Page 608 - Mathematical Techniques of Fractional Order Systems

578 Mathematical Techniques of Fractional Order Systems

q
D z 2 5 ae 3 2 ae 2 1 de 1 1 u 2
Here, e 3 5 α 2 , is the virtual controller
q
D z 2 5 aα 2 2 az 2 1 dz 1 1 u 2 ð19:33Þ
The new Lyapunov function for the above system can be written as,
1
V 2 5 V 1 1 z 2 2
2
q 2 q
.D V 2 #2 az 2 z 1 z 2 1 z 2 D z 2
1
2
#2 az 2 z 1 z 2 1 z 2 faα 2 2 az 2 1 dz 1 1 u 2 g
1
For α 2 5 0 and u 2 5 z 1 2 dz 1 ,
2
q
D V 2 #2 az 2 az 2
1 2
The above expression satisfies the stability condition. Next, for
z 3 5 e 3 2 α 2 ,
q
D z 3 5 ce 2 2 e 2 e 4 2 ~ x 4 e 2 2 e 4 ~ x 2 2 e 3 1 u 3
After substitution,
q
.D z 3 5 cz 2 2 z 2 e 4 2 ~ x 4 z 2 2 e 4 ~ x 2 2 z 3 1 u 3 ð19:34Þ
and the Lyapunov function up to this stage is:
1
V 3 5 V 2 1 z 2 3
2
q 2 2 q
.D V 3 #2 az 2 az 1 z 3 D z 3
1 2
Now, for e 4 5 α 3 , as the virtual controller,
q 2 2
D V 3 #2 az 2 az 1 z 3 cz 2 2 z 2 e 4 2 ~ x 4 z 2 2 e 4 ~ x 2 2 z 3 1 u 3 Š
½
1 2
Keeping α 3 5 0, the controller can be chosen as:
u 3 52 cz 2 1 ~ x 4 z 2
which leads to
2
2
q
D V 3 #2 az 2 az 2 z 2 3
1
2
Next from the last equation with z 4 5 e 4 2 α 3
q
D z 4 5 e 2 e 3 1 ~ x 3 e 2 1 ~ x 2 e 3 2 be 4 1 u 4
q ð19:35Þ
.D z 4 5 z 2 z 3 1 ~ x 3 z 2 1 ~ x 2 z 3 2 be 4 1 u 4
Lyapunov function will be:
1
V 4 5 V 3 1 z 2 4
2
q 2 2 2
.D V 4 #2 az 2 az 2 z 1 z 4 ðz 2 z 3 1 ~ x 3 z 2 1 ~ x 2 z 3 2 be 4 1 u 4 Þ
1 2 3