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

Rice. - Example of classification of types of modeling

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

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

For incomplete modeling this identity is not preserved.

At the core approximate modeling lies in 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 represents only the aspect of the system being studied well. For example, to assess the noise immunity of discrete information transmission channels, functional and information models of the system may not be developed. To achieve the goal of modeling, an event model described by a matrix of conditional probabilities of transitions of the i-th character of the alphabet to the j-th 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 depicts 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 over time. In this case, they operate with analog (continuous), discrete and mixed models.

Depending on the form of implementation of the medium 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, a microworld situation). Mental simulation real systems implemented in the form of visual, symbolic and mathematical. A significant number of tools and methods have been developed to represent functional, information and event models of this type of modeling.

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

The basis hypothetical modeling, a hypothesis is laid down about the patterns of the process in a real object, which reflects the researcher’s level of knowledge about the object and is based on cause-and-effect relationships between the input and output of the object being studied. This type of modeling is used when knowledge about an object is not enough to build formal models. Analog Modeling based on the use of analogies different levels. For fairly simple objects, the highest level is complete analogy. As the system becomes more complex, analogies of subsequent levels are used, when the analog model displays several (or only one) aspects of the functioning of the object.

Layout 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 models is also based on analogies, usually based on cause-and-effect relationships between phenomena and processes in an object.

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

At the core linguistic Modeling is based on a certain thesaurus, which is formed from a set of concepts of the subject area under study, and this set must be fixed. A thesaurus is a dictionary that reflects relationships between words or other elements of this language, designed to search for words by their meaning.

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

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

If you 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 compose 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 a correspondence between a given real object and some mathematical object called a mathematical model. In principle, to study the characteristics of any system using mathematical methods, including machine methods, it is necessary to formalize this process, i.e. a mathematical model was 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.

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

Invariant form is a recording of model relationships using traditional mathematical language, regardless of the method for solving the 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 in the form of the 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, which describe the law (algorithm) of its functioning, are written in the form of some analytical relations (algebraic, integrodifferential, finite-difference, etc.) or logical conditions. The analytical model is studied using several methods:

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

Currently, 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 construct an appropriate modeling algorithm.

Algorithmic form - recording the relationships between the model and the selected numerical solution method in the form of an algorithm. Among algorithmic models important class make up simulation models designed to simulate physical or information processes under different external influences. The actual imitation of these processes is called simulation modeling.

At imitation modeling reproduces the algorithm for the functioning of the system over time - the behavior of the system, and simulates the elementary phenomena that make up the process, preserving their logical structure and sequence of occurrence, which allows, from the source 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 tasks. Simulation models make it possible to quite simply take into account factors such as the presence of discrete and continuous elements, nonlinear characteristics of system elements, numerous random influences and others, which often create difficulties in analytical studies. Currently simulation- the most effective method of studying systems, and often the only practically accessible method of obtaining information about the behavior of a system, especially at the stage of its design.

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

Monte Carlo method - numerical method, which is used to model random variables and functions, probabilistic characteristics which coincide with the solutions of analytical problems. It consists of repeated reproduction of processes that are implementations 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 functioning processes 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 necessary to create a system with specified characteristics under certain restrictions.

Combined (analytical-simulation) modeling allows you to combine the advantages of analytical and simulation modeling. When constructing combined models, a preliminary decomposition of the process of object functioning is carried out into its constituent subprocesses, 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 strive to display only a certain 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 a 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 function of the real object under study, try to formalize this function in the form of some communication operators between input and output, and reproduce this function on a simulation model, and on a completely different mathematical language and, naturally, a different physical implementation of the process. For example, expert systems are models of decision makers.

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, 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 systems analysis includes:

• methods network modeling;
• 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 this case, the term “model structure” can be applied to both functions and elements of the system. The corresponding structures are called functional and morphological. Object-oriented modeling combines both types of structures into a hierarchy of classes that include both elements and functions.

In structural modeling for last decade a new CASE technology was formed. The abbreviation CASE has a double meaning, corresponding to two directions 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 called rapid application development (RAD) environments. The second - Computer-Aided System Engineering - emphasizes the focus on supporting the conceptual modeling of complex systems, mainly semi-structured ones. Such CASE systems are often called BPR (Business Process Reengineering) systems. In general, CASE technology is a set of methodologies for analysis, design, development and maintenance of 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 design and development of complex systems, including software.

Situational modeling is based on a model theory of thinking, within the framework of which the basic mechanisms for regulating decision-making processes can be described. At the center of the model theory of thinking is the idea of ​​the formation in the structures of the brain of an information model of an object and outside world. This information is perceived by a person on the basis of his existing knowledge and experience. Expedient human behavior is built by forming a target situation and mentally transforming the initial situation into a target one. The basis for constructing a model is the description of an 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 take place. The richer information model 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 opportunity to study characteristics either on a real object as a whole or on part of it. Such studies are carried out both on objects operating in normal modes and when special modes are organized to assess the characteristics of interest to the researcher (with other values ​​of variables and parameters, on a different time scale, etc.). Real modeling is the most adequate, but its capabilities are limited.

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

Another type of real simulation is physical, which differs from full-scale in that the research is carried out in installations that preserve the nature of the phenomena and have a physical similarity. In the process of physical modeling, certain characteristics of the external environment are specified and the behavior of either a real object or its model is studied under given or artificially created environmental influences. Physical modeling can take place on real and model (pseudo-real) time scales or be considered without taking into account time. In the latter case, the so-called “frozen” processes recorded at a certain point in time are subject to study.

Process and system modeling

1. Basics of system modeling.. 4

1.1. Models and simulation. 4

1.2. Applied aspects of modeling. 14

1.3. Basic properties of model and simulation. 16

2. Mathematical and computer modeling. 19

2.1. Classification of types of modeling. 19

2.2. Math modeling complex systems.. 21

2.3. Simulation of random variables and processes. 25

2.4. Basics 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. Generating random sequences. 48

4.1. Generating uniformly distributed random numbers. 48

4.2. Basic criteria for testing 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. Method least squares. 74

6. Markov chains. 77

6.1. Markov process with discrete time.. 78

6.2. Markov random processes with continuous time.. 87

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

6.4. Typical tasks application of Markov chains. 93

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

7. Canonical decomposition 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 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. Solving control problems using the Riccati equation. 121

10. General principles construction of modeling algorithms. 134

10.1. Δt principle. 135

10.2. Principle special conditions. 140

10.3. The principle of sequential posting of orders. 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 SP.. 150

11.3. Simulation of stationary normal SP.. 151

12. Processing of simulation results. 153

12.1. Probability assessment. 153

12.2. Estimation of mathematical expectation and variance. 154

12.3. Estimation of 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 stochastic linear optimal control problems. 159

Literature. 166

1. Basics 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, knowledge of a system, process, phenomenon.

View models and the methods of its research depend more on the information and 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 one to delve into the structure and internal logic of the modeled system.

