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The synchronization of nonlinear uncertain chaotic systems is investigated. We propose a sliding mode state observer scheme which combines the sliding mode control with observer theory and apply it into the uncertain chaotic system with unknown parameters and bounded interference. Based on Lyapunov stability theory, the constraints of synchronization and proof are given. This method not only can realize the synchronization of chaotic systems, but also identify the unknown parameters and obtain the correct parameter estimation. Otherwise, the synchronization of chaotic systems with unknown parameters and bounded external disturbances is robust by the design of the sliding surface. Finally, numerical simulations on Liu chaotic system with unknown parameters and disturbances are carried out. Simulation results show that this synchronization and parameter identification has been totally achieved and the effectiveness is verified very well.

Chaos has the class-random characteristics, so it has the great application prospects in the secure communications. There has been significant interest in using chaotic dynamics to realize secure communications and cryptography during the last two decades. Since Carroll and Pecora [

Most of the work in this area focused on synchronization of chaotic systems to recover information signals [

Sliding mode variable structure control [

In this paper, based on sliding mode variable structure control theory and observer, a sliding mode variable structure observer is obtained to achieve the synchronization of uncertain chaotic systems with unknown parameters and external disturbances. According to the Lyapunov stability theory, the condition of synchronization and proof is presented. We point out the capability of the sliding mode observer to handle disturbances, as these have shown to be a challenge for other observers. It also has stronger robustness for chaotic systems and could accurately identify the unknown parameters with external disturbances. The observer has fewer constraints and wide application, which is suitable for most common chaotic systems. Finally, numerical simulation by Liu chaotic system verifies the effectiveness of this method.

Consider the following nonlinear chaotic system model:

We make the following assumptions.

The known nonlinearity

where

There is

There are positive definite matrixes

Based on the previous assumptions, for the drive system (

The state error vector is

If the error states in system (

In order to design the sliding mode variable structure observer (

Considering the system (

Define the parameter error vector

According to Lyapunov stability theory, the original Lyapunov function is positive

In the design of the observer, it is very important that the assumption 4 is satisfied or not. When the matrix

Discontinuous switch

In order to verify the correctness of the above method, Liu chaotic system is an example to simulate. In this paper, we study the Liu which is a typical chaotic system with characteristics of chaos and its expression is

Assume that the parameters

Equation (

In simulation, the initial value of Liu drive system and its observer are chosen as

Graph of error state

Graph of parameter identification.

The state observer based on sliding mode control is proposed to synchronize a class of chaotic systems with uncertain parameters and external disturbance and identification method for unknown parameter is given. Nonlinear parts satisfy the Lipschitz condition and external disturbance is bounded in this uncertain nonlinear system which is discussed. Depending on characteristics of the system, four assumptions are given. In the condition of satisfying the assumptions, a sliding mode observer is designed by combining sliding mode variable structure control and observer methods, and the synchronization of chaotic systems is realized finally. By means of Lyapunov stability theory, the conditions to realize the synchronization of the disturbed chaotic system with uncertain parameters is given and the correctness of the observer is verified. The numerical simulations by Liu chaotic system show that the designed sliding mode observer can still effectively realize the observation of the state variables and identification of the unknown parameters. Simulation results verify the effectiveness of the proposed methods. The observer has stronger robustness to chaotic systems with unknown parameters or external disturbance and has fewer constraints and wide application, which is suitable for most common chaotic systems.

This research is supported by the Chinese National Natural Science Foundation (no. 60772025) and the National Natural Science Foundation of Heilongjiang province (Grant no. F201220).