Examples of modeling technical systems and processes. "Systems Theory and System Analysis

Classification of types of modeling can be carried out according to different grounds. One of the classification options is shown in the figure.

Rice. - An example of the classification of types of modeling

In accordance with the classification sign of completeness, modeling is divided into: complete, incomplete, approximate.

At complete modeling models are identical to the object in time and space.

For incomplete modeling this identity is not preserved.

At the core approximate Simulation lies in the similarity, in which some aspects of the real object are not modeled at all. The theory of similarity states that absolute similarity is possible only when one object is replaced by another exactly the same. Therefore, when modeling, absolute similarity does not take place. Researchers strive to ensure that the model well reflects only the studied aspect of the system. For example, to assess the noise immunity of discrete information transmission channels, the functional and information models of the system may not be developed. To achieve the goal of modeling, the event model described by the matrix of conditional probabilities of transitions of the i-th character of the alphabet to the j-th one is quite sufficient.

Depending on the type of media and model signature, the following types of modeling are distinguished: deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous.

deterministic modeling displays processes in which the absence of random influences is assumed.

Stochastic modeling takes into account probabilistic processes and events.

Static Simulation serves to describe the state of an object at a fixed point in time, and dynamic - to study the object in time. At the same time, they operate with analog (continuous), discrete and mixed models.

Depending on the form of implementation of the carrier and signature, modeling is classified into mental and real.

mental modeling is used when models are not realizable in a given time interval or there are no conditions for their physical creation (for example, the situation of the microworld). Mental modeling of real systems is realized in the form of visual, symbolic and mathematical. A significant number of tools and methods have been developed to represent functional, informational and event models of this type of modeling.

At visual modeling on the basis of human ideas about real objects, visual models are created that display the phenomena and processes occurring in the object. An example of such models are educational posters, drawings, charts, diagrams.

The basis hypothetical modeling, a hypothesis is laid about the patterns of the process in a real object, which reflects the level of knowledge of the researcher about the object and is based on cause-and-effect relationships between the input and output of the object under study. This type of modeling is used when knowledge about the object is not enough to build formal models. Analog modeling is based on the application of analogies of various levels. For sufficiently simple objects, the highest level is complete analogy. With the complication of the system, analogies of subsequent levels are used, when the analog model displays several (or only one) aspects of the object's functioning.

Prototyping is used when the processes occurring in a real object are not amenable to physical modeling or may precede other types of modeling. The construction of mental layouts is also based on analogies, usually based on causal relationships between phenomena and processes in an object.

Symbolic modeling is an artificial process of creating a logical object that replaces the real one and expresses its main properties using a certain system of signs and symbols.

At the core linguistic modeling lies some thesaurus, which is formed from a set of concepts of the studied subject area, and this set must be fixed. A thesaurus is a dictionary that reflects the relationships between words or other elements of a given language, designed to search for words by their meaning.

A traditional thesaurus consists of two parts: a list of words and set phrases grouped according to semantic (thematic) headings; an alphabetical dictionary of keywords defining classes of conditional equivalence, an index of relations between keywords, where the corresponding headings are indicated for each word. Such construction allows defining semantic (semantic) relations of hierarchical (genus/species) and non-hierarchical (synonymy, antonymy, associations) type.

There are fundamental differences between a thesaurus and a regular dictionary. Thesaurus is a dictionary that has been cleared of ambiguity, i.e. in it, only a single concept can correspond to each word, although in an ordinary dictionary, several concepts can correspond to one word.

If you enter symbol individual concepts, i.e. signs, as well as certain operations between these signs, then you can implement iconic modeling and using signs to display a set of concepts - to make separate chains of words and sentences. Using the operations of union, intersection and addition of set theory, it is possible to give a description of some real object in separate symbols.

Mathematical modeling is the process of establishing correspondence to a given real object of some mathematical object, called a mathematical model. In principle, to study the characteristics of any system by mathematical methods, including machine methods, this process must be formalized, i.e. a mathematical model is built. The type of mathematical model depends both on the nature of the real object and on the tasks of studying the object, on the required reliability and accuracy of solving the problem. Any mathematical model, like any other, describes a real object with a certain degree of approximation.

To represent mathematical models can be used various forms records. The main ones are invariant, analytical, algorithmic and circuit (graphic).

An invariant form is a record of model relations using a traditional mathematical language, regardless of the method for solving model equations. In this case, the model can be represented as a set of inputs, outputs, state variables and global equations of the system. Analytical form - recording the model as a result of solving the initial equations of the model. Typically, models in analytical form are explicit expressions of output parameters as functions of inputs and state variables.

For analytical modeling is characterized by the fact that basically only the functional aspect of the system is modeled. In this case, the global equations of the system that describe the law (algorithm) of its functioning are written in the form of some analytical relations (algebraic, integro-differential, finite-difference, etc.) or logical conditions. The analytical model is studied by several methods:

analytical, when they strive to get into general view explicit dependencies linking the desired characteristics with the initial conditions, parameters and state variables of the system;
numerical, when, not being able to solve equations in a general form, they strive to obtain numerical results with specific initial data (recall that such models are called digital);
qualitative, when, without having a solution in an explicit form, you can find some properties of the solution (for example, evaluate the stability of the solution).

At present, computer methods for studying the characteristics of the process of functioning of complex systems are widespread. To implement a mathematical model on a computer, it is necessary to build an appropriate modeling algorithm.

Algorithmic form - a record of the relationship between the model and the selected numerical solution method in the form of an algorithm. Among the algorithmic models, an important class is made up of simulation models designed to simulate physical or informational processes under various external influences. Actually, the imitation of these processes is called simulation modeling.

At imitation Simulation reproduces the algorithm of the system functioning in time - the behavior of the system, and the elementary phenomena that make up the process are simulated, with the preservation of their logical structure and sequence of flow, which allows, according to the initial data, to obtain information about the states of the process at certain points in time, making it possible to evaluate the characteristics of the system. The main advantage of simulation modeling compared to analytical modeling is the ability to solve more complex problems. Simulation models make it possible to easily take into account such factors as the presence of discrete and continuous elements, non-linear characteristics of system elements, numerous random effects, and others that often create difficulties in analytical studies. Currently, simulation modeling is the most effective method for studying systems, and often the only practically accessible method for obtaining information about the behavior of a system, especially at the stage of its design.

In simulation, a distinction is made between the method of statistical tests (Monte Carlo) and the method of statistical modeling.

The Monte Carlo method is a numerical method that is used to simulate random variables and functions whose probabilistic characteristics coincide with the solutions of analytical problems. It consists in multiple reproduction of processes that are realizations of random variables and functions, with subsequent processing of information by methods of mathematical statistics.

If this technique is used for machine simulation in order to study the characteristics of the processes of functioning of systems subject to random influences, then this method is called the method of statistical modeling.

The simulation method is used to evaluate options for the system structure, the effectiveness of various system control algorithms, and the impact of changing various system parameters. Simulation modeling can be used as the basis for structural, algorithmic and parametric synthesis of systems, when it is required to create a system with specified characteristics under certain restrictions.

Combined (analytical and simulation) modeling allows you to combine the advantages of analytical and simulation modeling. When building combined models, a preliminary decomposition of the Object Functioning process into constituent subprocesses is carried out, and for those of them, where possible, analytical models are used, and simulation models are built for the remaining subprocesses. This approach makes it possible to cover qualitatively new classes of systems that cannot be studied using analytical or simulation modeling separately.

informational (cybernetic) modeling is associated with the study of models in which there is no direct similarity of the physical processes occurring in the models to real processes. In this case, they seek to display only some function, consider the real object as a “black box” with a number of inputs and outputs, and model some connections between outputs and inputs. Thus, information (cybernetic) models are based on the reflection of some information management processes, which makes it possible to evaluate the behavior of a real object. To build a model in this case, it is necessary to isolate the investigated function of a real object, try to formalize this function in the form of some communication operators between the input and output, and reproduce this function on a simulation model, moreover, in a completely different mathematical language and, of course, a different physical implementation of the process. So, for example, expert systems are models of decision makers.

Structural system analysis modeling is based on some specific features of the structures a certain kind, which are used as a means of studying systems or serve to develop, on their basis, specific approaches to modeling using other methods of formalized representation of systems (set-theoretic, linguistic, cybernetic, etc.). The development of structural modeling is object-oriented modeling.

Structural modeling of system analysis includes:

network modeling methods;
combination of structuring methods with linguistic ones;
a structural approach in the direction of formalizing the construction and study of structures of various types (hierarchical, matrix, arbitrary graphs) based on set-theoretic representations and the concept of a nominal scale of measurement theory.

At the same time, the term "model structure" can be applied both to functions and to system elements. The corresponding structures are called functional and morphological. Object-oriented modeling combines structures of both types into a class hierarchy that includes both elements and functions.

In structural modeling, a new CASE technology has emerged over the past decade. The abbreviation CASE has a double interpretation, corresponding to two areas of use of CASE systems. The first of them - Computer-Aided Software Engineering - translates as computer-aided design software. The corresponding CASE systems are often referred to as Rapid Application Development (RAD) tooling environments. The second - Computer-Aided System Engineering - emphasizes the focus on supporting the conceptual modeling of complex systems, mostly semi-structured. Such CASE systems are often referred to as BPR systems ( business process Reengineering). In general, CASE technology is a set of methodologies for analyzing, designing, developing and maintaining complex automated systems supported by a complex of interconnected automation tools. CASE is a toolkit for system analysts, developers and programmers, which allows you to automate the process of designing and developing complex systems, including software.

situational modeling is based on the model theory of thinking, within which it is possible to describe the main mechanisms for regulating decision-making processes. At the center of the model theory of thinking lies the idea of the formation of an information model of an object and the external world in the structures of the brain. This information is perceived by a person on the basis of the knowledge and experience he already has. Expedient human behavior is built by forming the target situation and mentally transforming the initial situation into the target one. The basis for constructing the model is the description of the object in the form of a set of elements interconnected by certain relationships that reflect the semantics of the subject area. The object model has a multi-level structure and represents the information context against which management processes proceed. The richer the information model of the object and the higher the possibility of manipulating it, the better and more diverse the quality of decisions made in management.

At real modeling uses the possibility of studying the characteristics either on a real object as a whole or on its part. Such studies are carried out both on objects operating in normal modes and when organizing special modes to assess the characteristics of interest to the researcher (for other values of variables and parameters, on a different time scale, etc.). Real simulation is the most adequate, but its possibilities are limited.

Natural modeling is called conducting a study on a real object with subsequent processing of the results of the experiment based on the theory of similarity. Full-scale simulation is subdivided into a scientific experiment, complex tests and a production experiment. scientific experiment characterized by the widespread use of automation tools, the use of very diverse means of information processing, the possibility of human intervention in the process of conducting an experiment. One type of experiment complex tests, during which, due to the repetition of tests of objects as a whole (or large parts of the system), general patterns about the characteristics of quality, reliability of these objects. In this case, modeling is carried out by processing and generalizing information about a group of homogeneous phenomena. Along with specially organized tests, it is possible to implement full-scale simulation by summarizing the experience gained during the production process, i.e. can talk about production experiment. Here, on the basis of the theory of similarity, statistical material on the production process is processed and its generalized characteristics are obtained. It is necessary to remember about the difference between the experiment and the real course of the process. It lies in the fact that individual critical situations may appear in the experiment and the boundaries of the stability of the process can be determined. In the course of the experiment, new factors of perturbing influences are introduced into the process of the object's functioning.

Another kind of real simulation is physical, which differs from natural in that the study is carried out in installations that preserve the nature of phenomena and have a physical similarity. In the process of physical modeling, some characteristics of the external environment are set and the behavior of either a real object or its model is studied under given or artificially created environmental influences. Physical modeling can proceed in real and simulated (pseudo-real) time scales or be considered without regard to time. In the latter case, the so-called "frozen" processes, fixed at some point in time, are subject to study.

