Fuzzy Adaptive Controller for a DFI-Motor - Complex System Modelling and Control Through Intelligent Soft Computations - page 96

a 7 C l

e 1 ¼ _ a 5 x 4 e 3 e 2 ð c 1 þ a 6 Þ e 1 ð 24 Þ

where C l ¼ C l C l is the load torque estimation error, and e 2 is the
tracking error of the variable a 5 x 5 x 2 .
e 2 ¼ a 5 x 5 x 2 t 2 ð 25 Þ

Consider the following Lyapunov function candidate for the e 1 -subsystem

1
2

1
c

C

e 1 þ

2
l

V 1 ¼ ð 26 Þ

where 0 is a design constant.

By assuming that the load torque is slowly time-varying ( C l ¼

c [

0), the time-

derivative of ( 26 ) along ( 24 ) is given by

1
c

V 1 ¼

e 1 C l a 7 e 1 þ

C l

e 1 e 2 þ a 5 x 4 e 3 e 1 ð c 1 þ a 6 Þ ð 27 Þ

If the load torque adaptation law is designed as

C l ¼ bC l c

a 7 e 1 ð 28 Þ

where 0 is a design parameter.

Then, ( 27 ) can be written as

b [

Þ e 1 b

V 1 ¼ e 1 e 2 þ a 5 x 4 e 3 e 1 c 1 þ a 6 C

2
l

ð ð 29 Þ

c

The next step consists in stabilizing the tracking error e 2 .
Step 2. The time-derivative of ( 25 ) is given by

e 2 ¼ _ a 5 x 5 _ x 2 þ a 5 _ x 5 x 2 t 2 ð 30 Þ

From the second subsystem of ( 1 ), ( 18 ) and ( 23 ), we can write

e 2 ¼ _ f 1 ð z 1 Þþ e 1 þð a 5 a 2 x 2 a 5 x 5 x r a 5 c 1 x 4 Þ e 3

ð 31 Þ

c 1 ÞC l þ

a 7 ðb þ a 5 x 5 u 1

with

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

þ a 5 x s x 4 x 2 þ a 5 a 1 x 3d x 4 þ a 5 x s x 3d x 5 a 5 x 3d u s a 5 x 4 _ x 3d

a 7 c þ a 6 _ x 1d þ € x 1d þ c 1 e 2 þ ð ð c 1 þ a 6 Þ e 1 Þ e 1

þ a 7 ðb þ c 1 ÞC l þ a 5 x 5 d 1 ð x 1 ; x 2 Þ