## 4 Advanced Development of ANNs
### Neuro-Fuzzy Systems  
Fuzzy logic is a form of multi-valued logic derived from fuzzy set theory [79] to deal with reasoning that is approximate, rather than precise. In contrast to yes/no or 0/1 binary logic, Fuzzy logic provides a set of membership values inclusively between 0 and 1 to indicate the degree of truth. Fuzzy logic can be combined with ANNs to form a learning machine that is able to determine the parameters for a fuzzy system by ANN computing. A fuzzy system comprises of linguistic rules such as IF THEN from prior knowledge. Input and output variables of the system are described linguistically. If the knowledge is incomplete, wrong, or contradictory, the fuzzy system needs to be tuned. Fuzzy system is incapable of learning; it is a mechanism of straight interpretation and implementation. ANNs are capable of learning with a sufficient amount of observed examples for a problem. These observations are used to train the network model. The result can provide data for parameterization of fuzzy rules. The fuzzy-neural systems have been studied and applied. [PERSON] and [PERSON] [22] proposed a neural-fuzzy reasoning system that is capable of automatic determination of inference rules and adjustment according to the time-variant reasoning environment with the use of NN in fuzzy reasoning. [PERSON] _et al._[23] presented a fuzzy modeling method using fuzzy neural networks with the BP algorithm. The method can identify the fuzzy model of a nonlinear system automatically. [PERSON] and [PERSON] [24] approximated reasoning through a BP Neural Network with the aid of fuzzy set theory. An example of multivariable fuzzy control of blood pressure was examined for the underlying principles. [PERSON] and [PERSON] [25] introduced the fuzzy min-max function approximation neural network. The function approximation network is realized by modifying the developed fuzzy min-max clustering network [80] to include an output layer that sums and thresholds the hidden layer membership functions. [PERSON] [81] introduced the framework of ANFIS (Adaptive-Network-based Fuzzy Inference System). [PERSON] and [PERSON] [27] reviewed fundamental and advanced developments in neuro-fuzzy systems for modeling and control. They introduced design methods for ANFIS in modeling and control applications. A fuzzy-neural system can be represented as a MFN. The first layer is for input variables. The middle or hidden layer(s) are for encoding fuzzy rules and fuzzy inference. The last layer is for output variables. The fuzzy inferences are converted as network connection weights. Figure 7 shows the structure of a fuzzy-neural system based on the idea of ANFIS [81]. For n input variables {x\({}_{i}\) | i = 1, 2,..., n\({}_{j}\), suppose that there are m fuzzy if-then rules:  
Rule 1: If x\({}_{1}\) is M\({}_{11}\), x\({}_{2}\) is M\({}_{21}\),..., and x\({}_{n}\) is M\({}_{n1}\), then f\({}_{1}\)=a\({}_{11}\)x\({}_{1}\)+a\({}_{12}\)x\({}_{2}\)+...+a\({}_{1n}\)x\({}_{n}\)+f\({}_{1}\)  
Rule 2: If x\({}_{1}\) is M\({}_{12}\), x\({}_{2}\) is M\({}_{22}\),..., and x\({}_{n}\) is M\({}_{n2}\), then f\({}_{2}\)=a\({}_{21}\)x\({}_{1}\)+a\({}_{22}\)x\({}_{2}\)+...+a\({}_{2n}\)x\({}_{n}\)+f\({}_{2}\)  
.........  
Rule m: If x\({}_{1}\) is M\({}_{1m}\), x\({}_{2}\) is M\({}_{2m}\),..., and x\({}_{n}\) is M\({}_{nm}\), then f\({}_{m}\)=a\({}_{m1}\)x\({}_{1}\)+a\({}_{m2}\)x\({}_{2}\)+...+a\({}_{mn}\)x\({}_{n}\)+f\({}_{m}\)  
where M\({}_{ij}\) is a linguistic label (small, medium, and large) of a membership function, u\({}_{Mij}\)(x), of x\({}_{i}\) with the associated node j. The membership function can be in different forms with different parameter sets, such as the generalized bell-shaped function,  
\[\mu_{{}_{M_{ij}}}(x)=\frac{1}{1+[(\frac{x-c_{{}_{ij}}}{a_{{}_{ij}}})^{2}]^{b_ {{}_{3}}}}\]  
or the Gaussian function, \(\mu_{{}_{M_{ij}}}(x)=\exp[-(\frac{x-b_{{}_{ij}}}{a_{{}_{ij}}})^{2}]\).