Time series models characterizing the dependence of the resulting variable on time include:

a) a model of the dependence of the resulting variable on the trend component or a trend model;

b) result dependence model. variable from the seasonal component or seasonality model;

c) a model of the dependence of the resulting variable on the trend and seasonal components or a model of trend and seasonality.

If economic statements reflect the dynamic (time-dependent) relationship of the variables included in the model, then the values ​​of such variables are dated and called dynamic or time series. If economic statements reflect a static (relating to one period of time) relationship of all variables included in the model, then the values ​​of such variables are usually called spatial data. And there is no need to date them. Lagged are exogenous or endogenous variables of an economic model, dated to previous points in time and located in the equation with current variables. Models that include lagged variables belong to the class of dynamic models. Predestined called lagged and current exogenous variables, as well as lagged endogenous variables

23. Trend and spatiotemporal EM in economic planning

Statistical observations in socio-economic studies, they are usually carried out regularly at equal intervals of time and are presented in the form of time series xt, where t = 1, 2, ..., p. Trend regression models are used as a tool for statistical forecasting of time series, the parameters of which are estimated based on the available statistical base, and then the main tendencies (trends) are extrapolated to a given time interval.

Statistical forecasting methodology involves building and testing many models for each time series, comparing them based on statistical criteria, and selecting the best ones for forecasting.

When modeling seasonal phenomena in statistical studies, two types of fluctuations are distinguished: multiplicative and additive. In the multiplicative case, the range of seasonal fluctuations changes over time in proportion to the trend level and is reflected in the statistical model by a multiplier. With additive seasonality, it is assumed that the amplitude of seasonal deviations is constant and does not depend on the trend level, and the fluctuations themselves are represented in the model by a term.

The basis of most forecasting methods is extrapolation, associated with the dissemination of patterns, connections and relationships operating in the period under study beyond its borders, or - in a broader sense of the word - it is obtaining ideas about the future based on information related to the past and present.

The most famous and widely used are trend and adaptive forecasting methods. Among the latter, one can highlight methods such as autoregression, moving average (Box-Jenkins and adaptive filtering), exponential smoothing methods (Holt, Brown and exponential average), etc.

To assess the quality of the forecast model under study, several statistical criteria are used.

When presenting a set of observational results in the form of time series, the assumption is actually used that the observed values ​​belong to a certain distribution, the parameters of which and their changes can be estimated. Based on these parameters (usually the mean and variance, although sometimes more is used) full description) you can build one of the models for the probabilistic representation of the process. Another probabilistic representation is a model in the form of a frequency distribution with parameters pj for the relative frequency of observations falling into j-interval. Moreover, if no change in the distribution is expected during the accepted lead time, then the decision is made on the basis of the existing empirical frequency distribution.

When making forecasts, it is necessary to keep in mind that all factors influencing the behavior of the system in the base (studied) and forecast periods must be constant or change according to a known law. The first case is implemented in single-factor forecasting, the second - in multi-factor forecasting.

Multifactorial dynamic models should take into account spatial and temporal changes in factors (arguments), as well as (if necessary) the lag of the influence of these factors on the dependent variable (function). Multifactor forecasting allows you to take into account the development of interrelated processes and phenomena. Its basis is systematic approach to the study of the phenomenon under study, as well as the process of understanding the phenomenon, both in the past and in the future.

In multifactor forecasting, one of the main problems is the problem of choosing factors that determine the behavior of the system, which cannot be solved purely statistically, but only through an in-depth study of the essence of the phenomenon. Here it is necessary to emphasize the primacy of analysis (comprehension) over purely statistical (mathematical) methods of studying the phenomenon. In traditional methods (for example, in the least squares method), observations are considered to be independent of each other (by the same argument). In reality, there is autocorrelation and its failure to take it into account leads to suboptimal statistical estimates and makes it difficult to construct confidence intervals for regression coefficients, as well as to test their significance. Autocorrelation is determined by deviations from trends. It can occur if the influence of a significant factor or several less significant factors, but directed “in one direction,” is not taken into account, or the model that establishes the connection between the factors and the function is incorrectly selected. To identify the presence of autocorrelation, the Durbin-Watson test is used. To eliminate or reduce autocorrelation, a transition to a random component (detrending) or introducing time into the multiple regression equation as an argument is used.

