## 3 Results and Discussions
### Experimental Study
#### 3.2.1 Flow Process Model  
The experimental setup and piping and instrument diagram (P&ID) of the real-time flow process plant are illustrated in Ref. [40]. Within this system, the process tank, VE 420, has a capacity of 100 litres and receives liquid from tank VE 410 with the assistance of a centrifugal pump P412. The liquid level within VE 420 is effectively controlled by a hand valve HV 420, while the process control valve FCV 413 ensures a consistent flow at the designated level. A pressure transmitter FT 413 measures and regulates the flow, providing digital voltage signals within the range of 0 to 5 V.  
These voltage signals are then directed to the pressure-indicating controller FIC 413, which transmits the control signal to the host PC through dedicated I/O interface boards. This integration allows for real-time monitoring and control of the flow process, showcasing a comprehensive and sophisticated system architecture, from the open-loop response of the flow process followed by its numerical analysis in Table 5. The mathematical model derivation for the process plant involves analysing the open-loop step response characteristics. These characteristics encompass essential information about the behaviour and dynamics of the process, including parameters such as the process gain (\(K\)), process dead time (\(L_{p}\)), and process time constant (\(T\)). In this experimentation, an open-loop step response is performed for the flow process, yielding the following plant dynamics: \(K=0.072\), \(L_{p}=3\), and \(T=0.126\). Utilizing the characteristic equation for a first-order plus dead-time system, the process model for the flow process plant is established based on the acquired plant dynamics. The transfer function of the process is given as  
\[G(s)=\frac{K}{1+Ts}e^{-sL_{p}}=\frac{0.072}{0.126s+1}e^{-3s}. \tag{11}\]  
To ensure the system's stability, the plant is operated by controllers, which employ the same methodology as the previous system. However, fractional and complex-order controllers have been added to compare with other controllers. The parameters of these controllers and the step response and transient response characteristics are presented in the table and figure below. These figures provide a comprehensive overview of the control system's performance. Figures 12-14 show the step, set-point tracking, and disturbance rejection responses of all the controllers. The comparison of the traditional controller to the experimental analysis reveals that the PID controller exhibits an overshoot of \(6.1045\times 10^{-5}\%\), which is higher than the PID controller, which achieves zero overshoot. The PID controllers have smaller rise and settling time values, but it does not necessarily mean a better performance, as the overshoot is a crucial factor when evaluating control quality. In the case of fractional controllers, \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Controllers** & \(K_{p}\) & \(K_{i}\) & \(K_{d}\) & \(\lambda\) & \(\mu\) & \(\beta\) & \(\gamma\) & **OS (\%)** & **Settling Time (\(t_{s}\), Seconds)** & **Rise Time (\(t_{r}\), Seconds)** \\ \hline PID & 1 & 1 & 0.1 & - & - & - & - & \(6.1045\times 10^{-5}\) & 237.6946 & 78.3350 \\ PI-PD & 1 & 1 & 0.1 & - & - & - & - & 0 & 240.7698 & 80.6822 \\ FOPID & 1 & 1 & 0.1 & 0.98 & 0.02 & - & - & 1.2636 \(\times 10^{-4}\) & 239.3677 & 79.5915 \\ FOPI-FOPD & 1 & 1 & 0.1 & 0.98 & 0.02 & - & - & 0 & 241.8704 & 81.9465 \\ COPI-PD & 1 & 1 & 0.1 & 0.98 & - & 0.01 & - & 1.2549 \(\times 10^{-4}\) & 239.3861 & 79.6018 \\ PI-COPD & 1 & 1 & 0.1 & - & 0.02 & - & 0.01 & 1.2648 \(\times 10^{-4}\) & 239.3679 & 79.5916 \\ COPI-FOPD & 1 & 1 & 0.1 & 0.98 & 0.02 & 0.01 & - & 2.6519 \(\times 10^{-4}\) & 241.5152 & 82.1655 \\ FOPI-COPD & 1 & 1 & 0.1 & 0.98 & 0.02 & - & 0.01 & 0 & 241.8741 & 81.9517 \\ COPI-COPD & 1 & 1 & 0.1 & 0.98 & 0.02 & 0.01 & 0.01 & 2.3946 \(\times 10^{-4}\) & 241.5194 & 82.1699 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Controller parameters and transient response for real-time flow process plant.