Mathematical Problems in EngineeringVolume 2011 (2011), Article ID 521549, 11 pagesdoi:10.1155/2011/521549

Research Article

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

Ping Zhou1,2 and Rui Ding1,2

¹Center of System Theory and Its Applications, Chongqing University of Posts and Telecommunications, Chongqing 400065, China²Key Laboratory of Network Control and Intelligent Instrument of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Received 19 April 2011; Revised 22 August 2011; Accepted 22 September 2011

Academic Editor: J. Rodellar

Copyright © 2011 Ping Zhou and Rui Ding. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 as where is the smallest integer larger than , is the -order derivative in the usual sense, and stands for gamma function.

Now, we consider the fractional order system denoted as system (2.2) below where and are differentiable functions. Parameter is one of the system parameters.

When parameter is unknown, a controller is added to the original system (2.2), we obtain where is an unknown parameter to be estimated and is an real matrix to be designed.

Let be an arbitrary given reference signal. Our goal is to design the controller with an adaptive parameter such that the output signal of system (2.3) follows the reference signal ultimately and the uncertain parameter could be identified. That is where is the Euclidean norm and is the “true” value of the “unknown” parameter .

Remark 2.1. The reference signal may belong to other integer-orders chaotic system or belong to different fractional-orders chaotic system with different fractional orders.First, the controller is designed as follows: where and are real matrix. and are a compensation controller and a feedback controller, respectively. If the true value of the “unknown” parameter is chosen as , then the definition of the compensation controller is

Remark 2.2. For defining the controllers (2.6) one has to know already the correct value of the parameter , 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 and parameter error . So, (2.3) can be changed as In generally, we can assume where is an real matrix, , and is an real matrix.Now, the feedback controller is chosen as, where is an real matrix to be designed later. According to (2.8) and (2.9), (2.3) or (2.7) can be rewritten as Second, the parameter update laws is chosen as where is an real matrix. Because the Caputo derivative of a constant is zero, so (2.11) can be rewritten as According to (2.10) and (2.12), we can obtain where is an real matrix.By (2.13), we know that the fractional-order system (2.3) tracking an arbitrary given reference signal and the uncertain parameter that could be identified are transformed into the following problem: choose a suitable matrix and such that system (2.13) converges asymptotically to zero.

Theorem 2.3. For an arbitrary given reference signal , the synchronization between fractional-order systems (2.3) and reference signal will be obtained and the uncertain parameter will be estimated if where is a real symmetric positive definite matrix, is a real symmetric positive semidefinite matrix, and stands for conjugate transpose of a matrix.

Proof. Assume that is one of the eigenvalues of matrix and the corresponding nonzero eigenvector is , that is, Multiply the above equation left by , we derive that Then, by a similar argument, we also can obtain that From (2.16) and (2.17), we can obtain Since and , are real symmetric positive definite matrix and real symmetric positive semidefinite matrix, respectively, then so From (2.20), we have According to the stability theory of fractional-order systems [20], the equilibrium point in (2.13) is asymptotically stable.Therefore, which indicates that the output signal of fractional-order chaotic system (2.3) can track the reference signal ultimately and the uncertain parameter 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 by where system parameters . The fractional-order Lorenz system exhibits chaotic behavior for fractional-order . The chaotic attractor for is shown in Figure 1.

Figure 1: Chaotic attractors of the fractional-order Lorenz system for .

Let us assume that parameter is unknown in the fractional-order Lorenz system. The fractional-order Lorenz system with uncertain parameter is given by

Now, we investigate the fractional-order Lorenz chaotic system with uncertain system parameter tracking the arbitrary given reference signal. We take reference signal , for example. According to above mentioned, we can obtain

Now, the parameter update laws and real matrix are chosen as

Therefore,

Choosing real symmetric positive definite matrix , we can yield

Choosing symmetric positive semidefinite matrix , we can obtain

According to the above Theorem, the fractional-order Lorenz system (3.2) with uncertain parameters can track the reference signal ultimately, and the uncertain parameters will be estimated. The corresponding numerical result is shown in Figure 2, in which the initial conditions are , for system (3.2), and , the “true” value of the “unknown” parameter is chosen as , respectively.

(a)

(b)

Figure 2: (a) shows tracking errors between the fractional-order Lorenz system (3.2) with uncertain parameter and the reference signal . (b) shows the parameter error between uncertain parameter 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 by where system parameters . The chaotic attractor for is shown in Figure 3.

Figure 3: Chaotic attractor of a new fractional-order system (3.8) for .

