Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 732503, 6 pages

http://dx.doi.org/10.1155/2013/732503

## Synchronization of an Uncertain Fractional-Order Chaotic System via Backstepping Sliding Mode Control

^{1}College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China^{2}State Key Laboratory for Novel Software Technology, Nanjing University, Nanjing 210093, China

Received 19 March 2013; Accepted 7 June 2013

Academic Editor: Sridhar Seshagiri

Copyright © 2013 Zhen Wang. 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

Backstepping control approach combined with sliding mode control (SMC) technique is utilized to realize synchronization of uncertain fractional-order strict-feedback chaotic system. A backstepping SMC method is presented to compensate the uncertainty which occurs in the slave system. Moreover, the newly proposed control scheme is applied to implement synchronization of fractional-order Duffing-Holmes system. The simulation results demonstrate that the backstepping SMC method is robust against the modeling uncertainties and external disturbances.

#### 1. Introduction

Fractional calculus is a generalization of the ordinary integration and differentiation to noninteger-order case. In recent years, it has found widespread application in the fields of physics, applied mathematics, and engineering [1–3]. Moreover, it has been generally realized that fractional integrals and derivatives are more suitable to describe memory and hereditary properties of various materials and processes than its integer-order counterpart [4].

Chaotic behavior has been observed in the laboratory in a variety of systems, including electrical circuits, lasers, chemical reactions, fluid dynamics, and analog computers [5]. Over the past few years, more and more researchers have turned their attention to chaotic dynamics of fractional-order system. It has been shown that chaotic behavior of an integer-order nonlinear system is preserved when the order becomes fractional [6]. As a consequence, a number of fractional-order chaotic systems have been proposed, including fractional-order variants of Chua’s system [7], fractional-order Chen system [8], fractional-order Lorenz, Rössler, and Liu systems [9–11].

Synchronization control is one of the important research areas in chaos theory. During the past three decades, chaos synchronization has been attracting interest from researchers in various fields [12–14]. It should be pointed out that synchronization problem of fractional-order chaotic systems was first reported by Deng and Li [15]. Afterwards, different synchronization control methods have been successfully applied to chaos synchronization of factional order systems, such as sliding mode control [16, 17], adaptive fuzzy control [18, 19], nonlinear feedback control [20, 21], the open-plus-closed-loop control [22], observer-based control [23], and some other methods [24–26].

Backstepping approach was first proposed in [27] for designing stabilizing controls for a special type of nonlinear dynamical systems, which is referred to as the strict-feedback system. The technique consists in a recursive procedure that skillfully constructs the Lyapunov function and designs the virtual control input [28–30]. When this recursive procedure terminates, a feedback design for the real control input results [31]. On the other hand, sliding mode control (SMC) is one of the popular strategies to deal with uncertain systems. The main feature of SMC is the robustness against parameter uncertainties and external disturbances. In this paper, combining the merits of backstepping control and SMC, a backstepping sliding mode controller is devised to implement chaos synchronization of uncertain fractional-order strict feedback chaotic system.

The rest of this paper is organized as follows. In Section 2, definition and lemma in fractional calculus are presented. In Section 3, the backstepping sliding mode controller is systematically designed. In Section 4, synchronization problem of uncertain fractional-order Duffing-Holmes system is handled by the proposed method. Some concluding remarks are made in Section 5.

#### 2. Preliminaries

There are three most frequently used definitions for fractional derivatives [1], that is, Riemann-Liouville, Grünwald-Letnikov, and Caputo definitions. Caputo definition is widely used in engineering applications since it takes on the same form as for integer-order differential equations in the initial conditions. Therefore, the following sections are based on Caputo derivative.

*Definition 1 (see [1]). *The Caputo fractional derivative of order of a continuous function is defined as
where is -function, and

It should be noticed that the fractional integral of order is denoted by .

In what follows, we list several basic facts of fractional derivatives and integrals which will be used in the stability and stabilization analysis. The readers can refer to [1] for more details.

*Fact 1. * For , where is an integer, the operation gives the same result as classical calculus of integer-order . In particular, when , the operation coincides with the ordinary derivative .

*Fact 2. * For , the operation is the identity operation

*Fact 3. * Similary to integer-order calculus, fractional differentiation and fractional integration are both linear operations
where and are constants.

*Fact 4. * The additive law of exponents (semigroup property) holds

*Fact 5. * For , the following equation holds:
which means that the fractional differentiation operator is a left inverse to the fractional integration operator of the same order .

