§6. Heterogeneous system linear equations
If in the system of linear equations (7.1) at least one of the free terms V i is different from zero, then such a system is called heterogeneous.
Let a non-homogeneous system of linear equations be given, which can be represented in vector form as
, i = 1,2,.. .,To, (7.13)
Consider the corresponding homogeneous system
i = 1,2,...
,To.
(7.14)
Let the vector 
is the solution heterogeneous system(7.13), and the vector 
is a solution to the homogeneous system (7.14). Then it is easy to see that the vector 
is also a solution to the inhomogeneous system (7.13). Really
Now, using formula (7.12) for the general solution of the homogeneous equation, we have
Where 
any numbers from R, A 
– fundamental solutions homogeneous system.
Thus, the solution of an inhomogeneous system is the combination of its particular solution and the general solution of the corresponding homogeneous system.
Solution (7.15) is called general solution of an inhomogeneous system of linear equations. From (7.15) it follows that a simultaneous inhomogeneous system of linear equations has a unique solution if the rank r(A) main matrix A matches the number n unknown systems (Cramer system), if r(A) n, then the system has an infinite number of solutions and this set of solutions is equivalent to the subspace of solutions of the corresponding homogeneous system of equations of dimension n– r.
Examples.
1. Let a non-homogeneous system of equations be given, in which the number of equations To= 3, and the number of unknowns n = 4.
X 1 – X 2 + X 3 –2X 4 = 1,
X 1 – X 2 + 2X 3 – X 4 = 2,
5X 1 – 5X 2 + 8X 3 – 7X 4 = 3.
Let's determine the ranks of the main matrix A and expanded A * of this system. Because the A And A * non-zero matrices and k = 3 n, therefore 1 r (A), r * (A * ) 3. Consider the minors of the second order of matrices A And A * :
Thus, among the second-order minors of the matrices A And A * there is a minor other than zero, so 2 r(A),r * (A * ) 3. Now let's look at third order minors
, since the first and second columns are proportional. Likewise for minor 
.
And so all the third order minors of the main matrix A are equal to zero, therefore r(A) = 2. For the extended matrix A * there are also third order minors
Consequently, among the third order minors of the extended matrix A * there is a minor other than zero, so r * (A * ) = 3. This means that r(A) r * (A * ) and then, based on the Korneker–Capelli theorem, we conclude that this system is inconsistent.
2. Solve the system of equations
3X 1 + 2X 2 + X 3 + X 4 = 1,
3X 1 + 2X 2 – X 3 – 2X 4 = 2.
For this system 
and therefore 1 r(A),r *
(A *
) 2. Consider for matrices A And A *
second order minors
Thus, r(A)= r * (A * ) = 2, and, therefore, the system is consistent. As base variables, we choose any two variables for which the second-order minor, composed of the coefficients of these variables, is not equal to zero. Such variables could be, for example,
X 3 and X 4 because 
Then we have
X 3 + X 4 = 1 – 3X 1 – 2X 2 ,
– X 3 – 2X 4 = 2 – 3X 1 – 2X 2 .
Let's define a particular solution 
heterogeneous system. To do this, let's put X 1
= X 2
= 0.
X 3 + X 4 = 1,
– X 3 – 2X 4 = 2.
The solution to this system: X 3
= 4, X 4 = – 3, therefore, 
=
(0,0,4, –3).
Now let's define common decision corresponding homogeneous equation
X 3 + X 4 = – 3X 1 – 2X 2 ,
–X 3 – 2X 4 = – 3X 1 – 2X 2 .
Let's put: X 1 = 1, X 2 = 0
X 3 + X 4 = –3,
–X 3 – 2X 4 = –3.
The solution to this system X 3 = –9, X 4 = 6.
Thus 
Now let's put X 1 = 0, X 2 = 1
X 3 + X 4 = –2,
–X 3 – 2X 4 = –2.
Solution: X 3
= – 6, X 4 = 4, and then 
After a particular solution has been determined 
, inhomogeneous equations and fundamental solutions 
And 
the corresponding homogeneous equation, we write down the general solution of the inhomogeneous equation.
Where 
any numbers from R.
1st question Exam
1. System analysis methodology. The concept of a system. Static properties of the system. Openness. Difficulties in constructing a black box model. Heterogeneity of composition. Difficulties in constructing a composition model. Structure. Difficulties in constructing a structure model.
