1. Introduction

1.1 Essence of fuzzy set theory

Information accepted by methods based on conventional mathematics must be precise, for example, the speed of a car v = 111 (kmjh). Such information can be represented graphically by means of the so-called singleton, Fig. 1.1.

Exact information can only be delivered by precision engineered mea­suring devices, whereas a man can directly estimate the speed of a car by applying such terms as low, medium and high. These imprecise evaluations can also be represented graphically, Fig. 1.2.

The functions "low", "medium" and "high", called membership func­tions, tell us if the given precise speed value is respectively low, medium or high. A man observing a car running at a speed of v = 111 (kmjh) cannot evaluate its speed exactly, but he can estimate it roughly as a high speed, Fig. 1.2.

Such information can be defined as a granule of information (Zadeh 1979,1996). If 3 granules (low, medium, high) are not sufficient, one can in­crease the precision of evaluation applying, for example, 5 granules (very low, low, medium, high, very high), Fig. 1.3. A man can also decrease the precision of evaluation of the speed using only 2 granules (low, high). The granularity of information applied by a man is variable according to his requirements, mental powers or is otherwise dependent upon the context of information usage.

Information obtained from people is usually of less precision (large granu­larity), while information delivered by measuring devices is of higher precision

o 111 160 v (kmlh)

Fig. 1.1. Visualization of precise speed measurement

A. Piegat, Fuzzy Modeling and Control© Springer-Verlag Berlin Heidelberg 2001

2 1. Introduction

J.L(v) low medium high

O~-------+--r---~----;--+----------;---------~

o 50 60 80 100 111 160 v (kmIh)

Fig. 1.2. Visualization of imprecise, rough speed evaluation

J.L(v) very low low medium high veryhigh

O~~--~--~--L-~~~--~--------L-------~

o 20 40 60 70 80 90 100 120 160 v (kmIh)

Fig. 1.3. Evaluation of speed using 5 granules of information

(small granularity). The granularity of information is defined by the width of a granule (membership function). And so, the granule "medium" can have various widths, according to the total number of granules of information used by a man, Fig. 1.4.

As you can see in Fig. 1.4, by decreasing the granularity of information we approach a limit: a granule of an infinitely small width called the single­ton, which represents the precise information, Le. such information which is employed by conventional mathematics.

The information represented by the granule of a finite (greater than zero) width has been called by Prof. Lofti Zadeh, the discoverer and creator of the concept of granularity, fuzzy information. The mathematics field using such information has been called fuzzy set theory (Zimmerman 1994). The most important element of this theory is fuzzy logic, applied for fuzzy modeling and control. For science and technology fuzzy set theory has opened new exploratory possibilities, which are described below.

1. The possibility of creating artificial intelligence similar to human intelli­gence and providing automatons and robots with it. Today, the process of creating such intelligence is ongoing and ever increasing in bearing significant results which attest that artificial intelligence can be more ef-

1.1 Essence of fuzzy set theory 3

J.l(v) J.l(v) medium medium

o 50 75 100 v(krrVh) 60 75 90 v (krrVh)

J.l(v) J.l(v) medium medium

o ~----~ ______ ~~ 70 75 80 v (krrVh) 75 v (krrVh)

Fig. 1.4. Various width of the granule of information of the "medium" speed

fective than human intelligence in some weH defined applications, e.g. in respect of quantity and speed of information processing.

2. The creation of computers programmed with words (Zadeh 1996). The application of such computers in robots and automatons makes it possi­ble to control them and to "communicate with them" by means of human language using fuzzy notions. There presently exist devices for recogniz­ing a limited number of words or word associations.

3. The application of information of any granularity for modeling, control, optimization and diagnostics of systems and objects. The use of greater granularity allows for reducing processed and stored information and for accelerating the operation of algorithms.

4. The possibility of adapting granularity of information according to the required accuracy of modeling, control, optimization, diagnostics, etc. Such adaptation is applied by man. The illustration of this statement is shown in Figs. 1.5 - 1. 7.

Assurne for the moment that someone controls aplant by realizing the input/output mapping shown in Fig. 1.5. At the outset they will remember the extreme states of the plant and generate in their own mind a model based on two rules given in Fig. 1.6. The model, for argument's sake, is a necessary approximation of the plant.

