What models use systems of equations. Mathematical model in practice. Let us consider the features of the continuously deterministic approach using an example, using differential equations as a mathematical model

Mathematical models are distinguished mainly by the nature of the displayed properties of the system, the degree of their detail, methods of obtaining and formal representation.

Structural and functional models. If a MM displays elements and their connections in a system, then it is called structural mathematical model. If the MM reflects any processes occurring in the system, then it is classified as functional mathematical models. It is clear that mixed MMs can also exist , which describe both the functional and structural properties of the system. Structural MMs are divided into topological and geometric, which constitute two levels of the hierarchy of MMs of this type. The first ones reflect the composition of the system and the connections between its elements. Topological MM It is advisable to use it at the initial stage of studying a complex system. Such a MM has the form of graphs , tables, matrices, lists, etc., and its construction is usually preceded by the development of a structural diagram of the system.

Geometric MM in addition to the information presented in the topological MM, it contains information about the shape and size of the system and its elements, and their relative position. The geometric MM usually includes a set of equations of lines and surfaces and algebraic relations that determine the belonging of regions of space to the system or its elements. Geometric MMs are used in the design of elements of technical systems, development of technical documentation and technological processes for manufacturing products.

Functional MM consist of relationships connecting phase variables , those. internal, external and output parameters of the system. The functioning of complex systems can often be described only with the help of a set of its reactions to some known (or given) input influences. This type of functional MM is classified as black box and is usually called imitation a mathematical model, meaning that it only imitates the external manifestations of functioning, without revealing or describing the essence of the processes occurring in the system. Simulation MMs are widely used in the study of complex systems.

In terms of presentation form, a simulation MM is an example algorithmic MM, since the connection between the input and output parameters of the system can only be described in the form of an algorithm suitable for implementation as a program. The type of algorithmic MM includes a wide class of both functional and structural MM. If the connections between the parameters of the system can be expressed in analytical form, then we speak of analytical mathematical models . When creating a hierarchy of MMs for the same system, one usually strives to ensure that a simplified version of the MM is presented in an analytical form that allows for an exact solution that could be used for comparison when testing the results obtained using more complete and therefore more complex versions of the MM.

It is clear that the MM of a particular system, in terms of its presentation form, can include features of both analytical and algorithmic MM. Moreover, during the modeling process, the analytical MM is converted into an algorithmic one.

According to the method of obtaining mathematical models can be theoretical or empirical. The former are obtained as a result of studying the properties of the system, the processes occurring in it based on the use of known fundamental conservation laws, as well as equilibrium equations, and the latter are the result of processing the results of external observations of the manifestation of these properties and processes. One of the ways to construct empirical MMs is to conduct experimental studies related to the measurement of phase variables of the system, and to subsequently generalize the results of these measurements in an algorithmic form or in the form of analytical dependencies. Therefore, in terms of the form of presentation, an empirical MM may contain features of both algorithmic and analytical MM . Thus, the construction of an empirical MM comes down to solving identification problems.

Features of functional models. One of the characteristic features functional MM is the presence or absence of random variables among its parameters. In the presence of such quantities, MM is called stochastic(or probabilistic), and in their absence - deterministic.

Not all parameters of real systems can be characterized by well-defined values. Therefore, the MM of such systems, strictly speaking, should be classified as stochastic, since the output parameters of the system will be random variables. The values ​​of external parameters can also be random .

To analyze stochastic MMs, it is necessary to use the conclusions of probability theory, random processes and mathematical statistics. However, the main difficulty in their application is usually associated with the fact that the probabilistic characteristics of random variables (mathematical expectations, variances, distribution laws) are often unknown or known with low accuracy, i.e. MM does not satisfy the productivity requirement . In such cases, it is more effective to use a MM, which is rougher than the stochastic one, but also more resistant to the unreliability of the initial data.

An essential feature of the classification of MMs is their ability to describe changes in system parameters over time. If the MM reflects the influence of the inertial properties of the system, then it is usually called dynamic. In contrast, a MM that does not take into account changes in system parameters over time is called static.

Stationary They describe systems in which so-called steady-state processes occur , those. processes in which the output parameters we are interested in are constant over time. Periodic processes are also considered steady-state. , in which some output parameters remain unchanged, while others undergo fluctuations.

If the output parameters of the system change slowly and at the considered fixed point in time these changes can be neglected, then MM is considered non-stationary.

