An Adaptive Tracking Control of Fractional-Order Chaotic Systems with Uncertain System Parameter

Abstract

An adaptive tracking control scheme is presented for fractional-order chaotic systems with uncertain parameter. It is theoretically proved that this approach can make the uncertain parameter fractional-order chaotic system track any given reference signal and the uncertain system parameter is estimated through the adaptive tracking control process. Furthermore, the reference signal may belong to other integer-orders chaotic system or belong to different fractional-order chaotic system with different fractional orders. Two examples are presented to demonstrate the effectiveness of the proposed method.

1. Introduction

In nonlinear science, chaos synchronization is a hot topic, which has attracted much attention from scientists and engineers. In the past decades [1–13], various methodologies in control of chaotic system have been proposed, such as variable structure control approach, adaptive control approach, and time-delay feedback control. Up to date, the interest of the scientific community and engineering technological applications in controlling chaotic dynamics has increased after the development of the approach of “tracking,” which can be used to follow an unstable fixed point or an unstable periodic orbit embedded or an arbitrary given reference signal in different dynamical regimes [3, 8–11]. In electronic and laser systems, the experiments on tracking the unstable periodic orbits and steady states were carried out [12, 13]. Some researchers design a controller based on the reference signal and make the output of the chaotic system follow the given reference signal successfully [10, 11].

However, many existed tracking control methods mainly focus on chaotic systems of integer orders [11, 14–17]. Up to now, there have been only a few papers on tracking control for fractional-order chaotic systems [18, 19]. In [18], tracking control of the fractional-order hyper-chaotic Lü system was reported, in which all the system parameters were exactly known. In [19], an adaptive tracking control of the fractional-order hyper-chaotic Lorenz system with unknown parameters was also reported. But, the tracking control methods in [18, 19] were unusual and only suited for specific fractional-order chaotic system, and the given reference signals do not belong to different fractional-order chaotic systems.

Inspired by the above discussion, in this paper, a more universal adaptive tracking control method for fractional-order chaotic system with uncertain system parameter, which is different from the previous works, is given based on the stability theory of fractional-order system. Furthermore, the reference signal may belong to other integer-orders chaotic system or belong to different fractional-order chaotic system with different fractional orders. To illustrate the effectiveness of the proposed scheme, we take 3D fractional-order Lorenz chaotic system with uncertain system parameter tracking the arbitrary given reference signal and a new 3D fractional-order chaotic system with uncertain system parameter tracking the 3D fractional-order Lorenz chaotic system, for example. Numerical simulations are performed to verify the effectiveness of proposed scheme.

The paper is organized as follows. In Section 2, an adaptive tracking control scheme is presented. In Section 3, two groups of examples are used to verify the effectiveness of the proposed scheme. The conclusion is finally drawn in Section 4.

2. Adaptive Tracking Control Scheme for Fractional-Order Chaotic Systems with Uncertain System Parameter

There are several definitions of fractional derivatives. In this paper, the Caputo-type fractional derivative will be used. The Caputo definition of the fractional derivative, which sometimes is called smooth fractional derivative, is described as𝑑^𝑞𝑓(𝑡)𝑑𝑡^𝑞≡𝐷^𝑞𝑓(𝑡)=1Γ𝑚−𝑞^𝑡₀𝑓^(𝑚)(𝜏)(𝑡−𝜏)^{𝑞+1−𝑚}𝑑𝜏,𝑚−1<𝑞<𝑚, (2.1) where 𝑚 is the smallest integer larger than 𝑞, 𝑓^(𝑚)(𝑡) is the 𝑚-order derivative in the usual sense, and Γ(⋅) stands for gamma function.

Now, we consider the fractional order system denoted as system (2.2) below𝐷^𝑞𝐱=𝐟(𝐱,𝛼), (2.2) where 𝐱∈𝑅^𝑛 and 𝐟∶𝑅^𝑛→𝑅^𝑛 are differentiable functions. Parameter 𝛼 is one of the system parameters.

When parameter 𝛼 is unknown, a controller 𝜿 is added to the original system (2.2), we obtain𝐷^𝑞𝐱=𝐟(𝐱,𝛼)+𝜿, (2.3) where 𝛼 is an unknown parameter to be estimated and 𝜿 is an 𝑛×1 real matrix to be designed.

