Research On Nonlinear Control Of Fractional Order Dynamic Systems

Chaos control and synchronization has been an active research field because of its great potential applications in secure communication, biomedical science, sociolo-gy, information science, electronics and so on. Now, there are lots of results on the chaos synchronization of same dimensional fractional order determinate systems with-out time-delay. And the synchronization of fractional order chaotic systems with differ-ent dimensions, time-delays or uncertain parameters is still a challenge, interesting and meaningful problem. The main achievements of this paper are as follows:1. The synchronization of uncertain fractional order chaotic systems is considered. Due to the stability theory of fractional order systems, a nonlinear controller is pre-sented for projective lag synchronization of uncertain fractional order chaotic systems. And the update laws are devised to identify the unknown parameters of systems. It is worth noting that complete synchronization, anti-phase synchronization, projective synchronization and lag synchronization are special cases of projective lag synchro-nization, which also can be achieved based on the proposed controller. Both identical and different structural systems can be applied to realize the synchronization. Finally, the parameter identification and the projective lag synchronization of fractional order chaotic and hyperchaotic systems with unknown parameters are finished. The uncertain fractional order chaotic Chen and Lii systems, hyperchaotic Chen and Lorenz systems are respectively applied for the numerical simulations. The effectiveness and feasibility of the advised schemes are demonstrated by the numerical results.2. The synchronization of different dimensional fractional order chaotic systems is investigated. According to the stability theorem of linear fractional order autonomous systems, the controllers for generalized projective synchronization and function projec-tive lag synchronization are respectively proposed for the different dimensional frac-tional order chaotic systems. Both reduced and increased dimensional synchronization can be realized based on the advised controllers. Besides, the different dimensional fractional order chaotic and hyperchaotic systems are applied for the numerical simula-tions of the reduced and increased dimensional synchronization. The numerical results coincide with the theoretical analysis.3. The synchronization of time-delayed fractional order chaotic systems is stud-ied. According to the stability theorem of linear fractional order autonomous systems with multiple time-delays, the controllers are respectively designed for hybrid projec-tive synchronization, modified projective lag synchronization and generalized synchro-nization of fractional order chaotic systems with time-delays. Then, the fractional order time-delayed financial and Liu systems are used for the synchronization to verify the ef-fectiveness of the controllers. The synchronization speed can be improved by selecting an appropriate controller. And the synchronization of the fractional order time-delayed financial and Liu systems with the bounded noise is employed to explain the robustness of the controllers.