Backstepping-Based Adaptive Control for Uncertain Fractional-Order Nonlinear Systems

In this paper, we aim to solve the output tracking problem for a class of uncertain single-input-single-output (SISO) commensurate fractional-order uncertain nonlinear systems in the presence of bounded disturbances. To this end, an adaptive backstepping smooth control strategy is proposed, with two adaptive laws respectively estimating unknown system uncertainties and the upper bound of disturbances. Theoretical analysis is provided by employing fractional directed Lyapunov method to show that all the closed-loop signals are uniformly ultimately bounded. Subsequently, an adaptive smooth control scheme for second-order fractional systems is further proposed which guarantees asymptotically output tracking. Simulation studies are presented to verify the effectiveness of the proposed control methods.