In multifactor models, the problem of multicollinearity also arises - the presence of a strong correlation between factors, which can exist regardless of any dependence between the function and the factors. By identifying which factors are multicollinear, it is possible to determine the nature of the interdependence between the multicollinear elements of a set of independent variables.

In multivariate analysis, it is necessary, along with estimating the parameters of the smoothing (studied) function, to construct a forecast for each factor (based on some other functions or models). Naturally, the values ​​of factors obtained in the experiment in the base period do not coincide with similar values ​​found using predictive models for factors. This difference must be explained either by random deviations, the magnitude of which is revealed by the indicated differences and should be taken into account immediately when estimating the parameters of the smoothing function, or this difference is not random and no prediction can be made. That is, in a multifactor forecasting problem, the initial values ​​of the factors, as well as the values ​​of the smoothing function, must be taken with the corresponding errors, the distribution law of which must be determined by appropriate analysis preceding the forecasting procedure.

24. Essence and content of EM: structural and expanded

Econometric models are systems of interconnected equations, many of whose parameters are determined by methods of statistical data processing. To date, many hundreds of econometric systems have been developed and used abroad for analytical and forecasting purposes. Macroeconometric models, as a rule, are first presented in a natural, meaningful form, and then in a reduced, structural form. The natural form of econometric equations allows us to qualify their content and assess their economic meaning.

To build forecasts of endogenous variables, it is necessary to express the current endogenous variables of the model as explicit functions of predefined variables. The last specification, obtained by including random disturbances, is obtained as a result of the mathematical formalization of economic laws. This form of specification is called structural. In general, in a structural specification, endogenous variables are not expressed explicitly through predetermined ones.

In the equilibrium market model, only the supply variable is expressed explicitly through a predefined variable, so to represent endogenous variables through predefined ones, it is necessary to perform some transformations of the structural form. Let us solve the system of equations for the latter specification with respect to endogenous variables.

Thus, the endogenous variables of the model are expressed explicitly through predefined variables. This form of specification is called given. In a particular case, the structural and reduced forms of the model may coincide. With the correct specification of the model, the transition from the structural to the reduced form is always possible, but the reverse transition is not always possible.

A system of joint, simultaneous equations (or structural form of a model) typically contains endogenous and exogenous variables. Endogenous variables are denoted in the system of simultaneous equations presented earlier as y. These are dependent variables, the number of which is equal to the number of equations in the system. Exogenous variables are usually denoted as x. These are predetermined variables that influence but are not dependent on endogenous variables.

The simplest structural form of the model is:

where y are endogenous variables; x – exogenous variables.

The classification of variables into endogenous and exogenous depends on the theoretical concept of the adopted model. Economic variables can act as endogenous variables in some models and as exogenous variables in others. Non-economic variables (for example, climatic conditions) enter the system as exogenous variables. The values ​​of endogenous variables for the previous period of time (lag variables) can be considered as exogenous variables.

Thus, current year consumption (y t) may depend not only on the series economic factors, but also on the level of consumption in the previous year (y t-1)

The structural form of the model allows you to see the impact of changes in any exogenous variable on the values ​​of the endogenous variable. It is advisable to select as exogenous variables those variables that can be the object of regulation. By changing and managing them, it is possible to have target values ​​of endogenous variables in advance.

The structural form of the model on the right side contains the coefficients b i and a j for endogenous and exogenous variables (b i is the coefficient for the endogenous variable, a j is the coefficient for the exogenous variable), which are called the structural coefficients of the model. All variables in the model are expressed in deviations from the level, i.e. by x we ​​mean x- (and by y we mean y- (. Therefore, there is no free term in each equation of the system.

Using OLS to estimate the structural coefficients of the model gives, as is commonly believed in theory, biased structural coefficients of the model, structural coefficients of the model, the structural form of the model is transformed into the reduced form of the model.

The reduced form of the model is a system of linear functions of endogenous variables from exogenous ones:

In its appearance, the reduced form of the model is no different from the system of independent equations, the parameters of which are estimated by traditional least squares methods. Using OLS, one can estimate δ and then estimate the values ​​of endogenous variables through exogenous ones.

Deployed EV(her blocks)

Spatial integration of individual elements of a technical object is a widespread design task in any branch of technology: radio electronics, mechanical engineering, energy, etc. A significant part of spatial modeling is the visualization of individual elements and the technical object as a whole. Of great interest are the issues of constructing a database of graphical three-dimensional models of elements, algorithms and software implementation of graphical applications to solve this problem.

