Typov_ mathematical schemes of the model. IV

MODELING SYSTEMS

ROBOCH PROGRAM, Methodical instructions

FOR INDIVIDUAL ROBOTS THAT CONTROL PROGRAM

Faculty of ELECTRICAL ENERGY, zd

Specialty 220201 - MANAGEMENT TA INFORMATICS V

TECHNICAL SYSTEMS

Straight to bachelor 220200 - AUTOMATION AND CONTROL

System model: working program, Methodical instructions for independent robotics and control. - Vologda: Vogt, 2008 .-- 22 p.

The work program of discipline is guided by the values ​​of the topics of the main distributions, methodical instructions with the help of information on the list of information, control records and a list of literature.

Designed for students of the day and part-time forms of education, as they start directly: 220200 - automation and management and specialties 220201 - management and information science in technical systemsі for bachelor's degree: 220200 - automation and control.

Confirmed by the editorial board of the Vogt

Ukladach: V.N. Tyukin, Cand. tech. Sciences, Associate Professor

Reviewer: Є.V. Nezgovorov, Cand. tech. Sciences, Associate Professor

Department of UVS Vogt

As a basis, the programs are adopted by the State Educational Standard of Food Professional Education to the minimum of change and the level of training of engineers behind the fahom 210100 - management and informatics in technical systems, introduced from 10.03.

Vimogi to knowledge and min to discipline

As a result of discipline, students are guilty:

1. The student is guilty of the mother of the statement:

About models and models;

About the role of the model in the pre-monitoring, design and operation of systems;

About the recognition of EOM in the modeling of systems;

About software and technical features of systems modeling.

2. The student is guilty of the nobility:

Appointment and vimoga, scho presented to the model;

Classification of types of systems model;

Principles of approach to the model of systems;

Mathematical schemes for modeling systems;

The main stages of the system model.

3. The student is guilty of:

Otrimuvati mathematical models of systems;

Carry out formalization and algorithmization of the process of functioning of systems;

Buuvati conceptual and machine models of systems;

Optimize and interpret model results.

Vimogi to the minimum of discipline

Classification of models and types of models; butt models of systems; the main provisions of the theory of options; stages of mathematical modeling; the principle prompts that basic vimogi to mathematical models of systems; meti and pre-development of mathematical models of systems; out-of-the-box diagram of the distribution of mathematical models; formalization of the process of functioning of the system; understanding of the aggregate model; the formation of the submission of mathematical models; methods for the advancement of mathematical models of systems and processes; іmіtatsіyne modeluvannya; methods of simplifying mathematical models; technical and software for modeling.

Table 1

Rozpodil years of the beginning of the plan for the forms of development and views to take

see take	Denne Navchannya	Absentee navchannya
	this 7		all year long	this 9		all year long.
lectures
Practical busyness
Lab. robots
Samost. robot
all
bag control	s, f.			h, e, 2 c.r.

Table 2

Rozpodil years of independent student robots for types of robots

COURSE PROGRAM

JOIN

IN 1. Suchasny camp system model problems.

В 2. Vikoristannya modeliuvannya when advised, projected and

control systems.

Literature: st. 4-6.

1. Basic understanding of MODELING SYSTEMS

1.1. The design of the model and the model. Vimogi, we are presented before the model. Appropriate model.

1.2. Principles of approach to the model of systems.

1.3. Classification of types of systems model.

1.4. The flexibility and efficiency of the model of systems on calculating machines.

Literature: st. 6-34.

2. MATHEMATICAL SCHEMES OF MODELING SYSTEMS

2.1. Basically go to induce mathematical models of systems. The mathematical scheme of the zahalny viglyad.

2.2. Non-interruption-free determination of models (D - schemes).

2.3. Discretely determined models (F - schemes).

2.4. Discrete-stochastic models (R - schemes).

2.5. Interruption-free stochastic models (Q - schemes).

2.6. Uzagalnі models (A - diagrams).

Literature: st. 35-67, pp. 168-180.

3. FORMALIZATION AND ALGORITHMS TO THE PROCESS

FUNCTIONS OF SYSTEMS

3.1. Post-production and machine implementation of systems models.

3.2. Pobudova conceptual models of systems and formalization.

3.3. Algorithmization of the model and machine implementation.

3.4. Rejection and interpretation of model results.

Literature: st. 68-89.

4. MATHEMATICALLY MODELED SYSTEMS

4.1. Canonical forms of models of dynamical systems and methods of their lasting.

4.2. Імітаційне Model.

4.3. Statistical modeling.

4.4. Programming and technical for the model of systems.

Literature :.

META COURSE

"Intelligence means to build a model."

W. Thomson (Kelvin)

Realistic objects represent themselves, as a rule, great systems, which can be achieved by folding staff. The main methodology for the course is to implement a methodical approach to the tasks of modeling great systems and control systems. The main task can be divided into a number of tasks, as well as the goals of the course:

Knowledge of methods of analysis and principles of approach to model systems;

Vyvchennya foundations of mathematical model of systems;

Vivchennya principles and apparatus of model systems;

Knowledge of modeling methods in the design and operation of systems;

Vivchennya software and technical support for model systems;

Finding practical tips to inspire models of great systems and methods of processing the results of modeling.

METHODOLOGICAL VKAZIVKI

The course "Model of control systems" is responsible for giving the student a very hard working tool of an engineer for efficient development and exploitation of automation viral systems... The model itself is done in a way that allows, without capital vitrates, the problem of prompting great systems to be introduced and completely automated.

The importance of the up-to-date course of the field is also in the field of development and technology of practical revision of the model of the processes of functioning of systems on the EOM.

Students are guilty of the material course in the main independent. For the most folding food for the course, as well as for food, the lack of visibility in the literature, lectures are read. Practical tips for modeling students can be found on practical and laboratory work. In addition, in the course of the course, the correspondence students will be able to test the control robot.

JOIN

The development of the course will take some time to learn more about the current virobability, as it is possible to see how I will fold the system of interconnected and mutually modifying elements, in which, in the capacity of the technological form of the control system, the computer monitoring system Improvement of the efficiency of the implementation of control processes in the wide range of automated control systems in the field of automated control systems. In a given hour, there is no more and more general information about automated control systems at all stages of the process, by fixing the control system and the folding of the technical staff for the design and the implementation of the ineffective modernization of the system.

It is necessary to learn about the methodological basis of the model є the dialectical-material method of knowledge and scientific dosage. It is possible to use the model as a method of alternate recognition, in case of any ado

The basic principles of modeling є.

The principle of informative sufficiency. On the basis of the level of aprimary properties, in which case the model is adequate.

The principle of health. Visitors start to reach the end of the model for the last hour.

The principle of multiple models. The created model is guilty of transforming the power of the real system into the first place, as it infuses an indicator of efficiency on the other.

The principle of aggregation. The model of the object is represented with aggregates (subsystems), which can be described using standard mathematical schemes.

The principle of parametrization. The model is guilty of its mother in its warehouse of the subsystem, which is characterized by parameters.

Basic understanding of systems modeling

"Visnachte meaning of words,

І wi amuse people

See half of this pardon ".

Vivchayuchi tsei razdil it is important to learn the basic understanding, design, purpose and principle of the model.

The model is based on accepted hypotheses and analogies, and the model is based on the application of the model for rejection of information about the process of carrying out experiments with the model.

The main vimoga is guilty of the satisfaction of the model, the adequacy of the object. Adequacy of the model to cover all the models and accepted criteria. The model is adequate to the object, so that the results of the model are confirmed and can serve as a basis for predicting the processes that occur in the previous objects.

The model of the virginization of the development and the advancement of the objects, the transfer of the functions, the synthesis of the structure, the parameters and the algorithms of the behavior.

In the case of management models, it is possible to assess the lack of caution in changing the process, to predict the process in the case of explicit abortions of management and automatically synthesize the optimal management strategy.

During the design and operation of automated systems, the number of employees is determined, estimates of the most basic and precise patterns of processes in the functioning of systems, structural, algorithmic and parametric synthesis are carried out. Virіshennya Tsikh problems in Danian hour nemozhlivo without vikoristannya rіznih vidіv modelyuvannya scho obumovleno Especially great systems, such yak skladnіstyu structures stochasticity zv'yazkіv mіzh Some of the elements that zovnіshnіm seredovischem, neodnoznachnіstyu algoritmіv povedіnki, velikіy kіlkostі parametrіv zmіnnih i, i nepovnotoyu nedetermіnіrovannostyu vihіdnoї Informácie. Mathematical modeling allows you to change the hour of the project, but in some cases it is allowed to know optimal version, Viklyuchiti method of full-scale tests і grants, go to the parallel to the design process.

In a given hour, when analyzing and synthesizing great systems, having trimmed the development of the systemic signal, which is the transfer of the last transition from the outward to the concrete, if the basis of the view is meta, moreover, the adolescents seem to be seen from the mid-range. The model falls into a wide range of problems when a problem is posed, and the model of the field has problems, problems and motivates models, problems and robots from the model. For a correctly reversed model, we are characteristic of those who have shown themselves to be deprived of the regularities, which are necessary for a presenter, and do not look at the power of the system and are not essential for the given presenter.

In the basis of the classification of types of model systems, there are specific signs, such as the steps of the model, the nature of the mathematical description. The more important thing is to borrow a mathematical model, which is the process of establishing a view of the given real object of the deyakogo mathematical object, Called the mathematical model, and the subsequent development of the model, which allows you to correct the characteristics of the revealed real object. Mathematical modeling includes analytic and imitation. Іmіtatsіyne modeluvannya primed on the direct inventory of the object, how to model, vicorist and structural subdivision of the object and the model, so that the skin sutta from the point of view of the developmental tasks of the element needs to be set in the model.

technically solution engineering tasks based on the model є ЕОМ. A machine experiment with a model gives the ability to realize the process of functioning in any minds, quickly the triviality of testing in relation to a natural experiment, є a nasty variation of parameters, structure of processes, algorithms, how to modify

Food for self-revision

1. What model is it also?

2. Formulate the main options provided before the model.

3. What is the role of the model in the pre-development and design of systems and management?

4. Give the value of the system, the name of the middle, the function of the system.

5. Who has the system approach sensor in the model?

6. List the signs of the classification of types of model systems.

7. Tell us about the mathematical model and its types.

8. Who has the benefit of the analytical and analytical model?

9.What is the same cybernetic model?

10. The role and significance of EOM in modeling.

Mathematical schemes for modeling systems

"The essence of mathematics is

You know there is order in chaos,

Yakiy will leave us ".

When vivchennі tsyogo razdіlu persh for everything you need to pay respect to the understanding of mathematical schemes of modeling like a zalny viglyadu, so і typical.

I will describe the process of functioning of the system from the urahuvannya in the flow of the new middle-class, so that it’s possible to describe a model - a mathematical scheme - a mathematical model ”. The mathematical scheme allows one to look at mathematics not as a method of drawing, but as a method of misrepresentation, as a formulation is used to understand that we are most important in the transition from a verbal description of a system to a formal submission to the process of functioning at the viewer mathematical models.

The model of the object of the model, that is, the system, can be represented in the view of unimportant quantities, which describe the process of functioning of the real system and set up in the backward-looking type of the onset of the multiplicity: out-of-the-box characteristics of the system. The input to the flood, the inflow of the most recent middle-class, the internal parameters are independent (not to the genn and) change, and the out-of-the-box characteristics of the system are fallen (and not to the fore). The mathematical scheme for modeling a zagalny viglead is set by the operator, who will transform the exogenous changes into endogenous ones.

