Static and dynamic models. Spatial terrain models Accurate interpolation methods

Classification of types of modeling can be carried out on different grounds. Models can be distinguished by a number of characteristics: the nature of the objects being modeled, areas of application, and depth of modeling. Let's consider 2 classification options. First classification option. Based on the depth of modeling, modeling methods are divided into two groups: material (subject) and ideal modeling. Material modeling is based on a material analogy between an object and a model. It is carried out by reproducing the basic geometric, physical or functional characteristics of the object being studied. A special case of material modeling is physical modeling. A special case of physical modeling is analog modeling. It is based on the analogy of phenomena that have different physical natures, but are described by the same mathematical relationships. An example of analogue modeling is the study of mechanical vibrations (for example, an elastic beam) using an electrical system described by the same differential equations. Since experiments with an electrical system are usually simpler and cheaper, it is studied as an analogue of a mechanical system (for example, when studying the vibrations of bridges).

Ideal modeling is based on an ideal (mental) analogy. In economic research (at a high level of its implementation, and not on the subjective desires of individual managers), this is the main type of modeling. Ideal modeling, in turn, is divided into two subclasses: symbolic (formalized) and intuitive modeling. In symbolic modeling, models are diagrams, graphs, drawings, and formulas. The most important type of sign modeling is mathematical modeling, carried out by means of logical and mathematical constructions.

Intuitive modeling is found in those areas of science and practice where the cognitive process is at an initial stage or where very complex systemic relationships take place. Such studies are called thought experiments. In economics, sign or intuitive modeling is mainly used; it describes the worldview of scientists or the practical experience of workers in the field of its management. The second classification option is shown in Fig. 1.3. In accordance with the classification criterion of completeness, modeling is divided into complete, incomplete and approximate. In full modeling, the models are identical to the object in time and space. For incomplete simulations, this identity is not preserved. Approximate modeling is based on similarity, in which some aspects of a real object are not modeled at all. The theory of similarity states that absolute similarity is possible only when one object is replaced by another exactly the same. Therefore, when modeling, absolute similarity does not take place. Researchers strive to ensure that the model represents only the aspect of the system being studied well. For example, to assess the noise immunity of discrete information transmission channels, functional and information models of the system may not be developed. To achieve the goal of modeling, the event model described by the matrix of conditional probabilities ||рij|| transitions of the i-th character of the j-th alphabet. Depending on the type of media and model signature, the following types of modeling are distinguished: deterministic and stochastic, static and dynamic, discrete, continuous and discrete-continuous. Deterministic modeling depicts processes in which the absence of random influences is assumed. Stochastic modeling takes into account probabilistic processes and events. Static modeling is used to describe the state of an object at a fixed point in time, and dynamic modeling is used to study an object over time. In this case, they operate with analog (continuous), discrete and mixed models. Depending on the form of implementation of the medium, modeling is classified into mental and real. Mental modeling is used when models are not realizable in a given time interval or there are no conditions for their physical creation (for example, a microworld situation). Mental modeling of real systems is implemented in the form of visual, symbolic and mathematical. A significant number of tools and methods have been developed to represent functional, information and event models of this type of modeling. With visual modeling, based on human ideas about real objects, visual models are created that display the phenomena and processes occurring in the object. Examples of such models are educational posters, drawings, diagrams, diagrams. The basis of hypothetical modeling is a hypothesis about the patterns of the process in a real object, which reflects the researcher’s level of knowledge about the object and is based on cause-and-effect relationships between the input and output of the object being studied. This type of modeling is used when knowledge about an object is not enough to build formal models.

Dynamic modeling is a multi-step process, each step corresponds to the behavior of the economic system over a certain time period. Each current step receives the results of the previous step, which, according to certain rules, determines the current result and generates data for the next step.

Thus, a dynamic model in an accelerated mode makes it possible to study the development of a complex economic system, say, an enterprise, over a certain planning period in the conditions of changes in resource provision (raw materials, personnel, finance, technology), and obtain the results presented in the corresponding development plan of the enterprise for a given period.