The construction of models of elements is universal in nature and can be considered as an invariant part of many systems of spatial modeling and computer-aided design of technical objects.

Regardless of the capabilities of the graphical environment used, according to the nature of the formation of graphical models, three groups of elements can be distinguished:

1.Unique elements, the configuration and dimensions of which are not repeated in other similar parts.

2. Unified elements, including a certain set of configuration fragments characteristic of parts of a given class. As a rule, there is a limited range of standard sizes of a unified element.

3. Composite elements, including both unique and unified elements in an arbitrary set. The graphical tools used may allow some nesting of constituent elements.

Spatial modeling of unique elements is not very difficult. Direct generation of the model configuration is performed interactively, after which the software implementation is designed based on the model generation protocol or a text description of the resulting element.

2. Alternately selecting fragments of the spatial configuration and determining their sizes;

3. Linking the graphical model of an element to other elements, technical objects or systems;

4.Input of additional information about the modeled element

This approach to creating models of unified elements ensures reliable software implementation.

The composite element model consists of a set of models of both unique and unified elements. Procedurally, a model of a composite element is built similarly to a model of a unified element, in which ready-made models of elements act as graphic fragments. The main features are the method of mutual binding of included models and the mechanics of combining individual fragments into a composite element. The latter is determined mainly by the capabilities of graphical tools.

Integration of a graphical environment and a database management system (DBMS) of technical information ensures the openness of the modeling system for solving other design problems: preliminary design calculations, selection of element base, preparation of design documentation (text and graphic), etc. The structure of the database (DB) is defined as the requirements of graphical models and the information needs of related tasks. It is possible to use any DBMS that is interfaced with a graphical environment as tools. The most general character is the construction of models of unified elements. At the first stage, as a result of systematization of the nomenclature of elements that are of the same type in purpose and composition of graphic fragments, a hypothetical one is formed or an existing sample of a modeled element is selected, which has a full set of modeled parts of the object.

Methods of interpolation from discretely located points.

The general problem of interpolation by points is formulated as follows: given a number of points (interpolation nodes), the position and values ​​of the characteristics in which are known, it is necessary to determine the values ​​of the characteristics for other points for which only the position is known. At the same time, there are methods of global and local interpolation, and among them are exact and approximating.

Global interpolation uses a single calculation function for the entire area simultaneously z = F(x,y) . In this case, changing one value (x, y) at the input affects the entire resulting DEM. In local interpolation, a calculation algorithm is repeatedly used for some samples from a common set of points, usually closely located. Then changing the choice of points affects only the results of processing a small area of ​​the territory. Global interpolation algorithms produce smooth surfaces with few sharp edges; they are used in cases where the shape of the surface, such as a trend, is presumably known. When a large portion of the total data set is included in the local interpolation process, it essentially becomes global.

Accurate interpolation methods.

Accurate Interpolation Methods reproduce data at the points (nodes) on which the interpolation is based, and the surface passes through all points with known values. neighborhood analysis, in which all values ​​of the simulated characteristics are taken equal to the values ​​at the nearest known point. As a result, Thiessen polygons are formed with a sharp change in values ​​at the boundaries. This method is used in environmental studies, when assessing impact zones, and is more suitable for nominal data.

In method B-splines construct a piecewise linear polynomial that allows you to create a series of segments that ultimately form a surface with continuous first and second derivatives. The method ensures continuity of heights, slopes, and curvature. The resulting DEM is in raster form. This local interpolation method is used mainly for smooth surfaces and is not suitable for surfaces with distinct changes - this leads to sharp fluctuations in the spline. It is widely used in programs for interpolating general-purpose surfaces and smoothing contours when drawing them.

