## 1 Introduction
[PERSON], [PERSON], [PERSON], and [PERSON] developed a self-adaptive sliding mode control (SMC) method that introduces adaptive parameters for dynamic adjustment, significantly enhancing trajectory tracking and reducing jitter, as demonstrated in MATLAB/Simulink simulations [19]. Enhancement to high-order sliding mode control via the integration of a homogeneous continuous control law and a super-twisting adaptive algorithm have significantly increased system robustness. A radial basis function neural network, further optimized by an adaptive weight-update law, effectively mitigates modeling errors [20]. In [21], an adaptive sliding mode control strategy for hypersonic vehicles is presented, dynamically adjusting to the vehicle's changing dynamics through a quantitative model and an online method that estimates uncertainty bounds, thereby enhancing controller robustness in real time. In [22], the study explores sliding mode control (SMC) for nonlinear descriptor delay systems using a sliding mode observer (SMO), employing T-S fuzzy dynamic modeling to linearize the system and develop an integral-type sliding surface (ITSS). Through sufficient linear matrix inequality (LMI) conditions, the method validates sliding motions and stabilizes the system, with simulations demonstrating its effectiveness. [PERSON], [PERSON], and [PERSON] developed a robust control scheme for precise tip trajectory tracking of a flexible single-link arm, addressing tip payload variability and operational disturbances. The scheme employs sliding mode compensation to mitigate errors from the extended state observer (ESO), enhancing disturbance resistance and payload insensitivity [23]. In [24], an adaptive sliding mode control for robot manipulators integrates a novel fast terminal sliding mode with RBF neural network estimation of unknown terms, enhancing tracking accuracy. Theoretical and simulation analyses confirm that this nonsingular controller outperforms traditional methods in speed and precision, validated by Lyapunov stability. To enhance the performance of electro-hydraulic servo systems, differential evolution and genetic algorithms optimize the sliding mode factor and convergence rate, focusing on minimizing input-output error and control input, with penalties for overshoots [25]. A K-type observer and saturation function further ensure signal stability and reduce jitter. In [26], an active disturbance rejection sliding mode traction control system is introduced to address joint angular acceleration changes and jitter in manipulator terminal traction, incorporating a human-computer interaction (HCI) device for enhanced precision and speed. To mitigate chattering, the system employs a low-pass filter on the controller's output torque. In the realm of industrial automation, the challenge of formulating precise mathematical models frequently contributes to chattering issues in sliding mode control systems. To combat this, researchers have turned to artificial neural networks to effectively approximate the nonlinear dynamics of such systems, thereby enhancing their resistance to disturbances. However, it is difficult for Radial Basis Function (RBF) and backpropagation (BP) neural networks to obtain the optimal number of nodes in practical applications. To tackle this complexity, [PERSON] and [PERSON] presented the Stochastic Configuration Networks (SCNs), a novel incremental learning approach, in 2019 [27]. This approach uses a unique discriminating method to strategically augment the number of nodes in the hidden layers. Such enhancements not only significantly improve the algorithm's ability to approximate complex functions but also equip the SCN with virtually unlimited approximation potential, making it a robust tool for tackling the inherent uncertainties and dynamic challenges in modern industrial control systems. In [28], we introduce an enhanced Stochastic Configuration Network (SCN) with a novel inequality constraint for construction and an adaptive node selection from a candidate pool to accelerate efficiency.