Inhomogeneous systems of equations. Classification of heterogeneous systems

1st question Exam

1. Methodology of system analysis. The concept of a system. Static properties of the system. Openness. Difficulties in building a black box model. Inhomogeneity of the composition. Difficulties in building a composition model. Structured. Difficulties in building a structure model.

static properties Let us name the features of a particular state of the system. This is what the system has at any but a fixed point in time.

openness is the second property of the system. The isolated system, distinguishable from everything else, is not isolated from the environment. On the contrary, they are connected and exchange with each other any kind of resources (substance, energy, information, etc.). Let us note that the connections of the system with the environment are directional; according to one, the environment affects the system (they are called system inputs), according to others, the system affects the environment, does something in the environment, gives something to the environment (such connections are called system outputs). The list of inputs and outputs of the system is called black box model . This model lacks information about the internal features of the system. Despite the (apparent) simplicity and poor content of the black box model, this model is often quite sufficient to work with the system.

Difficulties in building a black box model . All of them stem from the fact that the model always contains a finite list of connections, while their number in a real system is not limited. The question arises: which of them should be included in the model, and which should not? We already know the answer: the model must reflect all the connections that are essential for

goal achievement.

Four types of errors when building a black box model:

An error of the first kind occurs when the subject regards the relationship as significant and decides to include it in the model, while in fact, in relation to the goal, it is insignificant and could be ignored. This leads to the appearance in the model of "superfluous" elements, which are essentially unnecessary.

An error of the second kind, on the contrary, is committed by the subject when he decides that this connection is insignificant and does not deserve to be included in the model, while in fact, without it, our goal cannot be fully or even completely achieved.

Type III error is considered to be the consequences of ignorance. In order to assess the significance of a certain connection, one must know that it exists at all. If this is not known, there is no question of including it in the model or not at all: the models contain only what we know. But from the fact that we do not suspect the existence of a certain connection, it does not cease to exist and manifest itself in reality. And then it all depends on how essential it is to achieve our goal. If it is insignificant, then in practice we will not notice its presence in reality and its absence in the model. If it is significant, we will experience the same difficulties as in the case of an error of the second kind. The difference is that the Type III error is more difficult to correct: new knowledge must be acquired.

An error of the fourth kind can occur when a known and recognized significant relationship is incorrectly assigned to the number of inputs or outputs.

Internal heterogeneity: distinguishability of parts (the third property of the system). If you look inside the "black box", it turns out that the system is not homogeneous, not monolithic; you can find that different qualities in different places are different. The description of the internal heterogeneity of the system is reduced to the isolation of relatively homogeneous areas, drawing boundaries between them. This is how the concept of the parts of the system appears. On closer examination, it turns out that the selected large parts are also not homogeneous, which requires the selection of even smaller parts. The result is a hierarchical list of system parts, which we will call the system composition model.

Difficulties in building a composition model , which everyone has to overcome, can be represented by three provisions:

First. The whole can be divided into parts in different ways (like cutting a loaf of bread into slices of different sizes and shapes). How exactly is it necessary? Answer: as you need to achieve your goal.

Second. The number of parts in the composition model also depends on the level at which the fragmentation of the system is stopped. The parts on the terminal branches of the resulting hierarchical tree are called elements .

Third. Any system is part of some larger system (and often part of several systems at once). And this metasystem can also be divided into subsystems in different ways. This means that the external boundary of the system has a relative, conditional character. Even the "obvious" boundary of the system (human skin, the fence of an enterprise, etc.) under certain conditions is not enough to determine the boundary under these conditions.

Structured The fourth static property is that the parts of the system are not independent, not isolated from each other; they are interconnected and interact with each other. At the same time, the properties of the system as a whole essentially depend on how exactly its parts interact. Therefore, information about the relationships of parts is so often important. The list of essential links between the elements of the system is called the system structure model. The indivisibility of any system by a certain structure will be called the fourth static property of systems - structuredness.

Difficulties in building a structure model . We emphasize that many different models of the structure can be proposed for this system. It is clear that in order to achieve a certain goal, one, specific, most suitable model of them is required. The difficulty of choosing from the available ones or building a model specifically for our case stems from the fact that, by definition, a structure model is a list of essential relationships.

The first difficulty is related to the fact that the structure model is determined after the composition model is chosen and depends on what exactly the composition of the system is. But even with a fixed composition, the structure model is variable - because of the possibility to determine the significance of relationships in different ways.

The second difficulty stems from the fact that each element of the system is a "little black box". So all four types of errors are POSSIBLE when determining the inputs and outputs of each element included in the structure model.

2. Methodology of system analysis. The concept of a system. Dynamic properties of the system: functionality, stimulation, variability of the system over time, existence in a changing environment. Synthetic properties of the system: emergence, inseparability into parts, inherence, expediency.

Dynamic properties of the system:

Functionality is the fifth property of the system. The processes Y(t) occurring at the outputs of the system (Y(1)^(yi(t), Yi(1), -, Yn(0) are considered as its functions. System functions - this is her behavior in the external environment; changes made by the system environment; the results of its activities; products produced by the system. From the multiplicity of outputs follows the multiplicity of functions, each of which can be used by someone and for something. Therefore, the same system can serve different purposes.

Stimulability - the sixth property of the system. Certain processes X(t) = (x^(t), X2 (t), x^(t)) also occur at the inputs of the system, affecting the system, turning (after a series of transformations in the system) into Y(t). Let's call the influences X(t) incentives, and the very susceptibility of any system to external influences and the change in its behavior under these influences - we will call stimulability.

System variability over time - the seventh property of the system. In any system, there are changes that must be taken into account; provide for and lay in the project of the future system; contribute to or counteract them, accelerating or slowing them down when working with existing system. Anything can change in the system, but in terms of our models, we can give a clear classification of changes: the values ​​of internal variables (parameters) Z(t), the composition and structure of the system, and any combination of them can change.

Existence in a changing environment - the eighth property of the system. Not only this system is changing, but also all the others. For this system, it looks like a continuous change in the environment. The inevitability of existence in a constantly changing environment has many consequences for the system itself, from the need to adapt to external changes in order not to perish, to various other reactions of the system. When considering a specific system for a specific purpose, attention is focused on some specific features of its response.

Synthetic properties of the system:

Synthetic . This term denotes generalizing, collective, integral properties, taking into account what has been said before, but emphasizing the interaction of the system with the environment, on integrity in the most general sense.

emergence - the ninth property of the system. Perhaps this property speaks more than any other about the nature of systems. Combining parts into a system gives rise to qualitatively new properties of the system that are not reduced to the properties of the parts, are not derived from the properties of the parts, are inherent only in the system itself and exist only as long as the system is one whole. A system is more than just a collection of parts. System qualities that belong only to her, are called emergents (from the English "to arise").

Inseparability into parts is the tenth property of the system. Although this property is a simple consequence of emergence, its practical importance is so great, and its underestimation is so common, that it is worth emphasizing it separately. If we need the system itself, and not something else, then it cannot be divided into parts. When a part is REMOVED from the system, two important events occur.

First, this changes the composition of the system, and hence its structure. It will be a different system, with different properties. Since the old system has many properties, some property associated with this particular part will disappear altogether (it may turn out to be emergent or not. Some property will change, but it will be partially preserved. And some properties of the system are generally unimportant We emphasize once again that whether or not the removal of a part from the system will have a significant impact is a matter of assessing the consequences.

The second important consequence of removing a part from a system is that a part in the system and outside it are not the same thing. Its properties change due to the fact that the properties of the object are manifested in interactions with the objects surrounding it, and when the element is removed from the system, the environment of the element becomes completely different.

ingrency - the eleventh property of the system. We will say that the system is the more iherent (from the English inherent - being an integral part of something), the better it is coordinated, adapted to the environment, compatible with it. The degree of inherence varies and can change (learning, forgetting, evolution, reforms, development, degradation, etc.). The fact that all systems are open does not mean that they are all equally well coordinated with the environment.

