# Robust Multi-Performances Control for Four-Link Manipulator Arm
## 5 Comparisons to Traditional Approaches  
In this section, we compare the performance of the proposed approach with the traditional fuzzy approach [39] and the singular perturbation method with high-gain feedback [36]. Figure 8 shows the general structure of the traditional fuzzy approach whose input variables of the IF-THEN rules are assigned as tracking error \(e(t)\) and its time derivative \(\dot{e}(t)\). The output variable is fuzzy control \(u_{fuzzy}\). To make the calculation easier, the desired membership functions of \(e_{track}(t)\), \(\dot{e}_{track}(t)\) and \(u_{fuzzy}\) are assigned as the triangular-shaped  
Figure 5: The tracking error before entering the global final attractor. Figure 6: The tracking error upon entering the global final attractor. Figure 7: The tracking error after entering the global final attractor. functions, as shown in Figures 9-11. The desired fuzzy control rule base for \(u_{fuzzy}\) is constructed in Table 1. The rule base, fuzzy inference engine and defuzzifier adopt the standard Macvicar-Whelan rule base, the Mamdani method and the centroid method, respectively. With the aid of the Matlab fuzzy toolbox, comparative tracking error responses of the proposed approach and traditional fuzzy controller design for the FLMA are shown in Figures 12-15. From Figures 12-15, it is obvious to see that the convergence rate of the proposed approach is faster than that of the traditional fuzzy approach. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(u_{fuzzy}\)} & \multicolumn{8}{c}{\(\epsilon_{track}(t)\)} \\ \cline{2-9}  & **NB** & **NB** & **NM** & **NS** & **ZE** & **PS** & **PM** & **PB** \\ \hline \multirow{5}{*}{\(\dot{e}_{track}(t)\)} & NB & PB & PB & PB & PM & PS & ZE & NS \\ \cline{2-9}  & NM & PB & PB & PB & PM & PS & ZE & NS \\ \cline{2-9}  & NS & PB & PB & PM & PS & ZE & NS & NM \\ \cline{2-9}  & ZE & PB & PM & PS & ZE & NS & NM & NB \\ \cline{2-9}  & PS & PM & PS & ZE & NS & NM & NB & NB \\ \cline{2-9}  & PM & PS & ZE & NS & NM & NB & NB & NB \\ \cline{2-9}  & PB & ZE & NS & NM & NB & NB & NB & NB \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Macvicar-Whelan fuzzy control rule base. Figure 8: General structure of traditional fuzzy approach. Figure 11: Triangular-shaped membership functions for \(u_{fuzzy}\). Figure 10: Triangular-shaped membership functions for \(\dot{e}(t)\). Figure 9: Triangular-shaped membership functions for \(\dot{e}(t)\). Following the second comparative example, we make some comparisons between the proposed approach and the famous singular perturbation method [35; 36]. The sufficient condition in [35; 36] requires that the nonlinearity multiplied by the disturbance meets the structural triangle criterion. Figure 14: The output tracking error for output 3. Figure 12: The output tracking error for output 1. Figure 13: The output tracking error for output 2. References [35; 36] have exploited the fact that the following system cannot achieve almost disturbance decoupling performance:  
\[ \begin{bmatrix}\dot{x}_{1}(t)\\ \dot{x}_{2}(t)\end{bmatrix} = \begin{bmatrix}\tan^{-1}(x_{2})\\ 0\end{bmatrix} + \begin{bmatrix}0\\ 1\end{bmatrix} u_{fb}+ \begin{bmatrix}1\\ 0\end{bmatrix} w_{noise}(t) \tag{169}\]  
\[y_{o1}=x_{1}\equiv\sigma_{o1},w_{noise}(t)=0.1\sin t \tag{170}\]  
It is easy to derive the following items: \(L_{f}^{0}\sigma_{o1}=\sigma_{o1}=x_{1}\), \(L_{g_{fh}}L_{f}^{0}\sigma_{o1}=0\), \(L_{f}^{1}\sigma_{o1}=\tan^{-1}(x_{2})\), \(L_{g_{fh}}L_{f}^{1}\sigma_{o1}=\frac{1}{1+x_{2}^{2}}\) and  
\[\widetilde{g}_{fb}\equiv \begin{bmatrix}0\\ \frac{1}{L_{g_{fh}}L_{f}^{1}\sigma_{o1}}\end{bmatrix} = \begin{bmatrix}0\\ 1+x_{2}^{2}\end{bmatrix}  \tag{171}\]  
Hence, the sufficient condition of [35; 36] is not satisfied since \(\widetilde{g}\) is not complete, and the almost disturbance decoupling problem cannot be not solved.