Design an adaptive sliding mode controller for drive-response synchronization of two different uncertain fractional-order chaotic systems with fully unknown parameters

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Journal of the Franklin Institute 349 (2012) 3078–3101 www.elsevier.com/locate/jfranklin

Design an adaptive sliding mode controller for drive-response synchronization of two different uncertain fractional-order chaotic systems with fully unknown parameters$ Chun Yina,b,n, Sara Dadrasb,c, Shou-ming Zhonga,d a

School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 611731, PR China b Center for Self-Organizing and Intelligent Systems (CSOIS), Electrical and Computer Engineering Department, Utah State University, Logan, UT 84322-4160, USA c Automation and Instruments Lab, Electrical Engineering Department, Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran d Key Laboratory for Neuroinformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu 611731, PR China Received 26 September 2011; received in revised form 8 May 2012; accepted 15 September 2012 Available online 28 September 2012

Abstract In this paper, design an adaptive sliding mode controller (ASMC) for master–slave synchronization of two different fractional-order chaotic systems with fully unknown parameters, uncertainties and external disturbances is proposed. The bounds of the unknown parameters, uncertainties and external disturbances are assumed to be unknown in advance. Appropriate adaptive laws are designed to tackle the unknown parameters, uncertainties and external disturbances. Based on the adaptive laws, the ASMC is constructed in order to ensure the occurrence of the sliding motion and synchronization of two different fractional-order systems. The analytical conditions for synchronization of the systems are

$ This work was supported by National Basic Research Program of China (2010CB732501) and the National Natural Science Foundation of China (NSFC-60873102). n Corresponding author at: School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 611731, PR China. Tel.: þ86 13880 488086. E-mail address: [email protected] (C. Yin).

0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2012.09.009

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obtained by utilizing Laplace transform. Finally, numerical examples are provided to illustrate the effectiveness of the proposed ASMC scheme. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction Fractional calculus, which was introduced in the early 17th century, deals with integration and derivatives of arbitrary noninteger orders [1–6]. Although fractional calculus has a long history, the applications of fractional calculus to physics and engineering have attracted lots of attention only in the recent decades. Fractional calculus has been reported in many areas, such as control theory, diffusion, signal processing, turbulence and bioengineering. Besides fractional differential equations have been widely applied in modeling of dynamical systems [7–10], such as electrode–electrolyte polarization and quantum evolution of complex systems. Chaos is very interesting nonlinear phenomenon and has been investigated in many fields of science and technology over the last years. It is also demonstrated that some fractional-order differential systems behave chaotically or hyper-chaotically, such as the fractional-order Chen system [11,12], the fractional-order financial system [13,14]. Recently, there has been increasing interest in the subject of chaotic synchronization [15–17], Pourmahmood et al. have designed a RAMSC law to realize chaos synchronization between two different systems with uncertainties, external disturbances and fully unknown parameters. Generally speaking, chaos synchronization can be thought as the design problem of a feedback law for full observer using the known information of the plant, in order to ensure that the controlled receiver is synchronized with the transmitter. Now, synchronization of fractional-order systems has started to attract many attention [18–25]. Several methods have been proposed to achieve chaos synchronization. One of the methods is based on the sliding mode control (SMC) approach [13,14,26–36]. The main feature of SMC is to switch the control law to force the states of the system from the initial states onto some predefined sliding surface. The system on the sliding surface has desired properties such as stability, disturbance rejection capability, and tracking ability. Dadras et al. [13] have presented a SMC technique for the chaos control of the fractional-order financial system. Yin et al. [14] have solved the control problem for a class of fractional-order systems via the SMC control law. In addition, adaptive control is a suitable approach to overcome uncertainties and external disturbances in the system. So the ASMC has the advantage of combining the robustness of the sliding mode control with the tracking facilities of the adaptive control. In this paper, design an adaptive sliding mode controller for synchronization of two different fractional-order systems with fully unknown parameters is proposed. The slave system is assumed to be perturbed by uncertainties and external disturbances. The ASMC law is obtained to stabilize the states of the fractional-order error system. In addition, chaos synchronization is implemented in three different fractional-order chaotic systems (Rossler-financial, financialLorenz, Lorenz–Rossler) by utilizing the law. Some numerical simulations are given to demonstrate the robustness and efficiency of the proposed ASMC method. The paper is presented as follows: in Section 2, basic definitions in fractional calculus and notations are given. In Section 3, the general form description of fractional-order chaotic systems is presented. Section 4 proposes the employment of the adaptive sliding

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mode control method to synchronization. Numerical simulations results are shown in Section 5. Finally, conclusion is addressed in Section 6. 2. Basic definition and preliminaries Definition 2.1 (Monje [37], Podlubny [38]). The Caputo fractional derivative of order q of a continuous function f(t) is defined by 8 R t f ðnÞ ðtÞ 1 > > > dt, n1oqon, < q GðnqÞ 0 ðttÞqþ1n d f ðtÞ ð1Þ Dqt f ðtÞ ¼ ¼ > dtq d n f ðtÞ > > , q ¼ n, : dtn where n is the first integer which is not less than q. The Laplace transform of the Caputo fractional derivative is  q  n1 X d f ðtÞ q L Lff ðtÞg uq1k f ðkÞ ð0Þ, n1oqrn: ¼ u dtq k¼0

ð2Þ

We know that only integer orders derivatives of function appear in the Laplace transform of the Caputo derivative. Upon considering the initial conditions to zero, this formula reduces to  q  d f ðtÞ L ð3Þ ¼ uq Lff ðtÞg: dtq In this paper, we choose the Caputo derivative. Lemma 2.1 (Barbalat’s lemma [39,40]). If Z : R-R is a uniformly continuous function for Rt tZ0 and if the limit of the integral 0 ZðoÞ do exists and is finite, then limt-1 ZðtÞ ¼ 0. 3. Fractional-order system description and problem formulation In this paper, the n-dimensional fractional-order chaotic master and slave systems with fully unknown parameters, uncertainties and external disturbances are given as follows: Master system: d q1 x1 ~ ¼ m1 ðx1 ,x2 , . . . ,xn Þ þ M 1 ðx1 ,x2 , . . . ,xn Þx, dtq1 d q2 x2 ~ ¼ m2 ðx1 ,x2 , . . . ,xn Þ þ M 2 ðx1 ,x2 , . . . ,xn Þx, dtq2 ^ d qn xn ~ ¼ mn ðx1 ,x2 , . . . ,xn Þ þ M n ðx1 ,x2 , . . . ,xn Þx, dtqn Slave system: d q 1 y1 ¼ s1 ðy1 ,y2 , . . . ,yn Þ þ S 1 ðy1 ,y2 , . . . ,yn Þ~B þ Dg1 ðy1 ,y2 , . . . ,yn ,tÞ þ d1 ðtÞ þ u1 ðtÞ, dtq1

