback systems according to [
A functional system :
→ is mutually unambiguous if a number of conditions are met in terms of structure, goals, strategies: 2. a) the condition of the target functionality ( ): ∃ : ( ) → ⇒ 3. b) the functionality of the systems is determined according to the goals (( , , ) ∈ is ⇒ ( = ′) - unambiguity. 4.
⊂ ( 1, … , ) is functionally controlled if the condition of goals alignment with the structure and strategies is met: ∀ ∈ (∃ ∈ ) ( , ) ∈ → ∃ , A multidimensional system will be autonomous as a result of feedback closure only if the condition of structural and information-management resilience against the impact of complex threat factors (additive and multiplicative models) is met.
Accordingly, a description of the dynamic state will be the next: ∀ ⊂ ( 1,× … × ) × × ( 1,× … × ) ∃ ⊂ ( 1,× … … × ) × , if (7)
The concept of autonomy means that after the introduction of feedback, each component of
the output signal { } can be changed only after changing the input action { }, while the output
, ≠ the control action does not affect the change in the state of the system with the target
strategy [
The functional controllability of the system means that an appropriately selected input control action ( / /Start ( )), according to the target control strategy, can bring the system to the target area , i.e.
∀ ( )∃( ): ∃ Start ( / ); ∃ ≡ ; : → ∈ → [ Autonomous operation of the energy-active system.
If S is a multidimensional functional system ].
: ( ) → , = ( 1 × … × ) = ( 1 × … × ), = ( 1 × 2 × … × ), then there is a feedback given in the form of the structure , then it is autonomous and represented in the form of a parametric ( × ) description
(∀ ∈ ) ∃( × ) ; ( × ) ⇒ ( = ( , )), where : → is the substructure that ensures the autonomy of the system.
To implement the operation of mixing the input signal with the feedback signal, the element H is introduced, which is an operation (+, −, ) of positive and negative feedback and implements the input stage of the system with feedback (Fig. 5), which represents the dynamics of changes in the state trajectory (y is a parameter, ( ) ∈ , ∈ ) of an energy-active object due to the impact of threats or information disorientation on the control process.
In accordance with the target task, a structural diagram of an automatic tracking system with information feedback is formed, which transmits signals of changes in the state of the object under the influence of the input control signal and interference with the functioning of the control object (Fig. 5): (8) [ ⊗ ( − )] → ( ) → ( , ) → (State) = [ ( ) ⊗ ] → (shifting... trajectory)
To ensure functional stability, the trajectory stabilization condition is met according to the specified conditions: (∀ ∈ , ∀ 0 ∈ 0, ∃z ∈ x): [ 0 = ( , )], and [H( , ) = H( ′, )] ⇒ ( = ′), [H( , ) = H( 1, 1′ )] ⇒ ( = ′), ∀ , ∃( , ̂): ( = ( ◻ 0)( , ̂)),
∀ , ∃ 0 : = 0 0 .
Terminal dynamic - systems are functional and, due to the internal development of control actions, are determined on the basis of a representation in the form of a logical structure [14,15,16]: ( × ) = equation has the form:
∀ , ∀ , , X/ = / ⇒ (S0(x)/ = S0(x)/ ), (9) that is (S0/ ), the system is functional ∀t ∈/ .
For such systems, the unambiguity of functionality is determined in accordance with the condition of management sustainability and adequacy of the structure to the goals: If ∃ , ⊂ ( × ) × ) - the system, then according to goals ∃strat ∃ , ⊂ determines that is functional and unbiased, then the trajectory ∀ ∈ , (∃( , , ) ∈ ) (( , , ̂) ∈ ) (( , )| = ( ̂, )| ) ⇒ ( | = | ) (10) and the system ∣ is unambiguously functional.
For hierarchical systems, the condition of unambiguous functioning of all systems ensures the functional stability of the structure; if such conditions are violated, the system will experience limit and emergency modes, structural collapse, loss of controllability, and disasters. For the functional controllability of the required system, it is enough to make the system ( S = H S )
0 autonomous with the help of a communication loop Sf.
Based on the above analysis of the structure and functionality of hierarchical systems and the impact of external and internal threats on control modes, there is a draw of conclusion about the training of operational personnel and their knowledge.
The modern development of the science of intelligence is based [
The blocks of knowledge necessary for performing professionally oriented activities are formed in the process of learning and work based on the ordering of the knowledge acquired in the past and the amount of new knowledge.
Gradation of stages of knowledge accumulation: 1. quality of school education as a basis for professional orientation; 2. technical education (vocational schools, colleges, workshops); 3. engineering and university education in the area of specialization of each student; 4. professional activity in the chosen field and assessment of compatibility with the requirements for efficiency and responsibility; 5. professional work of the highest rank, understanding of the independence of training, internships, doctoral studies for strategic level positions.
