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DEVELOPMENT OF THE METHOD TO DESIGN VERIFIABLE MODELS FOR MIVAR EXPERT SYSTEMS
Current state of expert systems (ES) was analyzed, which differs in representation and algorithms of knowledge processing. Solution algorithm of the certain problem is unknown and is constructed during the process of solution with the help of mechanisms of logical inference on the basis of the information accessible to the ES including various rules and input data. Before the 2002 logical inference was considered as a NP-complete task with a factorial computational complexity, but then the mivar approach, which integrated multidimensional evolutionary databases and rules in gnoseological formalism "Object-Property-Relation" and mivar networks in productional formalism "Object-Rule" with linear computational complexity of logical inference was introduced. Expert systems, capable of processing more than 5 billion rules in second even using personal computers, were created. It was proved that the design of big databases is a relevant task. Databases for each type and framework of an ES are different and non-interchangeable. As mivar approach is based on gnoseological representation of the data, so new method of knowledge base construction is used. Wi!Mi expert system designer, which uses mivar approach, is a good tool which has been thoroughly tested through 5 years of service.
This article shows that mivar ES might be used as a decision-making systems for an autonomous robot, UAVs, driverless cars and complex robotic multilevel systems. Such ES must be reliable and verifiable in various circumstances. Human expert can't design and test mivar model, which can contain up to 300 production rules by himself; such systems are usually made by an expert team, which can lead to a new set of problems. Beside that, the structure of the model may significantly differ when used in various subject domains, which leads to necessity to describe various unique scenarios for utilized models. Method to design verifiable model for mivar expert systems includes 3 sections: 1) Specification of model description, which contain the description itself in the form of text document; usage scenarios of the model; recommendation on how to name and comment objects of the model; 2) Decomposition of the model using classes; 3) Graph visualization of certain subject domain mivar model.
So the development of method to design verifiable models for mivar ES is a relevant, complex scientific task which has many practical purposes. This method can streamline, speed up and upscale the process of design and following verification, maintenance and further development of mivar ES.
Konstantin V. Nazarov,
Scientific Research Institute MIVAR, Moscow, Russia, [email protected]
Keywords: artificial intelligence; expert systems; expert system; systems based on knowledge; mivar networks; autonomous robots; mivar; self-driving cars; intellectual transport systems.
Oleg O. Varlamov,
Scientific Research Institute MIVAR, Moscow, Russia, [email protected]
Information about authors:
Nazarov Konstantin Vladimirovich, Graduate student, BMSTU, Research Officer, Scientific Research Institute MIVAR, Moscow, Russia.
Varlamov Oleg Olegovich, Dr. of Tech. Sci., Professor, BMSTU, Professor, MADI, Director, Scientific Research Institute MIVAR, Moscow, Russia.
Для цитирования:
Назаров К.В., Варламов О.О. Разработка методики создания верифицируемых моделей для миварных экспертных систем // T-Comm: Телекоммуникации и транспорт. 2017. Том 11. №4. С. 64-71.
For citation:
Nazarov K.V., Varlamov O.O. (2017). Development of the method to design verifiable models for mivar expert systems. T-Comm, vol. 11, no.4, рр. 64-71.
Modern Expert Systems
There is a trend in the field of artificial intelligenee research called "Systems, based on knowledge", which consists of software complexes, so-called "Expert systems" (ES). Currently ES term is used in different meanings, so it may be useful to remember that initially under the term "Experl system" we understood a software which could fill the role of human expert during the completion of some work in his field of competence [11. During such a work the ES uses a knowledge that was put in by the human expert. Term "knowledge" in certain subject domain is formalized and represented in the form of knowledge base, which can be changed through the process of development of ES [2], Expert systems are being actively developed and used for a wide spectrum of tasks such as a classification and analysis of data, advising, consulting and prediction. So, ES are oriented towards solution of the tasks usually requiring a work of human expert [2, 3],
It should be mentioned that ES were designed, first of all, to solve expertise tasks on the basis of deductive reasoning. "I leu-ristics", or rules, which w ere based on the intuition of the expert, could be put into the F,S. It was caused by the fact that heuristic approach is extremely useful on the occasions when incompleteness of knowledge or deficit of time excluded the possibility of conducting a full scientific analysis [4, 5],
It is important to point out the fact that there is a major difference between the ES and other software products is that ES utilize not only various types of data and knowledge (in the form of rules), but also special mechanisms of logical inference of solution. In the scientific literature you can find that knowledge is usually represented in the form which can be easily processed by the PC [ 1, 4]. So, ES have an algorithm of knowledge processing, not the algorithms for every specific task. Solution algorithm of the task is unknown and is constructed during the completion of the task by the "mechanism of logical inference'7 on the basis of rules and input data present in the ES.
