## 3 Finite-Time Disturbance Observer and Controller Design
### Finite-Time Disturbance Observer Design
\begin{split}\hat{\varpi}_{i1}(t)-\mu_{i1}(t)=&\hat {\varpi}_{i1}(t)+\lambda_{0}L_{i}^{1/3}|\hat{x}_{i}(t)-x_{i}(t)|^{2/3} \operatorname{sgn}(\hat{x}_{i}(t)\\ &-x_{i}(t))+\varrho_{0}(\hat{x}_{i}(t)-x_{i}(t))-\hat{\varpi}_{ i1}(t)\\ =&\lambda_{0}L_{i}^{1/3}|L_{i}\hat{x}_{i}(t)|^{2/3} \operatorname{sgn}(L_{i}\hat{x}_{i}(t))+\varrho_{0}(L_{i}\hat{x}_{i}(t))\\ =& L_{i}(\lambda_{0}|\hat{x}_{i}(t)|^{2/3} \operatorname{sgn}(\hat{x}_{i}(t))+\varrho_{0}(\hat{x}_{i}(t)))\\ =& L_{i}(\hat{\varpi}_{i1}(t)-\hat{x}_{i}(t))\end{split}  \tag{15}\]  
Substituting (15) into (14),  
\[ \begin{split}\hat{\varpi}_{i1}(t)=&\frac{\dot{\hat{ \varpi}}_{i1}(t)-\hat{\varpi}_{i1}(t)}{L_{i}}\\ =&\frac{1}{L_{i}}(-\lambda_{1}L_{i}^{1/2}\big{|}L_{ i}(\hat{\varpi}_{i1}(t)-\hat{x}_{i}(t))\big{|}^{1/2}\operatorname{sgn}(L_{i}( \hat{\varpi}_{i1}(t)-\hat{x}_{i}(t)))\\ &-\varrho_{1}(L_{i}(\hat{\varpi}_{i1}(t)-\hat{x}_{i}(t)))+L_{i} v_{i1}(t))\\ =&-\lambda_{1}\big{|}(\hat{\varpi}_{i1}(t)-\hat{x} _{i}(t))\big{|}^{1/2}\operatorname{sgn}(\hat{\varpi}_{i1}(t)-\hat{x}_{i}(t)) \\ &-\varrho_{1}(\hat{\varpi}_{i1}(t)-\hat{x}_{i}(t))+v_{i1}(t)\end{split}  \tag{16}\]  
Calculating the derivative of \(\hat{v}_{i}(t)\) from (12) yields  
\[ \begin{split}\hat{v}_{i1}(t)&=\frac{\dot{\hat{ \varpi}}_{i1}(t)-\hat{\varpi}_{i1}(t)}{L_{i}}\\ &=-\lambda_{2}\operatorname{sgn}((v_{i1}(t)-\hat{\varpi}_{i1}(t) )-\varrho_{2}(v_{i1}(t)-\hat{\varpi}_{i1}(t))-\frac{1}{L_{i}}\hat{\varpi}_{i1 }(t)\end{split}  \tag{17}\]  
Differentiating \(\hat{v}_{i}(t)\), one has  
\[ \begin{split}\hat{v}_{i}(t)=&\frac{\dot{\varrho} _{i}(t)-\hat{v}_{i}(t)}{L_{i}}\\ =&\frac{1}{L_{i}}(-\lambda_{3}L_{i}^{1/2}|\hat{v}_{ i}(t)-v_{i}(t)|^{1/2}\operatorname{sgn}(\hat{v}_{i}(t)-v_{i}(t))\\ &-\delta_{3}(\hat{v}_{i}(t)-v_{i}(t))+\hat{\varpi}_{i2}(t)-\hat{ \varpi}_{i2}(t))\\ =&\frac{1}{L_{i}}(-\lambda_{3}L_{i}^{1/2}|L_{i} \hat{v}(t)|^{1/2}\operatorname{sgn}(L_{i}\hat{v}(t))-\varrho_{3}(L_{i}\hat{v} (t))+L_{i}\hat{\varpi}_{i2}(t))\\ =&-\lambda_{3}|\hat{v}(t)|^{1/2}\operatorname{sgn} (\hat{v}(t))-\varrho_{3}(\hat{v}(t))+\hat{\varpi}_{i2}(t)\end{split}  \tag{18}\]  
And, the time derivative of \(\hat{\varpi}_{i2}(t)\) is  
\[ \begin{split}\hat{\varpi}_{i2}(t)&=\frac{\dot{ \hat{\varpi}}_{i2}(t)-\dot{\varpi}_{i2}(t)}{L_{i}}\\ &=\frac{1}{L_{i}}(-\lambda_{4}L_{i}\operatorname{sgn}(\hat{ \varpi}_{i2}(t)-\mu_{i2}(t))-\varrho_{4}(\hat{\varpi}_{i2}(t)-\mu_{i2}(t))-\hat{ \varpi}_{i2}(t))\end{split}  \tag{19}\]According to (12), one can obtain  
\[ \begin{split}\hat{\omega}_{l2}(t)-\mu_{l2}(t)=&\hat{ \omega}_{l2}(t)+\lambda_{3}L_{i}^{\frac{1}{2}}\Big{|}\hat{v}_{l}(t)-v_{i}(t) \Big{|}^{\frac{1}{2}}\operatorname{sgn}(\hat{v}_{i}(t)-v_{i}(t))\\ &+\varrho_{3}(\hat{v}_{i}(t)-v_{i}(t))-\hat{\omega}_{l2}(t)\\ =&\lambda_{3}L_{i}^{1/2}|L_{i}\hat{v}_{i}(t)|^{1/2} \operatorname{sgn}(L_{i}\hat{v}_{i}(t))+\varrho_{3}(L_{i}\hat{v}_{i}(t))\\ =& L_{i}(\lambda_{3}|\hat{v}_{i}(t)|^{1/2} \operatorname{sgn}(\hat{v}_{i}(t))+\varrho_{3}(\hat{v}_{i}(t)))\\ =& L_{i}(\hat{\omega}_{l2}(t)-\hat{v}(t))\end{split}  \tag{20}\]  
Substituting (20) into (19), we have  
\[ \begin{split}\hat{\omega}_{l2}(t)&=\frac{\hat{ \omega}_{l2}(t)-\hat{\omega}_{l2}(t)}{L_{i}}\\ &=\frac{1}{L_{i}}(-\lambda_{4}L_{i}\operatorname{sgn}(\hat{ \omega}_{l2}(t)-\mu_{l2}(t))-\varrho_{4}(\hat{\omega}_{l2}(t)-\mu_{l2}(t))- \hat{\omega}_{l2}(t))\\ &=\frac{1}{L_{i}}(-\lambda_{4}L_{i}\operatorname{sgn}(L_{i}( \hat{\omega}_{l2}(t)-\hat{v}(t)))-\varrho_{4}(L_{i}(\hat{\omega}_{l2}(t)- \hat{v}(t)))-\hat{\omega}_{l2}(t))\\ &=-\lambda_{4}\operatorname{sgn}(\hat{\omega}_{l2}(t)-\hat{v}(t ))-\varrho_{4}(\hat{\omega}_{l2}(t)-\hat{v}(t))-\frac{1}{L_{i}}\hat{\omega}_{ l2}(t)\end{split}  \tag{21}\]  
Then, the observation error can be expressed as  
\[