What is called the basic state of the system. Basic definitions. Systematic approach to modeling
| Parameter name | Meaning |
| Article topic: | State of the system |
| Rubric (thematic category) | Education |
Definition 1.6 System state call a set of parameters that at each considered moment in time reflect the most significant, from a certain point of view, aspects of the behavior of the system and its functioning.
The definition is very general. It emphasizes that the choice of state characteristics depends on the objectives of the study. In the simplest cases, the state can be assessed by one parameter that can take two values (on or off, 0 or 1). In more complex studies, it is necessary to take into account many parameters that can take on a large number of values.
A system whose state changes over time under the influence of certain cause-and-effect relationships is usually called dynamic system, in contrast to a static system, the state of which does not change over time.
The desired state of the system is achieved or maintained by appropriate control actions.
Control
In cybernetics, control is perceived as a process of purposefully changing the state of a system. Sometimes control is the process of processing perceived information into signals that direct the activities of machines and organisms. And the processes of information perception, its storage, transmission and reproduction belong to the field of communication. There is also a broader interpretation of the concept of management, which includes all elements of management activity, united by unity of purpose and commonality of tasks to be solved.
Definition 1.7 Management It is customary to call the information process of preparing and maintaining a purposeful impact on objects and processes of the real world.
This interpretation covers all the issues that the governing body has to resolve, from collecting information, system analysis, making decisions, planning measures to implement decisions, to generating control signals and communicating them to executive bodies.
State of the system - concept and types. Classification and features of the “System State” category 2017, 2018.
The concept of the external environment The system exists among other material objects that are not included in it. They are united by the concept of “external environment” - objects of the external environment. The external environment is a set of objects (systems) existing in space and time, which... [read more] .
Lecture 2: System properties. System classification
Properties of systems.
So, the state of a system is the set of essential properties that the system possesses at each moment in time.
A property is understood as a side of an object that determines its difference from other objects or its similarity to them and manifests itself when interacting with other objects.
A characteristic is something that reflects some property of the system.
What properties of systems are known.
From the definition of “system” it follows that the main property of the system is integrity, unity, achieved through certain relationships and interactions of the system elements and manifested in the emergence of new properties that the system elements do not possess. This property emergence(from English emerge - arise, appear).
- Emergence is the degree to which the properties of a system are irreducible to the properties of the elements of which it consists.
- Emergence is a property of systems that causes the emergence of new properties and qualities that are not inherent in the elements that make up the system.
Emergence is the opposite principle of reductionism, which states that a whole can be studied by dividing it into parts and then, by determining their properties, determining the properties of the whole.
The property of emergence is close to the property of system integrity. However, they cannot be identified.
Integrity system means that each element of the system contributes to the implementation of the target function of the system.
Integrity and emergence are integrative properties of the system.
The presence of integrative properties is one of the most important features of the system. Integrity is manifested in the fact that the system has its own pattern of functionality, its own purpose.
Organization- a complex property of systems, consisting in the presence of structure and functioning (behavior). An indispensable part of systems is their components, namely those structural formations that make up the whole and without which it is not possible.
Functionality- this is the manifestation of certain properties (functions) when interacting with the external environment. Here the goal (purpose of the system) is defined as the desired end result.
Structurality- this is the orderliness of the system, a certain set and arrangement of elements with connections between them. There is a relationship between the function and structure of a system, as between the philosophical categories of content and form. A change in content (functions) entails a change in form (structure), but also vice versa.
An important property of a system is the presence of behavior - actions, changes, functioning, etc.
It is believed that this behavior of the system is associated with the environment (surrounding), i.e. with other systems with which it comes into contact or enters into certain relationships.
The process of purposefully changing the state of a system over time is called behavior. Unlike control, when a change in the state of the system is achieved through external influences, behavior is implemented exclusively by the system itself, based on its own goals.
The behavior of each system is explained by the structure of the lower order systems that make up the system and the presence of signs of equilibrium (homeostasis). In accordance with the sign of equilibrium, the system has a certain state (states) that are preferable for it. Therefore, the behavior of systems is described in terms of the restoration of these states when they are disrupted by environmental changes.
