Modified Projective Synchronization For Fractional Order Chaos With Unknown Parameters Based On Adaptive Control

Chaos is a kind of dynamic behavior which similar to the random movement with no rules.With the profound study of chaos theory,The fractional-order chaotic system,which combines the chaos-type phenomenon with the fractional-order calculus theory,breaks limitations of the integer-order chaotic system,which make it more prominent in Concealment,Unpredictability,Complexity and easier to implement and so on.The fractional-order Chaos is very suitable for the security field such as secret communication and information encryption,which is the most important research on chaos system at present.The paper mainly researches on the problem of the modified projective synchronization of fractional-order chaotic systems with unknown parameters based on the adaptive control method and the Lyapunov stability theory.The main work of this paper is as follows:The active adaptive controller is designed taking the example of 3-D fractional order Chen chaotic system.Firstly,the active control method is used to linearize the error system.Then the adaptive controller is designed by using the adaptive control method and the stability theory of the fractional-order linear system.The modified projective synchronization of the fractional-order chaotic system is realized,while the unknown parameters in the system are also estimated in real time in this paper.The synchronization of Systems and the estimation of parameters can be achieved simultaneously.Finally,the effectiveness of the designed controller is verified by matlab simulation.The modified projective synchronization of fractional-order chaotic systems with different dimensions is realized,taking the 3-D fractional order Lorenz chaotic system and the 4-D fractional order Chen chaotic system as the response and drive systems respectively.Firstly,the transformation matrix is introduced for different dimensions of these two fractal order chaotic systems,the 4-D fractional order Chen chaotic system is transformed into a 3-D chaotic system,which is consistent with the dimension of the 3-D fractional order Lorenz chaotic system.Then the controller is designed based on the active adaptive control method to realize the modified projective synchronization and the estimation of parameters of different dimensions of the drive and response fractional order chaotic systems synchronously.After the interchange of systems,the active adaptive controller is designed considering the dimension of the drive system is less than the dimension of the response system.The simulation results validate the effectiveness of the designed adaptive controller.In order to improve the anti-interference ability of the systems,the sliding-mode adaptive control method is used to realize the projective synchronization and the 3-D fractional order Lorenz chaotic system and the 4-D fractional order Chen chaotic system are respectively used as the response system and drive system for Simulation.In this paper,the error system is simplified to meet the condition of consistent exponential stability,then the sliding mode surface and single sliding mode controller are designed.The Lyapunov method is theoretically validated the single sliding mode controller,and after the interchange of the system,it's also theoretically feasible.Finally,the feasibility of the single sliding mode adaptive controller is verified by matlab simulation.