Annotation: The first topic has an introductory, mostly terminological character. The concepts of model and modeling, their purpose as the main, and sometimes the only method of analysis and synthesis, are disclosed in detail. complex systems and processes. An overview of the classification of models and modeling is given, somewhat simplified, but sufficient for a complete understanding of the essence of modeling, both in general and mathematical in particular.
The modeling process itself is not fully formalized, a large role in this belongs to the experience of an engineer. But, nevertheless, the process of creating a model considered in the topic in the form of six stages can become the basis for beginners and, with the accumulation of experience, can be individualized.
Mathematical model, being an abstract image of a simulated object or process, cannot be its complete analogue. There is enough similarity in those elements that determine the purpose of the study. For a qualitative assessment of similarity, the concept of the adequacy of the model to the object is introduced and, in this regard, the concepts of isomorphism and isofunctionalism are revealed. There are no formal techniques that allow you to automatically, "thoughtlessly", create adequate mathematical models. The final judgment on the adequacy of the model is given by practice, that is, a comparison of the model with the existing object. Nevertheless, mastering all subsequent topics of the manual will allow the engineer to cope with the problem of ensuring the adequacy of the models.
The topic ends with a statement of the requirements for models that were formulated by R. Shannon at the dawn of computer modeling thirty years ago in the book " Simulation systems - art and science. "The relevance of these requirements is preserved at the present time.
1.1. General Model Definition
Practice shows that the most the best remedy to define object properties - natural experiment, i.e., the study of the properties and behavior of the object itself under the right conditions. The fact is that when designing it is impossible to take into account many factors, the calculation is carried out according to averaged reference data, new, insufficiently tested elements are used (progress is impatient!), environmental conditions change and much more. Therefore, a full-scale experiment is a necessary link in the study. The inaccuracy of the calculations is compensated by an increase in the volume of full-scale experiments, the creation of a number of prototypes and the "finishing" of the product to the desired state. This is what they did and continue to do when creating, for example, a new type of TV or radio station.
However, in many cases a full-scale experiment is impossible.
For example, a war can give the most complete assessment of a new type of weapon and methods of its use. But won't it be too late?
A full-scale experiment with a new aircraft design can cause the death of the crew.
Field study of a new drug is dangerous for human life.
Field experiment with elements space stations can also cause death.
The time for preparing a full-scale experiment and carrying out safety measures often significantly exceeds the time for the experiment itself. Many tests close to the boundary conditions can proceed so violently that accidents and destruction of part or the entire object are possible.
From what has been said, it follows that a full-scale experiment is necessary, but at the same time impossible or inappropriate.
There is a way out of this contradiction and it is called "modeling".
Modeling- this is the replacement of one object by another in order to obtain information about the most important properties of the original object.
This implies.
Modeling- this is, firstly, the process of creating or finding an object in nature, which in a sense can replace the object under study. This intermediate object is called model. The model can be a material object of the same or different nature in relation to the object under study (original). A model can be a mental object that reproduces the original in logical constructions or mathematical formulas and computer programs.
Modeling, secondly, this is a test, a study of the model. That is, modeling is associated with an experiment, which differs from the natural one in that an "intermediate link" - a model - is included in the process of cognition. Consequently, model is at the same time means of experiment and the object of the experiment, replacing the studied object .
Modeling, thirdly, it is the transfer of information obtained on the model to the original or, in other words, the attribution of the properties of the model to the original. For such a transfer to be justified, there must be a similarity between the model and the original, similarity.
Similarity can be physical, geometric, structural, functional, etc. degree of similarity can be different - from identity in all aspects to similarity only in the main. Obviously, models should not fully reproduce all aspects of the objects under study. Achieving absolute uniformity reduces modeling to a full-scale experiment, the possibility or expediency of which has already been mentioned.
Let's focus on the main modeling purposes.
Forecast- evaluation of the system behavior with some combination of its controlled and unmanaged parameters. Forecast - home modeling goal.