Another property is the property of growth (development). Development can be seen as an integral part of behavior (and the most important one at that).
One of the primary, and, therefore, fundamental attributes of the systems approach is the inadmissibility of considering an object outside of it. development, which is understood as an irreversible, directed, natural change in matter and consciousness. As a result, a new quality or state of the object arises. The identification (maybe not entirely strict) of the terms “development” and “movement” allows us to express it in such a sense that without development the existence of matter, in this case a system, is unthinkable. It is naive to imagine development occurring spontaneously. In the vast variety of processes that seem at first glance to be something like Brownian (random, chaotic) movement, with close attention and study, the contours of tendencies first appear, and then quite stable patterns. These laws, by their nature, act objectively, i.e. do not depend on whether we desire their manifestation or not. Ignorance of the laws and patterns of development is wandering in the dark.
He who does not know which harbor he is sailing to has no favorable wind.
The behavior of the system is determined by the nature of the reaction to external influences.
The fundamental property of systems is sustainability, i.e. the ability of the system to withstand external disturbances. The lifespan of the system depends on it.
Simple systems have passive forms of stability: strength, balance, adjustability, homeostasis. And for complex ones, active forms are decisive: reliability, survivability and adaptability.
If the listed forms of stability of simple systems (except for strength) concern their behavior, then the determining form of stability of complex systems is mainly structural in nature.
Reliability- the property of preserving the structure of systems, despite the death of its individual elements through their replacement or duplication, and survivability- as active suppression of harmful qualities. Thus, reliability is a more passive form than survivability.
Adaptability- the ability to change behavior or structure in order to preserve, improve or acquire new qualities in conditions of changing external environment. A prerequisite for the possibility of adaptation is the presence of feedback connections.
Every real system exists in an environment. The connection between them can be so close that it becomes difficult to determine the boundary between them. Therefore, the isolation of a system from its environment is associated with one degree or another of idealization.
Two aspects of interaction can be distinguished:
- in many cases it takes on the character of an exchange between the system and the environment (matter, energy, information);
- the environment is usually a source of uncertainty for systems.
The influence of the environment can be passive or active (antagonistic, purposefully opposing the system).
Therefore, in the general case, the environment should be considered not only indifferent, but also antagonistic in relation to the system under study.
Rice. — System classification
| Basis (criterion) of classification | System classes |
|---|---|
| By interaction with the external environment | Open Closed Combined |
| By structure | Simple Complex Large |
| By nature of functions | Specialized Multifunctional (universal) |
| By the nature of development | Stable Developing |
| By degree of organization | Well organized Poorly organized (diffuse) |
| According to the complexity of behavior | Automatic Decisive Self-organizing Foresighted Transforming |
| By the nature of the connection between elements | Deterministic Stochastic |
| By the nature of the management structure | Centralized Decentralized |
| By purpose | Producing Managers Attendants |
Classification called division into classes according to the most essential characteristics. A class is understood as a collection of objects that have certain characteristics of commonality. A characteristic (or a set of characteristics) is the basis (criterion) of classification.
A system can be characterized by one or more characteristics and, accordingly, a place can be found in various classifications, each of which can be useful when choosing a research methodology. Typically, the purpose of classification is to limit the choice of approaches to displaying systems and to develop a description language suitable for the corresponding class.
Real systems are divided into natural (natural systems) and artificial (anthropogenic) systems.
Natural systems: systems of inanimate (physical, chemical) and living (biological) nature.
Artificial systems: created by humanity for its own needs or formed as a result of deliberate efforts.
Artificial ones are divided into technical (technical and economic) and social (public).
A technical system is designed and manufactured by a person for a specific purpose.
Social systems include various systems of human society.
The identification of systems consisting of technical devices alone is almost always conditional, since they are not capable of generating their own state. These systems act as parts of larger organizational and technical systems that include people.
An organizational system, for the effective functioning of which a significant factor is the way of organizing the interaction of people with a technical subsystem, is called a human-machine system.
