## Appendix A
\right] \tag{132}\]  
\[\frac{d}{dt}\Big{(}L_{f,\_}\Big{)}=\frac{k}{2}\Bigg{[}\left(\overline{c_{tr \_track}^{\bullet}}\right)^{T}E_{1}^{1}\overline{e_{tr\_track}^{\overline{1}}}+ \left(\overline{c_{tr\_track}^{\overline{1}}}\right)^{T}E_{1}^{1}\Bigg{(} \overline{c_{tr\_track}^{\bullet}}\Bigg{)}\] \[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\] \[=\frac{k}{2}\Bigg{[}\overline{c_{tr\_track}^{\overline{1}}} \right)^{T}\Big{[}E_{1}^{1}\big{(}A_{L}^{1}\big{)}+\big{(}A_{L}^{1}\big{)}^{T }E_{1}\Big{]}\overline{c_{tr\_track}^{\bullet}}+\cdots+\frac{k}{2}\Bigg{(} \overline{c_{tr\_track}^{\overline{5}}}\Bigg{)}^{T}\Big{[}E_{L}^{5}\big{(}A_{ L}^{5}\big{)}+\big{(}A_{L}^{5}\big{)}^{T}E_{1}^{5}\Big{]}\overline{c_{tr\_track}^{ \overline{5}}}\] \[+\frac{k}{2}\Bigg{\{}\Bigg{(}\widetilde{\Omega}_{un}+\widetilde{ \Omega}_{n}\Bigg{)}^{T}\Big{[}\Big{(}\varphi_{T_{1}}^{1}\Big{)}^{T}E_{1}^{1} \overline{c_{tr\_track}^{\overline{1}}}+\cdots+\left(\varphi_{T_{1}}^{5} \right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\overline{5}}}\Bigg{]}\Bigg{\}}\] \[+\frac{k}{2}\Bigg{\{}\Bigg{(}\widetilde{\Omega}_{un}+\widetilde{ \Omega}_{n}\Bigg{)}^{T}\Big{[}\Big{(}\varphi_{T_{1}}^{1}\Big{)}^{T}E_{1}^{1} \overline{c_{tr\_track}^{\overline{1}}}+\cdots+\left(\varphi_{T_{1}}^{5} \right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\overline{5}}}\Bigg{]}\Bigg{\}}\] \[\leq-\frac{k}{2}\Bigg{[}\|\overline{c_{tr\_track}^{\overline{1}}} \|^{2}+\cdots+\|\overline{c_{tr\_track}^{\overline{5}}}\|^{2}\Bigg{]}+\frac{k}{ 2}\Bigg{\{}\Bigg{(}\widetilde{\Omega}_{un}+\widetilde{\Omega}_{n}\Bigg{)}^{T }\Big{[}\Big{(}\varphi_{T_{1}}^{1}\Big{)}^{T}E_{1}^{1}\overline{c_{tr\_track }^{\overline{1}}}+\cdots+\left(\varphi_{T_{1}}^{5}\right)^{T}E_{1}^{5}\overline {c_{tr\_track}^{\overline{5}}}\Big{]}\Bigg{\}}\] \[\leq-\frac{k}{2}\Bigg{[}\|\overline{c_{tr\_track}^{\overline{1}}} \|^{2}+\cdots+\|\overline{c_{tr\_track}^{\overline{5}}}\|^{2}\Bigg{]}+\frac{k}{ 2}\Big{[}\|\widetilde{\Omega}_{un}+\widetilde{\Omega}_{n}\Big{)}\|\] (A92) \[\Big{(}\|\varphi_{T_{1}}^{1}\|\|E_{1}^{1}\|\overline{c_{tr\_track }^{\overline{1}}}\|+\cdots+\|\varphi_{T_{1}}^{5}\|\|\overline{c_{1}^{5}}\| \overline{c_{tr\_track}^{\overline{5}}}\|\Big{)}\Bigg{]}\] \[\leq-\frac{k}{2}\bigg{[}\frac{L_{f,\_}^{\prime}}{\lambda_{\max}( E_{1}^{1})}+\cdots+\frac{L_{f,\_}^{\prime}}{\lambda_{\max}(E_{1}^{1})}\bigg{]}+ \frac{529}{46}\frac{k^{2}}{2^{\prime}}\|\varphi_{T_{1}}^{1}\|^{2}\|E_{1}^{1}\| ^{2}\overline{c_{tr\_track}^{\overline{1}}}\|^{2}+\frac{1}{46}\|\Big{(} \widetilde{\Omega}_{un}+\widetilde{\Omega}_{n}\Big{)}\|^{2}\] \[+\cdots+\frac{529}{46}\frac{k^{2}}{2^{\prime}}\|\varphi_{T_{1}}^{ 5}\|^{2}\|E_{1}^{5}\|^{2}\|\overline{c_{tr\_track}^{\overline{5}}}\|^{2}+\frac{ 1}{46}\|\Big{(}\widetilde{\Omega}_{un}+\widetilde{\Omega}_{n}\Big{)}\|^{2}\] \[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\]0 \[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\]1 where \[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\]2 (A93) i.e., \[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\]3 (A94) Define \[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\]4 (A95) \[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\]5 (A96) \[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\]6 (A97) \[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\]7 (A98) Then \[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\]8 (A99)Let the tracking error \(\overline{e_{tr\_track}}\) of the transformed system be  
