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Chapter 13
Hybrid Systems: Fuzzy Neural Integration

I. Burhan Türks,en


CONTENTS

1. Introduction
1.1. Knowledge Base
1.2. Inference Engine
2. Membership Functions
2.1. Meaning of Membership
2.2. Grade of Membership
2.3. Fuzzy Clustering
2.4. Modification of FCM Algorithm
2.5. Input Selection
2.6. Fuzzified Neural Network
2.6.1. Input-Output Relation
    2.6.2. An Example
    2.6.3. Learning Algorithm
3. Unified Fuzzy Model
3.1. Approximate Reasoning
3.2. Crisp Connectives of Fuzzy Theory
3.3. Implication and Aggregation
3.4. Inference with a Rule Set
3.4.1. Defuzzification
4. Case Study
4.1. A Nonlinear System
4.2. Gas Furnace Model
4.3. Industrial Process Model
5. Research Issues
References

1. INTRODUCTION

In this chapter, hybrid system model developments are discussed with a perspective of fuzzy neural integration. In particular, a fuzzy system is a set of rules that maps inputs to outputs. The set of rules defines a many-to-many map F: X --> Y. The set consists of linguistic "IF ... THEN" rules in the form of "IF X isr A THEN Y isr B," where X and Y are input and output base variables, respectively, of a system, and A and B are corresponding fuzzy sets that contain the elements x [element of] X and y [element of] Y to some degree (Zadeh, 1965); that is:

A = {(x, µA(x))| x [element of] X, µA(x) [element of] [0,1]} and B = {(y, µB(y))| y [element of] Y, µB(y) [element of] [0,1]}

and "isr" is a shorthand notation that may stand for "is contained in," "is a member of," "is compatible with," "belongs to," "is a possible member of ," etc., depending on the context (Zadeh, 1996).

Fuzzy rules represent our approximate knowledge extracted from a systems input-output analysis and identified by combinations of membership functions in the input-output space of XxY. They define information granules (Zadeh, 1996) or subsets of the state space. Less certain, more imprecise rules specify large information granules. More certain and more precise rules define smaller information granules. In practice, however, we do not know the exact shape of the information granules, i.e., the membership functions or the structure of the rules that connect the inputs to outputs of a system. Hence, we ask an expert or implement supervised or unsupervised learning techniques to find and/or modify fuzzy information granules as close as we can for a given system with the available expertise and/or sample data. In this context, supervised means that we have a target membership functions, whereas unsupervised means we do not have a target membership function. In the supervised learning, the aim is to minimize the error between the target and the initial membership functions. In the unsupervised learning, the aim is to determine the initial membership function that can be obtained from a given raw data set. The best learning schemes ideally provide a good coverage of the optimal information granules of a given system's hidden-inherent rules. In current literature it is found that most adaptive fuzzy system models are built without an adequate knowledge of the systems goals. Hence, these are unsupervised learning schemes. This often creates the exponential complexity associated with many learning schemes.

Learning schemes may be either unsupervised clustering techniques (Bazdek, 1981; Sugeno and Yasukawa, 1993; Nakanishi, Türks¸en and Sugeno, 1993; Emani, Türks¸en, and Goldenberg, 1996) or supervised gradient decent methods (Ishibuchi, Morioka, and Türks¸en,1995). Unsupervised learning is faster but it produces a first level of approximation in knowledge representation. Usually, a good sample data set is satisfactory for this purpose. Supervised learning requires more knowledge of a system and/or a large sample data set in order to compute a more accurate representation and often requires orders of magnitude more training cycles. Learning schemes utilized in most neural network investigations are generally supervised schemes. Hence, we have to deal with computational complexity and/or various approximation heuristics.

Fuzzy rule bases have a natural explanatory power embeded within their structure and semantics, i.e., in their meaning representation via fuzzy set membership functions; whereas, neural networks are black-box approximators and have all the advantages or disadvantages that come from them. Both fuzzy systems and neural networks do not admit proof of concepts in traditional senses. The feedforward version of neural networks provides more control but less power than their feedback versions. However, we cannot be sure of functions and patterns that were learned and represented by them until a large number of input-output test cases have been tried and they have been verified and validated. On the other hand, fuzzy rules are linguistic representations that can be tuned and modified more readily.

Neural networks provide a distributed connectionist structure and hence cannot be easily decoded, but they are useful to tune membership functions of fuzzy sets or provide distributed computational speed after appropriate learning. However, most neural nets forget part of their learning each time they learn new patterns. In general, patterns are usually modified and changed after learning. Fuzzy neural systems usually learn with the same laws that a neural network learns. Such systems are usually called "fuzzy-neural" systems. In this sense, "fuzzy-neural" system models are fuzzy system models that have been subject to some supervised learning in their development.

Let us next review briefly two essential components of a fuzzy expert system: (1) a knowledge base consisting of a set of linguistic rules, and (2) an inference engine that executes an approximate reasoning algorithm.


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