МММ З 1
1.1. A historical sketch
Modelling-.vozniknovenie - a temple in Kizhi, etc., at fashion designers - tailoring дешовой models before release present. - natural modelling. With the beginning of development of technics - occurrence мат.описаний objects - possibility of calculation of their properties and behaviour for the purpose of acceptance of certain decisions and carrying out of influences on object. An example - a computer. Games, a floor-mat. Description ГТУ
The person faces necessity of decision-making, as in household, and office areas of the activity. Made decisions differ both on responsibility degree, and on degree of the importance of consequences - from personal to the state scales. Various aspects of the problems connected with development of administrative decisions, with "optimum" behaviour of people, with crossing of interests of the several parties, many sciences are engaged: economy, sociology, the right, etc.
Unlike other approaches мат.модел. Analyzes these problems by means of a mathematical apparatus. It means, that at least some data appearing in the formulation of a problem, should have quantitative expressions. Qualitative given (conditions) of a problem are considered in addition and are an original background for use of mathematical models.
1 Model. Modelling. Classification of kinds of modelling of systems
All that, on what human activity is directed, is called as object (an armour. objectum - a subject). Objects co-operate among themselves and an environment. The logic schemes constructed on certain predictions (hypothesis) or judgements about any similarity of two objects (analogy) which simplify reasonings and logic constructions or allow to make the experiments specifying the nature of the phenomena, are called as models. In other words, model (an armour. modulus the-measure) is the object-assistant of object-original providing studying of some properties of the original. One more conclusion from here follows: the model only approximately reflects considered properties of modelled object, and degree of this approach should be co-ordinated with accuracy the entrance information on the phenomenon.
Replacement of one object by another for the purpose of reception of the information on the major (from the point of view of the researcher) properties of the obekta-original by means of object-model is called as modelling. Thus,
Modelling can be defined as representation of object by model for reception of the information on this object by carrying out of experiments with its model. The theory of replacement of one objects other (models) and исслeдования properties of objects on their models is called as the modelling theory. The model should be reliable and good-quality. The same properties should possess and the initial information necessary for construction of model.
That you will enclose in model that and you will receive - as the computer.
At the heart of modelling the theory of similarity which asserts lays, that absolute similarity can take place only at replacement of one object with another precisely same.
At modelling absolute similarity has no place and aspire to that the model well enough displayed the investigated party of functioning of object. Therefore as one of first signs of classification of kinds of modelling it is possible to choose degree of completeness of model and to divide models according to this sign on full, incomplete and approached. At the heart of the approached modelling the approached similarity at which some parties of functioning of real object are not modelled absolutely lays.
Depending on character of studied processes in system all kinds of modelling can be divided on determined and stochastic, statistical and dynamic, discrete, continuous and is discrete-continuous, fig. 1.1.
The determined modelling displays processes in which the initial information is completely defined. At stochastic modelling the initial information contains elements of uncertainty or problem parametres have casual character with known likelihood characteristics - газодинамика ГПА - processes трещинообразования.
Static modelling serves for the object description during the fixed moment of time - work ГПА on the established mode.
Dynamic modelling serves for reflexion of behaviour of object in time - transients in ГПА.
Discrete modelling serves for the description of processes, which are assumed discrete (change in steps) - valve operation.
Continuous modelling reflects continuous processes in systems - the engine as a whole. Is discrete-continuous modelling is applied to systems in which wish to allocate both discrete, and continuous processes - the engine with valves of restart-up of air.
Depending on the form of representation of object or system it is possible to allocate mental and real modelling.
At real modelling possibility of research of various characteristics or on real object, or on its part is used.
On the basis of mental modelling situations which do not give in to physical experiment can be analysed.
Mathematical models. The aspiration to increase of a generality and universality of models and methods of modelling demands increase of level of abstraction. Thus the most abstract.и, hence, to the general is the symbolical description. In this case for the system description special symbols are entered and rules of operating with them are established. Set of symbols and instructions for use them (grammar) is abstract language. Some statement which has been written down in abstract language, forms the formula. As an example of abstract languages it is possible to result language of chemistry, mathematics etc. Thus the mathematics represents делый a class of abstract languages with various грамматиками and various levels of abstraction (generality).