Practical models use typical mathematical schemes that are not strong in spirit, but rather overwhelmed by simplicity and precision. Determination, stochastic aggregate types of models are referred to them. Both deterministic models are capable of differentiating, integrating, integrating and integrating, and for systems that function in a discrete hour - developing and adapting automatically. Yak stochastic models for the representation of systems with a discrete hour vikoristovuyutsya imovirnіsnі automatically, and for the representation of systems with a continuous hour - a system of mass service. Aggregate models represent the systemic nature of objects, which can be divided into a number of parts at the end of the day;

Typical mathematical schemes (D-, F-, P-, Q-, A-) allow formalization to reach a wide class of great systems, which can be brought to the mother in practice for the development and design of computer projects.

Food for self-revision

1. What is the role of the mathematical model of the model?

2. What is the mathematical scheme of a zahalny viglyad?

3. Name the main form of the submitted non-intermittently deterministic models.

4. Give a description of the discrete end machine.

5. List the methods of making the robot F - automatic.

6. The name of the automatic machine is set.

7. Who is the SMO? Name the main elements of the SMO.

8. What is a transaction?

9. Tell us about the symbolism of the Q-circuits. It is graphically displayed: request dzherelo, service channel, accumulator, valve, pod flows. To guide the butt of the image of the SMO in the symbol of the Q - schemes.

10. What is the structure of the aggregate system?

Mathematical schemes for modeling systems

Perevagi and shortcomings of the imitational model

Main gіnstі IMITATION MODELING WITH ADVANCED FOLDING SYSTEMS:

· The ability to experience the specialness of the process of functioning of the S system in any minds;

· The triviality of viprobuvan will quickly speed up in the course of a natural experiment;

· The results of full-scale testing of real systems and parts can be victorious for carrying out an imitational model;

· The dullness of the variation of the structure, algorithms and parameters of the modeled system when the optimal version of the system is suggested;

· For folding systems - a chain of practical implementations of the method of furthering the process of functioning of systems.

Main shortcomings Imitational model:

· For a repeated analysis of the characteristics of the process of functioning of the systems and the idea of ​​the optimal variant, it is necessary to develop a lot of development and adaptation experiments, varying data from the factory;

· Great vitrati of the machine hour.

The efficiency of the machine model. When modeling, it is necessary to ensure the maximum efficiency of the system model. efficiency Make a name for yourself as a deyak of growth, as indicators of the value of the results, taken during the operation of the model, and these vitrates, such as investments in the box and the stem.

The efficiency of the simulation model can be assessed by low criteria:

Accurate and reliable results in the model,

Wake up the robots for an hour with the model M,

Vitrato of machine resources (hour and memory),

· Participate in the distribution and exploitation of the model.

The best assessment of the effectiveness of the adjustment of the results from the real ones. For the additional statistical approach to the first step of accuracy (in terms of the number of implementations of the machine experiment), one can find the averaged characteristics of the behavior of the system.

The total vitality of the machine hour is stored in an hour according to the input and output for the skin modeling algorithm, in an hour for the numerical operations, operational memoryі outbuildings, As well as the folding of the skin model, the algorithm and the planning of experiments.

Mathematical schemes.Mathematical model- the chain of mathematical objects (numbers, changes, plurals, vectors, matrices, etc.) and one of them, adequately depicting the physical power of the developed technical object. The process of forming a mathematical model and registration for analysis and synthesis is called mathematical models.

When prompting the mathematical model of the system, it is necessary to update the nutrition about the problem. The turn of the model is regulated, in the main, by the vibration cordon "system S- the middle E". The problem of simplifying the model is also guilty, as it is to help the vision in the fallowness of the model of the main power of the system, having seen other rows.

In the transition from the zesty to the formal, I will describe the process of the functioning of the system and from the urahuvannya in the inflow of the new middle ground. mathematical scheme like a lanka in a lance "a descriptive model - a mathematical scheme - a mathematical (analytical or (i) imitation) model".

The model of the object is formal. The model of the object (system S) It is possible to imagine in the view of the empty quantities, how to describe the process of functioning of the real system:

Sukupn_st of input injections to the system

x i = X,i =;

Sukupn_st of inflows of the newest middle

v j = V, j= ;

Sukupn_st internal (power) parameters of systems

h k = H, k =;

Sukupn_st of the output characteristics of the system

y j = Y, j =.

Have a fancy vipadku x i, v j, h k, y jє The elements of non-overlapping subdivisions and the substitution of both deterministic and stochastic warehouses.

Into the flow, into the out-of-the-way middle Eі internal parameters of the system є square (exogenous( t) = (x 1 (t), x 2 (t), …, x nX(t)); (t) = (v 1 (t), v 2 (t), …, v nV(t)); (t) = (h 1 (t), h 2 (t), …, h NН(t)), And the output characteristics є fallow (endogenous) May be shown in vector form: ( t) = (at 1 (t), at 2 (t), …, at nY(t)). You can see kerovans and non-roughened wines.

System function process S described in hours by the operator F S, Which transforms exogenous changes in endogenous types

(t) = F S(,,, t). (2.1)

Abundance of deposits of the characteristic characteristics of the system per hour y j(t) For all types j = be called outbound traktorієyu (t). The fallowness (2.1) is called the law of function of the system F S, Yakiy to be asked in the view of the function, functional, logical minds, in algorithmic, tabular forms, or at the view of the verbal rule of view. Functional algorithm A S called the method of rejecting the input characteristics from the input fuel injection ( t), Injection of the last middle ground ( t) I power parameters of the system and ( t). One and the same law of function F S systems S can be implemented in different ways, Tobto for the extra powerless algorithms in the function A S.

Mathematical models are called dynamic(2.1), how to describe the behavior of the object (system) model in hours t, To represent the dynamic power.

for static models, a mathematical model is a representation of two different powers of a modeled object Yі ( X, V, H) v singing moment, Шо in vector form can be written as yak

= f(, , ). (2.2)

Relationship (2.1) and (2.2) can be given in different ways: analytically (using additional formulas), graphically, tabularly, etc. The process of communication can be rejected through the power of the system S at a specific moment of the hour, called camps. Stan Systems S characterized by vectors

" = (z " 1, z " 2, ..., z "k) і "" = (z "" 1 ,z "" 2 , ..., z "" k),

de z " 1 = z 1 (t "), z " 2 = z 2 (t "), …, z "k= z k(t ") at the moment t "Î ( t 0 , T); z "" 1 = z 1 (t ""), z "" 2 = z 2 (t ""), …, z "" k = z k(t "") at the moment t ""Î ( t 0 , T) etc. k =.

How to look at the process of functioning of the system S as the last change of staniv z 1 (t), z 2 (t), …, z k(t), Then the stench can be interpreted as the coordinates of a point in k-worldly phase space... Moreover, the dermal realization of the process leads to the appearance of the deyak phase trajectory. The number of possible meanings of stanіv () is called spaciousness of stan_v ob'єkta model Z, moreover
z kÎ Z.

I will become a system S at the moment of the hour t 0 < t * £ T on the cob start 0 = ( z 0 1 , z 0 2 , …, z 0 k) [De z 0 1 = z 1 (t 0),
z 0 2 = z 2 (t 0), …, z 0 k = z k(t 0)], input injections ( t), Internal parameters ( t) І injections of the zestful middle ( t), Yakі mali mіsce in the intervening hour t * – t 0, with the addition of two vector lines

(t) = Ф (0 ,,,, t); (2.3)

(t) = F (, t). (2.4)

Perse ryvnyannya on the cob mill 0 і exogenous wines, viznacha vector-function ( t). t) - endogenous changes on system outputs ( t). In such a rank, lantsyuzhok rivnyan ob'єkta "enter - stanu - vihid"

(t) = F [Ф (0 ,,,, t)]. (2.5)

Have an hour in the model system S You can look at the model interval (0, T) Yak without interruption, so і discrete, so that quantized on the basis of D t time clock odnitsya kozhen, if T = m D t, de m = - number of sampling intervals.

In such a rank, pid mathematical model ob'єkta (real system) intelligence kintsev pіdnіzhina zminnykh (( t), (t), (t)) Along with the mathematical links between them and the characteristics ( t).

As well as a mathematical description of the object of the model, do not take revenge on the elements of the badness, for the stench does not get lost, so that it is possible to take into account, in the whole range of stochastic influx of the great middle class ( t) I stochastic internal parameters ( t) Відсутні, then the model is called deterministic in that sense, the characteristics are unambiguously determined by the determined input injections

(t) = f(, t). (2.6)

Obviously, the model is determined є we will limit it to a stochastic model.

Typov_ mathematical schemes. Practical modeling of objects in the field of system technology and system analysis on cob stages of advanced systems and rationalization types of mathematical schemes: Differential Equivalents, Kintsev and Imovirnіsnі automatics, mass service systems, Peter's hedge, aggregate system, etc.

Typical mathematical schemes can be overwhelmed by simplicity and precision. As deterministic models, if, when they are up to date, officials do not work, for the presentation of systems that function in an uninterrupted hour, they can be differentiated, integral, integrative for differential systems, ... Yak stochastic models (with an array of vypadkovyh factors) for the representation of systems with a discrete hour vikoristovuyutsya imovirnіsnі automatic, and for the representation of systems with an uninterrupted hour - a system of mass service. For the analysis of causal-wearable links in folding systems, one hour at a time, parallel to the chain of processes, the stagnation of Peter's hedge. To describe the behavior of non-intermittent and discrete, deterministic and stochastic systems (for example, ASOIU), it is possible to set the definitions (universal) on the basis of the aggregate system. With an aggregate inventory, the folding object (system) is divided into the end number of parts (subsystems), which are secured at the same connection, so that the interconnection of the parts is avoided.

In such a rank, with the induction of mathematical models of the processes of the functioning of systems, it is possible to see the onset of the main approach: uninterrupted determinations ( D-schemi); discrete determinations ( F-schemi); discrete-stochastic ( R-schemi); intermittently stochastic ( Q-schemi); hedgehog ( N-schemi); uzagalnye abo unversal ( a-schemi).

2.2. Permanently determined models ( D-schemi)

Main sports... The special features of the uninterruptedly deterministic approach to the application of the victorian in the quality of mathematical models of differential equations are clearly visible. differential They are called such rivnyannya, in which it is unrecognizable that there will be functions of one or more decilkoh minions, and moreover, in the rivnyannya it is not only the functions, but rather the older ones. If the functions of the minions are unavailable, then they are called Rivnyannyi private old, When looking at the functions of one independent winter village, they are called very special differential equals.

Mathematical analysis for deterministic systems (2.6) in the external viewer

" (t) = (, t); (t 0) = 0 , (2.7)

de " = d/dt, = (y 1 , y 2 , …, y n) І = ( f 1 , f 2 , …, f n) – n-world vectors; (, t) is a vector-function, which is assigned to the deyakom ( n+1) -worldly (, t) Even more richly and without interruption.

Mathematical schemes of this kind are called D-circuits(Eng. Dynamic), the stench imitate the dynamics of a pre-juvenile system, and in the capacity of an independent change, in which it’s not possible to lie in a shukan function, let us serve for an hour t.

The simplest vipad has a very different differential of the maj viglyad:

y "(t) = f(y, t). (2.8)

The simplest but simplest butt of formalization to the process of functioning of two elementary schemes of the nature: mechanical S M (pendulum arm, fig. 2.1, a) І electrical S K (colival contour, Fig. 2.1, b).

Small. 2.1. elementary systems

The process of small pendulum swinging is described by very different differential equations

m M l M 2 ( d 2 F(t)/ dt 2) + m M gl M F(t) = 0,

de m M, l M - mass and dovzhina to the swing of the pendulum; g- accelerated vіlnogo padіnnya; F(t) - kut lifting the pendulum at the moment of the hour t.

