Stability Theoiy Of Fractional-order Nonlinear System And Its Applications To Synchronization Of Fractional-order Chaotic Systems

Fractional calculus has useful applications in physics, engineering, mathematical biologyand finance, is also applied to the theory of control of dynamical systems.Recently, studying fractional-order chaotic systems has become an active research field.Stability and synchronization of fractional-order nonlinear systems has been attracted lots ofattention. But due to a lack of stability theorems of nonlinear fractional-order systems, mostof existing methods for stability and synchronization of fractional-order nonlinear/chaoticsystems are currently based on an active control concept to eliminate nonlinear terms ofsynchronization error dynamics. However, the active control-based methods result nonlinearcontrol laws which are inconvenient in implementation. Especially, when the systems areinfluenced by some uncertainties and external disturbances, these methods are helpless. Themain contribution about fractional-order system is as follows:(1) Two stability theories offractional-order nonlinear system are proposed and proved, and two methods aboutconstruction of Lyapunov function are presented. The universal adaptive stability offractional-order nonlinear systems and the universal adaptive synchronization offractional-order chaotic systems are realized.(2) This paper investigates the problem ofpractical synchronization of fractional-order chaotic systems. Based on fractional-ordernonlinear system stability strategy, adaptive control theory and sliding mode control theory, aadaptive control scheme and adaptive laws of parameters are developed to robustlysynchronize fractional-order chaotic systems with unknown parameters and uncertainperturbations.(3) The linear feedback synchronization and adaptive synchronization offractional-order time-delayed chaotic systems is first realized by using the fractional-ordernonlinear system stability theory and constructing Lyapunov functions.This paper is organized as follows:Chapter one introduced the definitions and properties of fractional calculus, and expendedthe properties of Caputo derivative and proved.Chapter two is a brief introduction to calculation and the numerical simulation methods offractional calculus, and circuit experiment method of fractional-order chaotic system.In chapter three, two stability theories of fractional-order nonlinear system are proposedand proved, and two methods about construction of Lyapunov function are presented.Chapter four presents the methods of the universal adaptive stability of fractional-ordernonlinear systems and the universal adaptive synchronization of fractional-order chaoticsystems. In chapter five, the control and synchronization of fractional-order chaotic systems isrealized via a single controller with a drive variable.Chapter six proposes the complete synchronization and projective synchronization offractional-order chaotic system with unknown parameters and uncertain perturbations.In Chapter seven, Based on fractional-order nonlinear system stability strategy, adaptivecontrol theory and sliding mode control theory, we realized the complete synchronization andlag projective synchronization between different fractional-order chaotic systems.Chapter eight proposes two synchronization methods of fractional-order time-delayedchaotic systems: linear feedback synchronization and adaptive synchronization.Chapter nine concludes the paper with the outlook for the future.