Modeling. Mathematical modeling schemes

MATHEMATICAL SCHEMES FOR MODELING SYSTEMS

BASIC APPROACHES TO THE CONSTRUCTION OF MATHEMATICAL MODELS OF SYSTEMS

The initial information in the construction of mathematical models of the processes of the functioning of systems is the data on the purpose and operating conditions of the investigated (designed) system. S... This information defines the main purpose of the system modeling. S and allows you to formulate requirements for the developed mathematical model M. Moreover, the level of abstraction depends on the range of those questions to which the system researcher wants to get an answer using the model, and to some extent determines the choice of a mathematical scheme.

Mathematical schemes. The introduction of the concept of a mathematical scheme allows us to consider mathematics not as a method of calculation, but as a method of thinking, as a means of formulating concepts, which is most important in the transition from a verbal description of a system to a formal representation of the process of its functioning in the form of some mathematical model (analytical or imitation). When using a mathematical scheme, first of all, the researcher of the system S should be interested in the question of the adequacy of the mapping in the form of specific schemes of real processes in the system under study, and not the possibility of obtaining an answer (solution result) to a specific research question. For example, the representation of the process of functioning of a collective information-computing system in the form of a network of queuing schemes makes it possible to describe well the processes occurring in the system, but with complex laws of incoming flows and service flows, it does not make it possible to obtain results in an explicit form.

Mathematical scheme can be defined as a link in the transition from a meaningful to a formal description of the system's functioning process taking into account the impact of the external environment, that is, there is a chain "descriptive model - mathematical scheme - mathematical (analytical and / or imitation) model".

Each specific system S is characterized by a set of properties, which are understood as values that reflect the behavior of the modeled object (real system) and take into account the conditions of its functioning in interaction with the external environment (system) E. When constructing a mathematical model of the system, it is necessary to resolve the issue of its completeness. The completeness of the model is mainly regulated by the choice of the boundary “system S - environment E» . Also, the problem of simplifying the model should be solved, which helps to highlight the main properties of the system, discarding the secondary ones. Moreover, the assignment of the properties of the system to the main or secondary essentially depends on the purpose of modeling the system (for example, the analysis of the probabilistic-temporal characteristics of the process of the functioning of the system, the synthesis of the structure of the system, etc.).

Formal model of the object. The model of the object of modeling, i.e., the system S, can be represented as a set of quantities that describe the process of functioning of a real system and generally form the following subsets: input actions per system

;

aggregate environmental influences

;

aggregate internal, (own) parameters systems

;

aggregate output characteristics systems

Moreover, in the listed subsets, managed and unmanaged variables can be distinguished. In general , , , are elements of disjoint subsets and contain both deterministic and stochastic components.

When modeling the system S, the input actions, the effects of the external environment E and the internal parameters of the system are independent (exogenous) variables, which in vector form have the form,,, and the output characteristics of the system are dependent (endogenous) variables and in vector form have the form).

The process of functioning of the S system is described in time by the operator F s , which in the general case transforms exogenous variables into endogenous ones in accordance with relations of the form

. (1)

The set of dependences of the output characteristics of the system on time y j (t) for all kinds
called output trajectory
. Dependence (1) is called system functioning lawS and denoted F s . In the general case, the law of functioning of the system F s can be specified in the form of a function, functional, logical conditions, in algorithmic and tabular forms, or in the form of a verbal matching rule.

Very important for the description and study of the system S is the concept algorithm of functioningA s , which is understood as the method of obtaining the output characteristics taking into account the input influences
, environmental influences
and own parameters of the system
. It is obvious that the same law of functioning F s system S can be implemented in various ways, i.e., using many different algorithms for the functioning A s .

Relations (1) are a mathematical description of the behavior of the object (system) of modeling in time t, that is, they reflect its dynamic properties. Therefore, mathematical models of this type are usually called dynamic models(systems).

For static models mathematical model (1) is a mapping between two subsets of the properties of a modeled object Y and { X, V, H), which in vector form can be written as

. (2)

Relations (1) and (2) can be specified in various ways: analytically (using formulas), graphically, tabularly, etc. Such relationships in some cases can be obtained through the properties of the system S at specific times, called states. The state of the system S is characterized by vectors

and
,

where
,
, …,
at the moment
;
,
, …,
at the moment
etc.,
.

If we consider the process of functioning of the system S as a sequential change of states
, then they can be interpreted as the coordinates of a point in To-dimensional phase space. Moreover, each implementation of the process will correspond to a certain phase trajectory. The collection of all possible values of states called state space object of modeling Z, moreover
.

The states of the system S at the moment of time t 0 < t*T are completely determined by the initial conditions
[where
,
, …,
], input influences
, own system parameters
and environmental influences
, which took place over a period of time t*- t 0 , With using two vector equations

; (3)

. (4)

The first equation for the initial state and exogenous variables
defines a vector function
, and the second according to the obtained value of the states
- endogenous variables at the output of the system
. Thus, the chain of equations of the object "input-state-output" allows you to determine the characteristics of the system

. (5)

In the general case, the time in the model of the system S can be considered on the modeling interval (0, T) both continuous and discrete, i.e., quantized into segments of length
time units each when
, where
- number of sampling intervals.

Thus, under mathematical model of the object(real system) understand a finite subset of variables (
} together with mathematical relationships between them and characteristics
.

If the mathematical description of the object of modeling does not contain elements of randomness or they are not taken into account, that is, if it can be assumed that in this case the stochastic effects of the external environment
and stochastic internal parameters
are absent, then the model is called deterministic in the sense that the characteristics are uniquely determined by deterministic inputs

. (6)

Obviously, the deterministic model is a special case of the stochastic model.

Typical schemes. The given mathematical relations represent general mathematical schemes and allow describing a wide class of systems. However, in the practice of modeling objects in the field of systems engineering and systems analysis at the initial stages of system research, it is more rational to use typical mathematical schemes: differential equations, finite and probabilistic automata, queuing systems, Petri nets, etc.

Not possessing such a degree of generality as the considered models, typical mathematical schemes have the advantages of simplicity and clarity, but with a significant narrowing of the possibilities of application. Differential, integral, integro-differential and other equations are used to represent systems operating in continuous time as deterministic models, when random factors are not taken into account in the study, and finite automata and finite-difference schemes are used to represent systems operating in discrete time. ... Probabilistic automata are used as stochastic models (taking into account random factors) to represent systems with discrete time, and queuing systems are used to represent systems with continuous time, etc.

The listed typical mathematical schemes, of course, cannot pretend to be able to describe on their basis all the processes occurring in large information management systems. For such systems, in some cases, the use of aggregate models is more promising.

Aggregate models (systems) make it possible to describe a wide range of research objects with a reflection of the systemic nature of these objects. It is with an aggregate description that a complex object (system) is divided into a finite number of parts (subsystems), while maintaining the connections that ensure the interaction of parts.

Thus, when constructing mathematical models of the processes of functioning of systems, the following main approaches can be distinguished: continuous-deterministic (for example, differential equations); discrete-deterministic (finite automata); discrete stochastic (probabilistic automata); continuous-stochastic (queuing systems); generalized or universal (aggregate systems).

CONTINUOUS DETERMINATION MODELS (D-CIRCUITS)

Let us consider the features of the continuous-deterministic approach on the example of using differential equations as mathematical models. Differential Equations such equations are called in which functions of one or several variables are unknown, and the equation includes not only functions, but also their derivatives of various orders. If the unknowns are functions of several variables, then the equations are called partial differential equations; otherwise, when considering functions of only one independent variable, the equations are called ordinary differential equations.

Basic relationships. Usually, in such mathematical models, time is used as the independent variable on which unknown sought-for functions depend t. Then the mathematical relation for deterministic systems (6) in general form will be

, (7)

where
,
and
- P-dimensional vectors;
- vector-function, which is defined on some ( P+1) -dimensional
set and is continuous.

Since mathematical schemes of this type reflect the dynamics of the system under study, that is, its behavior in time, they are called D-schemes(eng. dynamic).

In the simplest case, the ordinary differential equation has the form

. (8)

The most important application for systems engineering D-scheme as a mathematical apparatus in the theory of automatic control. To illustrate the features of the construction and application of D-circuits, let us consider the simplest example of formalizing the process of functioning of two elementary systems of different physical nature: mechanical S M (oscillations of the pendulum, Fig. 1, a) and electric S K (oscillatory circuit, Fig. 1, b).

Rice. 1. Elementary systems

The process of small oscillations of the pendulum is described by the ordinary differential equation

where
- the mass and length of the suspension of the pendulum; g - free fall acceleration;
- the angle of deflection of the pendulum at the moment of time t.

From this equation of free oscillation of the pendulum, estimates of the characteristics of interest can be found. For example, the period of swing of a pendulum

Similarly, the processes in the electric oscillatory circuit are described by the ordinary differential equation

where L To , WITH To - inductance and capacitance of the capacitor; q(t) - capacitor charge at time t.

From this equation, you can get various estimates of the characteristics of the process in the oscillatory circuit. For example, the period of electrical oscillations

Obviously, introducing the notation
,
, ,
, we obtain an ordinary second-order differential equation describing the behavior of this closed-loop system:

where
- system parameters; z(t) - system state at time t.

Thus, the behavior of these two objects can be investigated on the basis of a general mathematical model (9). In addition, it should be noted that the behavior of one of the systems can be analyzed using the other. For example, the behavior of a pendulum (system S M) can be studied using an electric oscillatory circuit (system S K).

If the system under study S, i.e. a pendulum or a contour, interacts with the external environment E, then an input action appears X(t) (external force for the pendulum and the source of energy for the circuit) and the continuous-deterministic model of such a system will have the form

From the point of view of the general scheme of the mathematical model X(t) is the input (control) action, and the state of the system S in this case can be considered as an output characteristic, i.e., assume that the output variable coincides with the state of the system at a given time y =z.

