Sliding mode control for fractional nonlinear systems of order (0,2)

This paper proposes a sliding mode control (SMC) approach for fractional nonlinear systems (FNSs) of order (0,2) in the presence of external disturbances. The study encompasses two distinct cases: (0,1) and (1,2). In the study of (0,1) order FNSs, a novel hybrid integral sliding mode surface (SMS) combining classical integral functions with fractional integral functions is designed to reduce buffeting. The disturbance factor is incorporated into the sliding mode controller, and the asymptotic stabilization of FNSs is attained using the Lyapunov asymptotic stability theorem under both known and unknown disturbances. In the study of (1,2) order FNSs, a novel generalized asymptotic expansion principle of Mittag–Leffler (M‐L) functions is established. This principle is then utilized to establish a novel Lyapunov asymptotic stability theorem for (1,2) order systems. Additionally, it is demonstrated that the FNSs are asymptotically stable under a general integral SMC, as substantiated by the Lyapunov stability theorem. Two numerical examples are provided to verify the feasibility of the proposed methods.