# Change Control Design in Product Supply Chain System Based on Radial Basis Function Neural Network
## 3 Product and Supply Chain System Controller Design
### RBFNN-Based Composite Controller Design  
With the rapid development of network technology and the emergence of new network innovations, cyber-attacks have become more sophisticated and frequent, posing a significant threat to product supply chain systems. These cyber-attacks not only cause data breaches but also disrupt supply chain operations, potentially affecting the stability of the global market. Moreover, the supply chain system itself is vulnerable to fluctuations in user-side demand, which are often amplified layer-by-layer throughout the supply chain, creating a bullwhip effect. This effect can magnify the impact of demand fluctuations and lead to further instability. Therefore, ensuring the cybersecurity of product supply chain systems has become a critical and urgent issue across various industries. In this section, we discuss the solutions to cyber-attacks and changes in system (5). Radial basis function neural networks (RBFNNs) have shown significant advantages over other traditional machine learning methods when dealing with DoS attacks. Although methods such as Support Vector Machines (SVMs) and Decision Trees (DTs) can also work effectively under certain conditions, they tend to perform poorly when faced with complex nonlinear problems. SVM has a slow training process, especially with high-dimensional data, and may suffer from overfitting. Decision Trees, on the other hand, are prone to overfitting when handling complex patterns. Kalman filtering, a classical dynamic system estimation method, although effective in noise removal and system state prediction, does not perform as well as RBFNNs in handling highly complex nonlinear attack patterns. Furthermore, RBFNNs have shorter training times and require fewer computational resources compared to deep neural networks (DNNs), while also avoiding issues suchas gradient vanishing. This enables RBFNNs to adapt more quickly and respond more efficiently to DoS attacks. In view of the excellent performance demonstrated by radial basis function neural networks (RBFNNs) in dealing with complex nonlinear problems and pattern recognition [40], this study decided to adopt the RBFNN as an effective means to cope with DoS attacks. RBFNNs, with their powerful learning and generalization capabilities, are expected to accurately identify and respond to DoS attacks, thus enhancing the security and stability of product supply chain systems. The RBFNN is expected to be able to accurately recognize and effectively identify DoS attacks due to its strong learning and generalization capabilities, thereby enhancing the security and stability of product supply chain systems. Therefore, when a product supply chain system experiences a DoS attack, the time-series prediction function of the RBFNN is used to predict the inventory change trend after the system is attacked, as follows:  
\[x= \begin{cases}x&if\ \gamma_{k}=0\\ \bar{x}&if\ \gamma_{k}=1\end{cases}  \tag{10}\]  
where \(\bar{x}\) represents the forecast result of the RBFNN. When there is no DoS attack, the system inventory data can be transmitted normally, and the system operates as expected. Once the DoS attack occurs, the system inventory data cannot be transmitted. In this case, the RBFNN is used to transmit time-series-based forecast results to minimize the impact of the DoS attack on the product supply chain system. In addition to being affected by cyber-attacks, the product supply chain system may also undergo changes due to uncontrollable factors, such as environmental changes. The previously mentioned LQR control method can provide good control in the absence of disturbances, but it may not be effective when the supply chain system is subject to external disturbances. Therefore, we employed an adaptive RBFNN to help the system maintain continuous and stable operation in the presence of disturbances.