Fuzzy neural intelligent systems : mathematical foundation and the applications in engineering

Although fuzzy systems and neural networks are central to the field of soft computing, most research work has focused on the development of the theories, algorithms, and designs of systems for specific applications. There has been little theoretical support for fuzzy neural systems, especially their mathematical foundations.

Fuzzy Neural Intelligent Systems fills this gap. It develops a mathematical basis for fuzzy neural networks, offers a better way of combining fuzzy logic systems with neural networks, and explores some of their engineering applications. Dividing their focus into three main areas of interest, the authors give a systematic, comprehensive treatment of the relevant concepts and modern practical applications:

Fundamental concepts and theories for fuzzy systems and neural networks.
Foundation for fuzzy neural networks and important related topics
Case examples for neuro-fuzzy systems, fuzzy systems, neural network systems, and fuzzy-neural systems

Suitable for self-study, as a reference, and ideal as a textbook, Fuzzy Neural Intelligent Systems is accessible to students with a basic background in linear algebra and engineering mathematics. Mastering the material in this textbook will prepare students to better understand, design, and implement fuzzy neural systems, develop new applications, and further advance the field.

1. Foundation of Fuzzy Systems	p. 1
1.1 Definition of Fuzzy Sets	p. 1
1.2 Basic Operations of Fuzzy Sets	p. 6
1.3 The Resolution Theorem	p. 8
1.4 A Representation Theorem	p. 12
1.5 Extension Principle	p. 17
References	p. 22
2. Determination of Membership Functions	p. 23
2.1 A General Method for Determining Membership Functions	p. 23
2.2 The Three-phase Method	p. 27
2.3 The Incremental Method	p. 29
2.4 The Multiphase Fuzzy Statistical Method	p. 29
2.5 The Method of Comparisons	p. 31
2.5.1 Binary Comparisons	p. 31
2.5.2 Preferred Comparisons	p. 32
2.5.3 A Special Case of Preferred Comparisons	p. 33
2.5.4 An Example	p. 34
2.6 The Absolute Comparison Method	p. 35
2.7 The Set-valued Statistical Iteration Method	p. 38
2.7.1 Statement of the Problem	p. 38
2.7.2 Basic Steps of Set-valued Statistical Method	p. 38
2.8 Ordering by Precedence Relations	p. 40
2.8.1 Precedence Relations	p. 40
2.8.2 Creating Order	p. 40
2.8.3 An Example	p. 41
2.8.4 Generalizations	p. 41
2.9 The Relative Comparison Method and the Mean Pairwise Comparison Method	p. 42
2.9.1 The Relative Comparison Method	p. 42
2.9.2 The Mean Pairwise Comparison Method	p. 43
References	p. 46
3. Mathematical Essence and Structures of Feedforward Artificial Neural Networks	p. 47
3.1 Introduction	p. 47
3.2 Mathematical Neurons and Mathematical Neural Networks	p. 48
3.2.1 MP Model with Discrete Outputs	p. 48
3.2.2 MP Model with Continuous-valued Outputs	p. 48
3.3 The Interpolation Mechanism of Feedforward Neural Networks	p. 52
3.4 A Three-layer Feedforward Neural Network with Two Inputs One Output	p. 56
3.5 Analysis of Steepest Descent Learning Algorithms of Feedforward Neural Networks	p. 58
3.6 Feedforward Neural Networks with Multi-input One Output and Their Learning Algorithm	p. 62
3.7 Feedforward Neural Networks with One Input Multi-output and Their Learning Algorithm	p. 65
3.8 Feedforward Neural Networks with Multi-input Multi-output and Their Learning Algorithm	p. 67
3.9 A Note on the Learning Algorithm of Feedforward Neural Networks	p. 68
3.10 Conclusions	p. 70
References	p. 71
4. Functional-link Neural Networks and Visualization Means of Some Mathematical Methods	p. 72
