Fuzzy Adaptive Controller for a DFI-Motor - Complex System Modelling and Control Through Intelligent Soft Computations - page 93

Fuzzy Rule Base

f ˆ

x

Defuzzifier

Fuzzifier

Fuzzy Inference
Engine

Fig. 2 The basic con guration of a fuzzy logic system

P i¼1 f i Q j¼1 l A j ð

x j Þ

f

ð x Þ ¼

P i¼1 Q j¼1 l A j ð

ð 12 Þ

x j Þ

T

¼ h wð x Þ

is the degree of membership of x j to A j , m is the number of fuzzy

where l A j ð x j Þ

T

f 1 f 2 f m

rules, h ¼½ ; ; ...; is the adjustable parameter vector (composed of conse-

T 1 2 m

quent parameters), and w ¼½ w w ...w with

Q j¼1 l A j ð x j Þ

i

w ð x Þ ¼

P i¼1 Q j¼1 l A j ð

x j Þ

being the fuzzy basis function (FBF). Throughout the paper, it is assumed that the
FBFs are selected so that there is always at least one active rule (Wang 1994 ), i.e.
P i¼1 Q j¼1 l A j ð x j Þ

0.
It is worth noting that the fuzzy system ( 12 ) is commonly used in control
applications. Following the universal approximation results (Wang 1994 ; Azar
2010a , b , 2012 ), the fuzzy system ( 12 ) is able to approximate any nonlinear smooth
function f

[

on a compact operating space to an arbitrary degree of accuracy. Of
particular importance, it is assumed that the structure of the fuzzy system (i.e. the
pertinent inputs, the number of membership functions for each input and the
number of rules) and the membership function parameters are properly speci

ð x Þ

ed

beforehand. The consequent parameters h are then determined by appropriate

adaptation algorithms.
In the following section, the proposed fuzzy adaptive backstepping controller
will be presented.