Let 𝐲∈𝑅^𝑛 be an arbitrary given reference signal. Our goal is to design the controller 𝜿 with an adaptive parameter such that the output signal 𝐱 of system (2.3) follows the reference signal 𝐲 ultimately and the uncertain parameter 𝛼 could be identified. That islim_𝑡→+∞‖𝐞‖=lim_𝑡→+∞‖𝐱−𝐲‖=0,lim_𝑡→+∞𝑒_𝛼=lim_𝑡→+∞(𝛼−𝛼)=0, (2.4) where ‖⋅‖ is the Euclidean norm and 𝛼 is the “true” value of the “unknown” parameter 𝛼.

Remark 2.1. The reference signal 𝐲∈𝑅^𝑛 may belong to other integer-orders chaotic system or belong to different fractional-orders chaotic system with different fractional orders.
First, the controller 𝜿 is designed as follows: 𝜿=𝜿₁(𝐲)+𝜿₂, (2.5) where 𝜿₁(𝐲) and 𝜿₂ are 𝑛×1 real matrix. 𝜿₁(𝐲) and 𝜿₂ are a compensation controller and a feedback controller, respectively. If the true value of the “unknown” parameter 𝛼 is chosen as 𝛼, then the definition of the compensation controller 𝜿₁(𝐲) is 𝜿₁(𝐲)=𝐷^𝑞𝐲−𝐟(𝐲,𝛼). (2.6)

Remark 2.2. For defining the controllers (2.6) one has to know already the correct value of the parameter 𝛼, this point is the same as the master-slave (or drive-response) adaptive chaos synchronization with uncertain parameter. In master-slave (or drive-response) adaptive chaos synchronization with uncertain parameter, the “true” value of unknown parameter in slave system (or response system) is the correct value of parameter in master system (or drive system).
Let tracking errors 𝑒_𝑖=𝑥_𝑖−𝑦_𝑖(𝑖=1,2,…,𝑛) and parameter error 𝑒_𝛼=𝛼−𝛼. So, (2.3) can be changed as 𝐷^𝑞𝐞=𝐟(𝐱,𝛼)−𝐟(𝐲,𝛼)+𝜿₂(𝐱,𝐲). (2.7)
In generally, we can assume 𝐟(𝐱,𝛼)−𝐟(𝐲,𝛼)=𝝆₁()𝐞𝑒𝐱,𝐲,𝛼_𝛼, (2.8) where 𝝆₁(𝐱,𝐲,𝛼) is an 𝑛×(𝑛+1) real matrix, 𝐞=(𝑒₁𝑒₂⋯𝑒_𝑛)^𝐓, and (^𝐞_𝑒𝛼) is an (𝑛+1)×1 real matrix.
Now, the feedback controller 𝜿₂ is chosen as, 𝜿₂=𝝆₂()𝐞𝑒𝐱,𝐲,𝛼_𝛼, (2.9) where 𝝆₂(𝐱,𝐲,𝛼) is an 𝑛×(𝑛+1) real matrix to be designed later. According to (2.8) and (2.9), (2.3) or (2.7) can be rewritten as 𝐷^𝑞𝐞=𝝆₁(𝐱,𝐲,𝛼)+𝝆₂(𝐞𝑒𝐱,𝐲,𝛼)_𝛼. (2.10)
Second, the parameter update laws is chosen as 𝐷^𝑞𝛼=𝛼_𝑛+1,1𝑒₁+𝛼_𝑛+1,2𝑒₂+⋯+𝛼_𝑛+1,𝑛𝑒_𝑛+𝛼_{𝑛+1,𝑛+1}𝑒_𝛼𝐞𝑒=𝜶_𝛼, (2.11) where 𝜶 is an 1×(𝑛+1) real matrix. Because the Caputo derivative of a constant is zero, so (2.11) can be rewritten as 𝐷^𝑞𝑒_𝛼=𝐷^𝑞()𝐞𝑒𝛼−𝛼=𝜶_𝛼. (2.12)
According to (2.10) and (2.12), we can obtain 𝐷^𝑞𝐞𝐷^𝑞𝑒_𝛼=𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶𝐞𝑒𝐱,𝐲,𝛼_𝛼, (2.13) where (^{𝝆1(𝐱,𝐲,𝛼)+𝝆2}_𝜶^{(𝐱,𝐲,𝛼)}) is an (𝑛+1)×(𝑛+1) real matrix.
By (2.13), we know that the fractional-order system (2.3) tracking an arbitrary given reference signal 𝐲 and the uncertain parameter 𝛼 that could be identified are transformed into the following problem: choose a suitable matrix 𝝆₂(𝐱,𝐲,𝛼) and 𝜶 such that system (2.13) converges asymptotically to zero.

