## 3 Finite-Time Disturbance Observer and Controller Design
### FTDO-Based Sliding Mode Controller Design and Stability Analysis
\begin{cases}-\ell_{i}\epsilon_{i}(t)-\frac{1}{\gamma} \Big{(}\dot{\phi}_{i+1}(t)+\ell_{i+1}\epsilon_{i+1}(t)\Big{)}-\frac{1}{\gamma} \Big{(}k_{1}|S_{i}(t)|^{1/2}\operatorname{sgn}(S_{i}(t))\\ +E_{\hat{\omega}_{i}}(t)+\int_{0}^{t}k_{2}\operatorname{sgn}(S_{i}(\tau))d\tau \Big{)},i=1,\ldots,N-1\\ -\ell_{i}\epsilon_{i}(t)-\frac{1}{\gamma}\Big{(}k_{1}|S_{i}(t)|^{1/2} \operatorname{sgn}(S_{i}(t))+\int_{0}^{t}k_{2}\operatorname{sgn}(S_{i}(\tau) )d\tau\Big{)}\\ +E_{\hat{\omega}_{i}}(t),i=N\end{cases}  \tag{29}\]  
where  
\[E_{\varpi_{i}}(t)= \begin{cases}(\hat{\omega}_{i1}(t)-\varpi_{i1}(t))+h_{i} \Big{(}\hat{\omega}_{i2}(t)-\varpi_{i2}(t)+\hat{\omega}_{i1}(t)\\ -\hat{\omega}_{i1}(t)\Big{)},i=1\\ (\hat{\omega}_{i-1,1}(t)-\hat{\omega}_{i-1,1}(t))+(\hat{\omega}_{i1}(t)-\varpi _{i1}(t))+h_{i}\Big{(}\hat{\omega}_{i2}(t)\\ -\varpi_{i2}(t)+\hat{\omega}_{i1}(t)-\hat{\omega}_{i1}(t)\Big{)},i=2,\ldots,N \end{cases} \]  
According to Theorem 1, \(E_{\varpi_{i}}(t)\) is bounded for \(t\in[0,T_{d})\). Based on (27), (26) becomes  
\[\dot{S}_{i}(t)=-k_{1}|S_{i}(t)|^{1/2}\operatorname{sgn}\left(S_{i}(t)\right)- \int_{0}^{t}k_{2}\operatorname{sgn}\left(S_{i}(\tau)d\tau\right)+\gamma E_{ \varpi_{i}}(t) \tag{30}\]  
Define the Lyapunov function as \(V_{i}(t)=\frac{1}{2}\mathcal{S}_{i}^{2}(t)\), and its time derivative is  
\[\dot{V}_{i}(t) =\dot{S}_{i}(t)S_{i}(t) \tag{31}\] \[=S_{i}(t)(-k_{1}|S_{i}(t)|^{1/2}\operatorname{sgn}(S_{i}(t))- \int_{0}^{t}k_{2}\operatorname{sgn}(S_{i}(\tau))d\tau+\gamma E_{\varpi_{i}}(t))\] \[=-k_{1}|S_{i}(t)|^{3/2}-k_{2}S_{i}(t)\int_{0}^{t}\operatorname{ sgn}(S_{i}(\tau))d\tau+\gamma E_{\varpi_{i}}(t)S_{i}(t)\] \[\leq-k_{2}S_{i}(t)\int_{0}^{t}\operatorname{sgn}(S_{i}(\tau))d \tau+\gamma E_{\varpi_{i}}(t)S_{i}(t)\] \[\leq k_{2}|S_{i}(t)|\int_{0}^{t}|\operatorname{sgn}(S_{i}(\tau))| d\tau+|\gamma E_{\varpi_{i}}(t)||S_{i}(t)|\]By Lemma 2, we can obtain  
\[ \begin{split}\dot{V}_{i}(t)&\leq k_{2}T_{d}(\frac{1}{2}S _{i}^{2}(t)+\frac{1}{2})+|\gamma E_{\varpi_{i}}(t)|(\frac{1}{2}S_{i}^{2}(t)+ \frac{1}{2})\\ &\leq\frac{1}{2}(k_{4}T_{d}+\left|\gamma_{E_{\varpi_{i}}}(t) \right|)S_{i}^{2}(t)+\frac{1}{2}(k_{2}T_{d}+\left|\gamma E_{\varpi_{i}}(t) \right|)\\ &\leq\Gamma_{\max}V_{i}(t)+\Lambda_{\max}\end{split}  \tag{32}\]  
where \(\Gamma_{\max}=\)max\(\left\{k_{2}T_{d}+\left|\gamma E_{\varpi_{i}}(t)\right|\right\}\) and \(\Lambda_{max}=\)\(\frac{1}{2}\)max\(\left\{k_{2}T_{d}+\left|\gamma E_{\varpi_{i}}(t)\right|\right\}\). As \(E_{\varpi_{i}}(t)\) is bounded for \(t\in[0,T_{d})\), both \(\Gamma_{\max}\) and \(\Lambda_{max}\) are bounded. From (32), we can get that \(V_{i}(t)\) and \(S_{i}(t)\) are also bounded for \(t\in[0,T_{d})\)[30]. Define \(\Xi_{N}(t)=-\frac{1}{\gamma}(k_{1}|S_{i}(t)|)^{1/2}\text{sgn}(S_{i}(t))+\int_ {0}^{t}k_{2}\text{sgn}(S_{i}(\tau))d\tau)+E_{\varpi_{i}}(t)\). Then, (29) can be rewritten as  
\[\dot{e}_{i}(t)= \begin{cases}-\ell_{i}e_{i}(t)+\[PERSON]_{N}(t)-A_{i}(t),i=1,...N-1\\ -\ell_{i}e_{i}(t)+\[PERSON]_{N}(t),i=N\end{cases}  \tag{33}\]  
For \(i=N\), define \(V_{e_{N}}(t)=\frac{1}{2}e_{N}^{2}(t)\), and its differential is  
\[ \begin{split}\dot{V}_{e_{N}}(t)&=\dot{e}_{N}(t)e_{N}(t) \\ &=-\ell_{i}e_{N}^{2}(t)+\[PERSON]_{N}(t)e_{N}(t)\\ &\leq|\[PERSON]_{N}(t)||e_{N}(t)|\leq\frac{e_{N}^{2}(t)}{2}+\frac{\Xi_{N}^{ 2}(t)}{2}\leq V_{e_{N}}(t)+\frac{\Xi_{N}^{2}(t)}{2}\end{split}  \tag{34}\]  
As \(\Xi_{N}\) is bounded for \(t\in[0,T_{d})\), \(V_{e_{N}}\) and \(e_{N}\) are also bounded for \(t\in[0,T_{d})\)[30]. For \(i=N-1\), let \(\Xi_{N-1}(t)=-\frac{1}{\gamma}(e_{N}(t)+\ell_{N}e_{N}(t))+\Xi_{N-1}(t)\), which is bounded for \(t\in[0,T_{d})\). Then, we have \(\dot{e}_{N-1}(t)=-\ell_{N-1}e_{N-1}(t)+\dot{\Xi}_{N-1}(t)\). Similar to the derivation of \(e_{N}\) which is bounded, we can obtain that \(V_{e_{N-1}}(t)\) and \(e_{N-1}(t)\) are bounded for \(t\in[0,T_{d})\), and \(e_{i}(t)\) is also bounded for \(t<T_{d}\).