# 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
When the system is subjected to cyber-attacks, the RBFNN can act as a predictor and make inventory predictions based on historical data, ensuring that the system continues to function properly despite attack disturbances. **Remark 4**. : _For ease of reference and simplicity, we define_  
\[f=\triangle Ax(t)+\triangle Bu(t)+D\bar{d}(t) \tag{11}\]  
In order to solve the above problems, an RBFNN composed of input, hidden, and output layers is considered in this paper. Figure 3 shows the basic structure of a typical RBF neural network. Figure 3 provides a structure diagram of a three-layer neural network. The figure shows that \(\Phi(x,x^{p})\) is the activation function of the hidden layer, where the activation function is chosen to be a Gaussian basis function, with the following expression:  
\[\Phi(x_{i},c_{i})=\exp(-\frac{\|x_{j}-c_{i}\|}{2\sigma^{2}})\qquad j=1,2,\cdot \cdot\cdot,n \tag{12}\]  
where \(c\) represents the weights, \(c_{i}\) is the centroid of the \(i\)th neuron, \(\sigma\) is the width of the Gaussian kernel, and \(\|x_{j}-c_{i}\|\) is the Euclidean distance from the sample input \(x_{i}\) to the centroid \(c_{i}\). In the figure \(\omega_{n}\) is the weight information of the RBFNN, which can be expressed as  
\[W=[w_{1},w_{2}\cdot\cdot\cdot,w_{n}]. \tag{13}\]Then, the complete mapping from the input layer to the output layer can be obtained as  
\[Y=\sum_{j=1}^{m}w_{j}H \tag{14}\]  
We assume that the disruptions in the supply chain system approximated by the RBFNN are caused by complex environmental changes and uncertain user demands,  
\[f=W^{T}H+\epsilon \tag{15}\]  
and the best approximation is given by  
\[\hat{f}=\hat{W}^{T}H \tag{16}\]  
where \(\hat{W}^{T}\) is the weight of the neural network, and \(\epsilon\) is the minimum approximation error. Therefore, the specific form of our controller is designed as follows:  
\[ \begin{split} u(t)&=u_{lqr}+u_{f}=-Kx-B^{-1}\hat{f} \\ &=-Kx-B^{-1}\hat{W}^{T}H\end{split}  \tag{17}\]  
where \(H\) is the output of each hidden layer of the RBF network; \(\hat{W}\) is the estimated weight of the RBFNN, which has a renewal rate, as follows:  
\[\dot{\hat{W}}=\gamma Hx^{T}P \tag{18}\]  
where \(\gamma\) is the RBF network learning rate; \(P\) is the symmetric positive definite matrix matrix obtained by solving the Riccati equation in Equation (9). The design of the adaptive RBFNN law is based on the theory of Lyapunov stability. By constructing a suitable Lyapunov function, the system ensures that the tracking error and parameter estimation error gradually converge to zero or a bounded region during the control process, thereby guaranteeing the global stability of the closed-loop system. Figure 3: The scheme of an RBFNN. In summary, the RBFNN reduces not only the impact of external uncertainties on product supply chain systems by approximating unknown nonlinear functions, but also the impact of DoS attacks through time -eries prediction, which helps to ensure the stable operation of product supply chain systems. In this section, an adaptive LQR control method using an RBFNN is introduced, which utilizes the infinite approximation property of RBFNNs for linear parameters. By designing the adaptive parameters of the RBFNN, we can approximate the nonlinear parameters after the link changes. Theorem 1: _The proposed control rate (18) allows system (3) to achieve the stability of the rated system (5) under the influence of perturbations at the control rate._  
Proof of Theorem 1.: From (15) and (16), we have  
\[f-\hat{f}=W^{T}H-\hat{W}^{T}H=\tilde{W}^{T}H \tag{19}\]  
where \(\tilde{W}=W-\tilde{W}\)  
Construct the Lyapunov functions:  
\[V=x^{T}Px+\frac{1}{\gamma}tr(\tilde{W}^{T}\tilde{W}) \tag{20}\]  
Then,  
\[ \begin{split}\dot{V}&=\dot{x}^{T}Px+x^{T}P\dot{x}+ \frac{2}{\gamma}tr(\tilde{W}^{T}\dot{W})\\ &=((A+\triangle A)x+(B+\triangle B)u+D\bar{d})^{T}Px\\ &+x^{T}P((A+\triangle A)x+(B+\triangle B)u+D\bar{d})+\frac{2}{ \gamma}tr(\tilde{W}^{T}\dot{W})\\ &=(Ax+Bu+f)^{T}Px+x^{T}P(Ax+Bu+f)\\ &+\frac{2}{\gamma}tr(\tilde{W}^{T}\dot{\tilde{W}})\end{split}