To solve dynamic optimization problems in mathematical programming, a corresponding class of models called dynamic programming was formed, its founder was the famous American mathematician R. Bellman. He proposed a special method for solving problems of this class based on the “optimality principle”, according to which the optimal solution to the problem is found by dividing it into n stages, each of which represents a subproblem with respect to one variable. The calculation is performed in such a way that the optimal result of one subtask is the input data for the next subtask, taking into account the equations and communication constraints between them, the result of the last of them is the result of the entire problem. What all models in this category have in common is that current management decisions “manifest” both in the period immediately surrounding the moment the decision was made and in subsequent periods. Consequently, the most important economic effects occur over different periods, not just within one period. These kinds of economic consequences tend to be significant in cases where we are talking about management decisions related to the possibility of new capital investments, increasing production capacity or training personnel for the purpose. creating prerequisites for increasing profitability or reducing costs in subsequent periods.

Typical applications of dynamic programming models in decision making are:

Development of inventory management rules that establish the moment of replenishment and the size of the replenishing order.

Development of principles for production scheduling and employment equalization in conditions of fluctuating demand for products.

Determining the required volume of spare parts to ensure the efficient use of expensive equipment.

Distribution of scarce capital investments between possible new areas of their use.

In problems solved by the dynamic programming method, the value of the objective function (optimized criterion) for the entire process is obtained by simply summing the partial values fi(x) the same criterion at individual steps, that is

If a criterion (or function) f(x) has this property, then it is called additive.

Dynamic programming algorithm

1. At the selected step, we specify a set (defined by constraint conditions) of variable values ​​that characterize the last step, possible states of the system at the penultimate step. For each possible state and each value of the selected variable, we calculate the values ​​of the objective function. From these, for each outcome of the penultimate step, we select the optimal values ​​of the objective function and the corresponding values ​​of the variable under consideration. For each outcome of the penultimate step, we remember the optimal value of the variable (or several values, if there is more than one such value) and the corresponding value of the objective function. We receive and fix the corresponding table.

2. We proceed to optimization at the stage preceding the previous one (moving “backwards”), searching for the optimal value of the new variable with fixed previously found optimal values ​​of the following variables. The optimal value of the objective function at subsequent steps (with optimal values ​​of subsequent variables) is read from the previous table. If a new variable characterizes the first step, then proceed to step 3. Otherwise, repeat step 2 for the next variable.

H. Given the initial condition in the problem, for each possible value of the first variable, we calculate the value of the objective function. We select the optimal value of the objective function corresponding to the optimal value(s) of the first variable.

4. With the known optimal value of the first variable, we determine the initial data for the next (second) step and, according to the last table, the optimal value(s) of the next (second) variable.

5. If the next variable does not characterize the last step, then go to step 4. Otherwise, go to step 6.

6. We form (write out) the optimal solution.

List of used literature

1. Microsoft Office 2010. Tutorial. Y. Stotsky, A. Vasiliev, I. Telina. Peter. 2011, - 432 p.

2. Figurnov V.E. IBM PC for the user. 7th edition. - M.: Infra-M, 1995.

3. Levin A. Self-instruction manual for working on a computer. M.: Knowledge, 1998, - 624 p.

4. Computer science: workshop on the technology of working on a personal computer / Ed. prof. N.V. Makarova - M.: Finance and Statistics, 1997 - 384 p.

5. Computer Science: Textbook / Ed. prof. N.V. Makarova - M.: Finance and statistics, 1997 - 768 p.

Related information.

3D cartographic images are electronic maps of a higher level and represent spatial images of the main elements and objects of the area visualized using computer modeling systems. They are intended for use in control and navigation systems (ground and air) for terrain analysis, solving calculation problems and modeling, designing engineering structures, and environmental monitoring.

Simulation technology terrain allows you to create visual and measurable perspective images that closely resemble the real terrain. Their inclusion according to a certain scenario in a computer film allows, when viewing it, to “see” the terrain from different shooting points, in different lighting conditions, for different seasons and days (static model) or to “fly” over it along given or arbitrary trajectories of movement and speed flight - (dynamic model).