ð4Þ

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d q 2 y2 ¼ s2 ðy1 ,y2 , . . . ,yn Þ þ S 2 ðy1 ,y2 , . . . ,yn Þ~B þ Dg2 ðy1 ,y2 , . . . ,yn ,tÞ þ d2 ðtÞ þ u2 ðtÞ, dtq2 ^ d q n yn ¼ sn ðy1 ,y2 , . . . ,yn Þ þ S n ðy1 ,y2 , . . . ,yn Þ~B þ Dgn ðy1 ,y2 , . . . ,yn ,tÞ þ dn ðtÞ þ un ðtÞ, dtqn Dg ð5Þ where xðtÞ ¼ ½x1 ,x2 , . . . ,xn T is the state vector, mi ðxÞ, ði ¼ 1,2, . . . ,nÞ is a continuous nonlinear function of the fractional-order master system (4), M i ðxÞ, ði ¼ 1,2, . . . ,nÞ is the ith row of an n  r matrix MðxÞ, whose elements are continuous nonlinear functions, x~ is an r  1 unknown vector parameter of the fractional-order master system, yðtÞ ¼ ½y1 ,y2 , . . . ,yn T is the state vector of the fractional-order slave system (5), si ðyÞ, ði ¼ 1,2, . . . ,nÞ is a continuous nonlinear function of the fractional-order slave system, S i ðyÞ, ði ¼ 1,2, . . . ,nÞ is the ith row of an n  r matrix SðyÞ, whose elements are continuous nonlinear functions, B~ is an r  1 unknown vector parameter of the slave system, Dgðy,tÞ ¼ ½Dg1 ðy,tÞ,Dg2 ðy,tÞ, . . . ,Dgn ðy,tÞT and dðtÞ ¼ ½d1 ðtÞ,d2 ðtÞ, . . . ,dn ðtÞT are the vectors of unknown uncertainties and external disturbances of the slave system, respectively, and uðtÞ ¼ ½u1 ðtÞ, u2 ðtÞ, . . . ,un ðtÞT is the vector of control inputs, qi , ði ¼ 1,2, . . . ,nÞ are fractional-order satisfying 0oqi o1, respectively. Remark 3.1. The fractional-order system (4) is called a commensurate fractional-order system if qi ¼ q, ði ¼ 1,2, . . . ,nÞ, otherwise we call the system (4) as incommensurate fractional-order system. Remark 3.2. Note that most of the famous fractional-order systems belong to the class characterized by Eq. (4). Examples include that fractional-order Rossler’s system, the fractional-order financial system and the unified chaotic system of fractional-order version (including the fractional-order Chen system, fractional-order Lorenz system). Assumption 3.1. Since the trajectories of chaotic systems are always bounded, then the unknown uncertainties Dgi ðy,tÞ ði ¼ 1,2, . . . ,nÞ and external disturbances di ðtÞ, ði ¼ 1,2, . . . ,nÞ are assumed to be bounded. Furthermore, the derivative of the uncertainties Dgi ðy,tÞ, ði ¼ 1,2, . . . ,nÞ and external disturbances di ðtÞ, ði ¼ 1,2, . . . ,nÞ are supposed to be bounded. Therefore, there exist appropriate positive constants W~ i , r~ i ði ¼ 1,2, . . . ,nÞ such that ð6Þ jDgi ðy,tÞjoW~ i , jdi ðtÞjor~ ði ¼ 1,2, . . . ,nÞ: i

Assumption 3.2. The constants W~ i , r~ i ði ¼ 1,2, . . . ,nÞ are unknown. To solve the synchronization problem, the error between the master system (4) and the slave system (5) can be defined as e ¼ yx. Then the error system is obtained by subtracting Eq. (4) from Eq. (5) d q 1 e1 ¼ s1 ðyÞ þ S 1 ðyÞ~B þ Dg1 ðy,tÞ þ d1 ðtÞm1 ðxÞM 1 ðxÞx~ þ u1 ðtÞ, dtq1 d q 2 e2 ¼ s2 ðyÞ þ S 2 ðyÞ~B þ Dg2 ðy,tÞ þ d2 ðtÞm2 ðxÞM 2 ðxÞx~ þ u2 ðtÞ, dtq2 ^

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d q n en ¼ sn ðyÞ þ S n ðyÞ~B þ Dgn ðy,tÞ þ dn ðtÞmn ðxÞM n ðxÞx~ þ un ðtÞ: ð7Þ dtqn The purpose of this paper is to design a suitable controller to guarantee the fractionalorder error system (7) to be globally asymptotic stable in the sense that limt-1 JeðtÞJ ¼ 0 or equivalently yðtÞ-xðtÞ as t-1.

4. Design of adaptive sliding mode controller Sliding mode control is a robust control method which has many interesting features such as low sensitivity to external disturbances to the plant due to structural variations and un-modeled dynamics. In addition, adaptive control is a suitable approach to overcome system uncertainties. So the ASMC has the advantage of combining the robustness of the sliding mode control with the tracking facilities of the adaptive control. To design a adaptive sliding mode controller has two steps: first constructing a sliding surface that represents a desired system dynamics, and next develop a switching control to make the sliding mode possible on every point in the sliding surface.