4.1 Discussion of the XOR Problem	p. 72
4.2 Mathematical Essence of Functional-link Neural Networks	p. 75
4.3 As Visualization Means of Some Mathematical Methods	p. 79
4.4 Neural Network Representation of Linear Programming	p. 81
4.5 Neural Network Representation of Fuzzy Linear Programming	p. 86
4.6 Conclusions	p. 87
References	p. 89
5. Flat Neural Networks and Rapid Learning Algorithms	p. 90
5.1 Introduction	p. 90
5.2 The Linear System Equation of the Functional-link Network	p. 91
5.3 Pseudoinverse and Stepwise Updating	p. 93
5.4 Training with Weighted Least Square	p. 98
5.5 Refine the Model	p. 99
5.6 Time-series Applications	p. 100
5.7 Examples and Discussion	p. 102
5.8 Conclusions	p. 108
References	p. 110
6. Basic Structure of Fuzzy Neural Networks	p. 113
6.1 Definition of Fuzzy Neurons	p. 113
6.2 Fuzzy Neural Networks	p. 118
6.2.1 Neural Network Representation of Fuzzy Relation Equations	p. 118
6.2.2 A Fuzzy Neural Network Based on FN ([logical or], [logical and])	p. 119
6.3 A Fuzzy [delta] Learning Algorithm	p. 121
6.4 The Convergence of Fuzzy [delta] Learning Rule	p. 123
6.5 Conclusions	p. 124
References	p. 125
7. Mathematical Essence and Structures of Feedback Neural Networks and Weight Matrix Design	p. 126
7.1 Introduction	p. 126
7.2 A General Criterion on the Stability of Networks	p. 129
7.3 Generalized Energy Function	p. 131
7.4 Learning Algorithm of Discrete Feedback Neural Networks	p. 133
7.5 Design Method of Weight Matrices Based on Multifactorial Functions	p. 135
7.6 Conclusions	p. 137
References	p. 139
8. Generalized Additive Weighted Multifactorial Function and its Applications to Fuzzy Inference and Neural Networks	p. 140
8.1 Introduction	p. 140
8.2 On Multifactorial Functions	p. 141
8.3 Generalized Additive Weighted Multifactorial Functions	p. 141
8.4 Infinite Dimensional Multifactorial Functions	p. 145
8.5 M ([perpendicular, bottom], [intercal]) and Fuzzy Integral	p. 146
8.6 Application in Fuzzy Inference	p. 147
8.7 Conclusions	p. 150
References	p. 151
9. The Interpolation Mechanism of Fuzzy Control	p. 152
9.1 Preliminary	p. 152
9.2 The Interpolation Mechanism of Mamdanian Algorithm with One Input and One Output	p. 154
9.3 The Interpolation Mechanism of Mamdanian Algorithm with Two Inputs and One Output	p. 156
9.4 A Note on Completeness of Inference Rules	p. 157
9.5 The Interpolation Mechanism of (+, -)-Centroid Algorithm	p. 158
9.6 The Interpolation Mechanism of Simple Inference Algorithm	p. 159
9.7 The Interpolation Mechanism of Function Inference Algorithm	p. 161
9.8 A General Fuzzy Control Algorithm	p. 162
9.9 Conclusions	p. 163
References	p. 164
10. The Relationship between Fuzzy Controllers and PID Controllers	p. 165
10.1 Introduction	p. 165
10.2 The Relationship of Fuzzy Controllers with One Input One Output and P Controllers	p. 166
10.3 The Relationship of Fuzzy Controllers with Two Inputs One Output and PD (or PI) Controllers	p. 169
10.4 The Relationship of Fuzzy Controllers with Three Inputs One Output and PID Controllers	p. 173
10.5 The Difference Schemes of Fuzzy Controllers with Three Inputs and One Output	p. 177
10.5.1 Positional Difference Scheme	p. 177
10.5.2 Incremental Difference Scheme	p. 178
10.6 Conclusions	p. 179
References	p. 180
11. Adaptive Fuzzy Controllers Based on Variable Universes	p. 181
11.1 The Monotonicity of Control Rules and the Monotonicity of Control Functions	p. 181
11.2 The Contraction-expansion Factors of Variable Universes	p. 184
11.2.1 The Contraction-expansion Factors of Adaptive Fuzzy Controllers with One Input and One Output	p. 184
