1. Topic 1. MODELING and SIMULATION Olga Marukhina, Associate Professor, Control System Optimization Department, Tomsk Polytechnic University Marukhina@tpu.ru

2. Course structure Lectures 36 hours Labs 36 hours Exam

3. Main terms The word "model" (from Latin "modelium") means "measure", "method", "similarity to something". Model is an object or description of an object or system for substitution (under certain conditions, suggestions, hypotheses) of one system (i. e. the original) with another system for better study of the original or reproduction of some of its properties. A model is the result of reflection of one (well-studied) structure onto another (less- studied) one.

4. Main terms A model must be built so that it provides the fullest possible replication of those object properties, which need to be studied according to the set goal. Different models can exist for one and the same object in accordance with different goals of its study.

5. Fashion Films, animations Medicine Robots Sport

6. 2. What models can be created for? For the safety of human life and health Reduce the cost of material resources To understand the essence of the object studied In order to learn how to manage the object Forecasting the effects of For relaxation For applied task solving

7. 3. Modeling The process of building, study, and application of models will be referred to as modeling, i. e. it can be said that modeling is a method of studying an object by means of building and studying its model, which is performed for a definite purpose and consists in substitution of the original with a model. Modeling - the process of creating and using models.

8. 4. The adequacy of the models Adequacy (from Latin "adaequatus" – equated) is a degree of conformity of modelling to original object, which we obtained in the process of model investigation, testing tasks and experiments Adequate model is the model with certain approximation degree which reflects the process of original object functioning in the real conditions

9. 4. The adequacy of the models Types of adequacy Full Not full (partly) The model can also be inadequate. This means that the model does not correspond to the object, which it replaces. Magician wanted to make a thunderstorm but have got the goat (famous song in Russia)

10. 5. Model classification Cognitive and pragmatic model. The distinction between cognitive and pragmatic model: a) cognitive model (model fit the reality); b) the pragmatic model (reality to fit the model) MODEL reality MODEL

11. 5. Model classification Form of presentation the model Objective form (globe; human skeleton; children's toys) Figurative and symbolic form (photo; picture; computer game; description of the person) Mental (image of an object in the memory of human) Documentary (photo; picture; maps) Computers (computer game)

12. 5. Classification of models by degree of abstraction of model from original.

14. Full-scale models are real studied systems used as mock-ups and experi-mental prototypes. Quasi full-scale models (from Latin "quasi" – almost) are a totality of full-scale and mathematical models. Scale models are systems of the same physical nature as the original, but different in size. Analog modeling is based on analogy of processes and phenomena of different physical nature, but formally described in the same way (by the same mathematical equations, logical schemes etc.). For example, it is well-known that the mathematical equation of pendulum oscillation has its equivalent in writing current oscillation equations. Ideal modeling is of theoretical nature. It is subdivided into two main types: intuitive modeling and symbolic modeling.

15. Modeling is considered intuitive, when it is based on intuitive conception of the object of study, which is impossible or not necessary to formally characterize. In this sense, for example, life experience of every person can be considered one's intuitive model of the surrounding world. Modeling is considered symbolic when it uses models that are symbolic conversions of various kinds: schemes, graphs, plots, formulae, character sets etc., which include the body of laws that can be used to operate the selected symbolic elements. Symbolic models are further subdivided into linguistic, visual, graphic and mathematical models. A model is linguistic if it is represented by a certain linguistic object, formalized language system or structure. Sometimes such models are also called verbal; for example, traffic regulations are a verbal structural model of vehicular and pedestrian traffic on the road. A model is visual if it allows to visualize relations and connections of the modeled system, especially in its dynamics. For example, visual models of keyboard sections are often used on computer display in keyboard training programs.

16. The most important part of symbolic modeling is the mathematical modeling; a classic example of mathematical modeling is description and study of Newton's laws of motion using mathematical tools. Mathematical model - a set of mathematical expressions that describe the behavior (structure) of the system and the conditions (perturbation limit), in which it operates.

17. Classification of mathematical models By belonging to hierarchy level mathematical models are subdivided into microlevel, macrolevel, and metalevel models. Mathematical models at the microlevel of process reflect physical processes occurring, for example, during metal cutting. They describe processes on the transition (passage) level. Mathematical models at the macrolevel of process describe technological processes. Mathematical models at the metalevel of process describe technological systems (production areas, workshops, enterprise in whole).