Theorem 2.3. For an arbitrary given reference signal 𝐲, the synchronization between fractional-order systems (2.3) and reference signal 𝐲 will be obtained and the uncertain parameter 𝛼 will be estimated if 𝐏𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶+𝐱,𝐲,𝛼𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶𝐱,𝐲,𝛼^𝐇𝐏=−𝐐, (2.14) where 𝐏 is a real symmetric positive definite matrix, 𝐐 is a real symmetric positive semidefinite matrix, and 𝐇 stands for conjugate transpose of a matrix.

Proof. Assume that 𝜆 is one of the eigenvalues of matrix (^{𝝆1(𝐱,𝐲,𝛼)+𝝆2}_𝜶^{(𝐱,𝐲,𝛼)}) and the corresponding nonzero eigenvector is 𝛽, that is, 𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶𝐱,𝐲,𝛼𝛽=𝜆𝛽. (2.15)
Multiply the above equation left by 𝛽^𝐇𝐏, we derive that 𝛽^𝐇𝐏𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶𝛽𝐱,𝐲,𝛼=𝛽^𝐇𝐏𝜆𝛽. (2.16)
Then, by a similar argument, we also can obtain that 𝛽^𝐇𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶𝐱,𝐲,𝛼^𝐇𝐏𝛽=𝜆𝛽^𝐇𝐏𝛽. (2.17)
From (2.16) and (2.17), we can obtain 𝜆+𝛽𝜆=^𝐇𝐏^{𝝆1()𝐱,𝐲,𝛼+𝝆2()}_𝜶^{𝐱,𝐲,𝛼}+^{𝝆1()𝐱,𝐲,𝛼+𝝆2()}_𝜶^{𝐱,𝐲,𝛼𝐇}𝐏𝛽𝛽^𝐇𝐏𝛽. (2.18)
Since 𝐏^{𝝆1(𝐱,𝐲,𝛼)+𝝆2}_𝜶^{(𝐱,𝐲,𝛼)}+^{𝝆1(𝐱,𝐲,𝛼)+𝝆2}_𝜶^{(𝐱,𝐲,𝛼)𝐇}𝐏=−𝐐 and 𝐏, 𝐐 are real symmetric positive definite matrix and real symmetric positive semidefinite matrix, respectively, then 𝛽^𝐇𝐐𝛽≥0,𝛽^𝐇𝐏𝛽>0, (2.19) so 𝜆+𝛽𝜆=−^𝐇𝐐𝛽𝛽^𝐇𝐏𝛽≤0. (2.20)
From (2.20), we have ≥𝜋arg𝜆2>𝑞𝜋2. (2.21)
According to the stability theory of fractional-order systems [20], the equilibrium point in (2.13) is asymptotically stable.
Therefore, lim_𝑡→+∞𝑒_𝑖=lim_𝑡→+∞𝑥_𝑖−𝑦_𝑖=0(𝑖=1,2,…,𝑛),lim_𝑡→+∞𝑒_𝛼=lim_𝑡→+∞(𝛼−𝛼)=0, (2.22) which indicates that the output signal 𝐱 of fractional-order chaotic system (2.3) can track the reference signal 𝐲 ultimately and the uncertain parameter 𝛼 will be estimated. The proof is completed.

3. Illustrative Example

To illustrate the effectiveness of the proposed scheme, two examples are considered and their numerical simulations are performed.

3.1. The 3D Fractional-Order Lorenz Chaotic System with Uncertain Parameter Tracking the Arbitrary Given Reference Signal

The 3D fractional-order Lorenz [21] system is given by𝑑^𝑞𝑥₁𝑑𝑡^𝑞=𝜎𝑥₂−𝑥₁,𝑑^𝑞𝑥₂𝑑𝑡^𝑞=𝛼𝑥₁−𝑥₁𝑥₃−𝑥₂,𝑑^𝑞𝑥₃𝑑𝑡^𝑞=𝑥₁𝑥₂−𝛽𝑥₃, (3.1) where system parameters (𝜎,𝛼,𝛽)=(10,28,8/3). The fractional-order Lorenz system exhibits chaotic behavior for fractional-order 0.993≤𝑞<1. The chaotic attractor for 𝑞=0.998 is shown in Figure 1.