Possible applications. When solving problems of systems engineering, the problems of managing large systems are of great importance. Pay attention to systems automatic control- a special case of dynamical systems described D-schemes and highlighted in a separate class of models due to their practical specifics.

When describing automatic control processes, they usually adhere to the presentation of a real object in the form of two systems: control and controlled (control object). The structure of a general multidimensional automatic control system is shown in Fig. 2, where are designated endogenous variables:
- vector of input (master) influences;
- vector of disturbing influences;
- vector of error signals;
- vector of control actions; exogenous variables:
- the vector of states of the system S;
is a vector of output variables, usually
=
.

Rice. 2. The structure of the automatic control system

A modern control system is a set of software and hardware tools that ensure the achievement of a specific goal by the control object. How accurately the control object achieves a given goal can be judged for a one-dimensional system by the state coordinate at (t). The difference between the given at backside (t) and valid at (t) the law of change of the controlled variable is a control error . If the prescribed law of change of the controlled quantity corresponds to the law of change of the input (master) action, i.e.
, then
.

Systems for which control errors
at all times are called ideal. In practice, the implementation of ideal systems is impossible. So the error h"(t) - a necessary element of automatic control based on the principle of negative feedback, since to bring the output variable into conformity y(t) its specified value uses information about the deviation between them. The task of the automatic control system is to change the variable y(t) according to a given law with a certain accuracy (with an acceptable error). When designing and operating automatic control systems, it is necessary to select the following system parameters S, which would provide the required control accuracy, as well as the stability of the system in the transient process.

If the system is stable, then the behavior of the system in time is of practical interest, the maximum deviation of the controlled variable is at (t) in the transient process, the time of the transient process, etc. Conclusions about the properties of automatic control systems of various classes can be made in the form of differential equations that approximately describe the processes in the systems. The order of the differential equation and the values of its coefficients are completely determined by the static and dynamic parameters of the system. S.

So using D-scheme allows to formalize the process of functioning of continuously-deterministic systems S and evaluate their main characteristics using an analytical or simulation approach implemented in the form of an appropriate language for modeling continuous systems or using analog and hybrid computing facilities.

Classification in any area of expertise is essential. It allows you to generalize the accumulated experience, to streamline the concepts of the subject area. The rapid development of mathematical modeling methods and the variety of areas of their application led to the emergence of a large number of models of various types and to the need to classify models into those categories that are universal for all models or are necessary in the field of the constructed model, for example. Let's give an example of some categories: area of use; taking into account the time factor (dynamics) in the model; branch of knowledge; the way the models are presented; the presence or absence of random (or uncertain) factors; type of efficiency criterion and imposed restrictions, etc.

Analyzing the mathematical literature, we have identified the most common signs of classifications:

1. According to the implementation method (including the formal language), all mathematical models can be divided into analytical and algorithmic.

Analytical - Models that use a standard mathematical language. Simulation - models in which a special modeling language or a universal programming language is used.

Analytical models can be written in the form of analytical expressions, i.e. in the form of expressions containing a countable number of arithmetic operations and transitions to the limit, for example:. An algebraic expression is a special case of an analytic expression, it provides an exact meaning as a result. There are also constructions that allow you to find the resulting value with a given accuracy (for example, the expansion of an elementary function in a power series). Models using this technique are called approximate.

In turn, analytical models are broken down into theoretical and empirical models. Theoretical models reflect real structures and processes in the objects under study, that is, they are based on the theory of their work. Empirical models are built on the basis of studying the reactions of an object to changes in environmental conditions. In this case, the theory of the object's operation is not considered, the object itself is a so-called "black box", and the model is a certain interpolation dependence. Empirical models can be built from experimental data. These data are obtained directly on the objects under study or with the help of their physical models.

If a process cannot be described in the form of an analytical model, it is described using a special algorithm or program. This model is algorithmic. When constructing algorithmic models, numerical or simulation approaches are used. In the numerical approach, the set of mathematical relations is replaced by a finite-dimensional analogue (for example, the transition from a function of a continuous argument to a function of a discrete argument). Then a computational algorithm is constructed, i.e. sequences of arithmetic and logical operations. The found solution of the discrete analogue is taken as an approximate solution to the original problem. In the simulation approach, the modeling object itself is discretized, and models of individual elements of the system are built.

2. According to the form of presentation of mathematical models, there are:

1) An invariant model is a mathematical model that is represented by a system of equations (differential, algebraic) without taking into account the methods for solving these equations.

2) Algebraic model - the ratio of the models is associated with the chosen numerical solution method and written in the form of an algorithm (sequence of calculations).

3) Analytical model - is an explicit dependence of the desired variables on the given values. Such models are obtained on the basis of physical laws, or as a result of direct integration of the original differential equations using tabular integrals. They also include regression models obtained on the basis of experimental results.

4) The graphical model is presented in the form of graphs, equivalent circuits, diagrams and the like. To use graphic models, there must be a rule of unambiguous correspondence of the conditional images of the elements of the graphic and the components of the invariant mathematical model.

3. Depending on the type of efficiency criterion and imposed restrictions, the models are subdivided into linear and non-linear. In linear models, the efficiency criterion and imposed constraints are linear functions of the model variables (otherwise, nonlinear models). The assumption about the linear dependence of the efficiency criterion and the set of imposed constraints on the model variables is quite acceptable in practice. This makes it possible to use a well-developed linear programming apparatus for making decisions.

4. Taking into account the factor of time and area of use, they distinguish static and dynamic models... If all quantities included in the model do not depend on time, then we have a static model of an object or a process (a one-time slice of information on an object). Those. a static model is a model in which time is not a variable. A dynamic model allows you to see changes in an object over time.

5. Depending on the number of parties making a decision, there are two types of mathematical models: descriptive and normative... There are no decision makers in the descriptive model. Formally, the number of such sides in the descriptive model is zero. A typical example of such models is the queuing system model. Reliability theory, graph theory, probability theory, statistical test method (Monte Carlo method) can also be used to build descriptive models.

There are many aspects to the normative model. In principle, two types of normative models can be distinguished: optimization models and game-theoretic models. In optimization models, the main task of developing solutions is technically reduced to strict maximization or minimization of the efficiency criterion, i.e. such values of the controlled variables are determined at which the efficiency criterion reaches an extreme value (maximum or minimum).

To develop solutions displayed by optimization models, along with classical and new variational methods (extremum search), methods of mathematical programming (linear, nonlinear, dynamic) are most widely used. The game-theoretic model is characterized by a multiplicity of the number of sides (at least two). If there are two parties with opposite interests, then game theory is used, if the number of parties is more than two and coalitions and compromises are impossible between them, then the theory of non-coalition games is used n persons.

6. Depending on the presence or absence of random (or uncertain) factors, there are deterministic and stochastic mathematical models. In deterministic models, all relationships, variables and constants are specified precisely, which leads to an unambiguous definition of the resulting function. A deterministic model is constructed in cases where the factors influencing the outcome of the operation lend themselves to sufficiently accurate measurement or assessment, and random factors are either absent or they can be neglected.

If some or all of the parameters included in the model are by their nature random variables or random functions, then the model belongs to the class of stochastic models. In stochastic models, the distribution laws of random variables are set, which leads to a probabilistic estimate of the resulting function and reality is displayed as a certain random process, the course and outcome of which is described by certain characteristics of random variables: mathematical expectations, variances, distribution functions, etc. The construction of such a model is possible if there is sufficient factual material to assess the necessary probability distributions or if the theory of the phenomenon under consideration allows one to determine these distributions theoretically (based on the formulas of the probability theory, limit theorems, etc.).

7. Depending on the goals of modeling, there are descriptive, optimization and management models. In descriptive (from Latin descriptio - description) models, the laws of change of model parameters are investigated. For example, a model of motion of a material point under the influence of applied forces based on Newton's second law:. By specifying the position and acceleration of a point at a given moment in time (input parameters), mass (intrinsic parameter) and the law of variation of the applied forces (external influences), it is possible to determine the coordinates of the point and the speed at any moment in time (output data).

Optimization models are used to determine the best (optimal), based on a certain criterion, the parameters of the simulated object or methods of controlling this object. Optimization models are built using one or more descriptive models and have several criteria for determining the optimality. Restrictions in the form of equalities or inequalities related to the features of the object or process under consideration can be imposed on the range of values of the input parameters. An example of an optimization model is the compilation of a food ration in a certain diet (the calorie content of a product, price values of the cost, etc., act as input data).

Management models are used to make decisions in various areas of purposeful human activity, when several alternatives are selected from the whole set of alternatives and the general decision-making process is a sequence of such alternatives. For example, the choice of a report for promotion from several prepared by students. The complexity of the problem lies both in the uncertainty about the input data (a report was prepared independently or someone else's work was used) and goals (the scientific nature of the work and its structure, the level of presentation and the level of training of the student, the results of the experiment and the conclusions obtained). Since the optimality of the decision made in the same situation can be interpreted in different ways, the form of the optimality criterion in management models is not fixed in advance. Methods for the formation of optimality criteria depending on the type of uncertainty are considered in the theory of choice and decision making, based on game theory and operations research.

8.Distinguish by the research method analytical, numerical and simulation models. An analytical model is a formalized description of a system that allows one to obtain an explicit solution to an equation using a well-known mathematical apparatus. The numerical model is characterized by a dependence that allows only partial numerical solutions for specific initial conditions and quantitative parameters of the model. A simulation model is a set of descriptions of the system and external influences, algorithms for the functioning of the system or the rules for changing the state of the system under the influence of external and internal disturbances. These algorithms and rules do not make it possible to use the available mathematical methods of analytical and numerical solution, but they allow simulating the process of the system's functioning and fixing the characteristics of interest. Further, some analytical and simulation models will be considered in more detail, the study of these types of models is associated with the specifics of the professional activity of students in the indicated direction of training.

