## 3 Controller and Observer
### Disturbance Observer Based on SSCN  
In contexts where the disturbance upper limit is quantifiable, Objective Sliding Mode Control (OSMC) guarantees convergence within a finite period. However, accurately estimating system uncertainties remains a substantial challenge in practical implementations. Traditionally, high switching gains were employed to enhance robustness, but this often resulted in significant chattering effects. To address these challenges, this research integrates a neural network into the OSMC framework to improve the estimation and compensation of system uncertainties. Thisapproach enhances the controller's robustness while minimizing chattering. Furthermore, to advance the dynamic capabilities of the Standard Configuration Network (SCN), we introduce the Stochastic State Configuration Network (SSCN). A disturbance observer using SSCN technology has been developed to counteract system uncertainties more effectively, reduce chattering, and improve the overall robustness of the control system. This integration represents a meaningful step toward refining the operational efficiency of advanced control systems. The operational efficiency of the SSCN is divided into two primary phases: an initial training phase and a real-time adjustment phase. In the initial phase, the SSCN model is calibrated using historical input-output data to optimize the node count in the hidden layer and tailor input parameter configurations, thereby reducing computational complexity. During the real-time phase, an adaptive law modifies the output weights of the SSCN model, enhancing its compensatory capabilities. The proposed SSCN architecture consists of a three-layer network, with the hidden layer augmented by an external feedback loop. This work makes several key contributions. First, a mathematical model of a multi-jointed robotic arm with uncertain parameters was established using the Lagrangian method. Based on this model, an Optimal Sliding Mode Control (OSMC) system was developed to ensure zero-error convergence. Additionally, we designed an observer utilizing the SSCN, which operates through two phases: initial and real-time. During the initial phase, historical datasets network parameters, significantly reducing the training period. The SSCN employs a novel node selection strategy in the hidden layer, executed in two steps:'single time,' where weights and biases are randomly selected through a uniform distribution; and 'double time,' where parameters are chosen from a normal distribution centered around previously selected values. This strategy focuses on high-quality points, enhancing the model's accuracy and efficiency. After computing the output signal, it is fed back to the input layer via an external feedback mechanism, forming the Stepwise Stochastic Configuration Networks illustrated in Figures 2 and 3. Figure 2: Stepwise Stochastic Configuration Network. In this study, the input layer serves to transmit both the incoming signal and the feedback from the output of the preceding beat model. The data inputs for this research include the positional coordinates and velocity of the manipulator. Within the hidden layer, the input values are subjected to a nonlinear transformation, activating the selected hyperbolic tangent (tanh) function. The output from the j-th node in the hidden layer is denoted accordingly. \[\phi_{j}=\tanh\bigl{(}\mathbf{w}_{j}\mathbf{x}+b_{j}\bigr{)} \tag{37}\]  
In (37), \(\mathbf{w}_{j}\) and \(b_{j}\) are the input weight and offset. In the networks, the tanh function is chosen as the activation function mainly because its output ranges from \(-1\) to \(1\), providing a zero-centered property. This means its average output is zero, allowing subsequent layers to receive inputs that are both positive and negative. This promotes balanced weight updates and helps accelerate the model's convergence speed.