Fuzzy Adaptive Controller for a DFI-Motor - Complex System Modelling and Control Through Intelligent Soft Computations - page 100

f 2 ð

T
2

z 2 ; h 2 Þ ¼ h w 2 ð z 2 Þ ð 47 Þ

where fixed a priori by the designer, and

h 2 is the adjustable parameter vector of the fuzzy system. Furthermore, the
functions f 2 ð

w 2 ð z 2 Þ is the FBF vector, which is

z 2 Þ

can be approximated optimally (Wang 1993 , 1994 )as

follows

z 2 Þ ¼f 2 ð z 2 ; h 2 Þþe 2 ð

f 2 ð z 2 Þ

ð Þ

48

¼ h 2 w 2 ð z 2 Þþe 2 ð z 2 Þ

h 2 is the optimal parameter vector and

where is the unavoidable

fuzzy approximation error which is assumed to be bounded (Boulkroune
et al. 2008 , 2009 , 2010a , b ; Wang 1993 , 1994 ) as follows

e 2 ð z 2 Þ

j e 2 ð z 2 Þ j e 2 ; 8 z 2 2 X z 2 ;

where e 2 is an unknown constant.

From ( 47 ) and ( 48 ), we have

f 2 ð z 2 Þ ¼f 2 ð z 2 Þ f 2 ð z 2 ; h 2 Þþ f 2 ð z 2 ; h 2 Þ
¼ f 2 ð z 2 ; h 2 Þþ z 2 Þ f 2 ð z 2 ; h 2 Þ

ð Þ

f 2 ð 49

¼ h T
2

w 2 ð

z 2 Þþe 2 ð z 2 Þ

To stabilise the dynamics ( 46 ), the following fuzzy adaptive controller is
proposed

2 e 3
j 2 e jjþ r 3 e r 4 t

j

T

u 2 ¼ h

2 w 2 ð

z 2 Þk 2 e 3 ð 50 Þ

where r 3 and r 4 [ 0 are small design constants, k 2 is a positive design

j 2 ¼ e 2 . constant and j 2 is the estimate of the unknown bound

Replacing ( 50 ) into ( 46 ) and using ( 49 ) yields

a 5 x 4 e 1 h

T

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

2 w 2 ð

z 2 Þ

ð 51 Þ

2

2 e 3
j 2 e jj þ r 3 e r 4 t

j

þ e 2 ð z 2 Þk 2 e 3

where h 2 ¼ h 2 h 2 is the parameter error vector.
Multiplying ( 51 )bye 3 , we get