A First Course in FUZZY and NEURAL CONTROL

266 CHAPTER 8. APPLICATIONS

consequent membership function and afiring strength. For the Mamdani fuzzy
inference system with max-min composition, a corresponding ANFIS can be
constructed if discrete approximations are used to replace the integrals in the
centroid defuzzification scheme. In this example, only the ANFIS architecture
for thefirst-order Sugeno fuzzy model is used due to its transparency and ef-
ficiency. However, detailed discussion on the other ANFIS architecture can be
found in [36].

Hybrid-learning algorithm FromtheANFISarchitectureinFigure8.12
(a), we observe that when the values of the premise parameters arefixed, the
overall output can be expressed as a linear combination of the consequent pa-
rameters. The outputfin Figure 8.12 (b) can be written as

f =

w 1

w 1 +w 2

f 1 +

w 2

w 1 +w 2

f 2 (8.8)

= wb 1 (a^10 +a^11 x+a^12 y)+wb 2 (a^20 +a^21 x+a^22 y)

=(wb 1 )a 01 +(wb 1 x)a^11 +(wb 1 y)a^12 +(wb 2 )a^20 +(wb 2 x)a^21 +(wb 2 y)a^22

which is linear in the consequent parametersa^10 ,a^11 ,a^12 ,a^20 ,a^21 ,a^22.
The consequent parameters can be obtained by solving the following over-
constrained, simultaneous equations

     

wb(1) 1 wb 1 (1)x(1) wb(1) 1 y(1) wb 2 (1) wb(1) 2 x(1) wb(1) 2 y(1)

wb(2) 1 wb 1 (2)x(2) wb(2) 1 y(2) wb 2 (2) wb(2) 2 x(2) wb(2) 2 y(2)

..

.

..

.

..

.

..

.

..

.

..

.

wb 1 (n) wb 1 (n)x(n) wb( 1 n)y(n) wb( 2 n) wb 2 (n)x(n) wb( 2 n)y(n)

     

         

a^10

a^11

a^12

a^20

a^21

a^22

         

=











d(1)

d(2)

..

.

d(n)











(8.9)

where

£°

x(k),y(k)

¢

,d(k)

§

are thekthtraining pairk=1, 2 ,∑∑∑,n,andwb( 1 k)

andwb 2 (k)are the outputs of Layer 3 associated with the inputs

°

x(k),y(k)

¢

.

Equation 8.9 can be expressed in matrix-vector form as

Ax=d (8.10)

where

x=[a^10 ,a^11 ,a^12 ,a^20 ,a^21 ,a^22 ]T, d=[d(1),d(1),∑∑∑,d(n)]T

andAis a matrix formed by the elementswb( 1 k), wb( 2 k),x(k),y(k).Thereare
several approaches to solve these kinds of constrained equations that can be
used to obtain the consequent parameters. One of the most concise ways to
solve Equation 8.10 ifATAis nonsingular is to use the pseudoinverse technique

x∗=(ATA)−^1 ATd (8.11)