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.

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• Preface xiii Contributors xvii1 Introduction 11.1 Early History of Fuzzy Control 11.2 What Is a Type-1 Fuzzy Set? 21.3 What Is a Type-1 Fuzzy Logic Controller? 31.4 What Is a Type-2 Fuzzy Set? 71.5 What Is a Type-2 Fuzzy Logic Controller? 91.6 Distinguishing an FLC from Other Nonlinear Controllers 101.7 T2 FLCs versus T1 FLCs 111.8 Real-World Applications of IT2 Mamdani FLCs 141.8.1 Applications to Industrial Control 141.8.2 Airplane Altitude Control 231.8.3 Control of Mobile Robots 241.8.4 Control of Ambient Intelligent Environments 271.9 Book Rationale 291.10 Software and How it Can Be Accessed 301.11 Coverage of the Other Chapters 302 Introduction to Type-2 Fuzzy Sets 322.1 Introduction 322.2 Brief Review of Type-1 Fuzzy Sets 322.2.1 Some Definitions 322.2.2 Set-Theoretic Operations 352.2.3 Alpha Cuts 362.2.4 Compositions of T1 FSs 392.2.5 Rules and Their MFs 402.3 Interval Type-2 Fuzzy Sets 422.3.1 Introduction 422.3.2 Definitions 432.3.3 Set-Theoretic Operations 512.3.4 Centroid of an IT2 FS 542.3.5 Properties of cl(k) and cr(k) 582.3.6 KM Algorithms as Well as Some Others 592.4 General Type-2 Fuzzy Sets 682.4.1 -Plane/zSlice Representation 682.4.2 Set-Theoretic Operations 722.4.3 Centroid of a GT2 FS 732.5 Wrapup 772.6 Moving On 793 Interval Type-2 Fuzzy Logic Controllers 803.1 Introduction 803.2 Type-1 Fuzzy Logic Controllers 803.2.1 Introduction 803.2.2 T1 Mamdani FLCs 813.2.3 T1 TSK FLCs 853.2.4 Design of T1 FLCs 863.3 Interval Type-2 Fuzzy Logic Controllers 863.3.1 Introduction 863.3.2 IT2 Mamdani FLCs 873.3.3 IT2 TSK FLCs 1033.3.4 Design of T2 FLCs 1053.4 Wu–Mendel Uncertainty Bounds 1053.5 Control Analyses of IT2 FLCs 1113.6 Determining the FOU Parameters of IT2 FLCs 1143.6.1 Blurring T1 MFs 1143.6.2 Optimizing FOU Parameters 1143.7 Moving On 122Appendix 3A. Proof of Theorem 3.4 1233A.1 Inner-Bound Set [ul(), ur()] 1233A.2 Outer-Bound Set [ul(), ur()] 1244 Analytical Structure of Various Interval Type-2 Fuzzy PI and PD Controllers 1314.1 Introduction 1314.2 PID, PI, and PD Controllers and Their Relationships 1344.2.1 Two Forms of PID Controller—Position Form and Incremental Form 1344.2.2 PI and PD Controllers and Their Relationship 1354.3 Components of the Interval T2 Fuzzy PI and PD Controllers 1364.4 Mamdani Fuzzy PI and PD Controllers—Configuration 1 1404.4.1 Fuzzy PI Controller Configuration 1404.4.2 Method for Deriving the Analytical Structure 1444.5 Mamdani Fuzzy PI and PD Controllers—Configuration 2 1544.6 Mamdani Fuzzy PI and PD Controllers—Configuration 3 1624.6.1 Fuzzy PI Controller Configuration 1624.6.2 Method for Deriving the Analytical Structure 1654.7 Mamdani Fuzzy PI and PD Controllers—Configuration 4 1694.7.1 Fuzzy PI Controller Configuration 1694.7.2 Method for Deriving the Analytical Structure 1714.8 TSK Fuzzy PI and PD Controllers—Configuration 5 1814.8.1 Fuzzy PI Controller Configuration 1814.8.2 Deriving the Analytical Structure 1844.9 Analyzing the Derived Analytical Structures 1854.9.1 Structural Connection with the Corresponding T1 Fuzzy PI Controller 1864.9.2 Characteristics of the Variable Gains of the T2 Fuzzy PI Controller 