In TIN models, the surface within each triangle is usually represented as a plane. Since for each triangle it is specified by the heights of its three vertices, then in a common mosaic surface the triangles for adjacent areas exactly adjoin the sides: the resulting surface is continuous. However, if horizontal lines are drawn on the surface, then in this case they will be rectilinear and parallel within the triangles, and at the boundaries there will be a sharp change in their direction. Therefore, for some TIN applications, a mathematical surface is constructed within each triangle, characterized by a smooth change in slope angles at the boundaries of the triangles. Trend analysis. The surface is approximated by a polynomial and the output data structure is an algebraic function that can be used to calculate values ​​at raster points or at any point on the surface. Linear equation, for example, z = a + bx + su describes an inclined flat surface, and the quadratic z = a + bx + cy + dx2 + yahoo + fy2 -a simple hill or valley. Generally speaking, any section of the surface t-th has no more order (T - 1) alternating highs and lows. For example, a cubic surface can have one maximum and one minimum in any section. Significant edge effects are possible because the polynomial model produces a convex surface.

Moving average and distance weighted average methods are most widely used, especially for modeling smoothly changing surfaces. The interpolated values ​​represent the average of the values ​​for n known points, or the average obtained from interpolated points, and in the general case are usually represented by the formula

Approximation interpolation methods.

Approximation interpolation methods are used in cases where there is some uncertainty regarding the available surface data; They are based on the consideration that many data sets show a slowly changing surface trend, overlaid with local, rapidly changing biases that lead to inaccuracies or errors in the data. In such cases, smoothing due to surface approximation makes it possible to reduce the influence of erroneous data on the nature of the resulting surface.

Methods of interpolation by area.

Interpolation by area consists of transferring data from one source set of areas (key) to another set (target) and is often used when zoning a territory. If the target habitats are a grouping of key habitats, this is easy to do. Difficulties arise if the boundaries of the target areas are not related to the original key areas.

Let's consider two options for interpolation by area: in the first of them, as a result of interpolation, the total value of the interpolated indicator (for example, population size) of the target areas is not fully preserved, in the second, it is preserved.

Let's imagine that we have population data for some regions with given boundaries, and they need to be extended to a smaller zoning grid, the boundaries of which generally do not coincide with the first.

The technique is as follows. For each source area (key area), population density is calculated by dividing the total number of residents by the area of ​​the site and assigning the resulting value to the central point (centroid). Based on this set of points, a regular grid is interpolated using one of the methods described above, and the population size is determined for each grid cell by multiplying the calculated density by the cell area. The interpolated grid is superimposed on the final map, the values ​​in each cell refer to the boundaries of the corresponding target area. The total population of each of the resulting areas is then calculated.

The disadvantages of the method include the not entirely clear choice of the central point; Point-by-point interpolation methods are inadequate, and most importantly, the total value of the interpolated indicator of key areas (in this case, the total population of census zones) is not preserved. For example, if the source zone is divided into two target zones, then the total population in them after interpolation will not necessarily be equal to the population of the source zone.

In the second version of interpolation, methods of GIS overlay technology or construction of a smooth surface based on the so-called adaptive interpolation are used.

In the first method, the key and target areas are superimposed, the share of each of the source areas in the target areas is determined, the indicator values ​​of each source area are divided proportionally to the areas of its areas in different target areas. It is believed that the density of the indicator within each area is the same, for example, if the indicator is the total population of the area, then the population density is considered a constant value for it.

The purpose of the second method is to create a smooth surface without ledges (attribute values ​​should not change sharply at the boundaries of areas) and maintain the total value of the indicator within each area. His technique is as follows. A dense raster is superimposed on the cartogram representing key areas, the total value of the indicator for each area is equally divided between the raster cells overlapping it, the values ​​are smoothed by replacing the value for each raster cell with the average for the neighborhood (over a window of 2 × 2, 3 × 3, 5 ×5) and sum the values ​​for all cells of each area. Next, the values ​​for all cells are adjusted proportionally so that the total value of the indicator for the area coincides with the original one (for example, if the sum is 10% less than the original value, the values ​​for each cell increase by 10%). The process is repeated until... changes will stop.

For the described method, homogeneity within areas is not necessary, but too strong variations of the indicator within their limits can affect the quality of interpolation.

The results can be represented on the map by contours or continuous halftones.

Application of the method requires setting some boundary conditions, since along the periphery of the original areas, raster elements may extend beyond the study area or be adjacent to areas that do not have the value of the interpolated indicator. You can, for example, set the population density to 0 (lake, etc.) or set it equal to the values ​​of the outermost cells in the study area.