An important property of the MM from the point of view of subsequent analysis is its linearity, in the sense of connecting the parameters of the system by linear relationships. This means that when any external (or internal) parameter of the system changes, the linear MM predicts a linear change in the output parameter that depends on it, and when two or more parameters change, the addition of their influences, i.e. such a MM has the property superpositions. If a MM does not have the property of superposition, then it is called nonlinear .

A large number of mathematical methods have been developed for the quantitative analysis of linear MMs, while the capabilities of analyzing nonlinear MMs are mainly associated with methods of computational mathematics. In order to use analytical methods to study a nonlinear MM system, it is usually linearized, i.e. nonlinear relationships between parameters are replaced by approximate linear ones and the so-called linearized MMsystems. Since linearization is associated with the introduction of additional errors, the results of the analysis of the linearized model should be treated with some caution. The fact is that linearization of the MM can lead to a loss of its adequacy. Taking into account nonlinear effects in MM is especially important, for example, when describing changes in forms of motion or equilibrium positions, when small changes in input parameters can cause qualitative changes in the state of the system.

Each parameter of the system can be of two types - continuously changing in a certain range of its values ​​or taking only some discrete values. An intermediate situation is also possible, when in one area a parameter takes all possible values, and in another - only discrete ones. In this regard, they highlight continuous discrete And mixed mathematical models . In the process of analysis, MMs of these types can be converted from one to another, but during such conversion the fulfillment of the requirement should be monitored adequacy MM of the system under consideration.

Forms of representation of mathematical models. When mathematically modeling a complex system, it is usually not possible to describe its behavior with one MM, and even if such a MM were constructed, it would turn out to be too complex for quantitative analysis. Therefore, such systems are usually applied decomposition principle. It consists of conditionally dividing the system into subsystems that allow their independent study with subsequent consideration of their mutual influence on each other. In turn, the decomposition principle can be applied to each selected subsystem down to the level of fairly simple elements. In this case there arises hierarchy MM interconnected subsystems. Hierarchical levels are also distinguished for individual types of MM. For example, among structural MM systems, topological MMs are classified at a higher level of the hierarchy , and to a lower level, characterized by greater detail, - geometric MM . Among the functional levels, hierarchical levels reflect the degree of detail in the description of the processes occurring in the system and its elements. From this point of view, three main levels are usually distinguished: micro - macro - and meta-level.

Micro-level mathematical models describe processes in systems with distributed parameters, and macro-level mathematical models- in systems with lumped parameters. In the first of them, phase variables can depend on both time and spatial coordinates, and in the second, only on time.

If in a macro-level MM the number of phase variables is of the order of 10 4 -10 5 , then a quantitative analysis of such a MM becomes cumbersome and requires significant computational resources. In addition, with such a large number of phase variables, it is difficult to identify the essential characteristics of the system and the features of its behavior. In this case, by combining and enlarging the elements of a complex system, they strive to reduce the number of phase variables by excluding the internal parameters of the elements from consideration, limiting themselves to only describing the mutual connections between the enlarged elements. This approach is typical for MM meta-level.

The most common form of presentation dynamic(evolutionary) Micro-level MM is the formulation of a boundary value problem for differential equations of mathematical physics. This formulation includes partial differential equations and boundary conditions. In turn, the boundary conditions contain initial and boundary conditions. The initial conditions include the distributions of the desired phase variables at some point in time. The boundaries of the spatial region, the configuration of which corresponds to the element under consideration or the system as a whole, are boundary conditions. When representing MM, it is advisable to use dimensionless variables and equation coefficients.

Micro-level MM is called one-dimensional, two-dimensional or three-dimensional , if the required phase variables depend on one, two or three spatial coordinates, respectively. The last two types of MM are combined into multidimensional mathematical models of the micro level .

Modeling, general concepts

The task of modeling is the study of complex objects or processes using their physical or mathematical models. The purpose of modeling is to find an optimal (best by any criteria) technical solution. Modeling types:

Ø physical;

Ø mathematical;

Ø graphic (geometric).

When modeling, the most important properties of the system being studied are replaced by strict, but simplified in relation to the original natural phenomenon, scientific formulations - models. The model provides the ability to accurately describe and predict the behavior of the system, but only in a strictly limited area of ​​application - as long as those initial simplifications on the basis of which the model was built are valid.

For example, when simulating the flight of a satellite around the Earth, its walls can be considered absolutely solid, and when simulating the collision of the same satellite with a micrometeorite, even superhard iron can be described with very high accuracy as an ideal incompressible fluid. This is a paradoxical feature of modeling - its accuracy, brought to life by fundamentally inaccurate, essentially approximate models, suitable only in a certain area of ​​phenomena, models of the real system.

