Fusion of Neural Networks, Fuzzy Systems and Genetic Algorithms: Industrial Applications Fusion of Neural Networks, Fuzzy Systems and Genetic Algorithms: Industrial Applications
by Lakhmi C. Jain; N.M. Martin
CRC Press, CRC Press LLC
ISBN: 0849398045   Pub Date: 11/01/98
  

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The structure of the chapter is as follows. In Section 2, the fuzzy system employed is characterized by its fuzzy logic operations. In Section 3, we review fuzzy modeling processes in the literature. Section 4 describes the neuro-fuzzy modeling algorithm. Section 5 presents the experimental system and the technique used to obtain a good training data set from the electro-hydraulic system. In Section 6, we extract the inverse-model of the actuator using the modeling algorithm presented in Section 4 and the training set of Section 5. Section 7 describes the neuro-fuzzy control system using the feedback-error-learning algorithms and presents some experimental tests.

2. The Fuzzy Logic System

Fuzzy sets establish a mechanism for representing linguistic concepts like big, little, small and, thus, they provide new directions in the application of pattern recognition based on fuzzy logic to automatically model drive systems [31], [32]. These computational models are able to recognize, represent, manipulate, interpret, and use fuzzy uncertainties through a fuzzy system.

A fuzzy logic system consists of three main blocks: fuzzification, inference mechanism, and defuzzification. The following subsections briefly explain each block, and characterize them with regard to the type of fuzzy system we used.

2.1 Fuzzification

Fuzzification is a mapping from the observed numerical input space to the fuzzy sets defined in the corresponding universes of discourse. The fuzzifier maps a numerical value denoted by into fuzzy sets represented by membership functions in U. These functions are Gaussian, denoted by μ(xj) as we expressed in Equation (1).

In this equation, 1≤jm refers to the variable (j) from m considered input variables; 1≤inj considers the i membership function among all nj membership functions considered for variable (j); aij defines the maximum of each Gaussian function, here aij = 1.0; bij is the center of the Gaussian function; and cij defines its shape width.

2.2 Inference Mechanism

Inference mechanism is the fuzzy logic reasoning process that determines the outputs corresponding to fuzzified inputs.

The fuzzy rule-base is composed by IF-THEN rules like

R(l): IF (x1 is A1(l) and x2 is A2(l) and... xm is THEN (y is ω(l)),

where: R(l) is the lth rule with 1≤lc determining the total number of rules; x1,x2,...xm and y are, respectively, the input and output system variables; are the antecedent linguistic terms (or fuzzy sets) in rule (l) with 1≤jm being the number of antecedent variables; and ω(l) is the rule conclusion being, for that type of rules, a real number usually called fuzzy singleton. The conclusion, a numerical value and not a fuzzy set, can be considered as a pre-defuzzified output that helps to accelerate the inference process.

Each IF-THEN rule defines a fuzzy implication between condition and conclusion rule parts and denoted by expression . The implication operator used in this work is the product-operator, as shown in expression (2). The right-hand term μ (x) represents the condition degree and is defined in Equation (3).

The symbol “ * ” in Equation (3) is the t-norm corresponding to the conjunction and in the rules. The most commonly used t-norms between linguistic expressions u and v are: fuzzy intersection defined by operation min(u,v), algebraic product uv, and the bounded sum max(0, u+v-1). This work uses algebraic product as the t-norm operator.

Since the rule conclusion ω(l) is considered a fuzzy singleton, the value of its membership degree in expression (2) stays 1.0. So, the final expression for fuzzy implication degree (2) results in multiplication of each condition membership degree (3) and equal to expression (4).

For this type of fuzzy system, the product inference in Equation (3) expresses the activation degree of each identified rule by measured condition variables, and equals the expression for implication degree in (4).

The reasoning process combines all rule contributions ω(l) using the centroid defuzzification formula in a weighted form, as indicated by inference function (5). This equation maps input process states (xj) to the value resulting from inference function Y(x). If we fix the structure made by the Gaussian membership functions, the parameters of the fuzzy logic system to be learned will be the rule conclusion value ω(l).


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