Page 598 - Mathematical Techniques of Fractional Order Systems

568 Mathematical Techniques of Fractional Order Systems

Lemma 1: (Ding et al., 2015) Power law for fractional order derivative:
Let xðtÞAR be a real continuously differentiable function. Then for any
n
p 5 2 ,nAN,
α
α p
D x tðÞ # px p21 ðtÞD xðtÞ ð19:12Þ
t t
where 0 , α , 1 is the fractional order.
From the above lemma, the following corollary is concluded:

Corollary 1: (Aguila-Camacho et al., 2014)Let xtðÞAR be a continuous and
derivable function. Then, for any time t
1 α 2 α
D x ðtÞ # xðtÞD xðtÞ; ’αAð0; 1Þ ð19:13Þ
t
t
2
Theorem 3: (Ding et al., 2015) The fractional order system (19.8), with
n
control law u 5 βðxÞ is stable if for a p 5 2 , nAN,
p21 α p21
x D xtðÞ 5 x fðx; βðxÞÞ # 0 ð19:14Þ
t
and the system with u 5 βðxÞ is asymptotically Mittag Leffler stable if
x p21 fx; β xðÞÞ , 0
ð
The conclusion can be summarized now in the form of a theorem as
given below:

Theorem 4: (Ding et al., 2015) The stability of the system (19.8) with a con-
troller defined as u 5 βðxÞ is guaranteed by the Lyapunov function Vt; xtðÞÞ
ð
and its fractional order derivative if following condition is met, where γ is
class K function.
α
D Vt; xtðÞð Þ #2 γgx; β xðÞÞ;
ð
t
also if
α
ð
D Vt; xtðÞð Þ ,2 γgx; β xðÞÞ;
t
then it can be concluded that, the corresponding controller u stabilizes the
system globally and asymptotically.

19.3 SYSTEM DESCRIPTION
The fractional order hyperchaotic system which is under focus in this chapter
is Lorenz Stenflo system. A hyperchaotic system generally exhibits chaotic
behavior with at least two positive Lyapunov exponents. Initially, let us have a
brief description of traditional (integer order) Lorenz Stenflo system. The
integer order version of this system was first proposed by Stenflo (1996)in