The functioning processes and structure of the system can be described through mathematical modeling. Mathematical modeling is the process of creating a mathematical model and acting on it in order to obtain information about a real system. A mathematical model is a set of mathematical objects and connections between them, which adequately reflects the most important properties of the system. Mathematical objects – numbers, variables, matrices, etc. Connections between mathematical objects - equations, inequalities, etc. Any scientific and technical calculations are specialized types of mathematical modeling.

A system is a set of elements naturally connected with each other, forming a single integrity, indicating the connections between them and the purpose of operation. The properties of a system differ from the sum of the properties of its elements. Examples: Machine ¹ å(parts + components); Human ¹ å(brain + liver + spine).

Classification of mathematical models

Based on the method of analysis, mathematical models are divided into analytical, algorithmic and simulation.

Analytical models can be:

1) qualitative, when the nature of the dependence of output parameters on input parameters, the very existence of a solution, etc. are determined. For example, will the cutting force increase or decrease with increasing speed, is it possible to move at a speed greater than the speed of light, etc. Building such a model is a necessary step when studying a complex system.

2) counting (analytical) models represent explicit mathematical dependencies between the input, internal and output characteristics of the system. Such models are always preferable, since they are most effective in analyzing the laws of system functioning, optimization, etc. Unfortunately, it is not always possible to obtain them and only with a significant simplification of the system under study. In addition to computational (analytical) models built on the basis of an understanding of the processes occurring in the system, these can also be models built on the basis of an analysis of the results of experiments with a “black box”. An example is the dependence of cutting force on speed, feed and depth of cut.

3) numerical, when they obtain numerical values ​​of output parameters for given input values. An example is finite element calculations. Numerical models are universal, but they give only partial results, from which it is difficult to draw general conclusions.

The algorithmic model is presented in the form of a computational algorithm. Unlike analytical models, the progress of the calculation depends on intermediate results.

Simulation modeling is based on a direct description of the modeled object. When constructing a simulation model, the laws of functioning of each element separately and the connections between them are described. Unlike the analytical one, it is characterized by structural similarity between the object and the model. Simulation modeling is most often used in the study of complex random processes. For example, blanks whose sizes have a random scatter are supplied to the input of an automatic line (AL) model. Moreover, the processing model on each AL machine is sensitive to the actual dimensions of the workpiece. After virtual “processing” of hundreds of thousands of workpieces, it is possible to find the set of circumstances in which the AL will stop and avoid it during design.

Based on the nature of functioning and the type of system parameters, mathematical models are also divided into

continuous and discrete;

static and dynamic;

deterministic and stochastic (probabilistic).

In continuous systems the parameters change gradually, in discrete systems they change abruptly and impulsively. For example, in the model of a turning cutter, wear constantly increases, and failure (chipping of the plate) occurs instantly - discretely.

In static models, all parameters included in the model have constant values ​​and the calculated parameters at the output of the system change simultaneously with the change in the parameters at the input. Such models describe systems with rapidly decaying transient processes.

Dynamic models take into account the inertia of the system. As a result, the change in the output parameter lags behind the change in the input parameter. Such models more accurately describe the real system, but are more difficult to implement.

The output of deterministic systems is uniquely determined by their input and current state. Possible random changes in system parameters or input parameters are neglected. In stochastic systems, on the contrary, the probabilistic nature of changes in system parameters is taken into account, taking random values ​​in accordance with some distribution law.

The main classification features and types of MM used in CAD are given in Table 1.

Table 1.

Classification sign	Mathematical models
The nature of the displayed object properties	Structural; functional
Belonging to a hierarchical level	Micro level; macro level; meta-level
The level of detail of the description within one level	Full; macromodels
Method for representing object properties	Analytical, algorithmic, simulation
Method for obtaining the model	Theoretical, empirical

By the nature of the displayed properties of the object MM are divided into structural And functional.

Structural MM are intended to display the structural properties of an object. There are structural MMs topological And geometric.

IN topological MM displays the composition and relationships of the elements of the object. Topological models can take the form of graphs, tables (matrices), lists, etc.

IN geometric MM displays the geometric properties of objects; in addition to information about the relative position of elements, they contain information about the shape of parts. Geometric MMs can be expressed by a set of equations of lines and surfaces; algebrological relations describing the areas that make up the body of an object; graphs and lists displaying structures from standard structural elements, etc.