4.1. Sliding surface design To propose an ASMC scheme to stabilize the error system (7), the sliding surface is designed as wi ðtÞ ¼ Dqi 1 ei ðtÞ þ ji ðtÞ,

i ¼ 1,2, . . . ,n,

ð8Þ

where wi ðtÞ 2 RðwðtÞ ¼ ½w1 ðtÞ,w2 ðtÞ, . . . ,wn ðtÞT Þ and j_ i ðtÞ, ði ¼ 1,2, . . . ,nÞ are given by j_ i ðtÞ ¼ ki ei ðtÞ,

i ¼ 1,2, . . . ,n,

ð9Þ

where the parameters ki , ði ¼ 1,2, . . . ,nÞ are positive constants. Remark 4.1. The goal of the SMC law is using a discontinuous control to force the system state trajectories to some predefined sliding surfaces on which the system has desired properties such as stability. To achieve this goal, the sliding surface is established to simplify the task of assigning the performance of the factional-order system in sliding mode. Having proposing the sliding surface, the SMC law is derived. The controller is composed of an equivalent control part that describes the behavior of the system when the trajectories stay over the sliding surface and a variable structure control part that enforces the trajectories to reach the sliding surface and remain on it evermore. The SMC law is a robust control technique which has many interesting features such as being robust to parameter uncertainties and insensitivity to external disturbances and guarantees the occurrence of sliding motion and chaotic control of the system. When the fractional-order error system operates in sliding mode, it satisfies the following equation: wðtÞ ¼ 0:

ð10Þ

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From Eq. (10), it is straightforward to verify that the sliding mode dynamics can be obtained as d q i ei ¼ ki ei , dtqi

i ¼ 1,2, . . . ,n:

ð11Þ

4.2. Sliding mode dynamics analysis To analysis the stability of the sliding mode dynamics (11) based on the Laplace transform theorem. Taking the Laplace transform in both sides of Eq. (11), letting Ei ðuÞ ¼ Lfei ðtÞg ði ¼ 1,2, . . . ,nÞ and utilizing Lfd qi ei =dtqi g ¼ uqi Ei ðuÞuqi 1 ei ð0Þ, one gets uqi Ei ðuÞuqi 1 ei ð0Þ ¼ ki Ei ðuÞ,

i ¼ 1,2, . . . ,n:

ð12Þ

Proposition 4.1. The fractional-order error system (11) is said to be stable if E1 ,E2 , . . . ,En are bounded and ki a0, ði ¼ 1,2, . . . ,nÞ under a suitable choice of k1 ,k2 , . . . ,kn . Proof. It follows from Eq. (12) that Ei ðuÞ ¼

uqi 1 ei ð0Þ , uq i þ k i

i ¼ 1,2, . . . ,n:

ð13Þ

By the final-value theorem of the Laplace transformation, one has lim ei ðtÞ ¼ lim uEi ðuÞ,

t-1

u-0

i ¼ 1,2, . . . ,n:

ð14Þ

Utilizing the above method to Ei ðuÞ ði ¼ 1,2, . . . ,nÞ, and one obtains uqi ei ð0Þ ei ð0Þ ¼ lim ¼ 0, ki u-0 uqi þ ki u-0 1þ q ui

lim ei ðtÞ ¼ lim uEi ðuÞ ¼ lim

t-1

u-0

i ¼ 1,2, . . . ,n:

ð15Þ

Since E1 ,E2 , . . . ,En are bounded and ki a0 ði ¼ 1,2, . . . ,nÞ, owing to the attractiveness of the attractors of the error system. Therefore, limt-1 e1 ðtÞ ¼ limt-1 e2 ðtÞ ¼    ¼ limt-1 en ðtÞ ¼ 0. This implies that lim ei ðtÞ ¼ 0,

t-1

i ¼ 1,2, . . . ,n:

Consequently, the stabilization of the fractional-order error system (11) is achieved.

ð16Þ &

Remark 4.2. The solution of Eq. (11) can be easily obtained as ei ðtÞ ¼ ei ð0ÞEqi ðki tqi Þ, where Eqi ðtÞ is Mittag–Leffler function. From Theorem 1.4 in [34], the following equation can be directly derived lim ei ðtÞ ¼ lim ei ð0Þ

t-1

t-1

1 ¼ 0: ki tqi Gð1qi Þ

Hence, the stabilization of the fractional-order error system also concluded.

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4.3. ASMC law synthesis To propose the equivalent control law, which is used in the sliding mode, the equivalent control law is calculated from Eqs. (7) and (11) ~ i ðyÞS i ðyÞ~B Dgi ðy,tÞdi ðtÞki ei uieq ðtÞ ¼ mi ðxÞ þ M i ðxÞxs

ði ¼ 1,2, . . . ,nÞ:

ð17Þ

The next step is to design reaching mode control scheme, which drives the system trajectories onto the sliding surface. In the proposed method, the reaching law is specified as uir ðtÞ ¼ gi sgnðwi Þ,

i ¼ 1,2, . . . ,n,

ð18Þ

where gi 40 is the switching gain and a constant. The total control law can be defined as follows: ~ i ðyÞS i ðyÞ~B Dgi ðy,tÞdi ðtÞki ei g sgnðwi Þ ui ðtÞ ¼ mi ðxÞ þ M i ðxÞxs i

ði ¼ 1,2, . . . ,nÞ:

ð19Þ In order to use this control law, the uncertainties Dgi ðy,tÞ and the disturbances di ðtÞ must be known. However, in practice the functions are fully known. To overcome this, the overall control law is modified to ui ðtÞ ¼ mi ðxÞ þ M i ðxÞxsi ðyÞS i ðyÞBðWi þ ri Þsgnðwi Þki ei gi sgnðwi Þ ði ¼ 1,2, . . . ,nÞ,

ð20Þ ~ B~ , W~ i , r~ , respectively, achieved by the following where x, B, Wi and ri are estimations for x, i equations: x_ ¼ ½MðxÞT Z, xð0Þ ¼ x0 , B_ ¼ ½SðyÞT Z, Bð0Þ ¼ B0 , W_ i ¼ pi jwi j, Wi ð0Þ ¼ Wi0 , r_ i ¼ li jwi j, ri ð0Þ ¼ ri0 ,

ð21Þ

T

where Z ¼ ½h1 w1 ,h2 w2 , . . . ,hn wn  , hi ,pi ,li , ði ¼ 1,2, . . . ,nÞ are positive constants and x0 , B0 , Wi0 , ri0 are the initial values of the update parameters x, B, Wi , ri , respectively. Remark 4.3. In this paper, an adaptive sliding mode controller is designed to realize synchronization between two different fractional-order chaotic systems with uncertainties, external disturbances and fully unknown parameters. The master system and slave system in [35] can be described by the master (4) system and slave system (5). The SMC law in this paper can be used to synchronize the fractional-order systems in [35]. On the other hand, the systems are perturbed by the uncertainties and external disturbances. They are assumed to be bounded. Many researches focused on the known bounds, which are assumed to be known constants, such as [13,14,26–36]. It should be worth noted that considering a known bound for the uncertainties and external disturbances of the system can be a very conservation limitation for the design scheme. However, when the bounds are difficult to be found, the bounds are not supposed to be known constant. The bounds of the uncertainties and external disturbances are estimated by the adaptive laws (21) in this paper. This solves the problem that the bounds are unknown. Theorem 4.1. Consider the fractional-order error system (7) and the sliding surface function (8), the trajectories of the system (7) under the controller (20) with the adaptive law (21) can be driven onto the sliding surface wðtÞ ¼ 0.

