Chapter 1
Introduction

1.1 Early History of Fuzzy Control

Fuzzy control (also known as fuzzy logic control) is regarded as the most widely used application of fuzzy logic and is credited with being a well-accepted methodology for designing controllers that are able to deliver satisfactory performance in the face of uncertainty and imprecision (Lee, 1990; Sugeno, 1985); Feng, 2006). In addition, fuzzy logic theory provides a method for less skilled personnel to develop practical control algorithms in a user-friendly way that is close to human thinking and perception, and to do this in a short amount of time. Fuzzy logic controllers (FLCs) can sometimes outperform traditional control systems [like proportional–integral–derivative (PID) controllers] and have often performed either similarly or even better than human operators. This is partially because most FLCs are nonlinear controllers that are capable of controlling real-world systems (the vast majority of such systems are nonlinear) better than a linear controller can, and with minimal to no knowledge about the mathematical model of the plant or process being controlled.

Fuzzy logic controllers have been applied with great success to many real-world applications. The first FLC was developed by Mamdani and Assilian (1975), in the United Kingdom, for controlling a steam generator in a laboratory setting. In 1976, Blue Circle Cement and SIRA in Denmark developed a cement kiln controller (the first industrial application of fuzzy logic), which went into operation in 1982 (Holmblad and Ostergaard, 1982). In the 1980s, several important industrial applications of fuzzy logic control were launched successfully in Japan, including a water treatment system developed by Fuji Electric. In 1987, Hitachi put a fuzzy logic based automatic train operation control system into the Sendai city's subway system (Yasunobu and Miyamoto, 1985). These and other applications of FLCs motivated many Japanese engineers to investigate a wide range of novel applications for fuzzy logic. This led to a “fuzzy boom” in Japan, a result of close collaboration and technology transfer between universities and industry.

According to Yen and Langari (1999), in 1988, a large-scale national research initiative was established by the Japanese Ministry of International Trade and Industry (MITI). The initiative established by MITI was a consortium called the Laboratory for International Fuzzy Engineering Research (LIFE). In late January 1990, Matsushita Electric Industrial (Panasonic) named their newly developed fuzzy-controlled automatic washing machine the fuzzy washing machine and launched a major commercial campaign of it as a fuzzy product. This campaign turned out to be a successful marketing effort not only for the product but also for fuzzy logic technology (Yen and Langari, 1999). Many other home electronics companies followed Panasonic's approach and introduced fuzzy vacuum cleaners, fuzzy rice cookers, fuzzy refrigerators, fuzzy camcorders (for stabilizing the image under hand jittering), fuzzy camera (for smart autofocus), and other applications. As a result, consumers in Japan recognized the now en-vogue Japanese word “fuzzy,” which won the gold prize for a new word in 1990 (Hirota, 1995). Originating in Japan, the “fuzzy boom” triggered a broad and serious interest in this technology in Korea, Europe, the United States, and elsewhere. For example, Boeing, NASA, United Technologies, and other aerospace companies developed FLCs for space and aviation applications (Munakata and Jani, 1994).

Today FLCs are used in countless real-world applications that touch the lives of people all over the world, including white goods (e.g., washing machines, refrigerators, microwaves, rice cookers, televisions, etc.), digital video cameras, cars, elevators (lifts), heavy industries (e.g., cement, petroleum, steel), and the like.

While this book focuses on type-2 fuzzy logic control, it will also provide background material about type-1 fuzzy logic control. Indeed, before we can explain what type-2 fuzzy logic control is we must briefly explain what type-1 fuzzy sets, type-1 fuzzy logic control, and type-2 fuzzy sets are. In this chapter we do this from a high-level perspective without touching on the mathematical aspects in order to give a feel for the nature of fuzzy sets and their applications. Later chapters in this book provide rigorous treatments of mathematical underpinnings of the subjects just mentioned.