Functional MM are intended to display physical or information processes occurring in an object during its operation or manufacture. Functional MMs are systems of equations connecting phase variables, internal, external and output parameters, i.e. algorithm for calculating the vector of output parameters Y for given element parameter vectors X and external parameters Q.

The number of hierarchical levels in modeling is determined by the complexity of the objects being designed and the capabilities of the design tools. However, for most subject areas, the existing hierarchical levels can be classified into one of three general levels, referred to below as micro-, macro- And meta-levels.

Depending on the place in the hierarchy of descriptions mathematical models are divided into MM related to micro-, macro- And meta-levels.

Feature MM at the micro level is a reflection of physical processes occurring in continuous space and time. Typical MMs at the micro level are partial differential equations (PDEs).

At the macro level they use an enlarged discretization of space according to a functional criterion, which leads to the representation of MM at this level in the form of systems of ordinary differential equations (ODE). ODE systems are universal models at the macro level, suitable for analyzing both dynamic and steady states of objects. Models for steady-state modes can also be represented in the form of systems of algebraic equations. The order of the system of equations depends on the number of selected elements of the object. If the order of the system approaches 10 3, then operating with the model becomes difficult and therefore it is necessary to move on to representations in meta level.

At the meta level Quite complex sets of parts are taken as elements. Meta level characterized by a wide variety of types of MMs used. For many objects, MMs at the meta level are still represented by ODE systems. However, since the models do not describe phase variables internal to elements, but only phase variables related to the mutual connections of elements appear, enlarging elements at the meta level means obtaining MM of an acceptable dimension for significantly more complex objects than at the macro level.

In a number of subject areas, it is possible to use specific features of the functioning of objects to simplify MM. An example is electronic digital automation devices, in which it is possible to use a discrete representation of phase variables such as voltages and currents. As a result, the MM becomes a system of logical equations that describe signal conversion processes. Such logical models are significantly more economical than electrical models that describe changes in voltages and currents as continuous functions of time. An important class of MM on meta level make up queuing models, used to describe the processes of functioning of information and computing systems, production areas, lines and workshops.

Structural models are also divided into models of different hierarchical levels. At the same time, the use of geometric models predominates at lower hierarchical levels, while topological models are used at higher hierarchical levels.

According to the level of detail of the description within each hierarchical level allocate full MM and macromodels.

Full MM is a model in which phase variables appear that characterize the states of all existing inter-element connections (i.e., the states of all elements of the designed object), describing not only the processes at the external terminals of the modeled object, but also the internal processes of the object.

Macromodel- MM, which displays the states of a significantly smaller number of interelement connections, which corresponds to the description of the object with an enlarged selection of elements.

Note. The concepts of “full MM” and “macromodel” are relative and are usually used to distinguish between two models that display different degrees of detail in describing the properties of an object.

By way of representing object properties functional MMs are divided into analytical And algorithmic.

Analytical MMs are explicit expressions of output parameters as functions of input and internal parameters. Such MMs are characterized by high efficiency, but obtaining an explicit expression is possible only in certain special cases, as a rule, when making significant assumptions and restrictions that reduce the accuracy and narrow the range of adequacy of the model.

Algorithmic MMs express connections between output parameters and internal and external parameters in the form of an algorithm.

Imitation MM is an algorithmic model that reflects the behavior of the object under study over time when external influences on the object are specified. Examples of simulation MMs include models of dynamic objects in the form of ODE systems and models of queuing systems specified in algorithmic form.

Usually in simulation models phase variables appear. Thus, at the macro level, simulation models are systems of algebraic-differential equations:

Where V- vector of phase variables; t- time; V o- vector of initial conditions. Examples of phase variables include currents and voltages in electrical systems, forces and speeds in mechanical systems, pressures and flow rates in hydraulic systems.

Output parameters of systems can be of two types. Firstly, these are functional parameters, i.e. dependency functionals V( t) in case of using (1). Examples of such parameters: signal amplitudes, time delays, dissipation powers, etc. Secondly, these are parameters that characterize the ability of the designed object to operate under certain external conditions. These output parameters are the boundary values ​​of the ranges of external variables in which the functionality of the object is maintained.

When designing technical objects, two main groups of procedures can be distinguished: analysis and synthesis. Synthesis is characterized by the use of structural models, and analysis is characterized by the use of functional models. The mathematical support for analysis includes mathematical models, numerical methods, and algorithms for performing design procedures. MO components are determined by a basic mathematical apparatus specific to each of the hierarchical design levels.

