Page 596 - Mathematical Techniques of Fractional Order Systems

566 Mathematical Techniques of Fractional Order Systems

Employing the above two definitions, the general numerical solution of
the nonlinear FODE of form,
q
t
a D ytðÞ 5 fðytðÞ; tÞ
can be expressed by using following description:
k
q
q ðÞ
yt k ðÞ 5 fy t k ðÞ; t k Þh 2 X c yðt k2j Þ ð19:7Þ
ð
j
j51
In the present work, the above description has been used for solving
FODE and also for the simulation of the proposed results on computer.

19.2.2 Stability of Fractional Order Nonlinear Dynamical
System
Owing to increasing application of fractional order nonlinear dynamical sys-
tems, their stability analysis has attracted a great amount of interest from the
research community. Although, the stability analysis is not as simple as it is
for their linear counterpart, still some development has been made in this
regard. Matignon (1996; Petras, 2011) has shown up that one cannot use
exponential stability to analyze asymptotic stability of fractional order sys-
tems. A definition given by Tavazoei and Haeri which uses matrix properties
is given below.

Definition 1: (Tavazoei and Haeri, 2007): (For commensurate order
systems): The equilibrium points are asymptotically stable for
q 1 5 q 2 5 ... 5 q n 5 q if all the eigen values λ i ði 5 1; 2; ... ; nÞ of the
T
Jacobian matrix J 5 @f=@x, where f 5 ½f 1 ; f 2 ; ... f n Š , evaluated at equilibrium

E , satisfy the condition
π

argðeigðJÞÞ 5 argðλ i Þ . q ; i 5 1; 2 ... ; n

2
Definition 2: (Tavazoei and Haeri, 2008): (For incommensurate order
systems): When incommensurate fractional order system is considered
q 1 6¼ q 2 6¼ ... 6¼ q n and suppose that m is the LCM of the denominators u i ’s
1
of q i ’s, where q i 5 , v i ; u i AZ , for i 5 1; 2; ... ; n and set γ 5 1=m.
v i
u i
The system is asymptotically stable if
π
argðλÞ . γ

2
For all the roots λ of the following equation:
det diag ½λ mq 1 mq 2 ... λ mq n ÞÞ 2 JÞ 5 0
λ
ð
ð