Lecture 9: Classification of types of system modeling

The classification of types of modeling can be carried out for various reasons. One of the classification options is shown in the figure.

Rice. — An example of the classification of types of modeling

In accordance with the classification sign of completeness, modeling is divided into: complete, incomplete, approximate.

At complete modeling models are identical to the object in time and space.

For incomplete modeling this identity is not preserved.

Depending on the type of media and model signature, the following types of modeling are distinguished: deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous.

deterministic modeling displays processes in which the absence of random influences is assumed.

Stochastic modeling takes into account probabilistic processes and events.

Depending on the form of implementation of the carrier and signature, modeling is classified into mental and real.

Symbolic modeling is an artificial process of creating a logical object that replaces the real one and expresses its main properties using a certain system of signs and symbols.

A traditional thesaurus consists of two parts: a list of words and set phrases grouped according to semantic (thematic) headings; an alphabetical dictionary of keywords that define classes of conditional equivalence, an index of relationships between keywords, where for each word the corresponding headings are indicated. Such construction allows defining semantic (semantic) relations of hierarchical (genus/species) and non-hierarchical (synonymy, antonymy, associations) type.

If we introduce a symbol for individual concepts, i.e. signs, as well as certain operations between these signs, then you can implement iconic modeling and using signs to display a set of concepts - to make separate chains of words and sentences. Using the operations of union, intersection and addition of set theory, it is possible to give a description of some real object in separate symbols.

Various notation forms can be used to represent mathematical models. The main ones are invariant, analytical, algorithmic and circuit (graphic).

analytical, when they strive to obtain explicit dependencies in a general form, connecting the desired characteristics with the initial conditions, parameters and state variables of the system;
numerical, when, not being able to solve equations in a general form, they strive to obtain numerical results with specific initial data (recall that such models are called digital);
qualitative, when, without having a solution in an explicit form, you can find some properties of the solution (for example, evaluate the stability of the solution).

Algorithmic form - record of relations between the model and the selected numerical method of solution in the form of an algorithm. Among the algorithmic models, an important class is made up of simulation models designed to simulate physical or informational processes under various external influences. Actually, the imitation of these processes is called simulation modeling.

At imitation Simulation reproduces the algorithm of the system functioning in time - the behavior of the system, and the elementary phenomena that make up the process are simulated, with the preservation of their logical structure and sequence of flow, which makes it possible to obtain information about the states of the process at certain points in time from the initial data, making it possible to evaluate the characteristics of the system. The main advantage of simulation modeling compared to analytical modeling is the ability to solve more complex problems. Simulation models make it possible to easily take into account such factors as the presence of discrete and continuous elements, non-linear characteristics of system elements, numerous random effects, and others that often create difficulties in analytical studies. Currently, simulation modeling is the most effective method for studying systems, and often the only practically accessible method for obtaining information about the behavior of a system, especially at the stage of its design.

In simulation, a distinction is made between the method of statistical tests (Monte Carlo) and the method of statistical modeling.

Structural modeling of system analysis is based on some specific features of structures of a certain type, which are used as a means of studying systems or serve to develop specific approaches to modeling based on them using other methods of formalized representation of systems (set-theoretic, linguistic, cybernetic, etc.) . The development of structural modeling is object-oriented modeling.

Structural modeling of system analysis includes:

network modeling methods;
combination of structuring methods with linguistic ones;
a structural approach in the direction of formalizing the construction and study of structures of various types (hierarchical, matrix, arbitrary graphs) based on set-theoretic representations and the concept of a nominal scale of measurement theory.

In structural modeling, a new CASE technology has emerged over the past decade. The abbreviation CASE has a double interpretation, corresponding to two areas of use of CASE systems. The first of them - Computer-Aided Software Engineering - translates as computer-aided software design. The corresponding CASE systems are often referred to as Rapid Application Development (RAD) tooling environments. The second - Computer-Aided System Engineering - emphasizes the focus on supporting the conceptual modeling of complex systems, mostly semi-structured. Such CASE systems are often referred to as BPR (Business Process Reengineering) systems. In general, CASE technology is a set of methodologies for analyzing, designing, developing and maintaining complex automated systems, supported by a set of interconnected automation tools. CASE is a toolkit for system analysts, developers and programmers that allows you to automate the process of designing and developing complex systems, including software.

Natural modeling is called conducting a study on a real object with subsequent processing of the results of the experiment based on the theory of similarity. Full-scale simulation is subdivided into a scientific experiment, complex tests and a production experiment. scientific experiment characterized by the widespread use of automation tools, the use of very diverse means of information processing, the possibility of human intervention in the process of conducting an experiment. One type of experiment is complex tests, during which, as a result of repeated testing of objects as a whole (or large parts of the system), general patterns are revealed about the quality characteristics and reliability of these objects. In this case, modeling is carried out by processing and generalizing information about a group of homogeneous phenomena. Along with specially organized tests, it is possible to implement full-scale simulation by summarizing the experience gained during the production process, i.e. can talk about production experiment. Here, on the basis of the theory of similarity, statistical material on the production process is processed and its generalized characteristics are obtained. It is necessary to remember about the difference between the experiment and the real course of the process. It lies in the fact that individual critical situations may appear in the experiment and the boundaries of the stability of the process can be determined. In the course of the experiment, new factors of perturbing influences are introduced into the process of the object's functioning.

Principles and approaches to the construction of mathematical models

Mathematical modeling is considered by many to be more of an art than a coherent and complete theory. Here the role of experience, intuition and other intellectual qualities of a person is very great. Therefore, it is impossible to write a sufficiently formalized instruction that determines how a model of a particular system should be built. Nevertheless, the lack of precise rules does not prevent experienced specialists from building successful models. To date, significant experience has already been accumulated, which gives grounds to formulate some principles and approaches to building models. When considered separately, each of them may seem rather obvious. But the totality of principles and approaches taken together is far from trivial. Many errors and failures in modeling practice are a direct consequence of the violation of this methodology.

The principles define those General requirements which a well-formed model must satisfy. Let's look at these principles.

Adequacy. This principle provides for the correspondence of the model to the objectives of the study in terms of the level of complexity and organization, as well as the correspondence of the real system with respect to the selected set of properties. Until the question of whether the model correctly displays the system under study is resolved, the value of the model is negligible.

Compliance of the model with the problem being solved. The model should be built to solve a certain class of problems or a specific problem of studying the system. Creation attempts universal model, aimed at solving a large number of various problems, lead to such complication that it turns out to be practically unusable. Experience shows that when solving each specific problem, you need to have your own model that reflects those aspects of the system that are most important in this problem. This principle is related to the principle of adequacy.

Simplification while maintaining the essential properties of the system. The model should be simpler than the prototype in some respects - that is the point of modeling. The more complex the system under consideration, the more simplified, if possible, its description should be, deliberately exaggerating typical ones and ignoring less essential properties. This principle can be called the principle of abstraction from secondary details.

Correspondence between the required accuracy of the simulation results and the complexity of the model. Models by their nature are always approximate. The question arises as to what this approximation should be. On the one hand, to reflect all any significant properties, the model must be detailed. On the other hand, it obviously does not make sense to build a model approaching the complexity of a real system. It should not be so complex that finding a solution is too difficult. A compromise between these two requirements is often achieved through trial and error. practical advice to reduce the complexity of the models are:

change in the number of variables, achieved either by eliminating irrelevant variables or by combining them. The process of transforming a model into a model with fewer variables and constraints is called aggregation. For example, all types of computers in the model of heterogeneous networks can be combined into four types - personal computers, workstations, large computers (mainframes), cluster computers;
changing the nature of the variable parameters. Variable parameters are considered as constants, discrete ones as continuous, etc. Thus, the conditions for the propagation of radio waves in the radio channel model can be taken constant for simplicity;
changing the functional relationship between variables. The non-linear dependence is usually replaced by a linear one, the discrete probability distribution function is replaced by a continuous one;
changing restrictions (adding, deleting or modifying). When restrictions are removed, an optimistic solution is obtained, when introduced, a pessimistic one. By varying the restrictions, one can find possible boundary values of efficiency. This technique is often used to find preliminary estimates of the effectiveness of solutions at the stage of setting tasks;
limiting the accuracy of the model. The accuracy of the model results cannot be higher than the accuracy of the original data.

Balance of errors various kinds. In accordance with the principle of balance, it is necessary to achieve, for example, a balance of the systematic error of modeling due to the deviation of the model from the original and the error of the initial data, the accuracy of individual elements of the model, the systematic error of modeling and the random error in interpreting and averaging the results.

Multivariance of implementations of model elements. A variety of implementations of the same element, differing in accuracy (and, consequently, in complexity), ensures the regulation of the "accuracy / complexity" ratio.

Block structure. If the principle of the block structure is observed, the development of complex models is facilitated and it becomes possible to use the accumulated experience and ready-made blocks with minimal connections between them. The allocation of blocks is carried out taking into account the division of the model into stages and modes of operation of the system. For example, when building a model For a radio intelligence system, one can single out a model for the operation of emitters, a model for detecting emitters, a direction finding model, etc.

Depending on the specific situation, the following approaches to building models are possible:

direct analysis of the functioning of the system;
conducting a limited experiment on the system itself;
use of analogue;
analysis of initial data.

There are a number of systems that allow direct research to identify significant parameters and relationships between them. Then either known mathematical models are applied, or they are modified or proposed new model. Thus, for example, it is possible to develop a model for the direction of communication in peacetime.

During the experiment, a significant part of the essential parameters and their influence on the efficiency of the system are revealed. Such a goal is pursued, for example, by all command post games and most exercises.

If the method of constructing a system model is not clear, but its structure is obvious, then you can use the similarity with a simpler system for which a model exists.

You can start building a model based on the analysis of initial data that are already known or can be obtained. The analysis allows us to formulate a hypothesis about the structure of the system, which is then tested. This is how the first models of a new model of foreign technology appear in the presence of preliminary data on their technical parameters.

Modelers are under the influence of two mutually contradictory tendencies: the desire for completeness of description and the desire to obtain the required results by the simplest possible means. A compromise is usually reached along the path of building a series of models, starting from extremely simple and ascending to high complexity (there is well-known rule: start with simple models, and then complicate). Simple models help to better understand the problem under study. Sophisticated models are used for impact analysis various factors on the simulation results. Such an analysis allows excluding some factors from consideration.

Complex systems require the development of a whole hierarchy of models that differ in the level of displayed operations. Allocate such levels as the whole system, subsystems, control objects, etc.

Consider one specific example— model of economic development (Harrod model). This simplified model of the development of the country's economy was proposed by the English economist R. Harrod. The model takes into account one determinable factor - capital investments, and the state of the economy is estimated through the size of the national income.

For the mathematical formulation of the problem, we introduce the following notation:

Y t is the national income in year t;
K t - production assets in year t;
K t is the volume of consumption in year t;
K t is the amount of savings in year t;
K t - capital investments in year t.

We will assume that the functioning of the economy occurs under the following conditions:

condition for the balance of income and expenses for each year

capital lien exclusion condition

condition of proportional division of the national annual income

Two conditions are accepted to characterize internal economic processes. The first condition characterizes the connection between capital investments and the total amount of production assets, the second - the connection between the national annual income and production assets.

Capital investments in year t can be considered as an increase in production assets or the derivative of the production assets function is taken as annual capital investments:

The national income in each year is taken as the return on production assets with the corresponding standard coefficient of return on assets:

Combining the conditions of the problem, we can obtain the following relation:

Y t = V t /a = dK/(a⋅dt) = b/a⋅dY/dt

From this follows Harrod's final equation:

Y t = b⋅dY/dt = a⋅Y

His solution is to change the national income exponentially over yearly intervals:

Y t = Y 0 ⋅e a⋅t/b

Despite the simplified form of the mathematical model, its result can be used for an enlarged analysis of the national economy. Parameters a and b can become control parameters when choosing a planned development strategy in order to get as close as possible to the preferred trajectory of changes in national income or to choose the minimum time interval for achieving a given level of national income.

