11R19. Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems. - Guanrong Chen (Univ of Houston, Houston TX) and Trung Tat Pham (Univ of Houston, Clear Lake, Houston TX). CRC Press LLC, Boca Raton FL. 2001. 316 pp. ISBN 0-8493-1658-8. $89.95.

Reviewed by NM Boustany (GM Tech Center, General Motors Energy Center, Eng Bldg, 30200 Mound Rd, 480-111-S31, Troy MI 48090).

In one of the debates of the 2000 presidential campaign, then Candidate George W Bush described Al Gore’s analysis of his tax refund proposals as Fuzzy Math. It is not clear whether or not Mr Bush was also taking a jab at the field, or even whether or not he was aware of its existence. At any rate, the popular press as well as TV and radio news shows made a lot of hay with the expression: Fuzzy Math, to the extent that Professor Bart Kosko (a major contributor to the field and author of the popular bestseller Fuzzy Thinking, Hyperion Press, NY, 1993) felt compelled to write a short editorial on the subject for the New York Times that same week.

The chances are that control theorists of this reviewer’s generation have had to study fuzzy analysis several years after graduating. To this crop of control engineers, fuzzy thinking required quite a paradigm shift. Therefore, the field was not readily accepted and was received, in the early years, with considerable skepticism; all this despite the write-ups in Time Magazine and all the touted successes by Japanese and Korean engineers in their implementations of fuzzy systems. Upon closer scrutiny, these successes did not appear to constitute definitive proof of the superiority of these approaches over the more classical ones. The detractors would always point to a sensor or actuator that had not been used in earlier implementations relying on more conventional techniques.

In addition, the claims that the methodology is mathematical model free seemed often exaggerated. Many reports on implementations relied on models for validation. Needless to say, the early years of fuzzy analysis witnessed a polarization of the control community. In recent years, this feud has considerably abated. This could be due to numerous factors such as the Old Guard reaching retirement and the wider acceptance of techniques with the promise of producing “intelligence” and “learning” in control systems.

Fuzzy analysis builds on fuzzy logic, which extends the classical logic handed down to us from the early days of Western thinking by Aristotle. In classical logic, something is true or false; there is no in between. An element either belongs to set or to its complement. This “black or white” of classical logic has led to paradoxes. Fuzzy thinking, on the other hand, introduces degrees of grayness, or degrees of belonging to a set. These gray scales have provided possible resolutions to these paradoxes. Quite often these features of fuzzy thinking are compared and likened to elements in Eastern philosophy. This is often cited as the reason why fuzzy thinking has found wider acceptance in the East.

Chapters 1 and 2 of this book lay the foundations. After introducing fuzzy logic and fuzzy set theory in Chapter 1, some results from measure theory are presented. The section on measure theory makes for difficult reading and could be relegated to an appendix in future editions. Interval arithmetic is then introduced, and many of the results on interval calculus are presented. This chapter is straightforward mathematically, but is nevertheless tedious to work through. The examples at the end do a good job of clarifying the theory. Chapter 2 takes the reader from classical logic to fuzzy logic via 2-valued and n-valued logic. Again, the examples at the end of the chapter do the reader a great service.

Chapter 3 builds on the foundations of Chapters 1 and 2 and develops the idea of fuzzy models moving from static models to dynamic models. The notion of least square parameter identification is extended to fuzzy models. In Chapter 4, fuzzy control is introduced beginning with a discussion of programable logic controllers. This provides a good starting point for the ensuing discussions on model-free and model-based fuzzy control methods. In Chapter 5, PID control is extended to the fuzzy case. Chapter 6 builds on the optimal parameter identification techniques developed earlier and extends notions from adaptive control to the fuzzy case. Chapter 7 discusses several case studies in detail.

In some cases, the mathematical development is tedious, especially the one on interval calculus which is crucial to the understanding of the rest of the text. This could be laid out in a more user-friendly way. Also, some ideas are presented without much motivation. One example is the discussion on defuzzification. Various alternatives for defuzzification are presented without much discussion on what these are attempting to do or why one would choose one over the other.

The above criticisms point to minor shortcomings that are relatively easy to amend in future editions. Overall, the text is very well written and provides a rigorous analytical approach to fuzzy systems. The topics are laid out in a logical sequence where later chapters build on the ideas of the earlier ones. Solved examples at the end of the chapters do a good job of clarifying the concepts in the body of these chapters. This reviewer recommends Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems either as a textbook on fuzzy control or as a companion reference for a more general course on intelligent control. The book also belongs on the shelves of engineering libraries of both industry and academia.