 # Course In Fuzzy Systems and Control, A, 1st edition

• LiXin Wang

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## Overview

Provides a comprehensive, self-tutorial course in fuzzy logic and its increasing role in control theory.KEY TOPICS:The book answers key questions about fuzzy systems and fuzzy control. It introduces basic concepts such as fuzzy sets, fuzzy union, fuzzy intersection and fuzzy complement. Learn about fuzzy relations, approximate reasoning, fuzzy rule bases, fuzzy inference engines, and several methods for designing fuzzy systems.MARKET:For professional engineers and students applying the principles of fuzzy logic to work or study in control theory.

1. Introduction.

Why Fuzzy Systems? What Are Fuzzy Systems? Where Are Fuzzy Systems Used and How? What Are the Major Research Fields in Fuzzy Theory? A Brief History of Fuzzy Theory and Applications. Summary and Further Readings. Exercises.

I. THE MATHEMATICS OF FUZZY SYSTEMS AND CONTROL.

2. Fuzzy Sets and Basic Operations on Fuzzy Sets.

From Classical Sets to Fuzzy Sets. Basic Concepts Associated with A Fuzzy Set. Operations on Fuzzy Sets. Summary and Further Readings. Exercises.

3. Further Operations on Fuzzy Sets.

Fuzzy Complement. Fuzzy Union —- The S-Norms. Fuzzy Intersection —- The T-Norms. Averaging Operators. Summary and Further Readings. Exercises.

4. Fuzzy Relations and the Extension Principle.

From Classical Relations to Fuzzy Relations. Compositions of Fuzzy Relations. The Extension Principle. Summary and Further Readings. Exercises.

5. Linguistic Variables and Fuzzy IF-THEN Rules.

From Numerical Variables to Linguistic Variables. Linguistic Hedges. Fuzzy IF-THEN Rules. Summary and Further Readings. Exercises.

6. Fuzzy Logic and Approximate Reasoning.

From Classical Logic to Fuzzy Logic. The Compositional Rule of Inference. Properties of the Implication Rules. Summary and Further Readings. Exercises.

II. FUZZY SYSTEMS AND THEIR PROPERTIES.

7. Fuzzy Rule Base and Fuzzy Inference Engine.

Fuzzy Rule Base. Fuzzy Inference Engine. Summary and Further Readings. Exercises.

8. Fuzzifiers and Defuzzifiers.

Fuzzifiers. Defuzzifiers. Summary and Further Readings. Exercises.

9. Fuzzy Systems as Nonlinear Mappings.

The Formulas of Some Classes of Fuzzy Systems. Fuzzy Systems As Universal Approximators. Summary and Further Readings. Exercises.

10. Approximation Properties of Fuzzy Systems I.

Preliminary Concepts. Design of A Fuzzy System. Approximation Accuracy of the Fuzzy System. Summary and Further Readings. Exercises.

11. Approximation Properties of Fuzzy Systems II.

Fuzzy Systems with Second-Order Approximation Accuracy. Approximation Accuracy of Fuzzy Systems with Maximum Defuzzifier. Summary and Further Readings. Exercises.

III. DESIGN OF FUZZY SYSTEMS FROM INPUT-OUTPUT DATA.

12. Design of Fuzzy Systems Using A Table Look-Up Scheme.

A Table Look-Up Scheme for Designing Fuzzy Systems from Input- Output Pairs. Application to Truck Backer-Upper Control. Application to Time-Series Prediction. Summary and Further Readings. Exercises and Projects.

13. Design of Fuzzy Systems Using Gradient Descent Training.

Choosing the Structure of Fuzzy Systems. Designing the Parameters by Gradient Descent. Application to Nonlinear Dynamic System Identification. Summary and Further Readings. Exercises and Projects.

14. Design of Fuzzy Systems Using Recursive Least Squares.

Design of the Fuzzy System. Derivation of the Recursive Least Squares Algorithm. Application to Equalization of Nonlinear Communication Channels.

15. Design of Fuzzy Systems Using Clustering.

An Optimal Fuzzy System. Design of Fuzzy Systems By Clustering. Application to Adaptive Control of Nonlinear Systems. Summary and Further Readings. Exercises and Projects.

