A Novel Control Method for Integer Orders Chaos Systems via Fractional-Order Derivative

Abstract

A fractional-order control method is obtained to stabilize the point in chaos attractor of integer orders chaos systems. The control law has simple structure and is designed easily. Two examples are also given to illustrate the effectiveness of the theoretical result.

1. Introduction

Chaotic phenomena has been observed in many areas of science and engineering such as mechanics, electronics, physics, medicine, ecology, biology, and economy. To avoid troubles arising from unusual behaviors of a chaotic system, chaos control has received a great deal of interest among scientists from various research fields in the past few decades [1–8]. On the other hand, derivatives and integrals of fractional order have been found in many applications in recent years in physics and engineering. Many systems [4–13] are known to display fractional-order dynamics, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, and electromagnetic waves.

In the recent years, emergence of effective methods in the differentiation and integration of non integer order equations makes fractional-order systems more and more attractive for the systems control community. It is verified that the fractional-order controllers can have better disturbance rejection ratios and less sensitivity to plant parameter variations compared to the traditional controllers [14]. However, few results on control the saddle point of index 2 in integer orders chaos system are presented via fractional-order derivative. A fractional-order control method [14] is presented to stabilize the unstable equilibrium of integer orders chaos systems, but the method cannot apply to the desired point in chaotic attractor. Motivated by that, in this paper, a very simple control method is presented for a class of integer orders chaotic systems via fractional-order derivative. The order of fractional-order derivative is only determined by the eigenvalues at the desired point in integer orders chaotic attractor. The control technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous.

2. Control Chaos via Fractional-Order Derivative

Consider the following 3D integer orders chaos system:𝑋̇=𝐹(𝑋), (2.1) where 𝑋=(𝑥₁,𝑥₂,𝑥₃)^𝑇 are real state variables and 𝐹∶𝑅³→𝑅³ is differentiable function.

Let 𝑋₀=(𝑥₁₀𝑥₂₀𝑥₃₀)^𝑇 be the any point in the chaos attractor of integer orders system (2.1), 𝐴₀ the Jacobian matrix of 𝐹(𝑋) at point 𝑋₀, and 𝜆_±=𝑎±𝑗𝑏, 𝜆₃=−𝑐 the eigenvalues of 𝐴₀, respectively. In this paper, we always assume that 𝑎>0, 𝑐>0, and 𝑏≠0, especially 𝑋₀  is called the saddle point of index  2 if it also is the equilibrium point of system (2.1). In order to stabilize the point 𝑋₀  via fractional-order derivative, we introduce the following controller:𝑉(𝑡)=−𝐹𝑋₀−𝑑^𝑞𝑋𝑑𝑡^𝑞+𝑋̇, (2.2) where 0<𝑞<1 is the fractional order and will be determined later. Then, we can construct the control system for (2.1) as follows:𝑋̇=𝐹(𝑋)+𝑉(𝑡). (2.3)

Hence, the problem of stabilization of 𝑋₀ in chaos attractor of system (2.1) is shifted into the stability of 𝑋₀ in system (2.3). So, the stabilization problem of 𝑋₀ in system (2.1) can be solved if 𝑞 is suitably designed such that 𝑋₀ in system (2.3) is asymptotically stable.

Now, we state the main result in this paper as follows.

Theorem 2.1. If  0<𝑞<𝑞=₀^Δ(2/𝜋)arctan|𝑏/𝑎| in 𝑉(𝑡), then the point 𝑋₀ is asymptotically stable in control system (2.3); that is, the point 𝑋₀in chaos system (2.1) can be stabilized via fractional-order derivative.

Proof. According to system (2.3) and controller (2.2), we can obtain that 𝑑^𝑞𝑋𝑑𝑡^𝑞=𝐹(𝑋)−𝐹𝑋₀. (2.4)
It can be seen that the point 𝑋₀  is one equilibrium point of system (2.4) and the Jacobian matrix at this point for system (2.4) is the same as the Jacobian matrix at this point for system (2.1); therefore, the eigenvalues of the Jacobian matrix of (2.4) at point 𝑋₀ are 𝜆_±=𝑎±𝑗𝑏, 𝜆₃=−𝑐.
For 𝜆₃=−𝑐<0, we have arg𝜆₃=𝜋>𝑞𝜋2. (2.5) For 𝜆_±=𝑎±𝑗𝑏, we also derive that arg𝜆_±𝑏=arctan𝑎=𝑞₀𝜋2>𝑞𝜋2. (2.6) In term of (2.5) and (2.6), for any eigenvalue 𝜆 of 𝐴₀, we obtain that arg𝜆>𝜋𝑞2, (2.7) which implies that the equilibrium point 𝑋₀ of system (2.4) is asymptotically stable [15, 16]; that is, the point 𝑋₀ in the chaos attractor of integer orders chaos system (2.1) can be stabilized via fractional-order derivative; the proof is completed.