If the model represented in Fig. 1.6 is insufficiently exact, a man will try to increase its accuracy, bearing in mind the essential (Babuska 1995b), Fig. 1.7,

4 1. Introduction

y

7

3 -, , , ,

5 x

Fig. 1.5. Input/output characteristic of the plant to be controlled

J1(y)

y y

L .....-___ -+ ________________ 7

s 4--___ -+ ________________ )

J1(X)

S L

Rl : IF (x small) THEN (y small) R2: IF (x large) THEN (y large)

5 x

x

Fig. 1.6. Model of the plant based on two granules of information: small and large

medium state, thereby creating a new rule determining the operation of the plant and progressively introducing new, smaller granules of information. Moreover, if the model represented in Fig. 1.7 proves insufficient, a man can examine the next essential state of the plant, decrease the granularity of information, increase the number of verbal rules characterizing the operation of the plant and subsequently obtain a higher accuracy of modeling.

As psychological studies (Kruse 1994) have shown, an average capable man can remember only 5 to 9 characteristic states of a plant. Therefore, for

1.1 Essence of fuzzy set theory 5

J1.(y)

y y

7 L ------------------

M ---------------- ---------------~~~

s '----------------t----------------J

J1.(x)

s

:3 5 I I I I

i I I I I I :M L

Rl : IF (x smalI) THEN (y smalI) Rl : IF (x medium) THEN (y medium) R3 : IF (x large) THEN (y large)

x

x

Fig. 1.7. Model of the plant based on three granules of information: smalI, medium, large

each variable, maximally 5 to 9 granules of information are applied. We note that in general, this kind of granularity is totally sufficient for controlling aircraft, vehicles and many other different objects and for solving thousands of everyday problems.

Since computer technology makes it possible to apply any granularity of information practically, we can get models of considerably higher accuracy. Retrospectively, experiments with modeling real world systems show that there are nearly always thresholds to limits of accuracy and exceeding such thresholds should not be rendered as costly objectives. Such situations arise when certain effects occur in complicated systems. They are described as follows.

1. Existence of chaos Within the kernel of systems there are active disturbances which cannot be measured by us, or we don't know that they exist. Also, certain unexam­ined pro ces ses can occur in these systems. Their influences are dependent on magnitude and can render variable unforseeability of the system that can be attributed to chaotic events.

6 1. Introduction

2. Explosion of the number of possible solutions In a complicated system the number of reasons which could cause the ob­served operation of the system, increases abruptly with the level of its com­plexity. This effect is called "combinatorial explosion of solutions" and usually it is not possible to enclose it in the mathematical model. To create a model of such a system, only the most essential causal factors should be detected and involved in the model. This decreases its complexity. On rejection, however, less apparent essential factors could cause an error (indeterminacy margin) of the model.

3. Impossibility of precise measurement of some signals in a system This fact means that even though the model is very accurate, it can cal­culate an inaccurate output (information) not corresponding to the known behaviour of the real system if the inputs of the real system were measured inaccurately.

In recognition of the existence of the above described effects the progen­itor of fuzzy logie, Prof. L. Zadeh, has enunciated the statement called the principle of incompatibillity (Zadeh 1973): "as the complexity of a system increases, our ability to make precise and yet significant statements about its behaviour diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics".

Exact modeling using a very small granule of information is possible in the case of simple systems with a small number of inputs. In non-trivial systems, especially those having a greater number of inputs, we are enforced to apply information of a larger granule (fuzzy information).

1.2 Development of fuzzy set theory

Today fuzzy set theory attracts a high level of interest. In 1993 the number of related publications had been estimated at 15,000 to 16,000 (Altrock 1993). Now, in 2000, this number exceeds 27,000 and is increasing rapidly. As sci­entific conferences are organized there paralleis an increase in the number of industrial applications. What causes are endemie to the great popularity of fuzzy set theory in the world of present-day science?

The development of fuzzy set theory was initiated by the American profes­sor, Lofti Zadeh, who discovered the existence of fuzzy sets and in his seminal paper "Fuzzy Sets" (Zadeh 1965), presented the idea and first conception of the theory which has created possibilities for the fuzzy description of real systems. The main division of fuzzy set theory is fuzzy logic (Zimmermann 1994a) applied for prototype modeling and mainstream control of systems.