18. Classification of mathematical models By nature of reflected object properties models can be classified into structural and functional

19. Classification of mathematical models Model is considered structural if it is expressible by data structures and relations between them; for example, a description (tabular, graph, functional, or other) of trophic structure of an ecosystem can serve as a structural model. In turn, a structural model can be hierarchical or network. A model is hierarchical (tree-type) if it is expressible by a certain hierarchical structure (tree); for example, to solve the task of finding a route in a search tree a tree-type model, can be built. A network model is expressible by a certain network structure.

20. Classification of mathematical models network project schedule a tree-type model

21. Classification of mathematical models A model is functional if it is represented as a system of functional relations. For example, Newton law and product manufacturing model are functional.

22. Classification of mathematical models By method of object properties representation models are subdivided into analytical,numerical, algorithmic and simulation models

23. Classification of mathematical models Analytical mathematical models represent explicit mathematical expressions of output parameters as functions of input and internal parameters and have unique solutions under any initial conditions. For example, a quadratic equation having one or multiple solutions is considered an analytical model. A model is algorithmic if it is described by a certain algorithm or a complex of algorithms, defining its functioning and development. For example, an algorithm of calculating the finite sum of number sequence to a certain desired degree of accuracy can be used as a model of calculating the sum of an infinite decreasing number sequence. A simulation model is intended for testing and study of possible development paths and behavior of an object by means of varying certain or all of model's parameters, for example, an economic system for production of two kinds of merchandise.

24. SIMULATION Model “What … if…” ResultsParameters ?

25. Classification of mathematical models By model acquisition method models are subdivided into theoretical and empirical/ Theoretical mathematical models are created as a result of studying objects (processes) at the theoretical level. For example, there exist expressions for cutting force, acquired through generalization of physical laws. But they are unacceptable for practical use, because they are too unwieldy and not exactly adapted to real material processing. Empirical mathematical models are created as a result of conducting experiments (studying external manifestations of object properties by measuring its input and output parameters) and processing their results by methods of mathematical statistics.

26. Classification of mathematical models By form of object properties representation models are subdivided into logical, set-theoretical, and graph models A model is logical, if it is represented by predicates, logical functions; for example, a sum of two logical functions can serve as a mathematical model of a one-digit adder. A model is set-theoretical if it is represented by means certain sets and relations of membership in them an between them. A graph model is represented by a graph or graphs and relations between them.

27. Classification of mathematical models By content of irregular components models are subdivided into deterministic and stochastic. If a model does not contain any irregular (stochastic) components, it is considered deterministic. However, sets of systems are modeled with several random input values; as a result, stochastic (probabilistic) models are created. Examples of such models are queuing systems and inventory control systems. Stochastic models return result which is in itself random, and therefore can be considered as assessment of a model's true characteristics.

28. Classification of models by degree of stability. All models can be divided into stable and unstable. A system is called stable when after being lead out of its original state it tends back to it. It can oscillate about the initial point for some time like a common pendulum set in motion, but disturbances in it damp and disappear with time. In an unstable system, initially in the state of rest, an emerging disturbance intensifies, causing the values of corresponding variables to increase or oscillate with increasing amplitude.

29. Classification of models by relation to external factors. By their relation to external factors models can be subdivided into open and closed. A closed model functions irrelatively to external (exogenous) variables. In a closed model change of variable values in time is defined by internal interaction of the variables themselves. A closed model can identify system behavior without introduction of external variable. Example: information feedback systems are closed systems. These are self-adjusting systems and their characteristics follow from internal structure and interactions reflecting external information input. A model connected to external (exogenous) variables is called open.

30. Classification of models by relation to time. There exist two classifications of models by their relation to time factor. Models can be: 1) continuous or discrete; 2) static or dynamic. A continuous model describes a system in time through a representation in which state variables change continuously in relation to time. An example of continuous model is a complex system of differential equations.

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32. Спасибо за внимание ! http://www.youtube.com/watch?v= M0iZ52kUOiQ http://www.youtube.com/watch?v= M0iZ52kUOiQ http://www.youtube.com/watch?v= M0iZ52kUOiQ