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Journal of Control Science and Engineering
Volume 2015, Article ID 703753, 7 pages
http://dx.doi.org/10.1155/2015/703753
Research Article

A Robust Control Method for Synchronization between Different Dimensional Integer-Order and Fractional-Order Chaotic Systems

1Department of Mathematics and Computer Science, University of Tebessa, 12002 Tebessa, Algeria
2Department of Health Informatics, College of Public Health and Health Informatics, King Saud Bin Abdulaziz University for Health Science, Riyadh 11481, Saudi Arabia

Received 8 November 2015; Accepted 17 December 2015

Academic Editor: Xiao He

Copyright © 2015 Adel Ouannas and Raghib Abu-Saris. 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

A robust control approach is presented to study the problem of Q-S synchronization between Integer-order and fractional-order chaotic systems with different dimensions. Based on Laplace transformation and stability theory of linear integer-order dynamical systems, a new control law is proposed to guarantee the Q-S synchronization between -dimensional integer-order master system and -dimensional fractional-order slave system. This paper provides further contribution to the topic of Q-S chaos synchronization between integer-order and fractional-order systems and introduces a general control scheme that can be applied to wide classes of chaotic and hyperchaotic systems. Illustrative example and numerical simulations are used to show the effectiveness of the proposed method.

1. Introduction

Fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. The advantages or the real objects of the fractional-order systems are that we have more degrees of freedom in the model and that a “memory” is included in the model [16].

Recently, more and more attentions were paid to synchronization of integer-order chaotic systems and fractional-order chaotic systems. Many control methods have been proposed and different synchronization types have been studied between integer-order and fractional-order chaotic system. For example, general control schemes have been described in [7, 8]. A sliding mode method has been designed in [911]. A synchronization method of a class of hyperchaotic systems is given in [12]. In [13], a nonlinear feedback control method has been introduced and some robust observer techniques have been used in [14, 15]. Also, complete synchronization and antisynchronization have been observed, for example, in [1618], and function projective synchronization has been studied in [19]. Until now, a variety of control schemes have been proposed to study the problem of chaos synchronization between different dimensional systems such as modified function projective synchronization [20], generalized matrix projective synchronization [21], generalized synchronization [2224], inverse generalized synchronization [25], full state hybrid projective synchronization [26], Q-S synchronization [27], increased order synchronization [28, 29], and reduced order generalized synchronization [30]. Amongst all kinds of synchronization, Q-S synchronization has been extensively considered [3139], due to its universality and its great potential applications in applied sciences and engineering.

Motivated by the above discussions, the main aim of this paper is to present constructive scheme to investigate Q-S synchronization between -dimensional integer-order master system and -dimensional fractional-order slave system in -D. By using Laplace transformation, stability property of integer-order linear systems, and suitable fractional-order control law, a new criterion is derived to achieve the Q-S chaos synchronization between -dimensional integer-order master system and -dimensional fractional-order slave system in -D. The proposed control method is simple, efficient, and easy to implement in applications. The outline of the rest of this paper is organized as follows. First, Section 2 provides some basic concepts of fractional derivative. In Section 3, the problem of Q-S synchronization between integer-order master system and fractional-order slave system is formulated. Our main result is presented in Section 4. In Section 5, numerical example is used to verify the effectiveness and feasibility of the proposed control scheme. Finally, Section 6 is the brief conclusion.

2. Basic Concepts of Fractional Derivative

There are several definitions of a fractional derivative of order [4042]. The two most commonly used definitions are the Riemann-Liouville and Caputo definitions. Each definition uses Riemann-Liouville fractional integration and derivatives of whole order. The difference between the two definitions is in the order of evaluation. The Riemann-Liouville fractional integral operator of order of the function is defined asSome properties of the operator can be found, for example, in [43, 44]. We recall only the following, for , , and ; we haveIn this study, Caputo definition is used and the fractional derivative of is defined asfor , ,  . The fractional differential operator is left-inverse (and not right-inverse) to the fractional integral operator ; that is, , where is the identity operator. The Laplace transform of the Caputo fractional derivative rule readsParticularly, when , we have . The Laplace transform of the Riemann-Liouville fractional integral rule satisfiesCaputo fractional derivative appears more suitable to be treated by the Laplace transform technique in that it requires the knowledge of the (bounded) initial values of the function and of its integer derivatives of order , in analogy with the case when .

