## 5 Comparisons to Traditional Approaches  
We make some comparisons between the new feedback linearization approach and the traditional singular perturbation method that pioneered the almost disturbance decoupling issue [23,24] in this section. The impractical shortcoming of the traditional singular perturbation method requires to meet the sufficient condition that the system dynamics multiplied by the external disturbance should satisfy the annoying \"structural triangle condition\" for the almost disturbance decoupling issue. The pioneering work carried out by [23,24] points out the fact that the following nonlinear control system cannot well address the almost disturbance decoupling issue:  
\[ \begin{bmatrix}\dot{x}_{se\_1}(t)\\ \dot{x}_{se\_2}(t)\end{bmatrix} = \begin{bmatrix}\tan^{-1}(x_{se\_2})\\ 0\end{bmatrix} + \begin{bmatrix}0\\ 1\end{bmatrix} u_{ip}+ \begin{bmatrix}1\\ 0\end{bmatrix} \Omega_{n} \tag{177}\]  
\[u_{op\_1}=x_{se\_1},\Omega_{n}(t)=0.8\sin 2t \tag{178}\]  
From (177) and (178), we can apply Lie differentiation to derive the following results: \(L_{f}^{0}u_{op\_1}=u_{op\_1}=x_{se\_1}\), \(L_{S_{\text{sip}}}L_{f}^{0}u_{op\_1}=0\), \(L_{f}^{1}u_{op\_1}=\tan^{-1}(x_{se\_2})\), \(L_{S_{\text{sip}}}L_{f}^{1}u_{op\_1}=\frac{1}{1+x_{se\_2}^{2}}\) and  
\[\widetilde{g}_{\text{uip}}\equiv \begin{bmatrix}0\\ \frac{1}{\frac{1}{\widetilde{s}_{\text{sip}}}\frac{1}{\widetilde{f}_{\text{sip }}}u_{op\_1}}\end{bmatrix} = \begin{bmatrix}0\\ 1+x_{se\_2}^{2}\end{bmatrix}  \tag{179}\]  
Since \(\widetilde{g}_{\text{uip}}\) is not a complete distribution, the critical condition of [23,24] is not well addressed. In contrast, the following proposed feedback linearization control can well solve the almost disturbance decoupling issue:  
\[u_{ip}=\left(x_{se\_2}^{2}+1\right)[(\cos t)-256(x_{se\_1}-(-\cos t))-16(\tan^ {-1}x_{se\_2}-(\sin t))] \tag{180}\]  
The output state trajectory of the above investigated control system with the proposed feedback linearization control described by (180) is shown in Figure 8. Based on the observation of Figure 8, the proposed robust control can indeed drive the output state trajectory to track the desired signal \(-\cos t\). Figure 7: Output state values after entering the convergent radius. To show the superiority of the proposed feedback linearization control, we compare the convergence rate performance with traditional PID control [34] shown by (181)  
\[\overset{\rightarrow}{u}_{PID}=K_{P} \begin{bmatrix}y_{o1}-y_{od}^{1}\\ y_{o2}-y_{od}^{2}\\ y_{o3}-y_{od}^{3}\\ y_{o4}-y_{od}^{4}\end{bmatrix} +K_{I} \begin{bmatrix}\int\left(y_{o1}-y_{od}^{1} \right)dt\\ \int\left(y_{o2}-y_{od}^{2}\right)dt\\ \int\left(y_{o3}-y_{od}^{3}\right)dt\\ \int\left(y_{o4}-y_{od}^{4}\right)dt\end{bmatrix} +K_{D} \begin{bmatrix}\frac{d}{ dt}\left(y_{o1}-y_{od}^{1}\right)\\ \frac{d}{dt}\left(y_{o2}-y_{od}^{2}\right)\\ \frac{d}{dt}\left(y_{o3}-y_{od}^{3}\right)\\ \frac{d}{dt}\left(y_{o4}-y_{od}^{4}\right)\end{bmatrix}  \tag{181}\]  
Next, we compare the proposed feedback linearization approach with the traditional PID control. In what follows, manual adjusting of the traditional PID control for the FLHBR system is shown. The manual adjusting of the related KP, KI, KD gains is executed by trial and error. We first set the related KP, KI, KD gains to be zero and then increase the proportional gain KP until the output of the loop is motivated. This is followed by the adjustment for the integral gain KI to optimize the output tracking error response. Finally, the differential gain KD is adjusted, together with the optimized KP, KI gains until a desired output tracking error response is achieved. Output tracking errors responses for pre-specified outputs x1 to x5 are shown in Figure 9.  
Comparing Figure 9 with Figure 3 proves the fact that the convergence rate with our proposed feedback linearization controller is better than the conventional PID control. From  
Figure 8: The system output trajectory for (177). Figure 9: The output trajectories by using PID controller.