Stages of building a mathematical model

The essence of building a mathematical model is that the real system is simplified, schematized and described using one or another mathematical apparatus. The following main stages of model building can be distinguished.

Meaningful description of the modeled object. Modeling objects are described from the standpoint of a systematic approach. Based on the purpose of the study, a set of elements, the relationship between the elements, the possible states of each element, the essential characteristics of the states and the relationship between them are established. For example, it is fixed that if the value of one parameter increases, then the value of another parameter decreases, etc. Issues related to the completeness and uniqueness of the choice of characteristics are not considered. Naturally, in such a verbal description, logical contradictions and uncertainties are possible. This is the original natural-science concept of the object under study. Such a preliminary, approximate representation of the system is called a conceptual model. In order for a meaningful description to serve as a good basis for subsequent formalization, it is required to study the modeled object in detail. Often, the natural desire to speed up the development of the model leads the researcher away from this stage directly to formal questions. As a result, a model built without a sufficient meaningful basis turns out to be unsuitable for use. At this stage of modeling are widely used qualitative methods descriptions of systems, sign and language models.

Formalization of operations. Formalization comes down to in general terms to the next. Based on a meaningful description, the initial set of system characteristics is determined. To highlight the essential characteristics, at least an approximate analysis of each of them is necessary. When conducting an analysis, they rely on the formulation of the problem and understanding the nature of the system under study. After the exclusion of non-essential characteristics, controlled and unmanaged parameters are isolated and symbolization is performed. Then the system of restrictions on the values of controlled parameters is determined. If the restrictions are not of a fundamental nature, then they are neglected.

Further actions are related to the formation of the objective function of the model. In accordance with the known provisions, the indicators of the outcome of the operation are selected and the approximate view utility functions on outcomes. If the utility function is close to the threshold (or monotonic), then the evaluation of the effectiveness of decisions is possible directly by the indicators of the outcome of the operation. In this case, it is necessary to choose the method of convolution of indicators (the method of transition from a set of indicators to one generalized indicator) and perform the convolution itself. Based on the convolution of indicators, an efficiency criterion and an objective function are formed.

If at qualitative analysis In the case of the utility function, it turns out that it cannot be considered a threshold (monotonic), a direct assessment of the effectiveness of decisions through indicators of the outcome of the operation is not authorized. It is necessary to determine the utility function and, on its basis, to form the efficiency criterion and the objective function.

In general, replacing a meaningful description with a formal one is an iterative process.

Checking the adequacy of the model. The requirement of adequacy is in conflict with the requirement of simplicity, and this must be taken into account when checking the model for adequacy. The initial version of the model is preliminarily checked for the following main aspects:

Are all relevant parameters included in the model?
Are there any irrelevant parameters in the model?
Are the functional relationships between the parameters correctly reflected?
Are the restrictions on parameter values correctly defined?

For verification, it is recommended to involve specialists who did not participate in the development of the model. They can view the model more objectively and notice it weaknesses than its developers. This preliminary check of the model reveals gross mistakes. After that, they begin to implement the model and conduct research. The obtained simulation results are analyzed for compliance known properties object under study. To establish the conformity of the created model with the original, the following ways are used:

comparison of simulation results with individual experimental results obtained under the same conditions;
use of other similar models;
comparison of the structure and functioning of the model with the prototype.

The main way to check the adequacy of the model to the object under study is practice. However, it requires the accumulation of statistics, which is not always sufficient to obtain reliable data. For many models, the first two are less acceptable. In this case, there is only one way: to make a conclusion about the similarity of the model and prototype based on a comparison of their structures and implemented functions. Such conclusions are not formal in nature, since they are based on the experience and intuition of the researcher.

Based on the results of checking the model for adequacy, a decision is made on the possibility of its practical use or about making adjustments.

Model correction. When adjusting the model, essential parameters, restrictions on the values of controlled parameters, indicators of the outcome of the operation, links between the indicators of the outcome of the operation with essential parameters, and the efficiency criterion can be specified. After making changes to the model, the adequacy assessment is performed again.

Model optimization. The essence of model optimization is their simplification at a given level of adequacy. The main indicators by which the optimization of the model is possible are the time and cost of research on it. Optimization is based on the ability to transform models from one form to another. The conversion can be done either using mathematical methods, or heuristically.

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1. System modeling. Approaches to the study of modeling

simulation mathematical simulation algorithm

Modeling is the main research method in all fields of knowledge and a scientifically based method for assessing the characteristics of complex systems used to make decisions in various fields of engineering. Model ( measure h-l) is an object-substitute for a real object (original) that provides the study of some properties of the original. The epistemological role in M- lies in its significance in the process of cognition and it is inherent in relation to all models, regardless of their nature. In the process of studying, the model acts as a relatively independent quasi-object, in the study of which one can obtain knowledge about the object itself. If the results of testing are confirmed and can serve as a basis for predicting the processes occurring in the objects under study, then they say that the model is adequate to the object. Historically, there have been two main approaches to modeling processes and systems.

Classical (inductive) considers the system by moving from the particular to the general, i.e. the model of the system is synthesized by merging the models of its components, developed separately. For example, when buying a TV, the buyer is only interested in its external characteristics - diagonal, design, etc., but not its internal part. With a systematic approach, a consistent transition from the general to the particular is assumed, when the model is built on the basis of the goal of the study. It is from it that they proceed, creating a model. The similarity of the process occurring in the model to the real process is not the goal, but only a condition for the correct functioning of the model, therefore, the goal should be to study some aspect of the functional object. For example, when developing a TV, the developer is only interested in inner part, from what details he will assemble it, in what sequence, but not the external characteristics of the TV.

2. Hypothesis, analogy, similarity theory

An object (object) is everything that human activity is aimed at. The purpose of M-I is to obtain knowledge, streamline and process information about objects that exist outside of our consciousness and interact with each other and with the external environment. A hypothesis is a certain prediction, which is based on a small number of experiments, observations, conjectures. Analogy is a judgment about the similarity of two objects, it can be. essential and non-essential. Earth- ball, space. An analogy connects a hypothesis with an experiment. Properties of the hypothesis and analogy: 1. D. have visibility, 2. Convenience of reduction to logical circuits. A model (measure h-l) is an object-substitute for a real object (original) that provides the study of some properties of the original. The theory of M-ia is based on the theory of similarity. Similarity is a characteristic of an object or Comparative characteristics showing the degree and quality of similarity between the object of the original and the model. The theory of similarity asserts that absolute similarity can take place only when one object is replaced by another exactly the same. There is no absolute similarity in the practice of M-iya, they strive for the model to reflect the essential side of the functioning of the object. The full similarity lies in the composition of the complete M-th, which is manifested both in space and in time. When it comes to approximate M-ii, then some properties of the object are not M-at all.

3. Classification of models

Types of models: 1. The pragmatic model is a means of organizing practical action and a working representation of the system chain for its management, such models are usually applied, i.e. reality is adjusted to a certain pragmatic model. Example, working drawings, SNiPs, code of laws, charters of the organization. 2. A cognitive model is a form of organization and representation of knowledge = it is a means of connecting new and old knowledge. Example, geographical maps, toys, production instructions. 3. An instrumental model is a means of constructing a study or using models 1 and 2. According to the "depth" of the model, there are: 1. Empirical (sensory experience) - based on empirical facts and dependencies. 2. Theoretical - based on mathematical descriptions. 3. mixed type(semiempirical) -empirical experience + mathematical description.

4. Model properties

Goal-oriented - the model has a goal.

Finiteness - the characteristics of the model are finite.

Simplicity - any model is simplified, highlighting the main properties of the object.

Approximation - how close the model is to the original object, roughly or approximately.

Adequacy - the model describes the real object well.

Completeness - the model should reflect the studied properties in full.

Stability - the model should describe the stable behavior of the system.

Integrity - the model implements the entire system as a whole.

Adaptability - the model must be adapted to various input parameters, as well as to environmental influences.

Manageability - the model must have at least one parameter that can be varied.

Evolvability - the ability to move the model from one level to another.

5. Stages and scheme of building a model

Model M describing the system S(x1, x2, ..., xn; R) has the form:

М=(z1, z2, ..., zm; Q),

where zi Z, i=1, 2, ..., n, Q, R - sets of relations over X - a set of input, output signals and system states, Z - a set of descriptions, representations of elements and subsets of X. Scheme for constructing a model M of the system S with input signals X and output signals Y is shown in fig. one

Rice. 1 Scheme of building a model

If signals from X are received at the input M and signals Y appear at the input, then the law, the rule f of the functioning of the model, system is set.

Stages of building a mathematical model

Formalization of operations. Based on a meaningful description, the initial set of system characteristics is determined. After the exclusion of non-essential characteristics, controlled and unmanaged parameters are isolated and symbolization is performed. Then the system of restrictions on the values of controlled parameters is determined. Further actions are related to the formation of the objective function of the model. In accordance with well-known provisions, indicators of the outcome of the operation are selected and an approximate form of the utility function on the outcomes is determined. According to the convolution of indicators,

Checking the adequacy of the model. The requirement of adequacy is in conflict with the requirement of simplicity, and this must be taken into account when checking the model for adequacy. The initial version of the model is preliminarily checked for the following main aspects:

Are all relevant parameters included in the model?

Are there any irrelevant parameters in the model?

Are the functional relationships between the parameters correctly reflected?

Are the restrictions on parameter values correctly defined?

Based on the results of checking the model for adequacy, a decision is made on the possibility of its practical use or on an adjustment.

Model correction. When adjusting the model, essential parameters, restrictions on the values of controlled parameters, indicators of the outcome of the operation, links between the indicators of the outcome of the operation with essential parameters, and the efficiency criterion can be specified. After making changes to the model, the adequacy assessment is performed again.

Model optimization. The essence of model optimization is their simplification at a given level of adequacy. The main indicators by which the optimization of the model is possible are the time and cost of research on it. Optimization is based on the ability to transform models from one form to another.

6. Life cycle of the simulated system

Collection of information about the system - hypotheses, pre-model analysis.

Structural design - determination of the composition of models and the relationship of submodels.

Model research - the choice of research method and the development of a modeling algorithm.

Study of adequacy, stability and other properties.

Cost estimate or simulation resource estimate.

Create report and design solutions.

Modification of the model (adding new knowledge or application in another area).