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Proof. Consider the following Lyapunov–Krasovskii functional candidate  n  X hi 2 hi ^ 2 þ J^B J2 , hi w2i þ W^ i þ r^ 2i þ JxJ V ðtÞ ¼ p l i i i¼1 ~ B^ ¼ B~B . Define W^ ¼ ½W^ 1 , . . . , W^ n T , where W^ i ¼ Wi W~ i , r^ i ¼ ri r~ i , x^ ¼ xx, T ½r^ 1 , . . . , r^ n  . Taking its derivative with respect to time, one has  n  X 2hi 2hi ~ _ ~ T x_ þ 2ðB~B ÞT B_ : _ 2hi wi w_ i þ ðWi W i ÞW i þ ðr r~ Þr_ þ 2ðxxÞ V ðtÞ ¼ pi li i i i i¼1

ð22Þ r^ ¼

ð23Þ

Introducing the adaptive law (21) into the right side of Eq. (23), one has n X ~ ½hi wi ðS i ðyÞB þ Dgi ðy,tÞ þ di ðtÞM i ðxÞxÞ þ hi wi ðM i ðxÞxS B V_ ðtÞ ¼ 2 i ðyÞ~ i¼1

ðWi þ ri Þ sgnðwi Þgi sgnðwi ÞÞ þ hi ðWi W~ i Þjwi j þ hi ðri r~ i Þjwi j ~ T ½MðxÞT Z þ 2ðB~B ÞT ½SðyÞT Z: 2ðxxÞ

ð24Þ

P P T Using the facts x~ ½MðxÞT Z ¼ ni ¼ 1 hi wi M i ðxÞx~ and B~ T ½SðyÞT Z ¼ ni ¼ 1 hi wi S i ðyÞ~B , one gets n X V_ ðtÞ ¼ 2 ½hi wi ðDgi ðy,tÞ þ di ðtÞÞ þ hi wi ðM i ðxÞxS i ðyÞBðWi þ ri Þ sgnðwi Þ i¼1

gi sgnðwi ÞÞ þ hi ðWi W~ i Þjwi j þ hi ðri r~ i Þjwi j2xT ½MðxÞT Z þ 2BT ½SðyÞT Z: ð25Þ Hence, one can derive by utilizing Assumption 3.1 n X ½hi wi ðM i ðxÞxS i ðyÞBðWi þ ri Þ sgnðwi Þgi sgnðwi ÞÞ þ hi Wi jwi j V_ ðtÞr2 i¼1

þhi ri jwi j2xT ½MðxÞT Z þ 2BT ½SðyÞT Z:

ð26Þ

P P From the facts xT ½MðxÞT Z ¼ ni ¼ 1 hi wi M i ðxÞx and BT ½SðyÞT Z ¼ ni ¼ 1 hi wi S i ðyÞB, one obtains n n X X V_ ðtÞr 2hi wi gi sgnðwi Þ ¼  2hi gi jwi j ¼ Ujwj, ð27Þ i¼1

i¼1

where U ¼ ½2h1 g1 ,2h2 g2 , . . . ,2hn gn 40 and jwj ¼ ½jw1 j,jw2 j, . . . ,jwn jT . Therefore, V_ ðtÞ becomes V_ ðtÞrUjwj ¼ CðtÞr0, ð28Þ where CðtÞ ¼ UjwjZ0. By using Lyapunov’s direct method, since V(t) is clearly positive-definite, V_ ðtÞ is ^ x^ and B^ tends to infinity, then negative semi-definite and V(t) tends to infinity as wðtÞ, W^ i , r, ^ ^ ^ xðtÞ, B^ ðtÞ ¼ ½0,0,0,0,0 is globally stable. the equilibrium at the origin ½wðtÞ, WðtÞ, rðtÞ,

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^ ^ ^ Therefore, the variables wðtÞ, WðtÞ, rðtÞ, xðtÞ, B^ ðtÞ are bounded. Since wðtÞ is bounded then it is deduced that ei , ði ¼ 1, . . . ,nÞ are bounded. On the other hand, taking the derivative of Eq. (8), it is obtained from w_ i ðtÞ ¼ Dqi ei ðtÞ þ ki ei ðtÞ

ð29Þ

then, substituting Eq. (7) into Eq. (29), w_ i ðtÞ ¼ S i ðyÞ~B þ Dgi ðy,tÞ þ di ðtÞM i ðxÞx~ þ ui ðtÞ þ ð1 þ ki Þei ðtÞ:

ð30Þ

_ is bounded. According to Eq. (25), it is deduced from Therefore, we can conclude that wðtÞ Assumption 3.1 that V€ ¼ 2

  n   X d dwi d hi wi ðDgi þ di Þ þ ðDgi þ di Þ hi ðW~ i þ r~ i Þ jwi j , dt dt dt i¼1

ð31Þ

_ which is bounded quantity since wðtÞ and wðtÞ are bounded. Hence, V_ is a uniformly continuous function according to Lyapunov-like analysis based on Barbalet’s lemma in [40], Then V_ -0 as t-1, which implies that w-0 by application of Lemma 2.1. Thus, the trajectories of the system can be driven onto the predefined sliding surface in a finite time. & Remark 4.4. The problem of designing a system (slave or response system), which mimics the behave of another one (master or drive system) is called synchronization. The aim in synchronizing master and slave systems is to find a suitable mechanism to control the slave system such that x(t) and y(t) asymptotically coincide. It is felt that the ASMC law, which has the advantage of combining the robustness of the sliding mode control with the tracking facilities of the adaptive control, is a useful tool to address the problem of synchronization of two different fractional-order chaotic systems. Remark 4.5. The synchronization between two identical fractional-order systems with unknown parameters and different initial values can be obtained by utilizing the proposed ASMC law, when the fractional-order systems (4) and (5) satisfy mi ðÞ ¼ si ðÞ, Mi ðÞ ¼ Si ðÞ ði ¼ 1, 2, . . . ,nÞ. Remark 4.6. The proposed controller (20) with the adaptive laws (21) can make the fractional-order system (4) reach the fractional-order system (5).

