16 Mathematical schemes for modeling systems.

Basic approaches to constructing mathematical models of a system. Continuously deterministic models. Discrete-deterministic models. Discrete-stochastic models. Continuous-stochastic models. Network models. Combined models.

Basic approaches to constructing mathematical models of a system.

The initial information when constructing mathematical models of systems functioning processes is data on the purpose and operating conditions of the system being studied (designed) S.

Mathematical schemes

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

Formal object model

Simulation object model,

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

describing the process of functioning of a real system and forming

in general the following subsets:

· totality input influences per system

Xi,еХ,(e-character belongs)i=1; nx

· totality environmental influences

vl eVl=1;nv

· totality internal (own) parameters systems

hkeHk=1;nh

· totality output characteristics systems

yJeYj=1;ny

Controllable and uncontrollable variables can be distinguished.

When modeling systems, input influences, external 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 general transforms exogenous variables into endogenous ones in accordance with relations of the form:

y(t)=Fs(x,v, h,t) – all with vekTori.

The operating law of the system 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 input influences, external environmental influences and the system’s own parameters.

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

The set of all possible state values ​​constitutes the state space of an object.

Thus, the chain of equations of the object “input - states - output” allows us 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 connections 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.

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

As stochastic models (taking into account random factors), probabilistic automata are used to represent discrete-time systems, and queuing systems, etc. are used to represent continuous-time systems.

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

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 the functions of one variable or several variables are unknown, and the equation includes not only their functions but their derivatives of different orders.

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

Mathematical relation for deterministic systems in general form:

Discrete-deterministic models.

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

State machine is an automaton whose set of internal states and input signals (and therefore the set of output signals) are finite sets.

State machine has a set of internal states and input signals, which are finite sets. Machine is 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) - output function.

The automaton operates in discrete automaton time, the moments of which are clock cycles, i.e. 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 specify 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 specify the operation of F-automata, the tabular, graphical and matrix methods are most often used.

In the tabular method of setting, tables of transitions and outputs are used, the rows of which correspond to the input signals of the machine, and the columns correspond to its states.

Description of work F- automatic machine Mili tables of transitions j and outputs y are illustrated by table (1), and the description of F - a Moore machine - by table of transitions (2).

Table 1

Transitions
…………………………………………………………
…………………………………………………………

table 2

…………………………………………………………

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

Table 3

Transitions

Table 4

Another way of specifying a finite automaton uses the concept of a directed graph. The graph of an automaton is a set of vertices corresponding to various 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 state zi to state zj, then on the automaton graph the arc connecting vertex zi to vertex zj is denoted xk. In order to specify the transition function, the arcs of the graph must be marked with the corresponding output signals.

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

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

Example. For the previously considered Moore automaton F2, we write the state matrix and 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 induce

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, characteristic of

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

maintenance and completion of service in random moments time,

i.e. the stochastic nature of the process of their functioning.

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

Rice. 3.1. SMO scheme.

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

Applications are being processed m channels.

There are universal and specialized service channels. For a universal channel of type j, the distribution functions Fji(t) of the duration of servicing requests of an arbitrary type are considered known. For specialized channels, the functions for distributing the duration of servicing channels of requests of some types are uncertain, the assignment of these requests to a given channel.

Q-circuits can be studied analytically and with simulation models. The latter provides greater versatility.

Let's consider the concept of queuing.

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

Rice. 3.2. SMO device diagram

Each element of the servicing device Pi receives streams of events: the drive Hi receives a stream of requests wi, and the channel ki receives a service stream ui.

The flow of events(PS) is a sequence of events occurring one after another at some random moments in time. There are 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. OPS can also be specified as a sequence of time intervals between the n-th and n-1st events (tn).

Heterogeneous A PS is called a sequence (tn, fn), where tn are the causing moments; fn is a set of event attributes. For example, belonging to a particular source of requests, the presence of priority, the ability to be served by a particular type of channel, etc. can be specified.

Requests served by channel ki and requests that left device 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 storage Hi). That. the state vector for Pi has the form: , where are the drive states, (https://pandia.ru/text/78/362/images/image010_20.gif" width="24 height=28" height="28">=1 - there is one request in the drive..., =- the drive is completely occupied; - channel state ki (=0 - channel is free, =1 channel is busy).

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

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

Depending on the location of such situations, there are algorithms (disciplines) for waiting for requests in the Hi storage tank and servicing requests by 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 that describes the process of functioning of a QS of any complexity is uniquely specified as a set of sets: Q = .

Network models.

To formally describe the structure and interaction of parallel systems and processes, as well as to analyze 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),

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

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

transition dj to the set of output positions bj O(dj). For each 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

positions 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 vertices of two types - 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 indicate transitions, black circles indicate marks.