Expediency - the twelfth property of the system. In systems created by man, the subordination of everything (both composition and structure) to the set goal is so obvious that it must be recognized as a fundamental property of any artificial system. The goal for which the system is created determines which emergent property will ensure the realization of the goal, and this, in turn, dictates the choice of the composition and structure of the system. One of the definitions of a system states that a system is a means to an end. It is understood that if the put forward goal cannot be achieved at the expense of existing opportunities, then the subject composes from the objects surrounding him new system specially designed to help achieve this goal. It is worth noting that the goal rarely unambiguously determines the composition and structure of the system being created: it is important that the desired function be implemented, and this can often be achieved in different ways.

3. Methodology of system analysis. Models and modeling. The concept of a model as a system. Analysis and synthesis as methods of building models. Artificial and natural classification of models. Consistency of models with the culture of the subject.

Depending on what we need to know, explain - how the system works or how it interacts with the environment, there are two methods of cognition: 1) analytical; 2) synthetic.

The analysis procedure consists in the sequential execution of the following three operations; 1) divide the complex whole into smaller parts, presumably simpler ones; 2) give a clear explanation of the received fragments; 3) combine the explanation of the parts into an explanation of the whole. If some part of the system is still incomprehensible, the decomposition operation is repeated and we again try to explain new, even smaller fragments.

The first product of the analysis is, as can be seen from the diagram, a list of system elements, i.e. . system composition model . The second product of the analysis is a system structure model . The third product of the analysis is black box model for each element of the system.

Synthetic method consists in the successive performance of three operations: 1) selection of a larger system (metasystem), in which the system of interest to us is included as a part; 2) consideration of the composition and structure of the metasystem (its analysis): 3) explanation of the role that our system occupies in the metasystem through its connections with other subsystems of the metasystem. The end product of the synthesis is the knowledge of the connections of our system with other parts of the metasystem, i.e. black box model. But in order to build it, we had to create models of the composition and structure of the metasystem along the way as by-products.

Analysis and synthesis are not opposites, but complement each other. Moreover, in analysis there is a synthetic component, and in synthesis there is an analysis of the metasystem.

There are two types of classifications: artificial and natural . With artificial classification the division into classes is carried out "as it should be", i.e. based on the goal - as many classes and with such boundaries as the goal dictates. A somewhat different classification is made when the set under consideration is clearly inhomogeneous. Natural groupings (they are called clusters in statistics) seem to suggest themselves to be defined as classes , (hence the name of the classification natural) . However, it should be borne in mind that natural classification is only a simplified, coarsened model of reality .

Consistency of models with the culture of the subject . In order for the model to realize its model function, it is not enough just to have the model itself. It is necessary that the model was compatible, consistent with the environment, which for the model is the culture (world of models) of the user. This condition, when considering the properties of systems, is called inherence: the inherence of a model to culture is a necessary requirement for modeling. The degree of model integrity can change: increase (user training, the emergence of a Rosetta Stone type adapter, etc.) or decrease (forgetting, destruction of culture) due to changes in the environment or the model itself. Thus, another element, culture, should be included in the modeling metasystem.

4. Methodology of system analysis. Control. Five components of management. Seven types of control.

Control - targeted impact on the system.

Five control components:

The first control component is the control object itself, the managed system.

The second mandatory component of the management system is the goal of management.

The control action U(t) is the third control component . The fact that the inputs and outputs of the system are interconnected by some relation Y(t)=S allows us to hope that there is such a control action under which the goal V*(t) is realized at the output.

The system model becomes the fourth component of the management process.

All actions necessary for management must be performed. This function is usually assigned to a system specially created for this. (the fifth component of the management process). Called a control unit or a control system (subsystem), a control device and so on. In real Control block it can be a subsystem of a controlled system (like;) an autopilot is part of a factory, an autopilot is part of an aircraft), but it can also be external system(as a ministry for a subordinate enterprise, as an airfield dispatcher for an aircraft coming in to land).

Seven types of control:

The first type of control is the control of a simple system, or program control.

The second type of management is the management of a complex system.

The third type of control is control by parameters, or regulation.

The fourth type of management is management by structure.

The fifth type of management is management by goals.

The sixth type of management is the management of large systems.

The seventh type of management. In addition to the first type of control, when everything necessary to achieve the goal is available, the other types of control considered are associated with overcoming factors that help to achieve the goal: lack of information about the control object (second type), third-party minor interference that slightly deviates the system from the target trajectory (third type ), discrepancy between the emergent properties of the system and the goal (fourth type), lack of material resources, making the goal unattainable and requiring its replacement (fifth type), lack of time to search best solution(sixth type).

5. Technology of system analysis. Conditions for the success of systematic research. Stages of systematic research: fixing the problem, diagnosing the problem, compiling a list of stakeholders, determining the problematic mess.

Conditions for the success of systems research :

guarantee of access to any necessary information (at the same time, the analyst, for his part, guarantees confidentiality);

guarantee of personal participation of the first persons of organizations - mandatory participants in a problem situation (heads of problem-containing and problem-solving systems);

abandoning the requirement to formulate the necessary result in advance (“terms of reference”), since there are many improving interventions and they are not known in advance, especially which one will be chosen for implementation.

Fixing the problem - the task is to formulate the problem and fix it documented. The formulation of the problem is developed by the client; the business of the analyst is to find out what the client complains about, what he is dissatisfied with. This is the client's problem as he sees it. At the same time, one should try not to influence his opinion, not to distort it.

Problem Diagnosis . Which of the methods of problem solving to apply to solve this problem depends on whether we choose to influence the most dissatisfied subject or to intervene in the reality with which he is dissatisfied (there are cases when a combination of both influences is advisable). The task of this stage is to make a diagnosis - to determine what type the problem belongs to.

Drawing up a list of stakeholders .Our ultimate goal is to implement an improving intervention. Each stage should bring us one step closer to it, but special care must be taken that this step is exactly in the right direction, and not in the other direction. In order to subsequently take into account the interests of all participants in the problem situation (namely, this is the basis for the concept of improving intervention), you must first find out who is involved in the problem situation, make a list of them. It is important not to miss anyone; after all, it is impossible to take into account the interests of someone who is unknown to us, and not taking into account someone threatens that our intervention will not be improving. Thus, the list of participants in the problem situation should be complete.

Identification of the problematic mess . Stakeholders have interests that we have to take into account. But for this they need to know. So far, we have only a list of owners of interests. The first piece of information that needs to be obtained about the stakeholder is his own assessment of the situation that is problematic for our client. It can be different: some of the stakeholders may have their own problems (negative assessment), someone is quite satisfied (positive assessment), others may be neutral to reality. So it will clear up<выражение л ица:^ каждого стейкхолдера. По сути, мы должны выполнить работу, которую делали на первом этапе с клиентом, но теперь с каждым стейкхолдером в отдельности.

6. Technology of system analysis. Operations of system analysis. Stages of system research: definition of a configurator, goal identification, definition of criteria, experimental research.

System Analysis Operations . If the client agrees to the terms of the contract, the analyst proceeds to the first stage, after completing which, he begins the second and so on until the last stage, at the end of which the implemented improving intervention should be obtained.

Configurator Definition . A necessary condition for the successful solution of the problem is the availability of an adequate model of the problem situation, with its help it will be possible to test and compare options for the proposed actions. This model (or set of models) must inevitably be built using the means of some language (or languages). The question arises of how many and which languages ​​are needed to work on a given problem and how to choose them. The configurator is called the minimum set of professional languages ​​that allows you to give a complete (adequate) description of the problem situation and its transformations. All work in the course of solving the problem will take place in the languages ​​​​of the configurator. And only on them. Defining the configurator is the task of this step. We emphasize that the configurator is not an artificial invention of system analysts, invented to facilitate their work.. On the one hand, the configurator is determined by the nature of the problem. On the other hand, the configurator can also be considered as another PROPERTY of systems, as a means by which the system solves its problem.