In CAD, analysis is performed by mathematical modeling.

Math modeling- the process of creating a model and operating it in order to obtain information about a real object.

Modeling of most technical objects can be performed at micro-, macro- and meta-levels, differing in the degree of detail in the consideration of processes in the object.

micro level, called distributed, is a system of partial differential equations (PDDE), describing processes in a continuous medium with given boundary conditions. The independent variables are spatial coordinates and time. To models on micro level Many comparisons of mathematical physics apply. The objects of research are fields of physical quantities, which is required when analyzing the strength of building structures or engineering parts, studying processes in liquid media, modeling concentrations and flows of particles in electronic devices, etc. Using these equations, fields of mechanical stresses and deformations, and electrical potentials are calculated , pressures, temperatures, etc. The possibilities of using MM in the form of PDEs are limited to individual parts; attempts to analyze processes in multicomponent environments, assembly units, and electronic circuits with their help cannot be successful due to the excessive increase in the cost of computer time and memory.

The system of differential equations, as a rule, is known (Lame equations for the mechanics of elastic media; Navier-Stokes equations for hydraulics; heat equations for thermodynamics, etc.), but its exact solution can only be obtained for special cases, so the first problem that arises when modeling, consists in constructing an approximate discrete model. For this purpose, the methods of finite differences and integral boundary equations are used, one of the variants of the latter is the boundary element method.

The number of jointly studied different environments (number of parts, layers of material, phases of the state of aggregation) in practically used micro-level models cannot be large due to computational difficulties. The only way to dramatically reduce computational costs in multi-component environments is to take a different modeling approach based on certain assumptions.

The assumption expressed by the discretization of space allows us to move on to models macro level, called Withfocused. Mathematical model of a technical object on macro level is a system of algebraic and ordinary differential equations (ODE) with given initial conditions.

In these equations the independent variable is time t, and the vector of dependent variables V constitute phase variables characterizing the state of the enlarged elements of the discretized space. Such variables include forces and speeds of mechanical systems, voltages and currents of electrical systems, pressures and flow rates of hydraulic and pneumatic systems, etc.

MM is based on component equations of individual elements and topological equations, the form of which is determined by the connections between elements. A prerequisite for the creation of a unified mathematical and software analysis at the macro level are analogies of component and topological equations of physically homogeneous subsystems that make up a technical object. Formal methods are used to obtain topological equations.

The main methods for obtaining MM objects at the macro level are:

Generalized method

Table method

Nodal method

Method of state variables.

The methods differ from each other in the type and dimension of the resulting system of equations, the method of discretizing the component equations of the reactive branches, and the permissible types of dependent branches. Simplification of the description of individual components (parts) makes it possible to study process models in devices, devices, mechanical units, the number of components in which can reach several thousand. For complex technical objects, the MM dimension becomes excessively high, and for modeling it is necessary to move to the meta level.

On meta level mainly model two categories of technical objects: objects that are the subject of research in the theory of automatic control, and objects that are the subject of queuing theory. For the first category of objects, it is possible to use macro-level mathematical apparatus; for the second category of objects, event modeling methods are used.

When the number of components in the system under study exceeds a certain threshold, the complexity of the system model at the macro level again becomes excessive. Having accepted the appropriate assumptions, we move on to functional-logical a level where the apparatus of transfer functions is used to study analog (continuous) processes or the apparatus of mathematical logic and finite state machines, if the object of study is a discrete process.

To study even more complex objects (manufacturing enterprises and their associations, computer systems and networks, social systems, etc.), the apparatus of queuing theory is used; it is also possible to use some other approaches, for example Petri nets. These models belong to systemic modeling level.

The classification of mathematical models can also be approached from different points of view, basing the classification on different principles (see Table 20.1).

by branch of science : mathematical models in physics, biology, sociology, etc. Such a classification is natural for a specialist in one science or subject area.

Models can be classified according to the mathematical apparatus used : models based on the use of ordinary differential equations, partial differential equations, probabilistic-statistical methods, discrete algebraic transformations, etc. Such a classification is convenient for a specialist in the field of mathematical modeling.

Depending from modeling purposes The following classification can be given:

· descriptive (descriptive) models;

· single-criteria optimization models;

· optimization multicriteria models;

· game models;

· simulation models.

For example, when modeling the movement of a comet in the solar system, the trajectory of its flight is described (predicted), the distance at which it will pass from the Earth, etc., i.e., purely descriptive goals are set. The researcher has no opportunity to influence the movement of the comet or change anything.

