Subject:  What is fuzzy control?
The purpose of control is to influence the behavior of a system by changing an input or inputs to that system according to a rule or set of rules that model how the system operates. The system being controlled may be mechanical, electrical, chemical or any combination of these.
Classic control theory uses a mathematical model to define a relationship that transforms the desired state (requested) and observed state (measured) of the system into an input or inputs that will alter the future state of that system.
reference----->0------->( SYSTEM ) -------+----------> output
+--------( MODEL )<--------+feedback
The most common example of a control model is the PID (proportional-integral- derivative) controller. This takes the output of the system and compares it with the desired state of the system. It adjusts the input value based on the difference between the two values according to the following equation.
output = A.e + B.INT(e)dt + C.de/dt
Where, A, B and C are constants, e is the error term, INT(e)dt is the integral of the error over time and de/dt is the change in the error term.
The major drawback of this system is that it usually assumes that the system being modelled in linear or at least behaves in some fashion that is a monotonic function. As the complexity of the system increases it becomes more difficult to formulate that mathematical model.
Fuzzy control replaces, in the picture above, the role of the mathematical model and replaces it with another that is build from a number of smaller rules that in general only describe a small section of the whole system. The process of inference binding them together to produce the desired outputs.
That is, a fuzzy model has replaced the mathematical one. The inputs and outputs of the system have remained unchanged.
The Sendai subway is the prototypical example application of fuzzy control.
Yager, R.R., and Zadeh, L. A., "An Introduction to Fuzzy Logic Applications in Intelligent Systems", Kluwer Academic Publishers, 1991.
Dimiter Driankov, Hans Hellendoorn, and Michael Reinfrank, "An Introduction to Fuzzy Control", Springer-Verlag, New York, 1993. 316 pages, ISBN 0-387-56362-8. [Discusses fuzzy control from a theoretical point of view as a form of nonlinear control.]
C.J. Harris, C.G. Moore, M. Brown, "Intelligent Control, Aspects of Fuzzy Logic and Neural Nets", World Scientific. ISBN 981-02-1042-6.
T. Terano, K. Asai, M. Sugeno, editors, "Applied Fuzzy Systems", translated by C. Ascchmann, AP Professional. ISBN 0-12-685242-1.
Subject:  What are fuzzy numbers and fuzzy arithmetic?
Fuzzy numbers are fuzzy subsets of the real line. They have a peak or plateau with membership grade 1, over which the members of the universe are completely in the set. The membership function is increasing towards the peak and decreasing away from it.
Fuzzy numbers are used very widely in fuzzy control applications. A typical case is the triangular fuzzy number
1.0 + +
| / \
| / \
0.5 + / \
| / \
| / \
| | |
5.0 7.0 9.0
which is one form of the fuzzy number 7. Slope and trapezoidal functions are also used, as are exponential curves similar to Gaussian probability densities.
For more information, see:
Dubois, Didier, and Prade, Henri, "Fuzzy Numbers: An Overview", in Analysis of Fuzzy Information 1:3-39, CRC Press, Boca Raton, 1987.
Dubois, Didier, and Prade, Henri, "Mean Value of a Fuzzy Number", Fuzzy Sets and Systems 24(3):279-300, 1987.
Kaufmann, A., and Gupta, M.M., "Introduction to Fuzzy Arithmetic", Reinhold, New York, 1985.
Subject:  Isn't "fuzzy logic" an inherent contradiction?
Why would anyone want to fuzzify logic?
Fuzzy sets and logic must be viewed as a formal mathematical theory for the representation of uncertainty. Uncertainty is crucial for the management of real systems: if you had to park your car PRECISELY in one place, it would not be possible. Instead, you work within, say, 10 cm tolerances. The presence of uncertainty is the price you pay for handling a complex system.
Nevertheless, fuzzy logic is a mathematical formalism, and a membership grade is a precise number. What's crucial to realize is that fuzzy logic is a logic OF fuzziness, not a logic which is ITSELF fuzzy. But that's OK: just as the laws of probability are not random, so the laws of fuzziness are not vague.