Page 341 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 11 331

where ψ ; ψ ; ψ and ψ are the controllers which are to be determined
1 2 3 4
using active control technique. Our goal is to find suitable controllers
ψ ; ψ ; ψ and ψ such that the drive system (11.14) asymptotically
1 2 3 4
synchronizes with response system (11.16).
Out of various possible switches, in this chapter we present results for
three randomly selected error state vector combinations.
Let the switching error states be defined as :
8
e 1 5 x 2 2 y 1
>
>
<
e 2 5 y 2 2 z 1
Switch 1 ð11:17Þ
e 3 5 z 2 2 w 1
>
>
:
e 3 5 w 2 2 x 1
8
e 1 5 x 2 2 z 1
>
>
<
e 2 5 y 2 2 w 1
Switch 2 ð11:18Þ
e 3 5 z 2 2 x 1
>
>
:
e 3 5 w 2 2 y 1
where we refer to Eqs. (11.17) and (11.18) as switch (11.1) and switch
(11.2) respectively.
11.7.1 Switch 1
The error dynamical system for switch 1 is obtained as follows:
q q q
8
d e 1 d x 2 d y 1
> 5 2
>
dt dt dt
> q q q
>
>
>
>
q q q
>
d e 2 d y 2 d z 1
>
>
> 5 2
>
dt dt dt
> q q q
<
q q q ð11:19Þ
d e 3 d z 2 d w 1
>
> 5 2
> q q q
> dt dt dt
>
>
>
> q q q
>
> d e 4 d w 2 d x 1
> 5 2
>
dt dt dt
> q q q
:
Using Eqs. (11.14), (11.16) and (11.17) the error system (11.19) is
obtained as :
8 q
d e 1
>
1
> 52 a 1 x 2 1 a 2 y 2 1 ψ 2 dx 1 1 x 1 z 1 2 cy 1 1 w 1
> q
> dt
>
>
q
>
>
d e 2
>
>
>
2
> q 5 a 3 x 2 2 x 2 z 2 2 y 2 1 w 2 1 ψ 2 x 1 y 1 1 bz 1
dt
>
<
q ð11:20Þ
d e 3
> 2
> 5 x 2 a 4 ðx 2 1 z 2 Þ 1 ψ x 1 2 k
> q 2 3
> dt
>
>
>
> q
>
> d e 4
>
4
> 52 a 5 x 2 1 ψ 2 a 1 ðy 1 2 x 1 Þ
> q
: dt