Lemma 2 (see [32]). *Let be an equilibrium point for either Caputo or RL fractional nonautonomous system
**
where and satisfies the Lipschitz condition with Lipschitz constant . Assume that there exits a Lyapunov function satisfying
**
where , and are positive constants and denotes an arbitrary norm. Then the equilibrium point of system (7) is Mittag-Leffler (asymptotically) stable.*

#### 3. Main Results

In synchronization task, there are two dynamical systems, which are, respectively, called master system and slave system. From the view point of control, the task is to design a controller which obtains signals from the master system to tune the behavior of the slave system. In this paper, we consider synchronization of fractional-order strict feedback system described by which is regarded as master system and the slave system is defined as where , and are the states of systems (9) and (10), denotes uncertainty, is the external disturbance, and is the control input.

Defining the synchronization error as , then the error system is given by where . Suppose is the upper bound of function ; that is, .

*Step 1. * We start with the scalar system
where , and is a virtual controller selected as .

Choose the candidate Lyapunov function as
then the time derivative of along the trajectories of system (12) is
Choose ; that is, , where is a positive constant. Thus, we have .

*Step 2. * Let , so

Design the switching surface as
where is the sliding surface parameter to be designed later.

According to SMC method, the condition which guarantees the trajectory of the system arrives at the sliding surface is , and when in the sliding mode, the switching surface and its derivative must satisfy the following conditions:
In view of (15), it means that
Thus, the equivalent control in the case of uncertainty-free is calculated by

To satisfy the sliding mode condition, the reaching law is chosen as
where is a positive constant. Hence, the total control law is

Selecting
as a composite Lyapunov function, we have
Substituting the control law (21) into the right hand side of (23), we have
Hence, according to Lemma 2, system (15) is asymptotically stable.

From the above two steps, we obtain . The properties , imply that . Thus, the global synchronization of systems (9) and (10) is achieved.

#### 4. Applications

In order to demonstrate the effectiveness of the proposed control scheme, numerical simulations are made for synchronization of fractional-order Duffing-Holmes system. In the simulation, Adams-Bashforth-Moulton predictor-corrector algorithm is used, and the detailed descriptions of this algorithm are available in [33].

##### 4.1. System Description

In [20], an integer-order chaotic Duffing-Holmes system has been studied. Let us consider the fractional-order case described by Initial conditions of master system and slave system are set as , and , , respectively.

Phase portrait of the chaotic dynamics in system (25) when are shown in Figure 1. Parameters and are set as 0.25 and 0.3, respectively.

The slave system is perturbed by the uncertainty and disturbance, which is described by where and are chosen as and , respectively.

According to the scheme proposed in Section 3, the control law is designed as follows: in which . When the parameters are chosen as , , , the simulation results are shown in Figures 2–5. Figures 2 and 3 show the synchronization performance between drive-response system (25)-(26) and the time response of the error system, respectively. The sliding surface and the control input are displayed in Figures 4 and 5.

#### 5. Conclusions

In this paper, the backstepping SMC method is developed for fractional-order strict-feedback chaotic system. The Lyapunov function, the virtual control, the switching surface, and the actual control are systematically designed, respectively. The proposed approach, which combines the merits of backstepping control and SMC, is robust against the modeling uncertainties and external disturbances. Numerical results demonstrate that the convergence speed of the synchronized error system is satisfactory. In addition, the strategy can be generalized to the investigation of synchronization of other fractional-order strict feedback chaotic or hyperchaotic systems.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61004078, 61104117, the Foundation of State Key Laboratory for Novel Software Technology of Nanjing University under Grant KFKT2013B11, the China Postdoctoral Science Foundation funded project, and Special Funds for Postdoctoral Innovative Projects of Shandong Province.