Examples of human-machine systems: car - driver; airplane - pilot; Computer - user, etc.
Thus, technical systems are understood as a single constructive set of interconnected and interacting objects, intended for purposeful actions with the task of achieving a given result in the process of functioning.
Distinctive features of technical systems in comparison with an arbitrary set of objects or in comparison with individual elements are constructiveness (practical feasibility of relations between elements), orientation and interconnectedness of constituent elements and purposefulness.
In order for a system to be resistant to external influences, it must have a stable structure. The choice of structure practically determines the technical appearance of both the entire system and its subsystems and elements. The question of the appropriateness of using a particular structure should be decided based on the specific purpose of the system. The structure also determines the ability of the system to redistribute functions in the event of complete or partial waste of individual elements, and, consequently, the reliability and survivability of the system for the given characteristics of its elements.
Abstract systems are the result of the reflection of reality (real systems) in the human brain.
Their mood is a necessary step in ensuring effective human interaction with the outside world. Abstract (ideal) systems are objective in their source of origin, since their primary source is objectively existing reality.
Abstract systems are divided into direct mapping systems (reflecting certain aspects of real systems) and generalizing (generalizing) mapping systems. The former include mathematical and heuristic models, and the latter include conceptual systems (theories of methodological construction) and languages.
Based on the concept of the external environment, systems are divided into: open, closed (closed, isolated) and combined. The division of systems into open and closed is associated with their characteristic features: the ability to preserve properties in the presence of external influences. If a system is insensitive to external influences, it can be considered closed. Otherwise - open.
An open system is a system that interacts with its environment. All real systems are open. An open system is part of a more general system or several systems. If we isolate the system under consideration from this formation, then the remaining part is its environment.
An open system is connected to the environment by certain communications, that is, a network of external connections of the system. Identification of external connections and description of the mechanisms of “system-environment” interaction is the central task of the theory of open systems. Consideration of open systems allows us to expand the concept of system structure. For open systems, it includes not only internal connections between elements, but also external connections with the environment. When describing the structure, they try to divide external communication channels into input (through which the environment influences the system) and output (vice versa). The set of elements of these channels belonging to their own system are called the input and output poles of the system. In open systems, at least one element has a connection with the external environment, at least one input pole and one output pole, by which it is connected with the external environment.
For each system, communications with all subsystems subordinate to it and between the latter are internal, and all others are external. The connections between systems and the external environment, as well as between the elements of the system, are, as a rule, directional in nature.
It is important to emphasize that in any real system, due to the laws of dialectics on the universal connection of phenomena, the number of all interrelations is enormous, so it is impossible to take into account and study absolutely all connections, therefore their number is artificially limited. At the same time, it is impractical to take into account all possible connections, since among them there are many insignificant ones that practically do not affect the functioning of the system and the number of solutions obtained (from the point of view of the problems being solved). If a change in the characteristics of a connection, its exclusion (complete break) lead to a significant deterioration in the operation of the system, a decrease in efficiency, then such a connection is significant. One of the most important tasks of the researcher is to identify the systems that are essential for consideration in the conditions of the communication problem being solved and to separate them from the unimportant. Due to the fact that the input and output poles of the system cannot always be clearly identified, it is necessary to resort to a certain idealization of actions. The greatest idealization occurs when considering a closed system.
A closed system is a system that does not interact with the environment or interacts with the environment in a strictly defined way. In the first case, it is assumed that the system does not have input poles, and in the second, that there are input poles, but the influence of the environment is constant and completely (in advance) known. Obviously, under the last assumption, the indicated impacts can be attributed to the system itself, and it can be considered as closed. For a closed system, any element of it has connections only with elements of the system itself.
Of course, closed systems represent some abstraction of the real situation, since, strictly speaking, isolated systems do not exist. However, it is obvious that simplifying the description of the system, which involves abandoning external connections, can lead to useful results and simplify the study of the system. All real systems are closely or weakly connected with the external environment - open. If a temporary break or change in characteristic external connections does not cause deviations in the functioning of the system beyond predetermined limits, then the system is weakly connected with the external environment. Otherwise it’s cramped.