5. Numerical simulations In this section, we evaluate the performance of the ASMC scheme by using the method on three different fractional-order chaotic systems. Some numerical simulations are given to demonstrate the effectiveness of the proposed ASMC scheme. The fractional-order Rossler’s, financial and Lorenz systems are three well-known fractional-order chaotic

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systems which are written as 8 q1 d x > > ¼ yx, > > dtq1 > > < d q2 y ¼ x þ a1 y, Fractional-order Rossler’ssystem dtq2 > > > q > d 3z > > : q3 ¼ b1 þ zðxc1 Þ, dt

ð32Þ

8 q1 d x > > ¼ z þ ðya2 Þx, > > dtq1 > > < d q2 y ¼ 1b2 yx2 , Fractional-order financial system dtq2 > > > > d q3 z > > : q3 ¼ xc2 z, dt

ð33Þ

8 q1 d x > > ¼ a3 ðyxÞ, > > dtq1 > > < d q2 y ¼ xðb3 zÞy, Fractional-order Lorenz system dtq2 > > > > d q3 z > > : q3 ¼ xyc3 z, dt

ð34Þ

where ða1 ,b1 ,c1 Þ ¼ ð0:5, 0:2, 10Þ in the fractional-order Rossler’s system (32), ða2 , b2 , c2 Þ ¼ ð1, 0:1, 1Þ in the fractional-order financial system (33) ða3 , b3 , c3 Þ ¼ ð10, 28, 8=3Þ in the fractional-order Lorenz system (34) and qi , ði ¼ 1, 2, 3Þ is the fractional order. The fractional-order Rossler’s system (32) exhibits a chaotic behavior as shown in Fractional−order Rossler’s system

20

x3

15 10 5 0 20 x2

0

0 −20 −20

−10

10

20

x1

Fig. 1. The chaotic trajectories of the fractional-order Rossler’s system.

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Fractional−order financial system

2

x3

1 0 −1 −2 4 2 x2

0 −2

−4

2

0

−2

4

x1

Fig. 2. The chaotic trajectories of the fractional-order financial system.

Fractional−order Lorenz’s system

50

x3

40 30 20 10 0 50 x2

0 −50

−20

−10

0

10

20

x1

Fig. 3. The chaotic trajectories of the fractional-order Lorenz system.

Fig. 1. The fractional-order financial system (33) illustrates chaotic attractor seen as Fig. 2. Fig. 3 illustrates the chaotic behavior of the fractional-order Lorenz system (34). 5.1. Synchronization between the fractional-order Rossler’s system and the fractional-order financial system with uncertainties and external disturbances In this case, the efficiency of the proposed ASMC is verified by an example of synchronization between fractional-order chaotic systems. Here, the fractional-order financial system drives the fractional-order Rossler’s system. The systems can be written

C. Yin et al. / Journal of the Franklin Institute 349 (2012) 3078–3101

(4) and (5) as follows: 2 q1 3 mðxÞ MðxÞ d x1 ~ zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl3 ffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 2 3 zfflffl}|fflffl{ 2 3 x, 6 dtq1 7 2 0 0 1 y1 z1 0 6 q2 7 6 d y1 7 6 7 6 x 76 a 7 6 q 7¼4 0 y 0 1 1 5þ4 54 1 5 6 dt 2 7 6 q 7 b1 þ z1 x1 0 0 z1 c1 4 d 3 z1 5 q3 dt

3089

ð35Þ

3 B~ Dgðy,tÞ zfflffl}|fflffl{ sðyÞ SðyÞ d q1 x 2 2 3 zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 3 a2 2 3 6 dtq1 7 2 x y þ z 3 2 x 0:5 sinðpx2 Þ 0 0 6 7 2 2 2 2 6 q2 7 6 d y2 7 6 b 6 27 6 6 7 6 q 7 ¼ 4 1x22 7 y2 0 7 5þ4 0 5 6 7 þ 4 0:5 sinð2py2 Þ 5 6 dt 2 7 4 5 c 2 6 q 7 0:5 sinð3pz2 Þ 0 0 z2 x2 4 d 3 z2 5 q3 dt 2

dðtÞ

uðtÞ

zfflfflfflfflfflfflfflffl ffl{ zfflffl}|fflffl{ 2 ffl}|fflfflfflfflfflfflfflffl3 2 3 u1 0:3 cos t 6 7 6u 7 þ4 0:3 cos t 5 þ 4 2 5 , u3 0:3 cos t

ð36Þ

where q1 ¼ 0:9, q2 ¼ 0:85, q3 ¼ 0:95. Therefore, there exist appropriate positive constants W~ i , r~ i ði ¼ 1,2,3Þ such that jDgi jrW~ i ,jdi jrr~ i ði ¼ 1,2,3Þ. From Eq. (7), the error dynamics can be obtained 8 q1 d e1 > > ¼ ~B 1 e1 þ x2 y2 þ z2 ~B 1 x1 þ x~ 1 ðy1 þ z1 Þ þ 0:5 sinðpx2 Þ þ 0:3 cos t þ u1 , > > dtq1 > > < d q2 e 2 B 2 e2 þ 1x22 x~ 1 x1 ðx~ 2 þ B~ 2 Þy1 þ 0:5 sinð2py2 Þ þ 0:3 cos t þ u2 , ð37Þ q2 ¼ ~ dt > > > q > d 3 e3 > > : q3 ¼ ~B 3 e3 x2 ð~B 3 x~ 3 Þz1 b1 z1 x1 þ 0:5 sinð3pz2 Þ þ 0:3 cos t þ u3 : dt Consequently, three sliding surfaces are chosen as wi ðtÞ ¼ Dqi 1 ei ðtÞ þ D1 ðki ei ðtÞÞ,

i ¼ 1,2,3,

ð38Þ

where k1 ¼ 2:123, k2 ¼ 0:781, k3 ¼ 7:892. Subsequently, the control inputs are established according to Eq. (20) 8 u ¼ B1 e1 x2 y2 z2 þ B1 x1 x1 ðy1 þ z1 ÞðW1 þ r1 Þ sgnðw1 Þk1 e1 g1 sgnðw1 Þ, > < 1 u2 ¼ B2 e2 1 þ x22 þ x1 x1 þ ðx2 þ B2 Þy1 ðW2 þ r2 Þ sgnðw2 Þk2 e2 g2 sgnðw2 Þ, > : u ¼ B e þ x þ ðB þ x Þz þ b þ z x ðW þ r Þ sgnðw Þk e g sgnðw Þ, 3 2 1 1 1 3 3 3 3 3 3 3 3 3 1 3 3

