Synchronous Control Of Fractional Order Chaotic System

Chaos is a generalization of a kind of complex and disordered motion,which is extremely sensitive to small changes.The order of chaos is extended to the range of fraction,and the fractional order chaos model is obtained.The change of order makes the fractional order model show more complex evolution track,and evolves many special properties of fractional order chaos.This promotes the development of fractional chaos to some extent.Synchronization plays an important role in the practical application of chaos,which means controlling two or more chaotic systems to keep the same state trajectory.In view of the potential advantages of fractional chaotic model,it is more valuable to study and realize the chaotic synchronization with fractional model in theory and practice.In this paper,a new method of designing fractional order chaotic synchronization controller is proposed.Considering the asymptotic stability and finite time stability of the error system,the controller with different structure is designed by combining impulse and sliding mode control.Scientific and rigorous mathematical derivation process and simulation results are given,which provide theoretical support and practical verification for the effectiveness of the controller.The main contents and research results of this paper are as follows:1.The inconsistency between the driving system structure and the response system is considered.Drive system and response system are linearized by fuzzy model.Based on the stability theorem of fractional order impulsive differential equation,a fuzzy impulsive controller is designed to track the driving system effectively.2.The design of synchronous controller is based on the perturbed fractional order system.Considering that the disturbance of the system is unknown,a new state observer is designed based on the idea of high order sliding mode to obtain the state trajectory of the response system.The observer can effectively estimate the response system and replace the system with unknown state to design the controller.Finally,according to the fractional stability theory,the mathematical expression of the controller is obtained.3.Aiming at the finite time synchronization problem of fractional order chaotic system in presence of uncertainty and external disturbance,a new fractional order sliding surface is designed and its finite time convergence to zero is proved analytically.The boundary value of the external disturbance and the Lipschitz constant of the nonlinear term of the system are estimated by using the adaptive law.In order to ensure the occurrence of sliding mode motion in finite time,an adaptive fractional order sliding mode controller is designed.Based on Lyapunov stability theory,the existence of finite time sliding mode motion is proved analytically.Finally,the effectiveness of this method is verified by the experiment of fractional chaotic system synchronization.