When interpolating by area, very complex cases can arise, for example, when you need to create a map showing “areas of settlement” based on population data for individual cities, especially if these areas are shown as a dot at the scale of the map. The problem also occurs for small source areas when there are no boundary files and the data only indicates the location of the center point. Here, different approaches are possible: replacing the points to which the data is assigned with circles, the radius of which is estimated by the distances to neighboring centroids; determining the threshold population density for classifying an area as urban; distribution of the population of each city over its territory so that in the center the population density is higher, and towards the outskirts it decreases; At points with a threshold value of the indicator, lines are drawn that limit populated areas.

Often attempting to create a continuous surface using area interpolation from point-only data will produce incorrect results.

The user usually evaluates the success of the method subjectively and mainly visually. Until now, many researchers use manual interpolation or interpolation “by eye” (this method is usually not highly regarded by geographers and cartographers, but is widely used by geologists). Currently, attempts are being made to “extract” the knowledge of experts using methods for creating knowledge bases and introducing them into an expert system that performs interpolation.

Information

Features of space-time

INDICATOR RELATIONS

MULTIFACTOR DYNAMIC MODELS

Multifactor dynamic models of indicator relationships are built according to spatiotemporal samples, which represent a set of data about the values ​​of attributes of a set of objects over a number of periods (points) of time.

Spatial samples are formed by combining spatial samples over a number of years (periods), i.e. collections of objects belonging to the same periods of time. Used in case of small samples, i.e. brief background development of the facility.

Dynamic selections are formed by combining dynamic series of individual objects in the case long prehistory, i.e. large samples.

The classification of sampling methods is conditional, because depends on the purpose of the modeling, on the stability of the identified patterns, on the degree of homogeneity of objects, on the number of factors. In most cases, preference is given to the first method.

Time series with a long history are considered as series on the basis of which it is possible to build models of the relationship between indicators of various objects of sufficiently high quality.

Dynamic communication models indicators can be:

· spatial, i.e. modeling the relationships between indicators for all objects considered at a certain point (interval) in time;

· dynamic, which are built based on the totality of implementations of one object for all periods (moments) of time;

· spatial-dynamic, which are formed for all objects for all periods (moments) of time.

Dynamics models indicators are grouped into the following types:

1) one-dimensional dynamics models: characterized as models of some indicator of a given object;

2) multidimensional models of the dynamics of one object: they model several indicators of the object;

3) multidimensional models of the dynamics of a set of objects : model several indicators of a system of objects.

Accordingly, communication models are used to spatial extrapolation(for predicting the values ​​of performance indicators of new objects based on the values ​​of factor characteristics), dynamics models - for dynamic extrapolation(to predict dependent variables).

We can identify the main tasks of using spatiotemporal information.

1. In the case of a brief background: identifying spatial relationships between indicators, i.e. studying the structure of connections between objects to increase the accuracy and reliability of modeling these patterns.

2. In the case of a long history: approximation of patterns of changes in indicators in order to explain their behavior and predict possible states.

The shape of the spatial configuration of the cable-rope when towing an underwater vehicle depends on the mode of movement (speed relative to water, distribution of currents in depth), features

apparatus and characteristics of the cable rope (diameter, length, buoyancy, etc.). The peculiarity of the shape of the cable-rope when the complex moves along a given profile line is that along its length the radial angles vary within wide limits (as well as additional meridian angles), but the azimuthal angles and hydrodynamic velocity angles k at any point of the cable have small values. This assumption allows us to present the coupling equations of a flexible thread for a given case, expressed in projections of the tangent vector onto the fixed axes, as follows:

and the equations obtained from the condition of equilibrium of forces on an elementary segment of a flexible thread in a stationary mode are written in the form

Nonlinear ordinary differential equations (7.30) and (7.31) are mathematical description static spatial cable-rope configuration. Below are some results of studies performed by solving equations (7.30) and (7.31) on a digital computer.

In Fig. Figure 7.10 shows the dependence curves of tension T, depth and distance between the PA and the ship on the towing speed for a fixed cable length of 6000 m. The tension at the point of attachment to the ship (at the towing winch) decreases with increasing speed up to 4 m/s and increases with further increasing towing speed. In this case, the UAV emerges from a depth of 6000 to 1000 m, but the distance between the device and the ship increases.