Combined systems contain open and closed subsystems. The presence of combined systems indicates a complex combination of open and closed subsystems.
Depending on the structure and spatiotemporal properties, systems are divided into simple, complex and large.
Simple - systems that do not have branched structures, consisting of a small number of relationships and a small number of elements. Such elements serve to perform the simplest functions; hierarchical levels cannot be distinguished in them. A distinctive feature of simple systems is the determinism (clear definition) of the nomenclature, number of elements and connections both within the system and with the environment.
Complex - characterized by a large number of elements and internal connections, their heterogeneity and different quality, structural diversity, and perform a complex function or a number of functions. The components of complex systems can be considered as subsystems, each of which can be detailed by even simpler subsystems, etc. until the element is received.
Definition N1: a system is called complex (from an epistemological standpoint) if its cognition requires the joint involvement of many models of theories, and in some cases many scientific disciplines, as well as taking into account the uncertainty of a probabilistic and non-probabilistic nature. The most characteristic manifestation of this definition is multi-model.
Model- a certain system, the study of which serves as a means of obtaining information about another system. This is a description of systems (mathematical, verbal, etc.) reflecting a certain group of its properties.
Definition N2: a system is called complex if in reality the signs of its complexity clearly (significantly) appear. Namely:
- structural complexity - determined by the number of elements of the system, the number and variety of types of connections between them, the number of hierarchical levels and the total number of subsystems of the system. The following types of connections are considered the main types: structural (including hierarchical), functional, causal (cause-and-effect), informational, spatiotemporal;
- complexity of functioning (behavior) - determined by the characteristics of a set of states, the rules of transition from state to state, the impact of the system on the environment and the environment on the system, the degree of uncertainty of the listed characteristics and rules;
- the complexity of choosing behavior - in multi-alternative situations, when the choice of behavior is determined by the purpose of the system, the flexibility of reactions to previously unknown environmental influences;
- complexity of development - determined by the characteristics of evolutionary or discontinuous processes.
Naturally, all signs are considered in interrelation. Hierarchical construction is a characteristic feature of complex systems, and the levels of hierarchy can be both homogeneous and heterogeneous. Complex systems are characterized by factors such as the impossibility of predicting their behavior, that is, poor predictability, their secrecy, and various states.
Complex systems can be divided into the following factor subsystems:
- the decisive one, which makes global decisions in interaction with the external environment and distributes local tasks among all other subsystems;
- information, which ensures the collection, processing and transmission of information necessary for making global decisions and performing local tasks;
- manager for the implementation of global decisions;
- homeostasis, maintaining dynamic balance within systems and regulating the flow of energy and matter in subsystems;
- adaptive, accumulating experience in the learning process to improve the structure and functions of the system.
A large system is a system that is not simultaneously observable from the position of one observer in time or space, for which the spatial factor is significant, the number of subsystems of which is very large, and the composition is heterogeneous.
The system can be large and complex. Complex systems unite a larger group of systems, that is, large systems - a subclass of complex systems.
Fundamental to the analysis and synthesis of large and complex systems are the procedures of decomposition and aggregation.
Decomposition is the division of systems into parts, followed by independent consideration of individual parts.
It is obvious that decomposition is a concept associated with a model, since the system itself cannot be dismembered without violating the properties. At the modeling level, disparate connections will be replaced by equivalents, or the system model will be built in such a way that its decomposition into separate parts turns out to be natural.
When applied to large and complex systems, decomposition is a powerful research tool.
Aggregation is the opposite concept of decomposition. In the process of research, the need arises to combine elements of the system in order to consider it from a more general perspective.
Decomposition and aggregation represent two opposing approaches to the consideration of large and complex systems, applied in dialectical unity.
Systems for which the state of the system is uniquely determined by the initial values and can be predicted for any subsequent point in time are called deterministic.
Stochastic systems are systems in which changes are random. With random influences, data on the state of the system is not enough to make a prediction at a subsequent point in time.