1.4. Graphical representation of mathematical models

In mathematics, the forms of connection between quantities can be represented by equations of the form of an independent variable (argument), y- dependent variable (function). In the theory of mathematical modeling, the independent variable is called the factor, and the dependent variable is called the response. Moreover, depending on the area of constructing a mathematical model, the terminology is somewhat modified. Some examples of definitions of factor and response, depending on the field of study, are shown in Table 1.

Table 1. Some definitions of the concepts "factor" and "response"

Presenting a mathematical model graphically, we will consider factors and responses as variables, the values of which belong to the set of real numbers.

Graphical representation of the mathematical model is some response surface corresponding to the arrangement of points in k- dimensional factor space X... Only one-dimensional and two-dimensional response surfaces can be visualized. In the first case, this is a set of points on a real plane, and in the second, a set of points that form a surface in space (to represent such points, it is convenient to use level lines - a way to represent the surface relief of a space built in a two-dimensional factor space X(Fig. 8).

The area in which the response surface is defined is called domain of definition X *. This area is, as a rule, only a part of the total factor space. X(X*Ì X) and is allocated using constraints imposed on control variables x i written as equalities:

x i = C i , i = 1,…, m;

f j(x) = C j, j = 1,…, l

or inequalities:

x i min £ x i£ x i max, i= 1,…, k;

f j(x) £ C j, j = 1,…, n,

In this case, the functions f j(x) can depend both simultaneously on all variables and on some part of them.

Constraints such as inequalities characterize either physical constraints on the processes in the object under study (for example, temperature constraints), or technical constraints associated with the operating conditions of the facility (for example, the limiting cutting speed, limitations on raw materials reserves).

The possibilities of studying models essentially depend on the properties (relief) of the response surface, in particular, on the number of “vertices” available on it and its contrast. The number of peaks (valleys) determines modality response surfaces. If in the domain of definition on the response surface there is one vertex (valley), the model is called unimodal.

The nature of the function change in this case can be different (Fig. 9).

The model can have break points of the first kind (Fig. 9 (a)), break points of the second kind (Fig. 9 (b)). Figure 9 (c) shows a continuously differentiable unimodal model.

For all three cases presented in Figure 9, the general requirement of unimodality is met:

if W (x *) is an extremum of W, then from the condition x 1< x 2 < x* (x 1 >x 2> x *) it follows W (x 1)< W(x 2) < W(x*) , если экстремум – максимум, или W(x 1) >W (x 2)> W (x *), if the extremum is a minimum, that is, as the distance from the extremal point is increased, the value of the function W (x) continuously decreases (increases).

Along with unimodal models, polymodal models are considered (Fig. 10).

Another important property of the response surface is its contrast, which shows the sensitivity of the resulting function to changes in factors. The contrast is characterized by the values of the derivatives. Let's demonstrate the contrast characteristics using the example of a two-dimensional response surface (Fig. 11).

Dot a located on a "slope" characterizing equal contrast for all variables x i (i= 1,2), point b is located in a "ravine" in which different contrast for different variables (we have a poor conditionality of the function), point With is located on a "plateau" where the contrast is low for all variables x i indicates the proximity of the extremum.

1.5. Basic methods for constructing mathematical models

Let us give the classification of methods of formalized representation of modeled systems Volkova V.N. and Denisova AA. The authors highlight analytical, statistical, set-theoretic, linguistic, logical, graphic methods. The basic terminology, examples of theories developing on the basis of the described classes of methods, as well as the scope and possibilities of their application are proposed in Appendix 1.

In the practice of modeling systems, analytical and statistical methods are most widely used.

1) Analytical methods for constructing mathematical models.

The terminological apparatus of analytical methods for constructing mathematical models is based on the concepts of classical mathematics (formula, function, equation and system of equations, inequality, derivative, integral, etc.). These methods are characterized by the clarity and validity of terminology using the language of classical mathematics.

On the basis of analytical concepts, such mathematical theories as classical mathematical analysis (for example, methods for studying functions), and modern foundations of mathematical programming and game theory have arisen and developed. In addition, mathematical programming (linear, nonlinear, dynamic, integer, etc.) contains both means of setting the problem and expands the possibilities of proving the adequacy of the model, in contrast to a number of other areas of mathematics. Ideas of optimal mathematical programming for solving economic (in particular, solving the problem of optimal cutting of a plywood sheet) problems were proposed by L.V. Kantorovich.

Let us explain the features of the method using an example.

Example. Suppose that for the production of two types of products A and V you need to use three types of raw materials. At the same time, for the manufacture of a unit of production of the type A 4 units are consumed. raw materials of the first type, 2 units. 2nd and 3rd units 3rd type. For the manufacture of a unit of production of the type V 2 units are consumed. raw materials of the 1st type, 5 units. 2nd type and 4 units. 3rd type of raw materials. There are 35 units in the factory warehouse. raw materials of the 1st type, 43 - of the 2nd, 40 - of the 3rd type. From the sale of a unit of production of the type A the factory has a profit of 5 thousand rubles, and from the sale of a unit of production of the form V profit is 9 thousand rubles. It is necessary to draw up a mathematical model of the problem, which provides for maximum profit.

The consumption rates of each type of raw material for the manufacture of a unit of this type of product are given in the table. It also indicates the profit from the sale of each type of product and the total amount of raw materials of this type that can be used by the enterprise.

Let us denote by x 1 and x 2 volume of products manufactured A and V respectively. The cost of the first grade material for the plan will be 4x 1 + 2x 2, and they should not exceed stocks, i.e. 35 kg:

4x 1 + 2x 2 35.

Restrictions on material of the second grade are similar:

2x 1 + 5x 2 43,

and on the material of the third grade

3x 1 + 4x 2 40.

Profit from sales x 1 units of production A and x 2 units of production B will be z = 5x 1+ 9x 2(objective function).

We got the problem model:

A graphical solution to the problem is shown in Figure 11.

Optimal (best, i.e. the maximum of the function z) the solution to the problem is at point A (the solution is explained in Chapter 5).

Got that x 1=4,x 2= 7, function value z at point A:.

Thus, the value of the maximum profit is 83 thousand rubles.

In addition to the graphical one, there are also a number of special methods for solving the problem (for example, the simplex method) or applied software packages that implement them are used. Depending on the type of the objective function, linear and nonlinear programming are distinguished, depending on the nature of the variables, integer programming is distinguished.

The general features of mathematical programming can be distinguished:

1) the introduction of the concept of an objective function and constraints are means of setting the problem;

2) it is possible to combine dissimilar criteria in one model (different dimensions, in the example - stocks of raw materials and profit);

3) the mathematical programming model allows going to the border of the range of permissible values of variables;

4) the possibility of implementing a step-by-step algorithm for obtaining results (step-by-step approximation to the optimal solution);

5) clarity, achieved through the geometric interpretation of the problem, which helps in cases where it is impossible to solve the problem formally.

2) Statistical methods for constructing mathematical models.

Statistical methods for constructing mathematical models became widespread and began to be widely used with the development of probability theory in the 19th century. They are based on the probabilistic laws of random (stochastic) events, reflecting real phenomena. The term "stochastic" is a clarification of the concept of "random", indicates predetermined, definite reasons affecting the process, and the concept of "random" is characterized by independence from the impact or absence of such reasons.

Statistical patterns are presented in the form of discrete random variables and patterns of the appearance of their values or in the form of continuous dependences of the distribution of events (processes). The theoretical foundations of building stochastic models are described in detail in Chapter 2.

Control questions

1. Formulate the main problem of mathematical modeling.

2. Give the definition of a mathematical model.

3. List the main disadvantages of the experimental approach in the study.

4. List the main stages of building a model.

5. List the types of mathematical models.

6. Give a brief description of the types of models.

7. What form does the mathematical model take when presented geometrically?

8. How are mathematical models of analytical type specified?

Tasks

1. Make a mathematical model for solving the problem and classify the model:

1) Determine the maximum capacity of a cylindrical bucket, the surface of which (without a lid) is S.

2) The enterprise ensures regular production with a trouble-free supply of components from two subcontractors. Probability of refusal in delivery from the first of the subcontractors -, from the second -. Find the likelihood of an enterprise failure.

2. The Malthus model (1798) describes the reproduction of a population at a rate proportional to its size. In discrete form, this law is a geometric progression:; or. The law, written in the form of a differential equation, is a model of exponential population growth and describes well the growth of cell populations in the absence of any limitation:. Set initial conditions and demonstrate how the model works.

The initial information in the construction of MM of the processes of functioning of systems are data on the purpose and operating conditions of the investigated (projected) system S. This information determines the main goal of modeling, the requirements for MM, the level of abstraction, and the choice of a mathematical modeling scheme.

Concept mathematical scheme allows us to consider mathematics not as a method of calculation, but as a method of thinking, a means of formulating concepts, which is most important in the transition from a verbal description to a formalized representation of the process of its functioning in the form of some MM.

When using the mat. scheme, first of all, the researcher of the system should be interested in the question of the adequacy of the display in the form of specific schemes of real processes in the system under study, and not the possibility of obtaining an answer (solution result) to a specific research question.

For example, the representation of the process of functioning of an ICS for collective use in the form of a network of queuing schemes makes it possible to describe well the processes occurring in the system, but with complex laws of incoming flows and service flows, it does not make it possible to obtain results in an explicit form.

Mathematical scheme can be defined as a link in the transition from a meaningful to a formalized description of the process of functioning of the system, taking into account the impact of the external environment. Those. there is a chain: a descriptive model - a mathematical scheme - a simulation model.

Each specific system S is characterized by a set of properties, which are understood as values that reflect the behavior of the modeled object (real system) and the conditions of its functioning in interaction with the external environment (system) E.

When constructing the MM of the system S, it is necessary to solve the question of its completeness. The completeness of modeling is regulated mainly by the choice of the boundaries "System S - environment E". Also, the problem of simplifying the MM should be solved, which helps to highlight the main properties of the system, discarding the secondary goals of modeling.