Targeting . In seeking to implement an improvement intervention, we must ensure that none of the stakeholders view it negatively. People rate change positively if it brings them closer to their goal, and negatively if it moves them further away from it. Therefore, in order to design an intervention, it is necessary to know the goals of all stakeholders. Of course, the main source of information is the stakeholder himself.

Definition of criteria . In the course of solving the problem, it will be necessary to compare the proposed options, assess the degree of achievement of the goal or deviation from it, and monitor the course of events. This is achieved by highlighting some features of the objects and processes under consideration. These signs should be associated with the features of the objects or processes that are of interest to us, should be available for observation and measurement. Then, according to the obtained measurement results, we will be able to carry out the necessary control. Such characteristics are called criteria. Every study (including ours) will require criteria. How many, what and how to choose criteria? First, about the number of criteria. Obviously, the fewer criteria needed, the easier it will be to compare. That is, it is desirable to minimize the number of criteria, it would be good to reduce it to one. Choice of criteria . Criteria are quantitative models of qualitative goals. Indeed, the formed criteria in the future in a certain sense represent, replace the goals: optimization according to the criteria should provide the maximum approximation to the goal. Of course, the criteria are not identical to the goal, it is a similarity of the goal, its model. Determining the value of a criterion for a given alternative is essentially a measure of its suitability as a means to an end.

Experimental study of systems. Experiment and model. Often the missing information about the system can only be obtained from the system itself, by conducting an experiment specially designed for this purpose. The information contained in the protocol of the experiment is extracted by subjecting the obtained data to processing, transformation into a form suitable for inclusion in the system model. The final step is the correction of the model, which includes the information obtained in the model. It is easily perceived that the experiment is needed to improve the model. It is also important to understand that an experiment is impossible without a model. They are in the same cycle. However, rotation in this cycle does not resemble a spinning wheel, but a rolling snowball - with each revolution it becomes larger and more weighty.

7. Technology of system analysis. Stages of system research: construction and improvement of models, generation of alternatives, decision making, +.

Construction and improvement of models. In system analysis, the model is problematic and the situation is needed in order to "lose" possible options for interventions in order to cut off not only those that will not improve, but also to choose among those that improve the most (according to our criteria) improving ones. It should be emphasized that the contribution to the construction of the situation model is made at each previous and at all subsequent stages (both by one's own contribution and by the decision to return to some early stage to replenish the model with information). Therefore, in fact, there is no separate, special “stage of building a model”, And yet it is worth focusing on the features of building models, or rather, their "completion" (i.e. adding new elements or removing unnecessary ones).

Generation of alternatives . In the described technology, this action is performed in two stages:

identifying discrepancies between the problem and target messes. The distinction between the current (and unsatisfactory) state of the organization and the future, most desirable, ideal state to be strived for must be clearly articulated. These differences are the gaps, the elimination of which needs to be planned;

suggesting possible options for eliminating or reducing the discrepancies found. Actions, procedures, rules, projects, programs and policies, all components of management, must be devised.

2.4.1. Definition. Let an inhomogeneous system of linear equations be given

Consider a homogeneous system

for which the matrix of coefficients coincides with the matrix of coefficients of system (2.4.1). Then system (2.4.2) is called reduced homogeneous system (2.4.1).

2.4.2. Theorem. The general solution of an inhomogeneous system is equal to the sum of some particular solution of the inhomogeneous system and the general solution of the reduced homogeneous system.

Thus, to find the general solution of the inhomogeneous system (2.4.1), it suffices:

1) Examine it for compatibility. In case of compatibility:

2) Find the general solution of the reduced homogeneous system.

3) Find any particular solution to the original (non-homogeneous) one.

4) Having added the particular solution found and the general solution of the given one, find the general solution of the original system.

2.4.3. Exercise. Investigate the system for compatibility and, in the case of compatibility, find its general solution in the form of the sum of the quotient and the general reduced.

Solution. a) To solve the problem, we use the above scheme:

1) We examine the system for compatibility (by the method of bordering minors): The rank of the main matrix is ​​3 (see the solution of exercise 2.2.5, a), and the non-zero minor of the maximum order is composed of the elements of the 1st, 2nd, 4th rows and the 1st, 3 th, 4th columns. To find the rank of the expanded matrix, we border it with the 3rd row and the 6th column of the expanded matrix: =0. Means, rg A =rg=3, and the system is consistent. In particular, it is equivalent to the system

2) Find a general solution X 0 reduced homogeneous of this system

X 0 ={(-2a - b ; a ; b ; b ; b ) | a , b Î R}

(see the solution of exercise 2.2.5, a)).

3) Find some particular solution x h of the original system . To do this, in system (2.4.3), which is equivalent to the original one, the free unknowns x 2 and x We set 5 equal, for example, to zero (these are the most convenient data):

and solve the resulting system: x 1 =- , x 3 =- , x 4=-5. Thus, (- ; 0; - ; -5; 0) ¾ is a particular solution of the system.

4) We find the general solution X n of the original system :

X n={x h }+X 0 ={(- ; 0; - ; -5; 0)} + {(-2a - b ; a ; b ; b ; b )}=

={(- -2a - b ; a ; - + b ; -5+b ; b )}.

Comment. Compare your answer with the second answer in example 1.2.1 c). To obtain an answer in the first form for 1.2.1 c), we take as basic unknowns x 1 , x 3 , x 5 (the minor for which is also not equal to zero), and as free ¾ x 2 and x 4 .

§3. Some applications.

3.1. On the question of matrix equations. We remind you that matrix equation over the field F is an equation in which some matrix over the field acts as an unknown F .

The simplest matrix equations are equations of the form

AX=B , XA =B (2.5.1)

Where A , B ¾ given (known) matrices over field F , A X ¾ such matrices, when substituting which equations (2.5.1) turn into true matrix equalities. In particular, the matrix method of certain systems is reduced to solving a matrix equation.

When the matrices A in equations (2.5.1) are non-degenerate, they have solutions, respectively X =A B And X =BA .

In the case when at least one of the matrices on the left side of equations (2.5.1) is degenerate, this method is no longer suitable, since the corresponding inverse matrix A does not exist. In this case, finding solutions to equations (2.5.1) reduces to solving systems.

But first, let's introduce some concepts.

The set of all solutions of the system is called common solution . An individual solution of an indefinite system, let's call it private decision .

3.1.1. Example. Solve the matrix equation over the field R.

A) X = ; b) X = ; V) X = .

Solution. a) Since \u003d 0, then the formula X =A B not suitable for solving this equation. If in the work XA =B matrix A has 2 rows, then the matrix X has 2 columns. Number of lines X must match the number of rows B . That's why X has 2 lines. Thus, X ¾ is some second-order square matrix: X = . Substitute X into the original equation:

Multiplying the matrices on the left side of (2.5.2), we arrive at the equality

Two matrices are equal if and only if they have the same dimensions and their corresponding elements are equal. Therefore (2.5.3) is equivalent to the system

This system is equivalent to the system

Solving it, for example, by the Gauss method, we arrive at a set of solutions (5-2 b , b , -2d , d ), Where b , d run independently of each other R. Thus, X = .

b) Similarly to a) we have X = and.

This system is inconsistent (check it out!). Therefore, this matrix equation has no solutions.

c) Denote this equation by AX =B . Because A has 3 columns and B has 2 columns then X ¾ some 3´2 matrix: X = . Therefore, we have the following chain of equivalences:

We solve the last system using the Gauss method (we omit the comments)

Thus, we arrive at the system

whose solution is (11+8 z , 14+10z , z , -49+8w , -58+10w ,w ) Where z , w run independently of each other R.