Static properties Let us name the features of a specific state of the system. This is what the system has at any fixed point in time.
Openness - the second property of the system. An isolated system, distinguishable from everything else, is not isolated from the environment. On the contrary, they are connected and exchange any types of resources (matter, energy, information, etc.) with each other. Let us remember that the connections between the system and the environment are directional; according to some, the environment influences the system (they are called system inputs), according to others, the system influences the environment, does something in the environment, produces something into the environment (such connections are called system outputs). The list of system inputs and outputs is called black box model . This model lacks information about the internal features of the system. Despite the (apparent) simplicity and poverty of content of the black box model, this model is often quite sufficient for working 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 unlimited. 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 natural for
achieving the goal.
Four types of errors when building a black box model:
An error of the first type occurs when a subject evaluates a connection as significant and decides to include it in the model, when in fact it is insignificant in relation to the goal and could not be taken into account. This leads to the appearance of “extra” elements in the model, essentially unnecessary.
An error of the second type, on the contrary, is made by the subject when he decides that a given connection is insignificant and does not deserve to be included in the model, when in fact, without it, our goal cannot be achieved fully or even at all.
An error of the third kind is considered to be the consequences of ignorance. In order to assess the significance of a certain connection, you need to know that it exists at all. If this is unknown, the question of including or not including it in the model does not arise at all: the models contain only what we know. But because we do not suspect the existence of a certain connection, it does not cease to exist and manifest itself in reality. And then everything depends on how significant it is for achieving our goal. If it is insignificant, then in practice we will not notice its presence in reality and absence in the model. If it is significant, we will experience the same difficulties as with an error of the second type. The difference is that an error of the third type is more difficult to correct: new knowledge must be acquired.
An error of the fourth type can occur when a known and recognized essential connection to the number of inputs or outputs.
Internal heterogeneity: distinguishability of parts (third property of the system). If you look inside the “black box”, it turns out that the system is not homogeneous, not monolithic; one may find that different qualities vary from place to place. The description of the internal heterogeneity of the system comes down to isolating relatively homogeneous areas and drawing boundaries between them. This is how the concept of parts of the system appears. Upon closer examination, it turns out that the selected large parts are also not homogeneous, which requires identifying even smaller parts. The result is a hierarchical list of system parts, which we will call a system composition model.
Difficulties in building a composition model that everyone has to overcome can be represented in three positions:
First. The whole can be divided into parts in different ways (like cutting a loaf of bread into slices of different sizes and shapes). And how exactly is it necessary? Answer: the way 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 is relative, conditional. Even the “obvious” boundary of the system (human skin, fence of an enterprise, etc.) under certain conditions turns out to be insufficient to determine the boundary in these conditions.
Structurality ,The fourth static property is that the parts of the system are ,not independent or isolated from each other; they are interconnected and interact with each other. Moreover, the properties of the system as a whole depend significantly on how exactly its parts interact. This is why information about the connections between parts is so often important. The list of essential connections between system elements is called a 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 structure models can be proposed for a given system. It is clear that to achieve a certain goal, one specific, most suitable model of them is required. The difficulty of choosing from existing ones or constructing a model specifically for our case stems from the fact that, by definition, a structure model is a list of essential connections.
The first difficulty is related to the fact that the structure model is determined after the composition model is selected, and depends on what exactly the composition of the system is. But even with a fixed composition, the structure model is variable - due to the possibility of defining the significance of connections differently.
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. System analysis methodology. 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 - the fifth property of the system. The processes Y(t) occurring at the outputs of the system (Y(1)^(уi(t), Ур(1), -, Ун(0) are considered as its functions. System functions - this is her behavior in external environment; changes made by the system in 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. At the inputs of the system, certain processes X(t) = (x^(t), X2 (t), x^(t)) also occur, affecting the system, turning (after a series of transformations in the system) into Y(t). Let us call the influences X(t) stimuli, and the susceptibility of any system to external influences and the change in its behavior under these influences will be called stimulability.
System variability over time - the seventh property of the system. In any system, changes occur that must be taken into account; provide for and include in the design of the future system; promote or counteract them, speeding them up or slowing them down when working with the existing system. Anything can change in the system, but in terms of our models we can give a visual classification of changes: the values of internal variables (parameters) Z(t), the composition and structure of the system, and any combinations thereof can change.
