Page 335 - Mathematical Techniques of Fractional Order Systems

Multiswitching Synchronization Chapter | 11 325

and
5 n α11 α11 ; j 5 0
a j;n11 2 ðn 2 αÞðn11Þ
α11 α11 α11
5 ðn2j12Þ 2 ðn2jÞ 2 2ðn2j11Þ ;
1 # j # n
h α α α
5
b j;n11 ððn2j11Þ 2 ðn2jÞ Þ
α
where
θ
maxðjxðt j Þ 2 x h ðt j ÞÞjÞ 5 Oðh Þ; ðj 5 0; 1; 2; :::; NÞ
is the estimation error and θ 5 minð2; 1 1 αÞ

11.4 PROBLEM FORMULATION
In this section, we describe the problem formulation to achieve the multi-
switching complete synchronization between fractional order hyperchaotic
systems by applying active control technique.
Consider the following chaotic system having fractional order α as
master system:
α
D uðtÞ 5 MuðtÞ 1 FðuðtÞÞ ð11:11Þ
where α represents the fractional order, uðtÞAR n 3 1 is the state vector.
MAR n 3 n is a constant matrix, MuðtÞAR n 3 1 describes the linear terms and
FðuðtÞÞAR n 3 1 represents the nonlinear terms in the system (11.11).
Consider the subsequent fractional order system, which acts as slave
system as:
α
D vðtÞ 5 NvðtÞ 1 GðvðtÞÞ 1 ψ ðuðtÞ; vðtÞÞ ð11:12Þ
ij
where vðtÞAR n 3 1 is the state vector. NAR n 3 n is a constant matrix,
NvðtÞAR n 3 1 describes the linear term, GðvðtÞÞAR n 3 1 describes the nonlinear
terms in the system (11.12) and ψðuðtÞ; vðtÞÞAR n 3 1 is the real feedback con-
troller which is to be designed.

Definition 4: The two systems (11.11) and (11.12) are said to achieve com-
plete synchronization, if
lim Oe ij ðtÞO 5 lim Ov j ðtÞ 2 u i ðtÞO 5 0
t- 1 N t- 1 N
where the symbols O O symbolize the matrix norm.

Now the following result is established on the controller of the fractional
order chaotic systems, which give the final destination to the problem
formulation.