Simulation modeling is the most universal method for studying systems and quantifying the characteristics of their functioning. In simulation modeling, the dynamic processes of the original system are replaced by processes simulated in an abstract model, but maintaining the same ratios of durations and time sequences of individual operations. Therefore, the simulation method could be called algorithmic or operational. In the process of simulation, as in an experiment with the original, certain events and states are recorded or output influences are measured, from which the characteristics of the quality of the system’s functioning are calculated.

Simulation modeling allows you to consider the processes occurring in the system at almost any level of detail. Using the algorithmic capabilities of a PC, any algorithm for controlling or operating a system can be implemented in a simulation model. Models that can be studied by analytical methods can also be analyzed by simulation methods. All this is the reason that simulation modeling methods are becoming the main methods for studying complex systems.

Simulation modeling methods vary depending on the class of systems under study, the method of advancing the model time and the type of quantitative variables of the system parameters and external influences.

First of all, we can divide the methods of simulation of discrete and continuous systems. If all elements of the system have a finite set of states, and the transition from one state to another is instantaneous, then such a system belongs to systems with discrete changes of states, or discrete systems. If the variables of all elements of the system change gradually and can take on an infinite number of values, then such a system is called a system with a continuous change of states, or a continuous system. Systems that have variables of both types are considered discrete-continuous. In continuous systems, certain states of elements can be artificially isolated. For example, some characteristic values ​​of variables are recorded as the achievement of certain states.

One of the main parameters in simulation is model time, which reflects the operating time of the real system. Depending on the method of advancing model time, modeling methods are divided into methods with an increment of time interval and methods with time advancement to special states. In the first case, the model time advances by a certain amount Dt. Changes in the states of elements and output impacts of the system that occurred during this time are determined. After this, the model time again advances by the amount Dt, and the procedure is repeated. This continues until the end of the simulation period. T m. Time increment Dt is often chosen to be constant, but in the general case it can also be variable. This method is called the "principle Dt ».

In the second case, at the current moment of model time t first, those future special states are analyzed - arrival of a discrete input action (request), completion of service, etc., for which the moments of their occurrence are determined t i > t. The earliest special state is selected and the model time is advanced until that state occurs. It is assumed that the state of the system does not change between two adjacent special states. The system's response to the selected special condition is then analyzed. In particular, during the analysis the moment of onset of a new special state is determined. Future special states are then analyzed and the model time is advanced to the nearest one. The procedure is repeated until the end of the simulation period T m. This method is called the “principle of special states”, or the “principle dz" Thanks to its use, computer simulation time is saved. However, it is used only when it is possible to determine the moments of occurrence of future special conditions.

Of particular importance is the stationarity or non-stationarity of random, independent variables of the system and external influences. When the variables are nonstationary, primarily external influences, which is often observed in practice, special modeling methods should be used, in particular, the method of repeated experiments.

Another classification parameter should be considered the formalization scheme adopted when creating the mathematical model. Here, first of all, it is necessary to separate methods focused on an algorithmic (software) or structural (aggregate) approach. In the first case, processes manage the elements (resources) of the system, and in the second, the elements manage processes and determine the order of the system’s functioning.

From the above it follows that the choice of one or another modeling method is completely determined by the mathematical model and the initial data.

Continuous modeling is the modeling of a system over time using a representation in which state variables change continuously with respect to time. Typically, continuous simulation models use differential equations that establish relationships for the rates of change of state variables over time. If the differential equations are very simple, they can be solved analytically to represent the values ​​of the state variables for all values ​​of time as a function of the values ​​of the state variables at time 0. For large continuous models, an analytical solution is not possible, but for the numerical integration of differential equations in the case of given special values For state variables at time 0, numerical analysis techniques such as Runge-Kutta integration are used.