Answer: a) X = , b , d Î R.

b) There are no solutions.

V) X = z , w Î R.

3.2. On the question of permutability of matrices. In general, the product of matrices is nonpermutable, that is, if A And B such that AB And BA defined, then, generally speaking, AB ¹ BA . But the identity matrix example E shows that commutability is also possible AE =EA for any matrix A , if only AE And EA were determined.

In this subsection, we consider problems of finding the set of all matrices that commute with a given one. Thus,

Unknown x 1 , y 2 and z 3 can take any value: x 1 =a , y 2 =b , z 3 =g . Then

Thus, X = .

Answer. A) X d ¾ any number.

b) X ¾ set of matrices of the form , where a , b And g ¾ any numbers.

The most common feature of any inhomogeneous system is the presence of two ( or more) phases that are separated from each other by a pronounced interface. In this feature, heterogeneous systems differ from solutions, which also consist of several components that form a homogeneous mixture. One of the phases, continuous, will be called the dispersed phase, and the other, finely divided and distributed in the first, will be called the dispersed phase. Depending on the type of dispersion medium, heterogeneous mixtures are distinguished, liquid and gas. In table. 5.1 is a classification of inhomogeneous systems according to the type of dispersed and dispersed phases.

Table 5.1

Classification of heterogeneous systems

Classification and characteristics of heterogeneous systems

heterogeneous system is considered a system that consists of two or more phases. Each phase has its own interface and can be mechanically separated from the other.

An inhomogeneous system consists of an internal (dispersed) phase and an external phase (dispersion medium) containing particles of the dispersed phase. Systems in which liquids are the external phase are called inhomogeneous liquid systems, and if gases are called inhomogeneous gas systems . Heterogeneous systems called heterogeneous, and homogeneous - homogeneous. A homogeneous liquid system is understood as a pure liquid or a solution of any substances in it. An inhomogeneous, or heterogeneous, liquid system is a liquid in which there are any undissolved substances in the form of tiny particles. Heterogeneous systems are often called dispersed.

There are the following types of heterogeneous systems: suspensions, emulsions, foams, dusts, fumes, fogs.

Suspension is a system consisting of a continuous liquid phase in which solid particles are suspended. For example, sauces with flour, starched milk, molasses with sugar crystals.

Suspensions, depending on the particle size, are divided into coarse (particle size over 100 microns), fine (0.1-100 microns) and colloidal solutions containing solid particles with a size of 0.1 microns or less.

Emulsion- this is a system consisting of a liquid and drops of another liquid distributed in it, which did not dissolve in the first. This is, for example, milk, a mixture of vegetable oil and water. There are gas emulsions in which the dispersion medium is a liquid and the dispersed phase is a gas.

Foam is a system consisting of a liquid and gas bubbles distributed in it. For example, creams and other whipped products. Foams are close to emulsions in their properties.

Emulsions and foams are characterized by the possibility of the transition of the dispersed phase into the dispersion medium and vice versa. This transition, which is possible at a certain mass ratio of phases, is called phase inversion or simply inversion.

Aerosols called a dispersed system with a gaseous dispersion medium and a solid or liquid dispersed phase, which consists of particles from a quasi-molecular to a microscopic size, which have the property of being in suspension for a more or less long time. This concept combines dust, smoke, fog. For example, flour dust formed during grain grinding, sifting, transportation of flour; sugar dust formed during the drying of sugar, etc. Smoke is formed during the combustion of solid fuels, fog - during the condensation of steam.

In aerosols, the dispersion medium is gas or air, while the dispersed phase in dust and smoke is solids, and in fogs it is liquid.

Dust and smoke- systems consisting of gas and solid particles distributed in them with sizes of 5-50 microns and 0.3-5 microns, respectively. Fog is a system consisting of a gas and liquid droplets of 0.3-3 microns in size distributed in it, formed as a result of condensation.

A qualitative indicator characterizing the uniformity of aerosol particles in size is the degree of dispersity. An aerosol is called monodisperse when its constituent particles are of the same size, and polydisperse when it contains particles of different sizes. Monodisperse aerosols practically do not exist in nature. There are only some aerosols, which, in terms of particle size, only approach monodisperse systems (hyphae of fungi, specially obtained fogs, etc.).

Dispersed or heterogeneous systems, depending on the number of dispersed phases, can be single- and multi-component. For example, milk is a multicomponent system (it has two dispersed phases: fat and protein); sauces (dispersed phases are flour, fat, etc.).

Separation methods heterogeneous systems are classified depending on the size of suspended particles of the dispersed phase, the difference between the densities of the dispersed and continuous phases, as well as the viscosity of the continuous phase. The following main separation methods are used: sedimentation, filtration, centrifugation, wet separation, electropurification.

precipitation is a separation process in which solid or liquid particles of the dispersed phase suspended in a liquid or gas are separated from the continuous phase under the action of gravity, centrifugal or electrostatic. Settling under the action of gravity is called settling.

Filtering - process separation using a porous partition capable of passing liquid or gas and retaining solid particles suspended in the medium. Filtration is carried out under the action of pressure forces and is used for finer separation of suspensions and dusts than during precipitation.

centrifugation- the process of separating suspensions and emulsions under the action of centrifugal force.

Wet separation- the process of capturing particles suspended in a gas with the help of a liquid.

Electrocleaning- purification of gases under the influence of electric forces.

Methods for separating liquid and heterogeneous gas systems are based on the same principles, but the equipment used has a number of features.

Solving systems of linear algebraic equations (SLAE) is undoubtedly the most important topic of the linear algebra course. A huge number of problems from all branches of mathematics are reduced to solving systems of linear equations. These factors explain the reason for creating this article. The material of the article is selected and structured so that with its help you can

• choose the optimal method for solving your system of linear algebraic equations,
• study the theory of the chosen method,
• solve your system of linear equations, having considered in detail the solutions of typical examples and problems.

Brief description of the material of the article.

First, we give all the necessary definitions, concepts, and introduce some notation.

Next, we consider methods for solving systems of linear algebraic equations in which the number of equations is equal to the number of unknown variables and which have a unique solution. First, let's focus on the Cramer method, secondly, we will show the matrix method for solving such systems of equations, and thirdly, we will analyze the Gauss method (the method of successive elimination of unknown variables). To consolidate the theory, we will definitely solve several SLAEs in various ways.

After that, we turn to solving systems of linear algebraic equations of a general form, in which the number of equations does not coincide with the number of unknown variables or the main matrix of the system is degenerate. We formulate the Kronecker-Capelli theorem, which allows us to establish the compatibility of SLAEs. Let us analyze the solution of systems (in the case of their compatibility) using the concept of the basis minor of a matrix. We will also consider the Gauss method and describe in detail the solutions of the examples.

Be sure to dwell on the structure of the general solution of homogeneous and inhomogeneous systems of linear algebraic equations. Let us give the concept of a fundamental system of solutions and show how the general solution of the SLAE is written using the vectors of the fundamental system of solutions. For a better understanding, let's look at a few examples.

In conclusion, we consider systems of equations that are reduced to linear ones, as well as various problems, in the solution of which SLAEs arise.

Page navigation.

Definitions, concepts, designations.

We will consider systems of p linear algebraic equations with n unknown variables (p may be equal to n ) of the form

Unknown variables, - coefficients (some real or complex numbers), - free members (also real or complex numbers).

This form of SLAE is called coordinate.

IN matrix form this system of equations has the form ,
Where - the main matrix of the system, - the matrix-column of unknown variables, - the matrix-column of free members.

