Page 599 - Mathematical Techniques of Fractional Order Systems

Control and Synchronization Chapter | 19 569

1996. It is basically an extended version of the famous Lorenz system. Further
investigations and analysis were made in Zhou et al. (1997). The integer order
Lorenz Stenflo system can be described by following expressions.
_ xðtÞ 5 ayðtÞ 2 axðtÞ 1 dwðtÞ
_ yðtÞ 5 cxðtÞ 2 yðtÞ 2 xðtÞzðtÞ
ð19:15Þ
_ zðtÞ 5 xðtÞyðtÞ 2 bzðtÞ
_ wðtÞ 52 xðtÞ 2 awðtÞ
where x, y, z, and w are the system states and a, b, c, and d are the system
parameters. The hyperchaotic behavior of the above system is shown in
Fig. 19.1. The system parameters are taken as a 5 1, b 5 0:5, c 5 26, and
d 5 1:5. The simulation has been carried out for 200 seconds for the initial
conditions x 0ðÞ; y 0ðÞ; z 0ðÞ; wð0ÞÞ 5 ð2 0:592; 0:04; 0:72; 0:5Þ.
ð
The dynamic analysis of fractional order version of the hyperchaotic
Lorenz Stenflo system was first discussed in Wang et al. (2014). System
dynamics of fractional order Lorenz Stenflo system are as follows:

q 1
D t xðtÞ 5 ayðtÞ 2 axðtÞ 1 dwðtÞ
q 2
D t yðtÞ 5 cxðtÞ 2 yðtÞ 2 xðtÞzðtÞ
ð19:16Þ
q 3
D t zðtÞ 5 xðtÞyðtÞ 2 bzðtÞ
ðq 4 Þ
D t wðtÞ 52 xðtÞ 2 awðtÞ
where q 1 ; q 2 ; q 3 , and q 4 are the orders of the fractional derivatives. The
hyperchaotic behavior of the above system is depicted in Fig. 19.2 for the
system parameters as a 5 1, b 5 0:5, c 5 26, and d 5 1:5. The order of deri-
vatives is taken as q 1 5 q 2 5 q 3 5 q 4 5 0:98 and the simulation time is 200
seconds. The initial conditions are chosen as ð x 0ðÞ; y 0ðÞ; z 0ðÞ; wð0ÞÞ 5
ð1; 0:2; 0:2; 2 0:2Þ.

19.4 CONTROL AND SYNCHRONIZATION VIA
BACKSTEPPING TECHNIQUE
As the dynamics of the fractional order Lorenz Stenflo system given in
(19.16), are not in strict-feedback form, a simple transformation has to be
carried out. For, w 5 x 1 , x 5 x 2 , y 5 x 3 , z 5 x 4 and q 1 5 q 2 5 q 3 5 q 4 5 q, the
system is transformed to:
q
D t x 1 52 x 2 2 ax 1
q
D t x 2 5 ax 3 2 ax 2 1 dx 1
q ð19:17Þ
D t x 3 5 cx 2 2 x 2 x 4 2 x 3 1 u
q
D t x 4 5 x 2 x 3 2 bx 4
where, u is the controller to be designed. As the above system is in the
desired form, two different strategies can be applied for control and synchro-
nization. In this section, the backstepping strategy is presented and in the
next section active backstepping technique will be described.