Rice. 7.11 shows how the tension at the point of attachment to the vessel, the length of the cable rope and the distance between the PA and the vessel change with increasing towing speed while maintaining a constant

PA immersion depth by 6000 m. With an increase in towing speed to 2 m/s, it is necessary to increase the length of the cable rope to 13000 m. View of static configurations of a cable rope 6000 m long in the vertical plane at towing speeds (curves 1, 2, 3, respectively) illustrated in Fig. 7.12.

Rice. 7.10. Static parameters of cable-rope movement depending on towing speed.

Rice. 7.11. Static parameters of cable-rope movement at a constant immersion depth of the PA.

The peculiarity of the cable-rope movement when towing a PA is that it occurs at low lateral and vertical speeds compared to the speed of the longitudinal movement of the cable. For any of its points, the conditions are met and the speed of translational longitudinal motion almost never exceeds m/s. In addition, they strive to ensure that towing proceeds smoothly, without sudden forces in the cable. Under these conditions, separate analysis of the dynamics of the cable-rope movement in the vertical (longitudinal movement) and horizontal (lateral movement) planes is allowed. The equations of longitudinal motion are written in the form

and lateral

All coefficients are calculated at constant values ​​of the hydrodynamic speed and its tangential component and constant cable tension over time, determined by the expression

Partial differential equations (7.32) and (7.33) are solved for initial and boundary conditions at the lower and upper ends of the cable rope, the latter playing the role of control actions and consisting of the corresponding projections of the speed of the tugboat and the change in cable length as a result of the operation of the towing winch:

A dynamic object is a physical body, a technical device or a process that has inputs, points of possible application of external influences, and those that perceive these influences, and outputs, points, the values ​​of physical quantities in which characterize the state of the object. An object is capable of responding to external influences by changing its internal state and output values ​​that characterize its state. The impact on an object and its reaction generally change over time, they are observable, i.e. can be measured with appropriate instruments. The object has internal structure, consisting of interacting dynamic elements.

If you read and think about the loose definition given above, you can see that a separate dynamic object in a “pure” form, as a thing in itself, does not exist: to describe an object, the model must also contain 4 sources of influences (generators):

The environment and the mechanism for applying these influences to it

The object must have an extension in space

Function in time

The model must have measuring devices.

The impact on an object can be a certain physical quantity: force, temperature, pressure, electrical voltage and other physical quantities or a combination of several quantities, and the reaction, the response of the object to the impact, can be movement in space, for example, displacement or speed, change in temperature, current strength etc.

For linear models of dynamic objects, the principle of superposition (overlay) is valid, i.e. the response to a set of impacts is equal to the sum of the reactions to each of them, and a large-scale change in the impact corresponds to a proportional change in the response to it. One impact can be applied to several objects or several elements of an object.

The concept of a dynamic object contains and expresses the cause-and-effect relationship between the impact on it and its reaction. For example, between the force applied to a massive body and its position and movement, between the electrical voltage applied to the element and the current flowing in it.

In the general case, dynamic objects are nonlinear, including they can have discreteness, for example, quickly change the structure when the impact reaches a certain level. But usually, most of the time of operation, dynamic objects are continuous in time and with small signals they are linear. Therefore, below the main attention will be paid to linear continuous dynamic objects.

An example of continuity: a car moving along the road is an object continuously functioning in time, its position depends on time continuously. Much of the time, a car can be seen as linear object, an object operating in linear mode. And only in case of accidents, collisions, when, for example, a car is destroyed, it is required to describe it as a nonlinear object.

Linearity and continuity in time of the output value of an object is simply a convenient special, but important case, which makes it possible to quite simply consider a significant number of properties of a dynamic object.

On the other hand, if an object is characterized by processes occurring on different time scales, then in many cases it is acceptable and useful to replace the fastest processes with their discrete change in time.

This work is primarily devoted to linear models dynamic objects under deterministic influences. Smooth deterministic influences of an arbitrary type can be generated by discrete, relatively rare additive action on the minor derivatives of the influence by dosed delta functions. Such models are valid for relatively small impacts for a very wide class of real objects. For example, this is how control signals are generated in computer games simulating driving a car or airplane using a keyboard. Accidental impacts remain beyond the scope of consideration for now.

The consistency of a linear model of a dynamic object is determined, in particular, by whether its output value is sufficiently smooth, i.e. whether it and several of its lower derivatives are continuous in time. The fact is that the output quantities of real objects change quite smoothly over time. For example, an airplane cannot instantly move from one point in space to another. Moreover, like any massive body, it cannot change its speed abruptly; this would require infinite power. But the acceleration of an airplane or car can change abruptly.

