## 1 Introduction  
In the contemporary transportation and aerospace industries, the demand for multi-phase permanent magnet synchronous motors has surged, due to their demonstrated superiority in performance and durability [1]. Vehicles, ships, and aerospace applications mainly rely on these motors due to their higher attributes, such as efficiency, power density, and fault tolerance [2, 3]. One of the most favorable aspects of these motors for such applications is their ability to maintain essential performance characteristics during faults, which is directly related to the number of phases in the motor. The more phases the motor has, the more degrees of freedom it can exploit, allowing it to maintain its performance despite faults [4]. Although five-phase permanent magnet synchronous motors (FPPMSMs) are robust, open-circuit faults (OCFs) remain a significant concern. These faults usually occur due to disrupted or disconnected wires in the motor windings [5]. It is crucial to detect these faults quickly and accurately, to maintain motor performance and apply an effective fault-tolerant control (FTC) strategy. A widely used approach for identifying OCFs involves monitoring the motor currents with the help of current sensors. In [6], a fault detection method based on current spectrum analysis was developed with the help of fault signatures. The experimental results showed that OCFs could be quickly detected and localized using current sensors without requiring additional hardware. In the realm of FPPMSMs, a plethora of techniques have been proposed for FTC to counteract the unbalanced currents induced by OCFs [7, 8, 9]. These strategies revolve around maintaining the rotational movement of the fundamental magnetic motive force and back electromotive force (EMF) at levels consistent with healthy operation. Although traditional linear regulators, such as a proportional-integral (PI) controller, dominate field-oriented control, the inherent complexity and nonlinearity of FPPMSMs during OCFs pose a significant challenge. This complexity can lead to structural asymmetry in FPPMSMs, resulting in undesirable torque ripples. Extensive research efforts have been devoted to proposing solutions aimed at suppressing these torque ripples in the presence of OCFs, as exemplified in numerous studies, including [7; 8; 9; 10] and many others. Remarkably, research fields have not given adequate focus to nonlinear control techniques and compensations under OCFs of multiphase machinery. Highlighting this gap emphasizes the importance of continued exploration and advancement in this field. Adopting a nonlinear control method like sliding mode control (SMC) has garnered substantial interest [11]. Its ability to accommodate system uncertainties and disturbances with a high degree of tolerance for model imperfections makes it particularly well-suited for application to permanent magnet synchronous motors during fault conditions [12; 13]. For instance, in [12], the authors advocated for employing SMC in the speed loop of an FPPMSM under an open circuit fault (OCF), effectively mitigating the effects of unknown dynamics. Meanwhile, [13] proposed the application of SMC in the current control loops, maintaining the traditional linear PI regulator for speed control. Another study proposed a nonlinear control approach using a backstepping controller for FTC to enhance the five-phase induction motor drive dynamics under an OCF [14]. While backstepping controllers can be effective in certain scenarios, they have limitations. They require full information of states, which causes a high computational burden and potentially produces larger control signals [15]. In contrast, SMC features a simple and easily tunable controller structure and aids in eliminating steady-state errors [16]. Although SMC may suffer from the chattering effect, various methods have been proposed to mitigate this issue [17].