Example 1.3. Consider a continuous model of competition between two populations. Biological models of this type, called models predator-prey(or parasite-host), have been considered by many authors, including Brown and Gordon. The environment is represented by two populations - predators and prey, interacting with each other. The prey is passive, but predators depend on its population as a source of food for them. (For example, sharks can be predators, and the fish they feed on as prey) Let x(t) and y(t) denote the number of individuals in populations of prey and predators, respectively, at a point in time t. Let's say the prey population has an abundant food supply; in the absence of predators, its growth rate will be r x(t) for some positive value r(r- natural birth rate minus natural death rate). The existence of interactions between predators and prey suggests that the mortality rate of prey due to this interaction is proportional to the product of the sizes of both populations x(t)y(t). Therefore, the overall rate of change in the prey population dx /dt: can be represented as

Where A - positive proportionality coefficient. Since the existence of predators themselves depends on the prey population, the rate of change of the predator population in the absence of prey is -sу(t) for some positive s. Moreover, the interaction between the two populations leads to an increase in the population of predators, the rate of which is also proportional x(t)y(t). Therefore, the overall rate of change in the predator population dy/dt amounts to

(2)

Where b- positive proportionality coefficient. Under initial conditions x(0)> 0 and y(0) >0 the solution to the model defined by equations (1) and (2) has an interesting property: x(t)> 0 and y(t)> 0 for any t³0. Consequently, the prey population will never be completely destroyed by predators. Solution (x(t), y(t)) is also a periodic function of time. In other words, there is such a meaning T> 0, at which x(t + nT)=x(t) And y(t + nT)= y(t) for any positive integer P. This result is not unexpected. As the predator population increases, the prey population decreases. This leads to a decrease in the growth rate of the predator population and, accordingly, causes a decrease in their number, which, in turn, leads to an increase in the prey population, etc.

Let's consider individual values ​​g = 0.001, a = 2 * 10 –6; s = 0.01; b=10 -6 , the initial population sizes are X( 0) = 12,000 and y(0) = 600. In Fig. presents a numerical solution to equations (1) and (2), obtained using a computing package developed for the numerical solution of systems of differential equations (and not a continuous modeling language).

Note that the example above is completely deterministic, meaning there are no random components. However, the simulation model may also contain unknown quantities; for example, random variables that depend in some way on time can be added to equations (1) and (2), or constant factors can be modeled as quantities that randomly change their values ​​at certain points in time.

5.3 Combined continuous-discrete modeling

Since some of the systems are neither fully discrete nor fully continuous, it may be necessary to create a model that combines aspects of both discrete event and continuous modeling, resulting in combined continuous-discrete modeling. Three main types of interaction can occur between discrete and continuous changes in state variables:

A discrete event can cause a discrete change in the value of a continuous state variable;

At a given point in time, a discrete event can cause a change in the relationship governing a continuous state variable;

A continuous state variable that reaches a threshold can cause a discrete event to occur or be scheduled.

The following example of combined continuous-discrete modeling provides a brief description of a model discussed in detail by Pritzker, who provides other examples of this type of modeling in his work.

Example 1.4. Tankers carrying oil arrive at one unloading dock, replenishing a storage tank from which the oil is piped to the refinery. From an unloading tanker, oil is supplied to the storage tank at a constant rate (Tankers arriving at a busy dock form a queue.) At the refinery, oil is supplied from the tank at various set rates. The dock is open from 6.00 to 24.00. For safety reasons, tanker unloading stops when the dock is closed.

The discrete events in this (simplified) model are the arrival of the tanker for unloading, the closing of the dock at midnight and the opening at 6.00. Oil levels in the unloading tanker and storage tank are specified by continuous state variables, the rates of change of which are described using differential equations. Tanker unloading is considered complete when the oil level in the tanker is less than 5% of its capacity, but unloading must be temporarily stopped if the oil level in the storage tank reaches its capacity. Unloading can be resumed when the oil level in the tank drops below 80% of its capacity. If the oil level in the reservoir drops below 5,000 barrels, the refinery must be temporarily closed. To avoid frequent shutdowns and restarts of the plant, oil from the reservoir will not be restored to the plant until it has 50,000 barrels of oil. Each of five oil level events (for example, oil level falling below 5% of a tanker's capacity), according to Pritzker's definition, is state event. Unlike discrete events, state events are not scheduled; they occur when continuous state variables cross a threshold.