Explanation and better understanding of objects. Here, optimization and sensitivity analysis problems are more common than others. Optimization- this is the exact definition of such a combination of factors and their values, which provides the best indicator of the quality of the system, the best achievement of the goal by the modeled system by any criterion. Sensitivity analysis- detection from a large number those factors that most influence the functioning of the system being modeled. The initial data are the results of experiments with the model.
Often a model is created to be used as means of education: simulator models, stands, exercises, business games etc.
Modeling as a method of cognition has always been used by mankind - consciously or intuitively. On the walls of the ancient temples of the ancestors of the South American Indians, graphic models of the universe were found. The doctrine of modeling arose in the Middle Ages. An outstanding role in this belongs to Leonardo da Vinci (1452-1519).
The brilliant commander A. V. Suvorov, before attacking the fortress of Izmail, trained soldiers on a model of the Izmail fortress wall, built specifically in the rear.
Our famous self-taught mechanic IP Kulibin (1735-1818) created a model of a single-arch wooden bridge across the river. Neva, as well as a number of metal models of bridges. They were fully technically justified and received appreciated Russian academicians L. Euler and D. Bernoulli. Unfortunately, none of these bridges were built.
A huge contribution to strengthening the defense capability of our country was made by the work on explosion modeling - engineer-general N. L. Kirpichev, modeling in the aircraft industry - M. V. Keldysh, S. V. Ilyushin, A. N. Tupolev and others, modeling nuclear explosion- I.V. Kurchatov, A.D. Sakharov, Yu. B. Khariton and others.
The works of N. N. Moiseev on modeling control systems are widely known. In particular, to test one new method of mathematical modeling, a mathematical model Sinop battle- the last battle of the era of the sailing fleet. In 1833, Admiral P.S. Nakhimov defeated the main forces of the Turkish fleet. Modeling on a computer showed that Nakhimov acted almost flawlessly. He positioned his ships so faithfully and struck the first blow that the only salvation for the Turks was a retreat. They had no other choice. They did not retreat and were defeated.
The complexity and bulkiness of technical objects that can be studied by simulation methods are practically unlimited. AT last years all major structures were studied on models - dams, canals, the Bratsk and Krasnoyarsk hydroelectric power stations, long-distance power transmission systems, samples of military systems, and other objects.
An instructive example of the underestimation of modeling is the sinking of the English battleship Captain in 1870. In an effort to further increase its then naval power and reinforce imperialist aspirations, the super-battleship Captain was developed in England. Everything that is needed for "supreme power" at sea was invested in it: heavy artillery in rotating turrets, powerful side armor, reinforced sailing equipment and very low sides - for less vulnerability to enemy shells. Consultant engineer Reed built a mathematical model of the Captain's stability and showed that even with slight wind and waves, he was in danger of capsizing. But the Lords of the Admiralty insisted on building a ship. At the very first exercise after launching, a squall turned the armadillo over. 523 sailors were killed. In London, a bronze plaque is attached to the wall of one of the cathedrals, reminding of this event and, we will add, of the stupidity of the self-confident lords of the British Admiralty, who neglected the results of modeling.
1.2. Classification of models and simulations
Each model is created for a specific purpose and is therefore unique. However, the presence of common features makes it possible to group all their diversity into separate classes, which facilitates their development and study. In theory, many signs of classification are considered, and their number has not been established. However, the following are the most relevant signs of classification:
- the nature of the modeled side of the object;
- the nature of the processes occurring in the object;
- way to implement the model.
