## 1 Introduction  
Variable flux reluctance machines (VFRMs) represent a viable substitute for conventional induction machines due to their distinctive doubly salient architecture and uncomplicated configuration [1, 2]. They exhibit enhanced efficiency by eliminating rotor losses,possess a more straightforward design that minimizes expenses, and afford superior torque regulation across an extensive speed spectrum. With an increased torque density, VFRMs further expand their applicability to high-performance domains, such as electric vehicles (EV), where smooth torque output and minimal noise are critical for passenger comfort. To enhance performance and reliability while functioning within a constrained DC bus voltage, VFRMs are constructed to feature a DC field winding paired with an AC armature winding [2; 3]. In achieving a closed circuit for the zero-sequence current (DC component), a mutual DC link is utilized alongside the open winding [4; 5]. Among the most common issues in machine drives are coupling terms in dq0-axis and torque ripple, which have the potential to induce negative mechanical vibrations and auditory disturbances. Therefore, addressing torque ripple has evolved into a pivotal concern within both research endeavors and commercial applications. In order to diminish the impacts of torque ripple, two fundamental categories of approach can be implemented [6]. The initial category encompasses motor-design strategies, which include techniques such as skewing [7], the design of the pole arcs of the stator [8], and the optimization of permanent magnet configurations [9]. These methods tend to be costly, and they are predominantly chiefly appropriate for recently designed machinery. The second category consists of active control methodologies, which are favored due to their cost-effectiveness and adaptability. Among the various active control strategies, speed control techniques that facilitate smooth speed regulation through torque compensation based on detected disturbances are garnering increasing attention across numerous applications [10]. Observer-based control strategies, such as the dual variable bandwidth extended state observers (DVB-ESOs) proposed in [11], have shown promise in enhancing the dynamic performance of machine drives. In the context of finite-control-set model predictive torque control (FCS-MPTC), DVB-ESOs are used to improve disturbance rejection and address parameter mismatches in PMSM systems. Typically, there exist two methods for the observation of disturbances. One method involves the utilization of an integrator within the feedback control system. In cases where the reference functions as a direct signal, the establishment of a typical proportional-integral (PI) controller can secure the attainment of zero steady-state error. Conversely, in cases where the reference entails a resonant signal, it is necessary to apply a resonant controller or generalized integrator (GI) [12]. In the stationary reference frame, a resonant controller achieves a similar effect to an integrator in the synchronous reference frame, providing an effective method for harmonic suppression, albeit requiring additional frame transformations [13]. When both direct and resonant signals are present in the reference, a proportional-integral-resonant (PIR) controller is recommended [14]. For addressing multiple harmonics, multiple resonant controllers [15], a repetitive controller [16], or harmonic injection [17] can be utilized. Although these controllers demonstrate impressive abilities to reject periodic disturbances, they simultaneously hinder tracking performance owing to their single-degree-of-freedom nature. To mitigate this issue, the use of two-degree-of-freedom (TDOF) controllers is advisable. In [18], to prevent the PIR controller from amplifying high frequency elements, a tracking differentiator is used as a prefilter to refine the velocity reference.