5.4 Monte Carlo simulation. Statistical modeling of systems

Among the methods for modeling continuous electric drive control systems, two can be distinguished, based on the use of mathematical models of systems in the form of state models and structural models, each of which has its own specific advantages in solving specific problems of modeling automated control systems. It is most convenient to use the state model when modeling and synthesizing multidimensional linear control systems for electric vehicles using state space methods. When modeling nonlinear ED systems, as well as some specific elements of modern ED systems, such as thyristor converters and microprocessors, it is more effective to use structural models. It is especially convenient to use them in analysis in connection with the expressed structure of real electric drive systems. However, the effectiveness of using structural (topological) methods decreases significantly as the control systems of electrical equipment become more complex. Therefore, the choice of modeling method is determined by the feasibility of its application in a particular case.

Digital modeling of continuous control systems is based on the description of the system by ordinary differential equations in Cauchy form, where in the general case for a multidimensional element, each input variable is associated with each output variable. If the relationships along all channels are linear or linearized, then in the general case a multidimensional element can be described by a system of inhomogeneous differential equations. The system can be written more compactly as a single vector differential equation. A vector differential equation in Cauchy form, reflecting the dynamic properties of a multidimensional linear object, is an equation of state and is used as a mathematical model when modeling by state space methods. A complete mathematical model of a linear multidimensional object, in addition to state equations, also contains an output equation that connects state variables and control actions with output variables.

The equations described above can be solved by various methods, which can be classified into two groups: methods of numerical integration of differential equations and matrix methods based on the calculation of the transition state matrix.

Numerical integration methods include long-known and tested methods: Euler, Runge-Kutta, Adams-Bashforth, Adams-Moulton, etc. Analyzing the known results, we can conclude that, along with the recognized exact methods of high-order numerical integration, for example, the Runge-Kutta methods fourth order, Kutta-Merson fourth order, it is advisable to use less accurate numerical methods, for example second-order Euler and Adams-Bashforth, when developing non-standard methods for digital modeling of automated control systems, using which it is possible to ensure sufficient modeling accuracy with the appropriate integration step. When solving problems in real time, it is advisable to use the first-order Euler method for numerical integration, which is economical in terms of both memory capacity and solution time. This is of particular relevance in microprocessor control systems for electronic devices.

Matrix methods for calculating the transition process in linear systems are based on the calculation of the transition (exponential) state matrix, which is associated with the need to perform complex and cumbersome calculations, and are especially difficult in the absence of specialized application software packages (the most famous package of symbolic mathematics, focused on working with vectors and matrices, should be recognized as MatLab). Methods for calculating the transition state matrix can be classified as follows: direct, based on the Plant method, Padé approximation, Keley-Hamilton theorem. All of the listed methods for calculating the transition state matrix use a recurrent algorithm for its calculation. The transition state matrix is ​​represented by a matrix series expansion. To ensure the functionality of the algorithm for calculating the transition matrix, it is necessary to set the maximum number of terms of the series, if exceeded, the calculations stop. It should be noted that with the number of members of the series To=2, the accuracy of calculating the transition state matrix corresponds to the accuracy of the Euler method, with To=3 - accuracy of the improved Euler method, with To=5 - accuracy of the Runge-Kutta method. Obviously, the computational costs are significantly higher compared to numerical integration methods. In addition to performing calculations for the transition state matrix, it is necessary to calculate the input matrix, which mainly uses two methods: analytical, when it is known in advance that the transition process is stable; approximate, when the nature of the transition process is not determined in advance. The use of both methods involves cumbersome matrix operations. But it should be noted that the matrix method has its advantages over other methods when modeling multidimensional control systems with several inputs and outputs.

Digital modeling of continuous control systems based on topological representations (structural modeling) makes it possible to make maximum use of information about the structure of the system under study; here, each typical link corresponds to a specific model, which, in turn, can be implemented on the basis of two typical links.