Existence in a changing environment - the eighth property of the system. Not only this system is changing, but also all others. For a given system, this 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 for it 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 that take into account what was said earlier, but place emphasis on 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 about the nature of systems than any other. The combination of parts into a system gives rise to qualitatively new properties in the system, which are not reducible 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 while the system is one whole. A system is more than a simple collection of parts. Qualities of the system that are unique to it are called emergents (from the English “to arise”).
Inseparability into parts - the tenth property of the system. Although this property is a simple consequence of emergence, its practical importance is so great, and its underestimation so common, that it is advisable to emphasize 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.
Firstly, this changes the composition of the system, and therefore its structure. This will be a different system, with different properties. Since the previous system has many properties, some property associated with this particular part will disappear altogether (it may or may not be emergent. Some property will change, but will be partially preserved. And some properties of the system are generally unimportant are associated with the part being withdrawn. Let us emphasize once again that whether or not the withdrawal 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 the system is that the part in the system and outside it are not the same thing. Its properties change due to the fact that the properties of an object are manifested in interactions with the objects surrounding it, and when removed from the system, the environment of the element becomes completely different.
Inserency - the eleventh property of the system. We will say that the system is the more inherent (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, reform, development, degradation, etc.). The fact that all systems are open does not mean that they are all in to the same degree well coordinated with the environment.
Feasibility - 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 should 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 implementation of the goal, and this, in turn, dictates the choice of the composition and structure of the system. One of the definitions of the system is states: a system is a means to an end. It is understood that if the put forward goal cannot be achieved due to existing capabilities, then the subject composes from the objects surrounding him new system, specially created 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 is implemented, and this can often be achieved in different ways.
3. System analysis methodology. Models and simulation. The concept of a model as a system. Analysis and synthesis as methods for constructing 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 is structured or how it interacts with the environment, two methods of cognition are distinguished: 1) analytical; 2) synthetic.
The analysis procedure consists of sequentially performing the following three operations; 1) divide a complex whole into smaller parts, presumably simpler; 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 remains unclear, the decomposition operation is repeated and we again attempt 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 analysis is a model of the system structure . The third product of the analysis is black box model for each element of the system.
Synthetic method consists of sequentially performing three operations: 1) identifying a larger system (metasystem), of 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 final product of the synthesis is 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 simultaneously create models of the composition and structure of the metasystem as by-products.
Analysis and synthesis are not opposite, 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 set - for as many classes and with such boundaries as dictated by the goal. Classification is performed somewhat differently when the set under consideration is clearly heterogeneous. Natural groupings (in statistics they are called clusters) seem to be asking to be defined as classes , (hence the name of the classification natural) . However, it should be kept in mind that natural classification is just a simplified, roughened model of reality .
Consistency of models with the culture of the subject . In order for a model to realize its model function, the presence of the model itself is not enough. 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 inherence of the model can change: increase (user training, appearance of an adapter such as the Rosetta stone, etc.) or decrease (forgetting, destruction of culture) due to changes in the environment or the model itself. Thus, one more element must be included in the modeling metasystem - culture.
4. System analysis methodology. Control. Five control components. 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 management goal.
The control action U(t) is the third control component . The fact that the inputs and outputs of the system are interconnected by a certain relation Y(t)=S allows us to hope that there is a control action in which the goal V*(t) is realized at the output.
The system model becomes the fourth component of the management process.
All actions required for control must be completed. This function usually assigned to a system specially created for this purpose (the fifth component of the management process). Called a control unit or control system (subsystem), control device and so on. In real Control block can be a subsystem of a controlled system (like an avodouiravle1gae - part of a plant, an autopilot - a part of an airplane), but it can also be an external system (like a ministry for a subordinate enterprise, like an airfield dispatcher for an airplane landing).
Seven control types:
The first type of control is simple system control, or program control.
The second type of control is control 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 objectives.
The sixth type of management is management of large systems.
Seventh type of control. 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 prevent one from achieving the goal: lack of information about the control object (second type), external minor interference that slightly deviates the system from the target trajectory (third type ), discrepancy between the emergent properties of the system and the set goal (fourth type), lack of material resources, making the goal unattainable and requiring its replacement (fifth type), lack of time to find the best solution (sixth type).
5. System analysis technology. Conditions for the success of systems research. Stages of systemic research: fixing the problem, diagnosing the problem, compiling a list of stakeholders, identifying the problem mix.