#### References

- I. Podlubny,
*Fractional Differential Equations*, vol. 198, Academic Press, New York, NY, USA, 1999. View at Zentralblatt MATH · View at MathSciNet - J. Wang and Y. Zhou, “Complete controllability of fractional evolution systems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 11, pp. 4346–4355, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Wang, X. Huang, Y. Li, and X. Song, “A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system,”
*Chinese Physics B*, vol. 22, no. 1, Article ID 010504, 2013. View at Publisher · View at Google Scholar - R. Hilfer,
*Applications of Fractional Calculus in Physics*, World Scientific, Hackensack, NJ, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - G. R. Chen,
*Controlling Chaos and Bifurcations in Engineering Systems*, CRC Press, 1999. - W. M. Ahmad and J. C. Sprott, “Chaos in fractional-order autonomous nonlinear systems,”
*Chaos, Solitons and Fractals*, vol. 16, no. 2, pp. 339–351, 2003. View at Publisher · View at Google Scholar - T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua's system,”
*Transactions on Circuits and Systems*, vol. 42, no. 8, pp. 485–490, 1995. View at Publisher · View at Google Scholar - C. Li and G. Peng, “Chaos in Chen's system with a fractional order,”
*Chaos, Solitons & Fractals*, vol. 22, no. 2, pp. 443–450, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Yu, H.-X. Li, S. Wang, and J. Yu, “Dynamic analysis of a fractional-order Lorenz chaotic system,”
*Chaos, Solitons and Fractals*, vol. 42, no. 2, pp. 1181–1189, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,”
*Physica A*, vol. 341, no. 1–4, pp. 55–61, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - V. Daftardar-Gejji and S. Bhalekar, “Chaos in fractional ordered Liu system,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1117–1127, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Lu and J. Cao, “Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters,”
*Chaos*, vol. 15, no. 4, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao and J. Lu, “Adaptive synchronization of neural networks with or without time-varying delay,”
*Chaos*, vol. 16, no. 1, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Lu, J. Cao, and D. W. C. Ho, “Adaptive stabilization and synchronization for chaotic Lur'e systems with time-varying delay,”
*IEEE Transactions on Circuits and Systems*, vol. 55, no. 5, pp. 1347–1356, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - W. Deng and C. Li, “Chaos synchronization of the fractional order Lü system,”
*Physica A*, vol. 353, pp. 61–72, 2005. View at Publisher · View at Google Scholar - M. R. Faieghi and H. Delavari, “Chaos in fractional-order Genesio-Tesi system and its synchronization,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 2, pp. 731–741, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. S. Tavazoei and M. Haeri, “Synchronization of chaotic fractional-order systems via active sliding mode controller,”
*Physica A*, vol. 387, no. 1, pp. 57–70, 2008. View at Publisher · View at Google Scholar - T. C. Lin and T. Y. Lee, “Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control,”
*IEEE Transactions on Fuzzy Systems*, vol. 4, pp. 623–635, 2011. View at Google Scholar - T. C. Lin and C. H. Kuo, “${H}^{\infty}$ synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach,”
*ISA Transactions*, vol. 50, no. 4, pp. 548–556, 2011. View at Publisher · View at Google Scholar - L. Song, J. Yang, and S. Xu, “Chaos synchronization for a class of nonlinear oscillators with fractional order,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 72, no. 5, pp. 2326–2336, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Y. Wang and J. M. Song, “Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 8, pp. 3351–3357, 2009. View at Publisher · View at Google Scholar - J. W. Wang and Y. B. Zhang, “Synchronization in coupled nonidentical incommensurate fractional order systems,”
*Physics Letters A*, vol. 374, no. 25, pp. 202–207, 2009. View at Publisher · View at Google Scholar - X. Wu, H. Lu, and S. Shen, “Synchronization of a new fractional-order hyperchaotic system,”
*Physics Letters A*, vol. 373, no. 27-28, pp. 2329–2337, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Zhang, H. Wang, and H. Fang, “Feedback control and hybrid projective synchronization of a fractional-order Newton-Leipnik system,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 1, pp. 317–328, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Zhou and W. Zhu, “Function projective synchronization for fractional-order chaotic systems,”
*Nonlinear Analysis*, vol. 12, no. 2, pp. 811–816, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Wang, Y. G. Yu, and M. Diao, “Hybrid projective synchronization of chaotic fractional order systems with different dimensions,”
*Physica A*, vol. 389, no. 21, pp. 4981–4988, 2010. View at Publisher · View at Google Scholar - M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic,
*Nonlinear and Adaptive Control Design*, Wiley, New York, NY, USA, 1995. - Z. Zhang, S. Xu, and H. Shen, “Reduced-order observer-based output-feedback tracking control of nonlinear systems with state delay and disturbance,”
*International Journal of Robust and Nonlinear Control*, vol. 20, no. 15, pp. 1723–1738, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Zhang, S. Xu, and B. Wang, “Adaptive actuator failure compensation with unknown control gain signs,”
*IET Control Theory & Applications*, vol. 5, no. 16, pp. 1859–1867, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - H. N. Pishkenari, N. Jalili, S. H. Mahboobi, A. Alasty, and A. Meghdari, “Robust adaptive backstepping control of uncertain Lorenz system,”
*Chaos*, vol. 20, no. 2, Article ID 023105, 5 pages, 2010. View at Publisher · View at Google Scholar - H. Shi, “A novel scheme for the design of backstepping control for a class of nonlinear systems,”
*Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems*, vol. 35, no. 4, pp. 1893–1903, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Li, Y. Chen, and I. Podlubny, “Mittag-Leffler stability of fractional order nonlinear dynamic systems,”
*Automatica*, vol. 45, no. 8, pp. 1965–1969, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,”
*Nonlinear Dynamics*, vol. 29, no. 1–4, pp. 3–22, 2002, Fractional order calculus and its applications. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet