## 4 Controller Design  
Based on the NN attack reconstruction strategies (8) and (9), and under the framework of a backstepping recursive control design algorithm, an NN-adaptive control scheme is developed which can solve approximate learning for unknown adversarial attacks. To analyze the stability of the designed controller, we define the following error coordinate change as follows:  
\[ \begin{array}{l}z_{1}=x_{1}-y_{0}\\ z_{s}=x_{s}-v_{s}\\ \xi_{s}=v_{s}-\beta_{s-1}\end{array}  \tag{10}\]  
where \(\beta_{s-1}\in\mathbf{R}\) is the virtual controller, which is a virtual controller, which means that the value of \(\beta_{s-1}\in\mathbf{R}\) is not a virtual value (it represents the control parameters within the system); \(z_{s}\) is the error surface; \(\xi_{s}\in\mathbf{R}\) is a coordinate of the first-order filter output; \(v_{s}\in\mathbf{R}\) is a variable, which is obtained through a first-order filter on intermediate virtual control \(\beta_{s-1}\in\mathbf{R}\) with a constant \(\tau_{s}\) to obtain \(v_{s}\), i.e.,  
\[\tau_{s}\dot{v}_{s}+v_{s}=\beta_{s-1},\;v_{s}(0)=\beta_{s-1}(0) \tag{11}\]  
where \(\tau_{s}\) is positive constant. Step 1: Similarly to Equations (8) and (9), we can express the approximation of the NN as follows:  
\[f_{s}(\overline{x}_{s})={\theta_{s}^{*T}}\varphi_{s}(\overline{x}_{s})+ \varepsilon_{s} \tag{12}\]  
where \(\theta_{s}^{*T}=[\theta_{s,1}^{*},\cdots,\theta_{s,r}^{*}]\) is an ideal weight vector, which is the optimal weight vector found through training to minimize error; \(\varphi_{s}(\overline{x}_{s})=[\varphi_{s,1}(\overline{x}_{s}),\cdots,\varphi _{s,r}(\overline{x}_{s})]^{T}\) is an activation function vector, which is the Gaussian activation function, with parameters tuned during training for optimal performance; given the positive constant \(\varepsilon_{s}^{*}>0\), the error \(\varepsilon_{s}\) of NN and satisfies \(|\varepsilon_{s}|\leq\varepsilon_{s}^{*}\). For unknown nonlinear dynamics, a NN learns it by adjusting weights during training based on input-output data \(f_{s}(\overline{x}_{s})\). By taking the derivative of both sides of (10) simultaneously, and using (1), it can be derived that  
\[ \begin{array}{ll}\dot{z}_{1}&=\dot{x}_{1}=x_{2}+f_{1}(x_{1})\\ &=x_{2}+{\theta_{1}^{*T}}\varphi_{1}(x_{1})+\varepsilon_{1}\end{array}  \tag{13}\]To analyze the system stability, we have chosen the following quadratic Lyapunov function. This function is selected to effectively capture the system's dynamics and provide a suitable framework for evaluating its stability; we have  
\[V_{1}=\frac{1}{2}(z_{1}^{2}+\frac{1}{p_{1}}\widetilde{\theta}_{1}^{T} \widetilde{\theta}_{1}+\frac{1}{q_{1}}\widetilde{w}_{1}^{T}\widetilde{w}_{1}) \tag{14}\]  
where \(\widetilde{\theta}_{s}=\theta_{s}^{*}-\hat{\theta}_{s}\). To ensure stability, we design the virtual controller and the parameter adaptation laws as follows, and the stability will be proven in the next section. \[\beta_{1}=-c_{1}\hat{z}_{1}-3\hat{z}_{1}-\hat{\theta}_{1}^{T}\varphi_{1}( \widetilde{\times}_{1})-\hat{w}_{1}^{T}\varphi_{1}(\xi,\widetilde{\times}_{1} )+\dot{y}_{0} \tag{15}\]  
\[\dot{\hat{\theta}}_{1}=p_{1}\hat{z}_{1}\varphi_{1}(\widetilde{\times}_{1})+ \sigma_{1}\hat{\theta}_{1} \tag{16}\]  
\[\dot{\hat{w}}_{1}=q_{1}\hat{z}_{1}\varphi_{1}(\xi,\widetilde{\times}_{1})+ \overline{\sigma}_{1}\hat{w}_{1} \tag{17}\]  
where \(c_{1}>0\), \(p_{1}>0\), \(q_{1}>0\), \(\sigma_{1}>0\) and \(\overline{\sigma}_{1}>0\) are design parameters. \(\tilde{z}_{1}=\widetilde{\times}_{1}-w_{1}^{T}\varphi_{1}(\xi,\widetilde{ \times}_{1})\) and the vector \(\hat{\theta}_{s}\) provide estimates for the ideal learning weight of \(\theta_{s}^{*}\). **Remark 2**. : _Therefore, the main challenge of adversarial attack recovery algorithms lies in how to achieve complete compensation or suppression of attack signals.