Page 639 - Mathematical Techniques of Fractional Order Systems

610 Mathematical Techniques of Fractional Order Systems

20.6 SIMULATION RESULTS
The chaotic behaviors in a fractional order modified Duffing system (gyro
systems) studied numerically by phase portraits are given by Lin et al.
(2012) and Hosseinnia et al. (2010). In this section, we will apply our adap-
tive fuzzy robust H N controller via sliding mode to synchronize two differ-
ent fractional order chaotic gyro systems.
Consider the following two fractional order chaotic systems (Lin and
Balas, 2011, Hosseinnia et al., 2010; Kuo et al., 2011):
Response system:
q
D 5 x 2
x 1
2100 1
3
3
q
D x 2 5 x 1 1 x 2 0:7x 2 2 0:08x 1 sinðx 1 Þ 1 33sinð2tÞx 1
2
1
4 12
1 3
2 x 2 0:1sinðx 1 Þ 1 dðtÞ 1 uðtÞ
1
6
ð20:46Þ
Drive system:
q
D 5 y 2
y 1
100 1 1
3
q
3
3
D 52 y 1 1 y 2 0:5y 2 2 0:05y 1 sin y 1 1 35:5sin 2tðÞy 1 2 x 1 dtðÞ
ðÞ
4 12 6
y 2 1 2 1
ð20:47Þ
where the external disturbance dðtÞ 5 0:3 sin ðtÞ: The main objective is to
control the trajectories of the response system to track the reference trajecto-
ries obtained from the drive system. The initial conditions of the drive and
response systems are chosen as:

x 1 ð0Þ 0:25 y 1 ð0Þ 0:2
5 and 5 , respectively.
x 2 ð0Þ 0:25 y 2 ð0Þ 0:2
For the other constants of design are fixed as follows: k 1 5 k 2 5 1,
r 1 5 175, r 2 5 37, r 3 5 75; r 4 5 7, h 5 0:001; and Tsim 5 40s:
The simulations results for fractional order q 5 0:98 are illustrated as
follows:
Fig. 20.2 represents the 3D phase portrait of the drive and response sys-
tems without control input. It is obvious that the synchronization perfor-
mance is bad without a control effort supplied to the response system.
The different values of 0 , q , 1 are considered in order to show the
robustness of the proposed adaptive fuzzy H N control with our law.