Adaptive Backstepping Design for a Class of Non‐Commensurate Fractional‐Order Strict‐Feedback Nonlinear Systems With an External Disturbance

ABSTRACT

This paper develops an adaptive backstepping controller for a class of non‐commensurate Caputo fractional‐order strict‐feedback nonlinear systems with unknown constant parameters and an unknown bounded external disturbance. The revised design works directly with the original recursive errors and does not require a common auxiliary order, filtered transformed errors, or an assumed pointwise inequality between successive errors. Each backstepping step contains an explicit cancellation term for the preceding strict‐feedback interconnection. A sum‐separable multi‐order Lyapunov function, with fractional adaptation laws matched to the order of the corresponding channel, yields an exact dissipation inequality in which all recursive cross terms cancel. A multi‐order Lyapunov–LaSalle argument establishes boundedness of the adaptive closed‐loop signals and asymptotic convergence of every tracking error. The same construction is applied to master–slave synchronization. Recursive synchronization errors and the output synchronization error converge without an auxiliary compatibility condition; full‐state synchronization follows under a stated regularity condition on the generated virtual controls. The numerical studies implement the theoretical laws without saturation, projection, smoothing, or implementation‐only damping terms. Tracking, pulse‐disturbance response, and a non‐commensurate Chua‐form synchronization example illustrate the resulting closed‐loop behavior.