# Robust Multi-Performances Control for Four-Link Manipulator Arm
## Appendix B
\begin{array}{l}J_{lin+non}\\ =J_{non}+k(\varepsilon)\big{(}W_{lin}^{1}+W_{lin}^{2}+W_{lin}^{3}+W_{lin}^{4} \big{)}\\ \geq\omega_{non1}\Big{\|}\overrightarrow{n}_{non}\Big{\|}^{2}+k\frac{1}{2} \bigg{\{}\gamma_{\text{min}}(P_{Lyup}^{1})\Big{\|}\overrightarrow{e_{track} }\Big{\|}^{2}+\ldots+\gamma_{\text{min}}(P_{Lyup}^{4})\Big{\|}\overrightarrow {e_{track}}\Big{\|}^{2}\bigg{\}}\\ \geq\omega_{non1}\Big{\|}\overrightarrow{n}_{non}\Big{\|}^{2}+k\frac{1}{2} \gamma_{\text{min}}^{*}\bigg{\{}\bigg{\|}\overrightarrow{e_{track}}\Big{\|} ^{2}+\ldots+\bigg{\|}\overrightarrow{e_{track}}\Big{\|}^{2}\bigg{\}}\\ \geq\omega_{non1}\Big{\|}\overrightarrow{n}_{non}\Big{\|}^{2}+k\frac{1}{2} \gamma_{\text{min}}^{*}\|\vec{e}_{track}\|^{2}\end{array}  \tag{110}\]  
We define \(\Delta_{\text{min}}\equiv\text{min}\Big{\{}\omega_{non1},\frac{k}{2}\gamma_{ \text{min}}^{*}\Big{\}}\) and can obtain  
\[\Delta_{\text{min}}\bigg{(}\Big{\|}\overrightarrow{y}_{track- total}\Big{\|}^{2}\bigg{)}\leq J_{lin+non} \tag{111}\]  
Combining (109) and (111) results in  
\[\Delta_{\text{min}}\bigg{(}\Big{\|}\overrightarrow{y}_{track- total}\Big{\|}^{2}\bigg{)}\leq J_{lin+non}\leq\Delta_{\text{max}}\bigg{(}\Big{\|} \overrightarrow{y}_{track-total}\Big{\|}^{2}\bigg{)} \tag{112}\]  
From (106), (107) and (112) and the input-to-state stable theorem [38], we can conclude that the first almost disturbance decoupling requirement is achieved. Next, we achieve the first almost disturbance decoupling performance. Combining (102), (103), (112) and (125) results in  
\[\hat{J}_{lin+non}\leq-\frac{T\cdot T_{2}}{\Delta_{\text{max}}}J_{lin+non}+T_{1} \tag{113}\]  
Applying the comparison theorem to (113) yields  
\[J_{lin+non}(t)\leq J_{lin+non}(t_{0})e^{-\frac{T\cdot T_{2}}{\Delta_{\text{ max}}}(t-t_{0})}+\frac{\Delta_{\text{max}}T_{1}}{T\cdot T_{2}},t\geq t_{0} \tag{114}\]  
Then, we can obtain  
\[\Big{|}y_{o1}(t)-y_{track}^{1}(t)\Big{|}\leq\sqrt{\frac{2J_{lin+non}(t_{0})}{ k\gamma_{\text{min}}^{*}}}e^{-\frac{T\cdot T_{2}}{\Delta_{\text{max}}}(t-t_{0})}+ \sqrt{\frac{2\Delta_{\text{max}}T_{1}}{k\gamma_{\text{min}}^{*}T\cdot T_{2}}} \tag{115}\]  
and  
\[ \begin{array}{l}\big{|}y_{oi}(t)-y_{track}^{1}(t)\big{|}\leq\sqrt{\frac{2J_ {lin+non}(t_{0})}{k\gamma_{\text{min}}^{*}}}e^{-\frac{T\cdot T_{2}}{2\Delta_{ \text{max}}}(t-t_{0})}+\sqrt{\frac{2\Delta_{\text{max}}T_{1}}{k\gamma_{\text{ min}}^{*}T\cdot T_{2}}},\\ 2\leq i\leq 4\end{array}  \tag{116}\]  
Observing (116) verifies the second almost disturbance decoupling requirement and the convergence rate of tracking errors is \(T\cdot T_{2}/\Delta_{\text{max}}\). It is obvious to see that we can change the value of \(T\cdot T_{2}\) to adjust the convergence rate. Moreover, from (102), (103) and (125), we obtain  
\[\hat{J}_{lin+non}\leq-T\cdot T_{2}\bigg{(}\Big{\|}\overrightarrow{y}_{track- total}\Big{\|}^{2}\bigg{)}+T_{1} \tag{117}\]Considering the output trajectory for \(\left\|\overrightarrow{y}_{track-total}\right\|>\pounds\pounds\pounds\equiv \sqrt{\frac{T_{1}}{T\cdot T_{2}}}\) yields \(\dot{J}_{lin+non}<0\), and then any sphere  
\[B_{L}\equiv\left\{ \begin{bmatrix}\overline{\dot{e}}_{track}\\ \overrightarrow{\eta}_{non}\end{bmatrix} :\left\|\overline{e}_{track}\right\|^ {2}+\left\|\overrightarrow{\eta}_{non}\right\|^{2}\leq\pounds\right\} \tag{118}\]  
will be a global final attractor of the nonlinear FLMA system with the convergent radius \(\pounds\equiv\sqrt{\frac{T_{1}}{T\cdot T_{2}}}\). Next, we verify the globally exponential stability of the nonlinear FLMA system. Combining (112) and (114) yields  
\[J_{lin+non}(t)\leq J_{lin+non}(t_{0})e^{-\frac{T\cdot T_{2}}{\Delta\max}\left(t -t_{0}\right)}+\frac{\Delta_{\max}T_{1}}{T\cdot T_{2}},t\geq t_{0} \tag{119}\]  
and  
\[