Aiming at the abrupt faults of the chaotic system, an adaptive observer is proposed to trace the states of the master system. The sufficient conditions for synchronization of such chaotic systems are also derived. Then the feasibility and effectiveness of the proposed method are illustrated via numerical simulations of chaotic Chen system. Finally, the proposed synchronization schemes are applied to secure communication system successfully. The experimental results demonstrate that the employed observer can manage real-time fault diagnosis and parameter identification as well as states tracing of the master system, and so the synchronization of master system and slave system is achieved.

Chaotic systems are complex nonlinear systems whose behaviors are highly sensitive to initial conditions, and they are long-term unpredictable in general, resulting long-term unpredictable in general. The characteristics of chaos have been attracting researchers’ attention since 1990s. The realization of OGY (Ott-Grebogi-Yorke) chaos-control method [

For many dynamic systems, chaotic systems, for example, system parameters and models, may change with the environments. Because of the uncertainty of these parameters or model, it is usually very difficult to synchronize the slave system and the master system. However, observer-based synchronization control methods need not know all statuses of the original chaotic system, so it is more suitable for practical applications [

Works described above did not take system failures into account, and they all focused on the design of synchronization observers under normal operating conditions, while, in practical chaotic secure communication systems, sensors, actuators, and inner components may inevitably fail, which will lead to sharp performance decline of the chaotic secure circuits, and may even cause paralysis of the whole communication system and then lead to enormous economic losses. For this reason, the reliability of chaotic system is crucial in secure communication systems. Research on fault tolerant synchronization has become a hot topic in chaotic synchronization research area, and it is attracting more and more attentions.

Our work considers the abrupt fault in chaotic system, employs an observer-based active fault tolerant approach, diagnoses fault online using adaptive observer, and achieves synchronization of chaotic system effectively. We verified the effectiveness of our method through numerical simulations; experimental results showed that synchronization can be achieved no matter the system failed or not. Finally, we also applied our method to chaotic secure communications.

The rest of this paper is organized as follows. Section

Consider the following chaotic system as a master system:

Observer-based chaotic synchronization method builds observer for the master system, and the constructed observer is employed as slave system to trace the state of the master system, and then the synchronization of master system and slave system is achieved. The adaptive observer is designed as

Assuming that the error of the fault function and the synchronization error of the chaotic system are, respectively, defined as

we can get the following error system from the derivative of (

In order to facilitate design, we provide the following hypothesis and lemma.

To chaotic system formulated in (

To chaotic system formulated in (

To appropriately dimensioned matrices

To adaptive observer (

Select the following Lyapunov function:

From the derivative of

According to Hypothesis

The following inequality can be derived from Hypothesis

From Hypothesis

Substitute (

Substituting adaptive law (

From Theorem

In order to validate the correctness of Theorem

Let the initial state of the master system-equation (

The trajectories of a fault and the fault estimation of the system are shown in Figure

The trajectories of a fault (solid line) and the fault estimation (dotted line).

Figure

Three error trajectories between synchronized chaotic systems.

We apply the proposed chaotic synchronization scheme for secure communication. The plaintext signal is sinusoidal as shown in Figure

The original signal from transmitter.

The masked sinusoidal signal.

The recovered signal from receiver is shown in Figure

The recovered signal from receiver.

The error curve between the plaintext and recovered signal is demonstrated in Figure

The error curve between plaintext and recovered signal.

Adopting fault tolerant synchronization method, this paper discusses chaotic fault tolerant synchronization control based on adaptive observer. By utilizing Lyapunov theory of stability, it is proved that the error systems can still get asymptotic stability after system failure. The constructed adaptive observer can not only manage real-time fault diagnosis and parameter identification, but also trace the states of the master system, and so the synchronization between the master system and the slave system is achieved. Then the feasibility and effectiveness of the proposed method are illustrated via numerical simulations of chaotic Chen system.

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