7. Types of modeling

Deterministic modeling displays processes in which the absence of random influences is assumed. Stochastic modeling takes into account probabilistic processes and events. Static modeling is used to describe the state of an object at a fixed point in time, while dynamic modeling is used to study an object over time. Mental modeling is used when models are not realizable in a given time interval or there are no conditions for their physical creation (for example, the situation of the microworld). With visual modeling based on human ideas about real objects, visual models are created that display the phenomena and processes occurring in the object. An example of such models are educational posters, drawings, charts, diagrams. Hypothetical modeling is based on a hypothesis about the patterns of the process in a real object, which reflects the level of knowledge of the researcher about the object and is based on cause-and-effect relationships between the input and output of the object under study. This type of modeling is used when knowledge about the object is not enough to build formal models. Analog modeling is based on the application of analogies of various levels. For sufficiently simple objects, the highest level is complete analogy. Modeling is used when the processes occurring in a real object are not amenable to physical modeling or may precede other types of modeling. Symbolic modeling is an artificial process of creating a logical object that replaces the real one and expresses its main properties using a certain system of signs and symbols. The basis of language modeling is a certain thesaurus, which is formed from a set of concepts of the studied subject area, and this set must be cleared of ambiguity. If we introduce a symbol for individual concepts, i.e. signs, as well as certain operations between these signs, then you can implement sign modeling and use signs to display a set of concepts - to make separate chains of words and sentences. Mathematical modeling is the process of establishing correspondence to a given real object of some mathematical object, called a mathematical model. Analytical modeling is characterized by the fact that basically only the functional aspect of the system is modeled. Simulation modeling reproduces the algorithm of the system functioning in time - the behavior of the system, and the elementary phenomena that make up the process are simulated, while preserving their logical structure and sequence of flow, which makes it possible to obtain information about the states of the process at certain points in time from the initial data, making it possible to evaluate the characteristics systems. Combined (analytical-simulation) modeling allows you to combine the advantages of analytical and simulation modeling. Information (cybernetic) modeling is associated with the study of models in which there is no direct similarity of the physical processes occurring in the models to real processes. In this case, they seek to display only some function, consider the real object as a “black box” with a number of inputs and outputs, and model some connections between outputs and inputs. Structural modeling of system analysis is based on some specific features of structures of a certain type, which are used as a means of studying systems or serve to develop specific approaches to modeling based on them using other methods of formalized representation of systems (set-theoretic, linguistic, cybernetic, etc. ). Situational modeling is based on the model theory of thinking, within which it is possible to describe the main mechanisms for regulating decision-making processes. In real modeling, the possibility of studying the characteristics either on a real object as a whole or on its part is used. Full-scale simulation is called conducting a study on a real object with subsequent processing of the experimental results based on the theory of similarity. Full-scale simulation is subdivided into a scientific experiment, complex tests and a production experiment. A scientific experiment is characterized by the widespread use of automation tools, the use of a wide variety of information processing tools, and the possibility of human intervention in the process of conducting an experiment. One of the varieties of the experiment is complex tests, during which, as a result of repeating tests of objects as a whole (or large parts of the system), general patterns are revealed about the quality characteristics and reliability of these objects. In this case, modeling is carried out by processing and generalizing information about a group of homogeneous phenomena. Along with specially organized tests, it is possible to implement full-scale simulation by summarizing the experience gained during the production process, i.e. we can talk about a production experiment. Another type of real simulation is physical, which differs from natural in that the study is carried out in installations that preserve the nature of phenomena and have a physical similarity.

8. Model of physical, economic, physiological systems

Model - an object or description of an object, a system for replacing (under certain conditions, proposals, hypotheses) one system (i.e. the original) with another system for better study of the original or reproduction of any of its properties. The model is the result of mapping one structure (studied) to another (little studied). By mapping a physical system (object) onto a mathematical system (for example, the mathematical apparatus of equations), we obtain a physical and mathematical model of the system or a mathematical model of a physical system. Any model is built and studied under certain assumptions, hypotheses. Example. Consider a physical system: a body of mass m rolling down an inclined plane with an acceleration a, which is affected by a force F. Investigating such systems, Newton obtained a mathematical relation: F=ma. This is a physical and mathematical model of a system or a mathematical model of a physical system. When describing this system (building this model), the following hypotheses were adopted: 1) the surface is ideal (i.e., the coefficient of friction zero); 2) the body is in a vacuum (i.e., air resistance is zero); 3) body weight is unchanged; 4) the body moves with the same constant acceleration at any point. Example. The physiological system - the human circulatory system - obeys certain laws of thermodynamics. Describing this system in the physical (thermodynamic) language of balance laws, we obtain a physical, thermodynamic model of a physiological system. If we write down these laws in mathematical language, for example, write out the corresponding thermodynamic equations, then we will already get a mathematical model of the circulatory system. Let's call it a physiological-physical-mathematical model or a physical-mathematical model. Example. The set of enterprises operates in the market, exchanging goods, raw materials, services, information. If we describe economic laws, the rules of their interaction in the market using mathematical relationships, for example, a system of algebraic equations, where the unknowns will be the profits received from the interaction of enterprises, and the coefficients of the equation will be the values of the intensities of such interactions, then we get a mathematical model economic system, i.e. economic and mathematical model of the system of enterprises in the market. Example. If a bank has developed a lending strategy, was able to describe it with the help of economic and mathematical models and predicts its lending tactics, then it has greater stability and viability.

9. Classification of mathematical models

A model is called static if there is no time parameter among the parameters involved in its description. The static model at each moment of time gives only a "photo" of the system, its slice. Example. Newton's law F=am is a static model of a moving with acceleration a material point mass m. This model does not take into account the change in acceleration from one point to another. The model is dynamic if among its parameters there is a time parameter, i.e. it displays the system (processes in the system) in time. Example. Model S=gt2/2 - dynamic model of the path in free fall of the body. Dynamic model like Newton's law: F(t)=a(t)m(t). A model is discrete if it describes the behavior of the system only in discrete moments time. Example. If we consider only t=0, 1, 2, :, 10 (sec), then the model St=gt2/2 or the numerical sequence S0=0, S1=g/2, S2=2g, S3=9g/2, :, S10=50g can serve as a discrete model of the motion of a freely falling body. A model is continuous if it describes the behavior of the system for all points in time from some time interval. Example. Model S=gt2/2, 0 possible ways development and behavior of the object by varying some or all of the parameters of the model. The model is deterministic if each input set of parameters corresponds to a well-defined and uniquely determined set of output parameters; otherwise, the model is non-deterministic, stochastic (probabilistic). Example. The above physical models are deterministic. If in the model S=gt2/2, 0 traffic- linguistic, structural model traffic and pedestrians on the roads. The model is visual if it allows visualizing the relationships and connections of the modeled system, especially in dynamics. Example. On a computer screen, a visual model of an object is often used, for example, a keyboard in a keyboard training simulator. The model is full-scale if it is a material copy of the modeling object. Example. Globe - full-scale geographical model of the globe. The model is geometric, graphic, if it can be represented by geometric images and objects. Example. The layout of the house is a full-scale geometric model of a house under construction. A model is cellular automaton if it represents a system using a cellular automaton or a system of cellular automata. A cellular automaton is a discrete dynamic system, an analogue of a physical (continuous) field. Cellular automata geometry is an analogue of Euclidean geometry.

10. Requirements for mat. models

The main requirements for mathematical models are the requirements of accuracy, efficiency and versatility.

The MM accuracy is a property that reflects the degree of agreement between the values of the object parameters predicted by the model and the true values of these parameters.

The profitability of MM is estimated primarily by the cost of machine time Tm (its cost determines the main part of the cost costs). The contribution of the mathematical model to the costs Tm for solving problems can be estimated by the number of arithmetic operations performed with a single implementation of the model equations. The number of internal parameters used in it can also serve as an indicator of the efficiency of the MM. The more such parameters, the greater the cost of computer memory, therefore, the more effort is required to obtain information about the numerical values of the parameters and their spread.

The degree of MM universality is determined by their applicability to the analysis of a more or less numerous group of objects of the same type, to their analysis in one or many modes of operation. The use of machine methods will become inconvenient if, in the process of analyzing an object, each change in the operating mode requires a change in the MM.

11. The economic effect of the mat. modeling

MM is the process of establishing correspondence between the real system S and the mathematical model M and the study of this model, which makes it possible to obtain the characteristics of the real system.

The use of MM makes it possible to study objects on which real experiments are difficult or impossible. The economic effect of MM is that the cost of system design is reduced by an average of 50 times.

12. Mat. modeling sl. System

The element s is some object that has certain properties, internal structure which, for the purposes of the study, does not play a role (aircraft: for flight simulation it is not an element, but for airport operation simulation it is an element). Communication l between elements is the process of their interaction, which is important for the purposes of the study. System S - a set of elements with connections and the purpose of functioning F. A complex system - consisting of elements of different types with different types of connections.

big system- consisting of a large number of elements of the same type with the same type of connections.

System: An automated system is a complex system with a decisive role of two types of elements: technical means (primarily computers) and human actions:

here are the remaining elements of the system. The structure of the system is its division (decomposition) into elements or groups of elements, indicating the links between them, unchanged during the operation of the system. Almost all systems are considered to be functioning in time, so we will determine their dynamic characteristics. State - a set of characteristics of system elements that change over time and are important for the purposes of functioning. Process (dynamics) - a set of values of system states that change over time. The purpose of functioning is the task of obtaining the desired state of the system. Achieving the goal usually entails a targeted intervention in the process of functioning of the system, which is called management.

System Research Tasks:

analysis - the study of the properties of the functioning of the system;

synthesis - the choice of structure and parameters according to the given properties of the system.

13. The problem of assessing the external environment. Black box problem

14. Basic operations mat. modeling

The mathematical model is described (represented) by mathematical structures, mathematical apparatus (numbers, letters, geometric images, relations, algebraic structures, etc.).

Let us note the main operations (procedures) of mathematical modeling.

1. Linearization. Let the mathematical model M=M(X, Y, A) be given, where X is the set of inputs, Y is the set of outputs, A is the set of system states. Schematically, this can be depicted as follows: X->A->Y. If X, Y, A are linear spaces (sets), and are linear operators (i.e., any linear combinations of ax + by arguments and are converted into the corresponding linear combinations and, then the system (model) is called linear. All other systems (models ) are non-linear. They are more difficult to study, although they are more relevant. Non-linear models are less studied, therefore they are often linearized - they are reduced to linear models in some way, by some correct linearizing procedure.

Example. Let's apply the linearization operation to the model (what physical system, phenomenon?) y=at2/2, 0<=t<=4, которая является нелинейной (квадратичной). Для этого заменим один из множителей t на его среднее значение для рассматриваемого промежутка, т.е. на t=2. Такая (пусть простят меня знакомые с линеаризацией читатели, - хоть и очень наглядная, но очень грубая!) процедура линеаризации дает уже линейную модель вида y=2at. Более точную линеаризацию можно провести следующим образом: заменим множитель t не на среднее, а на значение в некоторой точке (это точка - неизвестная!); тогда, как следует из теоремы о среднем из курса высшей математики, такая замена будет достаточно точна, но при этом необходимо оценить значение неизвестной точки. На практике используются достаточно точные и тонкие процедуры линеаризации.

2. Identification. Let M=M(X, Y, A), A=(ai), ai=(ai1, ai2, ..., aik) be the object (system) state vector. If the vector ai depends on some unknown parameters, then the problem of identification (of the model, model parameters) is to determine, according to some additional conditions, for example, experimental data characterizing the state of the system in some cases. Identification is the task of constructing, based on the results of observations, mathematical models of a certain type that adequately describe the behavior of the system. If S=(s1, s2, ..., sn) is a certain sequence of messages received from a source of information about the system, M=(m1, m2, ..., mz) is a sequence of models describing S, among which, perhaps , contains an optimal (in some sense) model, then the identification of the model M means that the sequence S makes it possible to distinguish (according to the criterion of adequacy under consideration) two different models of the VM. A sequence of messages (data) S is called informative if it allows one to distinguish between different models in M. The purpose of identification is to build a reliable, adequate, efficiently functioning flexible model based on the minimum volume of an informative sequence of messages. The most commonly used methods for identifying systems (system parameters): least squares method, maximum likelihood method, Bayesian estimation method, Markov chain estimation method, heuristics method, expert evaluation and others.

Example. Let's apply the operation of identifying the parameter a in the model of the previous example. To do this, you must additionally set the value of y for some t, for example, y=6 for t=3. Then from the model we get: 6=9a/2, a=12/9=4/3. The identified parameter a defines the following model y=2t2/3. Model identification methods can be disproportionately more complicated than the above technique.

3. Assessment of the adequacy (accuracy) of the model.

Example. Let us estimate the adequacy (accuracy) of the model y=at2/2, 0<=t<=4, полученной в результате линеаризации выше. В качестве меры (критерия) адекватности рассмотрим привычную меру - абсолютное значение разности между точным (если оно известно) значением и значением, полученным по модели (почему берется по модулю?). Отклонение точной модели от линеаризованной будет в рамках этого критерия равно |at2/2-2at|, 0<=t<=4. Если a>0, then, as it is easy to estimate using the derivative, this error will be extreme at t=2a. For example, if a=1, then this value does not exceed 2. This is a rather large deviation, and we can conclude that our linearized model in this case is not adequate (both to the original system and to the non-linearized model).