3. Synchronization Problem for Integer-Order and Fractional-Order Chaotic System

We assume that the master chaotic systems can be considered in the following form:where is the state vector of the master system (6), is a constant matrix, and is a nonlinear vector function. Also, consider the slave chaotic system aswhere is the state vector of the slave system (7), , is a rational number between and ,   is the Caputo fractional derivative of order , and is a vector controller. The problem of Q-S synchronization for the master system (6) and the slave system (7) in dimension is to find the controller such that the synchronization error,satisfies thatwhere , are continuously differentiable functions. We assume that

4. General Control Scheme

The integer-order derivative of the error system (8) can be derived aswhere , are the Jacobian matrices of the functions and , respectively,

Hence, we have the following result.

Theorem 1. There exists a suitable feedback gain matrix to realize the Q-S synchronization between the master system (6) and the slave system (7) in -D under the following controller:where the vector quantity is defined aswhere is the inverse of matrix ,

Proof. By inserting the control law described by (12) into (7), we can rewrite the slave system as follows:Applying the Laplace transform to (17) and letting , we obtainmultiplying both the left-hand and right-hand sides of (18) by and applying the inverse Laplace transform to the result, we obtain a new equation for the slave systemNow, the error system (10) can be described aswhere is a feedback gain matrix to be chosen. The error system (20) can be simplified as follows:where was defined by (16). By using (14), the error system (21) can be written asHence, by substituting (13) into (22), we getWith respect to the asymptotic stability property of linear continuous-time dynamical systems, if the feedback gain matrix is selected such that all eigenvalues of are strictly negative, it is immediate that all solutions of the error system (23) go to zero as . Therefore, systems (6) and (7) are globally synchronized in .

5. Illustrative Example

In this section, to validate the synchronization method proposed in the previous section, we consider the Lorenz system as a master system and the fractional-order hyperchaotic Lorenz system as a slave system. The Lorenz system can be described aswhich has a chaotic attractor, for example, when [45]. Then the linear part and the nonlinear part of the Lorenz system (24) are given by

The Lorenz chaotic attractor is shown in Figure 1.

Figure 1: The chaotic attractor of Lorenz system when

The controlled fractional-order hyperchaotic Lorenz system can be described by the following nonlinear fractional-order ODE:where , , are the controllers. This system, as shown in [46], exhibits hyperchaotic behaviors when and Chaotic attractor projections of the fractional-order hyperchaotic Lorenz system are shown in Figure 2.

Figure 2: Chaotic attractor projections of fractional-order hyperchaotic Lorenz system when and .

According to synchronization control technique proposed in the previous section, the functions and are selected as follows:

Then,

It is easy to show that if we choose the feedback gain matrix asthen the eigenvalues of the matrix are strictly negative. Then the controllers , and can be designed as follows:and the error system can be written as

Therefore, according to Theorem 1, systems (24) and (26) are globally synchronized in 3D. The numerical simulations of the error functions evolution are shown in Figure 3.

Figure 3: Time evolution of synchronization errors between systems (24) and (26).

6. Conclusion

In this paper, new control scheme, based on fractional control law, Laplace transformation, and stability theory of linear integer-order dynamical system, for synchronization was presented between -dimensional integer-order master system and -dimensional fractional-order slave system. To observe synchronization with respect to dimension , the synchronization criterion was obtained via controlling the linear part of the master system. Numerical example and simulations result were used to verify the effectiveness of the proposed control method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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