Construction models- a system task that requires analysis and synthesis of initial data, hypotheses, theories, and specialist knowledge. 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 studying the original or reproducing any of its properties.

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

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

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

When describing this system, the following hypotheses are accepted:

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

Classification models carried out according to various criteria.

Model called static , if among the parameters involved in its description there is no temporary parameter. Static model at each moment of time it gives only a “photograph” of the system, its slice.

Example. Newton's law F=a*m is static model moving with acceleration a material point with 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 look like:

Model discrete , if it describes the behavior of the system only at discrete moments in time.

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

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

Model continuous , if it describes the behavior of the system for all moments of time over a certain period of time.

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

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

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

a1x1 + a2x2 = S,

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

Modeldeterministic if each input set of parameters corresponds to a completely definite and uniquely defined 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 in the form of a system of any functional relationships.

Model set-theoretic , if it is representable using certain sets and relations of membership to them and between them.

Example . Let a set be given

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

Nikolai - Elena's husband,

· Catherine - Peter's wife,

· Tatyana - daughter of Nikolai and Elena,

· Mikhail - son of Peter and Catherine,

· Mikhail and Peter's families are friends with each other.

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

Modelis called logical if it is representable 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="http://*****/img/symbols/or.gif" width="9 height=12" height="12"> x, p = x y!}

there 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) of tax evasion (on the one hand) and of revealing tax evasion (on the other hand). Players choose natural numbers i and j(i,jn), which can be identified, respectively, with the fine of 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 hiding taxes. If we take as a model a matrix game with a payoff matrix of 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 set of algorithms that determines the functioning and development of the system.

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

Example. A model for calculating the sum of an infinite decreasing series of numbers can be an algorithm for calculating the finite sum of a series to a certain specified degree of accuracy. Algorithmic model square root of a number x An algorithm for calculating its approximate value using a known recurrent formula can serve.

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

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

Modelis called hierarchical (tree-like) if it is representable 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-like model(Fig. 1.2):

MsoNormalTable">

Table of work during house construction

Operation

Lead time (days)

Previous operations

Arcs of the graph

Site clearing

Laying the foundation

Site clearing (1)

Walling

Laying the foundation (2)

Construction of walls (3)

Plastering works

Electrical wiring installation (4)

Landscaping

Construction of walls (3)

Finishing work

Plastering works (5)

Roof decking

Construction of walls (3)

Network model(network diagram) of house construction is given in Fig. 1.3.

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

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

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

Language model M word formation can be represented:

= + <сi>.

For bi - “fish(a)”, сi - “n(th)”, we obtain from this models pi - “fishy”, zi - “cooked from fish”.

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

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

Modelfull-scale, if it is a material copy of the modeling object.

For example, a globe is a full-scale geographical model globe.

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

For example, the model of a house is full-scale geometric model house under construction. A polygon inscribed in a circle gives model circles. This is what is used to depict a circle on a computer screen. A straight line is model number axis, and the plane is often depicted as a parallelogram.

Modelcellular automata if it is representable by a cellular automaton or a system of cellular automata.

Cellular automaton - 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; on its basis, segments, straight lines, planes, etc. are constructed.

An indivisible element of a cellular automata field is a cell; on its basis, clusters of cells and various configurations are built cellular structures. A 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 a cellular automata 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), that is, there are no cavities in it, then we can assume that its density does not depend on size. For example, when increasing an object parameter R before 2R the mass of the object will increase by R2 times if the object is a circle and in R3 times, if the object is a ball, i.e. there is a relationship 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 an 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- Cantor set. Let's consider the segment. Divide it into 3 parts and discard the middle section. 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 presented formally as: M =< O, А, Z, B, C > .

Basic propertiesany models:

purposefulness - the model always displays a certain system, i.e. it has a purpose for such display; limb - the model displays the original only in finite number its relationships and modeling resources are finite; simplicity - the model displays only the essential aspects of the object and it should be easy to study or reproduce; clarity, visibility of its main properties and relationships; accessibility and manufacturability for research or reproduction; information content - the model must contain sufficient information about the system (within the framework of the hypotheses adopted when constructing the model) and must provide the opportunity to obtain new information; completeness - the model must take into account all the basic connections and relationships necessary to achieve the purpose of the modeling; controllability - the model must have at least one parameter, changes in which can simulate the behavior of the simulated system under various conditions.

Life cycle of the simulated system:

collecting information about the object, putting forward hypotheses, preliminary model analysis; designing the structure and composition of models (submodels); building model specifications, developing and debugging individual submodels, assembling the model as a whole, identifying (if necessary) model parameters; model research - selection of a research method and development of a modeling algorithm (program); 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) solutions; refining, modifying the model if necessary, and returning to the system under study with new knowledge gained from the model and simulation.

Modeling is a method of system analysis.

Often in system analysis with a model approach to 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 provide guarantees correctness constructed in this way models the entire system.

A model constructed without taking into account the connections between the system and the environment can serve as a confirmation of Gödel’s theorem, or rather, its corollary, which states that in In a complex isolated system, there may exist truths and conclusions that are correct in this system and incorrect outside it.

The science of modeling consists of dividing the modeling process (system, model) into stages (subsystems, submodels), studying in detail each stage, the relationships, connections, relationships between them and then effectively describing them with the highest possible degree of formalization and adequacy.

If these rules are violated, we get 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. What is important here is the isomorphism of the original and model systems.

Isomorphism - equality, sameness, similarity.

ModelsAnd modelingare applied in the main areas:

in teaching, 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 (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 a similar one in the form of a layout, for example, aircraft in a wind tunnel.

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

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

The analytical model can be studied using the following methods:

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

· numerical(approximate solutions are obtained);

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

Numerical modeling uses methods of computational mathematics.

Statistical modeling uses system data processing to obtain statistical characteristics of the system.

Imitation 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 makes it possible to find out data about the state of the system or its individual elements at certain points in time.

The use of mathematical modeling allows one to study objects real experiments over which are difficult or impossible.

The economic effect of mathematical modeling is that the costs of system design are reduced by an average of 50 times.