4. Assessment of the sensitivity of the model (sensitivity to changes in input parameters).

Example. It follows from the previous example that the sensitivity of the model is y=at2/2, 0<=t<=4 такова, что изменение входного параметра t на 1% приводит к изменению выходного параметра y на более, чем 2%, т.е. эта модель является чувствительной.

5. Computational experiment on the model. This is an experiment carried out with the help of a model on a computer in order to determine, predict certain states of the system, and respond to certain input signals.

15. Computer simulation. Stages

Computer modeling is a formulation in the form of an algorithm for the modeling process that allows you to calculate over the resulting model. Experiment (computer program)

1st statement of the problem includes the stages: description of the problem, determination of the purpose of modeling, analysis of the object.

2 task formalization is associated with the creation of a formalized model, that is, a model written in some formal language. For example, census data presented in the form of a table or chart is a formalized model.

3, the development of a computer model begins with the choice of a modeling tool, in other words, a software environment in which the model will be created and studied.

4 computer experiment includes two stages: model testing and research.

5 analysis of the results is key to the modeling process. It is at the end of this stage that the decision is made: to continue the study or to end it.

16. Simulation

Simulation modeling is a research method in which the system under study is replaced by a model that describes the real system with sufficient accuracy, with which experiments are carried out in order to obtain information about this system. Experimenting with a model is called imitation (imitation is the comprehension of the essence of a phenomenon without resorting to experiments on a real object).

Simulation modeling is a special case of mathematical modeling. There is a class of objects for which, for various reasons, analytical models have not been developed, or methods for solving the resulting model have not been developed. In this case, the analytical model is replaced by a simulator or simulation model.

Simulation is used when:

it is expensive or impossible to experiment on a real object;

it is impossible to build an analytical model: the system has time, causal relationships, consequences, non-linearities, stochastic (random) variables;

it is necessary to simulate the behavior of the system in time.

The purpose of simulation modeling is to reproduce the behavior of the system under study based on the results of the analysis of the most significant relationships between its elements, or in other words, to develop a simulator of the studied subject area for conducting various experiments.

Simulation modeling allows you to simulate the behavior of a system over time. Moreover, the advantage is that the time in the model can be controlled: slow down in the case of fast processes and speed up for modeling systems with slow variability. It is possible to imitate the behavior of those objects with which real experiments are expensive, impossible or dangerous.

17. Problems of systems research

Analysis - the study of the properties of the functioning of the system.

Synthesis - the choice of structure and parameters according to the given properties of the system.

Let T = be the time interval for modeling the system S. The construction of the model begins with the determination of the parameters and variables that determine the process of the system functioning. System parameters Q1, Q2, . . ., Qm - characteristics of the system that remain constant over the entire interval T. For example, the parameters of the diameter of the gear wheel. Variables are either dependent or independent. Independent variables are input actions, incl. and controlling + environmental influences. The sequence of change x(t) at t1t2…tN is called the phase trajectory of the system, where xX, where X is the state space or phase space. Dependent variables are output characteristics (signals). The general scheme of the mathematical model (MM) of the system functioning can be represented as mathematical model of the system. If t is continuous, then the model is called continuous. If the model does not contain random elements, then it is called deterministic, otherwise - stochastic. If the mathematical description of the model is too complex and partially or completely uncertain, then in this case aggregative models are used. The essence of the aggregative model is to divide the system into a finite number of interconnected parts (subsystems), each of which allows a standard mathematical description. These subsystems are called aggregates.

18. Methods for simulating random variables. Monte Carlo method

Simulation modeling allows you to reproduce the process of functioning of the system in time with the preservation of elementary phenomena, their logical structure and the sequence of flow in time. This allows you to obtain information about the states of the process in the future at certain points in time from the source data. The Monte Carlo method is the general name for a group of numerical methods based on obtaining a large number of implementations of a stochastic (random) process, which is formed in such a way that its probabilistic characteristics coincide with similar values of the problem being solved.

The essence of the Monte Carlo method is as follows: it is required to find the value in some studied quantity. To do this, choose such a random variable X, the mathematical expectation of which is equal to a: M(X)=a. The numerical method that solves the problem of generating a sequence of random numbers with given distribution laws is called the static test method - the Monte Carlo method. H: after what time the milling machine will fail. Monte Carlo algorithm: 1. Formation of uniformly distributed random variables. 2. Transformation of uniformly distributed quantities into a sequence with a given law. 3. Calculation of the reaction of the process or system object to random influences using appropriate methods. 4. Static processing. Random is a quantity that, as a result of the test, will take one and only one possible value, which is not known in advance and depends on random variables that cannot be taken into account in advance. Random variables can be: discrete and continuous. x, y, z - random variables, xi, yi, zi - possible values of SW. Discrete (continuous) is a random variable. Which takes on separate possible values xi, i=1, n i=1,? with certain probabilities. Continuous is a random variable that can take on all values from some finite or infinite interval, and the value of this interval can take on infinite values. The distribution law of a discrete SW is the correspondence between its possible values and the probabilities of their occurrence. Z-n distributions can be created tabular, analytically (in the form of a f-ly), graphically (in the form of a distribution polygon). X1, X2,…Xn-possible values of SW. P1, P2…Pn-probabilities of SW occurrence. Binomial distribution defined by Bernoulli's law. Pn(k)=-Bernoulli distribution law. k is the number of possible occurrences of the event. q=1-p-probability of not occurrence of an event. The Poisson distribution is defined by the asymptotic Poisson formula. Pn(k)=(lk*e-l)/k!, where l is the intensity of the flow of events, shows with what interval the SWs go. Graphic way. More universal is the integral distribution function. It allows you to set both discrete and continuous CV. Integral The distribution function is a function F(x) that determines for each possible value x the probability that the CV x will take a value less than xi-1. IFR property: 1. The value of the IFR belongs to the interval 0?F(x)?1. 2. The probability that the random variable x will take a value from the interval is equal to the increment of the integral distribution function on this interval P(a? F(x)?b)=F(b)-F(a). 3. If all possible values of x CB belong to the interval , then F(x)=0 if x?a, and F(x)=1 if x?b. The geometric meaning of the integral is finding the area of a curvilinear trapezoid. The mathematical expectation of a random variable is a non-random, constant value, it characterizes the average value of a random variable. Holy Math. Expectations: 1. M(S)=S-mat. Waiting for a constant = the constant itself.2. M(CX)=C*M(X) 3. M(SU)=M(X)*M(Y) 4. M(X+Y)=M(X)+M(Y). The dual simplex method - the use of the ideas of duality in combination with the general idea of the simplex method allowed us to develop another method for solving linear programming problems - the duality of systems. Invented by Lemke in 1954. The solution by this method is reduced to finding the optimal plan of the direct problem by successive transition from one basis to another. Maximize linear programming problems in canonical form under boundary conditions. Max((x))=.

19. Basic sensor. Database Requirements

The base sensor is a kind of generator that produces random values. The database generates independent uniformly distributed random variables: continuous (0;1) and discrete. Database types: physical (practically not used due to the fact that the characteristics are unstable and the implementation cannot be repeated - voice recording on a voice recorder) and pseudo-random sensors based on a deterministic algorithm (the data obtained are indistinguishable from random ones). Database requirements: aperiodicity interval, uniformity and non-correlation.

20. Evolutionary modeling. Main attributes of EM

The need for a forecast and an adequate assessment of the consequences of activities carried out by a person leads to the need for a dynamic change in the main parameters of the system of the dynamics of the interaction of an open system with its environment with which the exchange of resources is carried out in conditions of hostile, competitive, cooperative or indifferent relationships. This requires a systematic approach, effective methods and criteria for assessing the adequacy of models, which are aimed not only at maximizing the criteria (profit, profitability), but also at optimizing the relationship with the environment.

For a long-term forecast, it is necessary to identify and study the complete and informative system of parameters of the system under study and its environment, develop a methodology for introducing measures of informativeness and proximity of the state of the system. However, some criteria and measures may often conflict with each other. Many such socio-economic systems can be described from a unified position by means and methods of a unified theory - Evolutionary. In EM, the process of M-ing of a complex system is reduced to creating a model of its evolution or to searching for admissible states of the system, or to a procedure (algorithm) for tracking a set of admissible states (trajectories). Attributes of logical evolutionary dynamics: 1. Community (corporation, corporate objects, subjects, environment). 2. Species diversity and distribution in the ecological lower (types of resource distribution, the structure of communication in a given corporation). 3. Ecological inferior (sphere of influence and functioning of evolution in the market and in business). 4. Birth and death rates (production and destruction).5. Variability (in the economic environment, resources).6. Competitive relations (market relations) .7. Memory (ability to reproduce cycles - archive, database). 8. Natural selection (penalty and incentive measures). 9. Heredity (production cycles and their background). 10. Regulation (investment). 11. Self-organization and the desire of the system to maximize contact with the environment, in order to self-organize, return to the trajectory of sustainable development. N: man.

21. Main trends in the study of evolutionary systems

When studying the evolution of a system, it is necessary to decompose (partition) it into subsystems in order to ensure effective interaction with the environment; optimal exchange of determining material, energy, information, organizational resources with subsystems; evolution of the system under conditions of dynamic change and reordering of goals, structural activity and complexity of the system; controllability of the feedback system. Activity might be structural and business. Let there be some system S with N subsystems, for each i subsystem we define the vector =(x1, x2, x3…xn) - the vector of the main parameters without which it is impossible to describe and study the functioning of the subsystem in accordance with the goals and available resources of the system. We introduce a certain function S=S(x), which we will call the function of system activity. For the whole system, the state vector of the system X, the activity of the system S(x), as well as the concept of the total potential of the system are defined. The activity potential might be determined using the integral of the activity on a given time interval M-s. These functions reflect the intensity of the processes, both in max subsystems and in the system as a whole. Three values of the activity of the i subsystem are important for the tasks of M-i: Smax, Smin, Sopt. If an open economic system is given, and H0 and H1 (this is the entropy of the system in the initial and final states, then the measure of information is defined as the difference of the form: DN = H0-H1. Entropy is a decrease in uncertainties. A decrease in DN indicates that the system is approaching a state of static equilibrium, with available resources, and the increase is about moving away from the state of static equilibrium.The value of DA is the amount of information needed to move from one level of organization of the system to another at DN>0 to a higher one, at DN<0 к более низкой организации. Рассмотрим подход с использованием меры по Моиссеиву. Пусть дана нек-рая управляемая система о состояниях к-рой известно лишь нек-рые оценки: нижняя Smin, верхняя Smax, известна целевая функция управления F (2 параметра: S(t)-состояние системы в момент времени t; U(t)-управление из нек-рого множества допустимых управлении, причем t00. 3. Stationarity - the choice or definition of a function of the state of the system is carried out in such a way that the system has points of equilibrium state, and Sopt would be achieved at stationary points Xopt for short periods of time, in long periods of time the system can behave chaotically, spontaneously generating regular , cyclic, ordered interactions (Deterministic house). Mutual activities of subsystems are not taken into account as a state function, it is efficient to use functions of the Cobb-Douglas type. In such functions, the parameter bi is important, reflecting the degree of self-regulation, adaptation of the system, as a rule, it must be identified. The principle of EM implies the necessity and efficiency of using the methods and technologies of artificial intelligence, in particular, expert systems. An adequate means of implementing EM procedures is genetic algorithms.

22. Genetic algorithms. Basic procedures

A genetic algorithm is an algorithm based on the imitation of genetic procedures for the development of a population in accordance with the principles of evolutionary dynamics. They are used to solve optimization problems, for search and control problems, these algorithms are adaptive, they develop solutions and develop themselves. Feature: successful use in solving complex problems.