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 top officials of organizations - obligatory participants in a problem situation (managers of problem-containing and problem-solving systems);
refusal of the requirement to formulate the necessary result in advance (“technical specifications”), since there are many improving interventions and they are unknown in advance, especially which one will be chosen for implementation.
Fixing the problem – the task is to formulate the problem and document it. The formulation of the problem is developed by the client himself; The analyst's job is to find out what the client is complaining about, what he is dissatisfied with. This is the client's problem as he sees it. At the same time, you should try not to influence his opinion or distort it.
Diagnosis of the problem . Which of the problem solving methods to use to solve a given problem depends on whether we choose to influence the most dissatisfied subject or intervene in the reality with which he is dissatisfied (there may be cases when a combination of both influences is advisable). The task of this stage is to make a diagnosis - to determine what type of problem it is.
Compiling a list of stakeholders .Our ultimate goal is to implement improvement interventions. Each stage should bring us one step closer to it, but we must take special care that this step is 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 (and this is precisely what the concept of improving intervention is based on), it is necessary to first find out who is involved in the problem situation and make a list of them. At the same time, 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 anyone into account threatens that our intervention will not be improving. Thus, the list of participants in the problem situation must be complete.
Identifying the problem mess . Stakeholders have interests that we have to take into account. But for this you need to know them. For now, we only have a list of interest holders. The first piece of information that needs to be obtained about a 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), some are completely satisfied (positive assessment), others may be neutral about reality. This way it will become clearer<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. System analysis technology. System analysis operations. Stages of system research: determination of the configurator, target identification, determination of criteria, experimental research.
System Analysis Operations . If the client agrees to the terms of the contract, the analyst proceeds to the first stage, having completed which, 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 a successful solution to a problem is the presence of an adequate model of the problem situation, with its help it will be possible to test and compare options for proposed actions. This model (or a set of models) must inevitably be constructed using the means of some language (or languages). The question arises of how many and what languages are needed to work on this problem and how to choose them. It's called a configurator. a minimum set of professional languages that allows you to give a complete (adequate) description of the problem situation and its transformations. All work during problem solving will take place in the languages of the configurator. And only on them. Defining the configurator is the task of this stage. 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 be considered as another PROPERTY of systems, as a means by which the system solves its problem.
Target detection . When seeking to implement an improvement intervention, we must ensure that none of the stakeholders view it negatively. People evaluate a change positively if it brings them closer to their goal, and negatively if it moves them away from it. Therefore, 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 a problem, it will be necessary to compare the proposed options, assess the degree to which the goal has been achieved or deviated from it, and monitor the progress of events. This is achieved by highlighting some features of the objects and processes under consideration. These signs must be related to the features of the objects or processes under consideration that interest us, and must be accessible to observation and measurement. Then, based on 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 you need, the easier it will be to make comparisons. That is, it is desirable to minimize the number of criteria; it would be nice to reduce it to one. Selection of criteria . The criteria are quantitative models of qualitative goals. In fact, the formed criteria in the future, in a sense, represent and replace the goals: optimization according to the criteria should ensure maximum approximation to the goal. Of course, the criteria are not identical to the goal, they are a semblance of the goal, its model. Determining the criterion value for a given alternative is essentially a measurement of the degree of its suitability as a means to an end.
Experimental study of systems. Experiment and model. Often, missing information about a system can only be obtained from the system itself by conducting an experiment specially designed for this purpose. The information contained in the experimental protocol is extracted, subjecting the resulting data to processing and transformation into a form suitable for inclusion in the system model. The final step is to correct the model, incorporating the received information into the model. It is easy to perceive that experimentation is needed to improve the model. It is also important to understand that experimentation is impossible without a model. They are in the same cycle. However, rotation through this cycle resembles not a spinning wheel, but a rolling snowball - with each revolution it becomes larger and more weighty.
7. System analysis technology. Stages of system research: building and improving models, generating alternatives, decision making, +.
Construction and improvement of models. In systems analysis, a problem model and a situation are needed in order to "lose" possible options for interventions in order to cut off not only those that will not improve, but also to select among those that improve the most (according to our criteria) those that improve. It should be emphasized that a contribution to the construction of a 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 "completing construction" (i.e. adding new elements or removing unnecessary ones).
Generating Alternatives . In the technology described, this action is performed in two stages:
identifying discrepancies between the problem and target mixtures. The differences between the current (and unsatisfactory) state of the organization and the future, most desirable, ideal state to which it is supposed to strive must be clearly formulated. These differences are the gaps whose elimination needs to be planned;
proposing possible options for eliminating or reducing detected discrepancies. Actions, procedures, rules, projects, programs and policies - all components of management - must be designed for implementation.