1904.10 Design Guidelines for the T2 Fuzzy PI and PD Controllers 1944.10.1 Determination of 1 and 2 Values 1964.10.2 Determination of the Remaining Nine Parameter Values 1974.11 Summary 198Appendix 4A 2005 Analysis of Simplified Interval Type-2 Fuzzy PI and PD Controllers 2055.1 Introduction 2055.2 Simplified Type-2 FLCs: Design, Computation, and Performance 2065.2.1 Structure of a Simplified IT2 FLC 2075.2.2 Output Computation 2085.2.3 Computational Cost 2095.2.4 Genetic Tuning of FLC 2105.2.5 Performance 2115.2.6 Discussions 2165.3 Analytical Structure of Interval T2 Fuzzy PD and PI Controller 2215.3.1 Configuration of Interval T2 Fuzzy PD and PI Controller 2215.3.2 Analysis of the Karnik–Mendel Type-Reduced IT2 Fuzzy PD Controller 2276.7 Robust Control Design 2776.7.1 System Description 2776.7.2 Disturbance Rejection Problem and Solution 2806.7.3 Robust Control Example 2846.8 Summary 285Appendix 2857 Looking into the Future 2907.1 Introduction 2907.2 William Melek and Hao Ying Look into the Future 2907.3 Hani Hagras Looks into the Future 2937.3.1 Nonsingleton IT2 FL Control 2937.3.2 zSlices-Based Singleton General T2 FL Control 2997.4 Woei Wan Tan Looks into the Future 3067.5 Jerry Mendel Looks into The Future 3077.5.1 IT2 FLC 3077.5.2 GT2 FLC 309Appendix A T2 FLC Software: From Type-1 to zSlices-Based General Type-2 FLCs 315A.1 Introduction 315A.2 FLC for Right-Edge Following 315A.3 Type-1 FLC Software 316A.3.1 Define and Set Up T1 FLC Inputs 316A.3.2 Define T1 FSs That Quantify Each Variable 316A.3.3 Define Logical Antecedents and Consequents for the FL Rules 318A.3.4 Define Rule Base of T1 FLC 318A.4 Interval T2 FLC Software 321A.4.1 Define and Set Up FLC Inputs 323A.4.2 Define IT2 FSs That Quantify Each Variable 323A.4.3 Define Logical Antecedents and Consequents for the FL Rules 323A.4.4 Define Rule Base of the IT2 FLC 323A.5 zSlices-Based General Type-2 FLC Software 327A.5.1 Define and Set Up FLC Inputs 327A.5.2 Define zSlices-Based GT2 FSs That Quantify Each Variable 327A.5.3 Define Logical Antecedents and Consequents for the FL Rules 335A.5.4 Define Rule Base of the GT2 FLC 335References 338Index 347

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.

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• Preface xiii Contributors xvii1 Introduction 11.1 Early History of Fuzzy Control 11.2 What Is a Type-1 Fuzzy Set? 21.3 What Is a Type-1 Fuzzy Logic Controller? 31.4 What Is a Type-2 Fuzzy Set? 71.5 What Is a Type-2 Fuzzy Logic Controller? 91.6 Distinguishing an FLC from Other Nonlinear Controllers 101.7 T2 FLCs versus T1 FLCs 111.8 Real-World Applications of IT2 Mamdani FLCs 141.8.1 Applications to Industrial Control 141.8.2 Airplane Altitude Control 231.8.3 Control of Mobile Robots 241.8.4 Control of Ambient Intelligent Environments 271.9 Book Rationale 291.10 Software and How it Can Be Accessed 301.11 Coverage of the Other Chapters 302 Introduction to Type-2 Fuzzy Sets 322.1 Introduction 322.2 Brief Review of Type-1 Fuzzy Sets 322.2.1 Some Definitions 322.2.2 Set-Theoretic Operations 352.2.3 Alpha Cuts 362.2.4 Compositions of T1 FSs 392.2.5 Rules and Their MFs 402.3 Interval Type-2 Fuzzy Sets 422.3.1 Introduction 422.3.2 Definitions 432.3.3 Set-Theoretic