Table of Contents

List of Figures xi

List of Tables xiii

Preface xv

1 Fuzzy Equations 1

1.1 Introduction 1

1.2 Fuzzy Equations 1

1.3 Algebraic Fuzzy Equations 3

1.4 Numerical Methods for Solving Fuzzy Equations 5

1.4.1 Newton Method 5

1.4.2 Steepest Descent Method 7

1.4.3 Adomian Decomposition Method 8

1.4.4 Ranking Method 9

1.4.5 Intelligent Methods 10

1.4.5.1 Genetic Algorithm Method 10

1.4.5.2 Neural Network Method 11

1.4.5.3 Fuzzy Linear Regression Model 14

1.5 Summary 20

2 Fuzzy Differential Equations 21

2.1 Introduction 21

2.2 Predictor–Corrector Method 21

2.3 Adomian Decomposition Method 23

2.4 Euler Method 23

2.5 Taylor Method 25

2.6 Runge–Kutta Method 25

2.7 Finite Difference Method 26

2.8 Differential Transform Method 28

2.9 Neural Network Method 29

2.10 Summary 36

3 Modeling and Control Using Fuzzy Equations 39

3.1 Fuzzy Modeling with Fuzzy Equations 39

3.1.1 Fuzzy Parameter Estimation with Neural Networks 45

3.1.2 Upper Bounds of the Modeling Errors 48

3.2 Control with Fuzzy Equations 52

3.3 Simulations 59

3.4 Summary 67

4 Modeling and Control Using Fuzzy Differential Equations 69

4.1 Introduction 69

4.2 Fuzzy Modeling with Fuzzy Differential Equations 69

4.3 Existence of a Solution 72

4.4 Solution Approximation using Bernstein Neural Networks 79

4.5 Solutions Approximation using the Fuzzy Sumudu Transform 83

4.6 Simulations 85

4.7 Summary 99

5 System Modeling with Partial Differential Equations 101

5.1 Introduction 101

5.2 Solutions using Burgers–Fisher Equations 101

5.3 Solution using Wave Equations 106

5.4 Simulations 109

5.5 Summary 117

6 System Control using Z-numbers 119

6.1 Introduction 119

6.2 Modeling using Dual Fuzzy Equations and Z-numbers 119

6.3 Controllability using Dual Fuzzy Equations 124

6.4 Fuzzy Controller 128

6.5 Nonlinear System Modeling 131

6.6 Controllability using Fuzzy Differential Equations 131

6.7 Fuzzy Controller Design using Fuzzy Differential Equations and Z-number 135

6.8 Approximation using a Fuzzy Sumudu Transform and Z-numbers 138

6.9 Simulations 139

6.10 Summary 151

References 153

Index 167

About the Author

Wen Yu, PhD, is Professor at CINVESTAV-IPN (National Polytechnic Institute), Mexico City, Mexico. He is Associate Editor of IEEE Transactions on Cybernetics, Neurocomputing, and Journal of Intelligent and Fuzzy Systems. Dr. Yu is a member of the Mexican Academy of Sciences.

Raheleh Jafari is a postdoctoral research fellow at Centre for Artificial Intelligence Research (CAIR), University of Agder, Grimstad, Norway. She is on the editorial board of the Journal of Intelligent and Fuzzy Systems, and served as a reviewer in various journals and conferences. Her research interest is in the field of artificial intelligence, fuzzy control, machine learning, nonlinear systems, neural networks, and fuzzy engineering.