According to the degree of organization: well organized, poorly organized (diffuse).
To present the analyzed object or process in the form of a well-organized system means to determine the elements of the system, their relationships, and the rules for combining into larger components. The problem situation can be described in the form of a mathematical expression. The solution of a problem, when presented in the form of a well-organized system, is carried out by analytical methods of a formalized representation of the system.
Examples of well-organized systems: the solar system, which describes the most significant patterns of planetary motion around the Sun; display of the atom as a planetary system consisting of a nucleus and electrons; description of the operation of a complex electronic device using a system of equations that takes into account the peculiarities of its operating conditions (presence of noise, instability of power supplies, etc.).
The description of an object in the form of a well-organized system is used in cases where it is possible to offer a deterministic description and experimentally prove the legitimacy of its application and the adequacy of the model to the real process. Attempts to apply the class of well-organized systems to represent complex multi-component objects or multi-criteria problems are not successful: they require an unacceptably large amount of time, are practically impossible to implement and are inadequate to the models used.
Poorly organized systems. When presenting an object in the form of a poorly organized or diffuse system, the task is not to determine all the components taken into account, their properties and the connections between them and the goals of the system. The system is characterized by a certain set of macro-parameters and patterns that are found on the basis of a study not of the entire object or class of phenomena, but on the basis of a selection of components determined using certain rules that characterize the object or process under study. Based on such a sample study, characteristics or patterns (statistical, economic) are obtained and distributed to the entire system as a whole. In this case, appropriate reservations are made. For example, when statistical regularities are obtained, they are extended to the behavior of the entire system with a certain confidence probability.
The approach to displaying objects in the form of diffuse systems is widely used in: describing queuing systems, determining the number of staff in enterprises and institutions, studying documentary information flows in management systems, etc.
From the point of view of the nature of the functions, special, multifunctional, and universal systems are distinguished.
Special systems are characterized by a unique purpose and narrow professional specialization of service personnel (relatively uncomplicated).
Multifunctional systems allow you to implement several functions on the same structure. Example: a production system that provides the production of various products within a certain range.
For universal systems: many actions are implemented on the same structure, but the composition of functions is less homogeneous (less defined) in type and quantity. For example, a combine.
According to the nature of development, there are 2 classes of systems: stable and developing.
In a stable system, the structure and functions practically do not change during the entire period of its existence and, as a rule, the quality of functioning of stable systems only worsens as their elements wear out. Remedial measures can usually only reduce the rate of deterioration.
An excellent feature of evolving systems is that over time, their structure and functions undergo significant changes. The functions of the system are more constant, although they are often modified. Only their purpose remains virtually unchanged. Evolving systems have higher complexity.
In order of increasing complexity of behavior: automatic, decisive, self-organizing, anticipatory, transformative.
Automatic: they unambiguously respond to a limited set of external influences, their internal organization is adapted to transition to an equilibrium state when withdrawn from it (homeostasis).
Decisive: have constant criteria for distinguishing their constant response to broad classes of external influences. The constancy of the internal structure is maintained by replacing failed elements.
Self-organizing: have flexible discrimination criteria and flexible responses to external influences, adapting to different types of influence. The stability of the internal structure of higher forms of such systems is ensured by constant self-reproduction.
Self-organizing systems have the characteristics of diffuse systems: stochastic behavior, nonstationarity of individual parameters and processes. Added to this are signs such as unpredictability of behavior; the ability to adapt to changing environmental conditions, change the structure when the system interacts with the environment, while maintaining the properties of integrity; the ability to form possible behavior options and choose the best one from them, etc. Sometimes this class is divided into subclasses, highlighting adaptive or self-adjusting systems, self-healing, self-reproducing and other subclasses corresponding to various properties of developing systems.
Examples: biological organizations, collective behavior of people, organization of management at the level of an enterprise, industry, state as a whole, i.e. in those systems where there is necessarily a human factor.
If stability in its complexity begins to exceed the complex influences of the external world, these are anticipatory systems: it can foresee the further course of interaction.