Remark 2.2. Especially, if 𝑋₀ is the saddle point of index 2 of system (2.1), then the control law obtained in this paper is similar with the controller obtained in [14]. So, Theorem 2.1 proposed by this paper extends the controller proposed by [14].

Remark 2.3. By Theorem 2.1 proposed in this paper, we know that it is possible to stabilize the closed-loop system without applying any changes on the places of the poles.

Remark 2.4. If 𝑋₀ is the same saddle point for hyperchaotic systems, which implies that the eigenvalues of 𝐴₀ are 𝜆_±=𝑎±𝑗𝑏(𝑎>0,𝑏≠0), 𝜆₃=−𝑐₁(𝑐₁>0), and 𝜆₄=−𝑐₂(𝑐₂>0), respectively, then this point 𝑋₀ in hyperchaotic systems can be stabilized via the proposed controller.

3. Applications

Now, we take Lü chaos system [17] and hyperchaotic Chen system [18] for numerical simulation, respectively. All the numerical simulation of fractional-order system in this paper is based on [19]. It is a direct time-domain approximation numerical simulation. Reference [20] has shown that using frequency-domain approximation in the numerical simulations of fractional systems may result in wrong consequences. This mistake has occurred in the recent literature that found the lowest-order chaotic systems among fractional-order systems.

We introduce the numerical solution of fractional differential equations in [19]. There have been several definitions of fractional derivatives. In the following, we introduce the most common one of them𝑑^𝑞𝑓(𝑡)𝑑𝑡^𝑞=𝐽^𝑚−𝑞𝑓^(𝑚)(𝑡),𝑞>0, (3.1) where m is the first integer which is not less than 𝑞, 𝑓^(𝑚)(𝑡) is the m-order derivative in the usual sense, and 𝐽^𝑞(𝑞>0) is the q-order Riemann-Liouville integral operator with the expression𝐽^𝑞𝑓(𝑡)=1Γ𝑞^𝑡₀(𝑡−𝜏)^𝑞−1𝑓(𝜏)𝑑𝜏. (3.2) Here, Γ(⋅) stands for gamma function, and 𝑑^𝑞/𝑑𝑡^𝑞 is generally called “q-order Caputo differential operator.”

Consider the following fractional-order system:𝑑^𝑞1𝑥𝑑𝑡^𝑞1,𝑑=𝑓𝑥,𝑦^𝑞2𝑦𝑑𝑡^𝑞2=𝑔𝑥,𝑦,0<𝑞₁,𝑞₂<1, (3.3) with initial condition (𝑥₀,𝑦₀). Now, set ℎ=𝑇/𝑁, 𝑡_𝑛=𝑛ℎ(𝑛=0,1,2…,𝑁). The above system can be discretized as follows:𝑥_𝑛+1=𝑥₀+ℎ^𝑞1Γ𝑞₁𝑓𝑥+2^𝑝𝑛+1,𝑦^𝑝_𝑛+1+^𝑛_𝑗=0𝛼_{1,𝑗,𝑛+1}𝑓𝑥_𝑗,𝑦_𝑗,𝑦_𝑛+1=𝑦₀+ℎ^𝑞2Γ𝑞₂𝑔𝑥+2^𝑝𝑛+1,𝑦^𝑝_𝑛+1+^𝑛_𝑗=0𝛼_{2,𝑗,𝑛+1}𝑔𝑥_𝑗,𝑦_𝑗, (3.4) where𝑥^𝑝_𝑛+1=𝑥₀+1Γ𝑞₁^𝑛𝑗=0𝛽_{1,𝑗,𝑛+1}𝑓𝑥_𝑗,𝑦_𝑗,𝑦^𝑝_𝑛+1=𝑦₀+1Γ𝑞₂^𝑛𝑗=0𝛽_{2,𝑗,𝑛+1}𝑔𝑥_𝑗,𝑦_𝑗, (3.5) and for 𝑖=1,2,𝛼_{𝑖,𝑗,𝑛+1}=𝑛^𝑞𝑖+1−𝑛−𝑞_𝑖(𝑛+1)^𝑞𝑖,𝑗=0,𝑛−𝑗+2^𝑞𝑖+1+𝑛−𝑗^𝑞𝑖+1−2𝑛−𝑗+1^𝑞𝑖+1,1≤𝑗≤𝑛,1,𝑗=𝑛+1, (3.6)𝛽_{𝑖,𝑗,𝑛+1}=ℎ^𝑞𝑖𝑞_𝑖𝑛−𝑗+1^𝑞𝑖−𝑛−𝑗^𝑞𝑖,0≤𝑗≤𝑛. (3.7)