There is an important term: "Expert system shell" that means expert systems with replaceable knowledge bases. In essence shells allow to create certain ES for different subjcct domains [6]. The key advantage of using such shells is that user don1! need to be engaged in programming, he just need to formalize and enter knowledge by the means of ES shell. The most popular are the following systems: CLIPS, Prolog, N experl Object, Exsys Corvid, Jess, «G2».
The ES researches are also conducting in Russia. For example, in paper [7] the analysis of decision support systems evolution and means for their developing was completed, the aspects of developing hybrid ES which are characterized by a set of applied methods and knowledge models: analytical, simulational statistical, productional, and also neural and semantic networks, fuzzy systems, genetic and other algorithms for the solution of various intellectual tasks united by a common goal were presented. The scientific research was conducted and decision support subsystem for complex technical object was developed 17]. Besides, neural networks, fuzzy clusterization and genetic algorithms |8) are used for ES development. Also hybrid ES based on determined probabilistic models were created [9].
The term "Mivar informational space" was introduced in Russia in 2002 which gave an opportunity to develop ES and knowledge-based systems [10] on the much higher scientific level by the use of logical inference with the linear computa-
tional complexity. Specifics of database development arc closely related with the model of data representation and rules in the certain ES. Knowledge bases for each type and "shell" of ES are different and non-interchangeable. Next we will describe specifics of development of mivar expert systems.
Mivar expert systems
Mivar approach is based on a new gnoseological representation of knowledge in a form of evolutionary multidimensional know ledge bases. This approach is based on gnoseological representation of data in a form of semantic graph "Objeet-Proper!y-Relation" (VSO) 110|. There are several approaches to data representation and processing nowadays, but it should be mentioned that mivar approach is becoming more popular. That is because mivar approach processes immense amount of data in a real time and allows to describe different subject domains most completely. Solutions are unique for each situation. They are constructed automatically without an expert being involved. It was found out in 2005 that intellectual system based on GRID, service-oriented architecture and mivar information space could be developed fl 1, 12], More than that it was proved in 2007 that on the basis of mivar approach theory of active reflection as a synthesis of artificial intelligence theory could be created [13J. Practical implementation of mivar ES in a form of software was described in 2013 [14]. Developing information systems and technologies cause need of existence of the new automated and intellectual software systems of ACS, PCS and DSS [15]. Mivar theory is wide and through its flexibility covers many objects of everyday life [16]. It should be mentioned that Russian Federation Patent for "Automated acquisition of a logical deduction path in a mivar knowledge base" was taken out in 2017 [ 17].
One of the main features of mivar theory is data representation in the fonn of semantic graph "Obj ect-Property-Relation" (VSO) [10-17]. To describe a data model in mivar information space it is necessary to identify three axes:
• The axis of relations «0»;
• The axis of attributes (properties) «S»;
• The axis of elements (objects) of subject domain «V».
The point with certain coordinates corresponds to each relation attribute value in multidimensional space. Relations connect elements of the space. The set of all multidimensional space points corresponds to the data model. The structure of the model in mivar approach is defined by space points that store corresponding relation attribute values.
Mivar networks can be represented in the form of a bipartite
graph consisting of objects-variables and rules-procedures. First, two lists are made which form two non-intersecting partitions of the graph: the list of objects and the list of rules. Objects are denoted by circles. Each rule in mivar network is extension of productions, hyper rules with multi-activators. It is proved that from the perspective of their further processing, all these formalisms are identical and in fact they are nodes of the bipartite graph which are denoted by rectangles [10,13].
There are following abstractions in the mivar information space:
Parameter is a single terminal object that has meaning on this abstraction level. The length of the side AB of the triangle ABC can be used as an example. Parameter is a leaf in the element hierarchy tree in the model. Parameter can be connected with only one internal node (class) [ 15].
Class is an internal node of the hierarchy tree. Class has no meaning and can contain other internal nodes (other classes) or/and leaves (parameters). The introduction of classes allows us to simplify description of the model containing several objects of the same type [15].
