Stable Fuzzy Control System by Design - Fuzzy Modeling and Control: Theory and Applications - page 82

is given by 3 :
IF x 1 is Gaussmfs(1.464; 9.987) and x 2 is Gaussmf(

− 1 . 211; 9.964) and

u 1 is Gaussmf( − 6 . 976; 49.67) and u 2 is Gaussmf( − 7.39; 49.98)

THEN x 1 =− 1 . 551 × 10 − 8 + 1 x 1 + 2 x 2 + 1 u 1 + 2 . 196 × 10 − 11 u 2

IF x 1 is Gaussmf( − 2.982; 9.986) and x 2 is Gaussmf(7.149; 9.96) and
u 1 is Gaussmf(30.65; 49.67) and u 2 is Gaussmf(17.5; 49.98)
THEN x 1 = 3 . 164 × 10 − 8

+ 1 x 1 + 2 x 2 + 1 u 1 − 2 . 148 × 10 − 10 u 2
IF x 1 is Gaussmf( − 3.811; 13.62) and x 2 is Gaussmf(0.06026; 8.578) and
u 1 is Gaussmf(11.03; 60.97) and u 2 is Gaussmf(10.77; 61.63)
THEN x 2 =− 6498 + 1 . 993 x 1 + 419 . 8 x 2 − 0 , 1538 u 1 + 0 . 9489 u 2
IF x 1 is Gaussmf(

−

3.812; 13.61) and x 2 is Gaussmf(4.095; 11.44) and
u 1 is Gaussmf(11.04; 60.97) and u 2 is Gaussmf(10.78; 61.63)
THEN x 2 = ˙ 7385 + 0 . 549 x 1 − 926 x 2 + 0 . 1001 u 1 + 1 . 027 u 2

Gaussmf ( c ,β) is a gaussian membership function:

x − c
β

2

e −

μ Gauss [ ,β ] ( ) =

c x (4.54)

Once initialization has been performed by the Algorithm 4.4, four regions are
defined, s 4, corresponding to the regions defined by the four rules of each fuzzy

controller output. After the initialization, the phases of design and final adjustments
are made, obtaining the following fuzzy controller:
IF x 1 is Gaussmf(1.464; 9.987) and x 2 is Gaussmf( − 1.211; 9.964)
THEN u 1 = 0 . 09714 + 0 . 5753 x 1 − 0 . 2195 x 2
IF x 1 is Gaussmf( − 3.811; 13.62) and x 2 is Gaussmf(0.06026; 8.578)
THEN u 1 =−

=

. − .

864 x 1 +

.

0 02684 2 1 148 x 2

−

IF x 1 is Gaussmf( 2.982; 9.986) and x 2 is Gaussmf(7.149; 9.96)

THEN u 1 = 0 . 1881 + 0 . 7072 x 1 + 0 . 6508 x 2

IF x 1 is Gaussmf( − 3.812; 13.61) and x 2 is Gaussmf(4.095; 11.44)

THEN u 1 3739 x 2

IF x 1 is Gaussmf(1.464; 9.987) and x 2 is Gaussmf(

=− 0 . 2111 − 34 . 79 x 1 + 0 .

− 1.211; 9.964)

THEN u 2 =− 3 . 25 + 0 . 293 x 1 − 0 . 01715 x 2

IF x 1 is Gaussmf( − 3.811; 13.62) and x 2 is Gaussmf(0.06026; 8.578)

THEN u 2 = 7 . 839 + 0 , 2617 x 1 − 2 . 995 x 2

IF x 1 is Gaussmf( − 2.982; 9.986) and x 2 is Gaussmf(7.149; 9.96)

THEN u 2 = 0 . 2527 + 13 . 86 x 1 + 0 . 08707 x 2

IF x 1 is Gaussmf( − 3.812; 13.61) and x 2 is Gaussmf(4.095; 11.44)

6936 x 2
As minimization algorithm has been used a hybrid scheme based on a basic
genetic algorithm for global exploration (Goldberg 1989 ), followed by the numerical
algorithm by Nelder-Mead (Nelder and Mead 1965 ) for local deepening (Andújar
et al. 2009 ). Comparing the antecedents of the plant and the controller can see how
the initialization based on the Algorithm 4.4 has been performed.

THEN u 2 = 1 . 16 − 18 . 54 x 1 + 0 .

3 The fuzzy model has been obtained from a set of system data, using the clustering algorithm (Chiu
1994 ) and adjusting the parameters using the ANFIS algorithm (Jang 1993 ).