If we add to the matrix A as the (n + 1)-th column the matrix-column of free terms, then we get the so-called expanded matrix systems of linear equations. Usually, the augmented matrix is ​​denoted by the letter T, and the column of free members is separated by a vertical line from the rest of the columns, that is,

By solving a system of linear algebraic equations called a set of values ​​of unknown variables , which turns all the equations of the system into identities. The matrix equation for the given values ​​of the unknown variables also turns into an identity.

If a system of equations has at least one solution, then it is called joint.

If the system of equations has no solutions, then it is called incompatible.

If a SLAE has a unique solution, then it is called certain; if there is more than one solution, then - uncertain.

If the free terms of all equations of the system are equal to zero , then the system is called homogeneous, otherwise - heterogeneous.

Solution of elementary systems of linear algebraic equations.

If the number of system equations is equal to the number of unknown variables and the determinant of its main matrix is ​​not equal to zero, then we will call such SLAEs elementary. Such systems of equations have a unique solution, and in the case of a homogeneous system, all unknown variables are equal to zero.

We started studying such SLAE in high school. When solving them, we took one equation, expressed one unknown variable in terms of others and substituted it into the remaining equations, then took the next equation, expressed the next unknown variable and substituted it into other equations, and so on. Or they used the addition method, that is, they added two or more equations to eliminate some unknown variables. We will not dwell on these methods in detail, since they are essentially modifications of the Gauss method.

The main methods for solving elementary systems of linear equations are the Cramer method, the matrix method and the Gauss method. Let's sort them out.

Solving systems of linear equations by Cramer's method.

Let us need to solve a system of linear algebraic equations

in which the number of equations is equal to the number of unknown variables and the determinant of the main matrix of the system is different from zero, that is, .

Let be the determinant of the main matrix of the system, and are determinants of matrices that are obtained from A by replacing 1st, 2nd, …, nth column respectively to the column of free members:

With such notation, the unknown variables are calculated by the formulas of Cramer's method as . This is how the solution of a system of linear algebraic equations is found by the Cramer method.

Example.

Cramer method .

Solution.

The main matrix of the system has the form . Calculate its determinant (if necessary, see the article):

Since the determinant of the main matrix of the system is different from zero, the system has a unique solution that can be found by Cramer's method.

Compose and calculate the necessary determinants (the determinant is obtained by replacing the first column in matrix A with a column of free members, the determinant - by replacing the second column with a column of free members, - by replacing the third column of matrix A with a column of free members):

Finding unknown variables using formulas :

Answer:

The main disadvantage of Cramer's method (if it can be called a disadvantage) is the complexity of calculating the determinants when the number of system equations is more than three.

Solving systems of linear algebraic equations by the matrix method (using the inverse matrix).

Let the system of linear algebraic equations be given in matrix form , where the matrix A has dimension n by n and its determinant is nonzero.

Since , then the matrix A is invertible, that is, there is an inverse matrix . If we multiply both parts of the equality by on the left, then we get a formula for finding the column matrix of unknown variables. So we got the solution of the system of linear algebraic equations by the matrix method.

Example.

Solve System of Linear Equations matrix method.

Solution.

We rewrite the system of equations in matrix form:

Because

then SLAE can be solved by the matrix method. Using the inverse matrix, the solution to this system can be found as .

Let's build an inverse matrix using a matrix of algebraic complements of the elements of matrix A (if necessary, see the article):

It remains to calculate - the matrix of unknown variables by multiplying the inverse matrix on the matrix-column of free members (if necessary, see the article):

Answer:

or in another notation x 1 = 4, x 2 = 0, x 3 = -1.

The main problem in finding solutions to systems of linear algebraic equations by the matrix method is the complexity of finding the inverse matrix, especially for square matrices of order higher than the third.

Solving systems of linear equations by the Gauss method.

Suppose we need to find a solution to a system of n linear equations with n unknown variables
the determinant of the main matrix of which is different from zero.

The essence of the Gauss method consists in the successive exclusion of unknown variables: first, x 1 is excluded from all equations of the system, starting from the second, then x 2 is excluded from all equations, starting from the third, and so on, until only the unknown variable x n remains in the last equation. Such a process of transforming the equations of the system for the successive elimination of unknown variables is called direct Gauss method. After the completion of the forward run of the Gaussian method, x n is found from the last equation, x n-1 is calculated from the penultimate equation using this value, and so on, x 1 is found from the first equation. The process of calculating unknown variables when moving from the last equation of the system to the first is called reverse Gauss method.

Let us briefly describe the algorithm for eliminating unknown variables.

We will assume that , since we can always achieve this by rearranging the equations of the system. We exclude the unknown variable x 1 from all equations of the system, starting from the second one. To do this, add the first equation multiplied by to the second equation of the system, add the first multiplied by to the third equation, and so on, add the first multiplied by to the nth equation. The system of equations after such transformations will take the form

where , a .

We would come to the same result if we expressed x 1 in terms of other unknown variables in the first equation of the system and substituted the resulting expression into all other equations. Thus, the variable x 1 is excluded from all equations, starting from the second.

Next, we act similarly, but only with a part of the resulting system, which is marked in the figure

To do this, add the second multiplied by to the third equation of the system, add the second multiplied by to the fourth equation, and so on, add the second multiplied by to the nth equation. The system of equations after such transformations will take the form

where , a . Thus, the variable x 2 is excluded from all equations, starting from the third.

Next, we proceed to the elimination of the unknown x 3, while acting similarly with the part of the system marked in the figure

So we continue the direct course of the Gauss method until the system takes the form

From this moment, we begin the reverse course of the Gauss method: we calculate x n from the last equation as , using the obtained value x n we find x n-1 from the penultimate equation, and so on, we find x 1 from the first equation.

Example.

Solve System of Linear Equations Gaussian method.

Solution.

Let's exclude the unknown variable x 1 from the second and third equations of the system. To do this, to both parts of the second and third equations, we add the corresponding parts of the first equation, multiplied by and by, respectively:

Now we exclude x 2 from the third equation by adding to its left and right parts the left and right parts of the second equation, multiplied by:

On this, the forward course of the Gauss method is completed, we begin the reverse course.

From the last equation of the resulting system of equations, we find x 3:

From the second equation we get .

From the first equation we find the remaining unknown variable and this completes the reverse course of the Gauss method.

Answer:

X 1 \u003d 4, x 2 \u003d 0, x 3 \u003d -1.

Solving systems of linear algebraic equations of general form.

In the general case, the number of equations of the system p does not coincide with the number of unknown variables n:

Such SLAEs may have no solutions, have a single solution, or have infinitely many solutions. This statement also applies to systems of equations whose main matrix is ​​square and degenerate.

Kronecker-Capelli theorem.

Before finding a solution to a system of linear equations, it is necessary to establish its compatibility. The answer to the question when SLAE is compatible, and when it is incompatible, gives Kronecker–Capelli theorem:
for a system of p equations with n unknowns (p can be equal to n ) to be compatible it is necessary and sufficient that the rank of the main matrix of the system is equal to the rank of the extended matrix, that is, Rank(A)=Rank(T) .

Let us consider the application of the Kronecker-Cappelli theorem for determining the compatibility of a system of linear equations as an example.

Example.

Find out if the system of linear equations has solutions.

Solution.

. Let us use the method of bordering minors. Minor of the second order different from zero. Let's go over the third-order minors surrounding it:

Since all bordering third-order minors are equal to zero, the rank of the main matrix is ​​two.

In turn, the rank of the augmented matrix is equal to three, since the minor of the third order

different from zero.

Thus, Rang(A) , therefore, according to the Kronecker-Capelli theorem, we can conclude that the original system of linear equations is inconsistent.

Answer:

There is no solution system.

So, we have learned to establish the inconsistency of the system using the Kronecker-Capelli theorem.

But how to find the solution of the SLAE if its compatibility is established?

To do this, we need the concept of the basis minor of a matrix and the theorem on the rank of a matrix.

