## 3 Control Law Design and Stability Analysis
### Adaptive Neural Network Finite-Time Dynamic Surface Control Law Design
Design adaptive laws \(\dot{\hat{\Xi}}_{i,m}\), \(\dot{\Phi}_{i,m}\) and \(\dot{\hat{\Pi}}_{i,m-1}\) as  
\[\[PERSON]_{i,1}=\left[\zeta_{i,1},\left.\hat{\[PERSON]}_{i,1}e_{i,1}\right/\sqrt{e_{i,1 }^{2}+\eta^{2}}-b_{i}\dot{y}_{d}\right]^{T} \tag{23}\]4  
\[\[PERSON]_{i,1}=\left[\zeta_{i,1},\left.\hat{\[PERSON]}_{i,1}e_{i,1}\right/\sqrt{e_{i,1 }^{2}+\eta^{2}}-b_{i}\dot{y}_{d}\right]^{T} \tag{23}\]5  
\[\[PERSON]_{i,1}=\left[\zeta_{i,1},\left.\hat{\[PERSON]}_{i,1}e_{i,1}\right/\sqrt{e_{i,1 }^{2}+\eta^{2}}-b_{i}\dot{y}_{d}\right]^{T} \tag{23}\]6  
where \(\gamma_{i,m}\), \(\rho_{i,m}\), and \(\omega_{i,m-1}\) are designed positive constants. Substituting (38) and (41)-(45) into (37) and applying the adaptive laws (46)-(48) yields  
\[ \begin{array}{l}\dot{V}_{i,m}\leq(\hbar_{i,m}\mathcal{N}(\zeta_{i,m})\!+\!1 )\dot{\zeta}_{i,m}-c_{i,m}e_{i,m}^{2\eta_{i}}-\frac{z_{i,m-1}^{2}}{\tau_{i,m-1 }}-\frac{\gamma_{i,m}}{\Gamma_{i,m}}\widetilde{\Xi}_{i,m}\hat{\Xi}_{i,m}-\frac {\rho_{i,m}}{\dot{\gamma}_{i,m}}\widetilde{\Phi}_{i,m}^{T}\dot{\Phi}_{i,m}\\ -\frac{\omega_{i,m-1}}{\Gamma_{i,m-1}}\hat{\Pi}_{i,m-1}+\eta\Xi_{i,m}+\eta \end{array}  \tag{49}\]  
Let \(\alpha_{i,m}\) pass through the following nonlinear filter with time constant \(\tau_{i,m}\) to obtain the new state variable \(s_{i,m}\), that is  
\[\tau_{i,m}\dot{s}_{i,m}=-z_{i,m}-\frac{\tau_{i,m}\hat{\Pi}_{i,m}^{2}z_{i,m}}{ \sqrt{\left(\hat{\Pi}_{i,m}z_{i,m}\right)^{2}+\eta^{2}}},\;s_{i,m}(0)=\alpha_{ i,m}(0) \tag{50}\]  
where \(\hat{\Pi}_{i,m}\) is the estimation of \(\Pi_{i,m}\), which will be given later. Considering (15) and (50), we have  
\[\dot{z}_{i,m}=\dot{s}_{i,m}-\dot{a}_{i,m}=-\frac{z_{i,m}}{\tau_{i,m}}-\frac{\dot{ \Pi}_{i,m}^{2}z_{i,m}}{\sqrt{\big{(}\dot{\Pi}_{i,m}z_{i,m}\big{)}^{2}+\eta^{2}}}+ M_{i,m}(\cdot) \tag{51}\]  
where \(M_{i,m}(\cdot)\) is a continuous function of variables \(e_{i,1},\cdots,e_{i,m+1}\), \(\Psi_{i,1},\cdots,\Psi_{i,m}\), \(\hat{\Xi}_{i,1},\cdots,\hat{\Xi}_{i,m}\), \(\hat{\Phi}_{i,1},\cdots,\hat{\Phi}_{i,m}\), \(\hat{\Pi}_{i,1},\cdots,\hat{\Pi}_{i,m-1},y_{d},\dot{y}_{d}\), and \(\hat{y}_{d}\). There is an unknown constant \(\Pi_{i,m}>0\), such that \(|M_{i,m}(\cdot)|\leq\Pi_{i,m}\) in a given compact set \(\Omega_{i,m}\). _Step n_: In this step, the adaptive neural network finite-time dynamic surface control law \(u_{i}(t)\) will be given. Similar to the above steps, we have  
\[\dot{e}_{i,n}=h_{i,n}u_{i}(t)+f_{i,n}+\frac{z_{i,n-1}}{\tau_{i,n-1}}+\frac{\Pi _{i,n-1}^{2}z_{i,n-1}}{\sqrt{\big{(}\dot{\Pi}_{i,n-1}z_{i,n-1}\big{)}^{2}+\eta^ {2}}} \tag{52}\]  
where \(h_{i,n}=g_{i,n}\). An RBFNN is utilized to approximate the unknown nonlinear function \(f_{i,n}\), yielding  
\[f_{i,n}=(\Phi_{i,n}^{*})^{T}\zeta_{i,n}+\varepsilon_{i,n}(X_{i,n}) \tag{53}\]  
where \(X_{i,n}=[x_{i,1},\cdots,x_{i,n}]^{T}\), and there exists a positive constant \(\Xi_{i,n}\), such that \(|\varepsilon_{i,n}(X_{i,n})|\leq\Xi_{i,n}\). Construct the following Lyapunov function candidate as  
\[V_{i,n}=\frac{1}{2}e_{i,n}^{2}+\frac{1}{2}z_{i,n-1}^{2}+\frac{1}{2\Gamma_{i,n }}\widetilde{\[PERSON]}_{i,n}^{2}+\frac{1}{2\Upsilon_{i,n}}\widetilde{\[PERSON]}_{i,n}^ {T}\widetilde{\[PERSON]}_{i,n}+\frac{1}{2\text{N}_{i,n-1}}\widetilde{\[PERSON]}_{i,n-1} ^{2} \tag{54}\]  
where \(\widetilde{\[PERSON]}_{i,n}=\hat{\[PERSON]}_{i,n}-\[PERSON]_{i,n}\), \(\widetilde{\[PERSON]}_{i,n}=\hat{\[PERSON]}_{i,n}-\Phi_{i,n}\) and \(\widetilde{\[PERSON]}_{i,n-1}=\hat{\Pi}_{i,n-1}-\Pi_{i,n-1}\), \(\hat{\[PERSON]}_{i,n}\), \(\hat{\[PERSON]}_{i,n}\), and \(\hat{\[PERSON]}_{i,n-1}\) are the estimations of \(\Xi_{i,n}\), \(\Phi_{i,n}\), and \(\Pi_{i,n-1}\), respectively. \(\Gamma_{i,n}\), \(\Upsilon_{i,n}\), and \(N_{i,n-1}\) are designed positive constants. Taking the derivative of (54) along with (52), and considering (53), we have  
\[