MM of the simulation object, i.e. of the system S can be represented as a set of quantities describing the process of functioning of a real system and in the general case forming the following subsets:

A set of X - input influences on Sх i Х, i = 1… n x;

The totality of external environment influences v l V, l = 1… n v;

The set of internal (intrinsic) parameters of the system h k H, k = 1… n h;

The set of output characteristics of the system y j Y, j = 1… n y.

In the listed sets, controlled and uncontrolled quantities can be distinguished. In general, X, V, H, Y are disjoint sets containing both deterministic and stochastic components. Input actions E and internal parameters S are independent (exogenous) variables.Output characteristics - dependent variables (endogenous)... The operation process S is described by the operator F S:

(1)

Output trajectory. F S - the law of functioning S.F S can be a function, functional, logical conditions, algorithm, table or verbal description of rules.

Algorithm of functioning A S - a method for obtaining output characteristics taking into account input influences Obviously, the same FS can be implemented in different ways, i.e. using many different A S.

Relation (1) is a mathematical description of the behavior of the object S modeling in time t, i.e. reflects it dynamic properties... (1) is a dynamic model of the system S. For static conditions MM there are mappings X, V, H into Y, i.e. (2)

Relationships (1), (2) can be specified by formulas, tables, etc.

Also, relationships in some cases can be obtained through the properties of the system at specific points in time, called states.

The states of the system S are characterized by vectors:

and , where at the moment t l  (t 0, T)

at time t ll  (t 0, T), etc. k = 1 ... n Z.

Z 1 (t), Z 2 (t)… Z k (t) are the coordinates of a point in the k-dimensional phase space. Each implementation of the process will correspond to a certain phase trajectory.

The set of all possible values of states () is called the state space of the object of modeling Z, and z k Z.

System state S in the time interval t 0 , where the input, internal parameters and the effects of the external environment, which took place during the time interval t * - t 0 using 2 vector equations:

; (3)

otherwise: . (5)

Time in mod. S can be considered on the simulation interval (t 0, T) both continuous and discrete, i.e. quantized on a segment of length t.

Thus, under the MM of an object we mean a finite set of variables () together with mathematical connections between them and characteristics.

Modeling is called deterministic if the operators F, Ф are deterministic, i.e. for a specific input, the output is deterministic. Deterministic modeling is a special case of stochastic modeling. In practice, modeling objects in the field of system analysis at the primary stages of research is more rational to use standard mathematical schemes: diff. equations, finite and probabilistic automata, QS, etc.

Not possessed. such a degree of generality as models (3), (4), typical mathematical schemes have the advantage of simplicity and clarity, but with a significant narrowing of the scope of application.

As deterministic models, when a random fact is not taken into account in the study, differential, integral and other equations are used to represent systems operating in continuous time, and finite automata and finite difference schemes are used to represent systems operating in discrete time.

At the beginning of stochastic models (taking into account a random factor), probabilistic automata are used to represent systems with discrete time, and queuing systems (QS) are used to represent systems with continuous time. The so-called aggregate models.

Aggregate models (systems) make it possible to describe a wide range of research objects with a reflection of the systemic nature of these objects. It is with an aggregate description that a complex object is divided into a finite number of parts (subsystems), while maintaining connections, ensuring the interaction of parts.

16 Mathematical schemes for modeling systems.

The main approaches to the construction of mathematical models of the system. Continuously deterministic models. Discrete-deterministic models. Discrete stochastic models. Continuous stochastic models. Network models. Combined models.

The main approaches to the construction of mathematical models of the system.

Mathematical schemes

Real processes are displayed in the form of specific diagrams. Mat. schemes - the transition from a meaningful description to a formal description of the system, taking into account the impact of the environment.

Formal Object Model

Model of the simulation object,

i.e. systems S, can be represented as a set of quantities,

describing the process of functioning of a real system and generating

in general the following subsets:

Aggregate input actions per system

Xi, ex, (e-character belongs)i=1; nx

Aggregate environmental influences

vl eVl = 1; nv

Aggregate internal (own) parameters systems

hkeHk = 1; nh

Aggregate output characteristics systems

yJeYj = 1; ny

You can distinguish between managed and unmanaged variables.

When modeling systems, input influences, environmental influences and internal parameters contain both deterministic and stochastic components.

input influences, environmental influences E and the internal parameters of the system are independent (exogenous) variables.

System operation process S described in time by the operator Fs, which in the general case transforms exogenous variables into endogenous ones in accordance with relations of the form:

y(t) = Fs (x, v, h, t) - all with vektori.

The system functioning law Fs can be specified in the form of a function, functional, logical conditions, in algorithmic and tabular forms, or in the form of a verbal correspondence rule.

The concept of the functioning algorithm As - a method for obtaining output characteristics taking into account the input actions, the effects of the external environment and the intrinsic parameters of the system.

The states of the system are also introduced - the properties of the system at specific points in time.

The totality of all possible values of states constitutes the state space of an object.

Thus, the chain of equations of the object "input - states - output" allows you to determine the characteristics of the system:

Thus, under mathematical model of the object(real system) understand a finite subset of variables (x (t), v (t), h(t)) together with mathematical relationships between them and characteristics y (t).

Typical schemes

At the initial stages of the study, standard schemes are used. : differential equations, finite and probabilistic automata, queuing systems, Petri nets, etc.

Differential, integral, integro-differential and other equations are used to represent systems operating in continuous time as deterministic models, when random factors are not taken into account in the study, and finite automata and finite-difference schemes are used to represent systems operating in discrete time. ...

Probabilistic automata are used as stochastic models (taking into account random factors) to represent systems with discrete time, and queuing systems are used to represent systems with continuous time, etc.

Continuously deterministic models

Let us consider the features of the continuously deterministic approach using an example, using Mat. models differential equations.

Differential equations are those equations in which functions of one variable or several variables are unknown, and the equation includes not only their functions, but their derivatives of various orders.

If the unknowns are functions of several variables, then the equations are called - partial differential equations. If unknown functions of one independent variable, then ordinary differential equations.

General mathematical relationship for deterministic systems:

Discrete-deterministic models.

DDM are subject to review automata theory (TA)... TA is a section of theoretical cybernetics that studies devices that process discrete information and change their internal states only at acceptable times.

State machine is called an automaton, in which the set of internal states and input signals (and, consequently, the set of output signals) are finite sets.

Finite state machine has many internal states and input signals, which are finite sets. Machine given by the F-scheme: F = ,

where z, x, y are, respectively, finite sets of input and output signals (alphabets) and a finite set of internal states (alphabet). z0ÎZ - initial state; j (z, x) - transition function; y (z, x) - exit function.

The automaton operates in discrete automaton time, the moments of which are cycles, that is, equal time intervals adjacent to each other, each of which corresponds to constant values of the input, output signal and internal state. An abstract automaton has one input and one output channel.

To define an F - automaton, it is necessary to describe all elements of the set F = , i.e. input, internal and output alphabets, as well as transition and output functions. To set the work of F - automata, the tabular, graphical and matrix methods are most often used.

In the tabular way of setting, transition and output tables are used, the rows of which correspond to the input signals of the automaton, and the columns - to its states.

Work description F- Miles submachine gun tables of transitions j and outputs y are illustrated by table (1), and the description of F - Moore's automaton - is illustrated by the table of transitions (2).

Table 1



Transitions


…………………………………………………………



…………………………………………………………

table 2






…………………………………………………………

Examples of the tabular way of specifying F - the Mealy automaton F1 with three states, two input and two output signals, are given in table 3, and for F - the Moore automaton F2 - in table 4.

Table 3



Transitions

Table 4

Another way of defining a finite state machine uses the concept of a directed graph. The automaton graph is a set of vertices corresponding to different states of the automaton and connecting the vertices of the graph arcs corresponding to certain transitions of the automaton. If the input signal xk causes a transition from the state zi to the state zj, then on the automaton graph the arc connecting the vertex zi with the vertex zj is denoted by xk. In order to set the transition function, the graph arcs must be marked with the corresponding output signals.

Rice. 1. Graphs of the automata of Mealy (a) and Moore (b).

When solving modeling problems, a matrix definition of a finite state machine is often a more convenient form. In this case, the matrix of connections of the automaton is a square matrix C = || cij ||, the rows of which correspond to the initial states, and the columns to the transition states.

Example. For the previously considered Moore automaton F2, we write the state matrix and the output vector:

;

Discrete stochastic models

Let Ф be the set of all possible pairs of the form (zk, yi), where уi is an element of the output

subset Y. We require that any element of the set G induces

on the set Ф some distribution law of the following form:

Elements from Ф (z1, y2) (z1, y2zk, yJ-1) (zK, yJ)

(xi, zs) b11 b1bK (J-1) bKJ

Information networks "href =" / text / category / informatcionnie_seti / "rel =" bookmark "> processing of computer information from remote terminals, etc.

At the same time, typical for

the operation of such objects is the random appearance of applications (requirements) for

service and termination of service at random times,

that is, the stochastic nature of the process of their functioning.

QS is understood as a dynamic system designed to efficiently service a random flow of applications with limited system resources. The generalized structure of the QS is shown in Figure 3.1.

Rice. 3.1. SMO scheme.

Homogeneous claims arriving at the input of the QS are divided into types, depending on the generating cause, the intensity of the flow of claims of type i (i = 1 ... M) is denoted by li. The totality of applications of all types is the incoming flow of the QS.

Service of applications is carried out m channels.

Distinguish between universal and specialized service channels. For a universal channel of type j, the distribution functions Fji (t) of the duration of servicing of claims of an arbitrary type are considered known. For specialized channels, the distribution functions for the service duration of channels of certain types of claims are undefined, the assignment of these claims to this channel.

Q - circuits can be investigated analytically and by simulation models. The latter provides great versatility.

Let's consider the concept of queuing.