Relations arc a renewed element of mivar space. Relation describes interconnection between abstract variables. For example, "a=b-c" is an abstract subtraction formula. Relation stores its type, the list of input variables and the list of output variables, types of used variables and description.
Relations can be as follows:
1. Mathematical. A simple formula "a=b-c" can be an example;
2. Conditional. For example, "if y is IO,thenxis 14, else x is 7";
3. Programmable. Software code with its inputs and outputs can be used as an example.
4. String. For example, "loves", "connected".
5. System. "The part-the whole".
6. Location. For example, "over", "right", etc.
The rule contains the link to the relation and connects particular objects from the model. It is designed to simplify description of subject domains and repeated use of the same even complex programmable properties.
Rules contain at least:
1. The list of input variables;
2. The list of output variables;
3. Relation identifier.
Models developed in expert system designer Wi!Mi allow to find solution of any task in certain subject domain easily and quickly. It is enough just to enter the values of known parameters and choose parameters to find to get solution of a task. After that system will automatically construct solution algorithm, find intermediate parameter values if necessary and give out the result -the values of required parameters. The system works completely transparent: every step of calculations which is the result of certain rule triggering presents in the form of text logical inference in WilMi.
Application of mivar expert systems
Over the last 5 years in a formalism of mivar modeling a large number of various ES, beginning from "school" subject domains "Geometry" [18] and "Physics" [19], to medical ES of diagnosis on symptoms [20] was designed. Also an opportunity of developing active multisubject domain ES was shown [21].
Developing autonomous intellectual robots takes a special
place in mivar modelling [22]. For example, a group of Russian scientists from MADI developed algorithms for automatic car control, proved the choice of key technologies of functioning in-terobject interaction of intellectual vehicles systems [23], designed prototypes of autonomous wheeled vehicles as a part of intellectual vehicle system [24] and suggested methodology of developing control systems of the autonomous wheel vehicles movement integrated into the intellectual transport environment [23]. Other paper [25] describes the necessity of developing ES for automation of dispatching control of city passenger transport. In 2015-2016 it was shown that on the basis of mivar FS robot decision support system for calculation of any algorithms of functioning by rnulti-agency robotic system [26], for service robots [27] in ES shell "WilMi" [28] and also for developing autonomous intellectual robot control system [29] could be designed.
As generalization of opportunities of mivar approach manuals of mivar modelling [31 ] and designing mivar expert systems [30] were prepared. Mivar logical inference method is universal. It is suitable not only for describing easy subject domains, but also for the solution of complex, multilevel challenges [10-22,25-31].
However, developing models with large amount of objects turned out a heavy task due to complication of testing. Besides, the structure of model for various subject domains might be significantly different that results in unique scenarios of use for each model and these scenarios should be described. On the basis of these problems the decision to describe a universal method to design verifiable mivar models that will allow to simplify and accelerate process of development and subsequent support and completion has been made.
The proposed solutions on verification improvement of the mivar models
1. Specification of model description
A large number of the WilMi models has already been developed. But preparation of its use takes a lot of time because there are no correct and clear documentation. Experience of use mivar models has shown the need of the following comments and the accompanying documentation:
a) Model description in a form of text (or hypertext) document. This description has to include:
I. The name of the model and the author
II. Description of subject domain
III. The list of classes with description
IV. The list of input parameters. Input parameter means parameter which value is obligatory to set and (or) necessary to get a special particular solution. The name, type, description and range of possible values (if it is specified) of parameter has to be described, This list could be presented in a form of table (table 1).
Table 1
Example of the list of input parameters
Ks # Name Type Description
11 Probability of a mutation number Probability of gene mutation in a chromosome. The probability can't be less than 0 anil more than 1.
22 Quantity of chromosomes number Quantity of chromosomes in each population. Only integer allowed!
33 Maximum number of iterations number Stops work of an algorithm at the specified number of iterations. Only integer allowed!
V. The list of output parameters. Output parameter means parameter which value has to be found. Description of this list is similar to p. IV.
b) Scenarios of model use. The scenario has to include:
I. Description of a task. Example: Determine the efficiency of the tractor engine, which required 1,5 kg of fuel with the specific heat of combustion of 4.2-107 J/kg to perform the work of 1.89- /(f Joules.
II. The list of parameters which have to be set. It is recommended to make out in the form of a table or screenshot of WilMi shell with specified parameters. See example on Fig, 2.