The highest order minor of the matrix A, other than zero, is called basic.

It follows from the definition of the basis minor that its order is equal to the rank of the matrix. For a non-zero matrix A, there can be several basic minors; there is always one basic minor.

For example, consider the matrix .

All third-order minors of this matrix are equal to zero, since the elements of the third row of this matrix are the sum of the corresponding elements of the first and second rows.

The following minors of the second order are basic, since they are nonzero

Minors are not basic, since they are equal to zero.

Matrix rank theorem.

If the rank of a matrix of order p by n is r, then all elements of the rows (and columns) of the matrix that do not form the chosen basis minor are linearly expressed in terms of the corresponding elements of the rows (and columns) that form the basis minor.

What does the matrix rank theorem give us?

If, by the Kronecker-Capelli theorem, we have established the compatibility of the system, then we choose any basic minor of the main matrix of the system (its order is equal to r), and exclude from the system all equations that do not form the chosen basic minor. The SLAE obtained in this way will be equivalent to the original one, since the discarded equations are still redundant (according to the matrix rank theorem, they are a linear combination of the remaining equations).

As a result, after discarding the excessive equations of the system, two cases are possible.

If the number of equations r in the resulting system is equal to the number of unknown variables, then it will be definite and the only solution can be found by the Cramer method, the matrix method or the Gauss method.

Example.

.

Solution.

Rank of the main matrix of the system is equal to two, since the minor of the second order different from zero. Extended matrix rank is also equal to two, since the only minor of the third order is equal to zero

and the minor of the second order considered above is different from zero. Based on the Kronecker-Capelli theorem, one can assert the compatibility of the original system of linear equations, since Rank(A)=Rank(T)=2 .

As a basis minor, we take . It is formed by the coefficients of the first and second equations:


The third equation of the system does not participate in the formation of the basic minor, so we exclude it from the system based on the matrix rank theorem:


Thus we have obtained an elementary system of linear algebraic equations. Let's solve it by Cramer's method:


Answer:

x 1 \u003d 1, x 2 \u003d 2.

If the number of equations r in the resulting SLAE is less than the number of unknown variables n , then we leave the terms that form the basic minor in the left parts of the equations, and transfer the remaining terms to the right parts of the equations of the system with the opposite sign.

The unknown variables (there are r of them) remaining on the left-hand sides of the equations are called main.

Unknown variables (there are n - r of them) that ended up on the right side are called free.

Now we assume that the free unknown variables can take arbitrary values, while the r main unknown variables will be expressed in terms of the free unknown variables in a unique way. Their expression can be found by solving the resulting SLAE by the Cramer method, the matrix method, or the Gauss method.

Let's take an example.

Example.

Solve System of Linear Algebraic Equations .

Solution.

Find the rank of the main matrix of the system by the bordering minors method. Let us take a 1 1 = 1 as a non-zero first-order minor. Let's start searching for a non-zero second-order minor surrounding this minor:


So we found a non-zero minor of the second order. Let's start searching for a non-zero bordering minor of the third order:


Thus, the rank of the main matrix is ​​three. The rank of the augmented matrix is ​​also equal to three, that is, the system is consistent.

The found non-zero minor of the third order will be taken as the basic one.

For clarity, we show the elements that form the basis minor:


We leave the terms participating in the basic minor on the left side of the equations of the system, and transfer the rest with opposite signs to the right sides:


We give free unknown variables x 2 and x 5 arbitrary values, that is, we take , where are arbitrary numbers. In this case, the SLAE takes the form


We solve the obtained elementary system of linear algebraic equations by the Cramer method:


Hence, .

In the answer, do not forget to indicate free unknown variables.

Answer:

Where are arbitrary numbers.

Summarize.

To solve a system of linear algebraic equations of a general form, we first find out its compatibility using the Kronecker-Capelli theorem. If the rank of the main matrix is ​​not equal to the rank of the extended matrix, then we conclude that the system is inconsistent.

If the rank of the main matrix is ​​equal to the rank of the extended matrix, then we choose the basic minor and discard the equations of the system that do not participate in the formation of the chosen basic minor.

If the order of the basis minor is equal to the number of unknown variables, then the SLAE has a unique solution, which can be found by any method known to us.

If the order of the basis minor is less than the number of unknown variables, then we leave the terms with the main unknown variables on the left side of the equations of the system, transfer the remaining terms to the right sides and assign arbitrary values ​​to the free unknown variables. From the resulting system of linear equations, we find the main unknown variables by the Cramer method, the matrix method or the Gauss method.

Gauss method for solving systems of linear algebraic equations of general form.

Using the Gauss method, one can solve systems of linear algebraic equations of any kind without their preliminary investigation for compatibility. The process of successive elimination of unknown variables makes it possible to draw a conclusion about both the compatibility and inconsistency of the SLAE, and if a solution exists, it makes it possible to find it.

From the point of view of computational work, the Gaussian method is preferable.

See its detailed description and analyzed examples in the article Gauss method for solving systems of linear algebraic equations of general form.

Recording the general solution of homogeneous and inhomogeneous linear algebraic systems using the vectors of the fundamental system of solutions.

In this section, we will focus on joint homogeneous and inhomogeneous systems of linear algebraic equations that have an infinite number of solutions.

Let's deal with homogeneous systems first.

Fundamental decision system A homogeneous system of p linear algebraic equations with n unknown variables is a set of (n – r) linearly independent solutions of this system, where r is the order of the basis minor of the main matrix of the system.

If we designate linearly independent solutions of a homogeneous SLAE as X (1) , X (2) , …, X (n-r) (X (1) , X (2) , …, X (n-r) are matrices columns of dimension n by 1 ) , then the general solution of this homogeneous system is represented as a linear combination of vectors of the fundamental system of solutions with arbitrary constant coefficients С 1 , С 2 , …, С (n-r), that is, .

What does the term general solution of a homogeneous system of linear algebraic equations (oroslau) mean?

The meaning is simple: the formula specifies all possible solutions to the original SLAE, in other words, taking any set of values ​​of arbitrary constants C 1 , C 2 , ..., C (n-r) , according to the formula we will get one of the solutions of the original homogeneous SLAE.

Thus, if we find a fundamental system of solutions, then we can set all solutions of this homogeneous SLAE as .

Let us show the process of constructing a fundamental system of solutions for a homogeneous SLAE.

We choose the basic minor of the original system of linear equations, exclude all other equations from the system, and transfer to the right-hand side of the equations of the system with opposite signs all terms containing free unknown variables. Let's give the free unknown variables the values ​​1,0,0,…,0 and calculate the main unknowns by solving the resulting elementary system of linear equations in any way, for example, by the Cramer method. Thus, X (1) will be obtained - the first solution of the fundamental system. If we give the free unknowns the values ​​0,1,0,0,…,0 and calculate the main unknowns, then we get X (2) . And so on. If we give the free unknown variables the values ​​0,0,…,0,1 and calculate the main unknowns, then we get X (n-r) . This is how the fundamental system of solutions of the homogeneous SLAE will be constructed and its general solution can be written in the form .

For inhomogeneous systems of linear algebraic equations, the general solution is represented as

Let's look at examples.

Example.

Find the fundamental system of solutions and the general solution of a homogeneous system of linear algebraic equations .

Solution.