Orienting 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-scheme graph is a multigraph because it

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

Decomposition" href="/text/category/dekompozitciya/" rel="bookmark">Decomposition represents a complex system as a multi-level structure of interconnected elements combined into subsystems of various levels.

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

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

and the input and output signals are 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 over a short time interval, i.e. there is a jump in δz.

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

At the initial moment of time t0, states z have values ​​equal to z0, i.e. z0=z(t0), specified by the distribution law of the process z(t) at time t0, namely J. Let us assume that the process of functioning of the unit in the event of an impact input signal xn is described by a random operator V. Then at the moment the input signal tnT enters the unit

xn you can determine the state

z(tn + 0) = V.

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

t1 ≤ t< t2 как .

The set 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 in 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). There are no restrictions imposed on the operator U, therefore jumps in states δz are allowed at moments of time that are not the moments of arrival of input signals x. In what follows, the moments of jumps δz will be called special moments of time tδ, and the states z(tδ) will be called special states of the A-scheme. To describe state jumps δz at special moments of time 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 allocated such that if z(tδ) reaches Z(Y), then this state is the moment of issuing an output signal determined by the output operator

y = G.

Thus, by an aggregate we will understand 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-circuit will be called an input message or x-message. We call the sequence of output signals, ordered relative to the time of issue, an output message or y-message.

IF IN BRIEF

Continuously deterministic models (D-schemes)

They are used to study systems operating in continuous time. To describe such systems, differential, integral, and integro-differential equations are mainly used. Ordinary differential equations consider a function of only one independent variable, while partial differential equations consider functions of several variables.

An example of the use of D-models is the study of the operation of a mechanical pendulum or an electric oscillatory circuit. The technical basis of D-models is analog computing machines(AVM) or the currently rapidly developing hybrid computers (HCM). As is known, the basic principle of computer research is that, using given equations, a researcher (computer user) assembles a circuit from individual standard units - operational amplifiers with the inclusion of scaling, damping, approximation circuits, etc.

The structure of the AVM changes in accordance with the type of reproducible equations.

In a digital computer, the structure remains unchanged, but the sequence of operation of its nodes changes in accordance with the program embedded in it. A comparison of AVM and CVM 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 simulation models are based on the study of random numbers and processes, i.e., on statistical modeling. Continuously deterministic models are widely used in mechanical engineering in the study of systems automatic control, selection of shock-absorbing systems, identification of resonance phenomena and vibrations in technology
and so on.

Discrete-deterministic models (F-schemes)

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 study, an independent mathematical apparatus of automata theory has been developed. Based on this theory, the system is considered as an automaton that processes discrete information and changes its internal states, depending on the results of its processing.

This model is based on the principles of minimizing the number of elements and nodes in a circuit, device, optimizing the device as a whole and the sequence of operation of its nodes. Along with electronic circuits, a prominent 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.

Machine with numerical program controlled is also described by this model. The selection of the sequence of processing parts on this machine is carried out by setting 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 with two possible signal values ​​0 and 1.

Automata are divided into automata without memory and automata with memory. Their operation is described using tables, matrices, and graphs that display the machine’s transitions from one state to another. Analytical estimates for any type of description of the operation of the machine are very cumbersome and, even with a relatively small number of elements and nodes forming the device, are practically impossible. Therefore the study complex circuits automatic machines, which undoubtedly include robotic devices, are produced using simulation modeling.

Discrete-stochastic models (P-schemes)

They are used to study the operation of probabilistic automata. In machines 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 machine. However, unlike G-automata, these transitions are not strictly deterministic, but can be carried out 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 comparative analysis simple devices, the behavior of which is described by F-schemes, it is advisable to use simulation modeling. An example of such modeling is given in paragraph 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 of different physical nature can be represented: flows of product deliveries 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 automated control system 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 moment. The analysis of such schemes is carried out on the basis of the mathematical apparatus of queuing theory. These include a continuous Markov chain. Despite the significant advances achieved in the development of analytical methods, queuing theory, and analysis of Q-schemes by analytical methods can only be carried out under significant simplifying assumptions and assumptions. A detailed study of most of these schemes, especially such complex ones as automated process control systems and robotic systems, can only be carried out using simulation modeling.

Generalized models (A-schemes)

Based on a description of the functioning processes of any system based on the aggregative 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 partitioning (decomposition), a complex system is presented as a multi-level system, the individual levels (aggregates) of which are amenable to analysis. Based on the analysis of individual units and taking into account the laws of interrelations of these units, it is possible to conduct a comprehensive study of the entire system.

, Yakovlev systems. 4th ed. – M.: Higher School, 2005. – P. 45-82.

To use a computer in solving applied problems, first of all, the applied problem must be “translated” into a formal mathematical language, i.e. for a real object, process or system it must be built mathematical model.

Mathematical models in quantitative form, using logical and mathematical constructs, describe the basic properties of an object, process or system, its parameters, internal and external connections.

For building a mathematical model necessary:

1. carefully analyze a real object or process;
2. highlight its most significant features and properties;
3. define variables, i.e. parameters whose values ​​affect the main features and properties of the object;
4. describe the dependence of the basic properties of an object, process or system on the values ​​of variables using logical-mathematical relationships (equations, equalities, inequalities, logical-mathematical constructions);
5. highlight internal communications object, process or system using restrictions, equations, equalities, inequalities, logical and mathematical constructions;
6. identify external connections and describe them using restrictions, equations, equalities, inequalities, logical and mathematical constructions.

Math modeling, in addition to studying an object, process or system and drawing up their mathematical description, also includes:

1. building an algorithm that models the behavior of an object, process or system;
2. examination adequacy of the model and an object, process or system based on computational and natural experiment;
3. model adjustment;
4. using the model.

The mathematical description of the processes and systems under study depends on:

1. the nature of a real process or system and is compiled on the basis of the laws of physics, chemistry, mechanics, thermodynamics, hydrodynamics, electrical engineering, plasticity theory, elasticity theory, etc.
2. the required reliability and accuracy of the study and research of real processes and systems.

At the stage of selecting a mathematical model, the following are established: linearity and nonlinearity of an object, process or system, dynamism or staticity, stationarity or nonstationarity, as well as the degree of determinism of the object or process under study. In mathematical modeling, one deliberately abstracts from the specific physical nature of objects, processes or systems and mainly focuses on the study of quantitative dependencies between quantities that describe these processes.

Mathematical model is never completely identical to the object, process or system in question. Based on simplification, idealization, it is an approximate description of the object. Therefore, the results obtained from the analysis of the model are approximate. Their accuracy is determined by the degree of adequacy (compliance) between the model and the object.

It usually begins with the construction and analysis of the simplest, most crude mathematical model of the object, process or system in question. In the future, if necessary, the model is refined and its correspondence to the object is made more complete.

Let's take a simple example. It is necessary to determine the surface area of ​​the desk. Typically, this is done by measuring its length and width, and then multiplying the resulting numbers. This elementary procedure actually means the following: a real object (table surface) is replaced by an abstract mathematical model - a rectangle. The dimensions obtained by measuring the length and width of the table surface are assigned to the rectangle, and the area of ​​such a rectangle is approximately taken to be the required area of ​​the table.

However, the rectangle model for a desk is the simplest, most crude model. If you take a more serious approach to the problem, before using a rectangle model to determine the area of ​​the table, this model needs to be checked. Checks can be carried out as follows: measure the lengths of the opposite sides of the table, as well as the lengths of its diagonals and compare them with each other. If, with the required degree of accuracy, the lengths of the opposite sides and the lengths of the diagonals are equal in pairs, then the surface of the table can really be considered as a rectangle. Otherwise, the rectangle model will have to be rejected and replaced with a quadrilateral model general view. With more high demands To improve accuracy, it may be necessary to refine the model even further, for example, to take into account the rounding of the corners of the table.

With this simple example it was shown that mathematical model is not uniquely determined by the object, process or system being studied. For the same table we can adopt either a rectangle model, or a more complex model of a general quadrilateral, or a quadrilateral with rounded corners. The choice of one model or another is determined by the requirement of accuracy. With increasing accuracy, the model has to be complicated, taking into account new and new features of the object, process or system being studied.

Let's consider another example: studying the movement of the crank mechanism (Fig. 2.1).

Rice. 2.1.

For the kinematic analysis of this mechanism, first of all, it is necessary to construct its kinematic model. For this:

1. We replace the mechanism with its kinematic diagram, where all links are replaced hard ties;
2. Using this diagram, we derive the equation of motion of the mechanism;
3. Differentiating the latter, we obtain the equations of velocities and acceleration, which are differential equations of the 1st and 2nd order.

Let's write these equations:

where C 0 is the extreme right position of the slider C:

r – crank radius AB;

l – connecting rod length BC;

– crank rotation angle;

Received transcendental equations present a mathematical model of the motion of a flat axial crank mechanism, based on the following simplifying assumptions:

1. we were not interested in the structural forms and arrangement of the masses included in the mechanism of bodies, and we replaced all the bodies of the mechanism with straight segments. In fact, all the links of the mechanism have mass and a rather complex shape. For example, a connecting rod is a complex assembly, the shape and dimensions of which, of course, will affect the movement of the mechanism;
2. When moving the mechanism under consideration, we also did not take into account the elasticity of the bodies included in the mechanism, i.e. all links were considered as abstract absolutely rigid bodies. In reality, all bodies included in the mechanism are elastic bodies. When the mechanism moves, they will somehow be deformed, and elastic vibrations may even occur in them. All this, of course, will also affect the movement of the mechanism;
3. we did not take into account the manufacturing error of the links, the gaps in the kinematic pairs A, B, C, etc.

Thus, it is important to emphasize once again that the higher the requirements for the accuracy of the results of solving a problem, the greater the need to take into account when building a mathematical model features of the object, process or system being studied. However, it is important to stop here in time, since it is difficult mathematical model can turn into a difficult problem to solve.

A model is most easily constructed when the laws that determine the behavior and properties of an object, process or system are well known, and there is extensive practical experience in their application.

A more complex situation arises when our knowledge about the object, process or system being studied is insufficient. In this case, when building a mathematical model it is necessary to make additional assumptions that are in the nature of hypotheses; such a model is called hypothetical. The conclusions obtained as a result of studying such a hypothetical model are conditional. To verify the conclusions, it is necessary to compare the results of studying the model on a computer with the results of a full-scale experiment. Thus, the question of the applicability of a certain mathematical model to the study of the object, process or system under consideration is not a mathematical question and cannot be solved by mathematical methods.

The main criterion of truth is experiment, practice in the broadest sense of the word.

Building a mathematical model in applied tasks – one of the most complex and critical stages of work. Experience shows that in many cases choosing the right model means solving the problem by more than half. The difficulty of this stage is that it requires a combination of mathematical and special knowledge. Therefore, it is very important that when solving applied problems, mathematicians have special knowledge about the object, and their partners, specialists, have a certain mathematical culture, research experience in their field, knowledge of computers and programming.

The greatest difficulties and most serious errors in modeling arise during the transition from a meaningful to a formal description of research objects, which is explained by the participation in this creative process of teams of different specialties: specialists in the field of systems that need to be modeled (customers), and specialists in the field of machine modeling (performers). ). An effective means for finding mutual understanding between these groups of specialists is the language of mathematical schemes, which allows us to put at the forefront the question of the adequacy of the transition from a meaningful description of the system to its mathematical scheme, and only then decide on a specific method for obtaining results using a computer: analytical or simulation, and possibly combined, i.e. analytical-simulation. In relation to a specific modeling object, i.e., a complex system, the model developer should be helped by specific mathematical schemes that have already been tested for a given class of systems, which have shown their effectiveness in applied research on a computer and are called standard mathematical schemes.

BASIC APPROACHES TO CONSTRUCTION OF MATHEMATICAL MODELS OF SYSTEMS

The initial information when constructing mathematical models of systems functioning processes is data on the purpose and operating conditions of the system under study (design) 5. This information determines the main goal of modeling the system £ and allows us to formulate requirements for the developed mathematical model A/. Moreover, the level of abstraction depends on the range of questions that the system researcher wants to answer using the model, and to some extent determines the choice of mathematical scheme.

Mathematical schemes.

The introduction of the concept of “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 simulation) . When using a mathematical scheme, a researcher of a 5* system should be primarily interested in the question of the adequacy of the representation in the form of specific diagrams of real processes in the system under study, and not in the possibility of obtaining an answer (solution result) to a specific research question. For example, representing the process of functioning of a shared information computing system in the form of a network of queuing schemes makes it possible to well describe the processes occurring in the system, but with complex laws of distribution of incoming flows and service flows, it does not make it possible to obtain results explicitly.

Mathematical scheme can be defined as a link in the transition from a meaningful to a formal description of the process of functioning of a system, taking into account the influence of the external environment, i.e., there is a chain “descriptive model - mathematical scheme - mathematical [analytical and/or simulation] model”.

Each specific L1 system is characterized by a set of properties, which are understood as quantities that reflect the behavior of the simulated 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 a 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.U-environment £>>. The problem of simplifying the model must also be solved, which helps to highlight the main properties of the system, discarding the secondary ones. Moreover, classifying the properties of the system as basic or secondary significantly depends on the purpose of modeling the system ( for example, analysis of probabilistic-time characteristics of the system functioning process, synthesis of the system structure, etc.).

Formal model of the object. The model of the modeling object, i.e., system 5, can be represented as a set of quantities that describe the process of functioning of the real system and form, in the general case, following subsets: collection input influences per system

totality environmental influences

totality internal (own) parameters systems

totality output characteristics systems

In this case, in the listed subsets, controlled and uncontrollable variables can be distinguished. In the general case x„ r/, A*,

at y are elements of disjoint subsets and contain both deterministic and stochastic components.

When modeling the system 5 input influences, external environmental influences E and the internal parameters of the system are independent (exogenous) variables, which in vector form have the corresponding form x (/) = (*! (O, x 2 (0> -" x *x(0)*

" (0=("1 (0. "2(0. . "^(0; l (/)=(*! (0. L 2 (0. ■ . L -N (0). and the output characteristics of the system are dependent (endogenous) variables and in vector form they look like y (0=(y 1 0), y 2 ( 0" > U.gSh

The process of functioning of system 5 is described in time by the operator /* 5, which in the general case transforms exogenous variables into endogenous ones in accordance with relations of the form

The set of dependences of the output characteristics of the system on time yDg) for all types y = 1, p y called output trajectory y ((). Dependency (2.1) is called law of functioning of system B and is designated G 5. In general, the law of system functioning E 5 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 system 5 is the concept functioning algorithm L 5, which is understood as a method for obtaining output characteristics taking into account input influences X(/), environmental influences V(d) and the system’s own parameters AND(/). It is obvious that the same law of operation of system 5 can be implemented different ways, i.e. using many different operating algorithms L$.

Relations (2.1) are a mathematical description of the behavior of a modeling object (system) in time /, i.e., they reflect its dynamic properties. Therefore, mathematical models of this type are usually called dynamic models (systems) .

For static models, the mathematical model (2.1) is a mapping between two subsets of the properties of the modeled object U And (X, V, I), which in vector form can be written as

Relations (2.1) and (2.2) can be specified in various ways: analytically (using formulas), graphically, tabularly, etc. Such relations in a number of cases can be obtained

through the properties of the system 5 at specific points in time, called states. The state of system 5 is characterized by the vectors

Where *; = *!(/"), *2=*2(0" " **=**(0 at the moment /"e(/ 0 , 7); *1 =^(0, *2=*2(P", *£=**(*") at the moment /"b(/ 0, 7), etc., £=1, p g.

If we consider the process of functioning of system 5 as a sequential change of states (/), r 2 (/), G Who are they

can be interpreted as the coordinates of a point in the ^-dimensional phase space, and each implementation of the process will correspond to a certain phase trajectory. The set of all possible state values (G) called state space modeling object Zt and g to e Z.

State of system 5 at the moment of time completely

are determined by the initial conditions 7° = (2° 1,. 2 2°, G° k) [where

*°1 = *1(*o)" *°g = *2 (^o)" -" *°*=**(*o)]" by input influences X(/), internal parameters To(/) and environmental influences V(0, which occurred during the period of time - / 0, using two vector equations

The first equation for the initial state g° and exogenous variables x, V, I determines the vector function (/), and the second one based on the obtained value of the states G(/) - endogenous variables at the system output at(/). Thus, the chain of equations of the object “input - states - output” allows define system characteristics

In general, time in the system model I can be considered over the modeling interval (O, T) both continuous and discrete, i.e. quantized in negative cutting d line A/ time units each, when T=tA1, Where T- 1, t T- number of sampling intervals.

Thus, under mathematical model of the object(real system) understand a finite subset of variables (X (/), b (/), AND(d)) together with mathematical connections between them and characteristics at (/) .

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 V(/) and stochastic internal parameters AND(/) are missing, then the model is called deterministic in the sense that the characteristics are uniquely determined by deterministic input influences

It is obvious that the deterministic model is a special case of the stochastic model.

Typical schemes.

The presented mathematical relationships represent general mathematical schemes and make it possible to describe a wide class of systems. However, in the practice of modeling objects in the field of systems engineering and system 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 having the same degree of generality as the models considered, typical mathematical schemes have the advantages of simplicity and clarity, but with a significant narrowing of application possibilities. As deterministic models, when random factors are not taken into account in the study, differential, integral, integro-differential and other equations are used to represent systems operating in continuous time, and finite-difference automata are used to represent systems operating in discrete time. scheme. As stochastic models (taking into account random factors), probabilistic automata are used to represent discrete-time systems, and queuing systems, etc. are used to represent continuous-time systems.

The listed standard mathematical schemes, naturally, cannot claim to be able to describe on their basis all the processes occurring in large information and control systems. For such systems, in some cases, the use of aggregative models is more promising. Aggregate models (systems) make it possible to describe a wide range of research objects, reflecting the systemic nature of these objects. It is with an aggregative description complex object(system) is divided into a finite number of parts (subsystems), while maintaining the connections that ensure the interaction of the parts.

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

The mathematical schemes discussed in subsequent paragraphs of this chapter should help to operate with various approaches in practical work when modeling specific systems.

Classification in any field of knowledge is necessary. It allows you to generalize the accumulated experience and organize the concepts of the subject area. The rapid development of mathematical modeling methods and the variety of areas of their application have 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. Here is an example of some categories: area of ​​use; taking into account the time factor (dynamics) in the model; branch of knowledge; way of presenting models; the presence or absence of random (or uncertain) factors; type of efficiency criterion and imposed restrictions, etc.

Analyzing the mathematical literature, we identified the most common classification features:

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

Analytical – models that use standard mathematical language. Simulation models are models that use a special modeling language or a universal programming language.

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 analytical expression; it provides an exact value as a result. There are also constructions that allow you to find the resulting value with a given accuracy (for example, expansion of an elementary function into a power series). Models that use this technique are called approximate.

