
Introduction To Type-2 Fuzzy Logic Control
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Contributors xvii
1 Introduction 1
1.1 Early History of Fuzzy Control 1
1.2 What Is a Type-1 Fuzzy Set? 2
1.3 What Is a Type-1 Fuzzy Logic Controller? 3
1.4 What Is a Type-2 Fuzzy Set? 7
1.5 What Is a Type-2 Fuzzy Logic Controller? 9
1.6 Distinguishing an FLC from Other Nonlinear Controllers 10
1.7 T2 FLCs versus T1 FLCs 11
1.8 Real-World Applications of IT2 Mamdani FLCs 14
1.8.1 Applications to Industrial Control 14
1.8.2 Airplane Altitude Control 23
1.8.3 Control of Mobile Robots 24
1.8.4 Control of Ambient Intelligent Environments 27
1.9 Book Rationale 29
1.10 Software and How it Can Be Accessed 30
1.11 Coverage of the Other Chapters 30
2 Introduction to Type-2 Fuzzy Sets 32
2.1 Introduction 32
2.2 Brief Review of Type-1 Fuzzy Sets 32
2.2.1 Some Definitions 32
2.2.2 Set-Theoretic Operations 35
2.2.3 Alpha Cuts 36
2.2.4 Compositions of T1 FSs 39
2.2.5 Rules and Their MFs 40
2.3 Interval Type-2 Fuzzy Sets 42
2.3.1 Introduction 42
2.3.2 Definitions 43
2.3.3 Set-Theoretic Operations 51
2.3.4 Centroid of an IT2 FS 54
2.3.5 Properties of cl(k) and cr(k) 58
2.3.6 KM Algorithms as Well as Some Others 59
2.4 General Type-2 Fuzzy Sets 68
2.4.1 -Plane/zSlice Representation 68
2.4.2 Set-Theoretic Operations 72
2.4.3 Centroid of a GT2 FS 73
2.5 Wrapup 77
2.6 Moving On 79
3 Interval Type-2 Fuzzy Logic Controllers 80
3.1 Introduction 80
3.2 Type-1 Fuzzy Logic Controllers 80
3.2.1 Introduction 80
3.2.2 T1 Mamdani FLCs 81
3.2.3 T1 TSK FLCs 85
3.2.4 Design of T1 FLCs 86
3.3 Interval Type-2 Fuzzy Logic Controllers 86
3.3.1 Introduction 86
3.3.2 IT2 Mamdani FLCs 87
3.3.3 IT2 TSK FLCs 103
3.3.4 Design of T2 FLCs 105
3.4 Wu-Mendel Uncertainty Bounds 105
3.5 Control Analyses of IT2 FLCs 111
3.6 Determining the FOU Parameters of IT2 FLCs 114
3.6.1 Blurring T1 MFs 114
3.6.2 Optimizing FOU Parameters 114
3.7 Moving On 122
Appendix 3A. Proof of Theorem 3.4 123
3A.1 Inner-Bound Set [ul(), ur()] 123
3A.2 Outer-Bound Set [ul(), ur()] 124
4 Analytical Structure of Various Interval Type-2 Fuzzy PI and PD Controllers 131
4.1 Introduction 131
4.2 PID, PI, and PD Controllers and Their Relationships 134
4.2.1 Two Forms of PID Controller-Position Form and Incremental Form 134
4.2.2 PI and PD Controllers and Their Relationship 135
4.3 Components of the Interval T2 Fuzzy PI and PD Controllers 136
4.4 Mamdani Fuzzy PI and PD Controllers-Configuration 1 140
4.4.1 Fuzzy PI Controller Configuration 140
4.4.2 Method for Deriving the Analytical Structure 144
4.5 Mamdani Fuzzy PI and PD Controllers-Configuration 2 154
4.6 Mamdani Fuzzy PI and PD Controllers-Configuration 3 162
4.6.1 Fuzzy PI Controller Configuration 162
4.6.2 Method for Deriving the Analytical Structure 165
4.7 Mamdani Fuzzy PI and PD Controllers-Configuration 4 169
4.7.1 Fuzzy PI Controller Configuration 169
4.7.2 Method for Deriving the Analytical Structure 171
4.8 TSK Fuzzy PI and PD Controllers-Configuration 5 181
4.8.1 Fuzzy PI Controller Configuration 181
4.8.2 Deriving the Analytical Structure 184
4.9 Analyzing the Derived Analytical Structures 185
4.9.1 Structural Connection with the Corresponding T1 Fuzzy PI Controller 186
4.9.2 Characteristics of the Variable Gains of the T2 Fuzzy PI Controller 190
4.10 Design Guidelines for the T2 Fuzzy PI and PD Controllers 194
4.10.1 Determination of 1 and 2 Values 196
4.10.2 Determination of the Remaining Nine Parameter Values 197
4.11 Summary 198
Appendix 4A 200
5 Analysis of Simplified Interval Type-2 Fuzzy PI and PD Controllers 205
5.1 Introduction 205
5.2 Simplified Type-2 FLCs: Design, Computation, and Performance 206
5.2.1 Structure of a Simplified IT2 FLC 207
5.2.2 Output Computation 208
5.2.3 Computational Cost 209
5.2.4 Genetic Tuning of FLC 210
5.2.5 Performance 211
5.2.6 Discussions 216
5.3 Analytical Structure of Interval T2 Fuzzy PD and PI Controller 221
5.3.1 Configuration of Interval T2 Fuzzy PD and PI Controller 221
5.3.2 Analysis of the Karnik-Mendel Type-Reduced IT2 Fuzzy PD Controller 227
6.7 Robust Control Design 277
6.7.1 System Description 277
6.7.2 Disturbance Rejection Problem and Solution 280
6.7.3 Robust Control Example 284
6.8 Summary 285
Appendix 285
7 Looking into the Future 290
7.1 Introduction 290
7.2 William Melek and Hao Ying Look into the Future 290
7.3 Hani Hagras Looks into the Future 293
7.3.1 Nonsingleton IT2 FL Control 293
7.3.2 zSlices-Based Singleton General T2 FL Control 299
7.4 Woei Wan Tan Looks into the Future 306
7.5 Jerry Mendel Looks into The Future 307
7.5.1 IT2 FLC 307
7.5.2 GT2 FLC 309
Appendix A T2 FLC Software: From Type-1 to zSlices-Based General Type-2 FLCs 315
A.1 Introduction 315
A.2 FLC for Right-Edge Following 315
A.3 Type-1 FLC Software 316
A.3.1 Define and Set Up T1 FLC Inputs 316
A.3.2 Define T1 FSs That Quantify Each Variable 316
A.3.3 Define Logical Antecedents and Consequents for the FL Rules 318
A.3.4 Define Rule Base of T1 FLC 318
A.4 Interval T2 FLC Software 321
A.4.1 Define and Set Up FLC Inputs 323
A.4.2 Define IT2 FSs That Quantify Each Variable 323
A.4.3 Define Logical Antecedents and Consequents for the FL Rules 323
A.4.4 Define Rule Base of the IT2 FLC 323
A.5 zSlices-Based General Type-2 FLC Software 327
A.5.1 Define and Set Up FLC Inputs 327
A.5.2 Define zSlices-Based GT2 FSs That Quantify Each Variable 327
A.5.3 Define Logical Antecedents and Consequents for the FL Rules 335
A.5.4 Define Rule Base of the GT2 FLC 335
References 338
Index 347
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...
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