The genetic algorithm can be built on the basis of the following enlarged procedure:

We generate an initial population (a set of feasible solutions to the problem) - I0 = (i1, i2, :, in), ij (0,1) and determine some criterion for achieving a "good" solution, the stopping criterion, the SELECTION procedure, the CROSSING procedure, the MUTATION procedure, and population update procedure UPDATE;

k = 0, f0 = max(f(i), i I0);

execute until() :

using a probabilistic operator (selection), we select two feasible solutions (parents) i1, i2 from the selected population (calling the SELECTION procedure);

based on these parents, we construct a new solution (calling the CROSSING procedure) and obtain a new solution i;

modify this solution (calling the MUTATION procedure);

if f0< f(i) то f0 = f(i);

update the population (calling the UPDATE procedure);

Similar procedures are defined using similar procedures of wildlife (at the level of knowledge about them that we have). The SELECTION procedure can select the element with the largest value of f(i) from random elements of the population. The CROSSING procedure (crossover) can build a vector i by vectors i1, i2, assigning with a probability of 0.5 the corresponding coordinate of each of these vectors - parents. This is the simplest procedure. More complex procedures are also used that implement more complete analogues of genetic mechanisms. The MUTATION procedure can also be simple or complex. For example, a simple procedure with a given probability for each vector reverses its coordinates (0 to 1, and vice versa). The UPDATE procedure is to update all elements of the population in accordance with the specified procedures.

Although genetic algorithms can be used to solve problems that cannot be solved by other methods, they are not guaranteed to find the optimal solution, at least not in a reasonable time. Criteria like "good enough and fast enough" are more appropriate here.

The main advantage is that they allow solving complex problems for which stable and acceptable methods have not yet been developed, especially at the stage of formalizing and structuring the system.

23. Basic rules and operators of the GPSS language

To describe the simulation model in the GPSS language, it is useful to present it in the form of a diagram that displays the elements of queuing systems - devices, drives, nodes and sources. The description in the GPSS language is a set of operators (blocks) that characterize the processing of applications. Operators are also available for displaying the occurrence of requests, their delay in service devices, memory occupancy, exiting queuing systems, changing the parameters of requests (for example, priorities), printing accumulated information characterizing the loading of devices, fullness of queues, etc. Each transaction present in the model can have up to 12 parameters. There are operators that can be used to change the values of any parameters of transactions, and operators, the nature of the execution of which depends on the values of one or another parameter of the serviced transaction. The paths for advancing requests between the service devices are displayed by a sequence of operators in the description of the model in the GPSS language by special control transfer (transition) operators. The event method is used for modeling. Compliance with the correct time sequence of simulation of events in the queuing system.

Main Operators:

ADVANCE - transaction delay, ASSIGN - assign a value to the transaction parameter, CLEAR - clear the model, transition to the initial state, COUNT - count the number of elements in the group, DELETE - delete the line (s) of the model, DEPART - exit the transaction from the queue, END - exit from GPSS\PC, ENTER - transaction exit from memory, GATHER - synchronization of the movement of transactions, GENERATE - generation of transactions,

LEAVE - exit the transaction from memory, LOOP - repeat the cycle, PLOT - display the NAV graph in the data window during simulation, QUEUE - enter the transaction into the queue, RELEASE - release the busy device,

SEIZE - occupation of the device by a transaction, SIMULATE - declarations of the model execution mode (rudiment from GPSS-360), STORAGE - description of the memory capacity, TERMINATE - destruction of the transaction, TRANSFER - transfer of the transaction,

24. Function Interpolation

Interpolation, interpolation - in computational mathematics, a way to find intermediate values of a quantity from an existing discrete set of known values.

Given a table function

Points with coordinates (xi, yi) are called nodal points or nodes.

The number of nodes in the table function is N = n+1.

The length of the segment is (xn - x0).

In the calculation practice of an engineer, problems often arise to find the value of a function for arguments that are not in the table. Such problems are called interpolation or extrapolation problems.

The problem of function interpolation (or interpolation problem) is to find the values yk of the table function at any intermediate point xk located inside the interval , i.e.

The function extrapolation task (or the extrapolation task) is to find the values yl of the table function at the point xl, which is not included in the interval , i.e.

This problem is often called the forecast problem.

Both of these problems are solved by finding an analytical expression for some auxiliary function F(x), which would approximate the given table function, i.e. at nodal points would take the value of table functions

For definiteness of the problem, the required function F(x) will be sought from the class of algebraic polynomials:

This polynomial must pass through all the nodal points, i.e.

Therefore, the degree of the polynomial n depends on the number of nodal points N and is equal to the number of nodal points minus one, i.e. n=N-1.

A polynomial of the form (1.2) that passes through all the nodal points of a table function is called an interpolation polynomial.

Interpolation using algebraic polynomials is called parabolic interpolation.

Thus, to solve the interpolation problem, first of all, it is necessary to solve the problem, which can be formulated as follows:

For a function given in a table, construct an interpolation polynomial of degree n that passes through all the node points of the table:

where n is the degree of the polynomial, equal to the number of nodal points N minus one, i.e. n=N-1.

As a result, at any other intermediate point xk located inside the segment, the approximate equality Pn(xk) = f(xk) yk is fulfilled.

25. Lagrange interpolation. Newton interpolation

...

Modeling of processes and systems

SIMULATION OF PROCESSES AND SYSTEMS

The textbook covers the basics of modeling processes and systems. The principles of mathematical and computer modeling of systems are stated. The basic theoretical information about the generation of random sequences, criteria for checking the randomness of observations are considered. The main aspects of statistical modeling, modeling of Markov random processes, identification of objects, solving problems of deterministic linear optimal control, principles of constructing modeling algorithms on the examples of queuing systems are outlined. The main provisions of the simulation of random processes, processing of simulation results and much more are given.

1. Fundamentals of system modeling .. 4

1.1. Models and modeling. 4

1.2. Applied aspects of modeling. 14

1.3. Basic properties of the model and simulation. sixteen

2. Mathematical and computer modeling. nineteen

2.1. Classification of types of modeling. nineteen

2.2. Mathematical modeling of complex systems.. 21

2.3. Simulation of random variables and processes. 25

2.4. Fundamentals of mathematical modeling. 27

2.5. Computer modelling. 32

3. Evolutionary modeling and genetic algorithms.. 39

3.1. Basic attributes of evolutionary modeling. 39

3.3. Genetic algorithms.. 45

4. Generation of random sequences. 48

4.1. Generating uniformly distributed random numbers. 48

4.2. Basic criteria for checking random observations. 56

4.3. empirical criteria. 60

4.4. Numerical distributions. 63

4.5. Signs of a random sequence. 67

5. Statistical modeling. 69

5.1. Introduction. 69

5.2. Normal distribution. 70

5.3. Maximum Likelihood Estimation. 73

5.4. Least square method. 74

6. Markov chains. 77

6.1. Markov process with discrete time.. 78

6.2. Markov stochastic processes with continuous time.. 87

6.3. Mathematical apparatus of the theory of Markov chains. 91

6.4. Typical problems of using Markov chains. 93

6.5. Determination of the matrix M of the average transition time. 97

7. Canonical expansion of a random process. 104

7.1. Theoretical information. 104

7.2. Canonical expansion of a random process in problems. 105

8. Identification of dynamic objects. 108

8.1. General provisions for the identification of mathematical models. 108

8.2. Generalized identification procedure. 109

9. Problems of deterministic linear optimal control. 120

9.1. Theoretical information. 120

9.2. Solution of control problems using the Riccati equation. 121

10. General principles for constructing modeling algorithms. 134

10.1. The Δt principle. 135

10.2. The principle of special states. 140

10.3. The principle of sequential posting of applications. 142

10.5. Object modeling principle. 147

11. Imitation of random processes. 149

11.1. Simulation of non-stationary random processes. 149

11.2. Imitation of stationary joint ventures.. 150

11.3. Imitation of stationary normal joint ventures.. 151

12. Processing of simulation results. 153

12.1. Probability estimation. 153

12.2. Estimation of mathematical expectation and variance. 154

12.3. Evaluation of the characteristics of a random process. 154

12.4. The number of implementations that provide the specified accuracy. 155

13. Stochastic linear optimal control. 157

13.1. Theoretical foundations of stochastic regulation. 157

13.2. Solving problems of stochastic linear optimal control. 159

Literature. 166

1. Fundamentals of system modeling

1.1. Models and Simulation

Model and modeling- universal concepts, attributes of one of the most powerful methods of cognition in any professional field, cognition of a system, process, phenomenon.

View models and the methods of its research depend more on the information-logical connections of the elements and subsystems of the modeled system, resources, connections with the environment, and not on the specific content of the system.

The model style of thinking allows you to delve into the structure and internal logic of the modeled system.

Building models- a system task that requires analysis and synthesis of initial data, hypotheses, theories, knowledge of specialists. A systematic approach allows not only to build model real system, but also use this model to evaluate (for example, the effectiveness of management or operation) of the system.

Model - this is an object or description of an object, a system for replacing one system (original) with another system for better study of the original or reproduction of any of its properties.

For example, by mapping a physical system onto a mathematical system, we obtain the mathematical model physical system. Any model is constructed and investigated under certain assumptions, hypotheses.

Example. Consider a physical system: a body with mass m rolls down an inclined plane with acceleration a , which is affected by the force F .

Investigating such systems, Newton obtained a mathematical relation: F = m*a. This is a physical and mathematical model systems or mathematical model the physical system of the rolling body.

When describing this system, the following hypotheses were adopted:

the surface is ideal (the coefficient of friction is zero);

classification models carried out according to different criteria.

Model called static , if there is no time parameter among the parameters participating in its description. Static model at each moment of time gives only a "photo" of the system, its slice.

Example. Newton's law F=a*m is static model moving with acceleration a material point mass m. This model does not take into account the change in acceleration from one point to another.

Model dynamic , if among its parameters there is a time parameter, i.e. it displays the system (processes in the system) in time.

Example. Dynamic Model Newton's law will be:

Model discrete , if it describes the behavior of the system only at discrete times.

Example. If we consider only t=0, 1, 2, …, 10 (sec), then model

or numerical sequence: S0=0, S1=g/2, S2=2g, S3=9g/2, :, S10=50g can serve as discrete model motion of a freely falling body.

Model continuous , if it describes the behavior of the system for all times of a certain time interval.

Example. Model S=gt2/2, 0< t < 100 непрерывна на промежутке времени (0;100).

Modelsimulation, if it is intended to test or study possible ways of development and behavior of an object by varying some or all of the parameters of the model.

Example. Let model economic system for the production of goods of two types 1 and 2, in the amount x1 and x2 units and the cost of each unit of goods a1 and a2 at the enterprise is described as a ratio:

a1x1 + a2x2 = S,

where S is the total cost of all products produced by the enterprise (types 1 and 2). Can be used as simulation model, by which it is possible to determine (variate) the total cost S depending on certain values of the volumes and cost of goods produced.

Modeldeterministic, if each input set of parameters corresponds to a well-defined and uniquely determined set of output parameters; otherwise, the model is non-deterministic, stochastic (probabilistic).

Example. The above physical models- deterministic. If in models S=gt2/2, 0< t < 100 мы учли бы случайный параметр - порыв ветра с силой p when the body falls:

S(p) = g(p) t2 / 2, 0< t < 100,

then we would get stochastic model(no longer free) fall.

Model functional , if it can be represented as a system of some functional relations.

Model set-theoretic , if it is representable with the help of some sets and relations of belonging to them and between them.

