# Comparative Real-Time Study of Three Enhanced Control Strategies Applied to Dynamic Process Systems
## 5 Experimental Results
### Results of Tank 2  
Finally, we applied the same step trajectory to Tank 2 to showcase the performance of the proposed controller. In this setup, Tank 1's controller calculates the reference level, and Tank 2's controller is what determines the pump voltage command to ensure that the actual liquid level follows the given reference. As shown in Figures 14-16, the proposed FO-APIDSMC controller provides a rapid response with minimal overshoot and precise trajectory tracking compared to the FF-PI and CNPIR controller. Similarly, real-time results indicate that the proposed controller performs well for the Tank 2 configuration. Table 6 presents the optimal controller parameters for Tank 2, determined through experimentation. Figure 13: Square wave responses during disturbance for Tank 1 configuration. \begin{table}
\begin{tabular}{c c c c} \hline  & **FF-PI** & **CNPIR** & **FO-APIDSMC** \\ \hline
**ISE** & 795.2 & 675.3 & 532.1 \\ \hline \end{tabular}
\end{table} 
Table 5: ISE values with disturbance according to controllers for Tank 1. Figure 14: Control performances of the FF-PI, CNPIR, and FO-APIDSMC controllers for Tank 2. Figure 15: Tracking error for Tank 2 configuration. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**FF-PI Parameters**} \\ \hline \(\text{K}_{\text{p}_{2}}=5.29\) & \(\text{K}_{\text{i2}}=1.68\) & \(\text{K}_{\text{ff2}}=1\) \\ \hline \multicolumn{3}{c}{**CNPIR control parameters**} \\ \hline
**Square Wave** G = 1.12 & \(\text{h}=0.11\)\(\text{K}_{\text{p}}=10\) & \(\text{K}_{\text{i}}=0.32\)\(\text{K}_{\text{ir}}=0.61\) \\ \hline \multicolumn{3}{c}{**FO-APIDSMC Control Parameters**} \\ \hline \(k_{p2}=8.42\), \(k_{i2}=6.36\), \(k_{\text{d2}}=1.40\), \(\rho_{2}=21\), \(\gamma_{2}=18.2\), \(\delta=0.02\), \(a_{1}=0.37\) \\ \hline \end{tabular}
\end{table} 
Table 6: Controller parameters for Tank 2. As can be seen in Figures 14 and 15, the proposed FO-APIDSMC method exhibited better performance than FF-PI and CNPIR controllers. The percentage overshoot value (Mp) was 0.11% in FO-APIDSMC and 0.47% in CNPIR, while it was 2.65% in FF-PI. The settling time (Ts) was 24.4 s in FO-APIDSMC and 27.4 s in CNPIR, while it was 32.98 s in FF-PI. To demonstrate the effectiveness of the proposed FO-APIDSMC controller compared to the CNPIR and FF-PI controllers, the integral square error (ISE) values, as formulated in Equation (62), were calculated for Tank 2. The ISE error values of the graph obtained in Figure 15 are provided in Table 7. As can be seen from Table 7, lower ISE values are obtained in FO-APIDSMC due to the robust control signal of the FO-APIDSMC controller shown in Figure 16. In the provided Figure 16, the control signals for Tank 2 under three different control strategies, FF-PI, CNPIR, and the proposed FO-APIDSMC, are shown over time. The FO-APIDSMC control demonstrates a significantly robust response, evidenced by smoother control signal transitions and better disturbance rejection compared to FF-PI and CNPIR. The FO-APIDSMC maintains more stable control inputs, leading to a lower Integral Squared Error (ISE), which reflects its superior performance in reducing oscillations and ensuring precise control of the system's dynamics. The signal output of FO-APIDSMC displays less variance and sharper correction when compared to the others, making it a more efficient choice for controlling nonlinear systems such as the coupled tank system. This is particularly important in minimizing energy consumption and wear on actuators, further highlighting the advantages of the proposed method. responses, its performance fell short when compared to the more advanced CNPIR and FO-APIDSMC techniques, particularly in dynamic scenarios. The FO-APIDSMC controller, which was introduced as a more advanced option, used fractional calculus and adaptive sliding mode control to perform a better job than both the CNPIR and FF-PI controllers. This method provided chattering-free control, finite-time convergence, and significantly increased trajectory tracking accuracy.