In the sixties the period of rapid development of computers and digital technology based on binary logie began. Generally, there was optimism that this logie would empower us to solve most technieal and scientific problems. For that reason the rise of fuzzy logie went by almost unnotieed despite the fact that its conception can be acknowledged as a turning point. Nevertheless,

1.2 Development of fuzzy set theory 7

some scientific circles understood its importance, developed it and brought it to fruition within the framework of industrial application. Some time later, interest in that school of thought on the part of binary technology advocates increased, because it has become evident that traditional models and mathe­matieal methods were unable to solve many practieal problems in spite of an enormous rate of calculation. A new methodology was needed whose requisite features are to be found in fuzzy logic.

As with roboties, fuzzy logic has met with greater interest outside its country of origin, the USA. And so, the first acknowledged application is the use of it in Europe, in the control of the steam generating system in apower plant, as realized by Assilian (Assilian 1974). The steam generator has been proved such a complicated non-linear system, that the application of various traditional, often very sophisticated methods have not given a solution for the problem of its contro!. Only fuzzy logic has allowed the synthesis of a controller satisfying all requirements. In 1976 it was applied in the automatie control system of a rotary furnace for cement production (Mamdani 1977). Even so, the first European and American applications did not result in any major change of interest in fuzzy logic. Similarly as in the case of roboties, the change has occurred in Japan, the first nation to become fuHy aware of the enormous potential of fuzzy logic and to introduce it widely (BeHon 1992).

The best known Japanese application of fuzzy logie was in the control system of the Sendai underground railway, utilized by the Hitachi company. The project was undertaken with the cooperation of an experienced driver whose knowledge provided parameters for its design. This system automati­cally decreased the speed of a train on entering astation, ensuring that the train stopped at a predetermined place. It also had the benefit of being a highly comfortable ride through mild acceleration and braking (Abel 1991). There were many other advantages in comparison with traditional control methods.

Tests and improvements of the system were carried out for two years, and their purpose was to check the new control method and to secure the greatest margin of safety for passengers. The success of the Sendai under­ground railway was such that 12 months later, 50 large Japanese companies worked on the development of their own fuzzy logie applications. In 1991 the contribution of Japan to the world production of articles based on this logic was estimated at billions of dollars. In absolute numbers this amounted to 80 per cent, according to the data of Market Intelligence Research. Start­ing from 1989, at least 5 scientific societies engaged in fuzzy logic have been established in Japan. They are:

1. Laboratory for International Fuzzy Engineering Research (LIFE), 2. Japan Society of Fuzzy Theory and Systems (SOFT), 3. Biomedical Fuzzy Systems Association (BMFSA), 4. Fuzzy Logic Systems Institute Iizuka (FLSI), 5. Center for Promotion of Fuzzy Logic.

8 1. Introduction

The Japanese division of the international organization IFSA (Interna­tional Fuzzy Systems Organization) has been carrying out its activity in Japan since 1986. Of these institutions, the most well known organization is LIFE.

The institute has been organized by the J apanese Ministry of Interna­tional Trade and Industry and large industrial enterprises (49 in 1991), such as Honda, Kawasaki Steel, Tokyo Electric and others. Their purpose was to work out fuzzy methods for the needs of industry, trade, and all purpose decision making (for example, currency operations) etc. The best specialists in the field of fuzzy logic from Japanese universities and enterprises have been engaged at LIFE. In addition, large companies from outside Japan, for example Bosh, Zeiss, Siemens, Audi and Volkswagen have begun to spon­sor the institute. The sponsors of LIFE send their engineers to the institute where they participate in training programs and carry out research under the guidance of specialists .

. The fast development offuzzy logie in Japan has resulted in practieal ap­plications, not only in industry but also in the manufacture of commonly used products. The Camcorder (video camera), for instance, was provided with a fuzzy pieture stabilizer (Abel 1991). This was implemented in order to com­pensate for undesirable movements due to unskilled operator handling. The problem was difficult to solve with traditional methods because very often mobile objects to be filmed, such as people in motion, must be distinguished from induced disturbances. Another case is that of the automatie washing machine, controlled with only one push-button action (Zimmerman 1994). It's holistic nature aroused interest and met with approbation. Fuzzy con­trol enabled the reduction of the number of push-buttons to one, resulting in considerable simplification of its operation and also in optimization of the washing process via recognition of type, quantity and grade of soiled clothing. Japanese companies have applied their fuzzy ingenuity to many other prod­ucts such as the mierowave cooker (Sanyo), anti-block system and automatie gear (Nissan ), integrated control of running dynamies of a car (INVEC) and hard disk controllers in computers (reducing access time to the information to be searched).

