## 4 Finite-Time Fault-Tolerant Control Using Synchronous Terminal Sliding Mode Control
\begin{array}{ll}\dot{V}&=S^{T}\dot{S}\\ &=S^{T}\Big{(}\ddot{E}+\Gamma\Lambda\|^{\Lambda-1}\cdot\Big{)}\\ &=S^{T}\Big{(}\alpha\Big{(}\ddot{q}_{d}-\ddot{q}\Big{)}+\beta\Lambda|\epsilon|^{ \Lambda-I}\cdot\dot{\varepsilon}+\Gamma\Lambda\|^{\Lambda-1}\cdot\Big{)}\\ &=S^{T}\Bigg{[}\begin{array}{c}\alpha\Bigg{(}\begin{array}{c}\ddot{q}_{d}- \mathbf{M}^{-1}(q)\Big{(}\tau-\mathbf{H}(q,\dot{q})\Big{)}\\ +\mathbf{M}^{-1}(q)\zeta(q,\dot{q},\tau,t)\end{array}\Bigg{)}+\beta\lambda| \epsilon|^{\Lambda-I}\cdot\dot{\varepsilon}\\ +\Gamma\Lambda\|^{\Lambda-I}\cdots\end{array}\Bigg{)}\end{array}  \tag{18}\]  
Substituting Equation (16) into Equation (18):  
\[ \begin{array}{ll}\dot{V}&=-S^{T}(K_{1}sign(S))\\ &\leq-\sigma_{1}V^{\frac{1}{2}}<0\end{array}  \tag{19}\]  
where \(\sigma_{1}=\lambda_{\text{min}}(K_{1})\). When \(S=0\) converges, then \(E=0\) and \(\dot{E}=0\), and we have:  
\[ \begin{array}{ll}\dot{e}_{i}=-\frac{\ddot{g}_{i}}{\alpha_{i}}\big{[}\tau_{i} \dot{\tau}_{i}\big{]}^{\lambda}\\ &=\frac{\ddot{g}_{i}}{\alpha_{i}}\big{|}(1+\psi_{i}\psi_{i-1})e_{i}-\psi_{i}e_{ i-1}-\psi_{i}e_{i+1}\big{|}^{\lambda}sign\Bigg{(}\begin{array}{c}(1+\psi_{i} \psi_{i-1})e_{i}\\ -\psi_{i}e_{i-1}-\psi_{i}e_{i+1}\end{array}\Bigg{)}\end{array}  \tag{20}\]  
The system in Equation (20) has equilibrium points at \(e_{i}=0\)\((i=1,2,\ldots,n;\)\(n+1=1)\). According to the definition of terminal attractors [29], we have:  
\[ \begin{array}{ll}\left|\frac{\partial\dot{e}_{i}}{\partial\dot{e}_{j}}\right| =\frac{\beta_{i}}{\alpha_{i}}\lambda\big{|}(1+\psi_{i}\psi_{i-1})e_{i}-\psi_{ i}e_{i-1}-\psi_{i}e_{i+1}\big{|}^{\lambda-1}\left|\frac{\partial\dot{e}_{i}}{ \partial\dot{e}_{j}}\right|=\infty\end{array}  \tag{21}\]  
where \(j=(i-1,i,i+1)\)\((i=1,2,\ldots,n)\). From Equation (5), we have \(e_{i}\to 0(i=1,2,\ldots,n;\)\()\) at a finite-time. Therefore, Theorem 1 is proven. In Figure 1, the block diagram of the proposed controller AFTC-S-TSMC1 is presented. The block T in the diagram is synchronization matrix in Equation (13). The block of Fault Estimation uses the extended state observer in Equation (8). The fault estimation results were used to compensate with a novel finite-time synchronous terminal sliding mode controller 1. Figure 1: The block diagram of the proposed controller active fault-tolerant control with synchronous terminal sliding mode control 1 (AFTC-S-TSMC1). **Remark 1**. : _Time convergence from initial state to zero:_  
\[t=t_{r}+t_{s_{i}}+t_{e_{i}} \tag{22}\]  
_Time convergence \(S\to 0\):_  
\[t_{r}\leq\frac{2}{\sigma_{1}}\Big{(}\ln\Big{(}\sigma_{1}V_{0}^{\frac{1}{2}}+ \sigma_{2}\Big{)}-\ln\sigma_{2}\Big{)} \tag{23}\]  
_Time convergence \(E\to 0\):_  
\[t_{s_{i}}(1)=\frac{1}{c_{i}(1-\lambda_{i})}\Big{(}\ln\Big{(}\gamma_{i}+\sigma_{ 2}E_{i}^{1-\lambda_{i}}(0)\Big{)}-\ln\gamma_{1}\Big{)} \tag{24}\]  
_Time convergence \(e_{i}\to 0\):_  
\[t_{e_{i}}=\frac{\alpha_{i}}{\beta_{i}(1-\lambda_{i})}(ax_{i}(0)+bx_{i-1}(0)+cx_ {i+1}(0))^{1-\lambda_{i}} \tag{25}\]  
_where \(a=1+\psi_{i}\psi_{i-1},b=-\psi_{i}\) and \(c=-\psi_{i}\)._  
The Proposed Active Fault-Tolerant Control with Synchronous Terminal Sliding Mode Control 2 (AFTC S-TSMC2)  
The novel synchronous terminal sliding surface:  
\[ \begin{array}{ll}S&=\dot{e}+\pi[\varepsilon]^{\Lambda}\\ &=\dot{e}+\pi[T\varepsilon]^{\Lambda}\end{array}  \tag{26}\]  
where \(c=diag(c_{i})\in\Re^{n\times n}\) is the positive matrix gain. The proposed active fault-tolerant control with synchronous terminal sliding mode 2 (AFTC S-FTSMC2) is given as:  
\[\tau=\tau_{eq}+\tau_{0}+\tau_{ob} \tag{27}\]  
where \(\tau_{eq}=\mathbf{M}(q)(\ddot{q}_{d}+\pi\Lambda|\varepsilon|^{\Lambda-I}\cdot \dot{\varepsilon})+\mathbf{H}(q,\dot{q})\), \(\tau_{o}=\mathbf{M}(q)K_{1}sign(S)\), \(\tau_{ob}=-\mathbf{M}(q)\dot{\phi}\) and where \(K_{1}=diag(k_{1i})\in\Re^{n\times n}\). **Theorem 2**. : _The system described in Equation (5), using the controller specified in Equation (27) guarantees that \(e\to 0\) as finite-time._  
**Proof of Theorem 2.