2.2. Mathematical modeling of complex systems

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

Connection l between the elements there is a process of their interaction that 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 is a system consisting of different types of elements with different types of connections.

Large system is a system consisting of a large number of similar elements with similar connections.

In general, the system can be represented mathematically as:

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

Here s0 are the remaining elements of the system.

System decomposition is a division of the system into elements or groups of elements indicating the connections between them that are unchanged during the operation of the system.

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

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

Process (dynamics) – this is a set of values ​​of system states that change over time.

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

The main method for studying complex systems and processes, which underlies system analysis, is the modeling method. The essence of the method is that a model is created
the system under study, with the help of which the process of functioning of the real system is studied. Note that the term “model” is currently widely used both in scientific language, and in everyday practice, and in different situations it has different meanings.

The concept of a model has an ambiguous interpretation in scientific practice, as a result of which it is, again, impossible to give a general definition of this concept (as in the case of the definition of the term “system”). In this case, modeling interests us only as a method of scientific knowledge, and the model, accordingly, as a means of scientific knowledge. In this regard, we make the following comments.

In progress cognitive activity a person gradually develops a system of ideas about certain properties of the object being studied and their relationships. This system of ideas is fixed, fixed in the form of a description of the object on ordinary language, in the form of a drawing, diagram, graph, formula, in the form of layouts, mechanisms, technical devices. All this is summarized in a single concept “model”, and the study of objects of knowledge on their models is called modeling.

Thus, a model is a specially created object on which very specific characteristics of the real object under study are reproduced for the purpose of studying it. Simulation is an important tool scientific abstraction, allowing to identify, justify and analyze the characteristics of an object that are essential for this study: properties, relationships, structural and functional parameters.

The modeling method as a method of scientific knowledge has a history that dates back thousands of years. Academician N.N. In this regard, Moiseev notes: “There is one circumstance that underlies any process of cognition: we can only operate with models, study only models, regardless of what language we use - Russian, French or the language of mathematics.

Thus, modeling cannot be considered a recently discovered method of scientific research, but only in the middle of the twentieth century. it has become the subject of both philosophical and special research. This is explained, in particular, by the fact that the modeling method is now experiencing a genuine revolution associated with the development of cybernetics and electronics. computer technology.

Currently, there is extensive scientific literature that discusses in detail the concept of a model, the classification of models according to certain characteristics, the essence of modeling as a method of scientific knowledge, and the application of this method in case studies(economic, social, technical, etc.).

The purpose and scope of this textbook do not allow us to consider these issues in detail and force us to dwell very briefly only on those of them that will be needed in further presentation. First of all, let us add a useful clarification of the concept “model”, which allows us to define a model as an object of any nature that can replace the object under study so that its study provides new information about this object. Obviously, models are chosen in such a way that they are much simpler and more convenient for research than the objects of interest to us (especially since there are also objects that cannot be actively studied at all, for example, various space objects).

Without going into a detailed classification of all possible types models, we emphasize that depending on the means by which the models are implemented, a distinction is made, first of all, between material (objective) and ideal (abstract) modeling (Fig. 1.1).

Material modeling is modeling in which research is carried out on the basis of a model that reproduces the basic geometric, physical, dynamic and functional characteristics the object being studied. A special case of material modeling is physical modeling, in which the modeled object and the model have the same physical nature.

Rice. 1.1. Model classification

Ideal models are associated with the use of any symbolic schemes (graphical, logical, mathematical, etc.). For us, the most important mathematical models are
displaying the objects under study using logical-mathematical symbols and relationships. There are definitions of mathematical models that use the concepts of isomorphism and homomorphism. We will not present them here.

Mathematical models have their own classification.

First, mathematical models are usually divided into analytical and simulation. In the case of analytical models, the studied system (object) and its properties can be described by relations-functions in explicit or implicit form (differential or integral equations, operators) in such a way that it becomes possible directly, using the appropriate mathematical apparatus, to draw the necessary conclusions about the system itself and its properties (and during synthesis, these properties are optimized in some sense). Simulation models are a set of computer programs with the help of which algorithms and procedures are reproduced that describe the process of functioning of the system under study. In this case, the activity of the system with its inherent features is simulated on a computer. Repeated machine experiments, the results of which are processed using methods of mathematical statistics, make it possible to study and analyze the properties of a given system. Simulation models are usually used in cases where it is not possible to build sufficiently simple and convenient analytical models for the system being studied (a combination of simple analytical and more complex simulation models is often used).

Simulation modeling, since the 60s, has been widely used in scientific research both in our country and abroad (in our country such research was first carried out at the Computing Center Russian Academy sciences). In 1972, academician N.N. Moiseev and his collaborators introduced the concept of a simulation system, which is understood as a combination of a system of models (main and auxiliary), a data bank (a common source of information) and tools for conducting simulation experiments, which include the appropriate mathematical support for the entire process of simulation experimentation.

Secondly, a distinction is made between deterministic and stochastic (probabilistic) models. The first of them describe uniquely defined processes, the course of which can be completely predicted, knowing initial conditions and the patterns of these processes; the latter are used to describe random processes, the course of which is described by the laws of probability distribution of the corresponding random variables and cannot be unambiguously predicted.

Finally, analyzing the ways in which mathematical models emerged, Academician N.N. Moiseev introduced the concept of phenomenological and asymptotic models, as well as ensemble models. Models obtained as a result of direct observation of a phenomenon or process, its direct study and comprehension are called phenomenological.

Models obtained as special case from some more general model (as a result of a deductive process) are called asymptotic. Models that arise in the process of generalizing “elementary” models (as a result of the induction process) are called ensemble models.

All of the above types of mathematical models can be used to solve problems of supply fire safety cities, settlements and national economic facilities.

Of course, the results of mathematical modeling have practical meaning only if the model is adequate to the real process, that is, it reflects reality quite well. Issues of checking the adequacy of models will be considered separately in the future.

As is known, in order to build a mathematical model of the process of functioning of any system, one must first give a meaningful description of this process, then formalize all the concepts and relationships associated with the system, parameters characterizing the process under study, and then find it mathematical description. The diagram for constructing mathematical models is shown in Fig. 1.2.

In conclusion, we will consider some issues related to the modeling of complex processes. There is another definition of this concept: a complex process is considered to be a process whose model description is not available to the technology of mathematical modeling (analytical) at the current level of its development. The only thing here possible method The study of such processes is simulation modeling.