J apanese engineers established an enormous number of patents in this field using their cutting edge position in research and applications of fuzzy logic. One company alone, Omron (Kyoto), was the owner of more than 700 patents in 1993.

Mass application of fuzzy logic in Japanese products became subject to worldwide scrutiny, especially in Europe, where mainly German industrial­ists and scientists resolved to oppose Japanese competitive dominance. The European foundation known as ELITE (European Laboratory for Intelligent Techniques Engineering Foundation) is based in Aachen, wherein develop­ment and promotion of artificial intelligence methods such as fuzzy logie and neural networks proceed with an emphasis on development of knowl-

1.2 Development of fuzzy set theory 9

edge and research. The foundation organizes many international conferences, among them the yearly European conference EUFIT (European Congress on Intelligent Techniques and Soft Computing) which is dedicated to artificial intelligence.

Apart from the above-mentioned applications, the early nineties saw in­tensive fuzzy development in many misceHaneous secondary fields as weH as non- technical ones. In the foHowing list, a selection of applications should enable you to evaluate some of the possibilities offered by fuzzy logic. They are:

• artificial pacemaker control system (Akaiwa 1990; Kitamura 1991; Sugiura 1991),

• automotive vehicle control system (Altrock 1992), • water boiler (Bien 1992), • chemical reactors and devices (Altrack 1995; Bork 1993; Hanakuma 1989;

Häck 1997; Höhmann 1993; Kolios 1994; Roffeld 1991), .cooling systems (Becker 1994; Hakata 1990), • conditioning and ventilation equipment (Tobi 1991; Watanabe 1990), • refuse incinerating equipment (Altrock 1993; Fujiyoshi 1992; Ohnishi 1991), • glass smelting furnace (Aoki 1990; Hishida 1992), • blood pressure contral system (Arita 1990), • tumor diagnostics devices (Arita 1991), • heart diseases warning system (Altrock 1993), • crane or overhead crane control system (Altrock 1993; Watanabe 1991), • pumping station (Chen 1992), • picture processing (Fijiwara 1991; Franke 1994), • fast battery charger (Altrock 1993), • word recognition (Fujimoto 1989), • diabetic therapy and blood glucose level contral (Jacoby 1994; Kageyama

1990), • power system (Hiyama 1991), • metalworking equipment (Hsieh 1994), • bioprocessor control (Hanss 1994), • heating devices (Heider 1994), • electric motor control (Kawai 1990; Lee 1992), • welding equipment and pracesses (Murakami 1989; ReshufHed 1994), • trafiic control (trafiic systems) (Sasaki 1988; Voit 1994), • biomedicine (Takahashi 1990), • raom cleaning equipment (Yamashita 1992), • desludging equipment (Yu 1990), • water cleaning equipment (Altrock 1995).

Many books on the subject of fuzzy sets theory, e.g. (Altrock 1993,1995; Brown 1994; Bezdek 1981; Driankov 1993,1996; Gottwald 1993; Hung 1995; Kahlert 1994,1995; Knappe 1994; Kandel 1994; Kruse 1994; Kiendl 1997;

10 1. Introduction

Kaufmann 1985; Koch 1996; Kacprzyk 1986,1992,1997; Nguyen 1995; Pedrycz 1993; Rutkowska 1997; Tilli 1991; Wang 1994a; Yager 1994,1995; Zimmer­mann 1994a,1994b), have already been published.

Several ready-for-use programs facilitating fuzzy modeling and control have been produced and are in distribution. Information about them can be found in (Ader 1996; Baldwin 1995a; Koch 1996; Kuhn 1994; Krieger 1994; Krone 1996c).

In Poland, initial research into fuzzy sets was carried out in the sev­enties (Kacprzyk 1977,1978). Amongst Polish scientists, Prof. E. Czogala, Prof. J. Kacprzyk and Prof. W. Pedrycz (listed in strict alphabeticalorder of surname) made significant contributions to worldwide development of the theory.

Although fuzzy set theory allows for the solving of problems which con­ventional methods could not cope with, it should not be considered as the only solution to the exclusion of all others. One would be mistaken were they to believe it could replace all known methods. Experience and practice have shown that fuzzy logic shall mainly be applied where hitherto known methods fail (Altrock 1993). If conventional methods yield good results they should be pursued.