## 7 Simulation Results
### Comparison with the Classical PID Controller  
Let us consider the development presented in Section 6.2 for the classical PID controller. The objective of this part is to compare the findings obtained from the PID controller with those obtained from the affine state-feedback controller employed in this research. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Indicator** & **Approach 1** & **Approach 2** & **Approach 3** & **Approach 4** & **PID Controller** \\ \hline \(\mathcal{K}^{\text{T}}\) & \(\left[\begin{array}{c}-52.87\\ -14.18\end{array}\right]\) & \(\left[\begin{array}{c}-1.96\\ -1.01\end{array}\right]\) & \(\left[\begin{array}{c}-32.25\\ -4.52\end{array}\right]\) & \(\left[\begin{array}{c}-3.24\\ -1.20\end{array}\right]\) & \(\left[\begin{array}{c}-6.89\\ -7.22\end{array}\right]\) \\ \(\delta\) & 0.80 & 0.90 & 0.81 & 0.84 & 0.81 \\ \(t_{s}\) & 1.8 & 2.8 & 1.0 & 1.4 & 2.5 \\ \(u_{t=0}\) & 37.0 & 1.4 & 22.0 & 2.2 & 4.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance metrics for the affine state-feedback control law and PID controller: feedback gain, disturbance torque, stabilization time, and initial control effort \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Indicator** & **Approach 1** & **Approach 2** & **Approach 3** & **Approach 4** & **PID Controller** \\ \hline MAE\((\phi)\) & 0.0205 & 0.0571 & 0.0138 & 0.0396 & 0.0417 \\ MAE\((\phi)\) & 0.0850 & 0.1093 & 0.1071 & 0.1092 & 0.1520 \\ MSE\((\phi)\) & 0.0073 & 0.0238 & 0.0054 & 0.0172 & 0.0134 \\ MSE\((\phi)\) & 0.0890 & 0.0365 & 0.1633 & 0.0521 & 0.1343 \\ RMSE\((\phi)\) & 0.0855 & 0.1544 & 0.0732 & 0.1310 & 0.1156 \\ RMSE\((\phi)\) & 0.2983 & 0.1912 & 0.4041 & 0.2283 & 0.3665 \\ CEI\((u)\) & 1.3360 & 0.1801 & 1.3460 & 0.2569 & 0.5182 \\ MATI\((u)\) & 36.9124 & 1.3684 & 22.5164 & 2.2610 & 4.8115 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Evaluation of the performance of the adopted affine state-feedback control law for the four approaches and the PID controller in terms of RMSE, MSE, MAE, CEI, and MATI. Then, as discussed at the end of Section 6.2, let us consider the LMI conditions designed in Section 5.2.4, then the LMI-based minimization issue proposed in Theorem 4. By considering the matrices in (113), the same value of the desired position \(\theta_{d}=40^{\circ}\), and the same value of the decay rate \(\zeta\) of the candidate Lyapunov function \(\zeta=2\), we obtain, by considering the PID controller, the following results of the gain \(\mathcal{K}\), the minimal value, \(\epsilon_{\text{min}}\) of the parameter \(\epsilon\), and the maximal value, \(\delta_{\text{max}}\), of the parameter \(\delta\):  
\[\mathcal{K}=\left[ \begin{array}{cc}-6.8920&-1.6166&-7.2216\end{array} \right],\quad\epsilon_{\text{min}}=16.5029,\quad\text{and}\quad\delta_{\text{max}}=0.8 146 \tag{118}\]  
Note that the three gains of the PID controller (104) are equal to \(\mathcal{K}_{p}=-6.8920\), \(\mathcal{K}_{\triangledown}=-1.6166\), and \(\mathcal{K}_{i}=-7.2216\). It is clear that compared to the gain \(\mathcal{K}\) in (117) of the affine state-feedback controller, the gains (the proportional and derivative parts) of the PID control law are slightly larger. Moreover, the maximum bound of the disturbance \(\bar{\xi}_{\text{max}}\) by considering the PID controller is found to be smaller than that obtained with the affine controller adopted in this work. Furthermore, the value of \(\epsilon_{\text{min}}\) is relatively larger than that in (117). The difference between the two values is around 10, which is considerably high. The clear difference in the values of \(\mathcal{K}_{p}\), \(\bar{\xi}_{\text{max}}\), and more obviously \(\epsilon_{\text{min}}\) reveals the superiority of the adopted affine state-feedback control law compared to the classical PID controller, and also compared to the classical PD controller as discussed in Section 6.1.