Rice. 1.2. Scheme for constructing mathematical models

At the same time, quite often a situation occurs when in the complex process being studied, among the interacting processes it is possible to distinguish not a large number of“main” ones, the characteristics of which interest us, and it is for the sake of predicting these characteristics that the model is developed. The characteristic time scale of the remaining processes is much smaller, and their characteristics interest us insofar as they affect the characteristics of the main processes.

Thus, the processes under study are divided into “slow”, the forecast of whose development interests us, and “fast”, the characteristics of which interest us significantly less, but their influence on slow processes must be taken into account.

Dividing the interacting processes under study into fast and slow when creating their mathematical model is a typical example of a situation when random factors appear in the model. In this case, the parameters of slow processes that interest us are considered as random variables, and to calculate them numerical characteristics it is necessary to perform simulation in the sense in which this term is understood in probability theory and mathematical statistics, i.e. by carrying out a series of simulation experiments, obtain realizations of random variables of interest to us and then process the results using the methods of mathematical statistics.

We will use everything that has been said when studying the processes of functioning of the GPS and RSChS.

Introduction.

1. Basic principles of modeling control systems.

1.1. Principles systematic approach in modeling control systems.

1.2. Approaches to the study of control systems.

1.3. Stages of model development.

2. general characteristics problems of modeling control systems.

2.1. Goals of control systems modeling.

3. Classification of types of system modeling.

Conclusion.

Bibliography.

1.1. INTRODUCTION

In this course work on the topic “Application of modeling in the study of control systems,” I will try to reveal the basic methods and principles of modeling in the context of the study of control systems.

Simulation (in in a broad sense) is the main research method in all fields of knowledge and a scientifically based method for assessing the characteristics of complex systems, used for decision making in various fields engineering activities. Existing and designed systems can be effectively studied using mathematical models (analytical and simulation), implemented on modern computers, which in this case act as an experimenter’s tool with a system model.

Currently, it is impossible to name an area of ​​human activity in which modeling methods would not be used to one degree or another. This is especially true in the area of ​​management various systems, where the main processes are decision-making based on the information received. Let us dwell on the philosophical aspects of modeling, or rather the general theory of modeling.

Methodological basis modeling. Everything that is aimed at human activity, is called an object (lat. objection - subject). The development of a methodology is aimed at streamlining the receipt and processing of information about objects that exist outside of our consciousness and interact with each other and the external environment.

In scientific research, hypotheses play an important role, i.e., certain predictions based on a small amount of experimental data, observations, and guesses. A quick and complete test of the hypotheses can be carried out during a specially designed experiment. When formulating and testing the correctness of hypotheses, analogy is of great importance as a method of judgment.

In general, modeling can be defined as a method of indirect cognition, in which the original object being studied is in some correspondence with another model object, and the model is capable in one way or another of replacing the original at some stages of the cognitive process. The stages of cognition at which such replacement occurs, as well as the forms of correspondence between the model and the original, can be different:

1) modeling as cognitive process, containing the processing of information coming from the external environment about the phenomena occurring in it, as a result of which images corresponding to objects appear in consciousness;

2) modeling, which consists in building a certain model system (second system), connected by certain similarity relations with the original system (first system), and in this case, mapping one system to another is a means of identifying dependencies between two systems reflected in similarity relations , and not the result of direct study of incoming information.

1. BASIC CONCEPTS OF SYSTEM MODELING THEORY

Modeling begins with the formation of the subject of research - a system of concepts that reflects the characteristics of the object that are essential for modeling. This task is quite complex, which is confirmed by different interpretations in scientific and technical literature such fundamental concepts as system, model, modeling. Such ambiguity does not indicate the fallacy of some terms and the correctness of other terms, but reflects the dependence of the subject of research (modeling) both on the object under consideration and on the goals of the researcher. A distinctive feature of modeling complex systems is its versatility and variety of uses; it becomes an integral part of the entire life cycle of the system. This is explained primarily by the manufacturability of models implemented on the basis of computer technology: enough high speed obtaining modeling results and their relatively low cost.

1.1. Principles of the systems approach in systems modeling.

Currently, in the analysis and synthesis of complex (large) systems, a systems approach has been developed, which differs from the classical (or inductive) approach. The latter considers the system by moving from the particular to the general and synthesizes (constructs) the system by merging its components, developed separately. In contrast, the systems approach involves a consistent transition from the general to the specific, when the basis of consideration is the goal, and the object under study is distinguished from the environment.

Modeling object. Specialists in the design and operation of complex systems deal with control systems at various levels that have common property- the desire to achieve some goal. We will take this feature into account in the following definitions of the system. System S is a purposeful set! interconnected elements of any nature. External environment E is a set of elements of any nature existing outside the system that influence the system or are under its influence. "

Depending on the purpose of the study, research may be considered different ratios between the object S itself and the external environment E. Thus, depending on the level at which the observer is located, the object of study can be distinguished in different ways and different interactions of this object with the external environment can take place.

With the development of science and technology, the object itself is continuously becoming more complex, and now they are talking about the object of research as some complex system that consists of various components interconnected with each other. Therefore, considering the systems approach as the basis for building large systems and as the basis for creating methods for their analysis and synthesis, it is first of all necessary to define the very concept of a systems approach.

The systems approach is an element of the doctrine of the general laws of development of nature and one of the expressions of the dialectical doctrine. You can cite different definitions systems approach, but the most correct one is one that allows one to evaluate the cognitive essence of this approach using such a method of studying systems as modeling. Therefore, it is very important to isolate the system S itself and the external environment E from objectively existing reality and describe the system based on system-wide positions.

With a systematic approach to modeling systems, it is necessary first of all to clearly define the purpose of the modeling. Since it is impossible to completely simulate a really functioning system (the original system, or the first system), a model (the model system, or the second system) is created for the problem at hand. Thus, in relation to modeling issues, the goal arises from the required modeling tasks, which allows one to approach the selection of a criterion and evaluate which elements will be included in the created model M. Therefore, it is necessary to have a criterion for selecting individual elements into the created model.

1.2. Approaches to systems research.

It is important for the systems approach to determine the structure of the system - the set of connections between the elements of the system, reflecting their interaction. The structure of a system can be studied from the outside from the point of view of the composition of individual subsystems and the relationships between them, as well as from the inside, when individual properties are analyzed that allow the system to achieve a given goal, that is, when the functions of the system are studied. In accordance with this, a number of approaches to studying the structure of a system with its properties have emerged, which should primarily include structural and functional.

With a structural approach, the composition of the selected elements of the system S and the connections between them are revealed. The set of elements and connections between them allows us to judge the structure of the system. The latter, depending on the purpose of the study, can be described at different levels of consideration. Most general description structure is a topological description that allows you to determine in the most general concepts components of the system and well formalized on the basis of graph theory.