Thus, the choice of a method for modeling continuous control systems of electric power plants, as well as methods for calculating transient processes, is determined by the effectiveness of use in solving a specific problem.

When modeling discrete electric control systems, it is necessary to solve the problem of constructing algorithms for digital modeling of the joint operation of digital and analog elements of the system, which has some specific features. One of them is the large expenditure of computer time on the joint reproduction of the dynamic properties of the digital and analog parts of the system under study, associated with the need to repeatedly solve the differential equations of the analog part in one clock cycle of the digital part. Another important feature is the specific mathematical apparatus for calculating digital control systems, using z-transformation.

The results of studying transient processes in physical systems based on methods in which continuous signals are replaced by temporary sequences of numbers during calculations show that this approach provides significant savings in computational costs. The relationships between time sequences of real numbers (lattice functions) are described by convenient recurrent difference equations, the coefficients of which depend on the parameters of physical systems. Some recurrent methods, in particular the Tustin method, make it possible to obtain effective algorithms for digital modeling of discrete systems. The essence of currently known recurrent difference methods is to replace processes occurring in continuous systems with processes in equivalent discrete systems. The mathematical apparatus in this case is the method z-transformations. The considered methods of Tustin and Boxer-Thaler for constructing digital modeling algorithms for control systems, specified in the form of block diagrams, have very few restrictions or no restrictions at all. They are universal in the sense of being used with analytical or arbitrary waveform input signals. The order of the recurrent equations coincides with the order of the linear part of the modeled system, regardless of the method used. No additional effort is required during preparatory work. However, the accuracy of these methods is fundamentally not as high as methods that use information about the entire continuous system as a whole (methods of invariant impulse functions, Tsypkin-Goldenberg, Ragazzini-Bergen).

Systems modeling methods

The formulation of any problem is to translate its verbal description into a formal one. In the case of relatively simple tasks, such a transition is carried out in the mind of a person, who cannot always even explain how he did it. If the resulting formal model (mathematical relationship between quantities in the form of a formula, equation, system of equations) is based on a fundamental law or is confirmed by experiment, then this proves its adequacy to the depicted situation, and the model is recommended for solving problems of the corresponding class.

As problems become more complex, obtaining a model and proving its adequacy becomes more difficult. Initially, the experiment becomes expensive and dangerous (for example, when creating complex technical complexes, when implementing space programs, etc.), and in relation to economic objects, the experiment becomes practically unrealizable, the problem becomes a class of decision-making problems, and the formulation of the problem, the formation of a model , i.e. the translation of a verbal description into a formal one becomes an important part of the decision-making process. Moreover, this component cannot always be distinguished as a separate stage, after completing which one can treat the resulting formal model in the same way as with an ordinary mathematical description, strict and absolutely fair. Most real situations in the design of complex technical complexes and economic management must be represented as a class of self-organizing systems, the models of which must be constantly adjusted and developed.

In this case, it is possible to change not only the model, but also the modeling method, which is often a means of developing the decision maker’s understanding of the simulated situation. In other words, the translation of a verbal description into a formal one, comprehension, interpretation of the model and the results obtained become an integral part of almost every stage of modeling a complex developing system.

Often, in order to more accurately characterize this approach to modeling decision-making processes, they talk about creating a “mechanism” for modeling, a “mechanism” for decision-making (for example, “economic mechanism”, “mechanism for design and development of an enterprise”, etc.).

The questions that arise are how to form such evolving models or “mechanisms”? how to prove the adequacy of models? – and are the main subject of system analysis.

To solve the problem of translating a verbal description into a formal one, special techniques and methods began to develop in various fields of activity. Thus, methods such as “brainstorming”, “scenarios”, expert assessments, “goal trees”, etc. arose.