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 connections 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 the features of a certain state of the system. This is what the system has at any given point in time.Integrity. Every system appears as something unified, whole, separate, 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 (matter, energy, information, etc.). This feature is designated by the term “openness”.
The connections between the system and the environment are directional: in some ways, the environment influences the system (system inputs), in others, the system influences the environment, does something in the environment, and outputs something to the environment (system outputs). The description of the inputs and outputs of a system is called a black box model. In such a model there is no information about the internal features of the system. Despite its apparent simplicity, such a model is often quite sufficient for working with the system.
In many cases, when managing equipment or people, information only about the inputs and outputs of the system allows you to successfully achieve the goal. However, for this, the model must meet certain requirements. For example, the user may experience difficulties if he does not know that on some TV models the power button must be pulled out rather than pressed. Therefore, for successful management, the model must contain all the information necessary to achieve the goal. When trying to satisfy this requirement, four types of errors can occur, which stem from the fact that the model always contains a finite number of connections, whereas in a real system the number of connections is unlimited.
An error of the first type occurs when a subject mistakenly views a relationship as significant and decides to include it in the model. This leads to the appearance of extra, unnecessary elements in the model. An error of the second type, on the contrary, is made when a decision is made to exclude a supposedly insignificant connection from the model, without which, in fact, achieving 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 using a model containing an error inevitably leads to losses. Losses can be small, acceptable, intolerable or unacceptable. The damage caused by a type 1 error is due to the fact that the information it contains is superfluous. When working with such a model, you will have to spend resources on recording and processing unnecessary information, for example, wasting computer memory and processing time on it. This may not affect the quality of the solution, but it will certainly affect the cost and timeliness. Losses from an error of the second type are 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 worse mistake is the one from which the losses are greater, and this depends on specific circumstances. For example, if time is a critical factor, then an error of the first type becomes much more dangerous than an error of the second type: a decision made on time, even if not the best, is preferable to an optimal, but late one.
An error of the third kind is considered to be the consequences of ignorance. In order to assess the significance of a certain 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 worth it at all. If such a connection is insignificant, then in practice its presence in reality and absence in the model will be unnoticeable. If the connection is significant, then difficulties will arise similar to those with a type II error. The difference is that a type 3 error is more difficult to correct: this requires acquiring new knowledge.
An error of the fourth kind occurs when a known essential connection is erroneously attributed 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 was significantly superior to that of men wearing caps. It hardly follows from this that the type of headdress can be considered as an input for a system for predicting health status.
Internal heterogeneity of systems, distinctness of parts. If you look inside the “black box”, it turns out that the system is heterogeneous, not monolithic. One may find that different qualities differ in different parts of the system. The description of the internal heterogeneity of the system comes down to isolating relatively homogeneous areas and drawing boundaries between them. This is how the concept of parts of the system appears. Upon closer examination, it turns out that the identified large parts are also heterogeneous, which requires identifying even smaller parts. The result is a hierarchical description of the parts of the system, which is called a composition model.
Information about the composition of the system can be used to work with the system. The goals of interaction with the system may be different, and therefore the composition models of the same system may also differ. At first glance, it is not difficult to distinguish the parts of the system; they “catch the eye.” 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 nature reserves and agricultural systems. On the other hand, from the point of view of the rector, student, accountant and business manager, the university consists of different parts. An airplane consists of different parts from the point of view of the pilot, flight attendant, and passenger. The difficulties of creating a composition model can be represented in three ways.
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 differently to novice car enthusiasts, 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 distinguishability, and not about the separability of parts. You can distinguish between the parts of the system needed to achieve the goal, but you 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 parts on the terminal branches of the resulting hierarchical tree are called elements. In different circumstances, decomposition is terminated at different levels. For example, when describing upcoming work, it is necessary to give an experienced worker and a novice instructions of varying degrees of detail. Thus, the composition model depends on what is considered elemental. 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 external boundary of the system is relative, conditional. The boundaries of the system are determined taking into account the goals of the subject who will use the system model.
Structure. The property of structuredness is that the parts of the system are not isolated, not independent of each other; they are interconnected and interact with each other. Moreover, the properties of the system significantly depend on how exactly its parts interact. Therefore, information about the connections of system elements is so important. The list of essential connections between system elements is called a system structure model. The endowment of any system with a certain structure is called structuring.