Less general is the functional description when considering individual functions, i.e., algorithms for the behavior of the system, and a functional approach is implemented that evaluates the functions that the system performs, and a function is understood as a property that leads to the achievement of a goal. Since a function reflects a property, and a property reflects the interaction of the system S with the external environment E, the properties can be expressed in the form of either some characteristics of the elements S iV) and subsystems Si of the system, or the system S as a whole.

If you have some standard of comparison, you can enter the quantitative and qualitative characteristics of the systems. For a quantitative characteristic, numbers are entered that express the relationship between this characteristic and the standard. The qualitative characteristics of the system are found, for example, using the method of expert assessments.

The manifestation of system functions in time S(t), i.e. the functioning of the system, means the transition of the system from one state to another, i.e. movement in state space Z. When operating a system S, the quality of its functioning is very important, determined by the efficiency indicator and which is the value of the performance evaluation criterion. Exist different approaches to the selection of performance evaluation criteria. System S can be evaluated either by a set of particular criteria or by some general integral criterion.

It should be noted that the created model M from the point of view of the systems approach is also a system, i.e. S"=S"(M), and can be considered in relation to the external environment E. The simplest in presentation are models in which a direct analogy is maintained phenomena. Models are also used in which there is no direct analogy, but only the laws and general patterns of behavior of the elements of the system S are preserved. A correct understanding of the relationships both within the model M itself and its interaction with the external environment E is largely determined by the level at which the observer is located .

A simple approach to studying the relationships between individual parts of the model involves considering them as a reflection of the connections between individual subsystems of the object. This classic approach can be used to create fairly simple models. The process of synthesis of model M based on the classical (inductive) approach is presented in Fig. 1.1, a. The real object to be modeled is divided into separate subsystems, i.e., the initial data D for modeling is selected and goals C are set that reflect individual aspects of the modeling process. By separate population With the initial data D, the goal of modeling a separate aspect of the system’s functioning is set; on the basis of this goal, a certain component K of the future model is formed. The set of components is combined into a model M.

Thus, developing a model M based on the classical approach means summing up individual components into a single model, with each component solving its own problems and isolated from other parts of the model. Therefore, the classical approach can be used to implement relatively simple models in which separation and mutually independent consideration are possible individual parties functioning of a real object. For a model of a complex object, such disunity of tasks to be solved is unacceptable, since it leads to significant expenditure of resources when implementing the model on the basis of specific software and hardware. Two distinctive aspects of the classical approach can be noted: there is a movement from the particular to the general, the created model (system) is formed by summing up its individual components and the emergence of a new systemic effect is not taken into account.

With the increasing complexity of modeling objects, the need arose to observe them from a higher level. In this case, the observer (developer) considers this system S as a certain subsystem of some metasystem, i.e., a system of a higher rank, and is forced to move to the position of a new systems approach, which will allow him to build not only the system under study, solving a set of problems, but also to create a system that is integral part metasystems.

The systems approach was used in systems engineering due to the need to study large real systems, when the insufficiency and sometimes erroneousness of making any particular decisions affected. The emergence of a systems approach was influenced by the increasing amount of initial data during development, the need to take into account complex stochastic relationships in the system and the influences of the external environment E. All this forced researchers to study a complex object not in isolation, but in interaction with the external environment, as well as in conjunction with other systems of some kind. metasystems.

The systems approach allows us to solve the problem of building a complex system, taking into account all the factors and possibilities, proportional to their significance, at all stages of the study of the 5" system and the construction of the M" model. The systems approach means that each system S is an integrated whole even when it consists of separate disconnected subsystems. Thus, the basis of the systems approach is the consideration of the system as an integrated whole, and this consideration during development begins with the main thing - the formulation of the purpose of operation. Based on the initial data D, which is known from the analysis external system, those restrictions that are imposed on the system from above or based on the possibilities of its implementation, and based on the purpose of operation, the initial requirements T for the system model S are formulated. On the basis of these requirements, approximately some subsystems P, elements E are formed and the most difficult stage synthesis - you-< бор В составляющих системы, для чего используются специальные критерии выбора КВ.

When modeling, it is necessary to ensure maximum efficiency of the system model, which is defined as a certain difference between some indicators of the results obtained as a result of operating the model and the costs that were invested in its development and creation.

1.3. Stages of model development.

Based on the systems approach, a certain sequence of model development can be proposed, when two main design stages are distinguished: macrodesign and microdesign.

At the macro-design stage, based on data about the real system S and the external environment E, a model of the external environment is built, resources and limitations for constructing a system model are identified, a system model and criteria are selected to assess the adequacy of the model M of the real system S. Having built the system model and the external environment model , based on the criterion of the efficiency of the system’s functioning, an optimal control strategy is selected during the modeling process, which makes it possible to realize the model’s ability to reproduce individual aspects of the functioning of the real system S.

The micro-design stage depends largely on the specific type of model chosen. In the case of a simulation model, it is necessary to ensure the creation of information, mathematical, technical and software modeling systems. At this stage, it is possible to establish the main characteristics of the created model, estimate the time of working with it and the cost of resources to obtain the specified quality of compliance of the model with the process of functioning of the system S.

Regardless of the type of model M used, when constructing it, it is necessary to be guided by a number of principles of a systematic approach: 1) proportional and consistent progress through the stages and directions of creating the model; 2) coordination of information, resource, reliability and other characteristics; 3) the correct relationship between individual hierarchy levels in the modeling system; 4) the integrity of individual separate stages of model construction.

Model M must meet the specified purpose of its creation, therefore the individual parts must be assembled mutually, based on a single system task. The goal can be formulated qualitatively, then it will have greater content and can reflect for a long time objective possibilities of this modeling system. When a goal is formulated quantitatively, a target function arises that accurately reflects the most significant factors influencing the achievement of the goal.

Model building is one of the system problems, when solving which solutions are synthesized based on huge number initial data, based on proposals from large teams of specialists. Using a systematic approach in these conditions allows not only to build a model of a real object, but also to choose based on this model required amount control information in a real system, evaluate its performance indicators and thereby, on the basis of modeling, find the most effective construction option and profitable mode of operation of the real system S.

2. GENERAL CHARACTERISTICS OF THE PROBLEM OF SYSTEMS MODELING

With the development of systemic research, with the expansion of experimental methods of studying real phenomena are becoming increasingly important abstract methods, new scientific disciplines appear, elements of mental work are automated. Mathematical methods of analysis and synthesis are important when creating real systems S; a number of discoveries are based on! purely theoretical research. However, it would be wrong to forget that the main criterion of any theory is practice, and even purely mathematical, abstract sciences are based on the foundation of practical knowledge.