ð39Þ

where g1 ¼ 3:184, g2 ¼ 3:184, g3 ¼ 3:184. The following adaptive laws are utilized to update the vector parameters x_ ¼ ½MðxÞT Z,

B_ ¼ ½SðyÞT Z,

W_ i ¼ pi jwi j,

r_ i ¼ li jwi j ði ¼ 1,2,3Þ,

ð40Þ

where Z ¼ ½w1 ,3w2 ,2:2w3 T , p1 ¼ l1 ¼ 1, p2 ¼ l2 ¼ 3, p3 ¼ l3 ¼ 2:2 and ½0:1,0:1,0:1T are selected as the initial values of the update vector parameters x, B, W, r.

C. Yin et al. / Journal of the Franklin Institute 349 (2012) 3078–3101

3090

5 x1,x2

x1 x2

0

−5

0

0.5

1

1.5

2 Time(sec)

2.5

3

3.5

4

4 y1

y1,y2

2

y2

0 −2 −4

0

0.5

1

1.5

2 Time(sec)

2.5

3

3.5

4

0.2 z1

z1,z2

0.1

z2

0 −0.1 −0.2

0

0.5

1

1.5

2 Time(sec)

2.5

3

3.5

4

Fig. 4. The time response of the signals in the system (35) and the signals in the system (36).

50

ξ1 ξ2

0

ξ3

ξ1,ξ2,ξ3

−50 −100 −150 −200 −250 −300 0

0.5

1

1.5

2 2.5 Time(sec)

3

3.5

Fig. 5. The time response of the update vector parameter x.

4

C. Yin et al. / Journal of the Franklin Institute 349 (2012) 3078–3101

3091

350 ς1

300

ς2 ς3

ς1,ς2,ς3

250 200 150 100 50 0

0

0.5

1

1.5

1 2.5 Time(sec)

3

3.5

4

Fig. 6. The time response of the update vector parameter B.

Fig. 7. The time response of the update vector parameter W.

The master system (35) and the slave system (36) are started with the initial conditions ðx1 ð0Þ, y1 ð0Þ, z1 ð0ÞÞ ¼ ð1:5,1:5,0:3Þ and ðx2 ð0Þ,y2 ð0Þ,z2 ð0ÞÞ ¼ ð6:5,3,0:1Þ. As we expected, one can observe that the trajectories of the fractional-order financial system (36) asymptotical synchronize the ones of the fractional-order Rossler’s system (35) under the adaptive sliding mode controller (39) from Fig. 4. The time responses of the update vector parameters x, B, W, r in Eq. (40) are shown in Figs. 5–8, respectively. Obviously, all of the update parameters approach to some bounded values. The simulation results illustrate that the obtained theoretic results are feasible and efficient for synchronization between the fractional-order Rossler’s system and the fractional-order financial system.

3092

C. Yin et al. / Journal of the Franklin Institute 349 (2012) 3078–3101

350 ρ1 ρ2

300

ρ3

ρ1,ρ2,ρ3

250 200 150 100 50 0

0

0.5

1

1.5

2 2.5 Time(sec)

3

3.5

4

Fig. 8. The time response of the update vector parameter r.

5.2. Synchronization between the fractional-order financial system and the fractional-order Lorenz system with uncertainties and external disturbances Here, to illustrate the efficiency of the proposed ASMC scheme in synchronizing the fractional-order financial system and the fractional-order Lorenz system with uncertainties, external disturbances. It is assumed that the fractional-order Lorenz system drives the fractional-order financial system. The drive and slave systems can be rewritten in the form of Eqs. (4) and (5) as follows: 2 q1 3 ~ mðxÞ MðxÞ x, d x1 zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 3 zfflffl}|fflffl{ 2 3 6 dtq1 7 2 x y þ z 3 2 x 0 0 a2 1 1 1 1 6 q2 7 6 d y1 7 6 7 6 7 6 2 6 q 7 ¼ 4 1x1 5 þ 4 0 y1 0 5 4 b2 7 ð41Þ 5 6 dt 2 7 6 q 7 0 0 z1 x1 c2 4 d 3 z1 5 q3 dt Dgðy,tÞ 2 q1 3 zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{ SðyÞ B~ sðyÞ d x2 2 3 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflffl}|fflfflffl{ zfflfflfflfflfflfflfflfflfflffl ffl }|fflfflfflfflfflfflfflfflfflffl ffl { 32 3 0:5 sinðpx2 Þ 3 2 6 dtq1 7 2 a3 0 0 y2 x2 0 6 q2 7 6 7 6 d y2 7 6 6 0 7 6 1 7 6 0:5 sinð2py2 Þ 7 6 q 7 ¼ 4 b3 x2 x2 z2 7 y 0 2 5þ4 54 5 þ6 7 6 dt 2 7 4 0:5 sinð3pz2 Þ 5 6 q 7 x2 y2 c3 0 0 z2 4 d 3 z2 5 dtq3 dðtÞ

uðtÞ

zfflfflfflfflfflfflfflffl ffl{ zfflffl}|fflffl{ 2 ffl}|fflfflfflfflfflfflfflffl3 2 3 u1 0:3 cos t 6 7 6u 7 0:3 cos t þ4 5 þ 4 2 5, u3 0:3 cos t where q1 ¼ 0:993, q2 ¼ 0:993, q3 ¼ 0:993.