\[+\cdots+\left(\overline{c_{tr\_track}^{\bullet}}\right)^{T}E_{1}^ {1}\overline{c_{tr\_track}^{\overline{5}}}+\left(\overline{c_{tr\_track}^{ \bullet}}\right)^{T}E_{1}^{5}\overline{c_{tr\_track}^{\bullet}}\Bigg{)}\]9  
Then we obtain  
\[=\frac{k}{2}\Bigg{[}\overline{c_{tr\_track}^{\overline{1}}} \right)^{T}\Big{[}E_{1}^{1}\big{(}A_{L}^{1}\big{)}+\big{(}A_{L}^{1}\big{)}^{T }E_{1}\Big{]}\overline{c_{tr\_track}^{\bullet}}+\cdots+\frac{k}{2}\Bigg{(} \overline{c_{tr\_track}^{\overline{5}}}\Bigg{)}^{T}\Big{[}E_{L}^{5}\big{(}A_{ L}^{5}\big{)}+\big{(}A_{L}^{5}\big{)}^{T}E_{1}^{5}\Big{]}\overline{c_{tr\_track}^{ \overline{5}}}\]0  
Firstly, applying (101) easily yields  
\[=\frac{k}{2}\Bigg{[}\overline{c_{tr\_track}^{\overline{1}}} \right)^{T}\Big{[}E_{1}^{1}\big{(}A_{L}^{1}\big{)}+\big{(}A_{L}^{1}\big{)}^{T }E_{1}\Big{]}\overline{c_{tr\_track}^{\bullet}}+\cdots+\frac{k}{2}\Bigg{(} \overline{c_{tr\_track}^{\overline{5}}}\Bigg{)}^{T}\Big{[}E_{L}^{5}\big{(}A_{ L}^{5}\big{)}+\big{(}A_{L}^{5}\big{)}^{T}E_{1}^{5}\Big{]}\overline{c_{tr\_track}^{ \overline{5}}}\]1  
i.e.,  
\[=\frac{k}{2}\Bigg{[}\overline{c_{tr\_track}^{\overline{1}}} \right)^{T}\Big{[}E_{1}^{1}\big{(}A_{L}^{1}\big{)}+\big{(}A_{L}^{1}\big{)}^{T }E_{1}\Big{]}\overline{c_{tr\_track}^{\bullet}}+\cdots+\frac{k}{2}\Bigg{(} \overline{c_{tr\_track}^{\overline{5}}}\Bigg{)}^{T}\Big{[}E_{L}^{5}\big{(}A_{ L}^{5}\big{)}+\big{(}A_{L}^{5}\big{)}^{T}E_{1}^{5}\Big{]}\overline{c_{tr\_track}^{ \overline{5}}}\]2  
Integrate both sides of (103) to obtain  
\[=\frac{k}{2}\Bigg{[}\overline{c_{tr\_track}^{\overline{1}}} \right)^{T}\Big{[}E_{1}^{1}\big{(}A_{L}^{1}\big{)}+\big{(}A_{L}^{1}\big{)}^{T }E_{1}\Big{]}\overline{c_{tr\_track}^{\bullet}}+\cdots+\frac{k}{2}\Bigg{(} \overline{c_{tr\_track}^{\overline{5}}}\Bigg{)}^{T}\Big{[}E_{L}^{5}\big{(}A_{ L}^{5}\big{)}+\big{(}A_{L}^{5}\big{)}^{T}E_{1}^{5}\Big{]}\overline{c_{tr\_track}^{ \overline{5}}}\]3  
i.e.,  
\[=\frac{k}{2}\Bigg{[}\overline{c_{tr\_track}^{\overline{1}}} \right)^{T}\Big{[}E_{1}^{1}\big{(}A_{L}^{1}\big{)}+\big{(}A_{L}^{1}\big{)}^{T }E_{1}\Big{]}\overline{c_{tr\_track}^{\bullet}}+\cdots+\frac{k}{2}\Bigg{(} \overline{c_{tr\_track}^{\overline{5}}}\Bigg{)}^{T}\Big{[}E_{L}^{5}\big{(}A_{ L}^{5}\big{)}+\big{(}A_{L}^{5}\big{)}^{T}E_{1}^{5}\Big{]}\overline{c_{tr\_track}^{ \overline{5}}}\]4  
hence  
\[=\frac{k}{2}\Bigg{[}\overline{c_{tr\_track}^{\overline{1}}} \right)^{T}\Big{[}E_{1}^{1}\big{(}A_{L}^{1}\big{)}+\big{(}A_{L}^{1}\big{)}^{T }E_{1}\Big{]}\overline{c_{tr\_track}^{\bullet}}+\cdots+\frac{k}{2}\Bigg{(} \overline{c_{tr\_track}^{\overline{5}}}\Bigg{)}^{T}\Big{[}E_{L}^{5}\big{(}A_{ L}^{5}\big{)}+\big{(}A_{L}^{5}\big{)}^{T}E_{1}^{5}\Big{]}\overline{c_{tr\_track}^{ \overline{5}}}\]5  
Similarly, we obtain the tracking error for \(u_{op\_i},2\leq i\leq 5\) as  
\[=\frac{k}{2}\Bigg{[}\overline{c_{tr\_track}^{\overline{1}}} \right)^{T}\Big{[}E_{1}^{1}\big{(}A_{L}^{1}\big{)}+\big{(}A_{L}^{1}\big{)}^{T }E_{1}\Big{]}\overline{c_{tr\_track}^{\bullet}}+\cdots+\frac{k}{2}\Bigg{(} \overline{c_{tr\_track}^{\overline{5}}}\Bigg{)}^{T}\Big{[}E_{L}^{5}\big{(}A_{ L}^{5}\big{)}+\big{(}A_{L}^{5}\big{)}^{T}E_{1}^{5}\Big{]}\overline{c_{tr\_track}^{ \overline{5}}}\]6  
Therefore, we verify the significant result that the third condition of the almost disturbance decoupling performance is well proved.