Let us assume that parameter is unknown in the fractional-order system (3.8). The fractional-order system with uncertain parameter is described by

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

In this case, matrix and can be written as and , respectively. According to above mentioned, we can obtain

The parameter update laws and real matrix are chosen as

Therefore,

Choosing real symmetric positive definite matrix , we can yield

Choosing symmetric positive semidefinite matrix , we can obtain

According to the above theorem, the fractional-order system (3.9) with uncertain parameter can track the fractional-order Lorenz system (3.1) ultimately and the uncertain parameter will be estimated. The corresponding numerical result is shown in Figure 4, in which the initial conditions are for fractional-order Lorenz system (3.1), and , for fractional-order system (3.9), and , the “true” value of the “unknown” parameter is chosen as , respectively.

(a)

(b)

Figure 4: (a) shows tracking errors between the fractional-order system (3.9) with uncertain parameter and the reference signal belongs to the fractional-order Lorenz system (3.1). (b) shows the parameter error between uncertain parameter 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.

References

L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, no. 8, pp. 821–824, 1990.

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, no. 11, pp. 1196–1199, 1990.

G. Chen and X. Dong, From Chaos to Order: Methodologies, Perspectives and Applications, vol. 24 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1998.

X. Wu and Y. Lu, “Generalized projective synchronization of the fractional-order Chen hyperchaotic system,” Nonlinear Dynamics, vol. 57, no. 1-2, pp. 25–35, 2009.

X. Wu and H. Wang, “A new chaotic system with fractional order and its projective synchronization,” Nonlinear Dynamics, vol. 61, no. 3, pp. 407–417, 2010.

Z. M. Odibat, “Adaptive feedback control and synchronization of non-identical chaotic fractional order systems,” Nonlinear Dynamics, vol. 60, no. 4, pp. 479–487, 2010.

Q. Zhu and J. Cao, “Adaptive synchronization of chaotic Cohen-Crossberg neural networks with mixed time delays,” Nonlinear Dynamics, vol. 61, no. 3, pp. 517–534, 2010.

H. Salarieh and A. Alasty, “Adaptive synchronization of two chaotic systems with stochastic unknown parameters,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 508–519, 2009.

X. Wang and M. Wang, “Adaptive robust synchronization for a class of different uncertain chaotic systems,” International Journal of Modern Physics B, vol. 22, no. 23, pp. 4069–4082, 2008.

M. T. Yassen, “Controlling, synchronization and tracking chaotic Liu system using active backstepping design,” Physics Letters A, vol. 360, no. 4-5, pp. 582–587, 2007.

X. Wang and Q. Zhao, “Tracking control and synchronization of two coupled neurons,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 849–855, 2010.

T. L. Carroll, I. Triandaf, I. Schwartz, and L. Pecora, “Tracking unstable orbits in an experiment,” Physical Review A, vol. 46, no. 10, pp. 6189–6192, 1992.

Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, and I. Triandaf, “Tracking unstable steady states: extending the stability regime of a multimode laser system,” Physical Review Letters, vol. 69, no. 22, pp. 3169–3172, 1992.

N. Noroozi, M. Roopaei, P. Karimaghaee, and A. A. Safavi, “Simple adaptive variable structure control for unknown chaotic systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 707–727, 2010.

E. J. Hwang, C. H. Hyun, E. Kim, and M. Park, “Fuzzy model based adaptive synchronization of uncertain chaotic systems: robust tracking control approach,” Physics Letters A, vol. 373, no. 22, pp. 1935–1939, 2009.

T. Zhang and G. Feng, “Output tracking and synchronization of chaotic Chua's circuit with disturbances via model predictive regulator,” Chaos, Solitons and Fractals, vol. 39, no. 2, pp. 810–820, 2009.

X. Li, W. Xu, and Y. Xiao, “Adaptive tracking control of a class of uncertain chaotic systems in the presence of random perturbations,” Journal of Sound and Vibration, vol. 314, no. 3–5, pp. 526–535, 2008.

F. H. Min, Y. Yu, and C. J. Ge, “Circuit implementation and tracking control of the fractional-order hyper-chaotic Lü system,” Acta Physica Sinica, vol. 58, no. 3, pp. 1456–1461, 2009.

L. D. Zhao, J. B. Hu, and X. H. Liu, “Adaptive tracking control and synchronization of fractional hyper-chaotic Lorenz system with unknown parameters,” Acta Physica Sinica, vol. 59, no. 4, pp. 2305–2309, 2010.

D. Matignon, “Stability results on fractional differential equations with applications to control processing,” in IEEE Conference on Computational Engineering in Systems Applacations, pp. 963–968, Lille, France, 1996.

I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional Lorenz system,” Physical Review Letters, vol. 91, no. 3, Article ID 034101, 4 pages, 2003.

L.-J. Sheu, H.-K. Chen, J.-H. Chen, and L.-M. Tam, “Chaos in a new system with fractional order,” Chaos, Solitons and Fractals, vol. 31, no. 5, pp. 1203–1212, 2007.

C.-M. Chang and H.-K. Chen, “Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen-Lee systems,” Nonlinear Dynamics, vol. 62, no. 4, pp. 851–858, 2010.