## Appendix A
\begin{array}{l}L_{f\_l}=k\Big{[}L_{f\_l}^{1}+\cdots+L_{f\_l}^{5}\Big{]}\leq k \frac{1}{2}\Big{\{}\lambda_{\max}(E_{L}^{1})\overline{e_{tr\_track}^{1}}+ \cdots+\lambda_{\max}(E_{L}^{5})\overline{e_{tr\_track}^{5}}\Big{\}}\\ \leq k\frac{1}{2}\lambda_{\max}^{*}\bigg{\{}\overline{e_{tr\_track}^{1}}+ \cdots+\overline{e_{tr\_track}^{5}}\bigg{\}}\end{array}  \tag{114}\]  
Let \(Q_{\max}\equiv\frac{k}{2}\lambda_{\max}^{*}\) and we obtain  
\[L_{f\_l}\leq Q_{\max}\bigg{(}\|\vv{u}_{op\_track-total}\|^{2}\bigg{)} \tag{115}\]  
Similarly, we obtain  
\[ \begin{array}{l}L_{f\_l}=k\Big{[}L_{f\_l}^{1}+\cdots+L_{f\_l}^{5}\Big{]}\geq k \frac{1}{2}\Big{\{}\lambda_{\min}(E_{L}^{1})\overline{e_{tr\_track}^{1}}+ \cdots+\lambda_{\min}(E_{L}^{5})\overline{e_{tr\_track}^{5}}\Big{\}}\\ \geq k\frac{1}{2}\lambda_{\min}^{*}\bigg{\{}\overline{e_{tr\_track}^{1}}+ \cdots+\overline{e_{tr\_track}^{5}}\bigg{\}}\end{array}  \tag{116}\]  
Let \(Q_{\min}\equiv\frac{k}{2}\lambda_{\min}^{*}\) and we obtain  
\[L_{f\_l}\geq Q_{\min}\bigg{(}\|\vv{u}_{op\_track-total}\|^{2}\bigg{)} \tag{117}\]  
Combining (115) and (117) yields  
\[Q_{\min}\bigg{(}\|\vv{u}_{op\_track-total}\|^{2}\bigg{)}\leq L_{f\_l}\leq Q_{ \max}\bigg{(}\|\vv{u}_{op\_track-total}\|^{2}\bigg{)} \tag{118}\]  
(112), (113) and (118) imply that the system is in the input-to-state stable state for the disturbance input. Then the input-to-state stable theorem in [32] concludes the significant result that the first condition of the almost disturbance decoupling performance is completely verified. Next, we need to prove that the second condition of the almost disturbance decoupling performance holds. Combining (108), (109), (118) and (107) yields  
\[\frac{d}{dt}\Big{(}L_{f\_l}\Big{)}\leq-\frac{S\cdot S_{2}}{Q_{\max}}L_{f\_l}+S _{1} \tag{119}\]  
Use the comparison theorem in [33] for (119) to obtain  
\[L_{f\_l}(t)\leq L_{f\_l}(t_{0})\exp\bigg{(}-\frac{S\cdot S_{2}}{Q_{\max}}(t-t_ {0})\bigg{)}+\frac{Q_{\max}S_{1}}{S\cdot S_{2}},t\geq t_{0} \tag{120}\]Then, we obtain the tracking error with integral sense to be  
\[\left|u_{op\_1}(t)-u_{op\_track}^{1}(t)\right|\leq\sqrt{\frac{2L_{f\_J}(t_{0})}{k \lambda_{\min}^{*}}}\exp\left(-\frac{S\cdot S_{2}}{2Q_{\max}}(t-t_{0})\right)+ \sqrt{\frac{2Q_{\max}S_{1}}{k\lambda_{\min}^{*}S\cdot S_{2}}} \tag{111}\]  
and  
\[\left|u_{op\_i}(t)-u_{op\_track}^{i}(t)\right|\leq\sqrt{\frac{2L_{f\_J}(t_{0})} {k\lambda_{\min}^{*}}}\exp\left(-\frac{S\cdot S_{2}}{2Q_{\max}}(t-t_{0})\right) +\sqrt{\frac{2Q_{\max}S_{1}}{k\lambda_{\min}^{*}S\cdot S_{2}}},2\leq i\leq 5 \tag{112}\]  
So, we can prove that the second condition of the almost disturbance decoupling performance holds, and the convergent rate is given by \(S\cdot S_{2}/2Q_{\max}\). Combining (108) and (109) yields  
\[\frac{d}{dt}\Big{(}\left.L_{f\_J}\right)\leq-S\cdot S_{2}\Big{(}\left\| \overrightarrow{u}_{op\_track-total}\right\|^{2}\Big{)}+S_{1} \tag{113}\]  
Let us consider the range \(\left\|\overrightarrow{u}_{op\_track-total}\right\|>L\,\mathcal{I}\equiv \sqrt{\frac{S_{1}}{S\cdot S_{2}}}\). It is an easy routine to obtain \(\frac{d}{dt}\Big{(}L_{f\_J}\Big{)}<0\), and then the global ultimate attractor of the transformed system is written by  
\[B_{\mathcal{I}}\equiv\Big{\{}[\overline{e_{tr\_track}}]:\left\|\overline{e_{ tr\_track}}\right\|^{2}\leq\underline{r}\Big{\}} \tag{114}\]  
with the convergent radius \(\underline{r}\equiv\sqrt{\frac{S_{1}}{S\cdot S_{2}}}\). Next, we need to prove the globally exponential stability of the transformed system. Combining (118) and (110) obtains  
\[L_{f\_l}(t)\leq L_{f\_l}(t_{0})\exp\left(-\frac{S\cdot S_{2}}{Q_{\max}}(t-t_{ 0})\right)+\frac{Q_{\max}S_{1}}{S\cdot S_{2}} \tag{115}\]  
and  
\[ \begin{array}{l}Q_{\min}\left\|\overrightarrow{u}_{op\_track-total}\right\| ^{2}\leq L_{f\_l}\leq L_{f\_l}(t_{0})\exp\left(-\frac{S\cdot S_{2}}{Q_{\max}} (t-t_{0})\right)+\frac{Q_{\max}S_{1}}{S\cdot S_{2}}\\ \leq Q_{\max}\left\|\overrightarrow{u}_{op\_track-total}(t_{0})\right\|^{2} \exp\left(-\frac{S\cdot S_{2}}{Q_{\max}}(t-t_{0})\right)+\frac{Q_{\max}S_{1}} {S\cdot S_{2}}\end{array}