The use of computer tools, which include vector or raster displays that allow the conversion of input digital information into a given frame in their buffer devices, requires the preliminary creation of digital spatial terrain models (STM) as such information.

Digital PMM in essence represent a set of digital semantic, syntactic and structural data recorded on computer media, intended for reproduction (visualization) of three-dimensional images of terrain and topographic objects in accordance with specified conditions of observation (review) of the earth's surface.

Initial data for creating digital PMMs may include photographs, cartographic materials, topographic and digital maps, city plans and reference information that provide data on the position, shape, size, color, and purpose of objects. In this case, the completeness of the PMM will be determined by the information content of the photographs used, and the accuracy - by the accuracy of the original cartographic materials.

Technical means and methods for creating PMM

Development of technical means and methods for creating digital PMMs is a difficult scientific and technical problem. The solution to this problem involves:

Development of hardware and software for obtaining primary three-dimensional digital information about terrain objects from photographs and map materials;
- creation of a system of three-dimensional cartographic symbols;
- development of methods for generating digital PMMs using primary cartographic digital information and photographs;
- development of an expert system for forming the content of the PMM;
- development of methods for organizing digital data in the PMM bank and principles for constructing the PMM bank.

Development of hardware and software obtaining primary three-dimensional digital information about terrain objects from photographs and map materials is due to the following fundamental features:

Higher, compared to traditional digital digital computers, requirements for digital digital digital computers in terms of completeness and accuracy;
- using as initial decoding photographs obtained by frame, panoramic, slit and CCD filming systems and not intended to obtain accurate measurement information about terrain objects.

Creation of a system of three-dimensional cartographic symbols is a fundamentally new task of modern digital cartography. Its essence is to create a library of symbols that are close to the real image of terrain objects.

Methods for generating digital PMMs using primary digital cartographic information and photographs must ensure, on the one hand, the efficiency of their visualization in the buffer devices of computer systems, and, on the other hand, the required completeness, accuracy and clarity of the three-dimensional image.

Research currently being carried out has shown that to obtain digital PMMs, depending on the composition of the source data, methods using:

Digital cartographic information;
- digital cartographic information and photographs;
- photographs.

The most promising methods seem to be, using digital cartographic information and photographs. The main ones may be methods for creating digital PMMs of varying completeness and accuracy: from photographs and DEMs; from photographs and digital digital materials; from photographs and DTM.

The development of an expert system for forming the content of the PMM should provide a solution to the problems of designing spatial images by selecting the object composition, its generalization and symbolization, and displaying the display in the required map projection. In this case, it will be necessary to develop a methodology for describing not only conventional signs, but also the spatial-logical relationships between them.

The solution to the problem of developing methods for organizing digital data in a PMM bank and the principles for constructing a PMM bank is determined by the specifics of spatial images and data presentation formats. It is quite possible that it will be necessary to create a space-time bank with four-dimensional simulations (X, Y, H, t), where PMMs will be generated in real time.

Hardware and software tools for displaying and analyzing PMM

The second problem is development of hardware and software display and analysis of digital PMMs. The solution to this problem involves:

Development of technical means for displaying and analyzing PMM;
- development of methods for solving calculation problems.

Development of hardware and software display and analysis of digital PMMs will require the use of existing graphic workstations, for which special software (SPO) must be created.

Development of methods for solving calculation problems is an applied problem that arises in the process of using digital PMMs for practical purposes. The composition and content of these tasks will be determined by specific PMM consumers.

The model is called static when the input and output influences are constant in time. Static model describes the steady state.

A model is called dynamic if the input and output variables change over time. Dynamic model describes the unsteady operating mode of the object being studied.

The study of the dynamic properties of objects allows, in accordance with the fundamental Huygens-Hadamard principle of certainty, to answer the question: how does the state of an object change under known influences on it and a given initial state.

