Fractional-order Nonlinear System Control

It has been proved by the morden physics that many real physical phenomena or processes can be described by fractional-order systems using fractional calculus more accurate than the classical ones. At present, fractional-order systems control has emerged as a hot research topic in the area of nonlinear systems.Supported by the national science foundation project "Uncertain Fractional-order Chaotic System Synchronization Control and Its Application in Secure Communications" and the provincial science foundation project of Zhejiang "On the Asymptotic Stability of Fractional-order Nonlinear Systems", the fractional-order nonlinear systems control has been studied as an initial work toward to a systematic control theory.At the beginning, some fundamental concepts of fractional-order systems are discussed in Chapter2, including state-space model, pseud-sate, state, initialization and energy. By use of fractional Lyapunov stability, a new fractional-order control theory is proposed, together with the verification a class of power functions as fractional Lyapunov functions and the sufficient conditions of the stabilizability of fractional-order nonlinear systems. Besides, the concept of control fractional Lyapunov functions is introduced with or without the adaptive mechanism.In Chapter3, the idea of Backstepping is introduced into fractional-order nonlinear systems to establish a new control scheme for fractional-order nonlinear systems:Fractional Backstepping Design. Using fractional Lyapunov functions, several sufficient conditions of the Mittag-Leffler stabilizability of fractional-order nonlinear systems are proved. As the main result, several fractional-order stabilizers are designed for deterministic fractional-order nonlinear systems and uncertain ones respectively, guaranteeing the global convergence of the closed-loop systems.Fractional-order nonlinear systems usually incur many distrubances outside or inside, uncertain parameters or noises, for instance. In Chapter4, the stabilization problems of uncertain fractional-order nonlinear systems with disturbances are formed. No matter the disturbance occurs at the input or other positions of the system, the continuous fractional-order controllers and the switching controllers were both constructed with the prior known or unknown upper bound of disturbances. It is proven that the convergence of the closed-loop systems to the equilibria can be guaranteed.