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

This article presents two adaptive fuzzy output-feedback dynamic surface control schemes for single-input-single-output (SISO) strict-feedback fractional-order uncertain nonlinear systems under the complete smooth nonlinearity and partial piecewise-smooth nonlinearity, respectively. For the former, complete smooth nonlinear functions are approximated by employing fuzzy logic systems, based on which a novel fractional-order reduced-order observer is constructed to estimate the unmeasurable states. For the latter, a new switched approximation and estimate strategy is proposed to deal with the problem of piecewise-smooth nonlinearity. Meanwhile, fuzzy basis function property is utilized to eliminate the assumption requirement that the nonlinear functions must satisfy the Lipschitz condition inherent in the output-feedback control, which is suitable for two cases. Under certain assumptions, the stability of two closed-loop systems are proved via fractional-order Lyapunov function stability criterion. Specifically, the stability proof of the second case is built on the foundation of the common Lyapunov function method. Finally, two diverse Chua's circuit simulation examples are provided to, respectively, validate the effectiveness of the proposed two control strategy.