An example of a static model is the dependence of the duration of a technological operation on resource costs. The static model is described by the algebraic equation

An example of a dynamic model is the dependence of the volume of output of commercial products of an enterprise on the size and timing of capital investments, as well as expended resources.

The dynamic model is often described by the differential equation

An equation relates an unknown variable Y and its derivatives with independent variable t and a given time function X(t) and its derivatives.

A dynamic system can operate in continuous or discrete time, quantized into equal intervals. In the first case, the system is described by a differential equation, and in the second case, by a finite-difference equation.

If the sets of input, output variables and times are finite, then the system is described finite state machine.

A finite state machine is characterized by a finite set of input states; finite set of states; a finite set of internal states; transition function T(x, q), which determine the order of change of internal states; output function P(x, q) setting the output state depending on the input state and internal state.

A generalization of deterministic automata are stochastic automata, which are characterized by the probabilities of transitions from one state to another. If the functioning of a dynamic system is in the nature of servicing emerging requests, then a model of the system is built using methods queuing theory.

The dynamic model is called stationary, if the transformation properties of the input variables do not change over time. Otherwise it is called non-stationary.

Distinguish deterministic and stochastic (probabilistic) models. A deterministic operator allows you to uniquely determine output variables from known input variables.

Determinism models only means non-randomness of transformation of input variables, which themselves can be either deterministic or random.

The stochastic operator allows you to determine the probability distribution of input variables from a given probability distribution of input variables and system parameters.

In terms of input and output variables, models are classified as follows:

1. Input variables are divided into managed And uncontrollable. The former can be changed at the discretion of the researcher and are used by the object. The latter are unsuitable for management.

2. Depending on the dimension of the vectors of input and output variables, they are distinguished one-dimensional and multidimensional models. By a one-dimensional model we mean a model in which the input and output variables are both scalar quantities. A model whose vectors is called multidimensional x(t) And y(t) have dimension n³ 2.

3. Models in which input and output variables are continuous in time and magnitude are called continuous. Models in which input and output variables are discrete either in time or in magnitude are called discrete.

Note that the dynamics of complex systems largely depends on the decisions made by humans. Processes occurring in complex systems are characterized by a large number of parameters - large in the sense that the corresponding equations and relationships cannot be resolved analytically. Often the complex systems studied are unique in comparison even with systems of similar purposes. The duration of experiments with such systems is usually long and often turns out to be comparable to their lifetime. Sometimes conducting active experiments with the system is generally unacceptable.

For a complex object, it is often impossible to determine the content of each control step. This circumstance determines such a large number of situations characterizing the state of the object that it is almost impossible to analyze the influence of each of them on the decisions made. In this situation, instead of a rigid control algorithm that prescribes a certain unambiguous solution at each step of its implementation, it is necessary to use a set of instructions corresponding to what is commonly called calculus in mathematics. Unlike an algorithm in calculus, the continuation of the process at each step is not fixed and it is possible to arbitrarily continue the process of finding a solution. Calculus and similar systems are studied in mathematical logic.

1.5. The concept of building a system model of complex objects

Complex objects are a collection of individual structurally isolated elements: technological units, transport highways, electric drives, etc., interconnected by material, energy and information flows, and interacting with the environment as a whole. Energy and mass transfer processes occurring in complex objects are directional and associated with the movement of fields and matter (heat exchange, filtration, diffusion, deformation, etc.). As a rule, these processes contain unstable stages of development, and the management of such processes is more an art than a science. Due to these circumstances, there is an unstable quality of management of such objects. The requirements for the qualifications of technological personnel are sharply increasing and the time for their training is significantly increasing.

An element of a system is a certain object (material, energy, information) that has a number of properties that are important to us, the internal structure (content) of which is not of interest from the point of view of the purpose of analysis.

We will denote elements by M, and their entire considered (possible) set – through (M). It is customary to record the belonging of an element to a population.

Communication Let us call the exchange between elements important for the purposes of consideration: matter, energy, information.