Experimental studies of systems. Simultaneously with the development theoretical methods analysis and synthesis, methods for experimental study of real objects are also being improved, and new research tools are appearing. However, experiment has been and remains one of the main and essential tools of knowledge. Similarity and modeling allow us to describe the real in a new way! process and simplify its experimental study. The concept of modeling itself is also being improved. If earlier modeling! meant a real physical experiment or the construction of a model simulating a real process, then now new types of modeling have appeared, which are based on the formulation of not only physical, but also mathematical experiments.

Understanding reality is a long and complex process. Determining the quality of functioning of a large system, choosing the optimal structure and algorithms! behavior, building a system S in accordance with the set! the goal in front of her is the main problem in designing modern systems Therefore, modeling can be considered as one of the methods used in the design and study of large systems.

The simulation is based on some analogy between a real and a thought experiment. Analogy is the basis for explaining the phenomenon being studied, but only practice, only experience, can serve as a criterion of truth. Although modern scientific hypotheses can be created purely theoretically, they are, in fact, based on broad practical knowledge. To explain the real ones; processes, hypotheses are put forward, to confirm which an experiment is carried out or such theoretical reasoning, which logically confirm their correctness. In a broad sense, an experiment can be understood as a certain procedure for organizing and observing certain phenomena that are carried out under conditions close to natural ones or imitate them. 3

A distinction is made between a passive experiment, when the researcher observes the ongoing process, and an active one, when the observer intervenes and organizes the process. Recently, active experimentation has become widespread, since it is on its basis that it is possible to identify critical situations, obtain the most interesting patterns, ensure the possibility of repeating the experiment at different points, etc.

Any type of modeling is based on a certain model that has a correspondence based on some general quality that characterizes the real object. An objectively real object has some formal structure, therefore any model is characterized by the presence of some structure corresponding to the formal structure of the real object, or the aspect of this object being studied.

Modeling is based on information gaps, since the creation of the M model itself is based on information about the real object. In the process of implementing the model, information about this object is obtained, at the same time, during the experiment with the model, control information is introduced, a significant place is occupied by the processing of the results obtained, i.e. information underlies the entire modeling process.

Characteristics of system models. The object of modeling is complex organizational and technical systems, which can be classified as large systems. Moreover, in terms of its content, the created model M also becomes a system S(M) and can also be classified as a class of large systems, which are characterized by the following.

1. The purpose of operation, which determines the degree of purposefulness of the behavior of the model M. In this case, models can be divided into single-purpose, designed to solve one problem, and multi-purpose, allowing to resolve or consider a number of aspects of the functioning of a real object.

2. Complexity, which, given that model M is a collection of individual elements and connections between them, can be assessed by the total number of elements in the system and connections between them. Based on the variety of elements, one can distinguish a number of hierarchy levels, individual functional subsystems in the M model, a number of inputs and outputs, etc., i.e., the concept of complexity can be identified by a number of characteristics.

3. Integrity, indicating that the created model M is one whole system S(M) includes a large number of components (elements) that are in a complex relationship with each other.

4. Uncertainty that manifests itself in the system: in terms of the state of the system, the possibility of achieving the goal, methods. problem solving, reliability of initial information, etc. The main characteristic of uncertainty is a measure of information such as entropy, which in some cases makes it possible to estimate the amount of control information necessary to achieve a given state of the system. When modeling, the main goal is to obtain the required correspondence between the model and the real object, and in this sense, the amount of control information in the model can also be estimated using entropy and the minimum amount necessary to obtain the required result with a given reliability can be found. Thus, the concept of uncertainty, which characterizes a large system, is applicable to the M model and is one of its main features.

5. Behavioral stratum, which allows you to evaluate the effectiveness of the system in achieving its goal. Depending on the presence of random influences, it is possible to distinguish between deterministic and stochastic systems, by their behavior - continuous and discrete, etc. The behavioral stratum of consideration of the system allows, in relation to model M, to evaluate the effectiveness of the constructed model, as well as the accuracy and reliability of the results obtained. It is obvious that the behavior of model M does not necessarily coincide with the behavior of a real object, and often modeling can be implemented on the basis of a different material medium.

6. Adaptability, which is a property of a highly organized system. Thanks to adaptability, it is possible to adapt to various external disturbing factors in a wide range of changes in environmental influences. When applied to the model, it is essential to be able to adapt it to a wide range of disturbing influences, as well as to study the behavior of the model in changing conditions close to real ones. It should be noted that the question of the stability of the model to various disturbing influences may be significant. Since the M model is a complex system, issues related to its existence are very important, i.e. issues of survivability, reliability, etc.

7. The organizational structure of the modeling system, which largely depends on the complexity of the model and the degree of sophistication of the modeling tools. One of latest achievements In the field of modeling, one can consider the possibility of using simulation models to conduct machine experiments. Optimal organizational structure complex of technical means, information, mathematical and software modeling system S"(M), optimal organization of the modeling process, since it is necessary to pay Special attention on the simulation time and the accuracy of the results obtained.

8. Controllability of the model, resulting from the need to provide control on the part of experimenters in order to be able to consider the course of the process in various conditions simulating real ones. In this sense, the presence of many controllable parameters and model variables in the implemented modeling system makes it possible to conduct a wide experiment and obtain a wide range of results.

9. The possibility of developing a model that, based on the current level of science and technology, allows the creation of powerful S(M) modeling systems for studying many aspects of the functioning of a real object. However, when creating a modeling system, you cannot limit yourself only to tasks today. It is necessary to provide for the possibility of developing the modeling system both horizontally in the sense of expanding the range of functions being studied, and vertically in the sense of expanding the number of subsystems, i.e. the created modeling system should allow the use of new modern methods and funds. It is natural that intelligent system modeling can only function together with a team of people, so it is subject to ergonomic requirements.

2.1. Goals of control systems modeling.

One of the most important aspects of building modeling systems is the problem of purpose. Any model is built depending on the purpose that the researcher sets for it, so one of the main problems in modeling is the problem of purpose. The similarity of the process occurring in model M to the real process is not a goal, but a condition for the correct functioning of the model, and therefore the goal should be to study any aspect of the functioning of the object.