16. The Trial-and-Error Approach to Fuzzy Controller Design.

Fuzzy Control Versus Conventional Control. The Trial-and-Error Approach to Fuzzy Controller Design. Case Study I: Fuzzy Control of A Cement Kiln. Case Study II: Fuzzy Control of A Wastewater Treatment Process. Summary and Further Readings. Exercises.

17. Fuzzy Control of Linear Systems I: Stable Controllers.

Stable Fuzzy Control of Single-Input-Single-Output Systems. Stable Fuzzy Control of Multi-Input-Multi-Output Systems. Summary and Further Readings. Exercises.

18. Fuzzy Control of Linear Systems II: Optimal and Robust Controllers.

Optimal Fuzzy Control. Robust Fuzzy Control. Summary and Further Readings. Exercises.

19. Fuzzy Control of Nonlinear Systems I: Sliding Control.

Fuzzy Control As Sliding Control: Analysis. Fuzzy Control As Sliding Control: Design. Summary and Further Readings. Exercises.

20. Fuzzy Control of Nonlinear Systems II: Supervisory Control.

Multi-level Control Involving Fuzzy Systems. Stable Fuzzy Control Using Nonfuzzy Supervisor. Gain Scheduling of PID Controller Using Fuzzy Systems. A Fuzzy System for Turning the PID Gains. Summary and Further Readings. Exercises.

21. Fuzzy Control of Fuzzy System Models.

The Takagi-Sugeno-Kang Fuzzy Systems. Closed-Loop Dynamics of Fuzzy Model with Fuzzy Controller. Stability Analysis of the Dynamic TSK Fuzzy System. Design of Stable Fuzzy Controllers for the Fuzzy Model. Summary and Further Readings. Exercises.

22. Qualitative Analysis of Fuzzy Control and Hierarchical Fuzzy Systems.

Phase Plane Analysis of Fuzzy Control Systems. Robustness Indices for Stability. Hierarchical Fuzzy Control. Summary and Further Readings. Exercises.

23. Basic Adaptive Fuzzy Controllers I.

Classification of Adaptive Fuzzy Controllers. Design of Indirect Adaptive Fuzzy Controller. Application to Inverted Pendulum Tracking Control. Summary and Further Readings. Exercises.

24. Basic Adaptive Fuzzy Controllers II.

Design of Direct Adaptive Fuzzy Controller. Design of Combined Direct/Indirect Adaptive Fuzzy Controller. Summary and Further Readings. Exercises.

State Boundedness By Supervisory Control. Parameter Boundedness By Projection. Summary and Further Readings. Exercises.

Stable Indirect Adaptive Fuzzy Control System. Adaptive Fuzzy Control of General Nonlinear Systems. Intuitive Concepts of Input-Output Linearization. Summary and Further Readings. Exercises.

VI. MISCELLANEOUS TOPICS.

27. The Fuzzy C-Means Algorithm.

Why Fuzzy Models for Pattern Recognition? Hard and Fuzzy c-Partitions. Hard and Fuzzy c-Means Algorithms. Convergence of the Fuzzy c-Means Algorithm. Summary and Further Readings. Exercises.

28. Fuzzy Relation Equations.

Introduction. Solving the Fuzzy Relation Equations. Solvability Indices of the Fuzzy Relation Equations. Summary and Further Readings. Exercises.

29. Fuzzy Arithmetic.

Fuzzy Numbers and the Decomposition Theorem. Addition and Subtraction of Fuzzy Numbers. Multiplication and Division of Fuzzy Numbers. Fuzzy Equations. Fuzzy Ranking. Summary and Further Readings. Exercises.

30. Fuzzy Linear Programming.

Classification of Fuzzy Linear Programming Problems. Linear Programming with Fuzzy Resources. Linear Programming with Fuzzy Objective Coefficients. Linear Programming with Fuzzy Constraint Coefficients. Comparison of Stochastic and Fuzzy Linear Programming. Summary and Further Readings. Exercises.

31. Possibility Theory.

Introduction. The Intuitive Approach to Possibility. The Axiomatic Approach to Possibility. Possibility versus Probability. Summary and Further Readings. Exercises.