.ett: Ttefnud phera™na l3
Object V Thermal phenomena Value Find
As Number □
All 189000000 □
Efficiency Number a
1 Number □
m 1,5 o
1 420000000 □
Q Number o
ft Number □
Qh Number □
Qm Number o
Qt Number □
11 Number □
ti Number a
V Number o
M Number o
\ Number o
p Number □
c Number □
Fig,2, Example of list of parameters tor ccrtain scenario of mode! use
III. A list of parameters that have to be found. This list is also recommended to make out in the form of a table or a screenshot of WilMi shell with parameters that have a tick in the "Find" checkbox.
IV. An example of the logical inference of a model. It can be made in the form of a table or a screenshot. It is sufficient to copy the information contained in the "Console" tab of the WiMi expert system shell after execution. Example for the same task:
SLep # 0 Relation: y=a* b
Rule description; Calculation of spent work (As = mq)
Input parameters:
m=1.5;
q=420000000;
Formula:
y=a*b
Results: As=630000G00;
Step # 1
Relation: y=IOO*a/b
Rule description: Calculation of efficiency (n=Au/As* 100)
Input parameters:
Au=l 89000000;
As=630000000;
Formula:
y=100*a/b
Results: Kfficiency=30;
c) Recommendation on how to name and comment objects of the model.
I, Classes. The class name should reflect the goal of the solution of a specific subtask or provide a semantic grouping of parameters (for example, it is possible to single out classes of initial and output parameters of the model or group them according to some other attribute). It is recommended to name the root class of the model similar to the name of the model. The description of each class have to be filled. It can, for example, briefly describe the feature of grouping parameters or other useful information, depending on the particular model.
II. Parameters. The name of the parameter should reflect its essence, it is not recommended to use short names ("P" instead of "Pressure of air", for example) for parameters that are not intermediate, that is, for the input and output parameters. The parameter description must be specified. In addition to the short characteristic, it is recommended to add the parameter type, as well as its possible range of values, to the description,
III. Relations. This type of object represents an abstract expression, depending on the logic of which, the source data is converted to the output. Depending on the type, the following names are recommended:
1. For simple relations (Formulas). The name can be this formula or its verbal form ("y = a + b" or "Sum of two numbers").
2. For conditional relations. The name should reflect the essence of the condition, for example, "if a> 0 then b else c" or "b if a is greater than 0, otherwise c", etc.
3. For restrictions. Similar to point 2. Example: "a > 0" or "Check for non-negativity".
4. For complex relations. A complex relation is usually a function, a subroutine that performs a certain transformation of parameters, so the name should reflect either the essence of the transformation, or the essence of the subroutine itself. For example, "Calculation ofthe root of a non-linear equation" and etc.
It is also necessary to fill the description ofthe relation. It should describe the abstract input and output parameters, their type and range of values, especially for complex relations. Complex relations also require a description of the used algorithm.
IV. Rules. This type of object links the abstract relation to the specific parameters ofthe model. The name should reflect the essence of the rule, for example, "Calculation of resistance according to the Joule-Lenz law" or "Determining the child's blood type according to the parents", etc. The description, unlike the relations, should describe the specific parameters involved in the rule, and the essence of their transformation.
V. Restrictions. Their name and description is similar to the description ofthe constraint type relation, taking into account that not an abstract but a certain parameter should be described. Example name: "Fuel tank is empty" (if the restriction uses the relation "a> 0", etc.).