The concept of a dynamic object does not at all comprehensively define a physical object. For example, describing a car as a dynamic object allows us to answer the questions of how quickly it accelerates and brakes, how smoothly it moves on uneven roads and bumps, what impacts the driver and passengers of the car will experience when driving on the road, what mountain it can climb, etc. p. But in such a model, it does not matter what color the car is, its price, etc. are not important, insofar as they do not affect the acceleration of the car. The model should reflect the main properties of the modeled object from the point of view of some criterion or set of criteria and neglect its secondary properties. Otherwise, it will be overly complex, which will complicate the analysis of the properties of interest to the researcher.

On the other hand, if the researcher is interested in the change in the color of the car over time, caused by various factors, for example, sunlight or aging, then for this case the corresponding differential equation can be compiled and solved.

Real objects, as well as their elements, which can also be considered as dynamic objects, not only perceive influences from some source, but also themselves influence this source and resist it. The output value of a control object in many cases is an input for another, subsequent dynamic object, which, in turn, can also influence the operating mode of the object. That. The connections between a dynamic object and the external world in relation to it are bidirectional.

Often, when solving many problems, the behavior of a dynamic object is considered only in time, and its spatial characteristics, in cases where they are not directly of interest to the researcher, are not considered or taken into account, with the exception of a simplified account of the signal delay, which may be due to the time of propagation of the influence in space from source to receiver.

Dynamic objects are described by differential equations (a system of differential equations). In many practically important cases this is a linear, ordinary differential equation (ODE) or system of ODEs. The variety of types of dynamic objects determines the high importance of differential equations as a universal mathematical apparatus for their description, which makes it possible to conduct theoretical studies (analysis) of these objects and, on the basis of such analysis, construct models and build systems, instruments and devices useful for people, explain the structure of the world around us, according to at least on the scale of the macrocosm (not micro- and not mega-).

A model of a dynamic object is valid if it is adequate and corresponds to a real dynamic object. This correspondence is limited to a certain spatio-temporal region and range of influences.

A model of a dynamic object is realizable if it is possible to construct a real object, the behavior of which under the influence of influences in a certain space-time domain and for a certain class and range of input influences corresponds to the behavior of the model.

The breadth of classes and the variety of structures of dynamic objects can lead to the assumption that all of them together have an innumerable set of properties. However, an attempt to embrace and understand these properties and the principles of operation of dynamic objects in all their diversity is not at all so hopeless.

The fact is that if dynamic objects are adequately described by differential equations, and this is exactly the case, then the set of properties characterizing a dynamic object of any kind is determined by the set of properties characterizing its differential equation. It can be argued that, at least for linear objects, there is a rather limited and relatively small number of such basic properties, and therefore the set of basic properties of dynamic objects is also limited. Based on these properties and combining elements that have them, it is possible to build dynamic objects with a wide variety of characteristics.

So, the basic properties of dynamic objects are derived theoretically from their differential equations and correlated with the behavior of the corresponding real objects.

A dynamic object is an object that perceives external influences that change over time and reacts to them by changing the output value. An object has an internal structure consisting of interacting dynamic elements. The hierarchy of objects is limited from below by the simplest models and is based on their properties.

The impact on an object, as well as its reaction, are physical, measurable quantities; it can also be a set of physical quantities, mathematically described by vectors.

When describing dynamic objects using differential equations, it is implicitly assumed that each element of a dynamic object receives and expends as much energy (such power) as it requires for normal operation in accordance with its purpose in response to incoming influences. The object can receive part of this energy from the input action and this is described explicitly by the differential equation; the other part can come from third-party sources and not appear in the differential equation. This approach significantly simplifies the analysis of the model without distorting the properties of the elements and the entire object. If necessary, the process of energy exchange with the external environment can be described in detail in explicit form and these will also be differential and algebraic equations.

In some special cases, the source of all energy (power) for the output signal of an object is the input action: lever, acceleration of a massive body by force, passive electrical circuit etc.

In the general case, the influence can be considered as controlling energy flows to obtain the required power of the output signal: a sinusoidal signal amplifier, just an ideal amplifier, etc.

Dynamic objects, like their elements, which can also be considered as dynamic objects, not only perceive impact from its source, but also act on this

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