Fuzzy logic

Perhaps the most striking feature of human intelligence is the ability to make the right decisions in an environment of incomplete and fuzzy information. Building models of human approximate reasoning and using them in computer systems of future generations is today one of the most important problems of science.

Significant progress in this direction was made 30 years ago by a professor at the University of California (Berkeley) Lotfi A. Zadeh. His work "Fuzzy Sets", which appeared in 1965 in the journal Information and Control, No. 8, laid the foundations for the simulation of human intellectual activity and was the initial impetus to the development of a new mathematical theory.

What did Zade offer? First, he expanded the classical Cantor concept of a set, assuming that the characteristic function (the function of belonging to an element to a set) can take any values ​​in the interval (0; 1), and not just the values ​​0 or 1. He called these sets fuzzy (fuzzy ). L.Zadeh also defined a number of operations on fuzzy sets and proposed a generalization of the known methods of logical inference modus ponens and modus tollens.

Having then introduced the concept of a linguistic variable and assuming that fuzzy sets act as its values ​​(terms), L. Zadeh created an apparatus for describing the processes of intellectual activity, including fuzziness and uncertainty of expressions.

Further work of Professor L. Zadeh and his followers laid a solid foundation for the new theory and created the prerequisites for the introduction of fuzzy control methods in engineering practice.

As early as 1990, over 10,000 papers were published on this subject, and the number of researchers reached 10,000, with 200-300 people in the USA, Europe and the USSR, about 1,000 in Japan, 2,000-3,000 in India and about 5,000 researchers in China.

In the last 5-7 years, the use of new methods and models in industry has begun. And although the first applications of fuzzy control systems took place in Europe, such systems are most intensively implemented in Japan. The range of applications is wide: from managing the process of sending and stopping a metro train, managing freight elevators and a blast furnace to washing machines, vacuum cleaners and microwave ovens. At the same time, fuzzy systems can improve product quality while reducing resource and energy costs and provide a higher resistance to the effects of interfering factors compared to traditional automatic control systems.

In other words, new approaches allow us to expand the scope of application of automation systems beyond the limits of applicability of the classical theory. In this regard, L. Zadeh’s point of view is curious: “I believe that an excessive desire for accuracy has had an effect that nullifies control theory and system theory, since it leads to the fact that research in this area focuses on those and only problems that can be precisely solved. As a result, many classes of important problems, in which the data, goals and constraints are too complex or poorly defined to allow accurate mathematical analysis, have remained and remain aloof for the reason they are not amenable to mathematical interpretation. In order to say something essential for problems of this kind, we must abandon our accuracy requirements and admit results that are somewhat vague or uncertain. "

The shift of the research center of fuzzy systems towards practical applications has led to the formulation of a number of problems such as new computer architectures for fuzzy computing, the element base of fuzzy computers and controllers, development tools, engineering methods for calculating and developing fuzzy control systems and much more.

The main goal of the textbook offered to the readers is to attract the attention of students, graduate students and young researchers to fuzzy problems and to give an accessible introduction to one of the most interesting areas of modern science.

The mathematical theory of fuzzy sets, proposed by L. Zade more than a quarter of a century ago, allows us to describe fuzzy concepts and knowledge, operate on this knowledge and make fuzzy conclusions. Based on this theory, methods for constructing computer fuzzy systems significantly expand the field of application of computers. Recently, fuzzy control is one of the most active and effective areas of research in the application of the theory of fuzzy sets. Fuzzy control is especially useful when technological processes are too complex to analyze using generally accepted quantitative methods, or when available sources of information are interpreted qualitatively, inaccurately or uncertainly. It is experimentally shown that fuzzy control gives better results than those obtained with conventional control algorithms.