High torque ripple can be observed with a 6/4 variable flux reluctance machine (VFRM). In order to minimize the torque ripple in VFRMs, this paper presents a harmonic injection method for 6/4 VFRMs with an open-winding configuration. By analyzing the impact of harmonics on VFRMs, the method involves detecting the third harmonic using a first-order low-pass filter (FLPF). Subsequently, the extracted harmonics are controlled and shifted to counteract the voltage harmonics in both inverters without inducing phase imbalance or overvoltage. With the proposed method, the torque ripple can be significantly reduced by about 50% under load conditions. The effectiveness of the harmonic injection method is validated through a prototype VFRM.

Chaotic systems appear in a wide range of natural and engineering contexts, making the design of reliable and flexible control strategies a crucial challenge. This work proposes a robust control scheme based on the Fractional-Order Backstepping Control (FOBC) method for the stabilization of non-commensurate fractional-order chaotic systems subject to bounded uncertainties and external disturbances. The method is developed through a rigorous stability analysis grounded in the Mittag–Leffler function, enabling the step-by-step stabilization of each subsystem. By incorporating fractional-order derivatives into carefully selected Lyapunov candidate functions, the proposed controller ensures global system stability. The performance of the FOBC approach is validated on fractional-order versions of the Duffing–Holmes system and the Rayleigh oscillator, with the results compared against those of a fractional-order PID (FOPID) controller. Numerical evaluations demonstrate the superior performance of the proposed strategy: the error dynamics converge rapidly to zero, the system exhibits strong robustness by restoring state variables to equilibrium quickly after disturbances, and the method achieves low energy dissipation with a high error convergence speed. These quantitative indices confirm the efficiency of FOBC over existing methods. The integration of fractional-order dynamics within the backstepping framework offers a powerful, robust, and resilient approach to stabilizing complex chaotic systems in the presence of uncertainties and external perturbations.

In this article, we present a novel approach: a fractional-order fuzzy observer, an extension of the Thau-Luenberger observer, specifically designed for nonlinear systems characterized by commensurate non-integer order Takagi-Sugeno models. This work makes significant contributions in two key areas. Firstly, both the activation functions of the model and the Lipschitz fractional-order fuzzy observer are dependent on unmeasurable variables, particularly the system's state. Secondly, our proposed fuzzy observer explicitly incorporates the system's initial conditions. The stability conditions of the fractional-order fuzzy observer are expressed through Bilinear Matrix Inequalities, which are then converted into linear matrix inequalities (LMIs). Subsequently, numerical simulations are conducted to demonstrate the efficacy of our proposed estimator.