1.2 What Is a Type-1 Fuzzy Set?

Suppose that a group of people is asked about the temperature values they associate with the linguistic concepts Hot and Cold. If crisp sets are employed, as shown in Fig. 1.1a, then a threshold must be chosen above which temperature values are considered Hot and below which they are considered Cold. Reaching a consensus about such a threshold is difficult, and even if an agreement can be reached—for example, 18°C—, is it reasonable to conclude that 17.99999°C is Cold whereas 18.00001°C is Hot?

Figure 1.1 Representing Cold and Hot using (a) crisp sets, and (b) type-1 fuzzy sets.

On the other hand, Hot and Cold can be represented as type-1 fuzzy sets (T1 FSs) whose membership functions (MFs) are shown in Fig. 1.1b. Note that, prior to the appearance of type-2 fuzzy sets, the phrase fuzzy set was used instead of the phrase T1 fuzzy set. Even today, in many publications that focus only on T1 FSs, such sets are called fuzzy sets. In this book we shall use the phrase type-1 fuzzy set. Returning to Fig. 1.1b, observe that no sharp boundaries exist between the two sets and that each value on the horizontal axis may simultaneously belong to more than one T1 FS but with different degrees of membership. For example, 26°C, which is in the crisp Hot set with a membership value of 1.0 (Fig. 1.1a), is now in that set to degree 0.8, but is also in the Cold set to degree 0.2 (Fig. 1.1b).

Type-1 FSs provide a means for calculating intermediate values between the crisp values associated with being absolutely true (1) or absolutely false (0). Those values range between 0 and 1 (and can include them); thus, it can be said that a fuzzy set allows the calculation of shades of gray between white and black (or true and false). As will be seen in this book, the smooth transition that occurs between T1 FSs gives a good decision response for a type-1 fuzzy logic control system in the face of noise and other uncertainties.

1.3 What Is a Type-1 Fuzzy Logic Controller?

With the advent of type-2 fuzzy sets and type-2 fuzzy logic control, it has become necessary to distinguish between type-2 fuzzy logic control and all earlier fuzzy logic control that uses type-1 fuzzy sets (the distinctions between such fuzzy sets are explained in Section 1.4). We refer to fuzzy logic control that uses type-1 fuzzy sets as type-1 fuzzy logic control. When it does not matter whether the fuzzy sets are type-1 or type-2, we just use fuzzy logic control or fuzzy control.

Fuzzy logic control aims to mimic the process followed by the human mind when performing control actions. For example, when a person drives (controls) a car, he/she will not think:

If the temperature is 10 degrees Celsius and the rainfall is 70.5 mm and the road is 40% slippery and the distance between my car and the car in front of me is 3 meters, then I will depress the acceleration pedal only 10%.

Instead, it is much more likely that he/she thinks:

If it is Cold and the rainfall is High and the road is Somewhat Slippery and the distance between my car and the car in front of me is Quite Close, then I will depress the acceleration pedal Slightly.

So, in systems controlled by humans, the control cycle starts by a person converting a physical quantity (e.g., a distance) from numbers into words or perceptions (e.g., Quite Close distance). The input words (or perceptions) then trigger a person's knowledge, accumulated through that person's experience, resulting in words representing actions (e.g., depress the acceleration pedal Slightly). The person then executes an action to actuate a given device that interfaces the person with the controlled system (e.g., depress the acceleration pedal only 10% might represent the person's implementation of “depress the accelerator pedal Slightly”). Because people think and reason by using imprecise linguistic information, FLCs try to mimic and convert linguistic control information into numerical control information that can be used in automatic control systems.

In its attempt to mimic human control actions, a type-1 FLC, whose structure is shown in Fig. 1.2, is composed of four main components: fuzzifier, rules, inference engine, and defuzzifier, where the operation of each component is summarized as follows:

• The fuzzifier maps each measured numerical input variable into a fuzzy set. One motivation for doing this is that measurements may be corrupted by noise and are somewhat uncertain (even after filtering). So, for example, a measured temperature of 26°C may be modeled as a triangular type-1 fuzzy set that is symmetrically centered around 26°C, where the base of the triangle is related to the uncertainty of this measurement. If, however, one believes that there is no measurement uncertainty, then the measurements can be modeled as crisp sets.
• Rules have an...