In any elementary act of servicing, two main components can be distinguished: the expectation of service by the claim and the actual servicing of the claim. This can be displayed in the form of some i-th service device Pi, consisting of a claim accumulator, in which there can be simultaneously li = 0 ... LiH claims, where LiH is the capacity of the i-th accumulator, and a claim service channel, ki.

Rice. 3.2. Schematic diagram of the CMO device

Each element of the servicing device Pi receives streams of events: the stream of claims wi to the accumulator Hi, and the stream of servicing ui to the channel ki.

By the flow of events(PS) is a sequence of events that occur one after another at some random moments in time. Distinguish between streams of homogeneous and heterogeneous events. Homogeneous The PS is characterized only by the moments of arrival of these events (causing moments) and is given by the sequence (tn) = (0 £ t1 £ t2… £ tn £…), where tn is the moment of arrival of the nth event - a non-negative real number. The TSA can also be specified as a sequence of time intervals between the n-th and n-1-th events (tn).

Heterogeneous PS is called a sequence (tn, fn), where tn - causing moments; fn - a set of event attributes. For example, it can be assigned to one or another source of claims, the presence of a priority, the ability to serve one or another type of channel, etc.

The claims served by the channel ki and the claims that left the server Pi for various reasons not serviced form the output stream yiÎY.

The process of functioning of the service device Pi can be represented as a process of changing the states of its elements in time Zi (t). The transition to a new state for Pi means a change in the number of requests that are in it (in the channel ki and the accumulator Hi). That. the vector of states for Pi has the form:, where are the states of the drive, (https://pandia.ru/text/78/362/images/image010_20.gif "width =" 24 height = 28 "height =" 28 "> = 1 - there is one request in the storage ..., = - the storage is fully occupied; - the state of the channel ki (= 0 - the channel is free, = 1 the channel is busy).

Q-diagrams of real objects are formed by the composition of many elementary service devices Pi. If ki different service devices are connected in parallel, then there is multi-channel service (multichannel Q-circuit), and if devices Pi and their parallel compositions are connected in series, then multi-phase service takes place (multi-phase Q-circuit).

To define a Q-scheme, it is also necessary to describe the algorithms for its functioning, which determine the rules for the behavior of claims in various ambiguous situations.

Depending on the place of occurrence of such situations, there are algorithms (disciplines) for waiting for claims in the accumulator Нi and for servicing claims on the channel ki. The heterogeneity of the flow of applications is taken into account by introducing a priority class - relative and absolute priorities.

That. A Q-scheme describing the process of functioning of a QS of any complexity is uniquely defined as a set of sets: Q = .

Network models.

For a formal description of the structure and interaction of parallel systems and processes, as well as for the analysis of cause-and-effect relationships in complex systems, Petri Nets, called N-schemes, are used.

Formally, the N-scheme is given by a quadruple of the form

N = ,

where B is a finite set of symbols called positions, B ≠ O;

D is a finite set of symbols called transitions D ≠ O,

B ∩ D ≠ O; I - input function (direct incidence function)

I: B × D → (0, 1); О - output function (inverse incidence function),

О: B × D → (0, 1). Thus, the input function I maps the transition dj to

the set of input positions bj I (dj), and the output function O maps

transition dj to the set of output positions bj О (dj). For every transition

dj https://pandia.ru/text/78/362/images/image013_14.gif "width =" 13 "height =" 13 "> B | I (bi, dj) = 1),

O (dj) = (bi B | O (dj, bi) = 1),

i = 1, n; j = 1, m; n = | B |, m = | D |.

Similarly, for each position bi B, the definitions are introduced

set of input transitions of position I (bi) and output transitions

position O (bi):

I (bi) = (dj D | I (dj, bi,) = 1),

O (bi) = (dj D | O (bi, dj) = 1).

A Petri net is a bipartite directed graph consisting of two types of vertices - positions and transitions, connected by arcs; vertices of the same type cannot be connected directly.

An example of a Petri net. White circles indicate positions, stripes - transitions, black circles - labels.

Orientation arcs connect positions and transitions, with each arc directed from an element of one set (position or transition) to an element of another set

(transition or position). An N-design graph is a multigraph, since it

admits the existence of multiple arcs from one vertex to another.

Decomposition "href =" / text / category / dekompozitciya / "rel =" bookmark "> decomposition a complex system is represented as a multilevel structure of interconnected elements combined into subsystems of various levels.

An aggregate acts as an element of the A-diagram, and the connection between the aggregates (inside the S system and with the external environment E) is carried out using the conjugation operator R.

Any unit is characterized by the following sets: times T, input X and output Y signals, states Z at each time moment t. The state of the unit at time tT is denoted as z (t) Z,

and the input and output signals as x (t) X and y (t) Y, respectively.

We will assume that the transition of the aggregate from the state z (t1) to the state z (t2) ≠ z (t1) occurs in a short time interval, i.e., there is a jump δz.

The transitions of the unit from the state z (t1) to z (t2) are determined by the intrinsic (internal) parameters of the unit itself h (t) H and the input signals x (t) X.

At the initial time moment t0, the states z have values equal to z0, i.e., z0 = z (t0), given by the distribution law of the process z (t) at time t0, namely J. Assume that the process of functioning of the unit in the case of action input signal xn is described by a random operator V. Then, at the moment when the input signal arrives at the unit tnT

xn you can determine the state

z (tn + 0) = V.

We denote the half-time interval t1< t ≤ t2 как (t1, t2], а полуинтервал

t1 ≤ t< t2 как .

The collection of random operators V and U is considered as an operator of transitions of the aggregate to new states. In this case, the process of functioning of the unit consists of jumps of states δz at the moments of arrival of input signals x (operator V) and changes in states between these moments tn and tn + 1 (operator U). No restrictions are imposed on the operator U; therefore, jumps of states δz at times that are not times of arrival of input signals x are admissible. In what follows, the moments of jumps δz will be called special moments of time tδ, and states z (tδ) - special states of the A-scheme. To describe the jumps of the states δz at special times tδ, we will use the random operator W, which is a special case of the operator U, i.e.

z (tδ + 0) = W.

In the set of states Z, a subset Z (Y) is distinguished such that if z (tδ) reaches Z (Y), then this state is the moment of issuing the output signal determined by the output operator

y = G.

Thus, by an aggregate we mean any object defined by an ordered collection of the considered sets T, X, Y, Z, Z (Y), H and random operators V, U, W, G.

The sequence of input signals, arranged in the order of their arrival in the A-scheme, will be called an input message or x-message. A sequence of output signals, ordered with respect to the time of issue, will be called an output message or y-message.

IF BRIEFLY

Continuously deterministic models (D-schemes)

They are used to study systems operating in continuous time. Differential, integral, integro-differential equations are mainly used to describe such systems. In ordinary differential equations, a function of only one independent variable is considered, and in partial differential equations, functions of several variables.

As an example of the application of D-models, one can cite the study of the operation of a mechanical pendulum or an electric oscillatory circuit. The technical basis of D-models is made up of analog computers (AVM) or the currently rapidly developing hybrid computers (GVM). As you know, the basic principle of research on a computer is that according to the given equations, the researcher (user of the AVM) assembles a circuit from separate typical nodes - operational amplifiers with the inclusion of circuits for scaling, damping, approximation, etc.

The structure of the ABM changes in accordance with the form of the reproducible equations.

In a digital computer, the structure remains unchanged, but the sequence of operation of its nodes changes in accordance with the program laid down in it. Comparison of AVM and digital computer clearly shows the difference between simulation and statistical modeling.

ABM implements a simulation model, but, as a rule, does not use the principles of statistical modeling. In digital computers, most of the simulation models are based on the study of random numbers, processes, i.e., on statistical modeling. Continuous-deterministic models are widely used in mechanical engineering in the study of automatic control systems, the choice of damping systems, the identification of resonance phenomena and oscillations in technology.
etc.

Discrete-deterministic models (F-circuits)

Operate with discrete time. These models are the basis for studying the operation of an extremely important and widespread class of discrete automata systems today. For the purpose of their research, an independent mathematical apparatus of the theory of automata has been developed. On the basis of this theory, the system is considered as an automaton that processes discrete information and changes, depending on the results of its processing, its internal states.

This model is based on the principles of minimizing the number of elements and nodes in a circuit, device, optimization of the device as a whole and the sequence of operation of its nodes. Along with electronic circuits, a striking representative of the machines described by this model is a robot that controls (according to a given program) technological processes in a given deterministic sequence.

The numerical control machine is also described by this model. The choice of the sequence of processing parts on this machine is carried out by setting up the control unit (controller), which generates control signals at certain points in time / 4 /.

Automata theory uses the mathematical apparatus of Boolean functions that operate on two possible values of the signals, 0 and 1.

Automata are divided into automata without memory, automata with memory. The description of their work is done using tables, matrices, graphs that display the transitions of the machine from one state to another. Analytical evaluations for any kind of description of the operation of the machine are very cumbersome and even with a relatively small number of elements, nodes that make up the device, they are practically impracticable. Therefore, the study of complex circuits of automata, which undoubtedly include robotic devices, is carried out using simulation.

Discrete stochastic models (P-schemes)

They are used to study the work of probabilistic automata. In automata of this type, transitions from one state to another are carried out under the influence of external signals and taking into account the internal state of the automaton. However, unlike T-automata, these transitions are not strictly deterministic, but can occur with certain probabilities.

An example of such a model is a discrete Markov chain with a finite set of states. The analysis of F-schemes is based on the processing and transformation of transition probability matrices and the analysis of probability graphs. Already for the analysis of relatively simple devices, the behavior of which is described by F-circuits, it is advisable to use simulation. An example of such a simulation is given in clause 2.4.

Continuous stochastic models (Q-schemes)

They are used in the analysis of a wide class of systems considered as queuing systems. As a service process, processes that are different in their physical nature can be represented: product supply flows to an enterprise, flows of custom-made components and products, flows of parts on an assembly line, flows of control actions from the control center of the ACS to workplaces and return requests for information processing in a computer etc.