Transformables are imaginary complex systems at the highest level of complexity, not bound by the constancy of existing media. They can change material media while maintaining their individuality. Examples of such systems are not yet known to science.
A system can be divided into types based on the structure of their construction and the significance of the role that individual components play in them in comparison with the roles of other parts.
In some systems, one of the parts may play a dominant role (its significance >> (symbol of the relationship of “significant superiority”) the significance of other parts). Such a component will act as a central one, determining the functioning of the entire system. Such systems are called centralized.
In other systems, all the components that make them up are approximately equally important. Structurally, they are not located around some centralized component, but are interconnected in series or in parallel and have approximately the same importance for the functioning of the system. These are decentralized systems.
Systems can be classified by purpose. Among the technical and organizational systems there are: producing, managing, servicing.
In production systems, processes for obtaining certain products or services are implemented. They, in turn, are divided into material-energy ones, in which the transformation of the natural environment or raw materials into the final product of a material or energy nature, or the transportation of such products is carried out; and information - for collecting, transmitting and converting information and providing information services.
The purpose of control systems is to organize and manage material, energy and information processes.
Servicing systems are engaged in maintaining the specified limits of performance of production and control systems.
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The state of the system is determined by levels.
A level is the amount of mass, energy, information contained in a variable (block) or in the system as a whole at a given moment in time.
Levels do not remain constant, they undergo certain changes. The speed at which these changes occur is called tempo.
Rates determine the activity, intensity and speed of the processes of transformation, accumulation, transmission, etc. matter, energy, information flowing within the system.
Tempos and levels are interrelated, but their relationship is not clear-cut. On the one hand, rates generate new levels, which in turn influence rates, i.e. regulate them.
For example, the process of substance diffusion determines the transition of the system from level x 1 to level x 2 (the driving force of the mass transfer process). At the same time, the speed of this process (rate of mass transfer) depends on the mass of the indicated levels in accordance with the expression:
where: a is the mass transfer coefficient.
One of the most important characteristics of the system state is feedback.
Feedback is the property of a system (block) to respond to a change in one or more variables caused by an input influence, in such a way that, as a result of processes within the system, this change again affects the same or the same variables.
Feedback, depending on the method of influence, can be direct (when the reverse influence occurs without the participation of variables (blocks) - intermediaries) or contour (when the reverse influence occurs with the participation of variables (blocks) - intermediaries) (Fig. 3).
Rice. 3. Feedback principle
a – direct feedback; b – loop feedback.
Depending on the impact on primary changes in variables in the system, two types of feedback are distinguished:
§ Negative feedback, i.e. when an impulse received from the outside forms a closed circuit and causes attenuation (stabilization) of the initial impact;
§ Positive feedback, i.e. when an impulse received from the outside forms a closed circuit and causes an increase in the initial impact.
Negative feedback is a form of self-regulation that ensures dynamic balance in the system. Positive feedback in natural systems usually manifests itself in the form of relatively short-term bursts of self-destructive activity.
The predominantly negative nature of the feedback indicates that any change in environmental conditions leads to a change in the variables of the system and causes the system to transition to a new equilibrium state, different from the original one. This process of self-regulation is commonly called homeostasis.
The system’s ability to restore equilibrium is determined by two more characteristics of its state:
§ System stability, i.e. a characteristic indicating what magnitude of change in external influence (impact impulse) corresponds to the permissible change in the system variables, at which equilibrium can be restored;
§ System stability, i.e. a characteristic that determines the maximum permissible change in system variables at which equilibrium can be restored.
The goal of regulation in the system is formulated in the form of an extreme principle (the law of maximum potential energy): the evolution of the system goes in the direction of increasing the total energy flow through the system, and in a stationary state its maximum possible value is achieved (maximum potential energy).
The state of any real system at any given moment in time can be described using a certain set that characterizes the system of quantities - parameter.
The number of parameters, even for a relatively simple system, can be very large, and therefore, in practice, only the most significant, characteristic parameters corresponding to the specific purposes of studying objects are used to describe systems. So, to study a person’s health status from the point of view of the need to relieve him from work, the values of parameters such as temperature and blood pressure are first taken into account.