VI. Additional information stored in .xml model files. It is possible to fill additional information of the model inside "Model" tag in .xinl model file. Here is an example of standard information stored in "Moder tag:
< mode I id= "{a43 fc8cf- OSOd-449a-ajfe-5 (1966185ada 6¡ " shortName= "Genetics" formatXmlVersion="2¿Q" description^'Model for determining a child based on the genetic data of the parents ">...<...></...>
It is recommended to specify a description of the model
COMPUTER SCIENCE
within the "description" attribute, as well as a name inside the "shortName" attribute. Also, in the ,xml model file, you can store any useful data. It's enough to describe additional attributes inside the tag. In the following example, information about the mode! version and the author is added inside the "Mode!" tag:
<model id-"{a43fc8cf-0S0d-449a-affe-5d966185ada6l" shortName-" Genetics "formatXml Version-"2,0" description-" Model for determining a child based on the genetic data of the parents " vershn= "1.1"author="O.Shinygma">...<...></...>
2. Decomposition of the model using classes
One of the most common problems in developing models of WilMi expert systems with a large number of parameters is the complexity of debugging. This is especially true when all the parameters are in the same class or the parameters of different classes are used in one rule. There is a tool of testing the mode! integrated in expert system shell. When using it, you need to fill values of all the key parameters of the model, but you can only test the final result. However, the shell also has a built-in test too! for a particular class, which tests the intermediate result of a solution for a particular class. Thus, with the correct decomposition of the mode! into subclasses, we get the possibility of testing the individual components of the system apart from the entire model, which simplifies the process. However, in order to create such models, the following conditions must be observed:
a) Rules, linked with parameters of a certain class have to use the minimum quantity of parameters from other classes (if it is not constant values), at best no more than one. and this parameter containing value of the solution of a certain subtask of a certain class has to be the resultant output parameter of that other class.
b) Rules, linked with the parameters of a certain class must contain a minimum number of output parameters, at best no more than one, according to the logic indicated in point "a". An exception can only be made by a class that collects the output parameters of all other classes that perform intermediate tasks, aggregating the whole solution.
c) Rules should use parameters that belong to only one class, except for the cases described in points "a" and "b".
When these conditions are fulfilled, the model of the following structure is obtained (Fig, 3), in which each part of the model could be debugged independently. This structure is optimal for designing models with a large number of objects, and is also easier to scale.
3. Graph visualization of mivar model
Mivar networks can be represented as a bipartite graph consisting of objects-parameters and rules-procedures. This graph is a powerful debugging tool, because with its help you can visually track all the connections between the objects of the model. If there is an error in the model and there are incorrect connections between the parameters and rules, using the graph will greatly reduce the time it lakes to find and fix the problem. That's why the WilMi expert system shell contains a built-in graph model building tool. To display the graph, you need to test the model, and then launch the tool from the context menu of the program.
However, if the connection between the source and target parameters is broken, the tool will not be able to build the graph, since the task of finding the value of the target parameter will not be completed. To solve this problem, an application was developed that builds the complete graph of the model, and not just the solution graph. A complete graph will allow you to find the missing link, as well as convenient for graphical representation of the model. The created tool supports several algorithms for arranging the graph, scaling, and saving the entire graph as an image. When you hover over any node in the graph, the associated nodes are highlighted. Violet color indicates the rules, green — prameters, the class is indicated by a red label above the parameter. The red arrows indicate the input parameters; the blue ones indicate the output. Below in Figure 5, an enlarged portion ofthe model graph consisting of more than a hundred objects is shown.
Conclusions
Expert systems (ES) have been developing for more than 30 years and are focused on solving such problems, which usually require the human expertise. Expert systems are distinguished by models of data representation and algorithms for processing knowledge. Solution algorithm of the certain problem is unknown and is constructed during the process of solution with the help of mechanism of logical inference on the basis ofthe information accessible to the ES including various rules and input data. Before the 2002 logical inference was considered as a NP-complete task with a factorial computational complexity, but then the mivar approach, which integrated multi-dimensional evolutionary databases and rules in gnoseological formalism "Object-Property-Relation" and mivar networks in prodnctional formalism "Object-Rule" with linear computational complexity of logical inference was introduced. On a new scientific level, mivar expert systems, capable of processing more than 5 billion rules in second even using personal computers, were created.
Developing of knowledge bases of large subject areas and mull ¡subject ES containing more than 3 thousand production rules is an actual task at present. The features of developing knowledge bases are closely related to the model of data representation and ruies in a certain ES. The knowledge bases for each type or "shell" of the ES are different and are not interchangeable. As mivar approach is based on gnoseological representation of the data, so new method of knowledge base construction is used. WilMi expert system designer, which uses mivar approach, is a good tool which has been thoroughly tested through 5 years of service.
Mivar logical inference method is universal. It is suitable not only for describing easy subject domains, but also for the solution of complex, multilevel challenges. For example, this paper shows, that mivar ES could be used as decision support system for autonomous robots, self-driven cars and complex robotic multilevel systems. Such ES have to be reliable and verifiable in different situations.
