7.3. BASIC PRINCIPLES OF NEURAL-FUZZY SYSTEMS 241
the domain ofA 1 andA 2. The weight adjustments in this case become
zi(t+1) = zi(t)−η°
Ok−yk¢
Ai(xk)
a(t+1) = a(t)−η°
Ok−yk¢
(z 1 −z 2 )bA 1 (xk)A 2 (xk)b(t+1) = b(t)+η∂Ek(a,b)
∂b°
Ok−yk¢
(z 1 −z 2 )°
xk−a¢
A 1 (xk)A 2 (xk)as shown in the following computations
zi(t+1) = zi(t)−η
∂Ek
∂z 1=zi(t)−η°
Ok−yk¢
Ai(xk)a(t+1) = a(t)−η∂Ek(a,b)
∂ab(t+1) = b(t)−η∂Ek(a, b)
∂bwhere
∂Ek(a,b)
∂a=
°
Ok−yk¢∂Ok
∂a=
°
Ok−yk¢∂
∂a°
z 1 A 1 (xk)+z 2 A 2 (xk)¢
=
°
Ok−yk¢∂
∂a°
z 1 A 1 (xk)+z 2°
1 −A 1 (xk)¢¢
=
°
Ok−yk¢∂
∂a°
(z 1 −z 2 )A 1 (xk)+z 2¢
=
°
Ok−yk¢
μ
(z 1 −z 2 )∂A 1 (xk)
∂a∂
=
°
Ok−yk¢
(z 1 −z 2 )b
eb(xk−a)
°
1+eb(x−a)¢ 2
=
°
Ok−yk¢
(z 1 −z 2 )bA 1 (xk)°
1 −A 1 (xk)¢
=
°
Ok−yk¢
(z 1 −z 2 )bA 1 (xk)A 2 (xk)and similarly,
∂Ek(a,b)
∂b=−
°
Ok−yk¢
(z 1 −z 2 )°
xk−a¢
A 1 (xk)A 2 (xk)The following example illustrates how to use the ANFIS algorithm with
Matlab. In practice, the situation is typically like the following. In system
identification for indirect neural control, or in direct neural control where we
seek to model the control law as a neural network, we are modeling an input-
output relationship ñ that is, approximating some functiony=f(x)from the
data expressed as a set of fuzzy rules. An appropriate structure of ANFIS is
chosen for the approximation problem, guided by a theorem on universal ap-
proximation. In simulation studies, various choices of membership functions in
fuzzy rules, as well as choices of the number of rules, illustrate the approximation