ð42Þ

C. Yin et al. / Journal of the Franklin Institute 349 (2012) 3078–3101

3093

Therefore, there exist appropriate positive constants W~ i , r~ i ði ¼ 1,2,3Þ such that jDgi jrW~ i ,jdi jrr~ i ði ¼ 1,2,3Þ. From Eq. (7), the error dynamics can be obtained 8 q1 d e1 > > > q1 ¼ B~ 1 ðe2 e1 Þx1 y1 z1 þ B~ 1 ðy1 x1 Þ þ x~ 1 x1 þ 0:5 sinðpx2 Þ þ 0:3 cos t þ u1 , > > dt > > > < d q2 e2 ¼ ~B 2 e2 þ b3 x2 x2 z2 1 þ x21 þ ðx~ 2 ~B 2 Þy1 þ 0:5 sinð2py2 Þ þ 0:3 cos t þ u2 , dtq2 > > > d q3 e3 > > ¼ ~B 3 e3 þ x2 y2 þ x2 þ ðx~ 3 ~B 3 Þz1 þ 0:5 sinð3pz2 Þ þ 0:3 cos t þ u3 : > > > dtq3 : ð43Þ Consequently, three sliding surfaces are chosen as wi ðtÞ ¼ Dqi 1 ei ðtÞ þ D1 ðki ei ðtÞÞ,

i ¼ 1,2,3,

ð44Þ

where k1 ¼ 2:941, k2 ¼ 4:981, k3 ¼ 1:526. Subsequently, the control inputs are established according to Eq. (20) 8 u ¼ B1 ðe2 e1 Þ þ x1 y1 þ z1 B1 ðy1 x1 Þx1 x1 ðW1 þ r1 Þ sgnðw1 Þk1 e1 g1 sgnðw1 Þ, > < 1 u2 ¼ B2 e2 b3 x2 þ x2 z2 þ 1x21 þ ðB2 x2 Þy1 ðW2 þ r2 Þ sgnðw2 Þk2 e2 g2 sgnðw2 Þ, > : u ¼ B e x y x ðx B Þz ðW þ r Þ sgnðw Þk e g sgnðw Þ, 3

3 3

2 2

2

3

3

1

3

3

3

3 3

3

3

ð45Þ where g1 ¼ 3:749, g2 ¼ 3:749, g3 ¼ 3:749. The following adaptive laws are utilized to update the vector parameters x_ ¼ ½MðxÞT Z,

B_ ¼ ½SðyÞT Z,

W_ i ¼ pi jwi j,

r_ i ¼ li jwi j,

i ¼ 1,2,3,

ð46Þ

where Z ¼ ½2:3w1 , 6:7w2 , 4:1w3 T , p1 ¼ l1 ¼ 2:3, P2 ¼ L2 ¼ 6:7, P3 ¼ L3 ¼ 4:1 and ½0:1,0:1,0:1T are selected as the initial values of the update vector parameters x, B, W, r. The master system (41) and the slave system (42) are started with the initial conditions ðx1 ð0Þ, y1 ð0Þ, z1 ð0ÞÞ ¼ ð2, 1, 1Þ and ðx2 ð0Þ, y2 ð0Þ, z2 ð0ÞÞ ¼ ð0:1, 0:1, 0:11Þ. As we expected, one can observe that the trajectories of the fractional-order Lorenz system (42) asymptotical synchronize the ones of the fractional-order financial system (41) via ASMC law (45) from Fig. 9. The time responses of the update vector parameters x,B,W,r in Eq. (46) are shown in Figs. 10–13, respectively. Obviously, all of the update parameters approach to some bounded values. The simulation results illustrate that the obtained theoretic results are feasible and efficient for synchronization between the fractional-order financial system and the fractional-order Lorenz system.

5.3. Chaos synchronization between the fractional-order Lorenz system and the fractionalorder Rossler’s system with uncertainties and external disturbances To show the efficiency of the proposed ASMC in synchronizing the fractional-order Lorenz system and the fractional-order Rossler’s system with uncertainties and external disturbances, it is assumed that the fractional-order Rossler’s system drives the fractionalorder Lorenz system. The reformulated form of the fractional-order Lorenz and Rossler’s

C. Yin et al. / Journal of the Franklin Institute 349 (2012) 3078–3101

x1,y1

3094

2

x1

0

y1

−2 0

1

2

3

4

5 Time(sec)

6

7

8

9

x2

x2,y2

2

y2

0 −2

10

0

1

2

3

4

5 Time(sec)

6

7

8

9

10

2 x3,y3

x3 y3

0

−2

0

1

2

3

4

5 Time(sec)

6

7

8

9

10

Fig. 9. The time response of the signals in the system (41) and the signals in the system (42).

500 ξ1

400

ξ2 ξ3

300

ξ1,ξ2,ξ3

200 100 0 −100 −200 −300 0

2

4

6

8

Time(sec) Fig. 10. The time response of the update vector parameter x.

10

C. Yin et al. / Journal of the Franklin Institute 349 (2012) 3078–3101

350

3095

ς1 ς2

300

ς3

ς1,ς2,ς3

250 200 150 100 50 0

0

2

4

6

8

10

Time(sec) Fig. 11. The time response of the update vector parameter B.

Fig. 12. The time response of the update vector parameter W.

systems is expressed as 2 q1 3 ~ MðxÞ x, mðxÞ d x1 zfflfflffl}|fflfflffl{ zfflfflfflfflfflfflfflfflfflffl ffl}|fflfflfflfflfflfflfflfflfflffl3 ffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ 2 3 2 3 q 2 1 6 dt 7 a3 0 0 y1 x1 0 6 q2 7 6 d y1 7 6 6 6 7 6 q 7 ¼ 4 b3 x1 x1 z1 7 y1 0 7 5þ4 0 5 4 1 5 6 dt 2 7 6 q 7 x1 y1 c3 0 0 z1 4 d 3 z1 5 q3 dt

ð47Þ

C. Yin et al. / Journal of the Franklin Institute 349 (2012) 3078–3101

3096

450 ρ1

400

ρ2 ρ3

350

ρ1,ρ2,ρ3

300 250 200 150 100 50 0

0

1

2

3

4

5

Time(sec)

Fig. 13. The time response of the update vector parameter r.

