## Introduction to Neuro-Fuzzy SystemsFuzzy sets were introduced by Zadeh (1965) as a means of representing and manipulating data that was not precise, but rather fuzzy. Fuzzy logic pro vides an inference morphology that enables approximate human reasoning capabilities to be applied to knowledge-based systems. The theory of fuzzy logic provides a mathematical strength to capture the uncertainties associ ated with human cognitive processes, such as thinking and reasoning. The conventional approaches to knowledge representation lack the means for rep resentating the meaning of fuzzy concepts. As a consequence, the approaches based on first order logic and classical probablity theory do not provide an appropriate conceptual framework for dealing with the representation of com monsense knowledge, since such knowledge is by its nature both lexically imprecise and noncategorical. The developement of fuzzy logic was motivated in large measure by the need for a conceptual framework which can address the issue of uncertainty and lexical imprecision. Some of the essential characteristics of fuzzy logic relate to the following [242]. • In fuzzy logic, exact reasoning is viewed as a limiting case of ap proximate reasoning. • In fuzzy logic, everything is a matter of degree. • In fuzzy logic, knowledge is interpreted a collection of elastic or, equivalently, fuzzy constraint on a collection of variables. • Inference is viewed as a process of propagation of elastic con straints. • Any logical system can be fuzzified. There are two main characteristics of fuzzy systems that give them better performance für specific applications. |

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### Contents

1 Fuzzy systems | 1 |

12 Operations on fuzzy sets | 11 |

13 Fuzzy relations | 18 |

14 The extension principle | 26 |

15 The extension principle for nplace functions | 29 |

16 Metrics for fuzzy numbers | 39 |

17 Measures of possibility and necessity | 41 |

18 Fuzzy implications | 45 |

24 The generalized delta learning rule | 154 |

25 Effectivity of neural networks | 157 |

26 Winnertakeall learning | 160 |

27 Applications of artificial neural networks | 164 |

Bibliography | 169 |

3 Fuzzy neural networks | 171 |

32 Fuzzy neurons | 175 |

33 Hybrid neural nets | 184 |

19 Linguistic variables | 49 |

191 The linguistic variable Truth | 50 |

110 The theory of approximate reasoning | 53 |

111 An introduction to fuzzy logic controllers | 71 |

112 Defuzzification methods | 78 |

113 Inference mechanisms | 81 |

114 Construction of data base and rule base of FLC | 86 |

115 The ball and beam problem | 91 |

116 Aggregation in fuzzy system modeling | 95 |

117 Averaging operators | 98 |

118 Fuzzy screening systems | 109 |

119 Applications of fuzzy systems | 115 |

Bibliography | 119 |

2 Artificial neural networks | 133 |

22 The delta learning rule | 143 |

23 The delta learning rule with semilinear activation function | 149 |

34 Computation of fuzzy logic inferences by hybrid neural net | 195 |

35 Trainable neural nets for fuzzy IFTHEN rules | 201 |

36 Implementation of fuzzy rules by regular FNN of Type 2 | 208 |

37 Implementation of fuzzy rules by regular FNN of Type 3 | 212 |

38 Tuning fuzzy control parameters by neural nets | 216 |

39 Fuzzy rule extraction from numerical data | 224 |

310 Neurofuzzy classifiers | 228 |

311 FULLINS | 235 |

312 Applications of fuzzy neural systems | 240 |

Bibliography | 245 |

4 Appendix | 255 |

411 Tuning the membership functions | 259 |

42 Exercises | 262 |

287 | |

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### Common terms and phrases

activation function aggregation algorithm antecedent belongs to Class Carlsson Class C1 control action crisp criteria decision defined Definition defuzzified delta learning rule denoted derived desired output error error function Example Exercise extension principle firing level Fullér fuzzy control fuzzy control rules fuzzy expert system fuzzy IF-THEN rules fuzzy implication fuzzy inference fuzzy logic fuzzy logic controllers fuzzy neural network fuzzy partition fuzzy relation fuzzy rules Fuzzy Sets fuzzy subsets fuzzy systems hybrid neural hyperboxes i-th IEEE implication operator individual rule outputs input variable layer linear programming linguistic terms linguistic variable meaning elements membership function min{a monotonicity neural nets neuro-fuzzy neuron node o-level sets otherwise output interval OWA operator parameters perceptron portfolio value R.R. Yager real numbers rule base rule of inference Sets and Systems sigmoidal Step t-conorm t-norm theorem training set triangular fuzzy number triangular norms universe of discourse