Topic 1.
MODELING and SIMULATION
Olga Marukhina,
Associate Professor,
Control System Optimization Department,
Tomsk Polytechnic University
[email protected]
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Course structure
Lectures 36 hours
Labs 36 hours
Exam
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 (lessstudied) one.
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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.
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Films, animations
Fashion
Medicine
Sport
Robots
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
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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.
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4. The adequacy of the models
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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
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)
5. Model classification
reality
MODEL
MODEL
reality
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)
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)
5. Classification of models by
degree of abstraction of model
from original.
5. Classification of models by
degree of abstraction of model
from original.
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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.
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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.
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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.
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).
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Classification of mathematical
models
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By nature of reflected object properties models
can be classified into structural and
functional
Classification of mathematical
models
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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.
Classification of mathematical
models
a tree-type model
network project schedule
Classification of mathematical
models
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A model is functional if it is represented as a
system of functional relations. For example,
Newton law and product manufacturing model
are functional.
Classification of mathematical
models
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By method of object properties representation
models are subdivided into analytical,numerical,
algorithmic and simulation models
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.
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SIMULATION
Model “What … if…”
Parameters
?
Results
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.
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Classification of mathematical
models
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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.
Classification of mathematical
models
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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.
Classification of models by degree
of stability.
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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.
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.
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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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Спасибо за внимание!
http://www.youtube.com/watch?v=
M0iZ52kUOiQ
Спасибо за внимание!
http://www.youtube.com/watch?v=
M0iZ52kUOiQ