Analysis and Synthesis of Stochastic Nonlinear SystemsView this Special Issue
Secure Communication Based on a Hyperchaotic System with Disturbances
This paper studies the problem on chaotic secure communication, and a new hyperchaotic system is included for the scheme design. Based on Lyapunov method and techniques, two kinds of chaotic secure communication schemes in the case that system disturbances exist are presented for the possible application in real engineering; corresponding theoretical derivations are also provided. In the end, some typical numerical simulations are carried out to demonstrate the effectiveness of the proposed schemes.
Since the pioneer work of Fujisaka and Yamada in 1983  and Pecora and Carroll in 1990 , chaos science has become an active research field with a wide range of applications including secure communication. The idea of the chaotic secure communication is that in transmitting terminal the chaotic signal is used as a carrier with hidden information, which will be restored in receiving terminal based on chaos synchronization. Generally chaos synchronization can be classified into four categories: (i) identical or complete synchronization, (ii) generalized synchronization, (iii) phase synchronization, and (iv) lag synchronization; see, for instance,  and the references therein. During last decades, many researches on secure communication have been carried out [4–10]. For instance, in , a reduced-order observer and a step-by-step sliding mode observer are given to realize the secure communication in the case that the observer’s matching condition cannot be satisfied. In , the modified generalized projective synchronization for the fractional-order chaotic systems is introduced, and the unpredictability of the scaling factors in projective synchronization is included to enhance the security of communication. In , a constrained regularized least square state estimator is designed for deterministic discrete-time nonlinear dynamical systems, being subject to a set of equality or inequality constraints, to design the secure communication scheme. In , based on the synchronization of the hyperchaotic Chen system and the unified chaotic system, the encrypting and decrypting of digital images are carried out to realize the secure communication. However, these investigations usually do not consider the case that system disturbances exist, and that limits their practical value because, in real engineering, system disturbances exist widely and will have some effects on system performance. In addition, featuring the chaotic attractor with more than one positive Lyapunov exponent and the complicated dynamical behavior, the hyperchaotic system, especially the new hyperchaotic system, can be used to improve the safety of secure communication. In this paper a hyperchaotic system  will be included for the secure communication scheme design.
Hence, inspired by the above discussion, we try to propose the secure communication schemes based on a hyperchaotic system. The rest of the paper is organized as follows. In Section 2, the model description and preliminaries will be given. In Section 3, a secure communication scheme will be constructed by using the single-dimensional controller. Then we will extend the theoretical results on the base of the multidimensional controller. Finally in Section 4, we will include some typical examples to demonstrate the correctness of the proposed schemes.
Notations used in this paper are fairly standard. Let be the -dimensional Euclidean space; denotes the set of real matrix, indicates the symmetric part in a matrix, stands for the identity matrix with appropriate dimensions, and denotes the diagonal matrix.
2. System Description and Preliminaries
First, based on the single-dimensional controller, we present chaotic secure communication Scheme 1, which is made of the master system, the slave system, the mixer, the controller, and the channel; see Figure 1.
Thereinto, the master hyperchaotic system in transmitting terminal is designed bywhere , , is the system state variable and , , is the system parameter. The chaotic system can produce the hyperchaotic dynamic behavior when , , , , , , , is the signal input, and is a positive scalar.
The chaotic encrypted signal to be transmitted is defined by where is the control parameter.
The slave hyperchaotic system in receiving terminal is designed bywhere is the signal input of the single-dimensional controller, is the disturbance input, and is the receiving signal.
Define the tracking error variable aswhereWe get the following error dynamical model:where , is a positive scalar, and is a positive scalar.
The recovered signal is define byNow, to present the main objective of this paper more precisely, we introduce the following lemma and definition, which are essential for the later development.
Lemma 1 (See ). Given any real vectors and with appropriate dimensions and any positive scalar , the following inequality holds:
3. Main Results
In this section, based on Lyapunov method and LMI technology, the following theoretical results can be concluded.
Theorem 3. For Scheme 1, if there exist scalars and , such thatthe input signal in transmitting terminal can be recovered in receiving terminal with the required performance index .
Proof. Choose the following Lyapunov functional candidate:The time derivative of along trajectories of the error model (6) is given byConsider control law (9); we havewhereConsider the performance index as follows:For and , we havewhere Consider LMI (10); we have for any nonzero . The proof of Theorem 3 is thus completed.
4. Further Result
Next, based on the multidimensional controller, we present chaotic secure communication Scheme 2; see Figure 2.
The master hyperchaotic system is constructed asThe signals to be transmitted in channels are designed bywhere , , are the control parameters.
The slave system is constructed as follows:where is the disturbance input vector of slave system and is the multidimensional controller.
Define the tracking error vector asWe get the error dynamical model asThe recovered signal is defined byBased on Lyapunov method and LMI technique, the following theoretical result can be concluded.
Theorem 4. For chaotic secure communication Scheme 2, if there exist positive scalars , , such thatthe signal input in transmitting terminal can be recovered in receiving terminal with the required performance index .
Proof. First choose the following Lyapunov function:The time derivative of along trajectories of error model (22) isWith condition (24), we haveConsider performance index as follows:Because and , one can obtain where With LMI (25), we have for any nonzero . According to Definition 2, the proof of Theorem 4 can be completed.
5. Example and Simulation
In this section, we include some examples to validate the effectiveness of two proposed secure communication schemes. The numerical simulation is with the step size of 0.001 second and the following initial parameters: denotes the digital binary signal that is switching between 0 and 1 randomly. First, we consider secure communication Scheme 1. Based on Theorem 3, we get . The numerical simulation results are shown in Figures 3, 4, 5, and 6.
Remark 5. From numerical simulation, we notice that the input signal in transmitting terminal can be restored precisely in receiving terminal at early stage; later when disturbance is added at 20th second, the synchronization error jitters in a small range, which satisfies the required control performance index, and there exists an error between the input signal and the recovered signal. In addition, it can be found that Scheme 2 has better synchronization performance at the cost of more complicated structure, such as the requirement for all system state information and the increase of transmission channels, which cause the generation of new disturbances , , and will have an effect on the system control performance. Therefore, it can be concluded that both secure communication schemes are meaningful and are chosen according to the actual requirement in real engineering.
In this paper, a new hyperchaotic system is included for the secure communication scheme design in the case that disturbances exist. Based on Lyapunov method and technology, two secure communication schemes have been presented for the possible application in real engineering; some typical numerical simulations have been carried out to demonstrate the effectiveness of our schemes.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by the Young Scholars Project of Xihua University (01201419), the Open Research Subject of Key Laboratory of Signal and Information Processing of Sichuan Province (szjj2014-018), the Open Research Fund of Key Laboratory of Fluid and Power Machinery of Ministry of Education (SZjj2011-006), and the National Natural Science Foundation of China (61174058, 61134001).
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