The error of this approximation is described as follows: 𝑥𝑡_𝑛−𝑥_𝑛=𝑜ℎ^𝑝1,𝑝₁=min2,1+𝑞₁,𝑦𝑡_𝑛−𝑦_𝑛=𝑜ℎ^𝑝2,𝑝₂=min2,1+𝑞₂. (3.8)

Now, we choose Lü chaos system for numerical simulation. The Lü chaos system [17] is𝑑𝑥₁𝑥𝑑𝑡=36₂−𝑥₁,𝑑𝑥₂𝑑𝑡=−𝑥₁𝑥₃+20𝑥₂,𝑑𝑥₃𝑑𝑡=𝑥₁𝑥₂−3𝑥₃. (3.9)

Now, we choose (𝑥₁₀,𝑥₂₀,𝑥₃₀)=(10,9,15) in chaos attractor of chaos system (3.9). Since the eigenvalues of the Jacobian matrix at 𝑋₀ for system (3.9) are 𝜆_±=4.5554±14.2605𝑗, 𝜆₃=−28.1108, we have 𝑞=₀^Δ(2/𝜋)arctan|𝑏/𝑎|=0.8032. According to Theorem 2.1, in order to stabilize this point, we choose 𝑞<𝑞₀ for controller (2.2).

For example, the simulation results are shown in Figure 1 for 𝑞=0.7, and the initial conditions are (15,14,20)^𝑇, where 𝑒₁=𝑥₁−𝑥₁₀, 𝑒₂=𝑥₂−𝑥₂₀, and 𝑒₃=𝑥₃−𝑥₃₀.

(a)

(b)

(c)

Figure 1 

The control simulation result for (10, 9, 15) when 𝑞=0.7.

Finally, we choose hyperchaotic Chen system for numerical simulation. The hyperchaotic Chen system [18] is𝑑𝑥₁𝑥𝑑𝑡=35₂−𝑥₁+𝑥₄,𝑑𝑥₂𝑑𝑡=7𝑥₁−𝑥₁𝑥₃+12𝑥₂,𝑑𝑥₃𝑑𝑡=𝑥₁𝑥₂−3𝑥₃,𝑑𝑥₄𝑑𝑡=𝑥₂𝑥₃+0.5𝑥₄. (3.10)

Now, we choose (𝑥₁₀,𝑥₂₀,𝑥₃₀,𝑥₄₀)=(0,0,19,0) in chaos attractor of the hyperchaotic Chen system (3.10). Since the eigenvalues of the Jacobian matrix at 𝑋₀ for system (3.10) are 𝜆_±=0.4537±3.0878𝑗,𝜆₃=−23.4074, and 𝜆₄=−3, respectively, we have 𝑞=₀^Δ(2/𝜋)arctan|𝑏/𝑎|=0.9071. According to Theorem 2.1, in order to stabilize this point, we choose 𝑞<𝑞₀ for controller (2.2).

For example, the simulation results are shown in Figure 2 for 𝑞=0.8 and the initial conditions are (10,14,49,12)^𝑇, where 𝑒₁=𝑥₁−𝑥₁₀, 𝑒₂=𝑥₂−𝑥₂₀, 𝑒₃=𝑥₃−𝑥₃₀, and 𝑒₄=𝑥₄−𝑥₄₀.

(a)

(b)

(c)

(d)

Figure 2 

The control simulation result for (0, 0, 19, 0) when 𝑞=0.8.

4. Conclusion

We construct a novel control law for integer orders chaos system via fractional-order derivative. Any desired point in chaos attractor of integer orders chaos system can be stabilized via the fractional-order derivative. The order of fractional-order derivative of chaos system is only determined by the eigenvalues at the point. By comparison with the traditional controllers, we know that this controller without causing any changes in the eigenvalues of the system at the desired fixed points. The proposed controller is employed to control Lü chaotic system and hyperchaotic Chen system, and some numerical simulation results are obtained. Theoretical analysis and simulation results show that the control method in this paper is effective.

Acknowledgments

The work is supported jointly by National Natural Science Foundation of China under Grant no. 61004042 and Natural Science Foundation Project of CQ CSTC 2009BB2417.