Back-stepping projective synchronization of fractional-order unified systems based on the lower triangular structure

Abstract

The synchronization of fractional-order chaotic systems (FOCSs) plays an important role in modern control theory, the projective synchronization (PS) as a class of synchronization problems, also has huge applications and has attracted much attention. It is, however, shown in the obtained literature that the results on the PS of FOCSs either loss the rigorous theoretical demonstration or verify from the viewpoint of numerical simulations. How to derive a necessary and sufficient condition to guarantee the PS of complex FOCSs by a simple controller is still open. To this end, this article is concerned with the PS of fractional-order unified systems (FO-USs) that are important in FOCSs covering fractional-order Lorenz, Chen and Lü systems, where the controller is presented based on the lower triangular structure by use of the back-stepping technique. The necessary and sufficient criterion for the PS of FO-USs is proposed by solving an algebraic equation, and the controller for the PS of FO-USs is derived based on the lower triangular structure combined with back-stepping approach. Finally, the simulation results are reported to verify the correctness and efficiency of the obtained results.