Thus mathematical modelling - process of an establishment of conformity to the given real object of some mathematical object named mathematical model, and researches of this model, allowing to receive characteristics of considered real object. Mathematical model - the approached description of any class of the phenomena of the external world, expressed by means of mathematical symbolics, system of the mathematical parities describing studied process. The kind of mathematical model as it has already been told, depends as by nature real object, and on object research problems. Mathematical modelling is subdivided on analytical, imitating and combined.
Analytical models allow to receive or obvious dependences for required sizes, or, in that case when it is not possible, to define numerical decisions for concrete entry conditions and quantitative characteristics of model. For the decision of such problems digital COMPUTERS are widely used. At analytical modelling functioning of elements of systems registers in the form of some functional parities: algebraic, differential, integrated, etc. the Analytical model can be investigated following methods: 1) analytical when aspire to receive in a general view obvious dependences of required characteristics; 2) numerical, when there is no the decision of the initial equation in a general view also receive numerical results at concrete initial data. This method is especially effective at COMPUTER use; 3) qualitative when, not having the decision in an explicit form, it is possible to find some properties of the decision, for example, to estimate stability of the decision. It is necessary to notice what to receive the obvious decision for analytical models probably for rather simple systems. Therefore for difficult systems go on essential simplification of their initial models that on the simplified model to study at least the general properties of system.
However synthesis of analytical models for the big systems, as a rule, is impossible. In communication е it, now the wide circulation has received imitating modelling which is considered as experiment with the difficult mathematical model describing behaviour of system, realised on the COMPUTER, i.e. Natural experiment is modelled.
Combined (hybrid)-... Can will join even the person (the operator, the pilot, etc.)
Questions mathematical and, first of all, imitating modelling will be considered further basically. The reason - complexity and integrated approach of problems, the requirement exact (the numerical decision.) nuclear projects, usual and gene engineering etc. Sharp decrease in expenses for working out and operational development of difficult production - Snecma, etc.
Imitating modelling allows:
Experimentally to investigate difficult internal interactions in considered system;
To study influence on functioning of system of information and organizational changes and changes of character of interaction with an environment;
It is better to understand system, to estimate what of variables are most essential and as they co-operate;
To estimate behaviour of system in new situations, to check new strategy and decision-making rules;
To spend stochastic modelling, in particular a method monte-karlo. -
At imitating modelling the algorithm realising model reproduces process of functioning of system in time, and the elementary phenomena making process, with preservation of their logic structure and sequence of course in time are simulated. It allows on initial given to receive data on process conditions in the certain moments of time, giving the chance to estimate the system characteristic. By means of imitating modelling it is possible to solve more challenges, than at use of analytical modelling. Most widely in imitating modelling the method of statistical modelling representing a numerical method which was applied to modelling of random variables and the functions which likelihood characteristics coincided with decisions of analytical problems (such procedure is used has received the name of a method of Monte-Carlo). The combined (analitiko-imitating) modelling at the analysis and synthesis of systems allows to unite advantages of analytical and imitating modelling.
Mathematical models are most extended in the mechanic. As the initial information for construction of such model data about appointment and working conditions of investigated (projected) system serve. This information defines a main objective of modelling of system and allows to formulate requirements to developed mathematical model. Abstraction level defines a choice of the mathematical scheme. The mathematical scheme is understood as a thinking method, means of a formulation of concepts that is important at transition from the verbal description of system to formal representation of process of its functioning in the form of some mathematical model. At construction of mathematical model it is necessary to solve a question on its completeness, i.e. To choose border "system-environment". Full model of gas-transport system - the pipeline +ГПА; ГПА - ГТУ + нагнетатель+автоматика + внеш.условия +... - How to divide on System - Environment
Fig. 1.1 - Classification of kinds of modelling of systems