A single act of communication is impact. Denoting all effects of an element M 1 per element M 2 through x 12 a element M 2 per M 1 – through x 21, the connection can be depicted graphically (Fig. 1.6).

Rice. 1.6. Relationship between two elements

System Let's call a set of elements that has the following characteristics:

a) connections that allow, through transitions along them from element to element, to connect any two elements of the set;

b) a property (purpose, function) different from the properties of individual elements of the aggregate.

Let's call the feature a) the connectivity of the system, b) its function. Using the so-called “tuple” (i.e. sequence in the form of an enumeration) definition of the system, we can write

where Σ is the system; ( M} – the totality of elements in it; ( x) – a set of connections; F – function (new property) of the system.

We will consider the entry as the simplest description of the system.

Almost any object from a certain point of view can be considered as a system. It is important to be aware of whether such a view is useful or whether it is more reasonable to consider the given object an element. So, the system can be considered a radio board , converting the input signal into an output signal. For a specialist in the element base, the system will be a mica capacitor in this board, and for a geologist, the system will be mica itself, which has a rather complex structure.

Big system let's call a system that includes a significant number of similar elements and similar connections.

Complex system Let's call a system consisting of elements of different types and having heterogeneous connections between them.

Often, only those that are large are considered complex systems. The heterogeneity of elements can be emphasized by writing

A large, but not complex from a mechanical point of view, system is a crane boom assembled from rods or, for example, a gas pipeline pipe. The elements of the latter will be its sections between welds or supports. For deflection calculations, gas pipeline elements will most likely be considered to be relatively small (on the order of a meter) sections of pipe. This is done in the well-known finite element method. The connection in this case is of a force (energy) nature - each element acts on the neighboring one.

The distinction between a system, a large system and a complex system is arbitrary. Thus, the hulls of missiles or ships, which at first glance are homogeneous, are usually classified as a complex system due to the presence of different types of bulkheads.

An important class of complex systems are automated systems. The word “automated” indicates human participation, the use of human activity within the system while maintaining a significant role of technical means. Thus, a workshop, site, assembly can be either automated or automatic (“automatic workshop”). For a complex system, the automated mode is considered more preferable. For example, landing an airplane requires human assistance, and the autopilot is usually used only for relatively simple movements. Also typical is the situation when a solution developed by technical means is approved for execution by a person.

So, an automated system is a complex system with a decisive role of elements of two types: a) in the form of technical means; b) in the form of human actions. Its symbolic notation (compare with and)

Where M T– technical means, primarily computers; M H – decisions and other human activities; M" – other elements in the system.

In total ( X) in this case the connections between man and technology can be highlighted ( x T - H}.

Structure A system is called its division into groups of elements, indicating the connections between them, unchanged for the entire duration of consideration and giving an idea of ​​the system as a whole.

The specified division may have a material (substantial), functional, algorithmic and other basis. Groups of elements in a structure are usually distinguished on the basis of simple or relatively weaker connections between elements of different groups. It is convenient to depict the structure of the system in the form of a graphical diagram consisting of cells (groups) and lines (connections) connecting them. Such schemes are called structural.

To symbolically record a structure, we introduce instead of a collection of elements ( M), a set of groups of elements ( M*) and the set of connections between these groups ( x*).Then the structure of the system can be written as

Structure can be obtained by combining elements into groups. Note that the function (assignment) F the system is down.

Let's give examples of structures. The material structure of a prefabricated bridge consists of its individual sections assembled on site. A rough block diagram of such a system will indicate only these sections and the order in which they are connected. The latter are the connections that here are of a forceful nature. An example of a functional structure is the division of an internal combustion engine into power systems, lubrication, cooling, power transmission, etc. An example of a system where material and functional structures are merged is the departments of a design institute dealing with different aspects of the same problem.

A typical algorithmic structure will be an algorithm (diagram) of a software tool indicating a sequence of actions. Also, the algorithmic structure will be an instruction that defines actions when finding a malfunction of a technical object.

