Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 732503, 6 pages http://dx.doi.org/10.1155/2013/732503

Research Article

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

Zhen Wang1,2

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

Correspondence should be addressed to Zhen Wang; wangzhen.sd@gmail.com 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.

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, Rossler, 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 offractional-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, Granwald-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 a of a continuous function f : R+ ^ R is defined as

DaJ(t) =

_f f(m) (r)

T(m-a) Jo (t - T)a-m+1 dm dtm

dr, m - 1 < a < m,

a = m,

where r is T-function, and

T (z) = J e-ttz-1dt,

T(z+1) = zT(z). (2)

It should be noticed that the fractional integral of order a > 0 is denoted by D-".

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 a = n, where n is an integer, the operation Dff(t) gives the same result as classical calculus of integer-order n. In particular, when a = 1, the operation Dltf{t) coincides with the ordinary derivative df(t)/dt.

Fact 2. For a = 0, the operation D"f(t) is the identity operation

D0J(t) = f(t).

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

D" [af (t) + bg (f)] = aD"f (t) + bD"g (t). (4)

where a and b are constants.

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

D"Dff (t) = D^Dlf (t) = Dat+Pf (t).

Fact 5. For a > 0, the following equation holds:

D?D-«f(t) = D0f(t) = f(t)

which means that the fractional differentiation operator is a left inverse to the fractional integration operator of the same order a.

Lemma 2 (see [32]). Let x = 0 be an equilibrium point for either Caputo or RL fractional nonautonomous system

Dfx (t) = f (x, t),

where q e (0,1) and f(x,t) satisfies the Lipschitz condition with Lipschitz constant I > 0. Assume that there exits a Lyapunov function V(t, x(t)) satisfying

ajxll" < V(t,x) < a2 ||x|| V (t, x) < -a31|%||,

where a1,a2, a3, and a 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

Dfxi = %2, DtX2 =f(t,Xi,X2),

which is regarded as master system and the slave system is defined as

Dfyi = y 2'

Dfy2 = f (t, yi,y2) + Af (t, yi,y2) + d(t) + u (t),

where 0 < q < 1, X = (xltx2)T and Y = (y\,y2)T are the states of systems (9) and (10), Af(t,y1,y2) denotes uncertainty, d(t) is the external disturbance, and u(t) is the control input.

Defining the synchronization error as = yi - xt (i = 1,2), then the error system is given by

Dfei = e2,

= f (t, yi, y 2) - f (t, xi, X2) + F(t,Y) + u (t),

where F(t,Y) = Af(t,y1,y2) + d(t). Suppose y is the upper bound of function F; that is, ||F(i, Y)|| < y.

Step 1. We start with the scalar system D^z, = e2,

where zx = e1, and e2 is a virtual controller selected as e2 = a(zx).

Choose the candidate Lyapunov function as

V, = -z2 > 0; 1 2 1

then the time derivative of V1 along the trajectories of system (12) is

V = ^¿1 = ^Dl- (D?zi) = Z1D1-ev (14)

Choose D1 qe2 = -k1z1; that is, a(z1) = -k1Dq 1z1, where k-1 is a positive constant. Thus, we have V = —k1

Step 2. Let z2 = e2 - a(z1), so

Dqz2 = Dqe2 -rfta(zi)

= f(t,y1,y2)-f(t,X1,X2) + F(t,Y) (15)

+ u(t) + k1D^q-1z1.

Design the switching surface as

S = ktD-1z1 + Dflz2, (16)

where k2 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 SS < 0, and when in the sliding mode, the switching surface and its derivative must satisfy the following conditions:

S (t) = 0, S (t) = 0. (17)

In view of (15), it means that

S = k2Z1 + DqtZ2

= k2z1 +f(t,Y)-f (t, X) + F (t, Y) + k1D2q lz1 + u (t)

Thus, the equivalent control ueq(t) in the case of uncertainty-free is calculated by

(t) -f(t,X)-f (t, Y) - k2Zi - kiD2q-lZi, (19)

To satisfy the sliding mode condition, the reaching law is chosen as

ur = -k3S-y sgn (S), (20)

where k3 is a positive constant. Hence, the total control law is

u - weq + ur

= f(t,X)-f (t, Y) - k2z, - k, D2tq 1z1 -k3 S-y sgn (S)

Selecting

V2 = \S\

as a composite Lyapunov function, we have

V2 = sgn (S)S

= sgn (S) (k2Z1 + DqZ2)

= sgn (S) (k2Z1 +f(t,Y)-f(t,X)

+ F(t,Y)+klD2q-1zl +u(tj).

