This fuzzy system for computing a value of f(x) is relatively straightforward. It allows for the use of linguistic rules, it is easy to program on a computer, and it allows for an accurate representation of the function f(x). Figure 1.7 shows the function g(x) as defined by the fuzzy system described above. Notice that the function is not described by a series of steps as was the case with traditional production rules.

Figure 1.6  Deffuzzification involves taking a weighted average of the truncated membership function values prescribed by applicable rules to compute a crisp value of f(x).

Figure 1.7  g(x) is an approximation of f(x) using five fuzzy production rules.

1.5 Review - Preview

In this chapter we have attempted to do a few things. First, we gave a general synopsis of the reasons we chose fuzzy mathematics to serve as the core of our adaptive process control systems. For the most part, we felt the complex mathematics associated with traditional control schemes was unnecessary. Next, we provided some background as to how we came to view fuzzy systems as extensions of traditional expert systems. Finally, we provided an example of an instance in which a traditional production rule-based system was inadequate to solve a problem that could easily be solved using fuzzy production rules. The example provided was intentionally simple, but necessary because all of the complex applications that appear in this book are based on the fundamentals set forth in this chapter.

The development of the simple system that was provided quite possibly raised a number of questions for the reader. For instance, why were the membership functions Gaussian? How were the membership functions chosen? Are there other ways to take weighted averages? How does one handle more than one input? More than one output? How do I build one of these for a real problem? Where is the “adaptive” part? Unfortunately, it takes a good portion of this book to answer all of these questions (we don’t get to the adaptive part until the second section, and we continue to build fuzzy systems all of the way through). However, we can begin to answer some of the questions, and we begin to do so in the next chapter. Chapter 2 presents a problem environment and an associated control problem. This problem environment is used to provide a step-by-step procedure for building fuzzy control systems; one that is used throughout the remainder of the book.

References

Davis, B. E., Jordan, C. E. and Stanley, D. A. (1990). Expert advisor for phosphate flotation. Control ’90 - Society for Mining, Metallurgy, and Exploration, 77–85.

Holmblad, L. P., and Ostergaard, J. J. (1982). Control of a cement kiln by fuzzy logic. In Fuzzy Information and Decision Processes (Eds. M. M. Gupta and E. Sanchez). Amsterdam: North-Holland Publishing Company, 389–399.

Larsen, P., M. (1980). Industrial applications of fuzzy logic control. International Journal of Man-Machine Studies, 12, 255–283.

Mamdani, E. H. (1981). A fuzzy rule-based method of controlling dynamic processes. Proceedings of the 20th IEEE Conference on Decision and Control, December, San Diego, CA.

Mamdani, E. H., and Assilian, S. (1975). An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man-Machine Studies, 21, 213–227.

Mamdani, E. H., and Pappis, C. P. (1977). A fuzzy logic controller for a traffic junction. IEEE Transactions on Systems, Man, and Cybernetics, SMC-7, 707–717.

Melama, H., Barker, D., Hirvonen, M., and Hartikainen, J. (1987). Experiences gained from using an expert system approach on process management and control applications. Proceedings of the Artificial Intelligence in Minerals and Materials Technology Conference, October, Tuscaloosa, AL, 66–94.

Procyk, T. J., and Mamdani, E. H. (1979). A linguistic self-organizing process controller. Automatica, 15(1), 15–30.

Tanimoto, S. L. (1987). The elements of artificial intelligence. Rockville, MD: Computer Science Press.

Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8, 338–353.

Zadeh, L. A. (1968). Probability measures of fuzzy events. Journal of Mathematical Analysis and Applications, 23, 421–427.

Zadeh, L. A. (1971a). Quantitative fuzzy semantics. Information Sciences, 3, 159–176.

Zadeh, L. A. (1971b). Towards a theory of fuzzy systems. In: Aspects of Networks and Systems Theory (Eds., R. E. Kalman and R. N. deClairis), New York: Holt, Rinehart & Winston, 469–490.