However, the development of models for the mivar expert systems revealed the problem: an expert is not able to quickly create and lest a mivar model containing more than 300 productional rules, in addition, multisubject ES are designed by several experts, which creates additional problems in the verification of ES. The development of the mivar models, consisting of more objects, turned out to be difficult in practice due to the complication of the verification process. In addition, because of the existence of different subject areas, the structure of the model may differ significantly, which leads to the need to describe unique scenarios for the use of models.
Thus, the development of the method to design verifiable model for mivar expert systems is an actual, complex scientific and practically meaningful task. This method will simplify, scale and accelerate the process of development, verification and support mivar knowledge models. Method to design verifiable model for mivar expert systems includes 3 sections: 1) Specification of model description, which contain ihe description itself in the form of text document; usage scenarios of the model; recommendation on how to name and comment objects of the model; 2) Decomposition of ihe model using classes; 3) Graph visualization of certain subject domain mivar model.
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РАЗРАБОТКА МЕТОДИКИ СОЗДАНИЯ ВЕРИФИЦИРУЕМЫХ МОДЕЛЕЙ ДЛЯ МИВАРНЫХ ЭКСПЕРТНЫХ СИСТЕМ
Назаров Константин Владимирович, МГТУ им. Н.Э. Баумана, НИИ МИВАР, Москва, Россия, [email protected]
Варламов Олег Олегович, МГТУ им. Н.Э. Баумана, МАДИ, Научно-исследовательский институт (НИИ) МИВАР,
Москва, Россия, [email protected]
Дннотация
Проанализировано современное состояние в области экспертных систем (ЭС), которые различаются моделями представления и алгоритмами обработки знаний. Алгоритм решения конкретной задачи заранее неизвестен и строится по ходу решения задачи механизмом логического вывода на основании имеющихся в ЭС правил и поступивших входных данных. До 2002 года логический вывод считался NP-полной задачей с факториальной вычислительной сложностью, а потом был предложен ми-варный подход, объединивший многомерные эволюционные базы данных и правил в гносеологическом формализме "Вещь-Свойство-Отношение" и миварные сети в продукционном формализме "Объект-Правило" с линейной вычислительной сложностью логического вывода. Созданы ЭС, в которых применяется миварный логический вывод, обрабатывающий более 5 млн правил в сек. на персональных компьютерах. Обосновано, что создание больших баз знаний является актуальной задачей. Базы знаний для каждого типа или "оболочки" ЭС являются различными и взаимно не заменяются. Миварный подход основан на гносеологическом представлении данных, поэтому используется новая методика создания баз знаний. Конструктор экспертных систем Wi!Mi, использующий миварный подход, является хорошим инструментом и за последние 5 лет накоплен достаточно большой опыт создания миварных ЭС. В работе показано, что миварные ЭС могут использоваться в качестве систем принятия решений для автономных интеллектуальных роботов, беспилотных автомобилей и сложных робототехнических многоуровневых систем. Такие ЭС должны быть надежными и верифицируемыми в различных ситуациях. Однако, человек-эксперт не в состоянии быстро создать и протестировать миварную модель, содержащую более 300 продукционных правил. Мультипредметные ЭС создаются несколькими экспертами, что создает дополнительные проблемы. Кроме того, для различных предметных областей, структура модели может существенно отличаться, что приводит к необходимости описания уникальных сценариев использования моделей. Методика создания верифицируемых моделей для миварных ЭС включает 3 основных раздела: 1) Стандарт комментирования модели, содержащий описание модели в виде текстового документа; сценарии использования модели; рекомендации к названиям и комментированию объектов модели; 2) Декомпозиция модели с помощью классов; 3) Визуализация графов миварной модели предметной области. Таким образом, разработка методики создания верифицируемых моделей для ми-варных экспертных систем является актуальной, сложной научной и практически значимой задачей. Эта методика позволит упростить, масштабировать и ускорить процесс разработки и последующей верификации, поддержки и развития миварных ЭС.
Ключевые слова: искусственный интеллект; экспертные системы; системы, основанные на знаниях; миварные сети; автономные роботы; мивар; беспилотные автомобили; интеллектуальные транспортные системы.
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Информация об авторах:
Назаров Константин Владимирович, магистр МГТУ им. Н.Э. Баумана, научный сотрудник, НИИ МИВАР, Москва, Россия. Варламов Олег Олегович, д.т.н., профессор, МГТУ им. Н.Э. Баумана, профессор, МАДИ, директор, Научно-исследовательский институт (НИИ) МИВАР, Москва, Россия.
