This paper deals with the synchronization of fractional-order chaotic systems with unknown parameters and unknown disturbances. An adaptive control scheme combined with fractional-order update laws is proposed. The asymptotic stability of the error system is proved in the sense of generalized Mittag-Leffler stability. The two fractional-order chaotic systems can be synchronized in the presence of model uncertainties and additive disturbances. Finally these new developments are illustrated in examples and numerical simulations are provided to demonstrate the effectiveness of the proposed control scheme.
Two different sliding mode controllers for a fractional order unified chaotic system are presented. The controller for an integer-order unified chaotic system is substituted directly into the fractional-order counterpart system, and the fractional-order system can be made asymptotically stable by this controller. By proving the existence of a sliding manifold containing fractional integral, the controller for a fractional-order system is obtained, which can stabilize it. A comparison between these different methods shows that the performance of a sliding mode controller with a fractional integral is more robust than the other for controlling a fractional order unified chaotic system.
A robust adaptive repetitive learning control method is proposed for a class of time-varying nonlinear systems. Nussbaum-gain method is incorporated into the control design to counteract the lack of a priori knowledge of the control direction which determines the motion direction of the system under any input. It is shown that the system state could converge to the desired trajectory asymptotically along the iteration axis through repetitive learning. Simulation is carried out to show the validity of the proposed control method.
