Page 336 - Mathematical Techniques of Fractional Order Systems

326 Mathematical Techniques of Fractional Order Systems

From Eqs. (11.11) and (11.12), the error dynamical system is obtained as
α
D e ij ðtÞ 5 Nv j ðtÞ 1 Gðv j ðtÞÞ 1 φÞijðu; vÞ 2 Mu i ðtÞ 2 FðuðtÞÞ ð11:13Þ

Our goal is to design a suitable controller φ ðu; vÞ, such that the system
ij
(11.11) and (11.12) achieve multiswitching complete synchronization
in accordance with definition (11.4). In this chapter, we are considering
a four-dimensional hyperchaotic system for which the possible switches
can be obtained by imposing the conditions on i; j 5 1; 2; 3; 4. The list
of the possible errors for hyperchaotic system whose possible
combinations can be used to form the switches is as follows : For i 5 j,
we have, e 11 ; e 22 ; e 33 ; e 44 For i 6¼ j we have, e 12 ; e 13 ; e 14 ; e 21 ; e 23 ;
e 24 ; e 31 ; e 32 ; e 34

11.5 STABILITY OF FRACTIONAL ORDER SYSTEMS

Stability of fractional order systems has been thoroughly investigated where
necessary and sufficient conditions have been derived in Wolf et al. (1985).
The stability region of a linear set of fractional order equations, each of order
q, such that 0 , q , 1 is shown in Fig. 11.1. An autonomous system is
asymptotically stable iff j arg λj . απ is satisfied for all eigenvalues λ of
2
matrix A. Also this system is stable iff jargλj $ απ is satisfied for all eigen-
2
values of a matrix A and those critical eigenvalues which satisfy
jargλj . απ , and have geometric multiplicity one.
2

FIGURE 11.1 Stability of fractional order systems such that 0 , q , 1.