# Robust Multi-Performances Control for Four-Link Manipulator Arm
## 2 Complete Mathematical Model for Four-Link Manipulator Arm
: _We consider nonlinear system \(\stackrel{{\rightharpoonup}}{{X}}=\stackrel{{ \rightharpoonup}}{{f}}(t,\stackrel{{\rightharpoonup}}{{X}}, \stackrel{{\rightharpoonup}}{{\widetilde{u}}}_{input})\) with Lipschitz input \(\stackrel{{\rightharpoonup}}{{\widetilde{u}}}_{input}\), where \(\stackrel{{\rightharpoonup}}{{X}}\) is the Lipschitz state variable and \(\stackrel{{\rightharpoonup}}{{f}}:(0,\infty]\times\Re^{n}\times\Re^{n} \to\Re^{n}\) is differentiable and infinitely continuous. This system is defined to be input-to-state stable if_  
\[\left\|\stackrel{{\rightharpoonup}}{{X}}(t)\right\|\leq\gamma \bigg{(}\left\|\stackrel{{\rightharpoonup}}{{X}}(t_{0})\right\|,t-t_{0} \bigg{)}+\kappa\bigg{(}\sup_{t_{0}\leq\tau\leq t}\left\|\stackrel{{ \rightharpoonup}}{{\widetilde{u}}}_{input}(\tau)\right\|\bigg{)} \tag{91}\]  
_where \(\kappa\) is a \(K\) class function and \(\gamma\) denotes a \(KL\) class function._  
**Definition 2**. : _A nonlinear system with noise input \(\stackrel{{\rightharpoonup}}{{\widetilde{u}}}_{noise}\) is determined to possess almost disturbance decoupling performance if the following properties hold:_  
_(i) The nonlinear system has an input-to-state stable property for noise input \(\stackrel{{\rightharpoonup}}{{\widetilde{u}}}_{noise}\)._  
_(ii) Output tracking errors meet the following two inequalities for initial time \(t_{0}\) and initial state \(\stackrel{{\rightharpoonup}}{{X}}(t_{0})\):_\[\left|y_{oi}(t)-y^{i}_{track}(t)\right|\leq\kappa_{1}\bigg{(}\left\|\overset{ \rightharpoonup}{X}(t_{0})\right\|,t-t_{0}\bigg{)}+\frac{1}{\sqrt{\kappa_{2}} }\kappa_{3}\bigg{(}\sup_{t_{0}\leq\tau\leq i}\left\|\overset{\rightharpoonup}{u _{noise}}(\tau)\right\|\bigg{)} \tag{92}\]  
_and_  
\[\int\limits_{t_{0}}^{t}\left[y_{oi}(\tau)-y^{i}_{track}(\tau)\right]^{2}\!d \tau\leq\left[\kappa_{5}\bigg{(}\left\|\overset{\rightharpoonup}{X}_{o0} \right\|\bigg{)}+\int\limits_{t_{0}}^{t}\kappa_{3}\bigg{(}\left\|\overset{ \rightharpoonup}{u_{noise}}(\tau)\right\|^{2}\bigg{)}\!d\tau\right] \tag{93}\]  
_where \(\kappa_{1}\) belongs to the \(KL\) class function, \(\kappa_{2}\) and \(\kappa_{4}\) are positive constants, and \(\kappa_{3}\), \(\kappa_{5}\) belong to \(K\) class functions. From (65), we obtain_  
\[ \begin{array}{l}\left[\overset{\rightharpoonup}{\xi}_{lin\_1}(t)-y^{i}_{ track}\right]\\ =\left[\overset{\rightharpoonup}{\xi}_{lin\_dai}^{i}-\overset{j}{y}_{track}^{ i(d_{\mathrm{oi}}-1)}\right]\\ =\left[1\right]\!\left(u_{vi}-y^{(d_{\mathrm{oi}})}_{track}\right)+\left[ \sum\limits_{j=1}^{p}\left(\frac{\partial}{\partial X}\underset{f}{L^{d_{ \mathrm{oi}}-1}}\sigma_{oi}\right)\overset{\rightharpoonup}{q}_{j}^{*}\! \left(\theta_{un\_j}+\theta_{noise\_j}\right)\right]\\ d_{\mathrm{oi}}=1,\ i=1,2,3,4\end{array}  \tag{94}\]  
_From (77), (79) and (94), we obtain_  
\[\left[\varepsilon^{1-d_{\mathrm{oi}}}\overset{\rightharpoonup}{e^{i}_{d_{ \mathrm{oi}}}}^{\prime}\right]=\left[1\right]\!\left(u_{vi}-y^{i(d_{\mathrm{oi }})}_{track}\right)+\left[\sum\limits_{j=1}^{p}\left(\frac{\partial}{\partial X }\underset{f}{L^{d_{\mathrm{oi}}-1}}\sigma_{oi}\right)\overset{\rightharpoonup }{q}_{j}^{*}\!\left(\theta_{un\_j}+\theta_{noise\_j}\right)\right] \tag{95}\]  
_Substituting (67) and (68) into (95) yields_  
\[ \begin{array}{l}\varepsilon\!\left[\overset{\rightharpoonup}{e^{i}_{d_{ \mathrm{oi}}}}^{\prime}\right]=\left[-a^{i}_{d_{\mathrm{oi}}}\right]\!\left( \overline{e^{i}_{d_{\mathrm{oi}}}}\right)+\left[\varepsilon\sum\limits_{j=1} ^{p}\left(\frac{\partial}{\partial X}\underset{f}{L^{d_{\mathrm{oi}}-1}} \sigma_{oi}\right)\overset{\rightharpoonup}{q}_{j}^{*}\!\left(\theta_{un\_j}+ \theta_{noise\_j}\right)\right]\\ i=1,\ldots,4\end{array}  \tag{96}\]  
_Then, from (51), (59), (68), (80) and (96), we obtain_  
\[ \begin{array}{l}\overset{\rightharpoonup}{e^{i}_{track}}=A^{i}_{c} \overline{e^{i}_{track}}+\varphi^{i}_{\xi_{lin}}\bigg{(}\overset{\rightharpoonup}{ \theta}_{noise}+\overset{\rightharpoonup}{\theta}_{un}\bigg{)},i=1,\ldots,4 \end{array}  \tag{97}\]  
\[