The rank of the main matrix of homogeneous systems of linear equations is always equal to the rank of the extended matrix. Let us find the rank of the main matrix by the method of fringing minors. As a nonzero minor of the first order, we take the element a 1 1 = 9 of the main matrix of the system. Find the bordering non-zero minor of the second order:

A minor of the second order, different from zero, is found. Let's go through the third-order minors bordering it in search of a non-zero one:

All bordering minors of the third order are equal to zero, therefore, the rank of the main and extended matrix is ​​two. Let's take the basic minor. For clarity, we note the elements of the system that form it:

The third equation of the original SLAE does not participate in the formation of the basic minor, therefore, it can be excluded:

We leave the terms containing the main unknowns on the right-hand sides of the equations, and transfer the terms with free unknowns to the right-hand sides:

Let us construct a fundamental system of solutions to the original homogeneous system of linear equations. The fundamental system of solutions of this SLAE consists of two solutions, since the original SLAE contains four unknown variables, and the order of its basic minor is two. To find X (1), we give the free unknown variables the values ​​x 2 \u003d 1, x 4 \u003d 0, then we find the main unknowns from the system of equations
.

The term "system" is used in various sciences. Accordingly, different definitions of the system are used in different situations: from philosophical to formal. For the purposes of the course, the following definition is best suited: a system is a set of elements united by links and functioning together to achieve a goal.

Systems are characterized by a number of properties, the main of which are divided into three groups: static, dynamic and synthetic.

1.1 Static properties of systems

static properties are called features of some state of the system. This is what the system possesses at any fixed point in time.

Integrity. Every system acts as something unified, whole, isolated, different from everything else. This property is called system integrity. It allows you to divide the whole world into two parts: the system and the environment.

Openness. The isolated system, distinguished from everything else, is not isolated from the environment. On the contrary, they are connected and exchange various types of resources (substance, energy, information, etc.). This feature is referred to as "openness".

The connections of the system with the environment are directional: according to one, the environment affects the system (system inputs), according to others, the system influences the environment, does something in the environment, gives something to the environment (system outputs). The description of the inputs and outputs of the system is called the black box model. In such a model, there is no information about the internal features of the system. Despite the apparent simplicity, such a model is often enough to work with the system.

In many cases, when controlling equipment or people, information only about the inputs and outputs of the system allows you to successfully achieve the goal. However, this model must meet certain requirements. For example, the user may experience difficulties if he does not know that in some TV models the power button does not need to be pressed, but pulled out. Therefore, for successful management, the model must contain all the information necessary to achieve the goal. When attempting to satisfy this requirement, four types of errors can arise, which stem from the fact that the model always contains a finite number of connections, while the number of connections in a real system is unlimited.

An error of the first kind occurs when the subject erroneously considers the relationship as significant and decides to include it in the model. This leads to the appearance of unnecessary, unnecessary elements in the model. An error of the second kind, on the contrary, is made when a decision is made to exclude an allegedly insignificant connection from the model, without which, in fact, the achievement of the goal is difficult or even impossible.

The answer to the question of which error is worse depends on the context in which it is asked. It is clear that the use of a model containing an error inevitably leads to losses. Losses can be small, acceptable, intolerable and unacceptable. The damage caused by a Type I error is due to the fact that the information introduced by it is redundant. When working with such a model, you will have to spend resources on fixing and processing unnecessary information, for example, spending computer memory and processing time on it. This may not affect the quality of the solution, but it will definitely affect the cost and timeliness. Losses from an error of the second kind - damage from the fact that there is not enough information to fully achieve the goal, the goal cannot be fully achieved.

Now it is clear that the worst mistake is the one, the losses from which are greater, and this depends on the specific circumstances. For example, if time is a critical factor, then an error of the first kind becomes much more dangerous than an error of the second kind: a decision made on time, even if not the best, is preferable to an optimal, but late one.

Type III error is considered to be the consequences of ignorance. In order to assess the significance of some connection, you need to know that it exists at all. If this is not known, then the question of including the connection in the model is not at all worth it. In the event that such a connection is insignificant, then in practice its presence in reality and its absence in the model will be imperceptible. If the relationship is significant, then there will be difficulties similar to those with a Type II error. The difference is that the Type III error is more difficult to correct: it requires the extraction of new knowledge.

An error of the fourth kind occurs when an erroneous assignment of a known significant connection to the number of inputs or outputs of the system. For example, it is well established that in 19th-century England, the health of men wearing top hats far exceeded that of men wearing caps. It hardly follows from this that the type of headgear can be considered as an input for a system for predicting the state of health.

Internal heterogeneity of systems, distinctness of parts. If you look inside the "black box", it turns out that the system is heterogeneous, not monolithic. It can be found that different qualities in different parts of the system are different. The description of the internal heterogeneity of the system is reduced to the isolation of relatively homogeneous areas, drawing boundaries between them. This is how the concept of the parts of the system appears. On closer examination, it turns out that the selected large parts are also inhomogeneous, which requires the selection of even smaller parts. The result is a hierarchical description of the parts of the system, which is called the composition model.

Information about the composition of the system can be used to work with the system. The goals of interaction with the system can be different, and therefore the models of the composition of the same system can also differ. At first glance, it is not difficult to distinguish the parts of the system, they are "striking". In some systems, parts arise arbitrarily, in the process of natural growth and development (organisms, societies, etc.). Artificial systems are deliberately assembled from previously known parts (mechanisms, buildings, etc.). There are also mixed types of systems, such as reserves, agricultural systems. On the other hand, from the point of view of the rector, student, accountant and business executive, the university consists of different parts. The plane consists of different parts from the point of view of the pilot, the stewardess, the passenger. The difficulties of creating a composition model can be represented by three provisions.

First, the whole can be divided into parts in different ways. In this case, the method of division is determined by the goal. For example, the composition of a car is presented in different ways to novice motorists, future professional drivers, mechanics preparing to work in a car service center, and salespeople in car dealerships. It is natural to ask whether parts of the system "really" exist? The answer is contained in the formulation of the property in question: we are talking about the distinguishability, and not about the separability of parts. One can distinguish between the parts of the system necessary to achieve the goal, but one cannot separate them.

Secondly, the number of parts in the composition model also depends on the level at which the fragmentation of the system is stopped. The pieces on the terminal branches of the resulting hierarchical tree are called elements. Under different circumstances, decomposition is terminated at different levels. For example, when describing upcoming work, you have to give instructions to an experienced worker and a novice in varying degrees of detail. Thus, the composition model depends on what is considered elementary. There are cases when an element has a natural, absolute character (cell, individual, phoneme, electron).

Thirdly, any system is part of a larger system, and sometimes several systems at once. Such a metasystem can also be divided into subsystems in different ways. This means that the outer boundary of the system has a relative, conditional character. The definition of the boundaries of the system is made taking into account the goals of the subject who will use the system model.

Structured. The property of structuredness lies in the fact that the parts of the system are not isolated, not independent of each other; they are interconnected and interact with each other. At the same time, the properties of the system essentially depend on how exactly its parts interact. Therefore, information about the connections of the elements of the system is so important. The list of essential links between the elements of the system is called the system structure model. Endowment of any system with a certain structure is called structuredness.

The concept of structuring further deepens the idea of ​​the integrity of the system: connections, as it were, hold the parts together, hold them as a whole. Integrity, noted earlier as an external property, receives a reinforcing explanation from within the system - through the structure.

When building a model of the structure, certain difficulties are also encountered. The first of these is related to the fact that the structure model is determined after the composition model is chosen, and depends on what exactly the composition of the system is. But even with a fixed composition, the structure model is variable. This is due to the possibility of different ways to determine the significance of relationships. For example, a modern manager is recommended, along with the formal structure of his organization, to take into account the existence of informal relations between employees, which also affect the functioning of the organization. The second difficulty stems from the fact that each element of the system, in turn, is a "little black box". So all four types of errors are possible when determining the inputs and outputs of each element included in the structure model.

1.2 DYNAMIC PROPERTIES OF SYSTEMS

If we consider the state of the system at a new point in time, then again we can find all four static properties. But if you superimpose the “photographs” of the system at different points in time on top of each other, then it will be found that they differ in details: during the time between two points of observation, some changes occurred in the system and its environment. Such changes may be important when working with the system, and, therefore, should be reflected in the descriptions of the system and taken into account when working with it. Features of changes over time inside the system and outside it are called the dynamic properties of the system. Generally, four dynamic properties of a system are distinguished.