The concept of structuring further deepens the idea of the integrity of the system: connections, as it were, hold the parts together and hold them together as a whole. Integrity, noted earlier as an external property, receives a supporting explanation from within the system - through structure.
When constructing a structure model, certain difficulties are also encountered. The first of these is due to the fact that the structure model is determined after the composition model is selected, 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 defining the significance of connections in different ways. For example, a modern manager is recommended, along with the formal structure of his organization, to take into account the existence of informal relationships 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 defining 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, we can again detect all four static properties. But if you superimpose “photographs” of the system at different points in time on top of each other, you will find that they differ in detail: during the time between the two moments of observation, some changes occurred in the system and its environment. Such changes may be important when working with the system, and, therefore, must be reflected in system descriptions and taken into account when working with it. The features of changes over time inside and outside the system are called the dynamic properties of the system. Typically, 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 a system are its behavior in the external environment, the results of its activities, and the products produced by the system.
From the multiplicity of outputs follows a multiplicity of functions, each of which can be used by someone and for something. Therefore, the same system can serve different purposes. A subject using a system for his own purposes will naturally evaluate its functions and organize 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 system inputs X(t), affecting the system and turning after a series of transformations in the system into Y(t). Impacts X(t) are called stimuli, and the very susceptibility of any system to external influences and the change in its behavior under these influences is called stimulability.
Variability of the system over time. In any system, changes occur 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), composition and structure of the system and any combinations thereof. The nature of these changes may also be different. Therefore, further classifications of changes may be considered.
The most obvious classification is by the speed of change (slow, fast. The speed of change is measured relative to any speed taken as a standard. It is possible to introduce a large number of gradations of speed. It is also possible to classify trends in changes in the system regarding its structure and composition.
We can talk about changes that do not affect the structure of the system: some elements are replaced by others that are equivalent; options Z(t) can change without changing the structure. This type of system dynamics is called its functioning. Changes can be quantitative in nature: the composition of the system increases, 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 landfill). Such changes are called system growth. With qualitative changes in the system, its essential properties change. If such changes go in a positive direction, they are called development. With the same resources, a developed system achieves better results, and new positive qualities (functions) may appear. This is due to an increase in the level of consistency and 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 can occur simultaneously, but they are not necessarily related. 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 training, but training cannot be carried out instead of the learner. Therefore, there is an internal limitation 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 opposite to growth are called decline, contraction, decrease. A change that is opposite to development is called degradation, loss or weakening of beneficial properties.
The changes considered are monotonic, that is, they are directed “in one direction.” Obviously, monotonous changes cannot last forever. In the history of any system, one can distinguish periods of decline and rise, stability and instability, the sequence of which forms the 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 monotonic, periodic, harmonic, pulsed, etc.
Existence in a changing environment. Not only this system is changing, but also all 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 characteristics 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 response to changes in the external environment should be minimal to ensure the preservation of information. On the other hand, the response speed of the control system must be many times greater than the rate of change in the environment, since the system must select a 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. The combination of 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 the plane can fly, but the plane, nevertheless, 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 - adding one to the number at the input. When connecting such elements according to the diagram shown in the figure, we obtain a system without inputs, but with two outputs. At each cycle of operation, the system will produce a larger number, while only even numbers will appear at one input, and only odd numbers at the other.
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| Fig.1.1. Connection of system elements: a) system with two outputs; b) parallel connection of elements |
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The emergent properties of a system are determined by its structure. This means that with different combinations of elements, different emergent properties will arise. For example, if you connect 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 is worth noting an important case when the elements of the system possess all its properties. This situation is typical for the fractal construction of a 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 structured identically at all levels of the hierarchy.
Inseparability into parts. This property is, in fact, a consequence of emergence. It is especially emphasized because its practical importance is great, and underestimation is very common.
When a part is removed from the system, two important events occur. Firstly, this changes the composition of the system, and therefore its structure. This will be a different system with different properties. Secondly, an element removed from the system will behave differently due to the fact that its environment will change. All of this is to say that caution should be used when considering an element in isolation from the rest of the system.
Inherence. The more integral a system is (from the English inherent - “being part of something”), the better it is coordinated, adapted to the environment, and compatible with it. The degree of inherence varies and can change. The expediency of considering inherence as one of the properties of the system is due to the fact that the degree and quality of the system’s implementation of the chosen function depends on it. In natural systems, inherency increases through natural selection. In artificial systems, inherence should be a special concern of the designer.