To simplify the M model, goals are divided into subgoals and more effective types of models are created depending on the obtained modeling subgoals. A number of examples of modeling objectives in the field of complex systems can be identified. For example, for an enterprise it is very important to study the processes of operational production management, operational scheduling, long-term planning, and modeling methods can also be successfully used here.

If the purpose of the modeling is clear, then the next problem arises, namely the problem of building a model M. Building a model is possible if information is available or hypotheses have been put forward regarding the structure, algorithms and parameters of the object under study. Based on their study, the object is identified. Currently widely used various ways parameter estimates: least squares method, maximum likelihood method, Bayesian, Markov estimates.

If model M has been built, then the next problem can be considered the problem of working with it, i.e., implementing the model, the main tasks of which are minimizing the time for obtaining final results and ensuring their reliability.

What is characteristic of a correctly constructed model M is that it reveals only those patterns that the researcher needs and does not consider the properties of the system S that are not essential for this study. It should be noted that the original and the model must be simultaneously similar in some characteristics and different in others, which allows us to highlight the most important properties being studied. In this sense, the model acts as a kind of “substitute” for the original, ensuring the fixation and study of only some properties of the real object.

In some cases, identification is the most difficult; in others, it is the problem of constructing the formal structure of an object. There may also be difficulties in implementing the model, especially in the case of simulation of large systems. At the same time, the role of the researcher in the modeling process should be emphasized. Statement of the problem, construction of a meaningful model of a real object in many ways represent creative process and are based on heuristics. And in this sense, there are no formal ways to choose the optimal type of model. Often there are no formal methods that allow a sufficiently accurate description of the real process. Therefore, the choice of one or another analogy, the choice of one or another mathematical modeling apparatus is entirely based on the existing experience of the researcher, and the researcher’s error can lead to erroneous modeling results.

Computer technology, which is currently widely used either for calculations in analytical modeling or for implementing a simulation model of a system, can only help in terms of implementation efficiency complex model, but do not allow you to confirm the correctness of the tone or other model. Only on the basis of processed data and the researcher’s experience can one reliably assess the adequacy of the model in relation to the real process.

If a real physical experiment occupies a significant place during the simulation, then the reliability of the tools used is also very important, since failures and failures of software and hardware can lead to distorted values ​​of the output data reflecting the progress of the process. And in this sense, when carrying out physical experiments, special equipment is needed, specially developed mathematical and information support, which make it possible to implement diagnostics of modeling tools in order to weed out those errors in the output information that are caused by malfunctions of the functioning equipment. During a machine experiment, erroneous actions by a human operator may also occur. Under these conditions, serious challenges arise in the field of ergonomic support for the modeling process.

3. CLASSIFICATION OF TYPES OF SYSTEMS MODELING.

Modeling is based on the theory of similarity, which states that absolute similarity can only occur when one object is replaced by another exactly the same. When modeling, absolute similarity does not exist and one strives to ensure that the model sufficiently well reflects the aspect of the object’s functioning being studied.

Classification characteristics. As one of the first signs of classification of types of modeling, you can select the degree of completeness of the model and divide the models in accordance with this sign into complete, incomplete and approximate. At the core full simulation there lies complete similarity, which manifests itself both in time and in space. Incomplete modeling is characterized by incomplete similarity of the model to the object being studied. Approximate modeling is based on approximate similarity, in which some aspects of the functioning of a real object are not modeled at all.

Depending on the nature of the processes being studied in the system S, all types of modeling can be divided into deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous. Deterministic modeling reflects deterministic processes, i.e. processes in which the absence of any random influences is assumed; stochastic modeling displays probabilistic processes and events. In this case, a number of realizations of a random process are analyzed and the average characteristics, i.e., a set of homogeneous realizations, are estimated. Static modeling is used to describe the behavior of an object at any point in time, while dynamic modeling reflects the behavior of an object over time. Discrete modeling is used to describe processes that are assumed to be discrete, respectively, continuous modeling allows us to reflect continuous processes in systems, and discrete-continuous modeling is used for cases when they want to highlight the presence of both discrete and continuous processes.

Depending on the form of representation of the object (system J), mental and real modeling can be distinguished.

Mental modeling is often the only way to model objects that are either practically unrealizable in a given time interval or exist outside the conditions possible for their physical creation. For example, on the basis of mental modeling, many situations of the microworld that cannot be analyzed can be analyzed. physical experiment. Mental modeling can be implemented in the form of visual, symbolic and mathematical.

Analog modeling is based on the use of analogies at various levels. The highest level is complete analogy, which occurs only for fairly simple objects. As the object becomes more complex, analogies of subsequent levels are used, when the analog model displays several or only one side of the object’s functioning.

Prototyping occupies an essential place in mental visual modeling. A mental model can be used in cases where the processes occurring in a real object are not amenable to physical modeling, or can precede other types of modeling. The construction of mental models is also based on analogies, but usually based on cause-and-effect relationships between phenomena and processes in an object. If you introduce a symbol for individual concepts, i.e. signs, as well as certain operations between these signs, you can implement sign modeling and use signs to display a set of concepts - to compose 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.

The basis of language modeling is a thesaurus. The latter is formed from a set of incoming concepts, and this set must be fixed. It should be noted that there are fundamental differences between a thesaurus and a regular dictionary. A thesaurus is a dictionary that is cleared of ambiguity, i.e. in it, each word can correspond to only a single concept, although in a regular dictionary several concepts can correspond to one word.

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

Math modeling. To study the characteristics of the functioning process of any system S using mathematical methods, including machine methods, a formalization of this process must be carried out, i.e., a mathematical model must be built.

By mathematical modeling we mean the process of establishing a correspondence between a given real object and a certain mathematical object, called a mathematical model, and the study of this model, which makes it possible to obtain the characteristics of the real object under consideration. The type of mathematical model depends both on the nature of the real object and the tasks of studying the object and the required reliability and accuracy of solving this problem. Any mathematical model, like any other,

Figure 1. Classification of types of system modeling.

describes a real object only with some degree of approximation to reality. Mathematical modeling for studying the characteristics of the process of functioning of systems can be divided into analytical, simulation and combined.

For analytical modeling It is characteristic that the processes of functioning of the system elements are written in the form of certain functional relations (algebraic, integer-differential, finite-difference, etc.) or logical conditions. The analytical model can be explored using the following methods: a) analytical, when they strive to obtain, in a general form, explicit dependencies for the desired characteristics; b) numerical, when, not being able to solve equations in general form, they strive to obtain numerical results with specific initial data; c) qualitative, when, without having an explicit solution, one can find some properties of the solution (for example, assess the stability of the solution).