Table 1.3. calendar of events
Table 6.1. Manual imitation of the work of a bank teller.
| Time | customer no. | Event | QS state | |
| Number of clients | Cashier's condition | |||
| 0,0 | - | - | free | |
| 3,2 | Coming | Busy | ||
| 7,0 | Care | free | ||
| 10,9 | Coming | Busy | ||
| 13,2 | Coming | Busy | ||
| 14,4 | Care | Busy | ||
| 14,8 | Coming | Busy | ||
| 17,7 | Coming | Busy | ||
| 18,6 | Care | Busy | ||
| 19,8 | Coming | Busy | ||
| 21,5 | Coming | Busy | ||
| 21,7 | Care | Busy | ||
| 24,1 | Care | Busy | ||
| 26,3 | Coming | Busy | ||
| 28,4 | Care | Busy | ||
| 31,1 | Care | Busy | ||
| 32,1 | Coming | Busy | ||
| 32,2 | Care | Busy | ||
| 35,7 | Care | free | ||
| 36,6 | Coming | Busy | ||
| 40,0 | Care | free |
Logics event handling customer arrival and departure depends on the state system at the time of these events.
When the event "arrival of the client" occurs, the further situation is determined by the state of the cashier. If the cashier is free, he goes into the busy state and starts serving the client. In this case, the event "leaving this client" is planned at a time equal to the current time plus the duration of its service. If the cashier is busy, the customer service cannot start and, therefore, he gets in line (the length of the queue increases by one). The logic for handling the "client leaving" event depends on the length of the queue. If there is at least one customer in the queue, the cashier remains in the "busy" state, the queue length is reduced by 1, and an exit event is scheduled for the first customer in the queue. If the queue is empty, the cashier is transferred to the "free" state.
Figure 6.2 shows charts changes in the values of these state variables over time.
results simulations show that during the first 40 minutes of working in a bank average was at the same time 1,4525 clients and the cashier was free 20% time.
To arrange events in chronological order, it is necessary to keep a record of events to be processed further (future events). This is done by writing in the list of moments of the next arrival event and the next departure event. The comparison of these moments then determines the choice of one of the events for processing. Such an ordered list of events is usually called event calendar.
| Event | Coming | Care | Coming | Coming | Care | Coming | Coming |
| Completion time | 3,2 | 7,0 | 10,9 | 13,2 | 14,4 | 14,8 | 17,7 |
System models are classified into discretely and continuously changing. Note that these terms refer to the model and not to the real system. Almost the same system can be represented as a discretely changing model, or continuously changing.
Usually in simulation time is the main independent variable. Other variables included in the simulation model are functions of time, that is, dependent variables. Terms discrete and continuous relate to behavior dependent variables.
At discrete simulation dependent variables change discretely at certain moments of simulation time, called moments of commission events .
time variable in the simulation model can be either discrete, or continuous depending on whether discrete changes in dependent variables can occur at any time or only at certain times.
Imitation of the banking system is example of a discrete imitations. The dependent variables in this example are the state of the cashier and the number of customers waiting in line. The moments of events correspond to the moments of time when the client arrives in the system, and the moments of time when the client leaves it after being served by the cashier.
As a rule, in discrete models, the values of dependent variables do not change in between between moments committing events. An example of changing dependent variables in a discrete model is shown in fig. 6.3.
At continuous imitation the dependent variables of the model change continuously in each simulation time point.
The continuous model can be either with continuous, or with discrete time depending on whether the values of the dependent variables are available at any point or only at certain moments of the simulation time.
Process models in most electrical and mechanical systems are examples of situations where a continuous representation is appropriate. In addition, in some cases it is useful to model a discrete system using a continuous representation. For example, the development of populations of individual fish species in a lake in ecological problems is modeled using a continuous representation, although in reality the change in the population occurs discretely.
At combined imitation dependent variables may change discrete, continuous, or continuous with superimposed discrete jumps. Time changes either discretely or continuously.
Most important aspect combined imitation lies in the possibility interactions between discrete and continuously changing variables.
The simplest example such a model is given by an electrical circuit containing a thyristor and a load resistance (Fig. 6.5.). The graph shows how the continuous variable voltage across the load varies abruptly depending on the value of a discrete variable - the state of the thyristor (“open” or “closed”).
