Michael Reinfrank

Stability of Fuzzy Control Systems

FKBC has been proven to be a powerful tool when applied to the control of processes which are not amenable to conventional, analytic design techniques. The design of most of the existing FKBC has relied mainly on the process operator’s or control engineer’s experience based heuristic knowledge. Hence, the controller’s performance is very much dependent on how good this expertise is. Thus, from the control engineering point of view, the major effort in fuzzy knowledge based control has been devoted to the development of particular FKBC for specific applications rather than to general analysis and design methodologies for coping with the dynamic behavior of control loops. The development of such methodologies is of primary interest for control theory and engineering. In particular, stability analysis is of extreme importance, and the lack of satisfactory formal techniques for studying the stability of process control systems involving FKBC has been considered a major drawback of FKBC.

Logical Foundations of Nonmonotonic Truth Maintencance

Nonmonotonic truth maintenance systems and nonmonotonic logics have now been coexisting for more than a decade. I survey some recent results on the relationships between these two fields. I discuss what has been achieved so far and suggest some problems that demand further investigation.

The Mathematics of Fuzzy Control

One of the major developments of fuzzy set theory, fuzzy logic was primarily designed to represent and reason with some particular form of knowledge. It was assumed that the knowledge would be expressed in a linguistic or verbal form, and also that the whole exercise should not be a mere intellectual undertaking, but must also be operationally powerful so that computers can be used. However, when using a language-oriented approach for representing knowledge about a certain system of interest, one is bound to encounter a number of nontrivial problems. Suppose, for example [119], that you are asked how strongly you agree that a given number x ∈ [0, 20] is a large number. One way to answer this question is to say that if x ≥ d then you agree it is a large number and if x < d then you disagree. Thus, if you place a mark on an agree—disagree scale, it might be distributed uniformly over the right half of the scale whenever x ≥ d and uniformly over the left half if x < d. Such a person is called a threshold person.

Adaptive Fuzzy Control

Most of the real-world processes that require automatic control are nonlinear in nature. That is, their parameter values alter as the operating point changes, over time, or both. As conventional control schemes are linear, a controller can only be tuned to give good performance at a particular operating point or for a limited period of time. The controller needs to be retuned if the operating point changes, or retuned periodically if the process changes with time. This necessity to retune has driven the need for adaptive controllers that can automatically retune themselves to match the current process characteristics. An excellent introduction to “conventional” adaptive control systems is by Astrom [8].

FKBC Design Parameters

This chapter introduces the principal design parameters of a FKBC. These include scaling factors, fuzzification and defuzzification methods, rule base and membership function construction and representation. Furthermore, we discuss the relevance of the different design parameters with respect to the performance of a FKBC. Different design options for particular parameters are presented. Choice of membership functions, defuzzification methods, inference engine, and form and meaning of rules are considered in detail. Design parameters like fuzzification, and checking the properties of the rule base are only mentioned, since these were already described in Chapter 2. Other design parameters like scaling factors, derivation of rules, tuning of the FKBC, and stability analysis, are only informally discussed or just mentioned. Thus, this chapter gives a general view of the FKBC design problem and prepares the reader for the formal treatment and/or presentation of systematic techniques for rule derivation and scaling factors determination (Chapter 4), FKBC tuning and adaptation (Chapter 5), and stability analysis (Chapter 6).

Nonlinear Fuzzy Control

The analytic functions employed in models of linear and nonlinear systems (processes) operate on the domain of crisp (point-wise) reals. In addition, we have the class of fuzzy systems whose models, in general, are algebraic mappings from the domain of crisp reals into a prespecified domain of fuzzy (set-wise defined) reals.