3 Dgðy,tÞ sðyÞ SðyÞ B~ d q1 x2 zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl3 ffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ q 2 3 zfflffl}|fflffl{ 2 3 zfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflffl{ 2 3 6 dt 1 7 2 0:5 sinðpx2 Þ 0 0 1 y2 z2 0 6 q2 7 6 d y2 7 6 7 6 x 6 7 6 7 6 q 7¼4 0 y2 0 7 2 5þ4 5 4 a1 5 þ4 0:5 sinð2py2 Þ 5 6 dt 2 7 6 q 7 0:5 sinð3pz2 Þ b1 þ z2 x2 0 0 z2 c1 4 d 3 z2 5 2

dtq3 dðtÞ

uðtÞ

zfflfflfflfflfflfflfflffl ffl{ zfflffl}|fflffl{ 2 ffl}|fflfflfflfflfflfflfflffl3 2 3 u1 0:3 cos t 6 7 6u 7 þ4 0:3 cos t 5 þ 4 2 5 , u3 0:3 cos t

ð48Þ

where q1 ¼ 0:992, q2 ¼ 0:989, q3 ¼ 0:993. Therefore, there exist appropriate positive constants W~ i , r~ i ði ¼ 1,2,3Þ such that jDgi jrW~ i ,jdi jrr~ i ði ¼ 1,2,3Þ. From Eq. (7), the error system can be obtained 8 q1 d e1 > > ¼ ~B 1 ðe2 e3 Þ þ x~ 1 x1 ð~B 1 þ x~ 1 Þy1 þ B~ 1 z1 þ 0:5 sinðpx2 Þ þ 0:3 cos t þ u1 , > > dtq1 > > < d q2 e 2 ~ 1 e1 þ B~ 2 e2 b3 x1 þ x2 z1 þ B~ 1 x1 ðx~ 2 ~B 2 Þy1 þ 0:5 sinð2py2 Þ þ 0:3 cos t þ u2 , q2 ¼ B dt > > > q3 > d e > > : q33 ¼ ~B 3 e3 þ b1 þ z2 x2 x1 y1 þ ðx~ 3 ~B 3 Þx2 þ 0:5 sinð3pz2 Þ þ 0:3 cos t þ u3 : dt ð49Þ Consequently, three sliding surfaces are chosen as wi ðtÞ ¼ Dqi 1 ei ðtÞ þ D1 ðki ei ðtÞÞ,

i ¼ 1,2,3,

where k1 ¼ 1:021, k2 ¼ 4:810, k3 ¼ 3:273.

ð50Þ

C. Yin et al. / Journal of the Franklin Institute 349 (2012) 3078–3101

3097

Subsequently, the control inputs are established according to Eq. (20) 8 u1 ¼ B1 ðe2 e3 Þx1 x1 þ ðB1 þ x1 Þy1 B1 z1 þ ðW1 þ r1 Þ sgnðw1 Þk1 e1 g1 sgnðw1 Þ, > > > > < u2 ¼ B1 e1 B2 e2 þ b3 x1 x2 z1 B1 x1 þ ðx2 B2 Þy1 þ ðW2 þ r2 Þ sgnðw2 Þ k2 e2 g2 sgnðw2 Þ, > > > > : u3 ¼ B3 e3 b1 z2 x2 þ x1 y1 ðB3 þ x3 Þz1 þ ðW3 þ r Þ sgnðw3 Þk3 e3 g sgnðw3 Þ, 3 3

ð51Þ

where g1 ¼ 4:813, g2 ¼ 4:813, g3 ¼ 4:813. The following adaptive laws are utilized to update the vector parameters x_ ¼ ½MðxÞT Z,

B_ ¼ ½SðyÞT Z,

W_ i ¼ pi jwi j,

r_ i ¼ li jwi j,

i ¼ 1,2,3,

ð52Þ

T

x1,y1

where Z ¼ ½0:57w1 , 0:48w2 , 0:32w3  , p1 ¼ l1 ¼ 0:57, p2 ¼ l2 ¼ 0:48, p3 ¼ l3 ¼ 0:32 and ½0:1,0:1,0:1T are selected as the initial values of the update vector parameters x, B, W, r. The master system (47) and the slave system (48) are started with the initial conditions ðx1 ð0Þ, y1 ð0Þ, z1 ð0ÞÞ ¼ ð0:1,0:1,0:1Þ and ðx2 ð0Þ, y2 ð0Þ, z2 ð0ÞÞ ¼ ð4:4, 1:6, 0:2Þ. As we expected, one can observe that the trajectories of the fractional-order Rossler’s system (48) asymptotical synchronize the ones of the fractional-order Lorenz system (47) via adaptive sliding control law (51) from Fig. 14. The time responses of the update vector parameters x, B, W, r in Eq. (52) are shown in Figs. 15–18, respectively. Obviously, all of the 20

x1

10

y1

0 −10 −20

0

1

2

3

4

5 Time(sec)

6

7

8

9

10

x2,y2

30 20

x2

10

y2

0 −10

x3,y3

0

1

2

3

4

5 Time(sec)

6

7

8

9

50 40 30 20 10 0

10

x3 y3

0

1

2

3

4

5 Time(sec)

6

7

8

9

10

Fig. 14. The time response of the signals in the system (47) and the signals in the system (48).

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update parameters approach to some bounded values. The simulation results illustrate that the obtained theoretic results are feasible and efficient for synchronization between the fractional-order Lorenz system and the fractional-order Rossler’s system. Remark 5.1. We have also studied the problem of synchronization of others fractionalorder systems. The synchronization can be achieved by using Proposition 4.1 and Theorem 4.1. The analytical process is similar to the above three example. Hence simulations about synchronization of these systems are omitted in this paper.

9000

ξ1 ξ2 ξ3

8000 7000 6000

ξ1,ξ2,ξ3

5000 4000 3000 2000 1000 0 −1000 −2000

0

2.5

5 Time(sec)

7.5

10

Fig. 15. The time response of the update vector parameter x.

10000

ς1

8000

ς3

ς2

ς1,ς2,ς3

6000 4000 2000 0 −2000 −4000 −6000 0

2.5

5 Time(sec)

7.5

Fig. 16. The time response of the update vector parameter B.

10

C. Yin et al. / Journal of the Franklin Institute 349 (2012) 3078–3101

3099

Fig. 17. The time response of the update vector parameter W.

2500 ρ1 ρ2

ρ1,ρ2,ρ3

2000

ρ3

1500

1000

500

0

0

2.5

5 Time(sec)

7.5

10

Fig. 18. The time response of the update vector parameter r.

6. Conclusion In this paper, the problem of synchronization between two different fractional-order chaotic systems with the fully unknown parameters, uncertainties and external disturbances has been investigated using the ASMC law. A sliding surface has been proposed and its convergence to the zero equilibrium has been proved. The bounds of the uncertainties and external disturbances have been assumed to be unknown. Then, suitable

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update strategies have been introduced to tackle the unknown parameters of the fractional-order systems. The ASMC has been derived to ensure the occurrence of the sliding motion and synchronization of two different fractional-order systems under the Laplace transform theorem. Some numerical simulations have been given to demonstrate the effectiveness of the proposed ASMC method. Acknowledgments The authors would like to gratefully acknowledge Professor YangQuan Chen for his technical guidance and constructive comments.

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