A First Course in FUZZY and NEURAL CONTROL

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

Figure 7.5. First-order Sugeno fuzzy model with two rules

The fuzzy neurons in Layer 4 output the values

(O 41 ,O 42 )=(O 31 f 1 ,O 32 f 2 )

=

≥

(A 1 (x 1 )B 1 (x 2 ))(z 11 x 1 +z 12 x 2 +z 13 )

A 1 (x 1 )B 1 (x 2 )+A 2 (x 1 )B 2 (x 2 ) ,

(A 2 (x 1 )B 2 (x 2 ))(z 21 x 1 +z 22 x 2 +z 23 )

A 1 (x 1 )B 1 (x 2 )+A 2 (x 1 )B 2 (x 2 )

¥

Finally, the output layer calculates the control action by summing:

y∗=O 41 +O 42 =(A^1 (x^1 )B^1 (x^2 ))(z^11 x^1 A+ 1 z(^12 x 1 x)B^2 + 1 (zx^132 )+)+(AA 22 (x(x 1 )^1 B)B 22 (x(x 22 )))(z^21 x^1 +z^22 x^2 +z^23 )

Of course, the above neural network type for representing the inference pro-
cedureforarulebaseoftworulescanbeextendedinanobviouswaytoan
arbitrary number of rules.

7.3.2 ANFIS learning algorithm

The representation in the preceding section of a neural network is simply a
graphical display of the computation steps in the Sugeno-Takagi procedure. In
order for this representation to be more useful in implementing the control law,
one needs to equip it with an efficient learning algorithm. In conventional neural
networks, the backpropagation algorithm is used to learn, or adjust, weights on
connecting arrows between neurons from input-output training samples. In the
ANFIS structure, the parameters of the premises and consequents play the role
of weights. Specifically, when the membership functionsAjiused in the ìIfî part
of the rules are specified parametrically ñ that is, the shape is specified and
the function is determined by afinite number of parameters, these parameters
are calledpremise parameters, whereas the parametersai,bi,ci,i=1, 2 in
the ìthenî part of the rules are referred to asconsequent parameters.The
ANFIS learning algorithm consists of adjusting the above set of parameters from
sample data

°°

xk 1 ,xk 2

¢

,yk

¢

,k=1,...,N.