The third level of the pendulum swing can be used to estimate the characteristics. For example, pendulum swing period

T M = 2p.

Similarly, the processes in the electric colival circuit are described by very different differential equations

L K ( d 2 q(t)/dt 2) + (q(t)/C K) = 0,

de L K, C K - capacitor inductance and capacity; q(t) - the charge of the capacitor at the time of the hour t.

It is possible to evaluate the evaluation of the characteristics of the process in the colival contour. For example, period of electric cables

T M = 2p.

Obviously, having entered the meaning h 2 = m M l M 2 = L K, h 1 = 0,
h 0 = m M gl M = 1 / C K, F(t) = q(t) = z(t), Otrimaєmo zvychayne differential equal to another order, which describes the behavior of a closed system:

h 2 (d 2 z(t)/dt 2) + h 1 (dz(t)/dt) + h 0 z(t) = 0, (2.9)

de h 0 , h 1 , h 2 - system parameters; z(t) - mill system at the moment
hour t.

Thus, the behavior of cich two objects can be interpreted on the basis of the foreign mathematical model (2.9). In addition, it is essential that the behavior of the pendulum (system S M) can be used behind the addition of an electric colonic contour (system S TO).

Yaksho doslijuvana system S(Pendulum or contour) together E That is the inlet inflow x(t) (The value of the force for the pendulum and the energy for the contour), and the model of such a system will be determined without interruption:

h 2 (d 2 z(t)/dt 2) + h 1 (dz(t)/dt) + h 0 z(t) = x(t). (2.10)

From the point of view of the intrinsic mathematical model (Div. P. 2.1) x(t) Є inlet (keruyuchim) injected, and the mill system S in this view, you can see how the characteristic is displayed, so that the selected change is set to the system at the given moment of the hour y = z.

you can add D-schemi... For describe line systems management, like a dynamic system, heterogeneous differential competitiveness may be consistent performance

de ,, ..., - the function of the hour is unacceptable; and - see functions.

Vikoristovuchi, for example, the VisSim software package, the features for the simulated model of processes in control systems, which can be described by differential differential equation

deyaka shukana function for an hour at a time with zero cob wins, it is acceptable h 3 =1, h 2 =3, h 1 =1, h 0 =3:

Having presented the task of the rivnyannya of the old

as it is possible to promodeluvati, vikoristovuchi set of standard blocks of the VisSim package: arithmetic blocks - Gain (multiply by a constant), Summing-Junction (sumator); blocks of integration - Integrator (numerical integration), Transfer Function (establishment of an equalization represented by the transfer function); signal input blocks - Const (constant), Step (a single function of the "downwind" viewer), Ramp (linearly increasing signal); blocks-receiving signals - Plot (displaying signals in the time zone, which can be analyzed beforehand in the course of the model).

In fig. 2.2 is a graphical representation of this differential equation. The input of the extreme left integrator is displayed, the input of the middle integrator is, and the input of the extreme right integrator is. View of the extreme right integrator of the change y.

Let's limit ourselves with a vipad of dynamical systems. D-scheme, є automatic control systems(SPG)and regulation(SAR). The real object is presented in the view of two systems: keruyuchoi and kerovanoi (control object). The structure of the bagatovimirnaya system and automatic control of the zagalny viglyad is shown in Fig. 2.3, definition endogenous zminni: ( t) - vector of input (set) injections; ( t) - vector of stormy injections; " (t) is the vector of signals of the grave; "" (t) is the vector of hard injections; exogenous zminni: ( t) is the vector of the system S; (t) is a vector of out-of-order changes, t) = (t).

Small. 2.2. Graphic presentation

Keruyucha system - the price of software and technical support, which can be used to secure the accessibility of the control system of the song. Naskіlki exactly about the reach of a given location, you can judge (for the same system) by the coordinates of the country y(t). Riznytsya mіzh set y ass ( t) І work y(t) The law of change of the kerovan value є the control " (t) = y ass ( t) – y(t). Similarly, the proponation of the law of change of the kerovan magnitude is consistent with the law of change of input (set) injected, tobto x(t) = y ass ( t), That " (t) = x(t) – y(t).

Systems for those who control " (t) = 0 at the moment of the hour, called ideal... In practice, the implementation of ideal systems is unwise. To the head of the automatic control system y(t) Subject to the specified by the law s singing accuracy (with a permissible mercy). The parameters of the system and the guilt to ensure the need for accuracy of control, as well as the strength of the system in the transition process. If the system is stable, then the behavior of the system is analyzed in the hour, the maximum display of the regulated change y(t) In the transition process, the hour of the transition process, etc. The order of differential comparison and the value of its performance is based on static and dynamic parameters of the system.

Small. 2.3. The structure of the automatic control system:

US - Keruyucha system; OU - ob'єkt management

In such a rank, vikorystannya D-scheme allowing formalization of the process of functioning without interruption of deterministic systems S and evaluation of the main characteristics, stagnant and analytical, or imitation, visualization, realizations of the visualization of the display for the model of uninterrupted systems, or of the common analogs and hybrid ones.

2.3. Discretely determined models ( F-schemi)

Main sports... The distinctive features of the discrete-deterministic approach to the application of the victorian in the capacity of the mathematical apparatus of the theory of automatics are discernible. The system is presented at the viewer of the automaton as an attachment with input and output signals; kintsevym machine An automaton is called, at which powerless internal stations, input and output signals є endsexual multipliers.

Abstract Finite automata can be defined as a mathematical scheme ( F-schema), It is characterized by a variety of elements: X input signals (input alphabet); kintsevim without Y outgoing signals (outgoing alphabet); kintsevim without Z internal stations (internal alphabet or alphabet of countries); cob pit z 0 , z 0 Î Z; transition function j ( z, x); function of inputs y ( z, x). Automatic machine F-scheme: F = á Z, X, Y, Y, j, z 0 ñ, the function in a discrete hour, moments of a certain tact, the skin of which is perceived as a permanent value of the input and output signals in the internal state. Apparently the camp, as well as the input and output signals, t-th clock at t= 0, 1, 2, ..., through z(t), x(t), y(t). With a tsom for the wash z(0) = z 0, and z(t)Î Z, x(t)Î X, y(t)Î Y.

Abstract Kintsevy automaton has one input and one output channel. Have a skinny moment t= 0, 1, 2, ... discrete hour F the machine is in the singing station z(t) S without Z automatic machine, and at the cob at the hour of the hour t= 0 win is always in the cob mill z(0) = z 0. At the moment t, Being in the camp z(t), Automatic input signal x(t)Î Xі see on the outgoing channel і signal y(t) = y [ z(t),x(t)], Go to camp z ( t+1) = j [ z(t), x(t)], z(t)Î Z, y(t)Î Y... Abstract kintseviy automaton realizes the deyake of imaging without words in the input alphabet X on the face of a wicked
alphabet Y... In other words, when at the entrance of the automatic machine installed in the cob z 0, send the letters of the input alphabet in actuality x(0), x(1), x(2), ..., that is the input word, then the input of the automaton will be the last time the letters of the input alphabet will appear y(0), y(1), y(2), ..., a fictitious word.

In such a rank, the robot of the automatic machine will follow this pattern: in the skin t-m clock at the input of the machine, which is located in the station z(t), A signal sounds x(t), I react to a wien with a transition ( t+1) th beat in new camp z(t+1) when you see a signal. The said vishche can be described by the advancing ryvnyans: for F machine gun of the first kind, also a yak automatic machine Mil,

z(t+1) = j [ z(t), x(t)], t= 0, 1, 2, …; (2.15)

y(t) = Y [ z(t), x(t)], t= 0, 1, 2, …; (2.16)

for F submachine gun

z(t+1) = j [ z(t), x(t)], t= 0, 1, 2, …; (2.17)

y(t) = Y [ z(t), x(t - 1)], t= 1, 2, 3,…. (2.18)

An automatic machine of a different kind, for what

y(t) = Y [ z(t)], t= 0, 1, 2, …, (2.19)

so that the function of the input is not stale from the input x(t), Call Moore's assault rifle.

In such a rank, equal to (2.15) - (2.19), we again set
F automaton, є we limit it with vyvnyan (2.3) and (2.4), if
system S- Determined and discrete signal to be sent X.

For a number of stanis to develop kintsev automatically from memory and without memory. Automata with memory may be more than one, and automatics without memory (combination abo logical diagrams) Lash out with one camp. At the same time, zgіdno s (2.16), the robot of the combinational circuit of the field is in the fact that it is possible to set the response to the skin input signal x(t) Singing out signal y(t), To implement a logical function in the form

y(t) = Y [ x(t)], t= 0, 1, 2, … .

The function is called Boolean, which is the alphabet Xі Y, Yakim follow the value of the signals xі y, Consists of two letters.

Due to the nature of the discrete hour, the endurance automatically moves to synchronous and asynchronous. Synchronous F the automatic moment the hour, in which the automatic machine "reads" the input signals, it starts with primus synchronized signals. When a clock synchronizing signal is sent from the signal to the "desired" one and as long as it is indicated before (2.15) - (2.19), it is sent to a new station and a signal is sent to the output, for which the automaton can send a signal as well. Thus, the reaction of the automaton on the skin value of the input signal will end in one cycle, the triviality of which will start with an interval between the suspended synchronous signals. asynchronous F the automaton reads the input signal without interruption; x, Wien may be, as it is accepted from (2.15) - (2.19), as the number of output signals is visible, as long as the number of output signals is not changed, as long as it cannot be changed by the given input signal.

you can add F-schemi. Shchob set kintseviy F automatic machine, it is necessary to describe all the elements without F= <Z, X, Y, Y, j, z 0>, so that the input, internal and external alphabets, as well as the functions of transitions and inputs, and the middle of the empty stations is necessary to see z 0, in which the machine is located in the t= 0 F automaton, ale naybilsh often vikoristoyutsya tabular, graphical and matrix.

In the tabular way, the tables of transitions and outputs are set, the rows of which correspond to the input signals of the machine, and the hundredths - the first station. The first wickedness of the stan z 0. On overretin i first row i k-th hundred of the tables of transitions to be accommodated according to the value j ( z k, x i) Functions of transitions, and in the table of inputs - by type of value y ( z k, x i) Functions of inputs. for F Moore's automaton of offense tables can be added separately.

Description of the robot F Automatic machine Mіlі with tables of transitions j і entries and іt is illustrated in the table. 2.1, but describe F Moore's automaton - the transition table (Table 2.2).

Table 2.1

X i	z k
z 0	z 1	…	z k
go over
x 1	j ( z 0 , x 1)	j ( z 1 , x 1)	…	j ( z k,x 1)
x 2	j ( z 0 , x 2)	j ( z 1 , x 2)	…	j ( z k,x 2)
…	…	…	…	…
x i	j ( z 0 , x i)	j ( z 1 , x i)	…	j ( z k,x i)
come in
x 1	y ( z 0 , x 1)	y ( z 1 , x 1)	…	y ( z k, x 1)
x 2	y ( z 0 , x 2)	y ( z 1 , x 2)	…	y ( z k, x 2)
…	…	…	…	…
x i	y ( z 0 , x i)	y ( z 1 , x i)	…	y ( z k, x i)

Table 2.2

x i	y ( z k)
y ( z 0)	y ( z 1)	…	y ( z k)
z 0	z 1	…	z k
x 1	j ( z 0 , x 1)	j ( z 1 , x 1)	…	j ( z k, x 1)
x 2	j ( z 0 , x 2)	j ( z 1 , x 2)	…	j ( z k, x 2)
…	…	…	…	…
x i	j ( z 0 , x i)	j ( z 1 , x i)	…	j ( z k, x i)

Apply the tabular way F automatic machine Mil F 1 hovering in table. 2.3, and for F Moore assault rifle F 2 - in table. 2.4.

