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This paper mainly investigates the projection synchronization of complex chaotic systems with both uncertainty and disturbance. Using the linear feedback method and the uncertainty and disturbance estimation- (UDE-) based control method, the projection synchronization of such systems is realized by two steps. In the first step, a linear feedback controller is designed to control the nominal complex chaotic systems to achieve projection synchronization. An UDE-based controller is proposed to estimate the whole of uncertainty and disturbance in the second step. Finally, numerical simulations verify the feasibility and effectiveness of the control method.

The chaotic synchronization phenomenon that caused a great sensation in academia was firstly proposed by Pecora and Carroll in early 1990 [

Many control methods about chaotic projection synchronization have been reported [

We note that most of the literature on solving the control problems of chaotic systems with external perturbations is generally complex and difficult to implement. Moreover, when designing the controller, the method to deal with the external disturbance is just simply to cancel the disturbance term from the formula of the controller, and it is not rigorous in nonlinear system control theory. In fact, in the field of nonlinear system control, the UDE-based controller can deal with many structured and unstructured robust control problems and has been applied to the engineering field in some literatures [

The main contribution of this paper is to design a physical controller, which is simple in form, to realize the projection synchronization of a complex chaotic system. A linear feedback UDE-based control method is proposed by combining the linear feedback controller and the UED-based controller in two steps. A linear feedback control controller is designed for the nominal complex chaotic system in the first step. In the second step, an UDE-based controller is proposed to estimate the whole of uncertainty and disturbance. In the end, two complex chaotic systems with numerical simulations are used to verify the validity and effectiveness of the proposed theoretical results.

Consider the following controlled chaotic system:

Let system (

Let

Consider the controlled error system (

According to the results in [

The projection synchronization of system (

An algorithm was also proposed in [

Note that the subsystem

Consider the following controlled system:

It is well known that model uncertainty and external disturbance are inevitable in actual control problem, and the UDE-based control method [

Consider the following system:

The stable linear reference model is given as

Consider system (

According to the existing result in [

The other is the secondary filter:

In this section, the UDE-based linear feedback control method is proposed in two steps. In the first step, the linear feedback control method is proposed for the nominal system. The UDE-based control method is given in the second step.

Consider the following nominal system:

If the projection synchronization of system (

The corresponding slave system is presented as follows:

Let

Consider error system (

Since the matrix

In this section, the UDE controller is proposed to cancel the uncertainty and disturbance of the complex chaotic system.

Consider the following controlled master system:

The corresponding salve system is

Let

The controller

Step one: according to Theorem

Step two: the controller

Consider error system (

Substituting

According to condition (

Thus,

In this section, one example with numerical simulations is used to demonstrate the effectiveness and validity of the proposed results.

Consider the following complex Lorenz system:

Separating the real and imaginary parts of complex variables

According to the results in [

It results in

It is easy to obtain that

Thus, the master system (

The UDE-based linear feedback controller is designed by the following two steps.

Step one:

Then, the corresponding slave system is given as follows:

Let

Note that if

is globally asymptotically stable.

Thus,

Numerical simulation is given, and the initial values of the master-slave systems of given complex Lorenz system are chosen as follows:

From Figures

Step two: consider the following master system with both model uncertainty and external disturbance:

where

The phase portrait of master subsystem and the slave subsystem.

The phase portrait of master subsystem and the slave subsystem.

The phase portrait of master subsystem and the slave subsystem.

The salve system is

Let

According to Theorem

Numerical simulation results are given with the following conditions:

Case 1:

Case 2:

It can be seen from Figures

The phase portrait of master subsystem and the slave subsystem.

The phase portrait of master subsystem and the slave subsystem.

The phase portrait of master subsystem and the slave subsystem.

The phase portrait of master subsystem and the slave subsystem.

In conclusion, the projective synchronization of a class of complex chaotic systems with both uncertainty and disturbance has been solved. First, the linear feedback control method is proposed for the nominal system (without uncertainty and disturbance), and projection synchronization of such system has been realized. Then, the UDE-based linear feedback control method is presented by two steps, by which the projection synchronization of the complex chaotic systems with both uncertainty and disturbance has been completed. Finally, an experimental simulation example has been used to verify the feasibility and effectiveness of the obtained results.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

This study was supported by the National Natural Science Foundation of Shandong Province (ZR2018MF016).