Fractional Order Chaotic System Control And Synchronization Theory

Chaotic phenomena and chaotic systems generally exist in physics, engineering, biology and finance. Fractional order differential equations provide a more accurate tool to model the complicate systems. Fractional order chaotic systems have both the complexity of fractional order systems and chaotic dynamical characteristics. Therefore, it is every meaningful theoretically and practically to investigate the control and synchronization of fractional order chaotic systems.Based on fractional calculus theory and chaotic systems characteristics, this thesis deals with the stability, control and synchronization of fractional order chaotic systems. The thesis includes the following presentations:The basic definitions and properties of the fractional calculus and fractional order systems are introduced. Fractional order linear systems and nonlinear systems stabilities are discussed. Dynamical characteristics, stability, control methods and system simulations of the fractional order chaotic systems are presented.Based on fractional order differential operator theory, the corresponding integer form of fractional order chaotic systems are obtained by introducing fractional order feedback controllers, and a sliding mode controller is proposed to achieve asymptotical stabilization of fractional order chaotic systems. Under the perspective of system energy, passivity is introduced into the fractional order systems to design fractional order passive controllers. Based on steepest decent method, a parameter identification update law is designed to synchronize fractional order chaotic systems.Sliding mode control is generalized into fractional order systems, and fractional order integral sliding surface and reaching law are proposed to achieve robust stabilization of fractional order chaotic systems with parameters disturbances and modeling uncertainties. Based on fractional Lyapunov theory, fractional order system states are introduced into chaotic systems, to achieve adaptive synchronization of fractional order chaotic systems with unknown parameters and unknown external disturbances.Based on fractional backstepping control design, fractional order parameter estimates update law and unknown upper bound estimates laws are designed, to realize an adaptive control of fractional chaotic systems in strict form with unknown parameters and unknown external disturbances.Dynamical surface control is generalized into fractional order systems, by introducing fractional order filters, and fractional order comparison principle and fractional Lyapunov functional are employed to analysis closed-loop system stability. An adaptive control combined with fractional order update laws is designed to achieve tracking control of fractional order chaotic systems with arbitrary uncertainty.