By M. Mizumoto (auth.)

One of the sights of fuzzy good judgment is its software in fixing many actual engineering difficulties. As many have realised, the main stumbling blocks in development a true clever computing device contain facing random disturbances, processing quite a lot of obscure information, interacting with a dynamically altering setting, and dealing with uncertainty. Neural-fuzzy options aid one to resolve a lot of those difficulties. *Fuzzy good judgment and clever Systems* displays the latest advancements in neural networks and fuzzy good judgment, and their program in clever platforms. moreover, the stability among theoretical paintings and functions makes the e-book appropriate for either researchers and engineers, in addition to for graduate scholars.

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**Extra resources for Fuzzy Logic and Intelligent Systems**

**Example text**

This issue has been investigated by many a u t h o r s including Carroll and Dickinson [6], Cybenko [5], Funahashi [7], Gallant and W h i t e [8], Heclit-Nielsen [10], and Hornik [15]. For function approximation, b o t h the series expansion approach and the Stone-Weierstrass theorem are very effective analytic tools. However, Hedit-Nielsen [10,11] found the relationship between the Kolmogorov's theorem and the approximation principle of the feedforward networks. Indeed, functional analytic m e t h o d s have been successfully used to show t h a t feedforward neural structures with at-Ieast one hidden layer are capable of simultaneously approximating continuous functions in several variables and their derivatives if the neural activation functions of the hidden neural imits are differentiable [15].

Based on this idea, t h e fuzzy system may be trained to realize desired i n p u t - o u t p u t relationship using various learning algorithms stich as fuzzy back-propagation algorithm. As pointed out by Wang and Mendel [36], the most i m p o r t a n t advantage of using fuzzy basis functions, rather t h a n polynomials, or radial basis fimctions, etc, is t h a t a linguistic fuzzy I F - T H E N rule is naturally related to a fuzzy basis function. A fuzzy system whose basic configuration is depicted in Figure 10 has four principal elements: fuzzifier, fuzzy rule base, fuzzy inference engine, a n d defuzzifier.

Cos nx, and sin nx. All these functions have the common period 27r. 1) 22 CHAPTER 2 which is simply obtained by replacing the sigmoid function with a trigonometric function in the conventional two-layered neural network as shown in Figure 1. The trigonometric activation function 4> may be chosen as: i) all 4>i{x) = cos{x), (cosine network); ii) all 4>i{x) = sin{x), (sine network); iii) (t>i(x) = cos{x), or sin{x), (trigonometric network). ) ^V^TV Figure 1 Block diagram of the trigonometric network.