Table 2.3

x i	z k
z 0	z 1	z 2
go over
x 1	z 2	z 0	z 0
x 2	z 0	z 2	z 1
come in
x 1	y 1	y 1	y 2
x 2	y 1	y 2	y 1

Table 2.4

	Y
x i	y 1	y 1	y 3	y 2	y 3
	z 0	z 1	z 2	z 3	z 4
x 1	z 1	z 4	z 4	z 2	z 2
x 2	z 3	z 1	z 1	z 0	z 0

With a graphical method of assigning a finite automaton, the witness of a straight graph is victorious. The graph of an automaton is a set of vertices that represent the different camps of the automaton and connect the vertices of the arcs of the graph, so that they represent the transitions of the automaton. As input signal x k wiklikak transition from becoming z i to the camp z j Then on the graph of the automaton there is an arc, z i with apex z j, mean x k... In order to set the function of the outputs, the graph arcs need to be related to the output signals. For automatіv Milі tsya, the layout is carried out as follows: when the input signal x k dіє to the camp z i, Then go out arc, like go out iz z i i marked x k; qiu arc pre-dodatkovo mean a signal y= Y ( z i, x k). For the Moore automaton, the graph is similar to the following: what is the input signal x k, Diyuchi to the deyakiy camp of the automaton, wiklikє transition to the camp z j, Then an arc, straightened in z i i will x k, Dodatkovo means wihіdnim
signal y= Y ( z j, x k).

In fig. 2.4. a, b hovering set earlier by tables F-automatic Milі F 1 i Mura F 2 for sure.

Small. 2.4. Graphs automatіv a - Mili i b - Mura

With a matrix setting of a finite automaton, the matrix of an automaton is square Z=||s ij||, rows appear outgoing countries, And a hundred percent - I’ll become a transition. element s ij = x k/y s, Stand on the peretin
i first row i j-th hundredth, in the case of the machine Mili responds to the input signal x k, Шо wiklikє transition from stanu z i to the camp z j, І output signal y s, Which is seen during the transition. For the Mili vending machine F 1, a peeped-out view, a matrix of z'єdnan maє viglyad:

x 2 /y 1 – x 1 /y 1

C 1 = x 1 /y 1 – x 2 /y 2 .

x 1 /y 2 x 2 /y 1 –

I will move from start to z i to the camp z j to be displayed on the basis of decile signals, an element of the matrix c ij is a helpless pair of "enter-vihid" for the transition, marked by the sign of disjunction.

for F Mura Element submachine gun s ij doorways with no input signals at the transitions ( z i, z j), And the output is described by the vector of outputs

= y ( z k) ,

i-th component of which is the output signal, which is z i.

For a peeped-out vische F Moore assault rifle F2 matrices z'єdnan і vector of vyhodіv mayut viglyad:

–x 1 –x 2 –at 1

–x 2 – –x 1 at 1

C 2 = –x 2 – –x 1 ; = y 3

x 2 –x 1 – –at 2

x 2 –x 1 – –at 3

For deterministic automatons, they must be aware of the unambiguity of transitions: an automaton that is located in the deyakom station, before any input signal cannot go more than one country. One hundred percent to the graphical way of creating F the automaton means that in the graph of the automaton from any vertices two or more edges cannot come out, seen by one and the same input signal. And in the matrix of the automatic machine Z in the skin row, whether the input signal is not guilty of being generated more than once.

for F machine gun z k be called stiykim, how to come in x i ÎX, For which j ( z k, x i) = Z k, maє mіsce j ( z k,x i) = Y k. F automaton name asynchronous, yaksho kozhne yogo stan z k ÎZ stiyko.

In such a rank, the discernment in a discretely determined way to the end of the life on the models of the authorities of objects is a mathematical abstraction, manual for describing a wide class of processes in the function of real objects in automated systems management. For help F- The automaton can be described as an object, which is characterized by the presence of discrete machines, and the discrete nature of the robot at the time of the elements and universities of the EOM, the attachment of control, regulation and control, clock systems and spacious communication in technology, etc.

2.4. Discrete-stochastic models ( R-schemi)

Main sports... The peculiarities of the mathematical schemes are easily discernible for the discrete-stochastic approach on the imovirnіsnyh (stochastic) automata. At the out-of-the-box viglyad imovirnіsny machine
R-diagrams(Eng. Probabijistic automat) it is possible to make a discrete, discrete reworking of information from memory, the function of which, in the skin tact, lies only as a memory in a new one, and it can be described statistically.

Introduced mathematical understanding R automaton, vikoristovuyuchi priest, introduced for F machine. Clearly bezl_ch G, The elements of what є all the bets ( x i, z s), De x iі z s- elements of the input submenu X and the number of stanovas Z is different. You can see two of these functions j and y G®Z i G®Y, then it seems like F = X, Y, j, y> is the initial automaton of the deterministic type.

I will more easily understand the mathematical scheme. hey
F - bezlich all kind of couples ( z k, y i), De i- element of the outrageous pedigree Y... Vimagatimemo, be-like an element without G Inducing a lawless law to such a view:

With a tsom b kj= 1, de b kj- the transition of the machine to the mill z kі appear on the go signal y j, Yaksho vіn buv in stanі z sі at the first entry at the time of the hour, the signal is needed x i... The number of such razpodіlіv, presented in the view of the tables, according to the number of elements without lіchі G... Apparently without the help of the table through V. Todi four elements P = to be called an automatic machine
(R machine).

you can add P-schemi. Leave me an element bezlichі G Induce actions of laws on the basis of subsets Yі Z, Which can be seen from the viewer:

With a tsom z k = 1 i q j = 1, de z kі q j - transition
R machine gun z k if there is a signal y k for wash, scho
R z s and at the first input, there is an input signal x i.

Yaksho for all kі j maє mіsce spіvvіdnoshennya q j z k = b kj, then such
R automaton name іmovіrnіsnym automatic machine Milі... Qia vimoga means a victory over independence for a new country R automaton of the first output signal.

Let the signal be detected now R- I’ll stay away from the machine, in which the machine is located at the given time of the robot. In other words, do not care for the skin element of the Yіnducu razpodіl ymіvіrnosti vyhodіv, so mayut such viglyad:

here s i = 1, de s i- the appearance of the incoming signal y i at at Words scho R automatic machine known in stanі z k.

Yaksho for all kі i maє mіsce spіvvіdnoshennya z k s i =b ki, Then such
R automaton name Moore's smart assault rifle. witness
R the Mili and Moore automatic machine was introduced for the analogue of determinations
F machine. let's round off vipadkom R- machine, how to ask a yak P=X, Y, B>, Є automatically, for those who either move to a new country, or an out-of-order signal begins to be determined. If there is an outgoing signal
R automaton starts deterministically, then such an automaton will be called
Y-... similarly,
Z-Determine with an automatic machine be called R automatic machine, at which new vibir I will become є determіnovanim.

Application 2.1. let the assignments Y-determinated P machine

In fig. 2.5 indications of the arrangement of the transition graph of the automaton. The vertices of the graph are positioned by the automaton station, and the arcs - by possible transitions from one camp to the other. The arcs are on the way, depending on the possibilities of the transition p ij And close to the vertices of the graph, the values ​​of the outgoing signals are written. It is necessary to assess the total financial performance P machine gun z 2 i z 3 .

Small. 2.5. Graph of an imovirnіsny automaton

In case of an analytical approach, it is possible to record the type of performance from the theory of Markov's lancers and revise the system of equivalents for the value of financial information. With a cob mill z 0 it is possible not to vrahovuvati, so as a cobbled rose does not pour on the value of the final values. todi maєmo

de s k- the final transfer rate R automatic machine z k.

Otrimuєmo system of rіvnyany

Dodamo do tsikh rivnyan umova normuvannya s 1 + s 2 + s 3 + s 4 = 1. Todi, virishuchi system of rivnyans, otrimaєmo s 1 = 5/23, s 2 = 8/23, s 3 = 5/23,
s 4 = 5/23. In such a rank, s 2 + s 3 = 13/23 = 0.5652. In other words, if there is no end-of-line robotics, Y-determinated
R the automatic machine on the yogo vyhodі is formed dvіykova last from ymіrnіstu appear odinitsі, equal 0.5652.

more R-automatically can be victorious as a generator of Markov messages, which are necessary when prompting and realizing processes in the function of systems S for the infusion of the outgoing middle E.

2.5. Interruption-free stochastic models ( Q-schemi)

Main sports... The peculiarities of the non-intermittently stochastic approach can be seen on the application of typical mathematical Q- schemes - mass service systems(English Queueing system).

Yak Process obslugovuvannya mozhut Buti predstavlenі rіznі on svoїy fіzichnіy prirodі Process funktsіonuvannya ekonomіchnih, virobnichih, tehnіchnih that іnshih systems napriklad: supply flows produktsії deyakogo pіdpriєmstvu, flows of parts i komplektuyuchih virobіv on skladalnomu konveєrі shop, application obrobku Informácie AMR od vіddalenih termіnalіv i t .d. At the same time, such objects are typical for robots - the appearance of applications (vimog) for service and completion of service at a certain moment of the hour, so that the stochastic nature of the process of these functions.

stream of pod_y to be called the last day, which is displayed one after the other in the past hour. Razrіznyayut streams of one-sided and non-uniform podіy. potik go be called one-day, as it is, it is characterized only by the moments of hopefulness of the cih podiy (how to wick the moments) and to ask the last ( t n} = {0 £ t£ 1 t 2 ... £ t n£ … }, de t n - moment nastaniya P- th podії is not a number for me. One-time course of the journey can also be done by looking at the last prominence in an hour. P- m і (n - 1) th podіami (t n), Yaka is unambiguously tied to the last moment of the moment ( t n} , de t n = t n– t n -1 ,P³ 1, t 0 = 0, tobto t 1 = t 1 . The flow of heterogeneous podiy be called after-life ( t n, f n} , de t n - wick moments; f n - set of signs of the podії. For example, one hundred percent before the process of servicing for a heterogeneous flow of requests, it is possible to set the affiliation to that number of requests, the appearance of the priority, the possibility of servicing this type of channel.

In any elementary service act, you can see two main warehouses: cleaning service by request and not service request. The price can be seen in the eyes of the deyakogo i I will come service P i(Fig. 2.6), where to store from the accumulation of orders H i, in which one can overnight j i= applications, de L i H– mnist
i-go accumulative, and the channel of service applications (or just the channel) K i. I will add service staff to the leather element P i come up streams of go: in nakuchuvach H i number of applications w i, per channel K i - potik service providers i i.

Small. 2.6. Attachment of service requests

Applications served by the channel K i,і applications that left the attachment P i for other reasons, unserved (for example, through re-stocking accumulated H i), Confirm the outgoing flow y i Î Y, to intervene an hour between the moments of the entry of applications, approve a subset of the entries.

Call, order of applications w i ÎW, to intervene an hour between the moments of applications for entry K i, I approve a lot of non-rooted winter, and because of the service u i ОU, to intervene an hour between the ear and the end of the service request, approve the number of kerovan changes.

Service process P i you can see how the process of changing the stanza of the elements at the same time z i(t). Moving to a new camp for P i means a change in the number of applications that are in the new (in the channel K i i in nakuchuvachi H i). In such a rank, the vector of stanіv for P i maє viglyad: , de z i H- stan nakuchuvach H i (z i H= 0 - accumulated empty, z i H= 1 - in accumulative є one request, ..., z i H = L i H – accumulate knowledge); L i H - mn_stopichuvach H i, how to be influenced by the number of applications, which in the new can be supported; z i k - mill channel K i(z i k = 0 – channel vilny, z i k= 1 - occupation channel).

you can add Q- schemes. Practical model systems that can use more folding structural links and behavioral algorithms, for formalization of the vicarious process, it is not necessary to add service personnel, but
Q- schematics , approved the composition of the bagatio of the elementary attachments of the service P i. yaksho canals Before i Once the service providers are installed in parallel, then there is a lot of channel service ( bagatokanal Q- scheme) , and yaksho put it on P iі іх parallel compositions are given last, then there is a lot of phase of service ( bogatophase Q- scheme) . In such a rank, for zavdannya Q- circuits are required vikoristovuvati operator of the receipt R, Which visualizes the interconnection of the elements of the structure (channels and accumulators) between themselves.

Outward information when prompting mathematical models of processes, functions of systems є data on the recognition and minds of robots and pre-developed (projected) systems, which start the basic meta of the model and allow formulating viable models to mathematically ... mathematical scheme it is possible to value the language in the transition from the zealous to the formal, I will describe the process of functioning of the system and the urahuvannya in the inflow of the new middle, so that it’s possible to “describe a model - a mathematical scheme - a mathematical [analitical, but] model.”

Model of the object of the model, i.e. the system S, it is possible to imagine in a vigorous way the functionless quantities, how to describe the process of functioning of the real system and make it look like the next step:

Sukupn_st inlet injection per system - x i;

Sukupn_st infusion of the newest middle– n l;

Sukupn_st internal (power) parameters systems - h k;

Sukupn_st output characteristics systems - y j.

At the same time, in the re-insurance subsets, you can see kerovans and non-corroded changes. Have a fancy vipadku x i, n l, h k, y jє elements of non-overlapping subdivisions X, V, H, Y and place both determinations and stochastic warehouses.

When modeling systems S in flowing in, in flowing in the outgoing middle Eі internal parameters of the system є square (exogenous) wintry, how the vector form may appear in the same view

and the specific characteristics of the system є fallow (endogenous) winter i in vector form

System function process S described in hours by the operator Fs , Which, in a zealous vipad, transforms exogenous changes in endogenous forms, according to the type:

. (2.1)

Abundance of deposits of the characteristic characteristics of the system per hour y j(t) For all types, be called vyhіdniy traєktorієyu. The fallowness (2.1) is called the law of function of the system S i know F s. The law of function of the system F s There can be tasks in the form of functions, functions, logical minds, in algorithmic and tabular forms, or in the form of a verbal rule of judgment.

Even more important for describing and maintaining systems Sє understanding function algorithm А s The method of rejecting the input characteristics from the output of the input injections should be considered. , infusion of the newest middle and power parameters of the system . Obviously, one and the same law of functioning of the system can be implemented in different ways, so that for additional useless algorithms And s.

Relationship (2.1) є mathematical description of the behavior of the object (system) of the model in hours , to represent this dynamic power. Mathematical models of this kind are accepted by dynamic models(By systems).

For static models, the mathematical description (2.1) is a representation of two different powers of the modeled object Yі [ X, V, H], in vector form it can be written as yak

. (2.2)

Relationship (2.1) and (2.2) can be given in different ways: analytically (for additional formulas), graphically, tabularly, etc. S at a specific moment of the hour, camps. Stan Systems S characterized by vectors

і ,

de z ’ 1 = z 1 (t ’),z ’ 2 = z 2 (t ’), …, z'K = z k ( t'), at the moment t ’’ Î( t 0 , T); z ’’ 1 =z 1 (t ’’), z ’’ 2 =z 2 (t ’’), …, z'' K = z k ( t'') u moment t ’’ Î( t 0 , T) і etc.,.

How to look at the process of functioning of the system S as the last change of staniv z 1 (t), z 2 (t), ..., z k ( t), Then the stench can be interpreted as the coordinates of a point in k vimir phase space, moreover, the dermal realization of the process of appearance of the phase trajectory. The number of the most powerful meanings of the spaciousness of stan_v ob'єkta model Z, moreover z k Î Z.

I will become a system S at the moment of the hour t 0<t *£ T I will start to be cob wits [de z 0 1 = z 1 (t 0), z 0 2 =z 2 (t 0), ..., z 0 k = z k ( t 0)], input inlets, internal parameters and injections of the most recent middle level, such as mali in an hour t *– t 0, for the help of two vector ryvnyans:

; (2.3)

. (2.4)

Perse ryvnyannya on the cob mill and exogenous winter viznachaє vector-function , and for the other, for the otrimanim values ​​of the stan_v - endogenous changes on the system inputs . In such a rank, the lantsyuzhok of the rivnyan ob'єkta "enter - stan - vihid" allows for the value of the characteristics of the system:

Have an hour in the model system S You can look at the model interval (0, T) Yak without interruption, so it is discrete, so that it is quantized for the duration of the clock, if the number of sampling intervals.

In such a rank, pid mathematical model of the object(Real systems) the minds of the children of the winter together with mathematical links between them and characteristics.

As well as a mathematical description of the object of the model, not to take revenge on the elements of the drop, for the stench does not get lost, as it is possible to take into account, but in the whole range of stochastic inflow of the most important middle-class and stochastic settings of the internal appearance are called deterministic in that sense, the characteristics are unambiguously determined by the determined input injections

. (2.6)

Obviously, the model is determined є we will limit it to a stochastic model.

The induction of mathematical communication is a mathematical scheme of the outward view and gives the possibility of enhancing a wide class of systems. However, at the practical modeling of objects in the field of system technology and system analysis on cob stages, the system and rationalization types of mathematical schemes: Differential Equivalents, Kintsev and Imovirnіsnі Automatics, Mass Service Systems, Heirlooms, etc.

Not volodyuchi by such a level of spirituality, as models, typical mathematical schemes, can pass the simplicity and precision, albeit with the purely ringing of the possibilities of stagnation. As deterministic models, if, when they are up to date, the officials do not support, for the presentation of systems that function in an uninterrupted hour, they will be able to differentiate, integrate, integrate, partly design diagrams. A kind of stochastic models (with a cloud of different factors) for the presentation of systems with a discrete hour is automatic, and for the presentation of a system with an uninterrupted hour - a system of mass service, etc.

Pererakhovani types of mathematical schemes, naturally, can not claim to be able to describe on the basis of all the processes that occur in the great informational systems. For such systems, in a number of types, it is more promising є the consolidation of aggregate models. The aggregate model (system) gives the possibility of enhancing a wide range of objects to the extent that the systemic nature of these objects is displayed. Itself with the aggregate inventory, the folding object (system) is divided into the end number of parts (subsystems), which is secured at the same connection, so that it is not possible to interconnect the parts.

Thus, with the prompting of mathematical models of the processes of the functioning of systems, it is possible to see the onset of the main steps: without interruption, determinations (for example, differential equations); discrete determinations (automatic control); discrete-stochastic (automatic); non-interruption-stochastic (system and mass service); uzagalny, abo unversal (aggregate system).

Lecture 5.

Non-interruption-free determination of models (D-schemes)

The special features of the uninterruptedly deterministic approach to the application of the victorian in the quality of mathematical models of differential equations are clearly visible. differential They call themselves such rivnyannya, in which they will be invisible functions of the same or some of the minions, moreover, the rivnyannya does not include only functions, but rather old ones. If unavoidable are the functions of the minions, then they are called rivnyannymi in private old, in the first place when looking at the functions of only one independent winter village, they are called very special differential equals(ODE) .

Call in such mathematical models in the capacity of an independent winter, in which it is not possible to lie down in an unknown function, to serve an hour t. Todi mathematical analysis for deterministic systems (2.6)

de і - n-world vectors; - vector-function, which is assigned to deyakom ( n + 1) -worldly arched richly and without interruption. So, both mathematical schemes of this kind represent the dynamics of a pre-juvenile system, that is, behavior in the hour, then the stench is called D-circuits(View English Dynamic).

The simplest type of ODE has a viglyad:

,

de h 0 , h 1 , h 2 - system parameters; z(t) – mill system at the moment of the hour t.

Yakshho doslіdzhuvana system of mutual modulation from zovnіshnіm middle E , then there is an inlet injection X(t) І uninterruptedly-determined model of such a system will be mati viglyad:

.

From the point of view of the original scheme of the mathematical model X(t) Є inlet (keruyuchim) injected, and the mill system S in this view, you can see how the characteristic is shown, to think about how the change is taken from the system at the moment of the hour y = z.

When the establishment of systems engineering is revived, it is more important to solve the problems of managing great systems. Slid respect for the automatic control systems - an example of the type of dynamic systems, which is described D- schemes і vіdіlenikh in the okremiy class of models by virtue of їх practical specifics. Describe the processes of automatic control, you can see the appearance of a real object in two systems: keruyuchoy and kerovano (control object).

. Lecture 6.

Discretely determined models (F-circuits)

The peculiarities of the discretely determined approach to the stage of formalization of the process of functioning of the systems can be seen on the application as a mathematical apparatus of the theory of automatons. The theory of automatics is a chain of theoretical cybernetics, in which mathematical models are used - automatics. On the basis of the central theory, the system is presented at the sight of an automaton, over-sampled with discrete information and changing its internal will become deprived at an admissible moment of the hour. The understanding of "automaton" is varied in the nature of the specifically pre-sensed systems, in the form of the accepted standard of abstraction and pre-social world of spirituality. The automatic machine can be detected as an attachment (black box), on the yak input signals are given and the signals are known as well as the mother of the internal ones. An automaton is called an automatic machine, which has no internal stations but, also, no signal signals are equipped with end units. Abstract Finite automat can be defined as a mathematical scheme, which is characterized by a lot of elements: X input signals (input alphabet); kintsevim without Y outgoing signals (outgoing alphabet); kintsevim without Z internal stations (internal alphabet or alphabet of countries); cob pit z 0 Î Z; transition function j(z, x); function of connections y(z, x).

Automatic machine F-scheme: - function in a discrete automatic hour, moments of a certain tact, so that one to one interval of an hour is added, the skin of which displays the permanent value of the input and output signals to the internal state. How to indicate the camp, as well as the input and output signals, t- to the clock at t= 0, 1, 2, ..., through z(t),x(t),y(t). z(0)=z 0 , z(t)Î Z, x(t)Î X, y(t)Î Y. Abstract Kintsevy automaton has one input and one output channel. Do kozhin moment discrete hour F the machine is in the singing station z(t) S without Z automatic machine, and at the cob at the hour of the hour t= 0 win is always in the cob mill z(0)=z 0. At the moment t, being in the camp z(t), automatic reception on the input channel signal x(t)Î Xі see on the outgoing channel і signal at(t)=y[z(t), X(t)], Go to camp z(t+1)= j[z (t), x (t)], x(t)Î X, y(t)Î Y. Abstract kintseviy automaton realizes the deyake of imaging without words in the input alphabet X on the face of the wicked alphabet Y... In other words, when at the entrance of the automatic machine installed in the cob z 0, send the letters of the input alphabet in actuality X(0),X(1),X(2), ..., i.e. input word, then on the input of the machine there will be letters of the input alphabet at(0), y(1), at(2), ..., a fictitious word. In such a rank, the robot of the automatic machine will follow this pattern: in the skin t- m tact at the input of the machine, which is located in the station z(t), A signal sounds x(t), to which wine I react by switching to ( t+1) -m cycle in noviy stan z(t+1) when you see a signal.

For a number of stanis to develop kintsev automatically from memory and without memory. Automata from memory may become more than one, and automata without memory (combined or logical schemes) will lose one camp. Due to the nature of the discrete hour, the endurance automatically moves to synchronous and asynchronous. Synchronous F the automatic moment the hour, in which the automatic machine "reads" the input signals, it starts with primus synchronized signals. asynchronous F- the automaton reads the input signal without interruption, and therefore reacts to the delivery of a new input signal of a constant value X, There can be no change in the number of outgoing signals, as long as you do not go into the line, as it cannot be changed by the given input signal.