11.2.2 The Contraction-expansion Factors of Adaptive Fuzzy Controllers with Two Inputs and One Output	p. 185
11.3 The Structure of Adaptive Fuzzy Controllers Based on Variable Universes	p. 186
11.4 Adaptive Fuzzy Controllers with One Input and One Output	p. 187
11.4.1 Adaptive Fuzzy Controllers with Potential Heredity	p. 187
11.4.2 Adaptive Fuzzy Controllers with Obvious Heredity	p. 191
11.4.3 Adaptive Fuzzy Controllers with Successively Obvious Heredity	p. 193
11.5 Adaptive Fuzzy Controllers with Two Inputs and One Output	p. 193
11.6 Conclusions	p. 195
References	p. 196
12. The Basic of Factor Spaces	p. 197
12.1 What are "Factors"?	p. 197
12.2 The State Space of Factors	p. 198
12.3 Relations and Operations of Factors	p. 200
12.3.1 The Zero Factor	p. 200
12.3.2 Equality of Factors	p. 200
12.3.3 Subfactors	p. 200
12.3.4 Conjunction of Factors	p. 201
12.3.5 Disjunction of Factors	p. 201
12.3.6 Independent Factors	p. 202
12.3.7 Difference of Factors	p. 202
12.3.8 Complement of a Factor	p. 202
12.3.9 Atomic Factors	p. 202
12.4 Axiomatic Definition of Factor Spaces	p. 203
12.5 A Note on The Definition of Factor Spaces	p. 204
12.6 Concept Description in a Factor Space	p. 205
12.7 The Projection and Cylindrical Extension of the Representation Extension	p. 207
12.8 Some Properties of the Projection and Cylindrical Extension	p. 209
12.9 Factor Sufficiency	p. 212
12.10 The Rank of a Concept	p. 215
12.11 Atomic Factor Spaces	p. 216
12.12 Conclusions	p. 217
References	p. 218
13. Neuron Models Based on Factor Spaces Theory and Factor Space Canes	p. 219
13.1 Neuron Mechanism of Factor Spaces	p. 219
13.2 The Models of Neurons without Respect to Time	p. 220
13.2.1 Threshold Models of Neurons	p. 220
13.2.2 Linear Model of Neurons	p. 221
13.2.3 General Threshold Model of Neurons	p. 221
13.2.4 The Models of Neurons Based on Weber-Fechner's Law	p. 223
13.3 The Models of Neurons Concerned with Time	p. 224
13.4 The Models of Neurons Based on Variable Weights	p. 225
13.4.1 The Excitatory and Inhibitory Mechanism of Neurons	p. 225
13.4.2 The Negative Weights Description of the Inhibitory Mechanism	p. 226
13.4.3 On Fukushimas Model	p. 227
13.4.4 The Model of Neurons Based on Univariable Weights	p. 228
13.5 Naive Thoughts of Factors Space Canes	p. 229
13.6 Melon-type Factor Space Canes	p. 231
13.7 Chain-type Factor Space Canes	p. 233
13.8 Switch Factors and Growth Relation	p. 234
13.9 Class Partition and Class Concepts	p. 236
13.10 Conclusions	p. 239
References	p. 240
14. Foundation of Neuro-Fuzzy Systems and an Engineering Application	p. 241
14.1 Introduction	p. 241
14.2 Takagi, Sugeno, and Kang Fuzzy Model	p. 242
14.3 Adaptive Network-based Fuzzy Inference System (ANFIS)	p. 243
14.4 Hybrid Learning Algorithm for ANFIS	p. 244
14.5 Estimation of Lot Processing Time in an IC Fabrication	p. 245
14.5.1 Algorithm 1: Gauss-Newton-based Levenberg-Marquardt Method	p. 246
14.5.2 Algorithm 2: Backpropagation Neural Network	p. 247
14.5.3 Algorithm 3: ANFIS Algorithm	p. 247
14.5.4 Simulation Result	p. 248
14.5.4.1 Gauss-Newton-based LM Model Construction	p. 249
14.5.4.2 BP Neural Network Model Construction	p. 249
14.5.4.3 ANFIS Model Construction	p. 250
14.6 Conclusions	p. 251
References	p. 253
15. Data Preprocessing	p. 255
15.1 Introduction	p. 255
15.2 Data Preprocessing Algorithms	p. 256
15.2.1 Data Values Averaging	p. 257
15.2.2 Input Space Reduction	p. 257
15.2.3 Data Normalization (Data Scaling)	p. 260
15.3 Conclusions	p. 263
15.4 Appendix: Matlab Programs	p. 263
15.4.1 Example of Noise Reduction Averaging	p. 263
15.4.2 Example of Min-Max Normalization	p. 264
15.4.3 Example of Zscore Normalization	p. 264
15.4.4 Example of Sigmoidal Normalization	p. 264