IN in some cases System studies can also satisfy the conclusions that can be drawn using a qualitative method of analyzing a mathematical model. Such qualitative methods are widely used, for example, in the theory of automatic control to evaluate the effectiveness of various options for control systems.

Conclusion.

In conclusion of this course work, I would like to draw several conclusions from the above material about modeling in the study of control systems. So let’s define the epistemological nature of modeling.

Defining the epistemological role of modeling theory, i.e. its significance in the process of cognition, it is necessary, first of all, to abstract from the variety of models available in science and technology and to highlight what is common to models of objects that are different in nature real world. This commonality lies in the presence of some structure (static or dynamic, material or mental), which is similar to the structure of a given object. In the process of studying, the model acts as a relative independent quasi-object, which allows one to obtain some knowledge about the object itself during the study.

IN modern Russia management and its research are moving along the path of complexity. By using modeling techniques such as analogy, you can achieve impressive results in economic activity enterprises. An analogy is a judgment about any particular similarity between two objects, and such similarity can be significant or insignificant. It should be noted that the concepts of significance and insignificance of the similarity or difference of objects are conditional and relative. The significance of the similarities (differences) depends on the level of abstraction and general case determined by the ultimate goal of the research. Modern scientific hypothesis created, as a rule, by analogy with proven in practice scientific principles.

In conclusion, we can summarize the above that modeling is the main path in the research system of control systems and is of extreme importance for a manager at any level.

Bibliography.

1. Ignatieva A.V., Maksimtsov M.M. RESEARCH OF CONTROL SYSTEMS, Moscow, 2000

2. Paterson J. Petri net theory and systems modeling. - M.: Mir, 1984.

3. Priiker A. Introduction to simulation modeling and the SLAMP language. - M.: Mir, 1987.

4.Sovetov B.Ya.. Yakovlev S.A. Modeling of systems. - M.: graduate School, 1985.

5. Sovetov B. Ya., Yakovlev S. A. Modeling of systems (2nd ed.). - M.: Higher School, 1998.

6.Sovetov B.Ya.. Yakovlev S.A. Modeling of systems: Course design. - M.: Higher School, 1988.

7. Korotkoe E.M. Research of control systems. - M.: “DeKA”, 2000.

Tutoring

Need help studying a topic?

Our specialists will advise or provide tutoring services on topics that interest you.
Submit your application indicating the topic right now to find out about the possibility of obtaining a consultation.

Modeling is based on the theory of similarity, which states that absolute similarity can only occur when one object is replaced by another exactly the same. When modeling, absolute similarity does not take place and one strives to ensure that the model sufficiently well reflects the aspect of the object’s functioning under study.

Classification characteristics. As one of the first signs of classification of types of modeling, you can select the degree of completeness of the model and divide the models in accordance with this sign into complete, incomplete and approximate. The basis of complete modeling is complete similarity, which manifests itself both in time and in space. Incomplete modeling is characterized by incomplete similarity of the model to the object being studied. Approximate modeling is based on approximate similarity, in which some aspects of the functioning of a real object are not modeled at all.

Depending on the nature of the processes being studied in the system S, all types of modeling can be divided into deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous. Deterministic modeling reflects deterministic processes, i.e. processes in which the absence of any random influences is assumed; Stochastic modeling reflects probabilistic processes and events. In this case, a number of realizations of a random process are analyzed and the average characteristics, i.e., a set of homogeneous realizations, are estimated. Static modeling serves to describe the behavior of an object at any point in time, and dynamic modeling reflects the behavior of an object over time. Discrete modeling is used to describe processes that are assumed to be discrete, respectively, continuous modeling allows us to reflect continuous processes in systems, and discrete-continuous modeling is used for cases when they want to highlight the presence of both discrete and continuous processes.

Depending on the form of representation of the object (system J), mental and real modeling can be distinguished.

Mental modeling is often the only way to model objects that are either practically unrealizable in a given time interval or exist outside the conditions possible for their physical creation. For example, on the basis of mental modeling, many situations in the microworld that are not amenable to physical experiment can be analyzed. Mental modeling can be implemented in the form of visual, symbolic and mathematical.

Analog modeling is based on the use of analogies at various levels. The highest level is complete analogy, which occurs only for fairly simple objects. As the object becomes more complex, analogies of subsequent levels are used, when the analog model displays several or only one side of the object’s functioning.

Prototyping occupies an essential place in mental visual modeling. A mental model can be used in cases where the processes occurring in a real object are not amenable to physical modeling, or can precede other types of modeling. The construction of mental models is also based on analogies, but usually based on cause-and-effect relationships between phenomena and processes in an object. If you introduce a conventional designation for individual concepts, i.e., signs, as well as certain operations between these signs, then you can implement sign modeling and, using signs, display a set of concepts - create 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 individual symbols.

The basis of language modeling is a thesaurus. The latter is formed from a set of incoming concepts, and this set must be fixed. It should be noted that there are fundamental differences between a thesaurus and a regular dictionary. A thesaurus is a dictionary that is cleared of ambiguity, i.e. in it, each word can correspond to only a single concept, although in a regular dictionary several concepts can correspond to one word.

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

Math modeling. To study the characteristics of the process of functioning of any system S using mathematical methods, including machine methods, a formalization of this process must be carried out, i.e., a mathematical model must be built.

By mathematical modeling we mean the process of establishing a correspondence between a given real object and a certain mathematical object, called a mathematical model, and the study of this model, which makes it possible to obtain the characteristics of the real object under consideration. The type of mathematical model depends both on the nature of the real object and the tasks of studying the object and the required reliability and accuracy of solving this problem. Any mathematical model, like any other,

Fig 1.

describes a real object only with a certain degree of approximation to reality. Mathematical modeling for studying the characteristics of the process of functioning of systems can be divided into analytical, simulation and combined.

Analytical modeling is characterized by the fact that the processes of functioning of system elements are written in the form of certain functional relationships (algebraic, integer-differential, finite-difference, etc.) or logical conditions. The analytical model can be studied by the following methods: a) analytical, when one strives to obtain, in a general form, explicit dependencies for the desired characteristics; b) numerical, when, not being able to solve equations in general form, they strive to obtain numerical results with specific initial data; c) qualitative, when, without having an explicit solution, one can find some properties of the solution (for example, assess the stability of the solution).

In some cases, system studies can also satisfy the conclusions that can be drawn using a qualitative method of analyzing a mathematical model. Such qualitative methods are widely used, for example, in the theory of automatic control to evaluate the effectiveness of various options for control systems.