Substituting the control law (21) into the right hand side of (23), we have

V2 = sgn (S) \k2Z1 +f(t,Y)-f(t,X)

+ F(t,Y)+k1D2tq-1z1 + (f(t,X)-f(t,Y)-k2Z1

-k^z1 -k3S-y sgn (S))] = sgn (S) [F(t,Y)-k3S-y sgn (S)] <y-k3 sgn (S)S-y sgn (S) sgn (S)

- -k3 \S\.

Hence, according to Lemma 2, system (15) is asymptotically stable.

From the above two steps, we obtain limt^TOzi = 0 (i = 1,2). The properties z1 = e1, z2 = e2 - a(z1) = e2 + k1Dq 1z1 imply thatlimt= 0 (i = 1,2). Thus, theglobal 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

Dqx2 - x, - ax2 - x33 +b cos t.

Initial conditions of master system and slave system are set as %,(0) - 0.2, x2(0) - 0.2 and y, (0) - 0.1, y2(0) - -0.2, respectively.

-0.4 -0.6 -0.8

Figure 1: Phase portrait of Duffing-Holmes system (25) with fractional-order q = 0.98.

Figure 2: Synchronization performance of fractional-order Duff-ing-Holmes systems (25) and (26).

¿T -0.5 -1 -1.5

0 20 40 60 80 100

0 20 40 60 80 100

Figure 3: Synchronization error between fractional-order Duffing-Holmes systems (25) and (26).

8 6 4 2 0

S -2 -4 -6 -8 -10 -12

Figure 4: Switching surface S of the sliding mode controller (27).

Phase portrait of the chaotic dynamics in system (25) when q = 0.98 are shown in Figure 1. Parameters a and b are set as 0.25 and 0.3, respectively.

The slave system is perturbed by the uncertainty and disturbance, which is described by

Dbi = yi>

= yi - ay2 - y\ cos t + Af (t, Y) + d (t) + u,

where Af(t, Y) and d(t) are chosen as 0.1 sin(i) ^y2 + y\ and 0.1 sin(i), respectively

According to the scheme proposed in Section 3, the control law is designed as follows:

u = ae2 - (1 + k2) e1 - k1D2tq 1e1 + yl - -y sgn (S) - k3S,

in which S = k2Dt1 z1 + Df lz2 = k2Dt1e1 + Df 1e2 + k1D^q 2e1. When the parameters are chosen as q = 0.98, k1 = k2 = k3 = 2, y = 0.2, 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 u are displayed in Figures 4 and 5.

Figure 5: Control input и of the slave system (26).

5. Conclusions

In this paper, the backstepping SMC method is developed for fractional-order strict-feedback chaotic system. The Lya-punov function, the virtual control, the switching surface, and the actual control are systematically designed, respectively. The proposed approach, which combines the merits of back-stepping 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

[1] I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, New York, NY, USA, 1999.

[2] 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.

[3] 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.

[4] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Hackensack, NJ, USA, 2001.

[5] G. R. Chen, Controlling Chaos and Bifurcations in Engineering Systems, CRC Press, 1999.

[13 [14

[17 [18

[19 [20

[22 [23 [24

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.

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. 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.

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. C. Li and G. Chen, "Chaos and hyperchaos in the fractional-order Rossler equations," Physica A, vol. 341, no. 1-4, pp. 55-61, 2004.

V. Daftardar-Gejji and S. Bhalekar, "Chaos in fractional ordered Liu system," Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1117-1127, 2010.

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. J. Cao and J. Lu, "Adaptive synchronization of neural networks with or without time-varying delay," Chaos, vol. 16, no. 1, 2006. 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.

W. Deng and C. Li, "Chaos synchronization of the fractional order Lu system," Physica A, vol. 353, pp. 61-72, 2005. 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.

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. 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.

T. C. Lin and C. H. Kuo, "Hœ synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach," ISA Transactions, vol. 50, no. 4, pp. 548-556, 2011. 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.

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. 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. 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.

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.

[25] P. Zhou and W. Zhu, "Function projective synchronization for fractional-order chaotic systems," Nonlinear Analysis, vol. 12, no. 2, pp. 811-816, 2011.

[26] 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.

[27] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, Wiley, New York, NY, USA, 1995.

[28] 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.

[29] 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.

[30] 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.

[31] 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.

[32] 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.

[33] 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.

Copyright of Discrete Dynamics in Nature & Society is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.