The system can be discrete or continuous in inputs, outputs and time, depending on whether the sets are discrete or continuous. u, U, T respectively. Discrete is understood as a finite or countable set. By continuous we mean a set of objects for which an adequate model is a segment, a ray or a straight line, that is, a connected numerical set. If the system has several inputs and outputs, then this means that the corresponding sets U, T lie in multidimensional spaces, i.e., continuity and discreteness are understood component by component.
The convenience of a numerical set as a model of real collections of objects lies in the fact that several relations are naturally defined on it, formalizing the actually occurring relations between real objects. For example, relations of proximity, convergence formalize the concepts of similarity, similarity of objects and can be specified using the distance function (metric) d(x, y)(for example, d(x, y)=І x-yІ . Number sets are ordered: order relation (X y) formalizes the preference for one object over another. Finally, natural operations are defined on elements of numerical sets, for example, linear ones: x+y, x-y. If similar operations also make sense for real objects at the input and output, then the requirements for models (2.1) -(2.3) naturally arise: to be consistent with these operations, to save their results. So we come, for example, to linear models: du/dt =ay+ bu etc., which are the simplest models of many processes.
As a rule, the discreteness of the set U entails discretion. Y. In addition, for static systems the difference between continuous and discrete time disappears. Therefore, the classification of deterministic systems on the basis of "static - dynamic", "discrete - continuous" includes six main groups, presented in Table. 1.3, where for each group the mathematical apparatus for describing systems, methods for numerical analysis and estimating their parameters, synthesis (optimization) methods, as well as typical applications are indicated.
Example 1 Consider the operation of the turnstile at the entrance to the subway. In the first, “rough” approximation, the set of input values of this system has two elements: a person with a token (u 1) and a person without a token, i.e. U=( u 1 ). After a little thought, it becomes clear that the absence of a passenger (u 0) should also be included, i.e. U=(u 0 , u 1 , ). The set of output values contains the elements "open" ( y 0) and "closed" ( y one). So Y=( y 0 , y 1 ) and the system is discrete. In the simplest case, the system memory can be neglected and described by a static model in the form of a table or graph:
If it is necessary to store the MM of the system in a computer, it can be represented (encoded) in the form of a matrix or, more economically, in the form of a list (0, 0, 1), in which i-th place is worth j if the value of the input corresponds to the value of the output y i.
Example 2 If we are interested in the device of the turnstile itself in more detail (i.e. the system is a turnstile), then we will have to take into account that the input actions (signals) for it are the lowering of the nickel and the passage of a person through the turnstile. Thus, the system has two inputs, each of which can take two values ("yes" or "no").
Neglecting the possibility of simultaneously lowering the token and passing, we enter three input values: and 0 - "no impact", and 1 - "lowering the token", and 2 - "passing". Lots of Y can be set in the same way as in example 1. However, now the output value y(t) is not determined only by the value of the input and(t), but it also depends on whether the token was lowered earlier, i.e. from values u(s) at s
x(k+1)= F(x(k), and(k)), y(k) = G(x(k), and(j)), (2.4]
where k- the number of the tact time point. We note that, having singled out the “current” and “next” moments of time, we imperceptibly introduced an assumption about the discreteness of time, which, upon a more detailed study, may turn out to be illegal (see Section 2.2.3 below). transition function F(X, h) and the function of the outputs G(x, and) can be specified in a table:
You can also build transition and exit graphs:
Example 3 Consider the simplest electrical circuit - RC-chain (Fig. 1.6). The system input is the source voltage u( t)=E 0 ( t), the output is the voltage across the capacitor y(t)=E 1 (t). Ohm's law gives the MM of the system as a 1st order differential equation
y=u - y,(2.5)
where -RC- chain time constant. MM (2.5) is completely continuous: U==Y=T=R 1 . If the researcher is interested in the behavior of the system in static modes, i.e. at E 0 (t)= const, then we need to put in (2.5) y= 0 and get static model
y(t)=u(t).(2.6)
Model (2.6) can be used as an approximation in case I, when the input E 0 (t) changes quite rarely or slowly (compared to ).
