Fuzzy Chaotic Systems

2 Fuzzy Logic and Fuzzy Control (Sp. 13-14)

Over the past few decades, there has developed a tremendous amount of literature on the theory of fuzzy set and fuzzy control. This chapter attempts to sketch the contours of fuzzy logic and fuzzy control for the readers, who may have no knowledge in this field, with easy-to-understand words, avoiding abstruse and tedious mathematical formulae.

2.1 Introduction

Fuzzy logic is in nature an extension of conventional (Boolean) logic to handle the concept of partial truth – truth values between “completely true” and “completely false”. It was introduced by Lotfi Zadeh in 1965 as a means to model the uncertainty of natural language.

Fuzzy logic in the broad sense, which has been better known and extensively applied, serves mainly as a means for fuzzy control, analysis of vagueness in natural language and several other application domains. It is one of the techniques of soft-computing, i.e. computational methods tolerant to suboptimality and impreciseness (vagueness) and giving quick, simple and sufficiently good solutions [48, 49, 50, 51].

Fuzzy logic in the narrow sense is a symbolic logic with a comparative notion of truth developed fully in the spirit of classical logic (syntax, semantics, axiomatization, truth-preserving deduction, completeness, and propositional and predicate logic). It is a branch of many-valued logic based on the paradigm of inference under vagueness. This fuzzy logic is a relatively young discipline, not only serving as a foundation for the fuzzy logic in a broad sense but also of independent logical interest, since it turns out that strictly logical investigation of this kind of logical calculi can go rather far [52, 53, 54, 55]. Fuzzy logic has several unique features that make it a particularly good choice for many control problems:

i) It is inherently robust, since it does not require precise, noise-free inputs and can, thus, be fault-tolerant if a feedback sensor quits or is destroyed. The output control is a smooth control function despite a wide range of input variations.

ii) Since a fuzzy logic controller processes user-defined rules governing the target control system, it can be modified and tweaked easily to improve or drastically alter system performance. New sensors can easily be incorporated into the system simply by generating appropriate governing rules.

iii) Fuzzy logic is not limited to a few feedback inputs and one or two control outputs, nor is it necessary to measure or compute rate-of-change of parameters. It is sufficient with any sensor data to provide some indication of a system’s actions and reactions. This allows the sensors to be inexpensive and imprecise thus keeping overall system cost and complexity low.

iv) Because of the rule-based operation, any reasonable number of inputs can be processed (1–8 or more) and numerous outputs (1–4 or more) generated, although defining a rule-base quickly becomes complex if too many inputs and outputs are chosen for a single implementation, since rules defining their interrelations must be defined, too. Then, it would be better to break the control system into smaller chunks and use several smaller fuzzy logic controllers distributed on the system, each one with more limited responsibilities.

v) Fuzzy logic can control nonlinear systems that would be difficult or impossible to model mathematically. This opens doors for control systems that would normally be deemed unfeasible for automation. In summery, fuzzy logic was conceived as a better method for sorting and handling data, and has proven to be an excellent choice for many control system applications, since it mimics human control logic. It can be built into anything from small, hand-held products to large computerized process control systems. It uses an imprecise but very descriptive language to deal with input data more like a human operator. It is very robust and forgiving of operator and data input, and often works when first implemented with little or no tuning.