JERRY M. MENDEL is Professor in the Ming Hsieh Department of Electrical Engineering at the University of Southern California, Life Fellow of the IEEE, and a Distinguished Member of the IEEE Control Systems Society.

HANI HAGRAS is Professor and Director of the Computational Intelligence Centre in the School of Computer Science and Electronic Engineering at the University of Essex, UK, and is a Fellow of the IEEE.

WOEI-WAN TAN is Associate Professor in the Department of Electrical Engineering at the National University of Singapore.

WILLIAM W. MELEK is Associate Professor in the Department of Mechanical and Mechatronics Engineering at the University of Waterloo.

HAO YING is Professor in the Department of Electrical and Computer Engineering at Wayne State University and a Fellow of the IEEE.

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

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