Natsuki Kawaguchi

Abstract <p>This study applies fractional-order sliding mode control (SMC) to the robust target tracking control of a levitated object in an attractive-force-type magnetic levitation system featuring nonlinear and inherently unstable dynamics. A design strategy is proposed for fractional-order SMC. The design ensures structural consistency between the switching function and control input by utilizing a mathematical model linearized near the equilibrium point as the design basis. First, a sliding surface is constructed using optimal feedback gains derived from fractional-order linear quadratic regulator theory. Second, an equivalent control input is obtained by imposing the condition that the 0.5th-order derivative of the switching function is zero. Third, to suppress chattering, a sigmoid function is employed in place of the sign function in the switching control input. This method realizes quasi-SMC that guarantees Lyapunov stability within the boundary layer while avoiding the Filippov framework through continuous approximation. The size of the boundary layer does not depend on the order of differentiation but rather on the control gain, the smoothing parameter, and the disturbance bound. This result provides a framework for a comprehensive understanding of the convergence properties of SMC in fractional-order systems. Verification on an actual magnetic levitation system demonstrates that the proposed method exhibits clear advantages over conventional integer-order SMC, including enhanced smoothness of the output response and reduced chattering in control voltage. The proposed design method for fractional-order SMC integrates fractional-order optimal control theory and SMC, achieving both theoretical consistency and implementation effectiveness.</p>