Operations 512.3.4 Centroid of an IT2 FS 542.3.5 Properties of cl(k) and cr(k) 582.3.6 KM Algorithms as Well as Some Others 592.4 General Type-2 Fuzzy Sets 682.4.1 -Plane/zSlice Representation 682.4.2 Set-Theoretic Operations 722.4.3 Centroid of a GT2 FS 732.5 Wrapup 772.6 Moving On 793 Interval Type-2 Fuzzy Logic Controllers 803.1 Introduction 803.2 Type-1 Fuzzy Logic Controllers 803.2.1 Introduction 803.2.2 T1 Mamdani FLCs 813.2.3 T1 TSK FLCs 853.2.4 Design of T1 FLCs 863.3 Interval Type-2 Fuzzy Logic Controllers 863.3.1 Introduction 863.3.2 IT2 Mamdani FLCs 873.3.3 IT2 TSK FLCs 1033.3.4 Design of T2 FLCs 1053.4 Wu–Mendel Uncertainty Bounds 1053.5 Control Analyses of IT2 FLCs 1113.6 Determining the FOU Parameters of IT2 FLCs 1143.6.1 Blurring T1 MFs 1143.6.2 Optimizing FOU Parameters 1143.7 Moving On 122Appendix 3A. Proof of Theorem 3.4 1233A.1 Inner-Bound Set [ul(), ur()] 1233A.2 Outer-Bound Set [ul(), ur()] 1244 Analytical Structure of Various Interval Type-2 Fuzzy PI and PD Controllers 1314.1 Introduction 1314.2 PID, PI, and PD Controllers and Their Relationships 1344.2.1 Two Forms of PID Controller—Position Form and Incremental Form 1344.2.2 PI and PD Controllers and Their Relationship 1354.3 Components of the Interval T2 Fuzzy PI and PD Controllers 1364.4 Mamdani Fuzzy PI and PD Controllers—Configuration 1 1404.4.1 Fuzzy PI Controller Configuration 1404.4.2 Method for Deriving the Analytical Structure 1444.5 Mamdani Fuzzy PI and PD Controllers—Configuration 2 1544.6 Mamdani Fuzzy PI and PD Controllers—Configuration 3 1624.6.1 Fuzzy PI Controller Configuration 1624.6.2 Method for Deriving the Analytical Structure 1654.7 Mamdani Fuzzy PI and PD Controllers—Configuration 4 1694.7.1 Fuzzy PI Controller Configuration 1694.7.2 Method for Deriving the Analytical Structure 1714.8 TSK Fuzzy PI and PD Controllers—Configuration 5 1814.8.1 Fuzzy PI Controller Configuration 1814.8.2 Deriving the Analytical Structure 1844.9 Analyzing the Derived Analytical Structures 1854.9.1 Structural Connection with the Corresponding T1 Fuzzy PI Controller 1864.9.2 Characteristics of the Variable Gains of the T2 Fuzzy PI Controller 1904.10 Design Guidelines for the T2 Fuzzy PI and PD Controllers 1944.10.1 Determination of 1 and 2 Values 1964.10.2 Determination of the Remaining Nine Parameter Values 1974.11 Summary 198Appendix 4A 2005 Analysis of Simplified Interval Type-2 Fuzzy PI and PD Controllers 2055.1 Introduction 2055.2 Simplified Type-2 FLCs: Design, Computation, and Performance 2065.2.1 Structure of a Simplified IT2 FLC 2075.2.2 Output Computation 2085.2.3 Computational Cost 2095.2.4 Genetic Tuning of FLC 2105.2.5 Performance 2115.2.6 Discussions 2165.3 Analytical Structure of Interval T2 Fuzzy PD and PI Controller 2215.3.1 Configuration of Interval T2 Fuzzy PD and PI Controller 2215.3.2 Analysis of the Karnik–Mendel Type-Reduced IT2 Fuzzy PD Controller 2276.7 Robust Control Design 2776.7.1 System Description 2776.7.2 Disturbance Rejection Problem and Solution 2806.7.3 Robust Control Example 2846.8 Summary 285Appendix 2857 Looking into the Future 2907.1 Introduction 2907.2 William Melek and Hao Ying Look into the Future 2907.3 Hani Hagras Looks into the Future 2937.3.1 Nonsingleton