Discrete-stochastic models (P-schemes)

The peculiarities of the mathematical schemes are easily discernible in the discrete-stochastic approach to the formalization of the process of functioning of the preliminarily juvenile system. So, as the day of sampling for an hour at any given time becomes similar to those seen in the automatic machines, then the injection of the factor of stochasticity is simplicity itself on the types of such automatic machines, and on the very same (stochastic) machines.

In a probabilistic automat, it is possible to create a discrete and tactical re-development of information from memory, which functions in the skin tact to lie just as described in the memory in a new and statistical way. The stagnation of the schemes of automatic automation is of great importance for the development of methods for the design of discrete systems; Satisfied with the assignment.

Introduced mathematical understanding R- machine gun , vikoristovuyuchi witness, introduced for F machine . Clearly bezl_ch G, The elements of what є all the bets ( x i, z s), De x i,і z s- elements of the input submenu Xі pіdnіnі stanіv Z for sure. There are two such functions jі y, then you can see the image G® Zі G® Y, then it seems like The automatic machine is of a deterministic type. Introduce a larger mathematical scheme. hey F- without any kind of couples ( z k, y i) de at j- element of the outrageous pedigree Y... Vimagatimemo, be-like an element without G indukuvav on no F This is the law of such a vigil:

Elementi s F … (z 1 , y 1) … (z 1 , y 2) … … (z K, y J -1) (z K, y J)

(x i z k) … b 11 b 12 … b K (J -1 ) B KJ

With a tsom,

de b kj- the transition of the machine to the mill z kі appear on the go signal at j, yaksho vin buv in stanі z sі at the first entry at the time of the hour, the signal is needed x i... The number of such razpodіlіv, presented in the view of the tables, according to the number of elements without lіchі G. Meaninglessly without a table through V, Todi chetvirka elements be called an imovirnіsny automaton ( R machine) .

Lecture 7.

Interruption-free stochastic models (Q-schemes)

The peculiarities of the intermittently stochastic approach can be seen on the application of the queue in the capacity of typical mathematical schemes of the queueing systems, which will be called Q-scheme . Mass service systems are a class of mathematical schemes that have been broken up in the theory of mass service and various supplements for the formalization of processes of functioning of systems, like for their own sutta, processes of service.

As the process of servicing, you can be presented with the development of their physical nature, the process of functioning of economical, virobnichi, technical and other systems, for example, applications for processing EOM information from the past, etc. At the same time, such objects are typical for robots - the appearance of applications (vimog) for service and completion of service at a certain moment of the hour, so that the stochastic nature of the process of these functions. In any elementary service act, you can see two main warehouses: cleaning service by request and not service request. The price can be seen in the eyes of the deyakogo i I will come service P i, Warehouse for accumulating orders H i, In which case, you can buy applications overnight, de L i H - mnist i-th accumulator, і channel of service applications (or just a channel) Before i. I will add service staff to the leather element P i come up streams of go: in nakuchuvach H i - number of applications w i per channel Before i - the reason for the service u i.

Practical model systems that can use more folding structural links and behavioral algorithms, for formalization of the vicarious process, it is not necessary to add service personnel, but Q- diagrams, approved by the composition of the bagatokh of the elementary attachments of the service P i(Merezhi mass service). yaksho canals K i Once the service providers have been installed in parallel, then there is no more channel service Q-scheme) , and yaksho put it on P iі іх parallel compositions are given last, then there is a lot of phase service Q- scheme). In such a rank, for zavdannya Q-schemy is necessary to vikoristovuvati the operator of the receipt R, it shows the interconnection of the elements of the structure (channels and accumulators) between themselves. Open and close Q-schemi . At rosіmknutoї Q-scheme of the outgoing flow of servicing requests, I can not enter into a certain element, i.e. Q- schemes є zvorotny zyvyaz, for yakim applications collapse in a straight line, protylezhny rukh vhіd-vikhіd.

Possibilities for assessing the characteristics of the analysis of analytical models of the theory of mass service є to complete the assessment of the practical experience of the support and design of systems, formalize in the viewer Q- schemes. Unintentionally greater possibilities may be able to adapt models, so that they allow Q- scheme, scho to be asked without obmezhen.

Merezhevі models (N-schemes)

In practice, the model of objects is often brought up to date, linked to a formalized description and analysis of causal-inherited links in folding systems, sometimes simultaneously parallel to the chain of processes. Nayposhirenіshim in Danish hour formalism, which describes the structure and interaction of parallel systems and processes, є Petri (in English. Petri Nets).

Formally, Petri ( N-scheme) to be set in a four-way view:

,

de V- kintseve without any symbols, called positions; D- kintseve bezlich symbols, called transitions; I- input function (direct function of incidence); O - outgoing function (reverse function of incidence). In such a rank, the input function I view transition d j in unalterable out-of-the-box positions b iÎ I(d j), And the output function About view transition d j in unalterable out-of-the-box positions b iÎ D(d j).

graphically N-circuit The image is displayed in the view of a dicotyledonous multigraph, which is a combination of positions and transitions. Graph N-circuits There are two types of universities: positions and transitions, images 0 and 1 are visible. Orintovnі arcs to set positions and transitions, moreover, the skin arc is straightened from an element of one deaf (position or transition) to an element of one deaf (transition or transition). Graph N-circuitsє a multigraph, so as it is allowed to move multiple arcs from one vertex to the last.

not directed N-circuits It can only be used to visualize the statics of the modeled system (interconnection of minds and minds), or it is not allowed to display the dynamics of the function of the modeled system in the model. For the submission of the dynamic authorities of the object, the function of markup (design) is introduced M: B® (0, 1, 2, ...). markuvannya Mє assignment of abstract objects, called mits (tricks), positions N-circuits, moreover, a number of mitoks, depending on the skin position, can change. With the graphic manager N-circuits The layout is displayed at the center of the vertices-positions of a given number of points (if the number of points is large, put numbers). Markovana (razmіchena) N-circuit can be described in viglyadi p'yatirka і sukupnіstyu heirloom Petrі and markuvannya M.

function N-circuits drive through the transition from design to design. Pochatkova rosmіtka means yak M 0:V® (0, 1, 2, ...). A change in the changes to be made as a result of the appointment of one of the transitions d jÎ D fathoms. Necessary mindset for the transition d jє b iÎ I (d j){M (b i) ³ 1), de M (b i)- position adjustment b i. transition d j For whomever the mind is meant to be, it must begin to be in the state of readiness before the dispatch, or as the transitions are made.

Combined models (A-schemes)

We will take a backward approach to a formal description of the processes of functioning of systems є pidhid, proponation of Ya.P. Buslenko. Tsei pidhid allows to describe the behavior of uninterrupted and discrete, deterministic and stochastic systems, i.e. aggregate system Aggregate system) A-scheme.

An analysis of the current processes of modeling systems and factories, which is based on an additional modeling method on the EOM, will inevitably bring up a complex solution of problems, as well as emerge during the process of the machine implementation, if I develop my own model a formal mathematical scheme, tobto A-diagram. Such a scheme is guilty at once of the viconuvati k_lka functions: but an adequate mathematical description of the object of the model, i.e. E. System S, serve as a basis for stimulating algorithms and programs in machine implementation of a model M, allowance in a simplified version (for okremikh vipadkiv) to carry out analytical dosage.

Guided by the singing world of supreme words. Protest, within the framework of a public approach on the basis of A-schemes vdaєatsya zniti mіzh them singing compromise.

According to tradition, it was established in mathematics in the background and in the application of mathematics in winter, with an aggregative approach, it is given a formal definition of a model object - an aggregative system, as in a mathematical scheme, as well as in the image of the image of the image system. With an aggregate inventory, the folding object (system) breaks into the final number of parts (subsystems), which is secured at the same time, so that it will not interfere with the modality. As soon as the deyak from the otrimanyh pid systems appear in their own room to complete the foldable, then the processes of their development are trivial until the pid systems are established, as in the minds of the given tasks of the model you can play with the help of the hand for a mathematical description. As a result of such decomposition, the folding system appears in the view of the bagger-type design of interconnected elements, which are common in the subsystem of the old equipments.

In the quality of the element A-diagrams the unit is connected, and the connection between the units (all the Sі from the zznіshnіm middle E) Get help from the operator R... Obviously, the unit itself can look like a yak A-diagram, T.E. Whether the unit is characterized by advancing multiplies: moments in an hour T, in X and outgoing Y signals, staniv Z at the skin moment one o'clock t... Stan to the unit at the moment of the hour tÎ T be called yak z(t)Î Z, And incoming and outgoing signals - yak X(t)Î Xі at(t)Î Y for sure.

It is a very class of great systems, because through this folding it is impossible to be formalized in the mathematical schemes of single units A n,, Yaku is called an aggregate system abo A-scheme... To describe the deyakoi real system S at the viglyad A-diagrams it is necessary for a mother to describe what kind of units A n, So and the sound between them.

function A-diagrams tied with the information processing. All information that circulates in A-schema, To go to the call and to the inside. The name of the information should be taken from the external objects, which are not elements of this circuit, but the internal information is circulated by the aggregates of the self A-diagrams... Information exchange A-schemeі zvnіshnіm middle E seen through units, which are called poles A-diagrams... At the same time the input poles A-diagrams, Which are units, on which one should X- new house, and other poles A-diagrams, Visited information of such є at-in the house. Aggregates, which are not poles, are called internal.

Classification in any area, knowledge is necessary. Vona allows you to share the accumulations of information, to order the understanding of the subject area. The rapid development of methods of mathematical modeling and development of regions and areas of development have led to the emergence of a great number of models of older types and to the need for classifying models according to the categories of requirements, as well as universal models suitable for Guided butt of deyakikh categories: vicoristanny area; an image in the model of a time official (dynamism); galuzi knowledge; way of presenting models; the manifestation of or the presence of certain (or non-significant) factors; type of criterion for efficiency and overlapping, etc.

Analyzing mathematical literature, we have seen most often signs of the classifications:

1. For the method of implementation (including formal translation), all mathematical models can be broken into analytical and algorithmic.

Analytical - models, in which the standard mathematical language is used. Імітаційні - models, in which vikoristany special mova model or universal mova programuvannya.

Analytical models can be recorded by the viewers of the analytic virases, so that at the viewers the virazs, so that the number of arithmetic steps and transitions to the border can be replaced, for example :. Algebraic viraz є we will limit it to an analytic viraz, so that in the result we will not get the exact meaning. There are also constructions that allow knowing and resulting values ​​from a given accuracy (for example, placing an elementary function in a statistic row). Models, scho vikoristovuyut a similar priyom, are called nudges.

Have your own house, analytical models break into theoretical and empirical models. Theoretical models represent real structures and processes in previously perceived objects, so that they spiral to the theory of their robots. Empirical models will be based on the development of reactions of the object to the change of minds in the midst of the community. At the same time, the theory of the robot cannot be looked at, the object itself is so called "black box", and the model is an interpolation deyak. Empirical models can be inspired by experimental evidence. Tsі danі will be recognized without preseeding on preliminarily objects or for additional support of їх physical models.

It’s like a certain process cannot be described in an analytical model, it can be described using an additional special algorithm or programs. This model is algorithmic. When prompting the algorithmic models, the numerical approach is abnormal. With a numerical approach, the supremacy of mathematical relations is replaced by a finite analogue (for example, the transition from the function of a continuous argument to the function of a discrete argument). We'll be able to see the reason for the calculating algorithm, so that after the arithmetic and logical actions. Knowing the solution of a discrete analogue to approach the solution of a specific problem. With an imitational approach, the object of the model itself is discretized, the model of the surrounding elements of the system will be generated.

