Study On Control And Synchronization Of Fractional-order Chaotic Systems

Chaos universally exists in many fields of natural sciences and social sciences. For humans, the chaos is a double-edged sword. For example, in the fields of signal processing and communications, people can take advantage of the characteristics of chaos to study secure communication and encryption of images. But in controlling area, the emergence of chaos will cut down the controllability of the equipment controlled and exacerbate the damage to equipment. Therefore, the human needs to learn how to take advantage of the chaos, or to avoid the generation of chaos in different areas. In recent years, as a special nonlinear system, the fractional order chaotic systems has attracted more and more attentions of many scholars in mathematics and control. The research of fractional order chaotic systems has been study deeply. At present, the research of fractional order chaotic systems mainly have the following several branches:the design of fractional order chaotic circuit; the application of fractional order chaos in digital signal processing; the study of fractional order hyperchaos and its dynamic characteristics; the application of fractional order chaos in secure communication; the study on synchronization and control of fractional order chaotic systems and so on.In this thesis, based on the stability theory of fractional order linear systems, the relative problems of fractional order chaos synchronization and control are studied using the methods of theoretical derivation and numerical simulation. We achieved the following results:Firstly, based on the stability theory of fractional order linear systems, a novel method combining feedback control and active control is proposed for the lag synchronization of fractional order chaotic systems. It made the lag synchronization of fractional order chaotic systems convert to asymptotic stability of the fractional order linear error systems at origin by designing a proper controller. The numerical simulation results on fractional order Chen system verify the effectiveness of the proposed method. Secondly, a kind of novel model, called generalized Takagi-Segno (T-S) fuzzy model, is first developed by extending the conventional T-S fuzzy model. Then, a method to control fractional order chaotic systems is proposed using the generalized T-S fuzzy model and adaptive adjustment mechanism (AAM). Sufficient conditions are derived to guarantee chaos control from the stability criterion of linear fractional order systems. The proposed approach offers a systematic design procedure for stabilizing fractional order chaotic systems. The effectiveness of the approach is tested on fractional order Rossler system and fractional order Lorenz system.Lastly, an adaptive feedback control method is presented to stabilize a class of chaotic systems, the structural function of whose is not necessarily to satisfy the Lipsichtz conditions, but bounded by a polynomial of the infinity norm of the system state with the gains unknown. The adaptive feedback controller uses a simple polynomial function of the system state, moreover only one component in each dimension. To check the theoretical results, we try to stabilize the Lorenz system on numerical simulations.