Adaptive Backstepping Control for Fractional-Order Nonlinear Systems With External Disturbance and Uncertain Parameters Using Smooth Control

In this paper, we consider controlling a class of single-input-single-output (SISO) commensurate fractional-order nonlinear systems with parametric uncertainty and external disturbance. Based on backstepping approach, an adaptive controller is proposed laws that are used to estimate the unknown system parameters bound Instead using discontinuous functions such as $\mathrm{sign}$ function, auxiliary function employed obtain smooth control input still able achieve perfect tracking in presence bounded disturbances. Indeed, global boundedness all closed-loop signals asymptotic output given reference trajectory proved by fractional directed Lyapunov method. To verify effectiveness method, simulation examples presented.