Typically, these flows depend on many factors and specific situations. Therefore, in most cases, these flows are random in time with the possibility of changes at any time. The analysis of such schemes is carried out on the basis of the mathematical apparatus of the queuing theory. These include a continuous Markov chain. Despite the significant advances made in the development of analytical methods, queuing theory, analysis of Q-schemes by analytical methods can be carried out only with significant simplifying assumptions and assumptions. A detailed study of most of these schemes, especially such complex ones as process control systems, robotic systems, can only be carried out using simulation.

Generalized models (A-diagrams)

Based on the description of the processes of functioning of any systems based on the aggregate method. With an aggregate description, the system is divided into separate subsystems, which can be considered convenient for mathematical description. As a result of such a division (decomposition), a complex system is presented in the form of a multilevel system, the individual levels (aggregates) of which are amenable to analysis. Based on the analysis of individual aggregates and taking into account the laws of interconnection of these aggregates, it is possible to carry out a comprehensive study of the entire system.

, Yakovlev Systems. 4th ed. - M .: Higher school, 2005 .-- S. 45-82.

Mathematical schemes for modeling systems

Pros and cons of simulation

The main dignity simulation in the study of complex systems:

· The ability to explore the features of the process of functioning of the system S in any conditions;

· Due to the use of a computer, the duration of tests is significantly reduced in comparison with a full-scale experiment;

· The results of full-scale tests of a real system or its parts can be used for simulation;

· Flexibility of varying the structure, algorithms and parameters of the modeled system when searching for the optimal version of the system;

· For complex systems - this is the only practically realizable method for studying the process of systems functioning.

The main limitations simulation modeling:

· For a complete analysis of the characteristics of the systems functioning process and the search for the optimal option, it is required to reproduce the simulation experiment many times, varying the initial data of the problem;

· Large expenditures of computer time.

The effectiveness of machine modeling. When simulating, it is necessary to ensure the maximum efficiency of the system model. Efficiency usually defined as some difference between some measure of the value of the results obtained during the operation of the model, and the costs that were invested in its development and creation.

The effectiveness of simulation modeling can be assessed by a number of criteria:

Accuracy and reliability of simulation results,

Time of building and working with the model M,

The expense of machine resources (time and memory),

· The cost of developing and operating the model.

The best measure of effectiveness is a comparison of the results obtained with real studies. Using a statistical approach, with a certain degree of accuracy (depending on the number of realizations of a machine experiment), averaged characteristics of the system's behavior are obtained.

The total costs of computer time are made up of the time for input and output for each simulation algorithm, the time for performing computational operations, taking into account the access to RAM and external devices, as well as the complexity of each modeling algorithm and the planning of experiments.

Mathematical schemes.Mathematical model Is a collection of mathematical objects (numbers, variables, sets, vectors, matrices, etc.) and relations between them, which adequately reflects the physical properties of the created technical object. The process of forming a mathematical model and using it for analysis and synthesis is called mathematical modeling.

When constructing a mathematical model of the system, it is necessary to resolve the issue of its completeness. The completeness of the model is regulated mainly by the choice of the boundary “system S- Wednesday E". Also, the problem of simplifying the model should be solved, which helps to highlight, depending on the purpose of modeling, the main properties of the system, discarding the secondary ones.

In the transition from a meaningful to a formal description of the process of functioning of the system, taking into account the impact of the external environment, apply mathematical scheme as a link in the chain "descriptive model - mathematical scheme - mathematical (analytical and / or simulation) model".

Formal model of the object. Object model (systems S) can be represented as a set of quantities that describe the process of functioning of a real system:

A set of input influences on the system

x i = X,i =;

A set of environmental influences

v j = V, j= ;

A set of internal (intrinsic) parameters of systems

h k = H, k =;

Set of output characteristics of the system

y j = Y, j =.

In general x i, v j, h k, y j are elements of disjoint subsets and contain both deterministic and stochastic components.

Input influences, environmental influences E and the internal parameters of the system are independent (exogenous) variables, which in vector form have, respectively, the form ( 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 are dependent (endogenous) variables and in vector form have the form: ( t) = (at 1 (t), at 2 (t), …, at nY(t)). You can distinguish between managed and unmanaged variables.

System operation process S described in time by the operator F S, which transforms exogenous variables into endogenous ones in accordance with relations of the form

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

The set of dependences of the output characteristics of the system on time y j(t) for all types j = called output trajectory (t). Dependence (2.1) is called the system functioning law F S, which is specified in the form of a function, functional, logical conditions, in algorithmic, tabular form or in the form of a verbal matching rule. Algorithm of functioning A S is called the method of obtaining the output characteristics taking into account the input influences ( t), environmental influences ( t) and the system's own parameters ( t). The same law of functioning F S systems S can be implemented in various ways, i.e. using many different algorithms of functioning A S.

Mathematical models are called dynamic(2.1) if mathematical relations describe the behavior of the object (system) of modeling in time t, i.e. reflect dynamic properties.

For static models, a mathematical model is a mapping between two subsets of the properties of a modeled object Y and ( X, V, H) at a certain moment, which in vector form can be written as

= f(, , ). (2.2)

Relations (2.1) and (2.2) can be specified in different ways: analytically (using formulas), graphically, tabularly, etc. These relationships can be obtained through the properties of the system S at specific points in time, called states. State of the system S characterized by vectors

" = (z " 1, z " 2, …, Z "k) and "" = (z "" 1 ,z "" 2 ,…, Z "" k),

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

If we consider the process of functioning of the system S as a sequential change of states z 1 (t), z 2 (t), …, z k(t), then they can be interpreted as the coordinates of a point in k-dimensional phase space... Moreover, each implementation of the process will correspond to a certain phase trajectory. The set of all possible values of states () is called state space object of modeling Z, and
z kÎ Z.

System states S at the moment t 0 < t * £ T are completely determined by the initial conditions 0 = ( z 0 1 , z 0 2 , …, z 0 k) [where z 0 1 = z 1 (t 0),
z 0 2 = z 2 (t 0), …, z 0 k = z k(t 0)], input actions ( t), internal parameters ( t) and the effects of the external environment ( t) that took place in the time interval t * – t 0, using two vector equations

(t) = Ф (0,,,, t); (2.3)

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

The first equation for the initial state 0 and exogenous variables,, determines the vector function ( t), and the second according to the obtained value of the states ( t) Are endogenous variables at the output of the system ( t). Thus, the chain of equations of the object "input - states - output" allows you to determine the characteristics of the system

(t) = F [Ф (0,,,, t)]. (2.5)

In general, the time in the system model S can be considered on the simulation interval (0, T) both continuous and discrete, i.e. quantized into segments of length D t time units each when T = m D t, where m = - the number of sampling intervals.

Thus, under mathematical model object (real system) understand a finite subset of variables (( t), (t), (t)) together with mathematical connections between them and characteristics ( t).

If the mathematical description of the modeling object does not contain random elements or they are not taken into account, i.e. if we can assume that in this case the stochastic influences of the external environment ( t) and stochastic internal parameters ( t) are absent, then the model is called deterministic in the sense that the characteristics are uniquely determined by deterministic inputs

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

Obviously, the deterministic model is a special case of the stochastic model.

Typical mathematical schemes. In the practice of modeling objects in the field of systems engineering and systems analysis at the initial stages of system research, it is more rational to use typical mathematical schemes: differential equations, finite and probabilistic automata, queuing systems, Petri nets, aggregate systems, etc.

Typical mathematical schemes have the advantages of simplicity and clarity. Differential, integral, integro-differential and other equations are used to represent systems operating in continuous time as deterministic models, when random factors are not taken into account in the study, and finite automata and finite-difference schemes are used to represent systems operating in discrete time. Probabilistic automata are used as stochastic models (taking into account random factors) to represent systems with discrete time, and queuing systems are used to represent systems with continuous time. Petri nets are used to analyze cause-and-effect relationships in complex systems, where several processes occur simultaneously in parallel. To describe the behavior of continuous and discrete, deterministic and stochastic systems (for example, ASOIU), a generalized (universal) approach based on an aggregate system can be applied. In an aggregate description, a complex object (system) is divided into a finite number of parts (subsystems), while maintaining the connections that ensure the interaction of parts.

Thus, when constructing mathematical models of the processes of the functioning of systems, the following main approaches can be distinguished: continuous-deterministic ( D-scheme); discrete-deterministic ( F-scheme); discrete stochastic ( R-scheme); continuous-stochastic ( Q-scheme); network ( N-scheme); generalized or universal ( a-scheme).

2.2. Continuously deterministic models ( D-scheme)

Basic relations... Let us consider the features of the continuous-deterministic approach on the example of using differential equations as mathematical models. Differential Equations are called equations in which functions of one or several variables are unknown, and the equation includes not only functions, but also their derivatives of various orders. If unknown functions of several variables, then the equations are called partial differential equations, otherwise, when considering a function of one independent variable, the equations are called ordinary differential equations.

The general mathematical relation for deterministic systems (2.6) will be

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

where " = d/dt, = (y 1 , y 2 , …, y n) and = ( f 1 , f 2 , …, f n) – n-dimensional vectors; (, t) Is a vector function that is defined on some ( n+1) -dimensional (, t) set and is continuous.

Mathematical schemes of this kind are called D-circuits(eng. dynamic), they reflect the dynamics of the system under study, and time usually serves as an independent variable on which unknown unknown functions depend t.

In the simplest case, an ordinary differential equation has the form:

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

Consider the simplest example of formalizing the process of functioning of two elementary circuits of a different nature: mechanical S M (swing of the pendulum, fig. 2.1, a) and electrical S K (oscillatory circuit, Fig. 2.1, b).