IT2 FL Control 2937.3.2 zSlices-Based Singleton General T2 FL Control 2997.4 Woei Wan Tan Looks into the Future 3067.5 Jerry Mendel Looks into The Future 3077.5.1 IT2 FLC 3077.5.2 GT2 FLC 309Appendix A T2 FLC Software: From Type-1 to zSlices-Based General Type-2 FLCs 315A.1 Introduction 315A.2 FLC for Right-Edge Following 315A.3 Type-1 FLC Software 316A.3.1 Define and Set Up T1 FLC Inputs 316A.3.2 Define T1 FSs That Quantify Each Variable 316A.3.3 Define Logical Antecedents and Consequents for the FL Rules 318A.3.4 Define Rule Base of T1 FLC 318A.4 Interval T2 FLC Software 321A.4.1 Define and Set Up FLC Inputs 323A.4.2 Define IT2 FSs That Quantify Each Variable 323A.4.3 Define Logical Antecedents and Consequents for the FL Rules 323A.4.4 Define Rule Base of the IT2 FLC 323A.5 zSlices-Based General Type-2 FLC Software 327A.5.1 Define and Set Up FLC Inputs 327A.5.2 Define zSlices-Based GT2 FSs That Quantify Each Variable 327A.5.3 Define Logical Antecedents and Consequents for the FL Rules 335A.5.4 Define Rule Base of the GT2 FLC 335References 338Index 347

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.

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• Preface xiii Contributors xvii1 Introduction 11.1 Early History of Fuzzy Control 11.2 What Is a Type-1 Fuzzy Set? 21.3 What Is a Type-1 Fuzzy Logic Controller? 31.4 What Is a Type-2 Fuzzy Set? 71.5 What Is a Type-2 Fuzzy Logic Controller? 91.6 Distinguishing an FLC from Other Nonlinear Controllers 101.7 T2 FLCs versus T1 FLCs 111.8 Real-World Applications of IT2 Mamdani FLCs 141.8.1 Applications to Industrial Control 141.8.2 Airplane Altitude Control 231.8.3 Control of Mobile Robots 241.8.4 Control of Ambient Intelligent Environments 271.9 Book Rationale 291.10 Software and How it Can Be Accessed 301.11 Coverage of the Other Chapters 302 Introduction to Type-2 Fuzzy Sets 322.1 Introduction 322.2 Brief Review of Type-1 Fuzzy Sets 322.2.1 Some Definitions 322.2.2 Set-Theoretic Operations 352.2.3 Alpha Cuts 362.2.4 Compositions of T1 FSs 392.2.5 Rules and Their MFs 402.3 Interval Type-2 Fuzzy Sets 422.3.1 Introduction 422.3.2 Definitions 432.3.3 Set-Theoretic Operations 512.3.4 Centroid of an IT2 FS 542.3.5 Properties of cl(k) and cr(k) 582.3.6 KM Algorithms as Well as Some Others 592.4 General Type-2 Fuzzy Sets 682.4.1 -Plane/zSlice Representation 682.4.2 Set-Theoretic Operations 722.4.3 Centroid of a GT2 FS 732.5 Wrapup 772.6 Moving On 793 Interval Type-2 Fuzzy Logic Controllers 803.1 Introduction 803.2 Type-1 Fuzzy Logic Controllers 803.2.1 Introduction 803.2.2 T1 Mamdani FLCs 813.2.3 T1 TSK FLCs 853.2.4 Design of T1 FLCs 863.3 Interval Type-2 Fuzzy Logic Controllers 863.3.1 Introduction 863.3.2 IT2 Mamdani FLCs 873.3.3 IT2 TSK FLCs 1033.3.4 Design of T2 FLCs 1053.4 Wu–Mendel Uncertainty Bounds 1053.5 Control Analyses of IT2 FLCs 1113.6 Determining the FOU Parameters of IT2 FLCs 1143.6.1 Blurring T1 MFs 1143.6.2 Optimizing FOU Parameters 1143.7 Moving On 122Appendix 3A. Proof of Theorem 3.4 1233A.1 Inner-Bound Set [ul(), ur()] 1233A.2 Outer-Bound Set [ul(), ur()] 1244 Analytical Structure of Various Interval Type-2 Fuzzy PI and PD Controllers 1314.1 Introduction 1314.2 PID, PI, and PD Controllers and Their