Software Enabled Control

Chapter 12.2.1 - The Fuzzy Neural Construct

12.2.1.   The Fuzzy Neural Construct

The fuzzy neural network (FNN) proposed in reference 12 consists of a fuzzy
rule base of Takagi-Sugeno fuzzy rules with the rule consequents being
linear polynomials of the input premise variables. Both structure learning
and parameter learning are used to adaptively develop the FNN construct.
Structure learning inserts new membership functions, creates new fuzzy
rules, and selects initial parameters of the new rules on the basis of the
desired output data. Parameter learning updates the consequent weights via
Kaczmarz’s algorithm.

This FNN construct realizes the fuzzification, fuzzy reasoning, and defuzzification
functionalities of a connectional fuzzy inference mechanism.

Let x = [x1, . . . , xm]T and y = [y, . . . , yp]T denote the input and output
vectors of the FNN, respectively. The fuzzy rule base of the FNN consists of a
collection of N fuzzy rules of the form

 

where fjldenotes the jth rule output associated with the lth output
component yl . w0lj , . . . , wilj , . . . , wmlj are the polynomial coefficients connect-
ing linearly the input variables to the fjl consequent function. Finally,
A1j, . . . , Aij, . . . , Amj are labels of the fuzzy sets in the premise space associated
with the jth rule R(j). Each linguistic label Aij is associated with a
Gaussian membership function, μAij (xi), which specifies the degree to which a
given xi satisfies the quantifier Aij:

 

where mij and σij denote the mean and standard deviation of the Gaussian
membership function, respectively. The degree of fulfillment (or the firing
strength) of each rule R(j) is taken as

 

Given an input vector x, the lth output component yl of the fuzzy system is
inferred as follows:

 

The inferred outputs result from the application of the weighted-average
defuzzification method.

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