Rice. 2.1. Elementary systems

The process of small oscillations of the pendulum is described by the ordinary differential equation

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

where m M, l M is the mass and length of the suspension of the pendulum; g- acceleration of gravity; F(t) Is the angle of deflection of the pendulum at the moment of time t.

From this equation of free oscillation of the pendulum, estimates of the characteristics of interest can be found. For example, the period of swing of a pendulum

T M = 2p.

Similarly, the processes in the electric oscillatory circuit are described by the ordinary differential equation

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

where L K, C K - inductance and capacitance of the capacitor; q(t) Is the charge of the capacitor at the moment of time t.

From this equation, you can get various estimates of the characteristics of the process in the oscillatory circuit. For example, the period of electrical oscillations

T M = 2p.

Obviously, introducing the notation 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), we obtain an ordinary second-order differential equation describing the behavior of this closed-loop system:

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

where h 0 , h 1 , h 2 - system parameters; z(t) Is the state of the system at the moment
time t.

Thus, the behavior of these two objects can be investigated on the basis of the general mathematical model (2.9). In addition, it should be noted that the behavior of the pendulum (system S M) can be studied using an electric oscillatory circuit (system S TO).

If the system under study S(pendulum or contour) interacts with the external environment E, then the input action appears x(t) (external force for the pendulum and the source of energy for the circuit), and the continuous-deterministic model of such a system will have the form:

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 general mathematical model (see clause 2.1) x(t) is the input (control) action, and the state of the system S in this case, it can be considered as an output characteristic, i.e. the output variable matches the state of the system at a given time y = z.

Possible applications D-scheme... To describe linear control systems, like any dynamical system, inhomogeneous differential equations have constant coefficients

where,,…, - unknown function of time and its derivatives; and are known functions.

Using, for example, the VisSim software package designed for simulation of processes in control systems that can be described by differential equations, we simulate the solution of an ordinary inhomogeneous differential equation

where is some required function of time on an interval with zero initial conditions, we take h 3 =1, h 2 =3, h 1 =1, h 0 =3:

Representing the given equation with respect to the highest of the derivatives, we obtain the equation

which can be modeled using a set of building blocks of the VisSim package: arithmetic blocks - Gain (multiplication by a constant), Summing-Junction (adder); integration blocks - Integrator (numerical integration), Transfer Function (setting an equation represented as a transfer function); blocks for setting signals - Const (constant), Step (unit function in the form of a "step"), Ramp (linearly increasing signal); blocks-receivers of signals - Plot (display in the time domain of signals that are analyzed by the researcher during the simulation).

In fig. 2.2 shows a graphical representation of this differential equation. The input of the leftmost integrator corresponds to a variable, the input of the middle integrator -, and the input of the rightmost integrator -. The output of the rightmost integrator corresponds to the variable y.

A particular case of dynamical systems described D-schemes are automatic control systems(SPG)and regulation(SAR). A real object is presented in the form of two systems: control and controlled (control object). The structure of a general multidimensional automatic control system is shown in Fig. 2.3, where indicated endogenous variables: ( t) Is the vector of input (master) influences; ( t) Is the vector of disturbing influences; " (t) Is the vector of error signals; "" (t) - vector of control actions; exogenous variables: ( t) Is the state vector of the system S; (t) Is a vector of output variables, usually ( t) = (t).

Rice. 2.2. Graphical representation of the equation

The control system is a set of software and hardware tools that ensure the achievement of a specific goal by the control object. How accurately an object reaches a given goal can be judged (for a one-dimensional system) by the state coordinate y(t). The difference between the given y ass ( t) and valid y(t) the law of change of the controlled variable is a control error " (t) = y ass ( t) – y(t). If the prescribed law of change of the controlled quantity corresponds to the law of change of the input (master) action, i.e. x(t) = y ass ( t), then " (t) = x(t) – y(t).

Systems for which control errors " (t) = 0 at all times are called ideal... In practice, the implementation of ideal systems is impossible. The task of the automatic control system is to change the variable y(t) according to a given law with a certain accuracy (with an acceptable error). The system parameters must ensure the required control accuracy, as well as the stability of the system in the transient process. If the system is stable, then analyze the behavior of the system in time, the maximum deviation of the controlled variable y(t) in the transient process, the time of the transient process, etc. The order of the differential equation and the value of its coefficients are completely determined by the static and dynamic parameters of the system.

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

УC - control system; OU - control object

So using D-schemes allows you to formalize the process of functioning of continuously deterministic systems S and evaluate their main characteristics using an analytical or simulation approach implemented in the form of an appropriate language for modeling continuous systems or using analog and hybrid computing facilities.

2.3. Discrete-deterministic models ( F-scheme)

Basic relations... Let us consider the features of the discrete-deterministic approach on the example of using the theory of automata as a mathematical apparatus. The system is represented in the form of an automaton as a device with input and output signals that processes discrete information and changes its internal states only at acceptable times. State machine an automaton is called, in which the sets of internal states, input and output signals are finite sets.

Abstractly finite automata can be represented as a mathematical scheme ( F-schema), characterized by six elements: a finite set X input signals (input alphabet); finite set Y output signals (output alphabet); finite set Z internal states (internal alphabet or alphabet of states); initial state z 0 , z 0 Î Z; transition function j ( z, x); output function y ( z, x). Automatic machine set F-scheme: F = á Z, X, Y, y, j, z 0 ñ, operates in discrete time, the moments of which are clocks, each of which corresponds to constant values of the input and output signals and internal states. We denote the state, as well as the input and output signals corresponding to t-th clock at t= 0, 1, 2, ..., through z(t), x(t), y(t). Moreover, by the condition z(0) = z 0, and z(t)Î Z, x(t)Î X, y(t)Î Y.

An abstract state machine has one input and one output channel. At every moment t= 0, 1, 2, ... discrete time F-the machine is in a certain state z(t) from the set Z states of the automaton, and at the initial moment of time t= 0 it is always in the initial state z(0) = z 0. In the moment t being able z(t), the automaton is able to perceive the signal on the input channel x(t)Î X and output the signal on the output channel y(t) = y [ z(t),x(t)], passing to the state z ( t+1) = j [ z(t), x(t)], z(t)Î Z, y(t)Î Y... An abstract finite state machine implements some mapping of the set of words of the input alphabet X on a lot of weekend words
alphabet Y... In other words, if the input of the state machine set to the initial state z 0, supply letters of the input alphabet in a certain sequence x(0), x(1), x(2), ..., i.e. input word, then the letters of the output alphabet will appear sequentially at the output of the machine y(0), y(1), y(2),…, forming an output word.

Thus, the work of the state machine occurs according to the following scheme: in each t-th clock to the input of the machine in the state z(t), some signal is given x(t), to which it reacts with the transition ( t+1) of the th clock to the new state z(t+1) and giving some output signal. The above can be described by the following equations: for F-automaton of the first kind, also called automatic Miles,

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-automaton of the second kind

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 automaton of the second kind, for which

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

those. the exit function is independent of the input variable x(t) is called Moore's assault rifle.

Thus, equations (2.15) - (2.19), which completely define
F-automaton are a special case of equations (2.3) and (2.4), when
system S- deterministic and a discrete signal arrives at its only input X.

By the number of states, finite state machines with memory and without memory are distinguished. Automata with memory have more than one state, and automata without memory (combinational or logic circuits) have only one state. In this case, according to (2.16), the operation of the combinational circuit is that it assigns to each input signal x(t) certain output signal y(t), i.e. implements a logical function of the form

y(t) = y [ x(t)], t= 0, 1, 2, … .

This function is called boolean if the alphabet X and Y to which the signal values belong x and y, consist of two letters.

By the nature of the counting of discrete time, finite state machines are divided into synchronous and asynchronous. In synchronous F-automatons the times at which the automaton "reads" the input signals are determined by compulsory synchronizing signals. After the next synchronizing signal, taking into account "read" and in accordance with equations (2.15) - (2.19), a transition to a new state occurs and a signal is issued at the output, after which the machine can perceive the next value of the input signal. Thus, the reaction of the machine to each value of the input signal ends in one cycle, the duration of which is determined by the interval between adjacent synchronizing signals. Asynchronous F- the machine reads the input signal continuously and therefore, responding to a sufficiently long input signal of a constant value x, it can, as follows from (2.15) - (2.19), change the state several times, giving the corresponding number of output signals, until it goes into a stable one, which can no longer be changed by this input signal.

Possible applications F-scheme. To set the final F-automaton, it is necessary to describe all the elements of the set F= <Z, X, Y, y, j, z 0>, i.e. input, internal and output alphabets, as well as functions of transitions and outputs, and among the set of states it is necessary to single out the state z 0, in which the automaton is in the state t= 0. There are several ways to set the job F-automatons, but the most commonly used are tabular, graphical and matrix.

In the tabular method, tables of transitions and outputs are set, the rows of which correspond to the input signals of the automaton, and the columns - to its states. The first column on the left corresponds to the initial state z 0. At intersection i th line and k-th column of the transition table, the corresponding value j ( z k, x i) function of transitions, and in the table of outputs - the corresponding value of y ( z k, x i) output functions. For F- Moore's automaton both tables can be combined.

Work description F-automaton Miles with tables of transitions j and outputs y is illustrated in Table. 2.1, and the description F-More's automaton - by the transition table (Table 2.2).

Table 2.1

X i	z k
z 0	z 1	…	z k
Transitions
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)
Outputs
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)

Examples of tabular way of setting F-automatic Miles F 1 are given in table. 2.3, and for F-moore machine F 2 - in table. 2.4.

