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This study addresses an adaptive two-stage sliding mode control (SMC) scheme for the state synchronization between two identical systems, which belong to a kind of

Since the concept of the drive-driven chaotic synchronization has been first introduced in the commonly cited study [

The object of state synchronization is to achieve that each of state variables of the driven chaotic system is tracking to each of states of the drive one, respectively. Nowadays, many studies have been reported to achieve state synchronization of chaotic systems, such as active control [

In [

The drawbacks of these antecedent studies are not considering the input nonlinearities attached to the control inputs of the driven system. Under the limitation of physical properties, the inputs of control systems are commonly involved in nonlinearity, such as sector nonlinearity, in practical usage. It is revealed that the system performance is caused by a serious degradation by the existence of input nonlinearity for control. Consequently, it would be necessary to design control inputs by taking into account the effects of input nonlinearity [

For the jerk chaotic system [

For solving the aforementioned control problem, the main contribution of the present study is to develop an adaptive two-stage SMC scheme for achieving the state synchronization. In comparison with the past studies [

The developed adaptive two-stage SMC scheme is derived based on the introduced sequence of two sliding functions. The first is named the stage 1 sliding function

The rest of this paper is organized as follows. The control problem of the state synchronization between two

In this study, the following class of

Considering the nonlinear chaotic system defined in equation (

The input nonlinear function

The error states between system equations (

Taking the time derivatives of equation (

Owing to that the chaotic system always exhibits the globally bounded state trajectories, it is fairly assumed that

It is clear to show that the control problem of state synchronization between system equations (

In the following, two steps are introduced to design the adaptive two-stage SMC scheme to accomplish the state synchronization between system equations (

The stage 1 sliding function

Equation (

The novel integral type of the stage 2 sliding function

Let

From equation (

At this point, it is concluded that the objective of control scheme design is to force that

If the following control scheme

The mathematical Lyapunov function is selected to be

Taking the time derivative of equation (

Moreover, from equation (

By substituting equations

Therefore, the condition of

In the literature, the related SMC schemes for solving the chaos control and state synchronization of the jerk chaotic systems were addressed in [

In comparison with the past studies [

In the following section, the proposed adaptive two-stage SMC scheme is carried out by the numerical simulations for two identical 3D jerk chaotic systems reported in [_{.} The 3D jerk chaotic system reported in [

The system in equation (

The 3D jerk chaotic system with

For the drive and driven 3D jerk chaotic systems (

The guideline for choosing the positive design parameters of the proposed control scheme involves three basic steps. Firstly, according to equation (

According to the aforementioned guideline, for the adaptive two-stage SMC scheme in equation (

Time responses for error states of system equation (

Time responses for error states of system (

Figure

Time responses of

Figure

Time responses of the adaptive feedback gains and the control input

Time responses of states for the drive and driven 3D jerk chaotic systems.

For the control scheme given by equations (

To demonstrate the effectiveness of the proposed method, the existing control method introduced in [

Time responses for error states

Time responses of error states and the control input for system (

In this study, the adaptive two-stage SMC scheme for achieving state synchronization between two systems, which are pertaining to a class of the

The proposed adaptive control scheme including time-variable feedback gains updated by the suitably adaptive rules can cope with the effect of sector input nonlinearity for achieving the goal of control. The mathematically sufficient conditions are given to guarantee the stability of synchronization by means of the Lyapunov stable theorem. Besides, numerical studies for two 3D jerk chaotic systems reported in [

The data used to support the findings of this study are available from the corresponding author upon request.

This study was carried out as part of the Intelligent Manufacturing Program coordinated by the beautiful China Research Institute of Sanming University.

The authors declare that they have no conflicts of interest.

This study was financially supported by the Operational Funding of the Advanced Talents for Scientific Research (19YG04) of Sanming University and the Science and Technology Department of Fujian Province (Grant no. 2017H6019). The authors also acknowledge the support from the College of Mechanical and Electrical Engineering in Sanming University.