The state of a certain economic system is characterized by such parameters as the quantity and quality of output, labor productivity, return fund, etc.
To describe the state and movement of a system, methods such as verbal descriptions, tabular or matrix descriptions, mathematical expressions, and graphical images can be used.
Verbal description comes down to a sequential listing and characteristics of the system parameters, trends in their changes, and the sequence of changes in the state of the system. The verbal description is very approximate and gives only general ideas about the system, in addition, it is largely subjective, because reflects not only the true characteristics of the system, but also the attitude of the person describing them to them.
Tables and matrices are most widely used for the quantitative characteristics of a system, expressed by the values of their parameters at some fixed point in time. Based on the data from a table or a set of tables, diagrams and graphs can be constructed corresponding to different moments in time, giving a visual representation of the dynamics of the system.
To describe the movement of a system and changes in its elements, they are used mathematical expressions, which in turn are interpreted by graphs showing the course of certain processes in the system.
However, the most profound and adequate is formalized geometric interpretation states and movements of the system in the so-called state space or phase space.
System state space
System state space is a space in which each point uniquely corresponds to a certain state of the dynamic system under consideration, and each process of changing the state of the system corresponds to a certain trajectory of movement of the representing point in space.
To describe the movements of dynamic systems, a method based on the so-called phase space(n-dimensional Euclidean space), along the axes of which the values of all n generalized coordinates of the dynamic system under consideration are plotted. In this case, a unique correspondence between the state of the system and the points of the phase space is achieved by choosing a number of dimensions equal to the number of generalized coordinates of the dynamic system under consideration.
Let us denote the parameters of a certain system by the symbols z1, z2…zn, which can be considered as the coordinates of the vector z, n of dimensional space. Such a vector is a collection of real numbers z=(z1,z2..zn). The parameters z1, z2…zn will be called the phase coordinates of the system, and the states (phase of the system) will be represented by the point z in the phase space. The dimension of this space is determined by the number of phase coordinates, that is, the number of its essential parameters selected by us to describe the system.
In the case when the states of the system can be characterized by only one parameter z1 (for example, the distance from the departure point of a train moving along a given route), then the phase space will be one-dimensional and displayed as a portion of the z-axis.
If the state of the system is characterized by two parameters z1 and z2 (for example, the movement of a car, expressed by an angle relative to some given direction and the speed of its movement), then the phase space will be two-dimensional.
In cases where the state of the system is described by 3 parameters (for example, speed and acceleration control), it will be represented by a point in three-dimensional space, and the trajectory of the system will be a spatial curve in this space.
In the general case, when the number of parameters characterizing the system is arbitrary and, as in most complex economic systems, is significantly greater than 3, the geometric interpretation loses its clarity. However, geometric terminology in these cases remains convenient for describing the state and movement of systems in the so-called n-dimensional or multidimensional phase space (hyperspace).
The number of independent parameters of the system is called number of degrees of freedom or system variations.
In real operating conditions of the system and its parameters (phase coordinates), as a rule, can change only within certain limited limits. Thus, the speed of a car is limited from 0 to 200 km per hour, the temperature of a person is limited from 35 degrees to 42, etc.
The region of phase space beyond which the representing point cannot go is called area of permissible system states. When researching and designing systems, it is always assumed that the system is within the range of its permissible states.
If the representing point goes beyond this area, then this threatens to destroy the integrity of the system, the possibility of its disintegration into elements, disruption of existing connections, that is, the complete cessation of its functioning as a given system.
The region of permissible states, which can be called the field of the system, includes all kinds of phase trajectories, that is, the lines of behavior of the systems. The set of phase trajectories is called phase portrait dynamic system under consideration. In all cases when the parameters of the system can take on any values in a certain interval, that is, the representing point changes smoothly, which can be located at any point within the region of permissible states, and we are dealing with the so-called continuous state space. However, there are a large number of technical, biological and economic systems in which a number of parameters - coordinates - can only take discrete values.
