A First Course in FUZZY and NEURAL CONTROL

230 CHAPTER 7. FUZZY-NEURAL AND NEURAL-FUZZY CONTROL

to aneural-fuzzy systemñ a fuzzy system represented as a modified neural
network, resulting in a fuzzy inference system that is enhanced by neural net-
work capabilities. Fuzzy systems are generally more ìuser friendlyî than neural
systems because their behavior can be explained based on fuzzy rules fashioned
after human reasoning. Although fuzzy logic can encode expert knowledge di-
rectly using rules with linguistic labels, it usually takes a lot of time to design
and tune the membership functions that quantitatively represent these linguistic
labels, and applications of pure fuzzy control systems are restricted mainly to
thosefields where expert knowledge is available and the number of input vari-
ables is small. Neural network learning techniques can automate this process
and substantially reduce development time and cost while improving perfor-
mance. Neural networks are also used to preprocess data and to extract fuzzy
control rules from numerical data automatically, as well as to tune membership
functions of fuzzy systems. In this chapter, wefirst address some issues of fuzzy-
neural systems for control problems, and then look at neural-fuzzy systems. Our
primary focus is on adaptive neuro-fuzzy systems for control.

7.1 Fuzzy concepts in neural networks

Fuzzy logic concepts can be incorporated into neural network structure at any
level. Recall that a Mamdani fuzzy rule is of the form

ìIfxisAthenyisBî

and a Sugeno fuzzy rule is of the form

ìIfxisAthenyisf(x)î

whereAandBare fuzzy sets or products of fuzzy sets, andfis a real-valued
function. IfAis the product of fuzzy subsetsA 1 ,...,AnofUi,i=1, 2 ,...,n,
then

A:

Yn

i=1

Ui→[0,1]:(u 1 ,...,un) 7 →min{A 1 (u 1 ),...,An(un)}

and forx=(x 1 ,...,xn),ìxisAî stands for ìx 1 isA 1 andx 2 isA 2 , ...,xn
isAn,î andfis a real-valued function onRn. The fuzzy inference THEN is
implemented most commonly by minimum, the ìMamdani implication,î or by
the product. Rules are combined by a fuzzy OR, namely by some t-conorm ñ
usually maximum.
One modification of a neural network structure is to replace some or all
components of a neuron by fuzzy logic operations. Such a neuron is called a
fuzzy neuron. For example, if addition (as an aggregation operation) in a
neuron is replaced by minimum, we have amin-fuzzy neuron. A neuron that
uses maximum as an aggregation operation is amax-fuzzy neuron.
A neural network with fuzzy neurons becomes amulti-layer fuzzy-neural
network(Figure 7.1). Note that conventional neural networks are used to