## 3 Control Law Design and Stability Analysis
### Adaptive Neural Network Finite-Time Dynamic Surface Control Law Design  
Similar to the backstepping control technique, the analysis process includes \(n\) steps. Firstly, the consensus error and coordinate transformation are defined as  
\[e_{i,1}=\sum_{j\in\mathcal{J}_{i}}a_{ij}(y_{i}-y_{j})+b_{i}(y_{i}-y_{d}) \tag{13}\]  
\[e_{i,m}=x_{i,m}-s_{i,m-1} \tag{14}\]  
\[z_{i,m-1}=s_{i,m-1}-\alpha_{i,m-1} \tag{15}\]  
where \(i=1,\cdots,N\) and \(m=2,\cdots,n\), \(s_{i,m-1}\) is the output of the given first-order filter with the virtual control law \(\alpha_{i,m-1}\) as the input, and \(z_{i,m-1}\) is called the output error of the first-order filter. _Step_ 1: Applying (5) and (13)-(15), the derivative of consensus error \(e_{i,1}\) is given as  
\[\dot{e}_{i,1}=(\sum_{j\in\mathcal{J}_{i}}a_{ij}+b_{i})\dot{y}_{i}-\sum_{j\in \mathcal{J}_{i}}a_{ij}\dot{y}_{j}-b_{i}\dot{y}_{d}=\dot{h}_{i,1}\alpha_{i,1}+F _{i,1}-b_{i}\dot{y}_{d} \tag{16}\]  
where \(\dot{h}_{i,1}=(\sum_{j\in\mathcal{J}_{i}}a_{ij}+b_{i})\dot{g}_{i,1}\) and  
\[F_{i,1}=\dot{h}_{i,1}(e_{i,2}+z_{i,1})+(\sum_{j\in\mathcal{J}_{i}}a_{ij}+b_{i} )f_{i,1}-\sum_{j\in\mathcal{J}_{i}}a_{ij}(g_{j,1}x_{j,2}+f_{j,1}) \tag{17}\]Considering the fact that the information of the nonlinear function \(F_{i,1}\) may be unavailable directly, an RBFNN is utilized to approximate the unknown nonlinear function \(F_{i,1}\). According to Lemma 1, one has  
\[F_{i,1}=\left(\Phi_{i,1}^{*}\right)^{T}\zeta_{i,1}+\varepsilon_{i,1}(X_{i,1}) \tag{18}\]  
where \(X_{i,1}=\left[x_{i,1},x_{i,2},x_{j,1},x_{j,2},y_{d}\right]^{T}\), and there exists a positive constant \(\Xi_{i,1}\), such that \(\left|\varepsilon_{i,1}(X_{i,1})\right|\leq\Xi_{i,1}\). Construct the following Lyapunov function candidate as  
\[V_{i,1}=\frac{1}{2}e_{i,1}^{2}+\frac{1}{2T_{i,1}}\widetilde{\Xi}_{i,1}^{2}+ \frac{1}{2Y_{i,1}}\widetilde{\Phi}_{i,1}^{T}\widetilde{\Phi}_{i,1} \tag{19}\]  
where \(\widetilde{\[PERSON]}_{i,1}=\hat{\[PERSON]}_{i,1}-\Xi_{i,1}\) and \(\widetilde{\[PERSON]}_{i,1}=\dot{\Phi}_{i,1}-\Phi_{i,1}\), \(\hat{\[PERSON]}_{i,1}\) and \(\dot{\[PERSON]}_{i,1}\) are the estimations of \(\[PERSON]_{i,1}\) and \(\[PERSON]_{i,1}\), respectively. \(\Gamma_{i,1}\) and \(\Upsilon_{i,1}\) are designed positive constants, respectively. Taking the derivative of (16) along with (16), and considering (18), we have  
\[ \begin{split}\dot{V}_{i,1}&=e_{i,1}\dot{e}_{i,1}+ \frac{1}{\Gamma_{i,1}}\widetilde{\[PERSON]}_{i,1}\dot{\hat{\[PERSON]}}_{i,1}+\frac{1}{ \Gamma_{i,1}}\widetilde{\[PERSON]}_{i,1}^{T}\dot{\[PERSON]}_{i,1}\\ &\leq\dot{h}_{i,1}e_{i,1}\dot{\alpha}_{i,1}+e_{i,1}\Big{(}(\[PERSON]_ {i,1}^{*})^{T}\zeta_{i,1}-b_{i}\dot{y}_{d}\Big{)}+\[PERSON]_{i,1}|e_{i,1}|+\frac{1}{ \Gamma_{i,1}}\widetilde{\[PERSON]}_{i,1}\dot{\hat{\[PERSON]}}_{i,1}+\frac{1}{\Gamma_{i,1} }\widetilde{\[PERSON]}_{i,1}^{T}\dot{\[PERSON]}_{i,1}\end{split}  \tag{20}\]  
Let  
\[\[PERSON]_{i,1}^{T}\[PERSON]_{i,1}=(\[PERSON]_{i,1}^{*})^{T}\zeta_{i,1}+\frac{\hat{\[PERSON]}_{i,1 }e_{i,1}}{\sqrt{e_{i,1}^{2}+\eta^{2}}}-b_{i}\dot{y}_{d} \tag{21}\]  
where  
\[\[PERSON]_{i,1}=\left[\[PERSON]_{i,1}^{*},1\right]^{T} \tag{22}\]  
\[\[PERSON]_{i,1}=\left[\zeta_{i,1},\left.\hat{\[PERSON]}_{i,1}e_{i,1}\right/\sqrt{e_{i,1 }^{2}+\eta^{2}}-b_{i}\dot{y}_{d}\right]^{T} \tag{23}\]  
Using Lemma 6, one obtains  
\[\Xi_{i,1}|e_{i,1}|\leq\frac{\Xi_{i,1}e_{i,1}^{2}}{\sqrt{e_{i,1}^{2}+\eta^{2}} }+\eta\Xi_{i,1} \tag{24}\]  
Design the virtual control \(\alpha_{i,1}\) as  
\[\alpha_{i,1}=\mathcal{N}(\zeta_{i,1})\beta_{i,1} \tag{25}\]  
\[\dot{\zeta}_{i,m}=e_{i,1}\beta_{i,1} \tag{26}\]  
\[\beta_{i,1}=c_{i,1}e_{i,1}^{2\mu_{i}-1}+\dot{\Phi}_{i,1}^{T}\Psi_{i,1} \tag{27}\]  
where \(c_{i,1}\) is a designed positive constant.