## 1 Introduction  
As an environment-friendly alternative to internal combustion engine vehicles, electric vehicles (EVs) have gained significant attention due to the high efficiency of electric motors and their minimal greenhouse gas emissions [1, 2]. Safety and sustainability are the primary objectives in the development of electric vehicles. Stability control aims to keep the vehicle's driving state within a controllable range, minimizing the risk of accidents due to insufficient driver control. However, accurately estimating a vehicle's stability remains a challenging task [3, 4, 5], which is a main focus of the present study. Improving vehicle range primarily depends on either increasing battery capacity or reducing energy consumption. Given the limitations of battery capacity, optimizing both vehicle stability and energy efficiency through strategic motor torque distribution offers an effective approach to reducing energy consumption. In EVs, hierarchical control is commonly employed to study lateral stability, where the upper-layer controller computes the required external yaw moment, and the lower-layer controller allocates the torque accordingly. The complex dynamic control capabilities of the four-wheel independent drive electric vehicle (4 WID-EV) offer significant advantages over other powertrain layouts in EVs, particularly in terms of lateral stability control and energy efficiency [6; 7; 8]. Lateral stability is a crucial component of vehicle stability control. By independently distributing torque to each of the four motors, the vehicle's stability is improved through the generation of external yaw moment [9]. Additionally, implementing appropriate torque distribution can reduce energy consumption. The 4 WID-EV uses a hierarchical control structure [10] to enhance lateral stability control by simplifying the optimization process. The upper-layer controller employs direct yaw moment control (DYC) to generate external yaw moment, thereby achieving the desired state through control methods such as the sliding mode control (SMC) [11; 12], linear quadratic regulator (LQR) [13; 14; 15], proportional-integral-derivative (PID) [16], and model predictive control (MPC) methods [17]. [PERSON] et al. [18] proposed a switching strategy based on a stability criterion that involved real-time observation of side slip angle. They also implemented a steering support system for the operation of the active steering system. To minimize actuator usage, the DYC method based on H\({}_{\infty}\) control theory was employed for lateral stability control. [PERSON] et al. [19] introduced an adaptive sliding mode control method for vehicle stability, where switching surfaces were designed using linear matrix inequalities, and fuzzy logic was applied to dynamically adjust the uncertain term and the switching control gain. [PERSON] et al. [20] utilized a dynamic sliding mode control to compute the external yaw moment and proposed an adaptive energy-efficient control allocation method based on the Karush-Kuhn-Tucker conditions for optimal torque distribution. [PERSON] et al. [21] introduced a hierarchical control concept to balance energy efficiency and vehicle motion, with a concise and modular framework that enabled superior dynamic trade-offs. The lower-layer controller typically applies the optimal torque distribution algorithm [22] to provide precise control and achieve the desired steering performance, including dynamic load distribution [23] and average distribution [24]. According to refs. [25; 26], the use of an appropriate torque control allocation method can help reduce energy consumption. In ref. [27], a torque control allocation method based on experimentally measured efficiency maps of motors was proposed to enhance motor efficiency. Additionally, [PERSON] et al. [28] considered torque distribution under tire-road friction conditions. [PERSON] et al. [29] used tire workload usage as the objective function for torque distribution and compared it with dynamic load distribution and average load distribution algorithms.