Example . Let the set

X = (Nikolai, Peter, Nikolaev, Petrov, Elena, Ekaterina, Mikhail, Tatyana) and relations:

Nikolai - Elena's husband,

Ekaterina - Peter's wife,

Tatyana - daughter of Nikolai and Elena,

Mikhail is the son of Peter and Catherine,

The families of Mikhail and Peter are friends with each other.

Then the set X and the set of enumerated relations Y can serve as set-theoretic model two friendly families.

Modelis called logical if it can be represented by predicates, logical functions.

For example, a set of logical functions of the form:

z = x https://pandia.ru/text/78/388/images/image004_10.png" alt="" width="9 height=12" height="12"> x, p = x y!}

is a mathematical logical model of the operation of a discrete device.

Modelgame, if it describes, implements some game situation between the participants in the game.

Example. Let the player 1 - conscientious tax inspector and player 2 - unscrupulous taxpayer. There is a process (game) on tax evasion (on the one hand) and on revealing the concealment of tax payments (on the other hand). Players choose natural numbers i and j(i,jn), which can be identified, respectively, with the penalty for player 2 for non-payment of taxes when player 1 discovers the fact of non-payment and with the temporary benefit of player 2 from tax evasion. If we take as a model a matrix game with a payoff matrix of the order n, then each element in it is determined by the rule aij = |i - j|. Model the game is described by this matrix and the strategy of evasion and capture. This game is antagonistic.

Modelalgorithmic, if it is described by some algorithm or a set of algorithms that determine the functioning, development of the system.

It should be remembered that not all models can be explored or implemented algorithmically.

Example. The model for calculating the sum of an infinite decreasing series of numbers can be an algorithm for calculating the finite sum of a series up to a certain specified degree of accuracy. algorithmic model the square root of x can serve as an algorithm for calculating its approximate value using a well-known recursive formula.

The model is calledstructural if it can be represented by a data structure or data structures and relationships between them.

Modelis called graph if it can be represented by a graph or graphs and relations between them.

Modelis called hierarchical (tree-like) if it can be represented by some hierarchical structure (tree).

Example. To solve the problem of finding a route in a search tree, you can build, for example, a tree model(Fig. 1.2):

MsoNormalTable">

Table of works during the construction of a house

Operation

Lead time (days)

Previous Operations

Count Arcs

Site clearing

Foundation laying

Site clearing (1)

Walling

Foundation laying (2)

Building walls (3)

Plaster work

Electrical wiring (4)

Landscaping

Building walls (3)

Finishing work

Plastering (5)

Roof decking

Building walls (3)

network model(network diagram) of building a house is given in fig. 1.3.

Syntax" href="/text/category/sintaksis/" rel="bookmark">syntactic .

For example, the rules of the road - linguistic, structural model traffic and pedestrians on the roads.

Let B be the set of generating stems of nouns, C the set of suffixes, P the adjectives, bi the root of the word; "+" - word concatenation operation, ":=" - assignment operation, "=>" - output operation (output of new words), Z - set of meanings (semantic) adjectives.

Language model M word formation can be represented by:

= + <сi>.

With bi - "fish (a)", ci - "n (th)", we get from this models pi - "fish", zi - "made from fish".

Modelvisual, if it allows you to visualize the relationships and connections of the simulated system, especially in dynamics.

For example, on a computer screen, visual model one object or another.

Modelnatural, if it is a material copy of the simulation object.

For example, a globe is a natural geographical model the globe.

Modelgeometric, graphic, if it can be represented by geometric images and objects.

For example, the layout of the house is full-scale geometric model house under construction. A polygon inscribed in a circle gives model circles. It is she who is used when depicting a circle on a computer screen. The straight line is model the numerical axis, and the plane is often depicted as a parallelogram.

Modelcellular automaton if it can be represented by a cellular automaton or a system of cellular automata.

A cellular automaton is a discrete dynamic system, an analogue of a physical (continuous) field. Cellular automata geometry is an analogue of Euclidean geometry. An indivisible element of Euclidean geometry is a point; segments, straight lines, planes, etc. are built on its basis.

An indivisible element of the cellular-automaton field is a cell, on the basis of which clusters of cells and various configurations of cellular structures are built. The cellular automaton is represented by a uniform network of cells ("cells") of this field. The evolution of a cellular automaton unfolds in a discrete space - a cellular field.

The change of states in the cellular automaton field occurs simultaneously and in parallel, and time passes discretely. Despite the apparent simplicity of their construction, cellular automata can demonstrate diverse and complex behavior of objects and systems.

Recently, they have been widely used in modeling not only physical, but also socio-economic processes.

1.2. Applied aspects of modeling

Modelis called fractal if it describes the evolution of the modeled system by the evolution of fractal objects.

If a physical object is homogeneous (solid), i.e., it has no cavities, then we can assume that its density does not depend on size. For example, when increasing the object parameter R before 2R the mass of the object will increase in R2 times if the object is a circle and in R3 times, if the object is a ball, i.e. there is a connection between mass and length. Let n- dimension of space. An object whose mass and size are related is called "compact". Its density can be calculated using the formula:

If the object (system) satisfies the relationM(R) ~ Rf(n), wheref(n)< n , then such an object is called fractal.

Its density will not be the same for all values of R, then it is scaled according to the formula:

Since f(n) - n< 0 по определению, то плотность фрактального объекта уменьшается с увеличением размера R, а ρ(R) является количественной мерой разряженности объекта.

Example fractal model is the Cantor set. Let's consider a segment. Divide it into 3 parts and discard the middle segment. We again divide the remaining 2 intervals into three parts and discard the middle intervals, etc. We obtain a set called the Cantor set. In the limit, we obtain an uncountable set of isolated points ( rice. 1.4)

DIV_ADBLOCK135">

The model can be formally represented as: M =< O, А, Z, B, C > .

Main propertiesany models:

purposefulness - the model always displays some system, that is, it has the purpose of such a display; finiteness - the model reflects the original only in a finite number of its relations and modeling resources are finite; simplicity - the model displays only the essential aspects of the object and it should be simple to study or reproduce; visibility, visibility of its main properties and relationships; availability and manufacturability for research or reproduction; information content - the model should contain sufficient information about the system (within the framework of the hypotheses adopted when building the model) and should make it possible to obtain new information; completeness - the model must take into account all the main connections and relationships necessary to ensure the purpose of modeling; controllability - the model must have at least one parameter, by changing which it is possible to simulate the behavior of the modeled system in various conditions.

Life cycle of the simulated system:

collection of information about the object, hypotheses, preliminary model analysis; designing the structure and composition of models (submodels); construction of model specifications, development and debugging of individual submodels, assembly of the model as a whole, identification (if necessary) of model parameters; model research - the choice of a research method and the development of an algorithm (program) for modeling; study of the adequacy, stability, sensitivity of the model; assessment of modeling tools (expended resources); interpretation, analysis of modeling results and establishment of some cause-and-effect relationships in the system under study; generation of reports and design (national economic) decisions; refinement, modification of the model, if necessary, and return to the system under study with new knowledge obtained using the model and simulation.

Modeling is a method of system analysis.

Often in system analysis with a model approach of research, one methodological error can be made, namely, the construction of correct and adequate models (submodels) of system subsystems and their logically correct linking does not guarantee correctness constructed in this way models the entire system.

A model built without taking into account the connections of the system with the environment can serve as a confirmation of Gödel's theorem, or rather, its corollary, stating that in a complex isolated system may have truths and conclusions that are correct in this system and incorrect outside of it.

The science of modeling consists in dividing the modeling process (systems, models) into stages (subsystems, submodels), a detailed study of each stage, relationships, connections, relationships between them and then effectively describing them with the highest possible degree of formalization and adequacy.

In case of violation of these rules, we obtain not a model of the system, but a model of "own and incomplete knowledge".

Modeling is considered as a special form of experiment, an experiment not on the original itself, i.e., a simple or ordinary experiment, but over a copy of the original. The isomorphism of the original and model systems is important here.

Isomorphism - equality, sameness, similarity.

Modelsand modelingapplied in the main areas:

in teaching, in cognition and development of the theory of the systems under study; in forecasting (output data, situations, system states); in management (of the system as a whole, its individual subsystems); in automation (of a system or its individual subsystems).

2. Mathematical and computer modeling

2.1. Classification of types of modeling

Rice. 2.1. Classification of types of modeling

At physical modeling the system itself is used, or similar to it in the form of a layout, for example, an aircraft in a wind tunnel.

Mathematical modeling there is a process of establishing correspondence with the real system S mathematical model M and the study of this model, which makes it possible to obtain the characteristics of a real system.

At analytical modeling the processes of functioning of the elements are written in the form of mathematical relations (algebraic, integral, differential, logical, etc.).

The analytical model can be explored by methods:

· analytical(explicit dependencies are established, mostly analytical solutions are obtained);

· numerical(approximate solutions are obtained);

Computer mathematical modeling is formulated in the form of an algorithm (computer program), which makes it possible to carry out computational experiments on the model.

Numerical simulation uses methods of computational mathematics.

statistical modeling uses the processing of data about the system in order to obtain statistical characteristics of the system.

simulation modeling reproduces on a computer (simulates) the process of functioning of the system under study, observing the logical and temporal sequence of the processes, which allows you to find out data about the state of the system or its individual elements at certain points in time.

The use of mathematical modeling makes it possible to investigate objects on which real experiments are difficult or impossible.

The economic effect of mathematical modeling is that the cost of system design is on average reduced by 50 times.

2.2. Mathematical modeling of complex systems

We will think that element s there is some object that has certain properties, the internal structure of which does not play a role for the purposes of the study, for example, an aircraft for flight simulation is not an element, but for airport operation simulation it is an element.

Connection l between the elements there is a process of their interaction, which is important for the purposes of the study.

System S is a set of elements with connections and the purpose of functioning F.

A complex system is a system consisting of elements of different types with different types of connections.

big system is a system consisting of a large number of elements of the same type with the same type of connections.

In general, the system can be mathematically represented as:

Automated system S A there is a complex system with the decisive role of elements of two types: technical means ST and human actions SH:

Here s0 are the remaining elements of the system.

System decomposition there is a division of the system into elements or groups of elements with an indication of the links between them, unchanged during the operation of the system.

Almost all systems are considered to be functioning in time, so we will determine their dynamic characteristics.

State – this is a set of characteristics of the elements of the system that change over time and are important for the purposes of its functioning.

Process (dynamics) – it is a set of system state values that change over time.

Purpose of operation is the task of obtaining the desired state of the system. Achieving the goal usually entails a targeted intervention in the process of the system functioning, which is called management.

1 - General concepts. Definitions.
Definitions
An object- all that is aimed at human activity.

Hypothesis- a prediction about the properties of an object based on incomplete data.

Analogy- a judgment about any particular similarity of objects. An analogy connects a hypothesis with an experiment.

Model- an object-substitute object, providing the study of some properties of the original. The model provides visualization of the study of the original object.

Model- a logical scheme that simplifies reasoning and logical constructions, allowing experiments to be carried out, and clarifying the nature of phenomena.

Modeling- replacing one object with another in order to obtain information about the most important properties of the original object using the model object (hereinafter, for simplicity, we replace the original object with an object, the model object with a model).

Adequacy of the model to the object- coincidence of the simulation results and the results of experiments with the object.
General concepts
Model- an object or description of an object, a system for replacing (under certain conditions, proposals, hypotheses) of one system (i.e. the original) of another system to study the original or reproduce its properties. A model is the result of a mapping from one structure to another. By mapping a physical system (object) onto a mathematical system (for example, the mathematical apparatus of equations), we obtain a physical and mathematical model of the system or a mathematical model of a physical system. In particular, the physiological system - the human circulatory system, obeys some laws of thermodynamics, and having described this system in physical (thermodynamic) language, we will obtain a physical, thermodynamic model of the physiological system. If we write down these laws in mathematical language, for example, write out the corresponding thermodynamic equations, then we will get a mathematical model of the circulatory system.