In other cases, you can influence processes in an attempt to achieve some goal.

For example, By changing the range of products that the enterprise produces and the volume of output of each type of product, one can find values ​​at which maximum profit is achieved, i.e. the optimal production plan is determined according to the criterion of profit maximization.

Often you have to find the optimal solution to a problem based on several criteria at once, and the goals can be very contradictory.

For example, knowing the prices of food and human needs for food, determine the diet of large groups of people (in the army, summer camp, etc.) that is the cheapest and most nutritious. It is obvious that these goals may contradict each other and it is necessary to find a compromise solution that satisfies all the criteria to a certain extent.

Game models can be related not only to children's games (including computer games), but also to very serious things.

For example, Before a battle, in the presence of incomplete information about the opposing army, the commander must develop a plan: in what order to introduce certain units into battle, etc., taking into account the possible reaction of the enemy.

Finally, it happens that the model largely imitates the real process, i.e. imitates him.

For example, By modeling the change (dynamics) in the number of microorganisms in a colony, you can consider many individual objects and monitor the fate of each of them, setting certain conditions for its survival, reproduction, etc. In this case, an explicit mathematical description of the process may not be used, being replaced by certain conditions (for example, after a given period of time, the microorganism is divided into two parts, and during the other period it dies).

Currently, modeling is widely used in the management of various systems, where the main processes are decision-making based on the information received. Modeling is used in research, design, and implementation of computer systems (CS) and automated control systems (ACS).

The choice of mathematical model depends on the stage of system development. At the stages of examining a control object (for example, an industrial enterprise) and developing technical specifications for the design of an aircraft, an automated control system, descriptive models are built and the goal is to most fully present in a compact form the information about the object necessary for the system developer.

At the stage of development of the technical design of the aircraft, the automated control system, modeling serves to solve the design problem, i.e. selection of the optimal option according to a certain criterion or set of criteria under given restrictions from a set of acceptable ones (construction of single-criteria and multi-criteria optimization models).

At the stage of implementation and operation of aircraft and automated control systems, simulation models are built to reproduce possible situations in order to make informed and promising decisions on the management of the facility. Game and simulation models are also widely used in teaching and training personnel.

Depending on the nature of the processes being studied , occurring in a system (object), all types of models can be divided into deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous.

Deterministic model displays deterministic processes, i.e. processes in which the absence of any random influences is assumed. In deterministic models, input parameters can be measured unambiguously and with any degree of accuracy, i.e. are deterministic quantities. Accordingly, the process of evolution of such a system is determined.

For example, deterministic models are used in physics (a model of the movement of a car during uniformly accelerated motion: by setting the initial speed and acceleration, you can accurately calculate the path traveled by the car from the moment it started moving under ideal conditions); deterministic models are also used to describe the movement of celestial bodies in astronomy.

Stochastic (probability-theoretic) models are used to display probabilistic processes and events. In this case, a number of realizations of a random process are analyzed and the average characteristics are estimated. In stochastic models, the values ​​of input parameters (variables) are known only with a certain degree of probability, i.e. these parameters are stochastic; Accordingly, the process of evolution of the system will be random.

For example, a model that describes changes in air temperature throughout the year. It is impossible to accurately predict the air temperature for a future period; only the range of temperature changes and the probability that the true air temperature will fall within this range are specified.

Stochastic models are used to study a system whose state depends not only on controlled but also on uncontrolled influences, or where there is a source of randomness in it. Stochastic systems include all systems that include humans, for example, factories, airports, computer systems and networks, shops, consumer services, etc.

Static models serve to describe the behavior of an object at any point in time, and dynamic models reflect the behavior of an object over time.

For example, a probabilistic-statistical model describing the relationship between annual performance indicators (profit, production volume, wages fund, etc.) of Novosibirsk trading enterprises over the past year - static. Annual indicators for one year, for example, for 100 trade enterprises, are used as initial data for modeling.

If the same problem is being solved, but indicators are studied over several years, then dynamic models must be used to describe the relationships. In the mathematical description of a dynamic model, the variable time is always present; in the mathematical description of a static model, time is either not introduced or is fixed at a certain level.

Discrete Models serve to describe processes that are assumed to be discrete, respectively continuous models allow you to reflect continuous processes in systems, and discrete-continuous simulation used for cases where they want to highlight the presence of both discrete and continuous processes.

For example, The operation of a differentiating filter is modeled: every time step, an input signal X(t) is supplied at equal intervals; at the output, the value of the derivative X"(t) is taken. In this case, the input and output signals are discrete in time and, accordingly, the model is discrete.

