## 4 Controller Design
Due to the unavailability of the tracking error \(z_{1}\) in (10), we introduce a novel error transformation to construct an equivalent \(\hat{z}_{1}=\widetilde{\times}_{1}-w_{1}^{T}\varphi_{1}(\zeta,\widetilde{\times }_{1})\) instead of \(z_{1}\). The equivalence between \(\hat{z}_{1}=\widetilde{\times}_{1}-w_{1}^{T}\varphi_{1}(\zeta,\widetilde{\times }_{1})\) and \(z_{1}\) depends on the performance of (8) and (9), respectively. The estimation performance of NNs can be supported within tight sets._  
Step \(s\) (\(2\leq s\leq n-1\)): By taking the derivative of both sides of (10) simultaneously, it can be obtained that  
\[ \begin{array}{rl}\dot{z}_{s}&=\dot{x}_{s}-\dot{\hat{\beta}}_{s-1}\\ &=x_{s+1}+f_{s}(\overline{x}_{s})-\dot{\hat{\beta}}_{s-1}\\ &=x_{s+1}+\theta_{s}^{*T}\varphi_{s}(\overline{x}_{s})+\varepsilon_{s}-\hat{ \beta}_{s-1}\end{array}  \tag{18}\]  
From (11), one has  
\[ \begin{array}{rl}\dot{\xi}_{s}&=\dot{v}_{s}-\dot{\hat{\beta}}_{s-1}\\ &=\dot{\hat{\beta}}_{s-1}-\frac{\hat{\beta}_{s-1}-\overline{\varepsilon}_{s}}{ \overline{\varepsilon}_{s}}=N_{s}(\cdot)-\frac{\zeta_{s}}{\overline{ \varepsilon}_{s}}\end{array}  \tag{19}\]  
where \(\hat{\beta}_{s-1}=N_{s}(\cdot)=N_{s}(z_{1},\cdots,z_{s},\theta_{1},\cdots, \theta_{s},w_{1},\cdots,w_{s},\zeta_{2}\cdots,\zeta_{s},y_{0},\dot{y}_{0})\) is a known continuous function. Similarly to the Lyapunov function in (14), we choose the following Lyapunov function in step \(s\); we obtain  
\[V_{s}=V_{s-1}+\frac{1}{2}(z_{s}^{2}+\frac{1}{p_{s}}\widetilde{\theta}_{s}^{T} \widetilde{\theta}_{s}+\frac{1}{q_{s}}\widetilde{w}_{s}^{T}\widetilde{w}_{s} +\xi_{s}^{2}) \tag{20}\]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. \[\beta_{s}=-c_{s}\hat{z}_{s}-3\hat{z}_{s}-\hat{\theta}_{s}^{T}\varphi_{s}(\widetilde {\widetilde{\mathbf{x}}}_{s})-\hat{w}_{s}^{T}\varphi_{s}(\varsigma,\widetilde{ \widetilde{\mathbf{x}}}_{s})+\dot{v}_{s} \tag{21}\]  
\[\dot{\hat{\theta}}_{s}=p_{s}\hat{z}_{s}\varphi_{s}(\widetilde{\widetilde{ \mathbf{x}}}_{s})+\sigma_{s}\hat{\theta}_{s} \tag{22}\]  
\[\dot{\hat{w}}_{s}=q_{s}\hat{z}_{s}\varphi_{s}(\varsigma,\widetilde{\widetilde{ \mathbf{x}}}_{s})+\overline{\sigma}_{s}\hat{w}_{s} \tag{23}\]  
where \(c_{s}>0\), \(p_{s}>0\), \(q_{s}>0\), \(\sigma_{s}>0\) and \(\overline{\sigma}_{s}>0\) are design parameters. And \(\hat{z}_{s}=\widetilde{\widetilde{\mathbf{x}}}_{s}-\hat{w}_{s}^{T}\varphi_{s }(\varsigma,\widetilde{\widetilde{\mathbf{x}}}_{s})\). **Remark 3**. : _The author has designed an anti-attack observer that can compensate for attacks to restore the system state in [19], which only requires the sensor output to be attacked. Unlike reference [19], this article studies all states being attacked, and all variables are unavailable. The estimation performance of NNs can be supported within tight sets._  
Step \(n\): Similarly to Step \(s\), from (1), (3) and (12) and (10), by taking the derivative of both sides of (10) simultaneously, it can be gathered that  
\[ \begin{array}{ll}\dot{z}_{n}&=\dot{x}_{n}-\dot{v}_{n}\\ &=\widetilde{u}+f_{n}(\widetilde{\mathbf{x}}_{n})-\dot{v}_{n}\\ &=u+a(t,\widetilde{\mathbf{x}}_{n})+f_{n}(\widetilde{\mathbf{x}}_{n})-\dot{v} _{n}\\ &=u+v^{*T}\phi(\varsigma,\widetilde{\mathbf{x}}_{n})+v_{0}+\theta_{n}^{*T} \varphi_{n}(\widetilde{\mathbf{x}}_{n})+\varepsilon_{n}-\dot{v}_{n}\end{array}  \tag{24}\]  
Similarly to the Lyapunov functions (14) and (20), we choose the following Lyapunov function in step \(s\), finding the following:  
\[V_{n}=V_{n-1}+\frac{1}{2}(z_{n}^{2}+\frac{1}{p_{n}}\widetilde{\theta}_{n}^{T }\widetilde{\widetilde{\mathbf{\sigma}}}_{n}+\frac{1}{q_{n}}\widetilde{w}_{n }^{T}\widetilde{\widetilde{\mathbf{\sigma}}}_{n}+\frac{1}{m}\widetilde{ \widetilde{\mathbf{\sigma}}}^{T}\widetilde{\widetilde{\mathbf{\sigma}}}) \tag{25}\]  
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.