In some cases, inherence is ensured 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 COM service in Windows) 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 achieving 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 clarification is carried out using an artificial system as an example.
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 unhappy with this condition and would like to change it. Let him be satisfied by the values of the state vector Y*. This is the first definition of the goal. Further, it is discovered 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 desired future state. It immediately becomes clear that the future is not limited. The third step in clarifying the concept of a goal is to estimate the time T* when the desired state Y* can be achieved under given conditions. Now the target becomes two-dimensional, it is a point (T*, Y*) on the graph. The task is to move from point (0, Y 0) to 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 fall on the trajectory Y*( t). Thus, the goal now means not only the final state (T*, Y*), but also the entire trajectory Y*( t) (“intermediate goals”, “plan”). So, the goal is the desired future states Y*( t).
After time T*, 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 position. The main difference between natural and artificial systems is that natural systems, obeying the laws of nature, realize objective goals, and artificial systems are created to realize subjective goals.
Walras' general equilibrium theory, which is the ideological basis of a centralized economy, has a number of undoubted advantages, namely: integrity and certainty of conclusions, making it very attractive for economic analysis.
However, within the framework of this theory it is impossible to adequately describe a decentralized economy. We are talking about the coordination mechanism, the time aspect of economic processes, the nature of flows and agents.
The practice of "groping" for equilibrium in Walras's theory essentially implies that no one in the market can influence prices, that each agent has perfect knowledge of supply and demand, that the process of "groping" occurs instantly, and, finally, that the execution of transactions absolutely unacceptable until “true prices” are established by “groping”, i.e. centralized control over all flows. Thus, this model, which involves very significant restrictions, is very reminiscent of the ideal image of the Soviet economy.
As the Polish economist Lange argued, “nothing is more important than understanding the laws of a decentralized economy. First of all, because it is the only reality with which we deal.”
French economist Jean-Paul Fitoussi argues that between the state and the market there is something intermediate, and by this intermediate he means the variety of forms of coordination of their relations and connections. These two-way connections are not limited to the transmission of an order, nor to direct contact between the exchange participants within the framework of a specific contract. An order has meaning only to the extent that it is executed. This creates some asymmetry between the positions of the superior and the subordinate in favor of the latter. It is in the power of the subordinate to carry out the order. Of course, the boss can check the execution of orders and, as Stalin did in his time, punish the executor. But verification is also an order that reproduces the original asymmetry. Each inspection is followed by an inspection inspection. Thus, already in the very basis of the centralized economy are the origins of decentralization - operational and information asymmetry - heterogeneity.
According to Jacques Sapir, five such forms of heterogeneity can be distinguished.
1. Heterogeneity of products associated with unequal possibilities for their substitution. This is determined not only by the nature of the product, but also by the specific method of its inclusion in a particular technological or economic process.
2. Heterogeneity of economic agents, which is not limited to differences between an employee, an entrepreneur and a capitalist. Dominance means a situation in which around some types of behavior or around some agents there is a spontaneous organization of other types of behavior or agents, i.e. the formation of a collective. The transition from the individual to the collective level is carried out through cooperation within a group of organizations that act as economic agents. They, in turn, imply heterogeneity in methods of interaction and coordination.
3. Heterogeneity of time. It can take two different and complementary forms. One of them is due to the fact that acts of consumption, saving or production for different agents have different time durations - continuum. This is the problem of “action time” heterogeneity. The emergence of another form of time heterogeneity is associated with what we call the time frame within which each agent's decision remains valid. In this case we can talk about “time intervals”.
4. Heterogeneity of enterprises as local production systems. Even if the products produced are identical, the behavior of a small enterprise differs significantly from the behavior of an enterprise with a large number of employees. In addition, there is a difference between the production of a simple and the production of a complex product, etc.
5. Heterogeneity of spaces in which economic actions take place. The unequal provision of different regions with factors of production, both material and human, naturally affects the relative price of these factors.
The typologization of heterogeneities by J. Sapir would be incomplete without two more heterogeneities:
6. Heterogeneity of the information space, due to the geographical and historical and cultural characteristics of the economic space.
7. Political heterogeneity of regions and countries, ensuring the security of investments and accessibility to information sources, and significantly influencing their investment attractiveness. The example of China's economic development very clearly illustrates this point.
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