In turn, the development of mathematics followed the path of expanding the means of posing and solving difficult-to-formalize problems. Along with the deterministic, analytical methods of classical mathematics, probability theory and mathematical statistics arose (as a means of proving the adequacy of a model based on a representative sample and the concept of probability of the legitimacy of using the model and modeling results). For problems with a greater degree of uncertainty, engineers began to use set theory, mathematical logic, mathematical linguistics, and graph theory, which largely stimulated the development of these areas. In other words, mathematics began to gradually accumulate means of working with uncertainty, with meaning, which classical mathematics excluded from the objects of its consideration.

Thus, between the informal, figurative thinking of a person and the formal models of classical mathematics, a “spectrum” of methods has developed that help obtain and clarify (formalize) a verbal description of a problem situation, on the one hand, and interpret formal models, connect them with reality, with another. This spectrum is conventionally presented in Fig. 2.1, a.

The development of modeling methods, of course, did not proceed as consistently as shown in Fig. 2.1, a. Methods arose and developed in parallel. There are various modifications of similar methods. They were grouped in different ways, i.e. researchers have proposed different classifications (mainly for formal methods, which will be discussed in more detail in the next paragraph). New modeling methods are constantly emerging, as if at the “intersection” of already established groups. However, this figure illustrates the main idea - the existence of a “spectrum” of methods between verbal and formal representation of a problem situation.

Initially, researchers developing systems theory proposed classifications of systems and tried to match them with certain modeling methods that would best reflect the characteristics of a particular class. This approach to the selection of modeling methods is similar to the approach of applied mathematics. However, unlike the latter, which is based on classes of applied problems, system analysis can represent the same object or the same problem situation (depending on the degree of uncertainty and as it is learned) by different classes of systems and, accordingly, different models, like would thus organize the process of gradual formalization of the task, i.e. “growing” its formal model. The approach helps to understand that an incorrectly chosen modeling method can lead to incorrect results, the inability to prove the adequacy of the model, an increase in the number of iterations and a delay in solving the problem.

The formulation of any problem is to translate its verbal, verbal description in formal.

In the case of relatively simple tasks, such a transition is carried out in the mind of a person, who cannot always even explain how he did it. If the resulting formal model (mathematical relationship between quantities in the form of a formula, equation, system of equations) is based on a fundamental law or is confirmed by experiment, then this proves it adequacy displayed situation, and the model is recommended for solving problems of the corresponding class.

Adequacy (models of the problem being solved)- the legitimacy of using the model to study the problem being solved and display the problem situation. In a narrower sense, the adequacy of a model is understood as its compliance with the modeled object or process. It should be borne in mind that there cannot be a complete correspondence between the model and the object. This means proving the correspondence of the model and the object in terms of the most essential properties of the object.

The adequacy of the model in the development and research of technical systems is proven by experiment.

As problems become more complex, obtaining a model and proving its adequacy becomes more difficult. At first, the experiment becomes expensive and dangerous (for example, when creating complex technical complexes, when implementing space programs, etc.), and in relation to economic objects, the experiment becomes practically impossible to implement, the task becomes a class decision making problems, and setting the problem, forming a model, i.e. the translation of a verbal description into a formal one becomes an important part of the decision-making process. Moreover, this component cannot always be distinguished as a separate stage, after completing which one can treat the resulting formal model in the same way as with an ordinary mathematical description, strict and absolutely fair. Most real-life situations in the design of complex technical complexes and economic management must be represented as a class self-organizing systems(see unit 1), the models of which must be constantly adjusted and developed. In this case, it is possible to change not only the model, but also the modeling method, which is often a means of developing the decision maker’s understanding of the simulated situation.

In other words, the translation of a verbal description into a formal one, comprehension, interpretation of the model and the results obtained become an integral part of almost every stage of modeling a complex developing system. Often, in order to more accurately characterize this approach to modeling decision-making processes, they talk about creating a kind of “mechanism” for modeling, a “mechanism” for making decisions (for example, “economic mechanism”, “mechanism for design and development of an enterprise”, etc.) .

The questions that arise are how to form such evolving models or “mechanisms”? how to prove the adequacy of models? - are the main subject of system analysis.