15.4.5 The Definitions of Mean and Standard Deviation	p. 265
References	p. 266
16. Control of a Flexible Robot Arm using a Simplified Fuzzy Controller	p. 267
16.1 Introduction	p. 267
16.2 Modeling of the Flexible Arm	p. 268
16.3 Simplified Fuzzy Controller	p. 270
16.3.1 Derivation of Simplified Fuzzy Control Law	p. 273
16.3.2 Analysis of Simplified Fuzzy Control Law	p. 275
16.3.3 Neglected Effect in Simplified Fuzzy Control	p. 279
16.4 Self-Organizing Fuzzy Control	p. 280
16.4.1 Reference Model	p. 281
16.4.2 Incremental Model	p. 283
16.4.3 Parameter Decision	p. 286
16.5 Simulation Results	p. 286
16.6 Conclusions	p. 288
References	p. 293
17. Application of Neuro-Fuzzy Systems: Development of a Fuzzy Learning Decision Tree and Application to Tactile Recognition	p. 295
17.1 Introduction	p. 295
17.2 Tactile Sensors and a Tactile Sensing and Recognition System	p. 297
17.2.1 Types of FSRs	p. 297
17.2.2 A Tactile Sensing System	p. 298
17.2.2.1 Hardware Devices	p. 298
17.2.2.2 Software Kernel	p. 299
17.2.2.3 Man-machine Interface	p. 299
17.2.3 Interpolation to Increase Resolution	p. 299
17.2.3.1 Linear Interpolation	p. 300
17.2.3.2 Polynomial Interpolation	p. 300
17.2.3.3 Fractal Interpolation	p. 301
17.2.3.4 Fuzzy Interpolation	p. 301
17.3 Development of a Fuzzy Learning Decision Tree	p. 302
17.3.1 Architecture of the Fuzzy Learning Decision Tree	p. 302
17.3.2 Features Selection	p. 303
17.3.3 Fuzzy Sets for Compressing Training Data	p. 305
17.3.4 Determining Several Points on a Fuzzy Set	p. 305
17.3.5 Identifying a LR Type Fuzzy Set	p. 306
17.3.6 Learning Procedure of a Decision Tree	p. 306
17.3.7 Comparing to Rule Based Systems	p. 308
17.3.8 Comparison with Artificial Neural Networks	p. 311
17.4 Experiments	p. 315
17.4.1 Experiment Procedures	p. 316
17.4.2 Experiment Results and Discussions	p. 317
17.5 Conclusions	p. 318
References	p. 320
18. Fuzzy Assesment Systems of Rehabilitive Process for CVA Patients	p. 322
18.1 Introduction	p. 322
18.2 COP Signals Feature Extraction	p. 324
18.2.1 Space Domain Analysis	p. 325
18.2.2 Time Domain Analysis	p. 327
18.2.3 Frequency Domain Analysis	p. 327
18.2.4 Force Domain Analysis	p. 331
18.3 Relationship between COP Signals and FIM Scores	p. 331
18.4 Construction of Kinetic State Assessment System	p. 341
18.4.1 Balance Indices Input	p. 341
18.4.2 Knowledge Base	p. 342
18.4.3 Fuzzy Inference Engine	p. 343
18.4.4 Defuzzification	p. 344
18.4.5 Parameters and Rules Setup	p. 344
18.5 Results of Kinetic State Assessment System	p. 347
18.6 Conclusions	p. 348
References	p. 349
19. A DSP-based Neural Controller for a Multi-degree Prosthetic Hand	p. 351
19.1 Introduction	p. 351
19.2 EMG Discriminative System	p. 352
19.2.1 EMG Signal Processing	p. 352
19.2.2 Pattern Recognition	p. 353
19.2.2.1 Feature Extraction	p. 353
19.2.2.2 Feature Selection	p. 355
19.2.2.3 Classification by Neural Network	p. 355
19.3 DSP-based Prosthetic Controller	p. 355
19.3.1 Hardware Architecture of the Controller	p. 356
19.3.1.1 The Off-line Stage of the Prosthetic Controller	p. 356
19.3.1.2 The On-line Stage of the Prosthetic Controller	p. 356
19.3.2 The Software System of the Controller	p. 357
19.3.2.1 Signal Collection	p. 357
19.3.2.2 Signal Processing	p. 358
19.3.2.3 Feature Extraction	p. 359
19.3.2.4 BPNN Classification	p. 361
19.4 Implementation and Results of the DSP-based Controller	p. 361
19.4.1 Off-line Stage Implementation	p. 362
19.4.2 On-line Stage Implementation	p. 362
19.4.3 On-line Analysis Results	p. 365
19.5 Conclusions	p. 366
References	p. 367
Index	p. 369

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