1.6. The main stages of an engineering experiment aimed at studying complex objects

Let us characterize the main stages of an engineering experiment aimed at studying complex objects.

1. Construction of the physical basis of the model.

Construction of the physical basis of the model, which allows us to identify the most significant processes that determine the quality of management and determine the relationships between deterministic and statistical components in the observed processes. The physical basis of the model is built using the “projection” of a complex object into various subject areas used to describe the object under study. Each subject area sets its own systems of restrictions on the possible “movements” of an object. Taking into account the totality of these restrictions allows us to justify the complex of models used and build a consistent model.

The construction of the “framework” of the model, i.e. its physical basis, comes down to describing the system of relations characterizing the object under study, in particular, the laws of conservation and kinetics of processes. Analysis of the system of relations characterizing an object makes it possible to determine the spatial and temporal scales of the mechanisms that initiate the observed behavior of processes, to qualitatively characterize the contribution of the statistical element to the description of the process, and also to identify the fundamental heterogeneity (if it exists!) of the observed time series.

The construction of a “framework” comes down to establishing, based on a priori data, cause-and-effect relationships between external and internal destabilizing factors and the efficiency of the system, and quantitative estimates of these relationships are specified by conducting experiments on the site. This ensures the generality of the results obtained for the entire class of objects, their consistency with respect to previously acquired knowledge, and ensures a reduction in the volume of experimental research. The “framework” of the model should be built using a structural-phenomenological approach that combines the study of an object based on its reactions to “external” influences and the disclosure of the internal structure of the object of study.

2. Checking the statistical stability of observation results and determining the nature of changes in controlled variables.

The empirical justification of statistical stability comes down to studying the stability of the empirical average as the sample size increases (lengthening series scheme). The unpredictability of experimentally obtained values, as is known, is neither a necessary nor a sufficient condition for the application of probability-theoretic concepts. A necessary condition for the application of probability theory is the stability of the averaged characteristics of the initial quantities. Thus, a test using empirical induction of statistical robustness is required n-dimensional empirical distribution function of the original random variable and probability distribution for sample estimates.

3. Formation and testing of hypotheses about the structure and parameters of the “movement” of the object under study.

Note that, as a rule, the motive for choosing a statistical approach is the lack of regularity of the observed process, chaotic nature and sharp breaks. In this case, the researcher cannot visually detect patterns in a series of observations and perceives it as the implementation of a random process. We emphasize that we are talking about the detection of the simplest patterns, since in order to detect complex patterns, directed mathematical processing of observational results is required.

4. Forecasting of output variables is carried out taking into account the contribution of deterministic and statistical components to the final result.

Note that using only a statistical approach for forecasting encounters serious difficulties. Firstly, to make decisions regarding minimizing current losses, it is important to know not how the process develops on average, but how it will behave over a specific period of time. Secondly, in the general case we have the task of predicting a non-stationary, random process with changing mathematical expectation, dispersion and the very type of distribution law.

5. Planning and implementation of a computational experiment aimed at assessing the regulatory characteristics of the object and the expected efficiency of the control system.

Problems of synthesizing the structure of complex systems can only be solved analytically in the simplest cases. Therefore, there is a need for simulation modeling (IM) of the elements of the designed system.

IM is a special way of studying objects of complex structure, which consists in numerically reproducing all input and output variables of each element of the object. IM allows, at the stage of analysis and synthesis of the structure, to take into account not only the statistical relationships between the elements of the system, but also the dynamic aspects of its functioning.

To compile an IM you need:

– highlight the simplest elements in the modeling object for which the method for calculating output variables is known;

– create connection equations that describe the order of connecting elements in an object;

– draw up a structural diagram of the object;

– select modeling automation tools;

– develop an IM program;

– conduct computational experiments to assess the adequacy of the IM, the stability of the simulation results and the sensitivity of the IM to changes in control and disturbing influences;

– solve the problem of control system synthesis using the model.

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 (instants) of time.

Spatial samples are formed by combining spatial samples over a number of years (periods), i.e. a collection 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.