In turn, analytical models are divided 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 operation. Empirical models are built based on studying the object's reactions to changes in environmental conditions. In this case, the theory of operation of the object is not considered; the object itself is a so-called “black box”, and the model is some kind of interpolation dependence. Empirical models can be built based on experimental data. These data are obtained directly from the objects under study or using them. 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 the 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 of 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, they are distinguished:

1) Invariant model – a mathematical model represented by a system of equations (differential, algebraic) without taking into account methods for solving these equations.

2) Algebraic model - the relationships of the models are associated with the selected numerical solution method and are written in the form of an algorithm (sequence of calculations).

3) Analytical model – represents explicit dependences of the sought variables on 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. These also include regression models obtained based on the results of the experiment.

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

3. Depending on the type of efficiency criterion and the imposed restrictions, models are divided into linear and nonlinear. In linear models, the performance criterion and the imposed constraints are linear functions of the model variables (aka nonlinear models). The assumption of a linear dependence of the efficiency criterion and the set of imposed restrictions on the model variables is quite acceptable in practice. This allows you to use a well-developed linear programming apparatus to develop solutions.

4. Considering the time factor and area of ​​use, there are static and dynamic models. If all the quantities included in the model do not depend on time, then we have a static model of an object or process (a one-time snapshot of information on the object). Those. a static model is a model in which time is not a variable. Dynamic model allows you to see changes in an object over time.

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

The normative model has many aspects. In principle, two types of normative models can be distinguished: optimization models and game-theoretic ones. 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 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), mathematical programming methods (linear, nonlinear, dynamic) are most widely used. The game-theoretic model is characterized by a multiplicity of parties (at least two). If there are two parties with opposing 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-cooperative games is used n persons

6. Depending on the presence or absence of random (or uncertain) factors, 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 can be measured or assessed fairly accurately, and random factors are either absent or can be neglected.

If some or all of the parameters included in the model are random variables or random functions by their nature, then the model is classified as a stochastic model. In stochastic models, the laws of distribution of random variables are specified, which leads to a probabilistic assessment of the resulting function and reality is displayed as a certain random process, the course and outcome of which are 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 estimate the necessary probability distributions or if the theory of the phenomenon under consideration allows these distributions to be determined theoretically (based on formulas of probability theory, limit theorems, etc.).

7. Depending on the purposes of modeling, there are descriptive, optimization and management models. In descriptive (from Latin descriptio - description) models, the laws of change in model parameters are studied. For example, a model of the movement of a material point under the influence of applied forces based on Newton’s second law: . Specifying the position and acceleration of a point in this moment time (input parameters), mass (own parameter) and the law of change of applied forces (external influences), you can determine the coordinates of the point and speed at any time (output data).

Optimization models are used to determine the best (optimal), based on some criterion, parameters of the modeled object or methods of controlling this object. Optimization models are built using one or more descriptive models and have several criteria for determining optimality. Restrictions in the form of equalities or inequalities related to the characteristics 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 preparation of a diet for a specific diet (the calorie content of the product, price values, etc. are the input data).

Management models are used to make decisions in various areas of purposeful human activity, when several are selected from the entire set of alternatives and the overall decision-making process is a sequence of such alternatives. For example, choosing a report for promotion from several prepared by students. The complexity of the task lies both in the uncertainty about the input data (whether 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 preparation of the student, the results of the experiment and the conclusions obtained). Since the optimality of a decision made in the same situation can be interpreted in different ways, the type of optimality criterion in management models is not fixed in advance. Methods for forming 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. According to the research method, they distinguish analytical, numerical and simulation models. An analytical model is a formalized description of a system that allows one to obtain an explicit solution to the 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 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 existing mathematical methods for analytical and numerical solutions, but they make it possible to simulate the process of functioning of the system and record the characteristics of interest. Next, some analytical and simulation models will be examined in more detail; the study of these particular types of models is related to the specifics of the professional activities of students in this area of ​​training.

1.4. Graphical representation of mathematical models

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

Table 1. Some definitions of the concepts “factor” and “response”

Representing the mathematical model graphically, we will consider factors and responses to be variables whose values ​​belong to the set of real numbers.

Graphical representation of the mathematical model is some response surface corresponding to the location 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 forming a surface in space (to depict such points it is convenient to use level lines - a way of depicting the surface relief of a space constructed in a two-dimensional factor space X(Fig. 8).

The region in which the response surface is defined is called domain of definition of X *. This region is, as a rule, only part of the complete factor space X(X*Ì X) and is highlighted using restrictions imposed on the control variables x i, written in the form of 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,

At the same time, the functions f j(x) can depend both on all variables simultaneously and on some of them.

Constraints of the type of inequalities characterize either physical restrictions on the processes in the object under study (for example, temperature restrictions), or technical restrictions associated with the operating conditions of the object (for example, top speed cutting, restrictions on raw material reserves).

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

The nature of the change in function may be different (Fig. 9).