Figure 1 

Chaotic attractors of the fractional-order Lorenz system for 𝑞=0.998.

Let us assume that parameter 𝛼 is unknown in the fractional-order Lorenz system. The fractional-order Lorenz system with uncertain parameter 𝛼 is given by𝑑^𝑞𝑥₁𝑑𝑡^𝑞=𝜎𝑥₂−𝑥₁,𝑑^𝑞𝑥₂𝑑𝑡^𝑞=𝛼𝑥₁−𝑥₁𝑥₃−𝑥₂,𝑑^𝑞𝑥₃𝑑𝑡^𝑞=𝑥₁𝑥₂−𝛽𝑥₃. (3.2)

Now, we investigate the fractional-order Lorenz chaotic system with uncertain system parameter tracking the arbitrary given reference signal. We take reference signal 𝐲=(10sin𝑡,5cos𝑡,5)^𝐓, for example. According to above mentioned, we can obtain𝝆₁(𝐱,𝐲,𝛼)=−101000𝛼−𝑥₃−1−10sin𝑡𝑥₁𝑥₂810sin𝑡−30. (3.3)

Now, the parameter update laws and real matrix 𝝆₂(𝐱,𝐲,𝛼) are chosen as𝐷^𝑞𝛼=−𝑥₁𝑒₂,𝝆₂(𝐱,𝐲,𝛼)=0𝑥₃−𝛼−10−𝑥₂000000000. (3.4)

Therefore,𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶=𝐱,𝐲,𝛼−10𝑥₃−𝛼−𝑥₂0𝛼−𝑥₃−1−10sin𝑡𝑥₁𝑥₂810sin𝑡−300−𝑥₁00. (3.5)

Choosing real symmetric positive definite matrix 𝐏=diag(1,1,1,1), we can yield𝐏𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶+𝐱,𝐲,𝛼𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶𝐱,𝐲,𝛼^𝐇𝐏=diag−20,−2,−163,0. (3.6)

Choosing symmetric positive semidefinite matrix 𝐐=diag(20,2,16/3,0), we can obtain𝐏𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶+𝐱,𝐲,𝛼𝝆₁(𝐱,𝐲,𝛼)+𝝆₂()𝜶𝐱,𝐲,𝛼^𝐇𝐏=−𝐐. (3.7)

According to the above Theorem, the fractional-order Lorenz system (3.2) with uncertain parameters 𝛼 can track the reference signal 𝐲=(10sin𝑡,5cos𝑡,5)^𝐓 ultimately, and the uncertain parameters 𝛼 will be estimated. The corresponding numerical result is shown in Figure 2, in which the initial conditions are 𝐱=(1,1,2)^𝐓, 𝛼(0)=20 for system (3.2), and 𝜀=(∑³_𝑖=1𝑒_𝑖²)^1/2, the “true” value of the “unknown” parameter is chosen as 𝛼=28, respectively.

(a)

(b)

Figure 2 

(a) shows tracking errors 𝜀 between the fractional-order Lorenz system (3.2) with uncertain parameter 𝛼 and the reference signal 𝐲=(10sin𝑡,5cos𝑡,5)𝐓. (b) shows the parameter error 𝜀𝛼 between uncertain parameter 𝛼 and the true value of the “unknown” parameter.

3.2. A New 3D Fractional-Order Chaotic System with Uncertain System Parameter Tracking the 3D Fractional-Order Lorenz Chaotic System

A new 3D fractional-order chaotic system [22, 23] is reported by Sheu et al. It is described by𝑑^𝑞′𝑧₁𝑑𝑡^𝑞′=𝛼𝑧₁−𝑧₂𝑧₃,𝑑^𝑞′𝑧₂𝑑𝑡^𝑞′=−𝛽𝑧₂+𝑧₁𝑧₃,𝑑^𝑞′𝑧₃𝑑𝑡^𝑞′=𝑧₁𝑧₂3−𝛾𝑧₃, (3.8) where system parameters (𝛼,𝛽,𝛾)=(5,10,3.8). The chaotic attractor for 𝑞^′=0.9 is shown in Figure 3.

Figure 3 

Chaotic attractor of a new fractional-order system (3.8) for 𝑞′=0.9.

