1. Introduction
Fractional calculus is a classical mathematical concept, with a history as long as calculus itself. It is a generalization of ordinary differentiation and integration to arbitrary order, and is the fundamental theories of fractional order dynamical systems. Fractional-order differential/integral has been applied in physics and engineering, such as viscoelastic system [1], dielectric polarization [2], electrode-electrolyte polarization [3] and electromagnetic wave [4], and so on.
The fractional order system and its potential application in engineering field become promising and attractive due to the development of the fractional order calculus. Typically, chaotic systems remain chaotic when their equations become fractional. For example, it has been shown that the fractional order Chua’s circuit with an appropriate cubic nonlinearity and with an order as low as 2.7 can produce a chaotic attractor [5].
However, there are essential differences between ordinary differential equation systems and fractional order differential systems. Most properties and conclusions of ordinary differential equation systems cannot be extended to that of the fractional order differential systems. Therefore, the fractional order systems have been paid more attention. Recently, many investigations were devoted to the chaotic dynamics and chaotic control of fractional order systems [6-12].
In this paper, practical scheme is proposed to eliminate the chaotic behaviors in fractional order system by extending the nonlinear feedback control in ODE systems to fractional-order systems. This paper is organized as follows. In Section 2, the numerical algorithm for the fractional order system is briefly introduced. In Section 3, Dynamics of the fractional order system is numerically studied. In section 4, general approach to feedback control scheme is given, and then we have extended this control scheme to fractional order system, numerical results are shown. Finally, in Section 5, concluding comments are given.
2. Fractional Derivative and Numerical Algorithm
There are two approximation methods for solving fractional differential equations. The first one is an improved version of the Adams-Bashforth-Moulton algorithm, and the rest one is the frequency domain approximation. The Caputo derivative definition involves a time-domain computation in which nonhomogenous initial conditions are needed, and those values are readily determined. In this paper, the Caputo fractional derivative defined in [13] is often described by
when 
is the first integer that is not less than
, 
is the α-order Riemann-Liouville integral operator which defined by
where 
is the Gamma function, 
Now we consider the fractional order system [14] which is given by
(1)
where 
is the fractional order, 
By exploiting the Adams-Bashforth-Moulton scheme [15], the fractional order system (1) can be discretized as followings:
3. Dynamic Analysis of the Fractional Order System
Theorem 1: The fractional linear autonomous system
is locally asymptotically stable if and only if
Theorem 2: Suppose 
be an equilibrium point of a fractional nonlinear system
If the eigenvalues of the Jacobian matrix 
satisfy
then the system is locally asymptotically stable at the equilibrium point 
The system (1) has five equilibrium points:
where 
When 
we obtain
First, we choose 
to study, the eigenvalues of the Jacobian matrix are 
and 
We can obtain 
and 
According to Theorem 2, we can easily conclude that the equilibrium 
of system (1) is unstable when 
and 
are all greater than zero.
We choose 
and 
to study, the eigenvalues of the Jacobian matrix are 
and 
We can obtain 
and 
According to Theorem 2, we can easily conclude that when 
and 
are all less than 
the equilibrium 
of system (1) is stable. On the contrary, when 
and 
are all great than
, the equilibrium 
of system (1) is unstable.
Finally , when choose 
and 
to study, the eigenvalues of the Jacobian matrix are 
and 
We can obtain 
and 
According to Theorem 2, we can easily conclude that when 
and 
are all great than
, the equilibrium 
of system (1) is unstable.
In sum, there exists at least one stable equilibrium 
and 
of system (1), when 
and 
are all less than
, i.e., the system (1) will be stabilized at one point 
finally; when 
and 
are all greater than
, all the equilibriums of system (1) are unstable, the system (1) will exhibit a chaotic behaviour; when 
the problem will be complicated, the system (1) may be convergent, periodic or chaotic. For example, when 
the value of the largest Lyapunov exponent is 0.1653. Obviously, the fractional order system (1) is chaotic. When 
the fractional order system (1) is not chaotic, but periodic orbits appear.
4. Feedback Control
Let us consider the fractional order system
(2)
where 
is the system state vector, and 
the control input vector. Given a reference signal 
the problem is to design a controller in the state feedback form:
where 
is the vector-valued function, so that the controlled system
can be driven by the feedback control g(x, t) to achieve the goal of target tracking so we must have
Let 
be a periodic orbit or fixed point of the given system (2) with
, then we obtain the system error
where 
and 
Theorem 3: If 
is a fixed point of the system (2) and the eigenvalues of the Jacobian matrix at the equilibrium point 
satisfies the condition
then the trajectory 
of system (2) converge to 
Let us consider the fractional order system (2), we propose to stabilize unstable periodic orbit or fixed point, the controlled system is as follows:
(3)
Since 
is solution of system (1), then we have:
(4)
Subtracting (4) from (3) with notation, 
we obtain the system error
(5)
We define the control function as follow
(6)
So the system error (5) becomes
(7)
The Jacobian matrix of system (7) is
so we have the eigenvalues 
and 
When 
all eigenvalues are real negatives, one has 
therefore 
for all 
satisfies 
it follows from Theorem 3 that the trajectory 
of system (2) converges to 
and the control is completed.
5. Numerical Simulation
In this section we give numerical results which prove the performance of the proposed scheme. As mentioned in Section 2 we have implemented the improved AdamsBashforth-Moulton algorithm for numerical simulation.
The control can be started at any time according to our needs, so we choose to activate the control when 
in order to make a comparison between the behavior before activation of control and after it.
For 
and q3 = 0.98, unstable point 
has been stabilized, as shown in Figure 1, note that 
The control is activated when 
and the evolution of 
is chaotic, then when the control is started at 
we see that 
is rapidly stabilized.
For 
and 
the unstable point 
has been stabilized, as shown in Figure 2.
For 
and 
the unstable point 
has been stabilized, as shown in Figure 3.
For 
the unstable point 
has been stabilized, as shown in Figure 4.
For 
the unstable point 
has been stabilized, as shown in Figure 5.
When 
is less than
, there is a chaotic behavior,
Figure 1. Stabilizing the equilibrium point S0 for q1 = 0.93, q2 = 0.95 and q3 = 0.98.
Figure 2. Stabilizing the equilibrium point S1 for q1 = 0.93, q2 = 0.95 and q3 = 0.98.
Figure 3. Stabilizing the equilibrium point S2 for q1 = 0.92 and q2 = q3 = 0.97.
Figure 4. Stabilizing the equilibrium point S3 for q1 = q2 = q3 = 0.94.
Figure 5. Stabilizing the equilibrium point S4 for q1 = q2 = q3 = 0.96.
but when the control is activated at
, the five points 
and 
are rapidly stabilized.
6. Conclusions
Chaotic phenomenon makes prediction impossible in the real world; then the deletion of this phenomenon from fractional order system is very useful, the main contribution of this paper is to this end.
In this paper, we investigate the system with fractional order applying the fractional calculus technique. According to the stability theory of the fractional order system, dynamical behaviors of the fractional order system are analyzed, both theoretically and numerically. Furthermore, nonlinear feedback control scheme has been extended to control fractional order system. The results are proved analytically by stability condition for fractional order system. Numerically the unstable fixed points have been successively stabilized for different values of 
and 
Numerical results have verified the effectiveness of the proposed scheme.
7. Acknowledgements
This work is supported by the Qing Lan Project of Jiangsu Province under the Grant Nos. 2010 and the 333 Project of Jiangsu Province under the Grant Nos. 2011.
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