2. For the form of the submission of mathematical models, the following are developed:

1) Invariant model - a mathematical model is represented by a system of equivalents (differential, algebraic) without urahuvannya methods of conversion of ciches.

2) Algebraic model - the comparison of the models of the linking with the numerical method of solving and writing in the form of the algorithm (the last count).

3) An analytical model is a clear presence of shukany different kinds of given values. Such models are designed on the basis of physical laws, or as a result of direct integration of different differential equations, vicorist tables of integration. They are also referred to as regressive models, obtained on the basis of the results of the experiment.

4) The graphical model is presented in the view of graphs, equivalent schemes, diagrams too. For the registration of graphic models, the rule of unambiguous presentation of the images of the elements of the graphic and components of the invariant mathematical model is clear.

3. In terms of fallowness, in view of the criterion of efficiency and overlapping, the models are subdivided into line and line. In the linear models, the criteria of efficiency and the imposition of interconnection with the linear functions of the winter models (in the case of non-linear models). The assumption about the lineage of the criterion of efficiency and the availability of overlapping between different models is quite acceptable in practice. Tse is permissible for viroblennya vikoristovuvati kind of breakdowns of the apparatus of the line program.

4. I’ll look at the official at the hour and the oblast’s victorious, I’ll see static and dynamic models... If all the values ​​entered into the model do not lie for an hour, then the model of the object is as static as the process (one-time view of information on the object). Tobto a static model is a whole model, at every hour it is not a changeable value. The dynamic model allows you to beat the object per hour.

5. In the presence of the number of sides, which the solution is to be taken, two types of mathematical models are seen: description and normative... In the description of the model, there are no sides, which accept the solution. Formally, the number of such sides in the description of the model is zero. Typical butt of sub-models є models of mass service systems. To induce descriptive models, one can also use the theory of hopes, the theory of graphs, the theory of imovings, the method of statistical viprobauan (Monte Carlo method).

The normative model is characterized by a weak side. In principle, it is possible to see two types of normative models: models of optimization and theoretical games. In models of optimization, it is mainly the decision to be made technically up to the maximum value of the criterion of efficiency, in order to start such a value of the change in the field, with some criteria for the maximum efficiency.

For the development of solutions, which are represented by models of optimization, the order of the classic and new alternative methods (pushing the extremum), the most widely used methods of mathematical programming (linear, non-linear, dynamic). The theoretical-game model is characterized by a multiplicity of the number of sides (not less than two). If there are two sides with other interests, then the igor theory will be victorious, if the number of sides is more than two and among them there are unfortunate coalitions and compromises, then the theory of noncoalistic n OSIB.

6. In the absence of evidence, or in the absence of a low (or irrelevant) official, they see determinations and stochastic mathematical models. In deterministic models, all interconnections, changes and constants are specified precisely, to bring the resulting function to an unambiguous value. The model will be determined in a quiet way, if the factor is injected into the result of the operation, it will be able to achieve the exact value of the estimate, and the official officials are either visible, or they can be zehtuvati.

If there is a part of all the parameters that are included in the model for its nature, such as values ​​or functions, then the model is brought to the class of stochastic models. In stochastic models, the laws of the distribution of different values ​​are set, so that the variance of the resulting functions and the reality is visualized as a certain type of process, the process and the result of which can be described by these mathematicians. Such models can be motivated, as well as sufficient factual material for assessing the necessary imaging problems, as well as the theory of this phenomenon permits the value of the theorems and the grains theoretically (on the basis of the formulas of theories).

7. In fallowness for the purposes of model development descriptive, optimization and management models. In descriptive (from Latin, Descriptio - description) models, the laws of change in the parameters of the model can be observed. For example, the model of a material point under the influence of the applied forces on the basis of another Newton's law: Setting the position and acceleration of the point at the given moment of the hour (input parameters), the mass (power parameter) and the law of change of forces (call flow), it is possible to value the coordinates of the point and the speed at any moment of the hour (output data).

Optimization of the model is fixed for the value of the most beautiful (optimal), on the basis of a certain criterion, the parameters of the modeled object, or methods of controlling the object. Optimization models will be based on the same number of descriptive models, and there may be several criteria for determining the optimum. On the area of ​​the value of the input parameters, there can be overlaps between the views of the equilibrium and the irregularities associated with the peculiarities of the viewed object or the process. The application of an optimized model is to serve as a way to improve the efficiency of food in the beginning of the day (in the capacity of the existing data, the calorific value of the product is introduced, the prices for the price, etc.).

Keruyuchі models are stagnant for taking decisions in the other areas of the straightforwardness of people, since there are no other alternatives to choose from, and the process of taking decisions is the last of such alternatives. For example, vibіr advanced training for learning about decals prepared by students. The flexibility of the knowledge of the field is in ignorance about the input tribute (it was independently prepared for the reason of the victorious practice), as well as the goals (the science of robotics and the structure, the time of the week of the week and the event of the day) So, as the optimum of the adopted solution in one and the same situation can be interpreted in a reasonable way, the type of criterion of optimality in management models does not come to a head in the long run. The method of formulating the criteria of optimality in terms of availability in the form of non-value is looked at in the theory of choice and acceptance of the decision, based on the theory of hypotheses and pre-existing operations.

8. For the method of further development analytical, numerical and imitational models. An analytical model is also called a formalized description of the system, as it allows you to re-form the solution of the relationship in an explicit view, as well as the kind of mathematical apparatus. The numerical model is characterized by fallowness, as it allows only a few private numerical solutions for specific cob minds and only a few parameters of the model. Іmіtation model - the cost of describing the system and the internal flow, the algorithms of the functioning of the system and the rules for changing the system and the flow of the internal flow. The algorithms and the rules do not give the ability to register explicit mathematical methods of analytical and numerical solution, but rather allow the processes of functioning of the system and fix the characteristics. The distance will be more detailed in the analysis of analytical and adaptive models, the development of the same types of models is tied to the specific professional activity of students in direct training.

1.4. Graphical representation of mathematical models

In mathematics, the form of a link between values ​​can be represented by the same kind as an independent change (argument), y- fallow winter (function). In the theory of mathematical model, I call it a factor, and I don’t call it a word. Moreover, in the fallowness of the region prompts the mathematical model of the terminology of the trochus species. Deyak, add the value of the factor to the reader, in the presence of the region before the diagnosis, as indicated in Table 1.

Table 1. Deyaki value to understand the "factor" and "vidguk"

Representing graphically, a mathematical model, we will take into account factors and readings of variable values, meaning that there are many different numbers.

Graphical representations of the mathematical model is some surface of the k- vimir factorial space X... It’s possible to see just the same and two-world surfaces of the viewer. For the first one, there are a number of pointless points on the working area, and in the other - pointless points, which can be used to create a surface in the open space (for the image of such points, it is manually X(Fig. 8).

The area in which it is designated on the surface of the vidguk is called area of ​​value X *. Qia area to become, as a rule, deprives a part of the main factorial space X(X *Ì X) І see for the help of others, superimposed on keruyuchі changes x i, Recorded by viglyadі іvnostі:

x i = C i , i = 1,…, m;

f j(x) = C j, j = 1,…, l

for any inconveniences:

x i min £ x i£ x i max, i= 1,…, k;

f j(x) £ C j, j = 1,…, n,

With a wide function f j(x) You can lie down at once from all the winter, and from the deyakoi їkh part.

The exchange of the type of irregularities characterizes either the physical exchange in the processes in the preliminaries (for example, the exchange of temperature), or the technical intercourse, connected with the minds of the robots and the stock (for example, the borderline

Possibilities of up to date models can lie in the form of authorities (relєfu) of the surface of the vidguk, zokrem, of the number of apparent ones on the “peaks” and її of contrast. Number of peaks (depressions) modality surface to see. There is one vertex (depression), the model is called unimodal.

The nature of the change in function can be flexible (Fig. 9).

The model can be used for cutting points of the first genus (Fig. 9 (a)), points of cutting for another genus (Fig. 9 (b)). On little 9 (c), a continuously differentiable unimodal model is shown.

For all three submissions on baby 9, the submission is unimodal:

if W (x *) is an extremum W, then wash x 1< x 2 < x* (x 1 >x 2> x *) track W (x 1)< W(x 2) < W(x*) , если экстремум – максимум, или W(x 1) >W (x 2)> W (x *), as the extremum is the minimum, that is, in the world from the extreme point of the value of the function W (x), in an uninterrupted way to change (change).

The order of unimodal viewing of polymodal models (Fig. 10).

With this important power of the surface, the contrast is shown, showing the sensitivity of the resultant function to the change of factors. The contrast is characterized by the magnitudes of the older ones. Demonstrated characteristics of contrast on the butt double-sided surface of the viewer (Fig. 11).

Speck a roztashovana on "shil", which characterizes the equal contrast in all winter x i (i= 1,2), point b roztashovana in the "yar", in which there is a distribution of the building of the young ones (for the damn good things in the function), point s roztasovana on the "plateau", for which there is a low contrast in all winter x i talk about the proximity of the extremum.

1.5. Basic methods inducing mathematical models

Probably the classification of methods of formalized representation of modeling systems Volkov V.N. and Denisova A.A. The basic terminology, applied theories, is developed on the basis of the described classes of methods, as well as the scope of the scope of the possibilities for the provision of additional information 1.

The practical model of systems of the greatest extension has gained analytical and statistical methods.

1) Analytical methods stimulate mathematical models.

The basis of the terminological apparatus of analytical methods is prompted and mathematical models are taught by the understanding of classical mathematics (formula, function, equivalence and system of equivalence, ineptitude, commonplace, integral, etc.). These methods are characterized by the cleanliness and well-structured terminology of the laws of classical mathematics.

On the basis of analytical findings, we have revised the development of such mathematical theories as classical mathematical analysis (for example, methods of advanced functions), as well as the modern foundations of mathematical programs and theory. Until then, the mathematical program (linear, non-linear, dynamic, integer, etc.) can be used as a way of setting problems, as well as expanding the possibility of proving the adequacy of the model, on the basis of a number of other mathematics. Ideas of an optimal mathematical program for the development of economics (seedlings, solving problems of optimal cutting of a sheet of plywood) Kantorovich.

The peculiarities of the method on the butt are understandable.

Butt. Supposedly, for the production of two types of products Aі V it is necessary to vikoristovuvat a syrovin of three types. When preparing one type of product A Vitracha to be 4 units. syruvini of the first kind, 2 od. 2nd and 3rd unit 3rd kind. For preparation of one type of product V Vitrachate 2 units. sirovini of the 1st type, 5 od. 2nd type that 4 od. 3rd type of sirovini. In the warehouse of the factory є 35 pcs. sirovini of the 1st type, 43 - of the 2nd, 40 - of the 3rd type. From the implementation of one product to the type A factory maє arrived 5 yew. rubles, and from the implementation of one product to the type V the arrival becomes 9 yew. rub. It is necessary to use a mathematical model of problems, in which to transfer the maximum income.

Standards for vitrati sirovini skin type on the preparation of one given type of product guidance in the table. At the same time, the influx is indicated in the implementation of the skin type of product and beyond the range of this type, which can be used as an industry in Ukraine.

meaningfully through x 1і x 2 obsyag products, how to release views Aі V for sure. Vitrati material of the first grade on the plan to store 4x 1 + 2x 2, And the stench is not guilty of changing the stocks, tobto 35 kg:

4x 1 + 2x 2 35.

Similar material of another grade:

2x 1 + 5x 2 43,

і for material of the third grade

3x 1 + 4x 2 40.