Models, if we ignore the areas, areas of their application, are of three types: cognitive, pragmatic and instrumental.

cognitive model- a form of organization and presentation of knowledge, a means of combining new and old knowledge. The cognitive model, as a rule, is adjusted to reality and is theoretical model.

Pragmatic model- a means of organizing practical actions, a working representation of the goals of the system for its management. Reality in them is adjusted to some pragmatic model. These are, as a rule, applied models.

instrumental model- is a means of constructing, researching and/or using pragmatic and/or cognitive models.

Cognitive reflect existing, and pragmatic - although not existing, but desired and, possibly, feasible relationships and connections.

According to the level, "depth" of modeling, the models are empirical- based on empirical facts, dependencies, theoretical- based on mathematical descriptions and mixed, semi-empirical- using empirical dependencies and mathematical descriptions.

Primary requirements to the model: visibility of construction; visibility of its main properties and relations; its availability for research or reproduction; ease of research, reproduction; saving the information contained in the original (with the accuracy of the hypotheses considered when building the model) and obtaining new information.

The modeling problem consists of three tasks:

model building(this task is less formalizable and constructive, in the sense that there is no algorithm for building models);
model study(this task is more formalizable, there are methods for studying various classes of models);
model usage(constructive and concretized task).

Model M is called static if x i no time parameter t. The static model at each moment of time gives only a "photo" of the system, its slice.

Model - dynamic if among x i there is a time parameter, i.e. it displays the system (processes in the system) in time.

Model - discrete, if it describes the behavior of the system only at discrete times.

Model - continuous, if it describes the behavior of the system for all moments of time from some time interval.

Model - imitation, if it is intended for testing or studying, playing the possible ways of development and behavior of the object by varying some or all of the parameters x i models M.

Model - deterministic, if each input set of parameters corresponds to a well-defined and uniquely determined set of output parameters; otherwise - model non-deterministic, stochastic (probabilistic).

We can talk about different modes of using models - about the simulation mode, about the stochastic mode, etc.

Model includes: object O, subject (optional) A, task Z, resources B, simulation environment CM.

The properties of any model are as follows:

limb: the model reflects the original only in a finite number of its relations and, in addition, the modeling resources are finite;
simplicity: the model displays only the essential aspects of the object;
approximation: reality is displayed roughly or approximately by the model;
adequacy: the model successfully describes the simulated system;
informative: the model must contain sufficient information about the system - within the framework of the hypotheses adopted in the construction of the model.

Life cycle of the simulated system:

Collection of information about the object, hypotheses, pre-model analysis;
Designing the structure and composition of models (submodels);
Construction of model specifications, development and debugging of individual submodels, assembly of the model as a whole, identification (if necessary) of model parameters;
Model research - choice of a research method and development of an algorithm (program) for modeling;
Study of the adequacy, stability, sensitivity of the model;
Assessment of modeling tools (resources expended);
Interpretation, analysis of modeling results and establishment of some cause-and-effect relationships in the system under study;
Generation of reports and project (national - economic) decisions;
Clarification, modification of the model, if necessary, and return to the system under study with new knowledge obtained through simulation.

The main operations used on models are:

Linearization. Let M=M(X,Y,A), where X- multiple inputs Y- exits, A- system states. This can be shown schematically:

X -> A -> Y

If X, Y, A- linear spaces (sets), and - linear operators, then the system (model) is called linear. Other systems (models) - non-linear. Nonlinear systems are difficult to study, so they are often linearized - reduced to linear in some way.

Identification. Let M=M(X,Y,A), A=(a i ), a i =(a i1 ,a i2 ,...,a ik ) - object (system) state vector. If the vector a i depends on some unknown parameters , then the problem of identification (model, model parameters) is to determine by some additional conditions, for example, experimental data characterizing the state of the system in some cases. Identification is a solution to the problem of constructing, based on the results of observations, mathematical models that adequately describe the behavior of a real system.
Aggregation. The operation consists in transforming (reducing) the model to a model (models) of a smaller dimension (X, Y, A).
Decomposition. The operation consists in dividing the system (model) into subsystems (submodels) while preserving the structures and belonging of some elements and subsystems to others.
Assembly. The operation consists in the transformation of a system, a model that realizes the set goal from given or defined submodels (structurally related and stable).
Prototyping. This operation consists in approbation, research of structural coherence, complexity, stability with the help of layouts or submodels of a simplified form, in which the functional part is simplified (although the input and output of submodels are preserved).
Expertise, expert assessment. An operation or procedure for using the experience, knowledge, intuition, intelligence of experts to study or model poorly structured, poorly formalized subsystems of the system under study.
Computational experiment. This is an experiment carried out with the help of a model on a computer with the aim of distributing, predicting certain states of the system, responding to certain input signals. The instrument of the experiment here is a computer (and a model!).

Models and modeling are applied in the following main and important areas.

Education(both models, modeling, and the models themselves).
Cognition and development of the theory of the systems under study- with the help of some models, simulations, simulation results.
Forecasting(output data, situations, system states).
Control(the system as a whole, individual subsystems of the system, development of management decisions and strategies).
Automation(system or individual subsystems of the system).

In the basic quadruple of computer science: "model - algorithm - computer - technology" in computer modeling, the main role is already played by the algorithm (program), computer and technology (more precisely, instrumental systems for a computer, computer technology).

For example, in simulation modeling (in the absence of a strict and formally written algorithm), the main role is played by technology and modeling tools; the same is true for cognitive graphics.

The main functions of a computer in modeling systems:

to perform the role aid for solving problems solved by conventional computing tools, algorithms, technologies;
play the role of a means of setting and solving new problems that cannot be solved by traditional means, algorithms, technologies;
play the role of a means of designing computer training and modeling environments;
play the role of a modeling tool for obtaining new knowledge;
play the role of "learning" new models (self-learning models).

Computer modeling is the basis for the representation of knowledge in a computer (building various knowledge bases). Computer modeling for the birth of new information uses any information that can be updated with the help of a computer.

A kind of computer simulation is a computational experiment.

Computer simulation, computational experiment is becoming a new tool, a method of scientific knowledge, a new technology also due to the growing need to move from the study of linear mathematical models of systems.

2 – System classification and models. Black box model.

Model classification

Models can be relatively complete or incomplete. The theory of similarity states that absolute similarity can only take place when the object is replaced by exactly the same. But then the meaning of modeling is lost.

complete model characterizes all the basic properties of an object in time and space.

incomplete model characterizes a limited part of the properties of the object.

The systematization of models is given in the following table.

System classification
Systems can be classified according to different criteria. It is often strictly impossible to implement and depends on the goal and resources. Let us present the main methods of classification (other criteria for classifying systems are also possible).

In relation to the system to the environment:
- open(there is an exchange of resources with the environment);
- closed(no exchange of resources with the environment).
By the origin of the system (elements, links, subsystems):
- artificial(tools, mechanisms, machines, machine guns, robots, etc.);
- natural(living, non-living, ecological, social, etc.);
- virtual(imaginary and, although they do not really exist, but function in the same way as if they really existed);
- mixed(economic, biotechnical, organizational, etc.).
According to the description of system variables:
- with qualitative variables(having only a meaningful description);
- with quantitative variables(having discretely or continuously quantitatively described variables);
- mixed(quantitative - qualitative) descriptions.
According to the type of description of the law (laws) of the functioning of the system:
- type "Black box"(the law of the system functioning is not completely known; only input and output messages of the system are known);
- not parameterized(the law is not described, we describe it using at least unknown parameters , only some a priori properties of the law are known);
- parameterized(the law is known up to parameters and it can be attributed to a certain class of dependencies);
- type “White (transparent) box”(the law is fully known).
By the method of system management (in the system):
- externally controlled systems(without feedback, regulated, managed structurally, informationally or functionally);
- controlled from within(self-managed or self-regulating - programmatically controlled, automatically regulated, adaptable - adaptable with the help of controlled state changes and self-organizing - changing their structure in time and space in the most optimal way, ordering their structure under the influence of internal and external factors);
- with combined control(automatic, semi-automatic, automated, organizational).

Under regulation is understood as the correction of control parameters based on observations of the trajectory of the system's behavior - in order to return the system to the desired state (to the desired trajectory of the system's behavior; in this case, the trajectory of the system is understood as a sequence of system states taken during the operation of the system, which are considered as some points in the set of system states).

Example. Consider the ecological system “Lake”. This is an open system of natural origin, the variables of which can be described in a mixed way (quantitatively and qualitatively, in particular, the temperature of a reservoir is a quantitatively described characteristic), the structure of the inhabitants of the lake can be described both qualitatively and quantitatively, and the beauty of the lake can be described qualitatively. According to the type of description of the law of the functioning of the system, this system can be classified as not parameterized as a whole, although it is possible to single out subsystems various types, in particular, a different description of the subsystem “Algae”, “Fish”, “Inflowing stream”, “Outflowing stream”, “Bottom”, “Coast”, etc. The “Computer” system is an open, artificial origin, mixed description, parameterized, externally controlled (programmatically). The “Logical Disk” system is an open, virtual, quantitative description, “White Box” type (at the same time, we do not include the contents of the disk in this system!), Mixed control. The “Firm” system is open, of mixed origin (organizational) and description, controlled from within (adaptable, in particular, system).

The system is called big, if its study or modeling is difficult due to the large dimension, i.e. the set of states of the system S has a large dimension. What dimension should be considered large? We can judge this only for a specific problem (system), a specific goal of the problem under study, and specific resources.

A large system is reduced to a system of a smaller dimension by using more powerful computing tools (or resources) or by dividing the problem into a number of problems of a smaller dimension (if possible).

Example. This is especially true when developing large computing systems, for example, when developing computers with a parallel architecture or algorithms with a parallel data structure and their parallel processing.

The system is called difficult, if it does not have enough resources (mainly informational) for an effective description (of states, laws of functioning) and control of the system - definitions, descriptions of control parameters or for making decisions in such systems (in such systems there should always be a decision subsystem) .

Example. Complex systems are, for example, chemical reactions, if considered at the molecular level; cell of biological formation, considered at the metabolic level; the human brain, if considered from the point of view of the intellectual actions performed by a person; economics considered at the macro level (i.e. macroeconomics); human society - at the political-religious-cultural level; Computers (especially - the fifth generation), if it is considered as a means of obtaining knowledge; language, in many ways.

The complexity of these systems is due to their complex behavior. The complexity of the system depends on the accepted level of description or study of the system - macroscopic or microscopic.

The complexity of the system can be external and internal.

Internal complexity is determined by the complexity of the set of internal states, potentially estimated by the manifestations of the system, by the complexity of control in the system.

External complexity determined by the complexity of relationships with the environment, the complexity of managing the system, potentially assessed by feedback systems and environments.

Complex systems are:

complexity of structural or static(not enough resources to build, describe, manage the structure);
dynamic or temporary(there are not enough resources to describe the dynamics of the system behavior and control its trajectory);
information or information - logical, infological(not enough resources for informational, informational-logical description of the system);
computing or implementation, research(there are not enough resources for effective forecasting, calculations of system parameters or their implementation is hampered by a lack of resources);
algorithmic or constructive(there are not enough resources to describe the algorithm of functioning or control of the system, for a functional description of the system);
development or evolution, self-organization(not enough resources for sustainable development, self-organization).

The system is called sustainable, if it retains the tendency to strive for the state that best suits the goals of the system, the goals of maintaining quality without changing the structure or not leading to strong changes in the structure of the system on some given set of resources (for example, on a time interval). The concept of “strong change” must be specified and determined every time.

The system is called liaison, if any two subsystems exchange a resource, i.e. between them there are some resource-oriented relations, connections.

Examples of modeling technical systems and processes. "Systems Theory and System Analysis

Lecture 9: Classification of types of system modeling

Principles and approaches to the construction of mathematical models

Stages of building a mathematical model

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