Study On Stability Control And Synchronization Of Fractional Order Chaotic Systems

With the development of fractional calculus and chaos,significant progress has been made in the control and synchronization of fractional order chaotic systems.Fractional order chaotic system not only has the characteristics of the chaotic system,but also has the advantages of the fractional order dynamical system.Fractional order chaotic system is of extremely significant value in the field of secure communication.Therefore,it is very important to study the synchronization and control of fractional order chaotic systems.This paper,based on the fractional calculus theory,combined with the concepts and properties of fractional order system stability theorem and finite time stability,analyzes the stability of the fractional order chaotic system,and studies the control and synchronization of the fractional order chaotic systems.The main contents are as follows:First,the basic theory of fractional calculus and the chaos theory are introduced,and we extend the stability of fractional order system to the finite time stability of fractional order chaotic system.For the four dimensional fractional hyperchaotic Lorenz system with uncertain parameters,the phase diagram of the system is obtained by MATLAB simulation,and we can determine the chaotic state of the system.Based on the concepts of the fractional calculus lemma and finite-time stability,and applying the adaptive rule at the same time,an adaptive finite-time controller is designed.By constructing a new Lyapunov function,the stability of the state variables of the system can be realized in a finite time.The numerical simulation and theoretical derivation verify the effectiveness of the proposed control method.Secondly,the finite time synchronization of a class of fractional order hyperchaotic systems with same structure is studied.By selecting the appropriate parameter function in the process of designing the controller,the controller can eliminate the useless nonlinear terms in the error systems while controlling them,so that the error systems tend to zero,that is,the driving system and the response system are synchronized.The integral calculation of the error systems is carried out,and we can get the stability time of the error systems.The numerical simulations show that this method achieves a faster steady-state and a faster synchronization speed and control efficiency than the complete synchronization method,which verifies the superiority of this method in the field of synchronization.Finally,the idea of sliding mode control theory is applied to the synchronization problem of fractional-order chaotic systems with different structures.The synchronization of two different fractional order chaotic systems can be realized by designing a terminal sliding mode controller.At the same time,a non-singular terminal sliding mode surface is designed in the process of constructing this controller,and the sliding surface can converge to zero point in finite time.Simulation results verify the effectiveness of this method.It can be found that the controller can still make the system synchronization error converge to zero in the case of the uncertainties and external disturbances.