## 2 Materials and Methods
### Control Development
#### 2.3.3 Stability Analysis  
To analyze the stability of the proposed control method, the Lyapunov function was selected as follows:  
\[V_{1}=\frac{1}{2}e^{\mathsf{T}}e+\frac{1}{2}r^{T}Mr+\frac{1}{2}\tilde{Y}^{ \mathsf{T}}\Gamma^{-1}\tilde{Y} \tag{31}\]  
where \(\beta_{1}\|z\|^{2}\leq V_{1}\leq\beta_{2}\|z\|^{2}\), \(z=[e^{\mathsf{T}}\), \(r^{\mathsf{T}}\), \(\tilde{Y}]^{T}\) and the constants \(\beta_{1}\) and \(\beta_{2}\) are defined as \(\beta_{1}=\min((1/2),(\underline{m}/2),(1/2)\lambda_{\min}\Gamma^{-1})\), \(\beta_{2}=\max((1/2),(\bar{m}/2),(1/2)\lambda_{\max}\Gamma^{-1})\), where \(\underline{m}\) and \(\bar{m}\) are bounds of \(M\), \(\lambda_{\min}(\cdot)\), and \(\lambda_{\max}(\cdot)\) denotes the minimum and maximum eigenvalues of \((\cdot)\), respectively. The notation \((\cdot)\) denotes the value of the variable. According to (27), the time derivative of (31) can be formulated as follows:  
\[\begin{split}\dot{V}_{1}=& e^{\mathsf{T}}(r- \alpha e)+r^{\mathsf{T}}(-k_{r}r+\tilde{Y}-e)\\ &+r^{\mathsf{T}}(\frac{1}{2}\dot{M}-C)r-\tilde{Y}^{\mathsf{T}} \Gamma^{-1}\dot{\tilde{Y}}\end{split} \tag{32}\]  
By substituting (22), (32) can be rewritten as follows:  
\[\dot{V}_{1}=-\alpha e^{\mathsf{T}}e-r^{\mathsf{T}}k_{r}r-k_{cl}\tilde{Y}^{ \mathsf{T}}(y^{\mathsf{T}}y)\tilde{Y} \tag{33}\]  
Then, (33) is bounded as follows:  
\[\dot{V}_{1}\leq-\alpha\|e\|^{2}-\|k_{r}\|\|r\|^{2}-k_{cl}\underline{\lambda} \|Y\|^{2} \tag{34}\]  
Assuming \(\Lambda=\min(\alpha,\ \|k_{r}\|,\ k_{cl}\underline{\[V_{1}(t+\Delta t)\leq V_{1}(t)\exp(-\frac{\Lambda}{\beta_{2}}\Delta t) \tag{36}\]  
where \(\Delta t\) denotes arbitrary time.  
The candidate Lyapunov function is extended as follows:  
\[V=V_{1}+\tau_{e}^{T}A^{T}MA\tau_{e} \tag{37}\]  
Substituting (30) into (37), we obtain the time derivative of \(V\) as follows:  
\[\dot{V}= \dot{V}_{1}-\tau_{e}^{T}(A+A^{T}-A^{T}\dot{M}A)\tau_{e} \tag{38}\] \[+\dot{\tau}_{e}^{T}A^{T}MA\tau_{e}+\dot{\tau}_{e}^{T}A^{T}MA\dot{ \tau}_{e}\]  
According to the Schwartz inequality and bound of \(\|\tau_{e}\|\), (38) can be deduced as follows:  
\[\dot{V} \leq\dot{V}_{1}-\lambda_{\min}(\Gamma)\|\tau_{e}\|^{2}+2\zeta \lambda_{\max}(M)\|A\|^{2}\|\tau_{e}\| \tag{39}\] \[=\dot{V}_{1}-(1-\rho)\lambda_{\min}(\Gamma)\|\tilde{\tau}_{e}\|^ {2}-\rho\lambda_{\min}(\Gamma)\|\tilde{\tau}_{e}\|^{2}\] \[+2\zeta\lambda_{\max}(M)\|A\|^{2}\|\tau_{e}\|\]  
where \(\rho\in(0,1)\). Because \(V-V_{1}=\tau_{e}^{T}A^{T}MA\tau_{e}\leq\lambda_{\max}(M)\|A\|^{2}\|\tau_{e}\|^ {2}\), when \(\|\tau_{e}\|\geq\frac{2\zeta\lambda_{\max}(M)\|A\|^{2}}{\rho\lambda_{\min}( \Gamma)}\), (39) can be deduced as follows:  
\[\dot{V} \leq\dot{V}_{1}-(1-\rho)\lambda_{\min}(\Gamma)\|\tau_{e}\|^{2} \tag{40}\] \[\leq\dot{V}_{1}-\frac{(1-\rho)\lambda_{\min}(\Gamma)}{\lambda_{ \max}(M)\|A\|^{2}}(V-V_{1})\]  
Then, we obtain the following:  
\[\dot{V}-\dot{V}_{1}\leq-\frac{(1-\rho)\lambda_{\min}(\Gamma)}{\lambda_{\max} (M)\|A\|^{2}}(V-V_{1}) \tag{41}\]  
And, the exponential convergence was formulated as follows:  
\[V(t+\Delta t) -V_{1}(t+\Delta t) \tag{42}\] \[\leq(V(t)-V_{1}(t))\exp(-\frac{(1-\rho)\lambda_{\min}(\Gamma)}{ \lambda_{\max}(M)\|A\|^{2}}\Delta t)\]  
where \(\Delta V=V-V_{1}\) achieved exponential convergence. According to (36), \(V_{1}\) also achieves exponential convergence. Hence, \(V=V_{1}+\Delta V\) also achieved exponential convergence. The stability of the closed-loop system is guaranteed, and exponential convergence is ensured.