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Fuzzy Neural Intelligent Systems

Mathematical Foundation and the Applications in Engineering

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## Book Description

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:

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.

## Table of Contents

FOUNDATION OF FUZZY SYSTEMS

Definition of Fuzzy Sets

Basic Operations of Fuzzy Sets

The Resolution Theorem

A Representation Theorem

Extension Principles

References

DETERMINATION OF MEMBERSHIP FUNCTIONS

A General Method for Determining Membership Functions

The Three-Phase Method

The Incremental Method

The Multiphase Fuzzy Statistical Method

The Method of Comparisons

The Absolute Comparison Method

The Set-Valued Statistical Iteration Method

Ordering by Precedence Relations

The Relative Comparison Method and the Mean Pairwise Comparison Method

References

MATHEMATICAL ESSENCE AND STRUCTURES OF FEEDFORWARD ARTIFICIAL NEURAL NETWORKS

Introduction

Mathematical Neurons and Mathematical Neural Networks

The Interpolation Mechanism of Feedforward Neural Networks

A Three-Layer Feedforward Neural Network with Two Inputs, One Output

Analysis of Steepest Descent Learning Algorithms of Feedforward Neural Networks

Feedforward Neural Networks with Multi-Input One Output and Their Learning Algorithm

Feedforward Neural Networks with One Input Multi-Output and Their Learning Algorithm

Feedforward Neural Networks with Multi-Input Multi-Output and Their Learning Algorithm

A Note on the Learning Algorithm of Feedforward Neural Networks

Conclusions

References

FUNCTIONAL-LINK NEURAL NETWORKS AND VISUALIZATION MEANS OF SOME MATHEMATICAL METHODS

Discussion of the XOR Problem

Mathematical Essence of Functional-Link Neural Networks

A Visualization Means of Some Mathematical Methods

Neural Network Representation of Linear Programming

Neural Network Representation of Fuzzy Linear Programming

Conclusions

References

FLAT NEURAL NETWORKS AND RAPID LEARNING ALGORITHMS

Introduction

The Linear System Equation of the Functional-Link Network

Pseudoinverse and Stepwise Updating

Training with Weighted Least Square

Refine the Model

Time-Series Applications

Examples and Discussion

Conclusions

References

BASIC STRUCTURE OF FUZZY NEURAL NETWORKS

Definition of Fuzzy Neurons

Fuzzy Neural Networks

A Fuzzy d Learning Algorithm

The Convergence of Fuzzy d Learning Rule

Conclusions

References

MATHEMATICAL ESSENCE AND STRUCTURES OF FEEDBACK NEURAL NETWORKS AND WEIGHT MATRIX DESIGN

Introduction

A General Criterion on the Stability of Networks

Generalized Energy Function

Learning Algorithm of Discrete Feedback Neural Networks

Design Method of Weight Matrices Based on Multifactorial Functions

Conclusions

References

GENERALIZED ADDITIVE MULTIFACTORIAL FUNCTION AND ITS APPLICATIONS TO FUZZY INFERENCE AND NEURAL NETWORKS

Introduction

On Multifactorial Functions

Generalized Additive Weighted Multifactorial Functions

Infinite Dimensional Multifactorial Functions

M (-,T) and Fuzzy Integral

Application in Fuzzy Inference

Conclusions

References

THE INTERPOLATION MECHANISM OF FUZZY CONTROL

Preliminary

The Interpolation Mechanism of Mamdanian Algorithm with One Input and One Output

The Interpolation Mechanism of Mamdanian Algorithm with Two Inputs and One Output

A Note on Completeness of Inference Rules

The Interpolation Mechanism of (+, o)-Centroid Algorithm

The Interpolation Mechanism of Simple Inference Algorithm

The Interpolation Mechanism of Function Inference Algorithm

A General Fuzzy Control Algorithm

Conclusions

References

THE RELATIONSHIP BETWEEN FUZZY CONTROLLERS AND PID CONTROLLERS

Introduction

The Relationship of Fuzzy Controllers with One Input One Output and P Controllers

The Relationship of Fuzzy Controllers with Two Inputs One Output and PD (or PI) Controllers

The Relationship of Fuzzy Controllers with Three Inputs One Output and PID Controllers

The Difference Schemes of Fuzzy Controllers with Three Inputs and One Output

Conclusions

References

ADAPTIVE FUZZY CONTROLLERS BASED ON VARIABLE UNIVERSES

The Monotonicity of Control Rules and the Monotonicity of Control Functions

The Contraction-Expansion Factors of Variable Universes

The Structure of Adaptive Fuzzy Controllers Based on Variable Universes

Adaptive Fuzzy Controllers with One Input and One Output

Adaptive Fuzzy Controllers with Two Inputs and One Output

Conclusions

References

THE BASICS OF FACTOR SPACES

What are "Factors"?

The State Space of Factors

Relations and Operations of Factors

Axiomatic Definition of Factor Spaces

A Note on the Definition of Factor Spaces

Concept Description in a Factor Space

The Projection and Cylindrical Extension of the Representation Extension

Some Properties of the Projection and Cylindrical Extension

Factor Sufficiency

The Rank of a Concept

Atomic Factor Spaces

Conclusions

References

NEURON MODELS BASED ON FACTOR SPACES THEORY AND FACTOR SPACE CANES

Neuron Mechanism of Factor Spaces

The Models of Neurons without Respect to Time

The Models of Neurons Concerned with Time

The Models of Neurons Based in Variable Weights

Naïve Thoughts of Factor Space Canes

Melon-Type Factor Space Canes

Chain-Type Factor Space Canes

Switch Factors and Growth Relations

Class Partition and Class Concepts

Conclusions

References

FOUNDATION OF NEURO-FUZZY SYSTEMS AND AN ENGINEERING APPLICATION

Introduction

Takagi, Sugeno, and Kang Fuzzy Model

Adaptive Network-Based Fuzzy Inference System (ANFIS)

Hybrid Learning Algorithm for ANFIS

Estimation of Lot Processing Time in an IC Fabrication

Conclusions

References

DATA PREPROCESSING

Introduction

Data Preprocessing Algorithms

Conclusions

Appendix: MATLAB® Programs

References

CONTROL OF A FLEXIBLE ROBOT ARM USING A SIMPLIFIED FUZZY CONTROLLER

Introduction

Modeling of the Flexible Arm

Simplified Fuzzy Controller

Self-Organizing Fuzzy Control

Simulation Results

Conclusions

References

APPLICATION OF NEURO-FUZZY SYSTEMS: DEVELOPMENT OF A FUZZY LEARNING DECISION TREE AND APPLICATION TO TACTILE RECOGNITION

Introduction

Tactile Sensors and a Tactile Sensing and Recognition System

Development of a Fuzzy Learning Decision Tree

Experiments

Conclusions

References

FUZZY ASSESSMENT SYSTEMS OF REHABILITATIVE PROCESS FOR CVA PATIENTS

Introduction

COP Signals Feature Extraction

Relationship between COP Signals and FIM Scores

Construction of Kinetic State Assessment System

Results of Kinetic State Assessment System

Conclusions

References

A DSP-BASED NEURAL CONTROLLER FOR A MULTI-DEGREE PROSTHETIC HAND

Introduction

EMG Discriminative System

DSP-Based Prosthetic Controller

Implementation and Results of the DSP-Based Controller

Conclusions

References

INDEX