Fuzzy Adaptive Controller for a DFI-Motor - Complex System Modelling and Control Through Intelligent Soft Computations - page 101

e 3 _ 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 e 3 a 5 x 4 e 1 e 3

2 e 3
j 2 e jjþr 3 e r 4 t

j

e 3 h

T

2 w 2 ð z 2 Þþ e 3 e 2 ð z 2 Þk 2 e 3

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

ð 52 Þ

2 e 3
j 2 e jjþ r 3 e r 4 t

2

j

e 3 h

T
2 w 2 ð

z 2 Þþj 2 e jj k 2 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 e 3 a 5 x 4 e 1 e 3

e 3 h

T

2 w 2 ð z 2 Þk 2 e 3 j 2 e jjþ r 3 e r 4 t

j 2 ¼ j 2 j 2 .
Define a Lyapunov function candidate as follows

where

1
2 e 3 þ

1
2

1
2 c 2 h

2 h 2 þ

T 2

2

g 2 j

V 3 ¼ V 2 þ ð 53 Þ

where 0 are design constants.

Take the derivative of V 3 with respect to time and using ( 52 ) and ( 44 ), one
can obtain

c 2 and g 2 [

1 1

g 2 j 2 j 2

V 3 ¼ V 2 þ

c 2 h

2 h 2 þ

T

e 3 _ e 3 þ

e 1 b

C

e 3 h

2
l

k 1 e 2

T
2

z 2 Þk 2 e 3 ð c 1 þ a 6 Þ w 2 ð

c

1 1

g 2 j 2 j 2

c 2 h

2 h 2 þ

e 3 jþr 1 e r 2 t þ r 3 e r 4 t

T

j 2 j þ

ð Þ

54

e 1 b
c

C

l

k 1 e 2 k 2 e 3 þ r 1 e r 2 t þ r 3 e r 4 t

¼ ð c 1 þ a 6 Þ

1

c 2 h

2 ½ h 2 c 2 e 3 w 2 ð

T

þ z 2 Þ

1
g 2 j 2

j 2 g 2 e jj

þ ½

If the adaptation laws are designed as

h 2 ¼ c 2 e 3 w 2 ð

z 2 Þ ð 55 Þ

j 2 ¼ g 2 e jj

ð 56 Þ