To solve the problem of translating a verbal description into a formal one, special techniques and methods began to develop in various fields of activity. Thus, methods such as “brainstorming”, “scenarios”, expert assessments, “goal trees”, etc. arose.

In turn, the development of mathematics followed the path of expanding the means of posing and solving difficult-to-formalize problems.

Along with the deterministic, analytical methods of classical mathematics, probability theory and mathematical statistics arose as a means of proving the adequacy of a model based on a representative sample and the concept of probability, the legitimacy of using the model and modeling results.

For problems with a greater degree of uncertainty, engineers began to use set theory, mathematical logic, mathematical linguistics, graph theory, which largely stimulated the development of these areas.

In other words, mathematics began to gradually accumulate means of working with uncertainty, with a meaning that classical mathematics excluded from the objects of its consideration.

Thus, between the informal, figurative thinking of a person and the formal models of classical mathematics, a “spectrum” of methods has developed that help obtain and clarify (formalize) a verbal description of a problem situation, on the one hand, and interpret formal models, connect them with reality - with another. This spectrum is conventionally presented in Fig. 2.1, A.

Rice. 2.1. Systems modeling methods

The development of modeling methods, of course, did not proceed as consistently as shown in the figure. Methods arose and developed in parallel. There are various modifications of similar methods. They were grouped in different ways, i.e. researchers have proposed different classifications (mainly for formal methods). New modeling methods are constantly emerging, as if at the “intersection” of already established groups. However, the main idea - the existence of a "spectrum" of methods between verbal and formal representation of a problem situation - is shown in this figure.

Initially, researchers developing systems theory proposed classifications of systems and tried to match them with certain modeling methods that would best reflect the characteristics of a particular class.

This approach to the choice of modeling methods is similar to the approach of applied mathematics. However, unlike the latter, which is based on classes of applied problems, system analysis can represent the same object or the same problem situation (depending on the degree of uncertainty and as it is learned) by different classes of systems and, accordingly, different models, organizing Thus, the process of gradual formalization of the task, i.e. “growing” its formal model. The approach helps to understand that an incorrectly chosen modeling method can lead to incorrect results, the inability to prove the adequacy of the model, an increase in the number of iterations and a delay in solving the problem.

There is another point of view. If you successively change the methods shown in Fig. 2.1, A“spectrum” (not necessarily using everything), then you can gradually, limiting the completeness of the description of the problem situation (which is inevitable when formalizing), but preserving the most significant components from the point of view of the goal (structure of goals) and the connections between them, move on to a formal model.

This idea was realized, for example, in the creation of computer software and automated information systems by sequentially translating the description of a task from natural language into a high-level language (task management language, information retrieval language, modeling language, design automation), and from there into one of programming languages ​​suitable for a given task (PL/1, LISP, PASCAL, SI, PROLOG, etc.), which, in turn, is translated into machine instruction codes that operate the computer hardware.

At the same time, an analysis of the processes of inventive activity and the experience of forming complex decision-making models showed that practice does not obey such logic, i.e. a person acts differently: he alternately selects methods from the left and right parts of the “spectrum” shown in Fig. 2.1, A.

Therefore, it is convenient to “break” this “spectrum” of methods approximately in the middle, where graphical methods merge with structuring methods, i.e. divide systems modeling methods into two large classes: methods of formalized representation of systems - MFPS And methods aimed at enhancing the use of intuition and experience of specialists or more briefly - methods for activating the intuition of specialists - MAIS.

Possible classifications of these two groups of methods are shown in Fig. 2.1, b.

This division of methods is in accordance with the main idea of ​​system analysis, which consists of combining formal and informal representations in models and techniques, which helps in the development of techniques, the selection of methods for the gradual formalization of mapping and analysis of the problem situation.

Note that in Fig. 2.1, b in the MAIS group, methods are arranged from top to bottom approximately in increasing order of formalization possibilities, and in the IPPS group - from top to bottom, attention to the substantive analysis of the problem increases and more and more tools appear for such analysis. This ordering helps to compare methods and select them when forming developing decision-making models and when developing methods of system analysis.

