The Adaptive Backstepping Synchronization Control for a Kind of Variable-Order Fractional Uncertain Nonlinear Systems

This paper is concerned with the adaptive backstepping synchronization control for a class of variable-order (VO) fractional uncertain nonlinear system with external disturbances and a dead zone. A kind of VO fractional command filter is employed to cope with “explosion of complexity”. The unknown nonlinear term in considered system is decomposed into unknown parameters and error functions by the Szász–Mirakyan operator theory. A VO fractional disturbance observer derived in this paper is used to simultaneously deal with the difficulties brought by dead zone and external disturbances. Thus, a VO fractional backstepping synchronization controller with adaptive laws for the system handled is proposed; moreover, the stability of system controlled is established. Finally, numerical examples are given to validate the theoretical results.