Let us assume that parameter 𝛼 is unknown in the fractional-order system (3.8). The fractional-order system with uncertain parameter 𝛼 is described by𝑑^𝑞′𝑧₁𝑑𝑡^𝑞′=𝛼𝑧₁−𝑧₂𝑧₃,𝑑^𝑞′𝑧₂𝑑𝑡^𝑞′=−𝛽𝑧₂+𝑧₁𝑧₃,𝑑^𝑞′𝑧₃𝑑𝑡^𝑞′=𝑧₁𝑧₂3−𝛾𝑧₃. (3.9)

Now, we investigate the fractional-order system (3.9) with uncertain system parameter 𝛼 tracking the fractional-order Lorenz system (3.1). That is the reference signal belongs to the fractional-order Lorenz system (3.1), that is, 𝐲=(𝑥₁,𝑥₂,𝑥₃)^𝐓. So, the reference signal may belong to different fractional-order chaotic system with different fractional orders.

In this case, matrix 𝝆₁(𝐱,𝐲,𝛼) and 𝝆₂(𝐱,𝐲,𝛼) can be written as 𝝆₁(𝐳,𝐱,𝛼) and 𝝆₂(𝐳,𝐱,𝛼), respectively. According to above mentioned, we can obtain𝝆₁(𝐳,𝐱,𝛼)=𝛼−𝑥₃−𝑧₂𝑧₁𝑥₃−10𝑧₁0𝑧₂3𝑥₁3−3.80. (3.10)

The parameter update laws and real matrix 𝝆₂(𝐱,𝐱,𝛼) are chosen as𝐷^𝑞𝛼=−𝑧₁𝑒₁,𝝆₂(𝐳,𝐱,𝛼)=−𝛼−102𝑧₂3000000−𝑧₁−𝑥₁300. (3.11)

Therefore,𝝆₁(𝐳,𝐱,𝛼)+𝝆₂()𝜶=𝐳,𝐱,𝛼−1−𝑥₃−𝑧₂3𝑧₁𝑥₃−10𝑧₁0𝑧₂3−𝑧₁−3.80−𝑧₁000. (3.12)

Choosing real symmetric positive definite matrix 𝐏=diag(1,1,1,1), we can yield𝐏𝝆₁(𝐳,𝐱,𝛼)+𝝆₂()𝜶+𝐳,𝐱,𝛼𝝆₁(𝐳,𝐱,𝛼)+𝝆₂()𝜶𝐳,𝐱,𝛼^𝐇𝐏=diag(−2,−20,−7.6,0). (3.13)

Choosing symmetric positive semidefinite matrix 𝐐=diag(2,20,7.6,0), we can obtain𝐏𝝆₁(𝐳,𝐱,𝛼)+𝝆₂()𝜶+𝐳,𝐱,𝛼𝝆₁(𝐳,𝐱,𝛼)+𝝆₂()𝜶𝐳,𝐱,𝛼^𝐇𝐏=−𝐐. (3.14)

According to the above theorem, the fractional-order system (3.9) with uncertain parameter 𝛼 can track the fractional-order Lorenz system (3.1) ultimately and the uncertain parameter 𝛼 will be estimated. The corresponding numerical result is shown in Figure 4, in which the initial conditions are 𝐱=(1,1,2)^𝐓 for fractional-order Lorenz system (3.1), and 𝑧=(3,3,4)^𝐓, 𝛼(0)=8 for fractional-order system (3.9), and 𝜀=(∑³_𝑖=1𝑒_𝑖²)^1/2, the “true” value of the “unknown” parameter is chosen as 𝛼=5, respectively.

(a)

(b)

Figure 4 

(a) shows tracking errors 𝜀 between the fractional-order system (3.9) with uncertain parameter 𝛼 and the reference signal belongs to the fractional-order Lorenz system (3.1). (b) shows the parameter error 𝜀𝛼 between uncertain parameter 𝛼 and the true value of the “unknown” parameter.

4. Conclusion

In this paper, a more universal adaptive tracking control scheme for fractional-order chaotic systems with uncertain parameter is addressed. Based on the stability theory of fractional-order system, an adaptive controller is designed and the parameter update law for estimating the unknown parameter of the systems is also gained. The 3D fractional-order Lorenz chaotic system with uncertain system parameter tracking the arbitrary given reference signal and a new 3D fractional-order chaotic system with uncertain system parameter tracking the 3D fractional-order Lorenz chaotic system are chosen to illustrate the proposed method. The numerical simulations demonstrate the validity and feasibility of the proposed method.

Acknowledgment

The work is supported by Foundation of Science and Technology project of Chongqing Education Commission under Grant KJ110525.