In accordance with the stages of knowledge accumulation, let’s build a table of information and cognitive suitability of operational personnel (Table 2).
S g KV (Аі ) ZpKV (Аj ))
Comprehensive operator confidence and intelligence (
Trust of external experts in the identity of the operator ( Rd (Аі ))
Self-confidence in the ability to solve the problem ( Кcogn(Аі ) Кdu (А ) ) Кd (D(Сі)) Determination to act in the face of risk ( Кd (Drisk ) α r > 0.2 α r > 0.3 where (α risk ) - risk assessment, ( K d ) - cognitive trust coefficient, ( КV ) - coefficients of knowledge requirements.
The article considers certain aspects of the use of logical and intellectual procedures that form the basis of the scheme for synthesizing hierarchical control systems.
Based on the construction of hierarchical systems with different functional structure, an approach using the logic of actions and the theory of situational management is proposed, models of the structure of systems for active control of technological processes under conditions of dynamic disturbances, both systemic, structural and cognitive-informational types, are developed.
The concept of goal orientation and coordination of the logical and cognitive model of forming control decisions of a system with a hierarchical structure under conditions of threats and information attacks as a basis for the synthesis of robust decision-making strategies in crisis emergency situations is substantiated. [7] J. O'Connor, I. McDermott, The Art of Systems Thinking: Essential Skills for Creativity and
Problem Solving, Thorsons, London, UK, 2018. [8] K. Vellani, Strategic Security Management: A Risk Assessment Guide for Decision Makers, 2st ed., CRC Press, Boca Raton, FL, USA, 2019. [9] V. Khoroshchko, B. Bobalo, V. Dudykevych, Procurement of Integrated Information
Security Systems, NU LP PH, Lviv, Ukraine, 2020. (In Ukrainian). [10] M. Groover, Automation, Production Systems, and Computer-Integrated Manufacturing, 5th ed., Pearson, London, UK, 2018. [11] B. Durniak, L. Sikora, M.Antonyk, R. Tkachuk, Cognitive Models of Formation of Strategies for Operational Management of Integrated Hierarchical Structures in Conditions of Risks and Conflicts, Ukrainian Academy of Printing PH, Lviv, Ukraine, 2013. (In Ukrainian). [12] J. Girdhar, Management information systems, New Delhi Oxford University Press, New
Delhi, India, 2013. [13] D. Tattam, A Short Guide to Operational Risk; 1st edn. Routledge, London, UK, 2011. doi: 10.4324/9781315263649 [14] R. Watt, Visual Processing: Computational, Psychophysical, and Cognitive Research: Computational Psychophysical and Cognitive Research; 1st edn., Psychology Press, London, UK, 1988; doi: 10.4324/9781315785080 [15] M. Wiggins, Introduction to Human Factors for Organizational Psychologists; CRC Press, Taylor & Francis Group, Boca Raton, FL, USA, 2022. doi: 10.1201/9781003229858 [16] P. Yu, Multiple-Criteria Decision Making: Concepts Techniques and Extensions; Plenum
Press, New-York-London, USA-UK, 1985 doi: 10.1007/978-1-4684-8395-6. [17] H. Yousef, Power System Load Frequency Control: Classical and Adaptive Fuzzy
Approaches, CRC Press, Boca Raton, FL, USA, 2017. doi: 10.1201/9781315166292 [18] P. Melin, O. Castillo, Modelling, Simulation and Control of Non-Linear Dynamical Systems: An Intelligent Approach Using Soft Computing and Fractal Theory, CRC Press, Boca Raton, FL, USA, 2019. doi: 10.1201/9781420024524 [19] V. Bhise, Decision-Making in Energy System, CRC Press, Boca Raton, FL, USA, 2022. doi: 10.1201/9781003107514 [20] H. Jantan, A. Hamdan, Z. Othman, Intelligent techniques for decision support system in human resource management, International Journal of Innovative Computing, Information and Control, 7(1) (2011)13-24. doi: 10.5772/39401 [21] A. Nejad, Fundamentals of System Analysis & Design Publisher, LAP LAMBERT Academic
Publishing, Saarbrücken, Germany, 2014. doi:10.13140/2.1.1469.7280 [22] S. Demri, V. Goranko, M. Lange, Temporal Logics in Computer Science, Cambridge
University Press, Cambridge, UK, 2016. doi: 10.1017/CBO9781139236119 [23] I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, Y. Bengio, Generative adversarial nets, Advances in neural information processing systems, 27 (2014) 2672–2680. doi: 10.48550/arXiv.1406.2661 [24] E. Manziuk, Intelligent information technology for obtaining trust decisions based on the ontology of trust using a human-centered approach, Computer Systems and Information Technologies, 1 (2022), 83-88. doi: 10.31891/CSIT-2022-1-11 (In Ukrainian).