Neuro-fuzzy system can be classified intoThe following categories:1.A fuzzy rule-based model constructed

using a supervised NN learning technique

2.A fuzzy rule-based model constructed using NN to construct its fuzzy partition of the input space

A class of adaptive networks that are functionally equivalent to fuzzy inference systems.

ANFIS architectures representing both the Sugeno and Tsukamoto fuzzy models

Assume - two inputs X and Y and one output ZRule 1: If x is A1 and y is B1,

then f1 = p1x + q1y +r1Rule 2: If x is A2 and y is B2,

then f2 = p2x + q2y +r2

Every node i in this layer is an adaptive node with a node functionO1,i = mAi (x), for I = 1,2, or O1,i = mBi-2 (y), for I = 3,4Where x (or y) is the input to node i and Ai (or Bi) is a linguistic label** O1,i is the membership grade of a fuzzy set and it specifies thedegree to which the given input x or y satisfies the quantifies

Typically, the membership function for a fuzzy set can be any parameterized membership function, such as triangle, trapezoidal, Guassian, or generalized Bell function.

Parameters in this layer are referred to as Antecedence Parameters

Every node i in this layer is a fixed node labeled P, whose output is the product of all the incoming signals:O2,i = Wi = min{mAi (x) , mBi (y)}, i = 1,2

Each node output represents the firing strength of a rule.

Every node in this layer is a fixed node labeled N. The ith node calculates the ratio of the ith rule’s firing strength to the sum of all rules’firing stregths:

O3,i = Wi = Wi /(W1+W2) , i =1,2 (normalized firing strengths]

Every node i in this layer is an adaptive node with a node function

__ __O 4,i = wi fi = wi (pix + qiy +ri) …Consequent parameters

The single node in this layer is a fixed node labeled S, which computes the overall output as the summation of all incoming signals:

__O 5,1 = Si wi fi

ANFIS architecture for the Sugeno fuzzy model, weight normalization is performed at the very last layer

Equivalent ANFIS architecture using the Tsukamoto fuzzy model