The model may have discontinuity points of the first kind (Fig. 9 (a)), discontinuity points of the second kind (Fig. 9(b)). Figure 9(c) shows the 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*) 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 we move away from the extreme point, the value of the function W(x) continuously decreases (increases).

Along with unimodal ones, 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. Contrast is characterized by the values ​​of its derivatives. Let us demonstrate the contrast characteristics using an 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 located in a “ravine” in which there is a different contrast for various variables (we have poor conditionality of the function), point With located on a “plateau” where there is low contrast for all variables x i indicates the proximity of the extremum.

1.5. Basic methods for constructing mathematical models

Let us present the classification of methods for formalized representation of simulated systems by V.N. Volkova. and Denisova A.A.. The authors identified analytical, statistical, set-theoretic, linguistic, logical, and graphical methods. 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 system modeling, analytical and statistical methods are most widely used.

1) Analytical methods for constructing mathematical models.

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

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

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

Example. Let us assume that for the production of two types of products A And IN three types of raw materials must be used. At the same time, for the production of a unit of product of the type A 4 units are consumed. raw materials of the first type, 2 units. 2nd and 3rd units. 3rd type. For the production of a unit of product of the type IN 2 units are consumed. raw materials of the 1st type, 5 units. 2nd type and 4 units. 3rd type of raw material. There are 35 units in the factory warehouse. raw materials of the 1st type, 43 - 2nd, 40 - 3rd type. From sales of a unit of product of the type A the factory has a profit of 5 thousand rubles, and from the sale of a unit of product of the type IN profit is 9 thousand rubles. It is necessary to create a mathematical model of the problem, which provides for obtaining maximum profit.

The consumption rates of each type of raw material for the manufacture of a unit of a given 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 produced A And IN respectively. The cost of first grade material for the plan will be 4x 1 + 2x 2, and they should not exceed reserves, i.e. 35 kg:

4x 1 + 2x 2 35.

The restrictions for second grade material are similar:

2x 1 + 5x 2 43,

and according to third grade material

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 task model:

Graphic solution The tasks are shown in Figure 11.

Optimal (best, i.e. maximum 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 method, there are a number of special methods for solving the problem (for example, the simplex method) or application packages that implement them are used. Depending on the type of objective function, linear and nonlinear programming are distinguished; depending on the nature of the variables, integer programming is distinguished.

We can highlight the general features of mathematical programming:

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

2) it is possible to combine heterogeneous criteria (different dimensions, in the example – raw material reserves and profit) in one model;

3) the mathematical programming model allows reaching the boundary of the region of permissible values ​​of variables;

4) possibility of implementation step-by-step algorithm obtaining results (step-by-step approach to optimal solution);

5) clarity achieved through geometric interpretation of the problem, helping 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 probabilistic patterns of random (stochastic) events that reflect real phenomena. The term “stochastic” is a clarification of the concept “random”, indicating predetermined, specific causes affecting the process, and the concept “random” is characterized by independence from the influence or absence of such causes.

Statistical patterns are presented in the form of discrete random variables and patterns of occurrence of their values ​​or in the form of continuous dependencies of the distribution of events (processes). Theoretical basis The construction of stochastic models is described in detail in Chapter 2.

Control questions

1. Formulate the main problem of mathematical modeling.

2. Define a mathematical model.

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

4. List the main stages of building a model.

5. List the types of mathematical models.

6. Give brief description types of models.

7. What form does a mathematical model, represented geometrically, take?

8. How are mathematical models of analytical type defined?

Tasks

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

1) Determine the maximum capacity of a cylindrical bucket whose surface (without lid) is equal to S.

2) The company ensures regular production of products with a trouble-free supply of components from two subcontractors. The probability of refusal of delivery from the first of the subcontractors is , and from the second - . Find the probability of failure in the operation of the enterprise.

2. 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 well describes the growth of cell populations in the absence of any limitation: . Set initial conditions and demonstrate the model.

MATHEMATICAL SCHEME FOR SYSTEM MODELING

BASIC APPROACHES TO CONSTRUCTION OF MATHEMATICAL MODELS OF SYSTEMS

The initial information when constructing mathematical models of systems functioning processes is data on the purpose and operating conditions of the system being studied (designed) S. This information defines the main purpose of 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 questions that the system researcher wants to answer using the model, and to some extent determines the choice of 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 simulation). 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 representation in the form of specific diagrams of real processes in the system under study, and not in the possibility of obtaining an answer (solution result) to a specific research question. For example, representing the process of functioning of a shared information computing system in the form of a network of queuing schemes makes it possible to well describe the processes occurring in the system, but given the 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 process of system functioning, taking into account the influence of the external environment, i.e. there is a chain “descriptive model - mathematical scheme - mathematical (analytical and/or simulation) model.”

