Page 346 - Mathematical Techniques of Fractional Order Systems

336 Mathematical Techniques of Fractional Order Systems

Using Eqs. (11.14) and (11.16) the error system (11.26) is transformed
into the following form:

8 q
d e 1
>
1
> 52 a 1 x 2 1 a 2 y 2 1 ψ 2 x 1 y 1 1 bz 1
dt
> q
>
>
>
>
q
>
d e 2
>
5 a 3 x 2 2 x 2 z 2 2 y 2 1 w 2 1 ψ 2 x 1 2 k
>
>
> 2
dt
> q
<
q ð11:27Þ
d e 3 2
>
> 5 x 2 a 4 ðx 2 1 z 2 Þ 1 ψ 2 a 1 ðy 1 2 x 1 Þ
> q 2 3
> dt
>
>
>
> q
d e 4
>
>
>
4
> 52 a 5 x 2 1 ψ 2 dx 1 1 x 1 z 1 2 cy 1 1 w 1
dt
> q
:
We define the active control functions ψ ðtÞ as follows :
i
8
ψ 5 V 1 1 a 1 z 1 2 a 2 y 2 1 x 1 y 1 2 bz 1
> 1
>
ψ 5 V 2 2 a 3 x 2 1 x 2 z 2 1 w 1 2 w 2 1 x 1 1 k
<
2
2 ð11:28Þ
ψ 5 V 3 2 x 1 a 4 x 2 1 a 4 x 1 1 a 1 y 1 2 a 1 x 1
> 3 2
>
:
ψ 5 V 4 1 a 5 x 2 1 dx 1 2 x 1 z 1 1 cy 1 2 w 1
4
The terms V i ðtÞ are linear functions of the error term e i ðtÞ. With the
choice of ψ ðtÞ given by (11.28), the error system (11.27) becomes
i
8 q
d e 1
52 a 1 e 1 1 V 1 ðtÞ
>
>
> q
> dt
>
>
q
>
>
d e 2
>
>
> 52 e 2 1 V 2 ðtÞ
dt
> q
>
<
q ð11:29Þ
d e 3
>
> 52 a 4 e 3 1 V 3 ðtÞ
> q
> dt
>
>
>
> q
>
> d e 4
>
> 5 V 4 ðtÞ
> q
: dt
The control terms V i ðtÞ are chosen so that the system (11.29) becomes
stable. There is not a unique choice for such functions.
We choose
2 3 2 3
V 1 e 1
V 2 e 2
6 7 6 7
5 A
6 7 6 7 ð11:30Þ
V 3 e 3
4 5 4 5
V 4 e 4
where A is a 4 3 4 real matrix, chosen so that all eigenvalues λ i of the sys-
tem (11.30) satisfy the condition