## 3 Position Tracking Control Design
\(b_{j}\) is a parameter that can be set manually, determining the width of the Gaussian function around the center point. \(m\) is the number of nodes in the hidden layer, and \(m\) determines the control effect of the controller and the calculation power required to be provided by the calculation module. Therefore, selecting an appropriate number of hidden layer nodes is imperative. The derivative of the activation function is:  
\[ \begin{array}{l}\frac{\partial R_{j}(x)}{\partial b_{j}}=exp\left(-\frac{ \left\|x-c_{j}\right\|^{2}}{b_{j}^{2}}\right)\frac{\left\|x-c_{j}\right\|^{2 }}{b_{j}^{3}}=R_{j}(x)\frac{\left\|x-c_{j}\right\|^{2}}{b_{j}^{3}}\\ \frac{\partial R_{j}(x)}{\partial c_{ji}}=exp\left(-\frac{\left\|x-c_{j} \right\|^{2}}{b_{j}^{2}}\right)\frac{\left(x_{i}-c_{ji}\right)}{b_{j}^{2}}=R_ {j}(x)\frac{\left(x_{i}-c_{ji}\right)}{b_{j}^{3}}\end{array}  \tag{15}\]  
The RBF training consists of the input signal's forward propagation and the error's back propagation process. P-1: The input signal (horizontal force provided by the propeller of the AUV) is propagated forward, and the output of the RBF network is calculated. The output of the neurons in the input layer is \(x=\left[x_{1}(k),x_{2}(k),\ldots,x_{n}(k)\right]^{T}\), and (14) is used to obtain the hidden layer neuron. The output of meta is:  
\[ \begin{array}{l}R_{j}(x(k))=exp\left(-\frac{\left\|x(k)-c_{j}(k-1)\right\|^ {2}}{b_{j}^{2}(k-1)}\right)\\ j=1,2,\cdots,n\end{array}  \tag{16}\]  
The input to the neurons in the output layer is:  
\[y_{m}(k)=\sum\limits_{m}^{j=1}w_{j}\big{(}k-1)R_{j}(x(k)\,\big{)} \tag{17}\]  
where \(w_{j}(k-1)\) is the weight from the \(j\) hidden layer neuron to the output layer neuron at \((k-1)\) time:  
\[e(k)=y(k)-y_{m}(k) \tag{18}\]  
where \(y(k)\) is the system's actual output and \(y_{m}(k)\) is the output of the reference model. The performance indicator function can be introduced:  
\[E(k)=\frac{1}{2}e^{2}(k) \tag{19}\]  
P-2: Error backpropagation. The \(\delta\) learning algorithm is used to adjust the weights between the layers of the RBF network, ensuring that AUVs adapt to time-varying disturbances. According to the gradient descent method, the weight learning algorithm is as follows:  
The weight from the hidden layer to the output layer is \(w_{j}(k)\). From (18) and (19) we can obtain:  
\[\delta^{(2)}=\frac{\partial E(k)}{\partial y_{m}(k)}=\frac{\partial E(k)}{ \partial e(k)}\frac{\partial e(k)}{\partial y_{m}(k)}=-e(k) \tag{20}\]From (17) and (20) we can obtain:  
\[\frac{\partial E(k)}{\partial w_{j}(k-1)}=\frac{\partial E(k)}{\partial y_{m}(k)} \frac{\partial y_{m}(k)}{\partial w_{j}(k-1)}=\delta^{(2)}R_{j}(x(k))=-e(k)R_{j }(x(k)) \tag{21}\]  
Therefore, the learning algorithm of the weight \(w_{j}(k)\) from the hidden layer to the output layer is:  
\[\triangle w_{j}(k)=-\eta\frac{\partial E(k)}{\partial w_{j}(k-1)}=\eta e(k)R_ {j}(x(k)) \tag{22}\]  
\[w_{j}(k)=\ w_{j}(k-1)+\
abla w_{j}(k)+\alpha\big{(}w_{j}(k-1)-w_{j}(k-2)\big{)} \tag{23}\]  
where \(\eta\) is the learning rate (\(\eta>0\)) and \(\alpha\) is the momentum factor (\(\alpha\in[0,1)\)). The hidden-layer Gaussian function parameters are \(b_{j}(k)\) and \(c_{ji}(k)\). From (20) and (17) we can obtain:  
\[\delta_{j}^{(1)}=\frac{\partial E(k)}{\partial R_{j}(x(k))}=\frac{\partial E (k)}{\partial y_{m}(k)}\frac{\partial y_{m}(k)}{\partial R_{j}(x(k))}=\delta^ {(2)}w_{j}(k-1) \tag{24}\]  
Then, we can use (24) and (15) to obtain:  
\[ \begin{array}{c}\frac{\partial E(k)}{\partial b_{j}(k-1)}=\frac{\partial E (k)}{\partial b_{j}(x(k))}\frac{\partial R_{j}(x(k))}{\partial b_{j}(k-1)}=-e (k)w_{j}(k-1)R_{j}(x(k))\frac{x(k)-c_{j}(k-1)^{2}}{b_{j}^{2}(k-1)}\\ \frac{\partial E(k)}{\partial c_{ji}(k-1)}=\frac{\partial E(k)}{\partial b_{j} (x(k))}\frac{\partial R_{j}(x(k))}{\partial c_{ji}(k-1)}=-e(k)w_{j}(k-1)R_{j}( x(k))\frac{x(k)-c_{ji}(k-1)}{b_{j}^{2}(k-1)}\end{array}  \tag{25}\]  
Then, the learning algorithm of \(b_{j}(k)\) and \(c_{ji}(k)\) is:  
\[