Table 2.3

x i	z k
z 0	z 1	z 2
Transitions
x 1	z 2	z 0	z 0
x 2	z 0	z 2	z 1
Outputs
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

In the graphical way of defining a finite state machine, the concept of a directed graph is used. The automaton graph is a set of vertices corresponding to different states of the automaton and connecting the vertices of the graph arcs corresponding to certain transitions of the automaton. If the input signal x k causes a transition from the state z i in a state z j, then on the graph of the automaton there is an arc connecting the vertex z i with top z j, denoted x k... In order to set the function of the outputs, the graph arcs must be marked with the corresponding output signals. For Miles machines, this marking is done as follows: if the input signal x k acts on the state z i, then we get an arc outgoing from z i and marked x k; this arc is additionally marked with an output signal y= y ( z i, x k). For a Moore automaton, a similar marking of the graph is as follows: if the input signal x k, acting on a certain state of the automaton, causes a transition to the state z j, then the arc directed to z i and marked x k, additionally celebrate the weekend
signal y= y ( z j, x k).

In fig. 2.4. a, b given earlier in the tables F-Mile machines F 1 and Moore F 2 respectively.

Rice. 2.4. Automata graphs a - Miles and b - Moore

For the matrix assignment of the finite automaton, the matrix of connections of the automaton is square WITH=||with ij||, rows correspond to initial states and columns correspond to transition states. Element with ij = x k/y s standing at the intersection
i th line and j-th column, in the case of the Miles automaton corresponds to the input signal x k causing the transition from the state z i in a state z j, and the output signal y s generated by this transition. For the Miles machine F 1, considered above, the matrix of compounds has the form:

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 –

If the transition from the state z i in a state z j occurs under the action of several signals, the element of the matrix c ij is a set of input-output pairs for this transition, connected by a disjunction sign.

For F-moore machine element with ij is equal to the set of input signals at the transition ( 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 indicating the state z i.

For the above F-moore machine F2 the matrix of connections and the vector of outputs are of the form:

–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 automata, the condition of uniqueness of transitions is satisfied: an automaton in a certain state cannot pass into more than one state under the action of any input signal. Applied to the graphical way of setting F-automaton, this means that in the automaton graph, two or more edges marked with the same input signal cannot go out of any vertex. And in the matrix of connections of the machine WITH any input signal must not occur more than once on each line.

For F-automatic condition z k called sustainable, if for any input x i ÎX for which j ( z k, x i) = z k, j ( z k,x i) = y k. F-the machine is called asynchronous, if every state z k ÎZ stable.

Thus, the concept in the discrete-deterministic approach to studying the properties of objects on models is a mathematical abstraction, convenient for describing a wide class of processes of functioning of real objects in automated control systems. Via F- of an automaton, it is possible to describe objects that are characterized by the presence of discrete states, and the discrete nature of work in time - these are elements and nodes of a computer, control, regulation and control devices, systems of time and space switching in information exchange technology, etc.

2.4. Discrete stochastic models ( R-scheme)

Basic relations... Let us consider the features of constructing mathematical schemes with a discrete-stochastic approach on probabilistic (stochastic) automata. In general probabilistic automaton
R-schemes(English probabijistic automat) can be defined as a discrete line-to-line converter of information with memory, the functioning of which in each cycle depends only on the state of the memory in it, and can be described statistically.

Let's introduce the mathematical concept R-automaton, using the concepts introduced for F-automat. Consider the set G, whose elements are all possible pairs ( x i, z s), where x i and z s- elements of the input subset X and subsets of states Z, respectively. If there are two such functions j and y that they are used to carry out the mappings G®Z and G®Y, then they say that F = X, Y, j, y> defines an automaton of deterministic type.

Let's consider a more general mathematical scheme. Let
Ф - set of all possible pairs of the form ( z k, y i), where i- element of the output subset Y... We require that any element of the set G induced on the set Ф some distribution law of the following form:

Wherein b kj= 1, where b kj- the probabilities of the transition of the automaton to the state z k and the appearance of the signal at the output y j if he was able z s and at its input at this moment in time the signal was received x i... The number of such distributions presented in the form of tables is equal to the number of elements of the set G... We denote the set of these tables by B. Then the four elements P = called a probabilistic automaton
(R-automaton).

Possible applications P-scheme. Let the elements of the set G induce some distribution laws on subsets Y and Z, which can be represented, respectively, in the form:

Wherein z k = 1 and q j = 1, where z k and q j - transition probabilities
R-automatic machine in state z k and the appearance of the output signal y k provided that
R z s and its input received an input signal x i.

If for everyone k and j the relation holds q j z k = b kj, then such
R-the machine is called Miles's probabilistic automaton... This requirement means the fulfillment of the condition of independence of distributions for the new state R-automatic device and its output signal.

Now let the definition of the output signal R- the automaton depends only on the state in which the automaton is in a given cycle of work. In other words, let each element of the output subset Y induces a probability distribution of outputs that has the following form:

Here s i = 1, where s i- the probability of the appearance of the output signal y i at at words and that R-the machine was in a state z k.

If for everyone k and i the relation holds z k s i =b ki then such
R-the machine is called Moore's probabilistic automaton. Concept
R-Miley and Moore's automata is introduced by analogy with the deterministic
F-automat. A particular case R- automaton defined as P=X, Y, B> are automata in which either the transition to a new state or the output signal are determined deterministically. If the output signal
R-automaton is determined deterministically, then such an automaton is called
Y-... Likewise,
Z-deterministic probabilistic automaton called R- an automaton in which the choice of a new state is deterministic.

Example 2.1. Let it be given Y-deterministic P-machine

In fig. 2.5 shows a directed transition graph of this automaton. The vertices of the graph are associated with the states of the automaton, and the arcs are associated with possible transitions from one state to another. The arcs have weights corresponding to the transition probabilities p ij, and the values of the output signals induced by these states are written near the vertices of the graph. It is required to estimate the total final probabilities of staying of this P-automaton in states z 2 and z 3 .

Rice. 2.5. Probability automaton graph

Using the analytical approach, one can write down the known relations from the theory of Markov chains and obtain a system of equations for determining the final probabilities. In this case, the initial state z 0 can be ignored, since the initial distribution does not affect the values of the final probabilities. Then we have

where with k- final probability of stay R-Automatic device in a state z k.

We get the system of equations

We add to these equations the normalization condition With 1 + With 2 + With 3 + With 4 = 1. Then, solving the system of equations, we obtain With 1 = 5/23, With 2 = 8/23, With 3 = 5/23,
With 4 = 5/23. In this way, With 2 + With 3 = 13/23 = 0.5652. In other words, with endless work given in this example Y-deterministic
R-automaton at its output a binary sequence is formed with the probability of occurrence of one equal to 0.5652.

Similar R-automatons can be used as generators of Markov sequences, which are necessary in the construction and implementation of processes for the functioning of systems S or environmental influences E.

2.5. Continuous stochastic models ( Q-scheme)

Basic relations... We will consider the features of the continuous-stochastic approach using the example of typical mathematical Q- schemes - queuing systems(English queueing system).

As a service process, various in their physical nature processes of functioning of economic, production, technical and other systems can be represented, for example: flows of supply of products to a certain enterprise, flows of parts and components on the assembly line of a workshop, requests for processing computer information from remote terminals and etc. In this case, a characteristic feature of the operation of such objects is the random appearance of claims (requirements) for servicing and completion of servicing at random times, i.e. the stochastic nature of the process of their functioning.

By the flow of events is called a sequence of events that occur one after another at some random moments in time. Distinguish between streams of homogeneous and heterogeneous events. Stream of events called homogeneous, if it is characterized only by the moments of arrival of these events (causing moments) and is given by the sequence ( t n} = {0 £ t£ 1 t 2 ... £ t n£ … }, where t n - moment of arrival P- th event is a non-negative real number. A homogeneous stream of events can also be specified as a sequence of time intervals between P- m and the (n - 1) th events (t n), which is unambiguously associated with the sequence of challenging moments ( t n} , where t n = t n– t n -1 ,P³ 1, t 0 = 0, those. t 1 = t 1 . A stream of heterogeneous events is called a sequence ( t n, f n} , where t n - challenging moments; f n - set of event signs. For example, in relation to the service process, for a non-uniform flow of claims, membership in a particular source of claims, the presence of a priority, and the ability to serve one or another type of channel can be specified.

In any elementary act of servicing, two main components can be distinguished: the expectation of service by the claim and the actual servicing of the claim. This can be depicted in the form of some i-th service device P i(Fig. 2.6), consisting of the accumulator of orders H i, which can simultaneously be j i= applications where L i H– capacity
i-go storage, and a channel for servicing requests (or just a channel) K i. For each element of the service device P i streams of events arrive: to the drive H i flow of applications w i, per channel K i - service flow and i.

Rice. 2.6. Application service device

Applications served by the channel K i, and requests that left the device P i unserved for various reasons (for example, due to an overflow of the drive H i), form an output stream y i Î Y, those. the time intervals between the moments of the exit of orders form a subset of the output variables.

Usually, the flow of applications w i ÎW, those. time intervals between the moments of appearance of orders at the entrance K i, forms a subset of unmanaged variables, and the service flow u i ОU, those. the time intervals between the beginning and the end of servicing a claim, forms a subset of controlled variables.

Service device operation process P i can be represented as a process of changing the states of its elements of time z i(t). Transition to a new state for P i means a change in the number of applications that are in it (in the channel K i and in the drive H i). Thus, the vector of states for P i looks like: , where z i H- drive state H i (z i H= 0 - the drive is empty, z i H= 1 - there is one request in the storage, ..., z i H = L i H – the drive is completely full); L i H - storage capacity H i, measured by the number of applications that can fit in it; z i k - channel state K i(z i k = 0 – the channel is free, z i k= 1 - the channel is busy).

Possible applications Q- schemes. In the practice of modeling systems with more complex structural relationships and behavior algorithms, for formalization, not separate service devices are used, but
Q- scheme , formed by the composition of many elementary service devices P i. If the channels K i different service devices are connected in parallel, then multichannel service takes place ( multichannel Q- scheme) , and if the devices P i and their parallel compositions are connected in series, then there is a multiphase service ( multiphase Q- scheme) . So for the job Q- schema must use the conjugate operator R, reflecting the interconnection of structure elements (channels and storage devices) with each other.