A First Course in FUZZY and NEURAL CONTROL

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

Example 7.4We illustrate the above tuning process by a simplified example.
Consider two fuzzy rules, with one input variablexand one output variabley,
of the form
R 1 :IfxisA 1 theny=z 1
R 2 :IfxisA 2 theny=z 2

where the fuzzy setsA 1 andA 2 have sigmoid membership functions defined by

A 1 (x)=

1

1+eb^1 (x−a^1 )

A 2 (x)=

1

1+eb^2 (x−a^2 )

Thena 1 ,a 2 ,b 1 ,andb 2 are the parameter set for the premises, and the an-
tecedent of the ruleRiis simply the value of the membership functionAi(x).
The outputO(x)of the system, computed by the discrete center-of-gravity
defuzzification method, is

O(x)=

A 1 (x)z 1 +A 2 (x)z 2

A 1 (x)+A 2 (x)

Given a training set

©°

x^1 ,y^1

¢

,...,

°

xK,yK

¢™

,wewanttoprovidethetwo
fuzzy rules with appropriate membership functions and consequent parts to
generate the given input-output pairs. That is, we want to learn the parameters
a 1 ,a 2 ,b 1 ,andb 2 of the sigmoid membership functions, and the valuesz 1 and
z 2 of the consequent parts.
The measure of error for thekthtraining pair is defined as

Ek=Ek(a 1 ,b 1 ,a 2 ,b 2 ,z 1 ,z 2 )=

1

2

°

Ok(a 1 ,b 1 ,a 2 ,b 2 ,z 1 ,z 2 )−yk

¢ 2

k=1,...,K,whereOkis the computed output from the fuzzy inference system
corresponding to the input patternxk,andykis the desired output.
The steepest descent method is used to learnziin the consequent part of
theithfuzzy rule, and the shape parameters of the membership functionsA 1
andA 2 .Thatis,

zi(t+1) = zi(t)−η

∂Ek

∂zi

=zi(t)−η

°

Ok−yk

¢ Ai(xk)

A 1 (xk)+A 2 (xk)

ai(t+1) = ai(t)−η

∂Ek

∂ai

bi(t+1) = bi(t)−η

∂Ek

∂bi

whereη> 0 is the learning constant andtindexes the number of adjustments
ofzi,ai,andbi.
Assuming further thata 1 =a 2 =aandb 1 =b 2 =bsimplifies the learning
rules because, in this case, the equationA 1 (x)+A 2 (x)=1holds for allxfrom