Arrival from implementation x 1 product unit A i x 2 Product unit In stock z = 5x 1+ 9x 2(complete function).

We took out the problem model:

The graphical solution of tasks is focused on little 11.

Optimal (most beautifully, to maximize the functionality z) The solution of the problem is at point A (the solution is explained in section 5).

Otrimal, scho x 1=4,x 2= 7, the value of the function z at point A :.

In this rank, the value of the maximum arrival is one 83 yew. Rub.

Apart from the graphical one, there are a number of special methods for the translation of tasks (for example, a simplex method), or it is possible to freeze packages of applied programs, and to implement them. In view of the whole function, the development of the program is linear and non-linear, in the fallowness due to the nature of the changes, they see an integral program.

You can see the general figures of the mathematical program:

1) the introduction of the understanding of the central function and the introduction of the by means of setting up the installation;

2) can be combined in one model of different criteria (small sizes, in the butt - stocks of syrovin and influx);

3) a model of the mathematical program of admission to the cordon of the region of the permissible values ​​of changes;

4) the possibility of realizing the algorithm for rejecting the results (covering the approximation to the optimal solution);

5) in fact, to reach beyond the aid of the geometric interpretation of the problem, as it is additional help in quiet situations, if it is unwise to submit the problem formally.

2) Statistical methods stimulate mathematical models.

Statistical methods prompted mathematical models to expand and began to expand widely in the development of the theory of definitions in 19 centuries. In these bases there are the imovirnіsnі regularities of the type (stochastic) podіy, so that they represent the real phenomena. The term "stochastic" is a clarification of the understanding of "vypadkovi", in order to be given in advance, the bureaucrats who are injected into the process, and the understanding of "vypadkovi" is characterized by the nexus of the given reasons, for such reasons.

Statistical regularities are represented in the view of discrete large values ​​and regularities have appeared, for example, in the view of uninterrupted deposits of growth rates (processes). The theoretical foundations of stochastic models are described in detail in section 2.

Control food

1. Formulate the basic knowledge of the mathematical model.

2. Give the value of the mathematical model.

3. To redistribute the main shortcomings of the experimental approach in the preliminaries.

4. Recover the basic steps to encourage the models.

5. Change types of mathematical models.

6. Give short description types of models.

7. What kind of method is the mathematical model presented geometrically?

8. How do you ask a mathematical model of an analogous type?

zavdannya

1. On the basis of the mathematical model of the solution of the problem and to carry out the classification of the model:

1) Visually, the most important place of the cylindrical core, the surface of the yak (without crush) road S.

2) The enterprise will secure regular releases of products when there is no uninstallation of complementary items of two summaries. Immovability of views in the output from the first from the summaries -, from the other -. To know the identity of an animal in the robotics of an enterprise.

2. The Malthus model (one thousand seven hundred and ninety times) describes the multiplication of the population of shvidkistyu, proportional number of. V discrete viewer tsei law є geometric progress :; Abo. The law, records in the view of differential equality, is a model of the exponential growth of the population and well describes the growth of the clientele populations at all times, whether there is any kind of intercourse :. Ask the cob and show the model to the robot.

At the proponation of your uvazi statti mi proponumo attach mathematical models. Krym, I have great respect at the stage of the models' development and the selection of the original ones, linked to the mathematical models.

Another one of our food is the whole of mathematical models in the economy, but the value of some of them is clear. Read our rosmovu mi proponmo from the very understanding of the "model", briefly discernible їkh classification and we pass on to our main food.

Understanding the "model"

The word "model" is often creepy. Well tse take? Daniy term is without value, the axis is only three of them:

• a specific object, which is set up for the rejection and collection of information, which shows the authorities and the characteristics and so given to the original this object(The whole specific object can be rotated in a different form: explicit, describe behind other signs and so far);
• Next to the model, be mindful of the image of any specific situation, life or management;
• The model can be served by changing a copy of some kind of object (the stench is created for a larger report and analysis, as the model depicts the structure and interconnection).

From the start, as it was said earlier, it is possible to create a small customization: the model allows detailed installation of a folding system, or an object.

All models can be classified for a number of signs:

• by region victorian (navchalny, prelude, scientific and technical, іgrovі, імітаційні);
• by dynamism (static and dynamical);
• for knowledge (physical, chemical, geographic, historical, sociological, economical, mathematical);
• for the method of submission (material and information).

Information models, in their turn, share signs and verbal. And signs - on computers and non-computers. Now we turn to the lecturer view of applications in the mathematical model.

Mathematical model

It’s not important to greet, the mathematical model is visualizing whether it’s a picture of an object, or it’s appearing behind the help of special mathematical symbols. Mathematics is necessary in order to model the regularities of the daily light on your own specific language.

The method of mathematical modeling was born a long time ago, thousands of years ago, at the same time with the advent of this science. Same shipment for sales given the way The model gave birth to EOM (electronic calculating machines).

Now let's move on to the classification. This is how you can carry it out behind the signs. The smell is presented in the table below.

Every presentation and presentation will show the rest of the classification, so as you can see the geographic patterns of modeling and the range of models.

descriptive models

In a number of distributions, it is possible to make reports on descriptive mathematical models. For that, everything was borderingly zozy, if the butt was aimed.

For some reason, we can describe the species as well. The price is tied to those who are simply robotized to predict and predict, but we cannot fit into the result of the story.

Let's use the butt of a descriptive mathematical model for calculating the trajectory of benefit, speed, comet from the Earth, as it invaded the vastness of our Sleeping System. The model is descriptive, so that all the results are rejected and they can only get in front of us about being of some kind. It’s a pity that we cannot fit into the result. However, primed on otrimanikh rozrakhunks, you can go in before you come in to save your life on Earth.

optimized models

Let's talk about economic and mathematical models, with the butts of which can serve the development of the situation. In the given vypadku move about models, which will help you to know the correct message from the singing minds. The stench is obov'yazkovo yakis parameters. The schob became borderline zzumilo, the butt from the agrarian part was discernible.

We have є grain holdings, ale grain is very dry. In general, we need to correctly adapt the temperature regime and optimize the recovery process.

In such a rank, we can give the understanding of the “optimization model”. In the mathematical sense, a system of equivalents (both linear, and not), the solution of which additional help is to know the optimal solution in a particular economic situation. We looked at the application of the mathematical model (optimization), but I still want to do it: the given form is brought to the class of extreme problems, it smelt in addition to describing the functionality of the economic system.

There is one more nuance: the models can be of a different nature (div. Table below).

bagatocriterial models

It’s easy for you to talk about the mathematical model of critical optimization. Until then, they brought up the butt of a mathematical model for optimizing the process according to one criterion, but more robust, how much?

Let us use the butt of various criterial tasks to serve as the organization of the correct, corny and one-hour economical feeding of great groups of people. Such enthusiasts are often found in the army, school days, summer camps, pharmacies and so far.

What criteria were given to us in the given tasks?

1. Kharchuvannya maє buti korisnim.
2. Vitrati for being guilty are the smallest.

Yak bachite, tsіlі zvsіm do not get lost. This means that in case of revision, it is necessary to make an optimal decision, the balance between two criteria.

Іgrovі models

Talking about іgrovі models, it is necessary to understand the understanding "theory of іgor". As simple as saying, these models represent mathematical models of reference conflicts. Tilki varto rosumiti, scho, in view of a real conflict, the mathematical model has its own rules.

Infection will be provided a minimum of information from the theory of igor, as it will help you with the intelligence, as well as the game model. And so, in the model, the presence of the sides (two or more), which are accepted as gravels, are usually present.

All models may be acts of power.

The game model can be double or multiple. As we have two sub'kti, then the conflict is guys, but more is a lot. It is also possible to see an antagonistic load, and call it a load with a zero sum. This is a model, in which I will play one of the participants in the road, I will program the one.

імітаційні models

V given given I have the utmost respect for the іmіtatsіynі mathematical models. You can use butts to serve:

• model of the dynamics of the number of microorganisms;
• a model to the collapse of molecules, and so far.

In this context, we are talking about models, which are as close as possible to real processes. Behind the great rakhunk, the stench is manifested in nature. In the first place, for example, we can model the dynamics of the number of ants in one colony. With this, it is possible to sposter for the share of the skin surrounding the individual. In this context, the mathematical description of viciousness is rather rare, often in the presence of letters, say:

• after five days of life, a person will deposit an egg;
• in twenty days the goose goose, and so far.

In such a rank, vikoristoyutsya for the description great system... Mathematical visnovok is a sequence of otriman statistical tributes.

vimogi

It is even more important for the nobility, before this kind The models show the actions of the vimogs, the middle ones - the hovers in the tables below.

universality	The power of power permits the vicorists to use the same model when listing the same type of groups of enterprises. It is important to note that universal mathematical models absolutely do not lie in the physical nature of a pre-existing object
adequacy	Here it is important to think about it, so that the power allows the most correct way to visualize the real processes. The managers of the exploitation have even more importantly given the power of the mathematical model. The application of the model can serve the process of optimizing the gas system. In this case, the parameters are put into place, and as a result, the correctness of the folded model is changed.
accuracy	Vimoga is given a value that can be recognized when developing a mathematical model and input parameters in our real object
economy	Vimoga economy, which is presented before any mathematical model, is characterized by vitrates for implementation. If the robot has to work with the model manually, then it is necessary to work out, for an hour before the completion of one problem for the additional given mathematical model. If I’m talking about automated design, then the vitratic indicators will be redeemed for the hour and memory of the computer

step-by-step model

All in the mathematical model it is accepted to see the chotiri stages.

1. Formula of laws, how to ring parts of the model.
2. Doslіdzhennya math problems.
3. Compilation of practical and theoretical results.
4. Analysis and modernization of the model.

Economic-mathematical model

At the same time, there is a short supply of food. The staff can serve:

• Formation of virobnichy programs for the release of meat products, which will ensure the maximum influx of virobnitstva;
• maximizing the arrival of organizing a noble house for the optimal number of tables and styles at a furniture factory, and so far.

The economic-mathematical model is a representation of an economic abstraction, as it is rotated beyond the additional mathematical terms and signs.

Computer mathematical model

Computer mathematical model applications є:

• running a guide for additional block diagrams, diagrams, tables, and so on;
• zavdannya on the mechanics of solid solid, and so far.

The computer model is an image of an object, either a system or a viewer:

• tables;
• block diagrams;
• diagrams;
• graph, and so far.

With a tsom given a model visualizing the structure and interconnection of the system.

Pobudova economic and mathematical model

We have already said about those who also have an economical-mathematical model. The butt of solving tasks will be viewed right at a time. We need to carry out an analysis of the virobnichoi programs for detecting the reserve for the arrival of the arrival in the assortment.

We will not look at the problems again, but only an economical and mathematical model. The criterion of our staff is to maximize the arrival. Todi funktsiya maє viglyad: L = p1 * x1 + p2 * x2 ..., scho pragne to the maximum. In a given model, r - tse arrival per unit, x - tse number of viral units. Dal, primed on the motivated model, it is necessary to carry out the racking and to bring the bags.

Butt and simplicity of mathematical model

Zavdannya. Ribak, turning with an offensive catch:

• 8 ribs - bags of wine seas;
• 20% of the catch - bags of new seas;
• from the small river there was no new ribini.

Having bought ribs in the store?

Also, the butt will inspire the mathematical model of the given tasks of the viglead to the next rank. Initially, the number of ribs is given per x. Dotrimuyuchis umovі, 0.2x - the number of ribs, scho lingering in the middle latitudes. Now all the obvious information and mathematical model of the problem is understood: x = 0.2x + 8. Virishuєmo rivnyannya and recognizable on the head of food: 10 ribs of wine bought in the store.