Functionality. Processes Y(t) occurring at the outputs of the system are considered as its functions. The functions of the system are its behavior in the external environment, the results of its activities, the products produced by the system.

From the multiplicity of outputs follows the multiplicity of functions, each of which can be used by someone and for something. Therefore, the same system can serve different purposes. The subject using the system for his own purposes will naturally evaluate its functions and arrange them in relation to his needs. This is how the concepts of main, secondary, neutral, undesirable, superfluous function, etc. appear.

Stimulability. Certain processes also occur at the inputs of the system. X(t), affecting the system and turning after a series of transformations in the system into Y(t). Impact X(t) are called incentives, and the susceptibility of any system to external influences and the change in its behavior under these influences are called stimulability.

Variability of the system over time. In any system, there are changes that must be taken into account. In terms of the system model, we can say that the values ​​of internal variables (parameters) can change Z(t), the composition and structure of the system, and any combination thereof. The nature of these changes can also be different. Therefore, further classifications of changes may be considered.

The most obvious classification is according to the rate of change (slow, fast. The rate of change is measured relative to some rate taken as a standard. A large number of gradations of rates can be introduced. It is also possible to classify trends in changes in the system regarding its structure and composition.

We can talk about such changes that do not affect the structure of the system: some elements are replaced by others, equivalent ones; options Z(t) can change without changing the structure. This type of system dynamics is called its functioning. Changes can be quantitative in nature: there is an increase in the composition of the system, and although its structure automatically changes, this does not affect the properties of the system until a certain point (for example, the expansion of a garbage dump). Such changes are called system growth. With qualitative changes in the system, its essential properties change. If such changes are in a positive direction, they are called development. With the same resources, a developed system achieves better results, new positive qualities (functions) may appear. This is due to an increase in the level of consistency, organization of the system.

Growth occurs mainly due to the consumption of material resources, development - due to the assimilation and use of information. Growth and development may occur simultaneously, but they are not necessarily linked. Growth is always limited (due to limited material resources), and development from the outside is not limited, since information about the external environment is inexhaustible. Development is the result of learning, but learning cannot be done instead of the learner. Therefore, there is an internal restriction on development. If the system “does not want” to learn, it cannot and will not develop.

In addition to the processes of growth and development, reverse processes can also occur in the system. Changes inverse to growth are called recession, contraction, decrease. The reverse development of the change is called degradation, loss or weakening of useful properties.

The considered changes are monotonous, that is, they are directed "in one direction". Obviously, monotonous changes cannot last forever. In the history of any system, periods of decline and rise, stability and instability can be distinguished, the sequence of which forms an individual life cycle of the system.

You can use other classifications of processes occurring in the system: according to predictability, processes are divided into random and deterministic; according to the type of time dependence, processes are divided into monotonous, periodic, harmonic, impulse, etc.

Existence in a changing environment. Not only this system is changing, but also all the others. For the system under consideration, this looks like a continuous change in the environment. This circumstance has many consequences for the system itself, which must adapt to new conditions in order not to perish. When considering a specific system, attention is usually paid to the features of a particular reaction of the system, for example, the reaction rate. If we consider systems that store information (books, magnetic media), then the speed of reaction to changes in the external environment should be minimal to ensure the preservation of information. On the other hand, the response rate of the control system must be many times greater than the rate of change in the environment, since the system must choose the control action even before the state of the environment changes irreversibly.

1.3 SYNTHETIC PROPERTIES OF SYSTEMS

Synthetic properties include generalizing, integral, collective properties that describe the interaction of the system with the environment and take into account integrity in the most general sense.

Emergence. Combining elements into a system leads to the emergence of qualitatively new properties that are not derived from the properties of the parts, inherent only in the system itself and existing only as long as the system is one whole. Such qualities of the system are called
emergent (from the English "to arise").

Examples of emergent properties can be found in various fields. For example, none of the parts of an airplane can fly, but the airplane still flies. The properties of water, many of which are not fully understood, do not follow from the properties of hydrogen and oxygen.

Let there be two black boxes, each of which has one input, one output and performs one operation - adds one to the number at the input. When connecting such elements according to the scheme shown in the figure, we get a system without inputs, but with two outputs. At each cycle of work, the system will give out a larger number, while only even numbers will appear on one input, and only odd numbers on the other.

A	b
Fig.1.1. Connection of system elements: a) system with two outputs; b) parallel connection of elements

The emergent properties of a system are determined by its structure. This means that different combinations of elements will produce different emergent properties. For example, if you connect the elements in parallel, then the functionally new system will not differ from one element. Emergence will manifest itself in increasing the reliability of the system due to the parallel connection of two identical elements - that is, due to redundancy.

It should be noted an important case when the elements of the system have all its properties. This situation is typical for the fractal construction of the system. At the same time, the principles of structuring the parts are the same as those of the system as a whole. An example of a fractal system is an organization in which management is built identically at all levels of the hierarchy.

Inseparability into parts. This property is, in fact, a consequence of emergence. It is emphasized especially because it practical importance large, and underestimation is very common.

When a part is removed from the system, two important events occur. First, the composition of the system changes, and hence its structure. It will be a different system with different properties. Secondly, the element withdrawn from the system will behave differently due to the fact that its environment will change. All this suggests that when considering an element separately from the rest of the system, care should be taken.

Inherence. The system is all the more integral (from the English inherent - “being part of something”), the better it is coordinated, adapted to the environment, compatible with it. The degree of inherence is different and may change. The expediency of considering inherence as one of the properties of the system is related to the fact that the degree and quality of the implementation of the chosen function by the system depend on it. In natural systems, inherence is increased by natural selection. In artificial systems, inherence should be a special concern of the designer.

In a number of cases, inherence is provided with the help of intermediate, intermediary systems. Examples include adapters for using foreign electrical appliances in conjunction with Soviet-style sockets; middleware (such as the Windows COM service) that allows two programs from different manufacturers to communicate with each other.

Expediency. In systems created by man, the subordination of both structure and composition to the achievement of the set goal is so obvious that it can be recognized as a fundamental property of any artificial system. This property is called expediency. The goal for which the system is created determines which emergent property will ensure the achievement of the goal, and this, in turn, dictates the choice of the structure and composition of the system. In order to extend the concept of expediency to natural systems, it is necessary to clarify the concept of purpose. The refinement is carried out on the example of an artificial system.

The history of any artificial system begins at some point in time 0, when the existing value of the state vector Y 0 turns out to be unsatisfactory, that is, a problematic situation arises. The subject is dissatisfied with this condition and would like to change it. Let him be satisfied with the values ​​of the state vector Y*. This is the first definition of purpose. Further, it turns out that Y* does not exist now and cannot, for a number of reasons, be achieved in the near future. The second step in defining a goal is to recognize it as a desirable future state. It immediately becomes clear that the future is not limited. The third step in refining the notion of goal is to estimate the time T* when the desired state Y* can be reached under given conditions. Now the target becomes two-dimensional, it is a point (T*, Y*) on the graph. The task is to move from the point (0, Y 0) to the point (T*, Y*). But it turns out that this path can be taken along different trajectories, and only one of them can be realized. Let the choice fell on the trajectory Y*( t). Thus, the goal is now understood not only as the final state (T*, Y*), but also as the entire trajectory Y*( t) (“intermediate goals”, “plan”). So the goal is the desired future states Y*( t).

After time T* the state Y* becomes real. Therefore, it becomes possible to define the goal as a future real state. This makes it possible to say that natural systems also have the property of expediency, which allows us to approach the description of systems of any nature from a unified standpoint. The main difference between natural and artificial systems is that natural systems, obeying the laws of nature, realize objective goals, while artificial systems are created to achieve subjective goals.