Relationships 1344.2.1 Two Forms of PID Controller—Position Form and Incremental Form 1344.2.2 PI and PD Controllers and Their Relationship 1354.3 Components of the Interval T2 Fuzzy PI and PD Controllers 1364.4 Mamdani Fuzzy PI and PD Controllers—Configuration 1 1404.4.1 Fuzzy PI Controller Configuration 1404.4.2 Method for Deriving the Analytical Structure 1444.5 Mamdani Fuzzy PI and PD Controllers—Configuration 2 1544.6 Mamdani Fuzzy PI and PD Controllers—Configuration 3 1624.6.1 Fuzzy PI Controller Configuration 1624.6.2 Method for Deriving the Analytical Structure 1654.7 Mamdani Fuzzy PI and PD Controllers—Configuration 4 1694.7.1 Fuzzy PI Controller Configuration 1694.7.2 Method for Deriving the Analytical Structure 1714.8 TSK Fuzzy PI and PD Controllers—Configuration 5 1814.8.1 Fuzzy PI Controller Configuration 1814.8.2 Deriving the Analytical Structure 1844.9 Analyzing the Derived Analytical Structures 1854.9.1 Structural Connection with the Corresponding T1 Fuzzy PI Controller 1864.9.2 Characteristics of the Variable Gains of the T2 Fuzzy PI Controller 1904.10 Design Guidelines for the T2 Fuzzy PI and PD Controllers 1944.10.1 Determination of 1 and 2 Values 1964.10.2 Determination of the Remaining Nine Parameter Values 1974.11 Summary 198Appendix 4A 2005 Analysis of Simplified Interval Type-2 Fuzzy PI and PD Controllers 2055.1 Introduction 2055.2 Simplified Type-2 FLCs: Design, Computation, and Performance 2065.2.1 Structure of a Simplified IT2 FLC 2075.2.2 Output Computation 2085.2.3 Computational Cost 2095.2.4 Genetic Tuning of FLC 2105.2.5 Performance 2115.2.6 Discussions 2165.3 Analytical Structure of Interval T2 Fuzzy PD and PI Controller 2215.3.1 Configuration of Interval T2 Fuzzy PD and PI Controller 2215.3.2 Analysis of the Karnik–Mendel Type-Reduced IT2 Fuzzy PD Controller 2276.7 Robust Control Design 2776.7.1 System Description 2776.7.2 Disturbance Rejection Problem and Solution 2806.7.3 Robust Control Example 2846.8 Summary 285Appendix 2857 Looking into the Future 2907.1 Introduction 2907.2 William Melek and Hao Ying Look into the Future 2907.3 Hani Hagras Looks into the Future 2937.3.1 Nonsingleton IT2 FL Control 2937.3.2 zSlices-Based Singleton General T2 FL Control 2997.4 Woei Wan Tan Looks into the Future 3067.5 Jerry Mendel Looks into The Future 3077.5.1 IT2 FLC 3077.5.2 GT2 FLC 309Appendix A T2 FLC Software: From Type-1 to zSlices-Based General Type-2 FLCs 315A.1 Introduction 315A.2 FLC for Right-Edge Following 315A.3 Type-1 FLC Software 316A.3.1 Define and Set Up T1 FLC Inputs 316A.3.2 Define T1 FSs That Quantify Each Variable 316A.3.3 Define Logical Antecedents and Consequents for the FL Rules 318A.3.4 Define Rule Base of T1 FLC 318A.4 Interval T2 FLC Software 321A.4.1 Define and Set Up FLC Inputs 323A.4.2 Define IT2 FSs That Quantify Each Variable 323A.4.3 Define Logical Antecedents and Consequents for the FL Rules 323A.4.4 Define Rule Base of the IT2 FLC 323A.5 zSlices-Based General Type-2 FLC Software 327A.5.1 Define and Set Up FLC Inputs 327A.5.2 Define zSlices-Based GT2 FSs That Quantify Each Variable 327A.5.3 Define Logical Antecedents and Consequents for the FL Rules 335A.5.4 Define Rule Base of the GT2 FLC 335References 338Index 347