Example continuous time model - a simulation model that describes the process of processing parts in the production area of ​​the workshop during a work shift. The model input receives requests (parts) at random time intervals, and the part processing interval is also set randomly. The output of the model is an estimate of the average processing time for a part, an estimate of the average waiting time in the queue for processing, the probability of equipment downtime, etc. The operation of the system is simulated continuously for a given period of time (work shift), i.e. At any moment, a part may arrive for processing or the processing of a part may be completed.

A mathematical model is a simplification of a real situation and is an abstract, formally described object, the study of which is possible using various mathematical methods.

Let's consider classification of mathematical models.

Mathematical models are divided into:

1. Depending on the nature of the displayed properties of the object:

· functional;

· structural.

Functional mathematical models are designed to display informational, physical, time processes occurring in operating equipment, during technological processes, etc.

Thus, functional models- display the processes of the object’s functioning. They most often take the form of a system of equations.

Structural models- can take the form of matrices, graphs, lists of vectors and express the relative position of elements in space. These models are usually used in cases where problems of structural synthesis can be formulated and solved, abstracting from the physical processes in the object. They reflect the structural properties of the designed object.

To obtain a static representation of the modeled object, a group of methods called schematic models - these are analysis methods that include a graphical representation of the system's operation. For example, flow charts, diagrams, multifunctional operation diagrams and flowcharts.

2. By methods of obtaining functional mathematical models:

· theoretical;

· formal;

· empirical.

Theoretical are obtained based on the study of physical laws. The structure of the equations and parameters of the models have a certain physical interpretation.

Formal are obtained based on the manifestation of the properties of the modeled object in the external environment, i.e. considering an object as a cybernetic “black box”.

The theoretical approach allows us to obtain more universal models that are valid for wider ranges of changes in external parameters.

Formal - more accurate at the point in the parameter space at which the measurements were made.

Empirical mathematical models are created as a result of conducting experiments (studying the external manifestations of the properties of an object by measuring its parameters at the input and output) and processing their results using the methods of mathematical statistics.

3. Depending on the linearity and nonlinearity of the equations:

· linear;

· nonlinear.

4. Depending on the set of domains and values ​​of model variables, there are:

· continuous

· discrete (domains of definition and values ​​are continuous);

· continuous-discrete (the domain of definition is continuous, and the range of values ​​is discrete). These models are sometimes called quantized;

· discrete-continuous (the domain of definition is discrete, and the range of values ​​is continuous). These models are called discrete;

· digital (domains of definition and values ​​are discrete)

5. According to the form of connections between output, internal and external parameters:

· algorithmic;

· analytical;

· numerical.

Algorithmic are called models presented in the form of algorithms that describe a sequence of uniquely interpreted operations performed to obtain the desired result.

Algorithmic mathematical models express connections between output parameters and input and internal parameters in the form of an algorithm.

Analytical mathematical models is a formalized description of an object (phenomenon, process) that represents explicit mathematical expressions of output parameters as functions of input and internal parameters.

Analytical modeling is based on an indirect description of the modeled object using a set of mathematical formulas. The analytical description language contains the following main groups of semantic elements: criterion (criteria), unknowns, data, mathematical operations, restrictions. The most significant characteristic of analytical models is that the model is not structurally similar to the object being modeled. Structural similarity here refers to the unambiguous correspondence of the elements and connections of the model to the elements and connections of the modeled object. Analytical models include models built on the basis of mathematical programming, correlation and regression analysis. An analytical model is always a construct that can be analyzed and solved mathematically. So, if a mathematical programming apparatus is used, then the model consists basically of an objective function and a system of restrictions on variables. The objective function, as a rule, expresses the characteristic of the object (system) that needs to be calculated or optimized. In particular, this may be the performance of the technological system. Variables express the technical characteristics of an object (system), including variable ones, restrictions – their permissible limit values.

Analytical models are an effective tool for solving problems of optimizing processes occurring in technological systems, as well as optimizing and calculating the characteristics of technological systems themselves.

An important point is the dimension of a specific analytical model. Often for real technological systems (automated lines, flexible production systems), the dimension of their analytical models is so large that obtaining an optimal solution turns out to be very difficult from a computational point of view. To increase computational efficiency in this case, various techniques are used. One of them is associated with dividing a large-dimensional problem into subproblems of smaller dimension so that autonomous solutions of subproblems in a certain sequence provide a solution to the main problem. In this case, problems arise in organizing the interaction of subtasks, which are not always simple. Another technique involves reducing the accuracy of calculations, thereby reducing the time required to solve the problem.

