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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.
IV. NONADAPTIVE FUZZY CONTROL.
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.
V. ADAPTIVE FUZZY CONTROL.
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.
25. Advanced Adaptive Fuzzy Controllers I.State Boundedness By Supervisory Control. Parameter Boundedness By Projection. Summary and Further Readings. Exercises.
26. Advanced Adaptive Fuzzy Controllers II.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.
The field of fuzzy systems and control has been making rapid progresses during recent years. Motivated by the practical successes of fuzzy control in consumer products and industrial process control, there has been an increasing amount of work on the rigorous theoretical studies of fuzzy systems an d fuzzy control.
Researchers are trying to explain why the practical results are good, to systematize the existing approaches, and to develop more powerful ones. As a result of these efforts, the whole picture of a fuzzy systems and fuzzy control theory is becoming clearer. Although there are many books on fuzzy theory, most of them are either research monographs which concentrate on special topics, or collections of papers, or books on fuzzy mathematics. We desperately need a real textbook on fuzzy systems and control which provides the skeleton of the field and summarizes the fundamentals.
This book, which is based on a course developed at the Hong Kong University of Science and Technology, is intended as a textbook for graduate and senior students, and as a self-study book for practicing engineers.
When writing this book, we keep the following requirements in mind: Well- Structuredness: This book is not intended as a collection of existing results on fuzzy systems and fuzzy control; rather, we first establish the structure which a reasonable theory of fuzzy systems and fuzzy control should follow, and then fill in the details. For example, when studying fuzzy control systems, we should consider the stability, optimality, and robustness of the systems, and classify the approaches according to whether the plant is linear, nonlinear, or modeled by fuzzy systems.
Fortunately, the major existing results fit very well into this structure and are therefore covered in details in this book. Because the field is not mature as compared with other mainstream fields, there are holes in the structure for which no results exist. For these topics, we either provide our preliminary approaches, or point out that the problems are open.
Clearity and Preciseness: Clear and logic presentation is crucial for any book, especially for a book associated with the word ``fuzzy.'' Fuzzy theory itself is precise; the ``fuzziness'' appears in the phenomena that fuzzy theory tries to study. Once a fuzzy description (for example, ``hot day'') is formulated in terms of fuzzy theory, nothing will be fuzzy anymore. We pay special attention to use precise languages to introduce the concepts, to develop the approaches, and to justify the conclusions.
Practicality: We do not forget that the driving force for fuzzy systems and control is practical applications. Most approaches in this book are tested for problems which have practical significance. In fact, a main objective of the book is to teach the students or practicing engineers how to use the fuzzy systems approaches to solve engineering problems in control, signal processing, and communications.
Richness and Rigor: This book should be intelligently challenging for students. In addition to the emphasis on practicality, many theoretical results are given (which, of course, have practical relevance and importance). All the theorems and lemmas are proven in a mathematically rigorous fashion, and some efforts have to be taken for an average student to comprehend the details.
Easy to Use as A Textbook: To facilitate its use as a textbook, this book is written in such a style that each chapter is designed for a one-and-half hour lecture. Sometimes, three chapters may be covered by two lectures, or vice versa, depending upon the emphasis of the instructor and the background of the students. Each chapter is attached with some exercises and mini-projects which, as in many other textbooks, form an integrated part of the text.
The book is divided into six parts.
The first part (Chapters 2-6) introduces the fundamental concepts and principles in the general field of fuzzy theory which are particular useful in fuzzy systems and fuzzy control.
The second part (Chapters 7-11) studies the fuzzy systems in details. The operations inside the fuzzy systems are carefully analyzed and certain properties of the fuzzy systems (for example, approximation capability and accuracy) are studied.
The third part (Chapters 12-15) introduces four methods for designing fuzzy systems from sensory measurements, and all these methods are tested for a number of control, signal processing, or communication problems.
The fourth (Chapters 16-22) and fifth (Chapters 23-26) parts concentrate on fuzzy control, where Part IV studies nonadaptive fuzzy control and Part V studies adaptive fuzzy control.
Finally, Part VI (Chapters 27-31) reviews a number of topics which are not included in the main structure of the book, but are important and strongly relevant to fuzzy systems and fuzzy control.
The book can be studied in many ways, according to the particular interests of the instructor or the reader. Chapters 1-15 cover the general materials which can be applied to a variety of engineering problems. Chapters 16-26 are more specialized in control problems. If the course is not intended as a control course, then some materials in Chapters 16-26 may be omitted, and the time saved may be used for a more detailed coverage of Chapters 1-15 and 27-31. On the other hand, if it is a control course, then Chapters 16- 26 should be studied in details. The book can also be used, together with a book on neural networks, for a course on neural networks and fuzzy systems. In this case, Chapters 1-15 and a selected topics from Chapters 16-31 may be used for the fuzzy system half of the course. If a practicing engineer wants to learn fuzzy systems and fuzzy control quickly, then the proofs of the theorems and lemmas may be skipped. This book has benefited from many colleagues, students, and friends. First of all, I would like thank my advisors Lotfi Zadeh and Jerry Mendel for their continued encouragements. I would like to thank Karl {\AA str\"{o m for sending his student Mikael Johansson to help me to prepare the manuscript during the summer of 1995. Discussions with Kevin Passino, Frank Lewis, Jyh-Shing Jang, Hua Wang, Hideyuki Takagi, John Yen and other researchers in fuzzy theory have helped the organization of the materials. The book also benefited from the students who took the course at HKUST. The supports for the author from the Research Grants Council of Hong Kong are greatly appreciated. Finally, I would like to express my gratitude to my Department at HKUST for providing the nice research and teaching environment. Especially, I would like to thank my colleagues Xiren Cao, Zexiang Li, Li Qiu, Erwei Bai, Justin Chuang, Philip Chan and Kwan-Fai Cheung for their collaboration and critical remarks on various topics in fuzzy theory. Li-Xin Wang The Hong Kong University of Science and Technology