Dynamic Analysis And Control Of Complex Lorenz Systems

With the continuous development of chaotic systems,the research of chaotic systems in complex domain attracts more attention.Because of the singularity,complexity and nonlinearity of complex chaotic systems,its theoretical research has become a hot topic.Complex Lorenz system is a typical complex chaotic system.Compared with traditional Lorenz system,complex Lorenz system is closer to the change process in the real world.Therefore,the in-depth study of complex Lorenz system will help us to better understand and grasp the laws of motion in the real world.Dynamic analysis of complex Lorenz system can reveal its complex nonlinear behavior patterns,including singular attractors,periodic orbits,chaos phenomena,etc.These behavior patterns have important theoretical and practical value,and have been widely used in communication,finance,biology and other disciplines.There are still some shortcomings in the existing research results of complex Lorenz system,such as not considering the influence of external disturbances,especially the influence of unbounded external disturbances on the system.Therefore,in this thesis,several control problems of complex Lorenz systems subjected to external perturbations are studied in depth,and some meaningful theoretical results are obtained.The main research work and innovation of this thesis are as follows:(1)Dynamic analysis of complex Lorenz system.The dynamic properties equilibrium of complex Lorenz systems are analyzed.The existence of chaotic attractors is confirmed by Lyapunov exponent and Poincare diagram.The phase diagram and state diagram of complex Lorenz system are drawn by MATLAB fourth-order Runge-Kutta method,and the chaotic behavior is further verified.(2)A filter which can estimate the external disturbance asymptotically is designed.The corresponding asymptotic estimation filters are designed for bounded disturbance and unbounded disturbance respectively,thus realizing the accurate suppression of unbounded disturbance and ensuring the control performance of the designed controller.The design of filters for unbounded external perturbations is an original feature of this thesis.(3)The stabilization problem of complex Lorenz system subjected to external disturbance is studied.A two-step controller is designed to stabilize the complex Lorenz system.In the first step,the stabilization problem of a nominal complex Lorenz system is completed according to the different controllers obtained by the linear-like feedback method and the linear feedback method.In the second step,a controller which can estimate the external disturbance asymptotically is designed according to the filter designed above.By combining the results of the two steps,a controller based on the estimation of external disturbances is obtained,which completes the stabilization of the complex Lorenz system subjected to external disturbances.Finally,the validity and correctness of the above theoretical methods are verified by MATLAB numerical simulation.The two-step design of controller based on estimation of external disturbance is another innovation of this thesis.(4)The complete synchronization problem of complex Lorenz system with external disturbance is studied.The controller is designed in two steps to realize the complete synchronization problem of complex Lorenz system.In the first step,the complete synchronization problem of a nominal complex Lorenz system is completed according to the different controllers obtained by the linear-like feedback method and the linear feedback method.In the second step,a controller which can estimate the external disturbance asymptotically is designed according to the filter designed above.By combining the results of the two steps,a controller based on the estimation of external disturbance is obtained,thus completing the complete synchronization problem of complex Lorenz system with external disturbance.Finally,the validity and correctness of the above theoretical methods are verified by MATLAB numerical simulation.(5)The synchronization and anti-synchronization of complex Lorenz systems with external disturbances are studied.The controller is designed in two steps to realize the synchronization and anti-synchronization of complex Lorenz system.In the first step,synchronization and anti-synchronization of a nominal complex Lorenz system are solved in terms of the different controllers obtained by the linear-like feedback method and the linear feedback method.In the second step,a controller which can estimate the external disturbance asymptotically is designed according to the filter designed above.By combining the results of the two steps,a controller based on the estimation of external disturbance is obtained,and the problem of synchronization and anti-synchronization of complex Lorenz system with external disturbance is solved.Finally,the validity and correctness of the above theoretical methods are verified by MATLAB numerical simulation.