Adaptive Fuzzy Variable Structure Control of Fractional-Order Nonlinear Systems with Input Nonlinearities

The unknown dead-zone input nonlinearities (DZINs) are considered in the Riemann–Liouville fractional-order nonlinear systems (FONSs) and the Caputo FONSs in this paper. The unknown DZINs in the FONSs will cause FONSs instability. In this paper, by using the fractional-order Lyapunov stability theory, a variable structure adaptive fuzzy control (AFC) scheme is designed to solve the unknown DZINs in the FONSs. The unknown terms of the FONSs and the uncertain terms of DZINs are handled by fuzzy logic systems (FLSs). The parameters boundedness of FLSs is guaranteed via the constructed fractional-order adaptive laws (FOALs). By using FLSs, this paper does not need to know the exact values of gain reduction tolerances (GRTs) in the unknown DZINs, which makes the constructed scheme more suitable for the actual system. The scheme proposed in this paper can be used to effectively control the Riemann–Liouville FONSs and the Caputo FONSs with/without unknown DZINs. Finally, three simulation results verify the AFCs we designed are effective for both Riemann–Liouville FONSs and Caputo FONSs with unknown DZINs.