Only discretely can one measure the number of machines in a workshop, the number of certain organs and cells in a living organism, etc.
The state space of such systems must be considered discrete, therefore their point representing the state of such a system cannot be located in any place in the region of permissible states, but only in certain fixed points of this region. A change in the state of such systems, that is, their movement, will be interpreted by jumps of the representing point from one state to another, to a third, etc. Accordingly, the trajectory of movement of the representing point will have a discrete, intermittent character.
State. The concept of state usually characterizes an instant photograph, a “slice” of the system, a stop in its development. It is determined either through input influences and output signals (results), or through properties, parameters of the system (for example, pressure, speed, acceleration - for physical systems; productivity, cost of production, profit - for economic systems).
Thus, a state is a set of essential properties that a system possesses at a given moment in time.
Possible states of a real system form the set of admissible system states.
The number of states (the power of a set of states) can be finite, countable (the number of states is measured discretely, but their number is infinite); power continuum (states change continuously and their number is infinite and uncountable).
States can be described through state variables. If the variables are discrete, then the number of states can be either finite or countable. If the variables are analog (continuous), then the power is continuum.
The minimum number of variables through which a state can be specified is called phase space. Changes in the state of the system are displayed in phase space phase trajectory.
Behavior. If a system is capable of transitioning from one state to another (for example, s 1 →s 2 →s 3 → ...), then they say that it has behavior. This concept is used when the patterns (rules) of transition from one state to another are unknown. Then they say that the system has some behavior and find out its nature.
Equilibrium. The ability of a system in the absence of external disturbing influences (or with constant influences) to maintain its state for an indefinitely long time. This state is called a state of equilibrium.
Sustainability. The ability of a system to return to a state of equilibrium after it has been removed from this state under the influence of external (and in systems with active elements - internal) disturbing influences.
The state of equilibrium to which the system is capable of returning is called a stable state of equilibrium.
Development. Development is usually understood as an increase in the complexity of a system, an improvement in adaptability to external conditions. As a result, a new quality or state of the object arises.
It is advisable to distinguish a special class of developing (self-organizing) systems that have special properties and require the use of special approaches to their modeling.
System inputsx i- these are various points of influence of the external environment on the system (Fig. 1.3).
The inputs of the system can be information, matter, energy, etc., which are subject to transformation.
Generalized input ( X) name some (any) state of all r system inputs, which can be represented as a vector
X = (x 1 , x 2 , x 3 , …, x k, …, x r).
System outputsy i- these are various points of influence of the system on the external environment (Fig. 1.3).
The output of the system is the result of the transformation of information, matter and energy.
Movement of the system is a process of consistent change in its state.
Let us consider the dependences of the system states on the functions (states) of the system inputs, its states (transitions) and outputs.
State of the system Z(t) at any time t depends on the function of the inputs X(t), as well as from its previous states at moments (t– 1), (t– 2), ..., i.e. from the functions of its states (transitions)
Z(t) = F c , (1)
Where F c– function of the state (transitions) of the system.
Relationship between input function X(t) and exit function Y(t) systems, without taking into account previous states, can be represented in the form
Y(t) = Fв [X(t)],
Where F in– function of system outputs.
A system with such an output function is called static.
If the system output depends not only on the functions of the inputs X(t), but also on functions of states (transitions) Z( t – 1), Z(t– 2), ..., then
systems with such an output function are called dynamic(or systems with behavior).
Depending on the mathematical properties of the functions of inputs and outputs of systems, discrete and continuous systems are distinguished.
For continuous systems, expressions (1) and (2) look like:
(4)
Equation (3) determines the state of the system and is called the equation of system states.
Equation (4) determines the observed output of the system and is called the observational equation.
Functions F c(function of system states) and F in(output function) take into account not only the current state Z(t), but also previous states Z(t – 1), Z(t – 2), …, Z(t – v) systems.
Previous states are a parameter of the system's "memory". Therefore, the value v characterizes the volume (depth) of system memory.
System processes is a set of successive changes in the state of the system to achieve a goal. System processes include:
– input process;
– output process;
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