## 5 Discussion  
\(PI\)-\(D^{\text{AE}}\) demonstrates significant advantages over the conventional \(PID\), particularly in scenarios with varying \(\tau_{D}\) values (Tables 3 and 4). For \(\tau_{D}\) = 0.03, \(PI\)-\(D^{\text{AE}}\) shows marginal improvement; however, its rapid response and steady error reduction are evident. As \(\tau_{D}\) increases to 0.1, \(PI\)-\(D^{\text{AE}}\) achieves better performance across most criteria, except for an 8% higher overshoot (\(M_{p}\)). Crucially, while \(PID\) fails to reach a steady state, \(PI\)-\(D^{\text{AE}}\) ensures stability and error convergence to zero. For higher values of \(\tau_{D}\) (0.2 and 0.4), the PID controllers diverge, whereas \(PI\)-\(D^{\text{AE}}\) remains stable and delivers substantial performance improvements. Error accumulation metrics improve by up to 23%, while \(t_{s}\) shows enhancements of up to 61%. Despite a slight increase in \(t_{r}\), the overall performance indicates faster and more-stable setpoint attainment with \(PI\)-\(D^{\text{AE}}\). Additionally, \(t_{p}\) shows minimal improvement at \(\tau_{D}\) = 0.2 but is notably reduced by 53% at \(\tau_{D}\) = 0.4. Importantly, \(M_{p}\) decreases by up to 11.3% with \(PI\)-\(D^{\text{AE}}\), ensuring better system stability. \(PI\)-\(D^{\text{AE}}\) dynamically adapts to \(D^{\text{AE}}\), effectively addressing issues such as instability and overshoot while accelerating response times. Figures 3-12 confirm the superior robustness and disturbance rejection of \(PI\)-\(D^{\text{AE}}\) compared with \(PID\). Optimizing \(D^{\text{AE}}\) mitigates instability, improves response acceleration, and ensures zero steady-state error, achieving enhanced system performance and reliability. \begin{table}
\begin{tabular}{c c c c} \hline  & **PID** & \(PI\)**-\(D^{\text{AE}}\)** & **Improvement** \\ \hline \(t_{d}\) (s) & 1.876 & 18.800 & \(-\)902.132 \\ \hline \(t_{r}\) (s) & 15.089 & & \\ \hline \(M_{p}\) & 1.019 \(\times\) 10\({}^{-1}\) & 9.807 \(\times\) 10\({}^{-2}\) & 3.759 \\ \hline \(t_{p}\) (s) & 26.193 & 34.588 & \(-\)32.051 \\ \hline \(t_{s}\) 2\% (s) & 43.795 & & \\ \hline \(t_{s}\) 5\% (s) & 7.280 & & \\ \hline \(IAE\) & 0.3216 & 1.99 & \(-\)518.781 \\ \hline \(ISE\) & 0.01672 & 0.1442 & \(-\)762.440 \\ \hline \(ITAE\) & 1.92 & 26.58 & \(-\)1284.375 \\ \hline \(ITSE\) & 0.0225 & 1.284 & \(-\)5606.667 \\ \hline \end{tabular}
\end{table} 
Table 10: Position control with disturbance comparison. \begin{table}
\begin{tabular}{c c c c} \hline  & **PID** & \(PI\)**-\(D^{\text{AE}}\)** & **Improvement** \\ \hline \(t_{d}\) (s) & & & \\ \hline \(t_{r}\) (s) & 1.343 & 25.172 & \(-\)1774.31\% \\ \hline \(M_{p}\) & \(5.176\times 10^{-3}\) & \(3.093\times 10^{-5}\) & 99.40\% \\ \hline \(t_{p}\) (s) & 0.618 & 0.835 & \(-\)35.11\% \\ \hline \(t_{s}\) 2\% (s) [0.002, \(-\)0.002] & 2.617 & & 100.00\% \\ \hline \(t_{s}\) 5\% (s) [0.005, \(-\)0.005] & 0.730 & & 100.00\% \\ \hline \(IAE\) & 0.02108 & 0.002263 & 89.26\% \\ \hline \(ISE\) & \(3.338\times 10^{-5}\) & \(8.439\times 10^{-7}\) & 97.47\% \\ \hline \(ITAE\) & 0.3341 & 0.05867 & 82.43\% \\ \hline \(ITSE\) & 0.000161 & 2.398 \(\times\) 10\({}^{-5}\) & 85.10\% \\ \hline \end{tabular}
\end{table} 
Table 9: Angle control with disturbance comparison. _PI-D_ demonstrates significant advantages over \(PID\), particularly in equilibrium control. With identical \(PID\) tuning, _PI-D\({}^{\text{AE}}\)_ achieves a near-zero steady-state error from the start, while \(PID\) requires multiple oscillations before reaching stability. Notably, _PI-D\({}^{\text{AE}}\)_ effectively neutralizes \(DK\) observed in \(PID\), as shown in Figure 20, maintaining control over the \(\theta\) angle with minimal oscillations and a stable signal. Furthermore, _PI-D\({}^{\text{AE}}\)_'s transient response exhibits unique characteristics. While its \(t_{r}\) is faster, it achieves \(t_{s}\) almost immediately, maintaining control near SP without oscillations. This contrasts with \(PID\), which takes longer to stabilize.