Fuzzy Adaptive Controller for a DFI-Motor - Complex System Modelling and Control Through Intelligent Soft Computations - page 99

1 1

g 1 j 1 j 1 c 1 h

1 h 1 þ

V 2 ¼ V 1 þ

T

e 2 _ e 2 þ

e 1 b

C

2
l

ð a 5 a 2 x 2 a 5 x 5 x r a 5 c 1 x 4 Þ e 2 e 3 þ a 5 x 4 e 1 e 3 ð c 1 þ a 6 Þ

c

1 1

g 1 j 1 j 1

e 2 h

c 1 h

1 h 1 þ

T
1 w 1 ðe

T

e 2 jk 1 e 2 þ r 1 e r 2 t

z 1 Þj 1 j þ ð 41 Þ

e 1 b
c

C

2
l

¼ ð a 5 a 2 x 2 a 5 x 5 x r a 5 c 1 x 4 Þ e 2 e 3 þ a 5 x 4 e 1 e 3 ð c 1 þ a 6 Þ

h i

1 1

g 1 j 1 c 1 h

h 1 c 1 e 2 w 1 ð^

T
1

k 1 e 2 þ r 1 e r 2 t

þ z 1 Þ þ ½ j 1 g 1 e jj

If the adaptation laws are designed as

h 1 ¼ c 1 e 2 w 1 ð^

z 1 Þ ð 42 Þ

j 1 ¼ g 1 e jj

ð 43 Þ

Then, ( 41 ) can be expressed as follows

V 2 ð

a 5 a 2 x 2 a 5 x 5 x r a 5 c 1 x 4 Þ e 2 e 3 þ a 5 x 4 e 1 e 3

ð 44 Þ

e 1 b

C

2

l k 1 e 2 þ r 1 e r 2 t

ð c 1 þ a 6 Þ

c

In the next step, we try to stabilize the tracking error e 3 :
Step 3. At this step, we will construct the control law u 2 . The time-derivative of
( 32 ) is given by

e 3 ¼ a 3 x 3 þ a 4 x 5 þ x r x 2 þ d 2 ð x 3 ; x 2 Þþ u 2 x 3d ð 45 Þ

We can rewrite ( 45 ) as follows

e 3 ¼ ð

_

a 5 a 2 x 2 a 5 x 5 x r a 5 c 1 x 4 Þ e 2 a 5 x 4 e 1 þ f 2 ð z 2 Þþ

ð Þ

u 2 46

with

f 2 ð z 2 Þ ¼ ð a 5 a 2 x 2 a 5 x 5 x r a 5 c 1 x 5 Þ e 2 þ a 5 x 4 e 1 a 3 x 3

þ a 4 x 5 þ x r x 2 þ d 2 ð x 3 ; x 2 Þ_ x 3d

T .
The uncertain continuous function f 2 ð z 2 Þ

where z 2 ¼½x 1 ; x 2 ; x 3 ; x 4 ; x 5

can be approximated by the fuzzy

system ( 12 ) as follows