Classifications of MAIS and especially MFPS may be different. In Fig. 2.1, b The classification of MPPS proposed by F.E. is given. Temnikov .

It should be noted that sometimes the terms MAIS and IPPS are used to name groups quality And quantitative methods. However, on the one hand, methods classified as MAIS group can also use formalized representations (when developing scenarios statistical data may be used and some calculations may be made; formalization is associated with obtaining and processing expert assessments, methods of morphological modeling); and, on the other hand, by virtue of Gödel’s theorem on incompleteness, within the framework of any formal system, no matter how complete and consistent it may seem, there are provisions (relations, statements), the truth or falsity of which cannot be proven by the formal means of this system, but to overcome In order to solve an insoluble problem, it is necessary to expand the formal system, relying on meaningful, qualitative analysis. Therefore, the names of the groups of methods MAIS and MFPS were proposed, which seems more preferable.

Gödel's results were obtained for arithmetic, the most formal branch of mathematics, and suggested that the process of logical, including mathematical proof, is not reduced to the use of only the deductive method, that informal elements of thinking are always present in it. Subsequent studies of this problem by mathematicians and logicians showed that “proofs do not at all have absolute, time-independent rigor and are only culturally mediated means of persuasion.”

In other words, there is no strict division between formal and informal methods. We can only talk about a greater or lesser degree of formalization or, on the contrary, a greater or lesser reliance on intuition and common sense.

A systems analyst must understand that any classification is conditional. It is just a tool that helps you navigate a huge number of different methods and models. Therefore, it is necessary to develop a classification taking into account specific conditions, the characteristics of the systems being modeled (decision-making processes) and the preferences of decision makers (DMs), who can be asked to choose a classification.

It should also be noted that new modeling methods are often created based on a combination of pre-existing classes of methods.

So, integratedmethods(combinatorics, topology) began to develop in parallel within the framework of linear algebra, set theory, graph theory, and then formed into independent directions.

There are also new methods based on a combination of MAIS and MFPS tools. This group of methods is presented in Fig. 2.1 as an independent group of modeling methods, generally called special methods.

The following special methods for modeling systems are most widely used.

Simulation dynamic modeling, proposed by J. Forrester (USA) in the 50s. XX century, uses a human-friendly structural language that helps express real relationships that display closed control loops in the system, and analytical representations (linear finite-difference equations) that make it possible to implement a formal study of the resulting models on a computer using the specialized DYNAMO language.

Idea situational modeling proposed by D.A. Pospelov, developed and put into practice by Yu.I. Klykov and L.S. Zagadskaya (Bolotova). This direction is based on displaying in computer memory and analyzing problem situations using a specialized language developed using the expressive means of set theory, mathematical logic and language theory.

Structural-linguistic modeling. The approach arose in the 70s. XX century in engineering practice and is based on the use of structural representations of various kinds, on the one hand, and the means of mathematical linguistics, on the other, to implement the ideas of combinatorics. In an expanded understanding of the approach, other methods of discrete mathematics, languages ​​based on set-theoretic representations, and the use of tools of mathematical logic, mathematical linguistics, and semiotics are also used as linguistic (linguistic) means.

Information field theory and information approach to modeling and analysis of systems. The concept of the information field was proposed by A.A. Denisov and is based on the use of the laws of dialectics to activate the decision maker’s intuition, and as a means of formalized mapping - the apparatus of mathematical field theory and circuit theory. For brevity, this approach is subsequently called informational, since it is based on the display of real situations using information models.

A method of gradual formalization of tasks and problem situations with uncertainty through the alternate use of MAIS and IPPS tools. This approach to modeling self-organizing (developing) systems was originally proposed based on the concept structural-linguistic modeling, but subsequently became the basis of almost all systems analysis techniques.

A classification of modeling methods, similar to the one discussed, helps to consciously choose modeling methods and should be part of the methodological support for work on the design of complex technical complexes, and on the management of enterprises and organizations. It can be developed and supplemented with specific methods, i.e. accumulate experience gained in the design and management process.