The analytical model can be studied by the following methods:

· analytical, when they strive to obtain in general terms the dependencies for the desired characteristics;

· numerical, when they strive to obtain numerical results with specific initial data;

· qualitative, when, having solutions in explicit form, one can find some properties of the solution (assess the stability of the solution).

However, analytical modeling gives good results in the case of fairly simple systems. In the case of complex systems, either a significant simplification of the initial model is required in order to study at least the general properties of the system. This allows you to obtain approximate results, and to determine more accurate estimates, use other methods, for example, simulation modeling.

Numerical model characterized by a dependence of this type, which allows only solutions obtained by numerical methods for specific initial conditions and quantitative parameters of the models.

6. Depending on whether the model equations take into account the inertia of processes in the object or do not take into account:

· dynamic or inertial models(written in the form of differential or integro-differential equations or systems of equations) ;

· static or non-inertial models(written in the form of algebraic equations or systems of algebraic equations).

7. Depending on the presence or absence of uncertainties and the type of uncertainties, models are:

· deterministic e (no uncertainties);

· stochastic (there are uncertainties in the form of random variables or processes described by statistical methods in the form of laws or distribution functionals, as well as numerical characteristics);

· fuzzy (to describe uncertainties, the apparatus of fuzzy set theory is used);

· combined (both types of uncertainties are present).

In general, the type of mathematical model depends not only on the nature of the real object, but also on the problems for which it is created, and the required accuracy of their solution

The main types of models are presented in Figure 2.5.

Let's consider another classification of mathematical models. This classification is based on the concept of controllability of a control object. We will conditionally divide all MMs into four groups.1. Forecast models (calculation models without control). They can be divided into static And dynamic.The main purpose of these models: knowing the initial state and information about the behavior at the boundary, to give a forecast about the behavior of the system in time and space. Such models can also be stochastic. As a rule, forecasting models are described by algebraic, transcendental, differential, integral, integro-differential equations and inequalities. Examples include models of heat distribution, electric field, chemical kinetics, hydrodynamics, aerodynamics, etc. 2.Optimization models. These models can also be divided into static And dynamic. Static models are used at the design level of various technological systems. Dynamic - both at the design level and, mainly, for optimal control of various processes - technological, economic, etc. There are two directions in optimization problems. The first includes deterministic tasks. All input information in them is completely determinable. The second direction relates to stochastic processes. In these problems, some parameters are random or contain an element of uncertainty. Many optimization problems for automatic devices, for example, contain parameters in the form of random noise with some probabilistic characteristics. Methods for finding the extremum of a function of many variables with various restrictions are often called mathematical programming methods. Mathematical programming problems are one of the important optimization problems. The following main sections are distinguished in mathematical programming.· Linear programming . The objective function is linear, and the set on which the extremum of the objective function is sought is specified by a system of linear equalities and inequalities.· Nonlinear programming . Nonlinear objective function and nonlinear constraints.· Convex programming . The objective function is convex and the set on which the extremal problem is solved.· Quadratic programming . The objective function is quadratic, and the constraints are linear.· Multiextremal problems. Problems in which the objective function has several local extrema. Such tasks seem to be very problematic.· Integer programming. In such problems, integer conditions are imposed on variables.

Rice. 4.8. Classification of mathematical models

As a rule, methods of classical analysis for finding the extremum of a function of several variables are not applicable to mathematical programming problems. Models of optimal control theory are among the most important in optimization models. The mathematical theory of optimal control is one of the theories that has important practical applications, mainly for optimal control of processes. There are three types of mathematical models of optimal control theory.· Discrete optimal control models. Traditionally, such models are called dynamic programming models, since the main method for solving such problems is the Bellman dynamic programming method.· Continuous models of optimal control of systems with lumped parameters (described by ordinary derivative equations).· Continuous models of optimal control of systems with distributed parameters (described by partial differential equations).3. Cybernetic models (game). Cybernetic models are used to analyze conflict situations. It is assumed that the dynamic process is determined by several subjects who have several control parameters at their disposal. A whole group of subjects with their own interests is associated with the cybernetic system. 4. Simulation modeling . The types of models described above do not cover a large number of different situations, such as those that can be fully formalized. To study such processes, it is necessary to include a functioning “biological” element – ​​a person – in the mathematical model. In such situations, simulation modeling is used, as well as methods of examination and information procedures.