A Fractional-Order Backstepping Controller with Quadratic Lyapunov Function for Rotor Displacement Control in Active Magnetic Bearings

In this paper, a fractional-order quadratic Lyapunov function (FQLF) controller is designed for an active magnetic bearings (AMB) model with uncertain parameters by adding a fractional-order term to the backstepping controller design derivation. By tuning the parameters and orders of the two fractional-order calculus terms, the controller can better approximate the system dynamics and more effectively regulate the rotor displacement, thereby achieving the desired control performance. Additionally, a generalized modified Oustaloup numerical approximation with endpoint compensation and fixed poles is introduced to solve for the optimal fractional-order ordinate. Considering the effects of unknown perturbations and system nonlinearity, an adaptive update law is introduced to mitigate the impact of perturbation variations on the system. Compared with existing control algorithms, the proposed control algorithm demonstrates superior performance and robustness, as validated through numerical simulations. The results show that the proposed method achieves smaller overshoot, stronger disturbance rejection, and improved stability, making it more effective in controlling the system.