Example 4 Consider an ecological system consisting of two interacting populations that exist in a certain territory. Let us assume that the system is autonomous, i.e. external influences (inputs) can be neglected; for the outputs of the system we take the number of populations (species) y 1 (t), y 2 (t). Let the 2nd species be food for the 1st, i.e. the system belongs to the class "predator - prey" (for example, at 1 - the number of foxes in the forest, and at 2 - number of hares; or at 1 - the concentration of pathogenic bacteria in the city, and at 2 - the number of cases, etc.). In this case at 1 ,at 2 are integers and, at first glance, in the MM system, the set Y must be discrete. However, to construct the MM, it is more convenient to assume that at 1 ,at 2 can take arbitrary real values, i.e. switch to a continuous model (for sufficiently large at 1 ,at 2 this transition will not introduce a significant error). In this case, we will be able to use such concepts as the rate of change of output variables at 1 ,at 2 . The simplest model of population dynamics is obtained by assuming that:
In the absence of predators, the number of prey grows exponentially;
In the absence of prey, the number of predators decreases exponentially;
The number of "eaten" victims is proportional to the value at 1 ,at 2 .
Under these assumptions, the dynamics of the system, as it is easy to see, is described by the so-called Lotka-Volterra model:
where a, b, c, d are positive parameters. If it is possible to change the parameters, then they turn into input variables, for example, when the birth and death rates of species, the multiplication rates of bacteria (during the introduction of drugs), etc. are changed.
Processes in linear pulse and digital automatic control systems are described by discrete-difference equations of the form:
where x(n) is the lattice function of the input signal; y(n) is the lattice function of the output signal, which is determined by the solution of equation (1.2); b k are constant coefficients; 
- difference to-th order; t=nT, where nt
– n– th point in time T is the discrete period (in expression (1.2) it is conditionally taken as unity).
Equation (1.2) can be represented in another form:
Equation (1.3) is a recursive relation that allows you to calculate any (i+1)-th member of the sequence by the values of its previous members i,i-1,... and meaning x(i+1).
The main mathematical tool for modeling digital automatic systems is the Z-transform, which is based on the discrete Laplace transform. To do this, it is necessary to find the impulse transfer function of the system, set the input variable, and by varying the system parameters, you can find the best version of the system being designed.
1.3.4. Discrete - stochastic models (p - schemes)
The discrete-stochastic model includes probabilistic automaton. In general, a probabilistic automaton is a discrete step-by-step information converter with memory, the operation of which in each cycle depends only on the state of the memory in it and can be described statistically. The behavior of the automaton depends on the random choice.
The use of schemes of probabilistic automata is important for the design of discrete systems in which statistically regular random behavior is manifested.
For the P-automaton, a similar mathematical concept is introduced, as for the F-automaton. Consider a set G whose elements are all possible pairs (x i ,z s )
, where x i and
z s
input subset elements X and subsets of states Z
respectively. If there are two such functions 
and 
that they are used to display 
and 
, then it is said to define an automaton of a deterministic type.
The transition function of a probabilistic automaton determines not one specific state, but the probability distribution on a set of states
(automaton with random transitions). The output function is also a probability distribution on the set of output signals (an automaton with random outputs).
To describe a probabilistic automaton, we introduce a more general mathematical scheme. Let Φ be the set of all possible pairs of the form (z k ,y j ) , where y j is an element of the output subset Y. Next, we require that any element of the set G induced on the set Φ some distribution law of the following form:
elements from f...
...
...
where 
are the probabilities of the transition of the automaton to the state z k and the appearance of a signal at the output y j if he was able z s and a signal was received at its input at this point in time x i .