Each specific system S is characterized by a set of properties, which are understood as quantities that reflect the behavior of the simulated 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 a system, it is necessary to resolve the issue of its completeness. The completeness of the model is regulated mainly by the choice of the “system S - environment” boundary E» . The problem of simplifying the model must also be solved, which helps to highlight the main properties of the system, discarding the secondary ones. Moreover, classifying the properties of a system as primary or secondary significantly depends on the purpose of modeling the system (for example, analysis of the probabilistic-time characteristics of the process of system functioning, synthesis of the structure of the system, etc.).

Formal model of the object. The model of the modeling object, i.e., 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: set input influences per system

;

totality environmental influences

;

totality internal (own) parameters systems

;

totality output characteristics systems

.

Moreover, in the listed subsets one can distinguish controlled and uncontrollable variables. In general , , , are elements of disjoint subsets and contain both deterministic and stochastic components.

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

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

. (1)

A set of dependencies of the system output characteristics on time y j (t) for all types
called exit path
. Dependency (1) is called law of system functioningS and is designated F s . In general, the law of system functioning 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 system S is the concept functioning algorithmA s , which is understood as a method for obtaining output characteristics taking into account input influences
, environmental influences
and own system parameters
. It is obvious that the same operating law F s system S can be implemented in various ways, i.e. using many different operating algorithms A s .

Relations (1) are a mathematical description of the behavior of the modeling object (system) in time t, i.e., 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 the modeled object Y And { X, V, N), 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 relations in a number of 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 the vectors

And
,

Where
,
, …,
at a point in time
;
,
, …,
at a point in time
etc.,
.

If we consider the process of functioning of 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 set of all possible state values called state space modeling object Z, and
.

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
, that 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 system output
. Thus, the chain of equations of the object “input-states-output” makes it possible to determine the characteristics of the system

. (5)

In general, time in the system model S can be considered over 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(of a real system) understand a finite subset of variables (
} along with mathematical connections between them and characteristics
.

If the mathematical description of the modeling object does not contain random elements or they are not taken into account, i.e. if it can be assumed that in this case the stochastic influences of the external environment
and stochastic internal parameters
are missing, then the model is called deterministic in the sense that the characteristics are uniquely determined by deterministic input influences

. (6)

It is obvious that the deterministic model is a special case of the stochastic model.

Typical schemes. The presented mathematical relationships represent general mathematical schemes and make it possible to describe a wide class of systems. However, in the practice of modeling objects in the field of systems engineering and system 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 having the same degree of generality as the models considered, typical mathematical schemes have the advantages of simplicity and clarity, but with a significant narrowing of application possibilities. As deterministic models, when random factors are not taken into account in the study, differential, integral, integrodifferential and other equations are used to represent systems operating in continuous time, and finite-difference schemes are used to represent systems operating in discrete time. . As stochastic models (taking into account random factors), probabilistic automata are used to represent discrete-time systems, and queuing systems, etc. are used to represent continuous-time systems.

The listed standard mathematical schemes, naturally, cannot claim to be able to describe on their basis all the processes occurring in large information and control systems. For such systems, in some cases, the use of aggregative models is more promising.

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

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

CONTINUOUS DETERMINISTIC MODELS (D-SCHEMS)

Let us consider the features of the continuously deterministic approach using the example of using differential equations as mathematical models. Differential equations These are equations in which functions of one or more variables are unknown, and the equation includes not only functions, but also their derivatives of various orders. If the unknowns are functions of many 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. Typically, in such mathematical models, time serves as the independent variable on which the unknown unknown functions depend. t. Then the mathematical relation for deterministic systems (6) in general form will be

, (7)

Where
,
And
- P-dimensional vectors;
- vector function that 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, i.e. its behavior in time, they are called D-schemes(English) dynamic).

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

. (8)

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

Rice. 1. Elementary systems

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

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

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

.

Similarly, processes in an 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 one can obtain various estimates of the characteristics of the process in the oscillatory circuit. For example, the period of electrical oscillations

.

It is obvious that by introducing the notation
,
, ,
, we obtain an ordinary differential equation of the second order describing the behavior of this closed system:

Where
- system parameters; z(t) - state of the system at a moment in time t.

Thus, the behavior of these two objects can be studied 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 being studied S, i.e. a pendulum or circuit, interacts with the external environment E, then the input influence appears X(t) (external force for the pendulum and energy source for the circuit) and the continuously 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 an 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 point in time y =z.

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

When describing automatic control processes, they usually adhere to the representation 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 indicated endogenous variables:
- vector of input (setting) influences;
- vector of disturbing influences;
- vector of error signals;
- vector of control actions; exogenous variables:
- system state vector S;
- vector of output variables, usually
=
.

Rice. 2. Structure of the automatic control system

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

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 match the output variable y(t) its set 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 system stability in the transient process.

If the system is stable, then the behavior of the system over time, the maximum deviation of the controlled variable, is of practical interest y(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 drawn from the type 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-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 tools.