###### Abstract
In [25], the problem of observer-based SMC for fractional-order singular fuzzy systems was studied, and a new fractional-order integral sliding function was constructed. Thus far, the research of the SMC has integrated fractional calculus, FLS, and adaptive control to deal with uncertainties. Additionally, researchers have made significant progress in choosing sliding surfaces. However, the fundamental problem of the SMC has not been solved. The presence of switching mechanisms will result in chattering. Typical switching strategies include power switching functions that contain both sign functions and sliding-mode surface variables, exponential switching functions that contain both sign functions and sliding-mode surface variables, and sign switching functions that only contain sign functions. These methods have been proved to be effective for system control, but there is chattering. Many studies on how to advance switching technology have been carried out in recent years. In [26], the authors applied the exponential switching function to the fractional-order system and constructed the fractional-order SMC. In fractional-order systems, it was merely a direct application of the exponential switching function. Fractional calculus had a greater impact than integer order because of its memory properties. In [27], the sliding surface in integer-order systems was determined by the system variables. However, the switching mechanism was incorporated with a complex function that combined the sign function and the fractional-order algorithm. The complex switching functions were used to improve the chattering of the SMC [28; 29; 30]. Even though they could successfully reduce chattering, the controller structure was complicated and more computation was required. In [31], the SMC switching mechanism was modified to reduce chattering by applying the rectifier principle in power systems. In [32], the TS-SMC was used to control single-input and single-output fractional-order systems with lower triangular structure. The FLSs have reached a relatively mature stage of development, their calculation complexity has decreased, and they can be effectively implemented in the control system. It is necessary to enhance the modeling and analysis of uncertainty in fractional-order systems. In this article, a better performing fractional-order integral AFSMC is proposed to achieve the state trajectory tracking. Additionally, the uncertain nonlinear functions in the system are estimated using FLSs. A review of the SMC literature makes it clear that optimizing SMC requires improving the switching function. Thus, an FLS-based improved switching function is presented. To validate the stability and robustness of the designed controller, disturbances are intentionally added to the simulation process. The primary contribution of this article can be summarized as follows:  
1. The dynamics of the fractional-order nonlinear PMSM system, which serves as a representative example of fractional order MIMO nonlinear dynamic systems, are demonstrated. Referring to [14; 33], the simulation experiments of special PMSM and PMSM with actual parameters are carried out, respectively. FLSs and adaptive control are used effectively to manage unknown nonlinear functions and uncertain disturbances in dynamic systems. 2. An enhanced integral AFSMC is created, and its capacity to follow the state trajectory with minimum errors is demonstrated mathematically. Simulation experiments further prove the superiority of the proposed method. 3. Compared with the typical switching function, the fuzzy switching function in AFSMC can effectively weaken the chattering and reduce the tracking error. Compared with [32], the system model of this paper is more universal.