The number of such distributions, presented in the form of tables, is equal to the number of elements of the set G. If we denote this set of tables by B, then the four elements 
called probabilistic automaton
(P - automatic). Wherein 
.
A special case of the P-automaton given as 
are automata in which either the transition to a new state or the output signal is determined deterministically ( Z–deterministic probabilistic automaton,Y–- deterministic probabilistic automaton respectively).
Obviously, from the point of view of the mathematical apparatus, the assignment of a Y - deterministic P - automaton is equivalent to the assignment of some Markov chain with a finite set of states. In this regard, the apparatus of Markov chains is the main one when using P-schemes for analytical calculations. Similar P-automata use generators of Markov sequences when constructing the processes of functioning of systems or environmental influences.
Markov sequences, according to the Markov theorem, is a sequence of random variables for which the expression
,
where N is the number of independent tests; D–- dispersion.
Such P-automata (P-schemes) can be used to evaluate various characteristics of the systems under study both for analytical models and for simulation models using statistical modeling methods.
Y - deterministic P-automaton can be specified by two tables: transitions (Table 1.1) and outputs (Table 1.2).
Table 1.1
Table 1.2
Where P ij is the probability of the transition of the P-automaton from the state z i to the state z j , while 
.
Table 1.1 can be represented as a square matrix of dimension 
. We will call such a table transition probability matrix or simply transition matrix of the P-automaton, which can be represented in a compact form:
To describe the Y-deterministic P-automaton, it is necessary to set the initial probability distribution of the form:
where d k is the probability that, at the beginning of the work, the P-automaton is in the state z k , while 
.
And so, before the start of work, the P-automaton is in the state z 0 and, at the initial (zero) time step, changes the state in accordance with the distribution D. After that, the change in the states of the automaton is determined by the transition matrix P. Taking into account z 0, the dimension of the matrix Р р should be increased before 
, while the first row of the matrix will be (d 0
,d 1
,d 2
,...,d k )
, and the first column will be null.
Example. Y-deterministic P-automaton is given by the transition table:
Table 1.3
and output table
Table 1.4
Taking into account Table 1.3, the graph of transitions of a probabilistic automaton is shown in Fig. 1.2.
It is required to estimate the total final probabilities of this automaton being in the state z 2 and z 3 , i.e. when units appear at the output of the machine.
Rice. 1.2. Transition graph
With an analytical approach, one can use known relations from the theory of Markov chains and obtain a system of equations for determining the final probabilities. Moreover, the initial state can be ignored since the initial distribution does not affect the values of the final probabilities. Then table 1.3 will take the form:
where 
is the final probability that the Y-deterministic P-automaton is in the state z k .
As a result, we obtain a system of equations:
(1.4)
The normalization condition should be added to this system:
(1.5)
Now, solving the system of equations (1.4) together with (1.5), we obtain:
Thus, with the infinite operation of a given automaton, a binary sequence will be formed at its output with a probability of occurrence of one, equal to: 
.
In addition to analytical models in the form of P-schemes, simulation models can also be used, implemented, for example, by the method of statistical modeling.
discrete models. However, the division of systems into continuous and discrete depends in many respects arbitrarily on the purpose and depth of the study. Continuous systems are often reduced to discrete ones, while continuous parameters are presented as discrete quantities by introducing various kinds of scoring scales, etc. Discrete systems are studied using the apparatus of the theory of algorithms and the theory of automata.
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Discrete Modelsrefer to systems, all elements of which, as well as the connections between them (i.e., information circulating in the system) are discrete in nature. Therefore, all parameters of such a system are discrete.
continuous models. The opposite concept is a continuous system. However, the division of systems into continuous and discrete is largely arbitrary, depending on the purpose and depth of the study. Continuous systems are often reduced to discrete ones (in this case, continuous parameters are presented as discrete quantities by introducing various kinds of scales, scoring, etc.). Discrete systems are studied using the apparatus of the theory of algorithms and the theory of automata. Their behavior can be described using difference equations.
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