## 5 Evaluation
### Example 1: Interacting a Two Tank Liquid Level System  
The liquid level system shown in Figure 7 was considered for validating the proposed tool [4]. This is one of the commonly used real-time examples of the chemical process [37]. In a liquid level system, at steady state, the inflow rate and outflow rate are equal to \(\overline{Q}\). The flow rate between the two tanks is also zero and the heads of the tank 1 and tank 2 are equal to \(\overline{H}\). At time \(t=0\), the inflow rate is changed from \(\overline{Q}\) to \(\overline{Q}+q\), where \(q\) is small. The resulting changes in the heads (\(h_{1}\) and \(h_{2}\)) and flow rates (\(q_{1}\)_and_\(q_{2}\)) are assumed to be small as well. The capacitances of tank 1 and tank 2 are \(C_{1}\) and \(C_{2}\) respectively. The resistance of the valve between the tanks is \(R_{1}\) and the outlet valve resistance is \(R_{2}\). From Figure 7, the transfer function between liquid level \(H_{2}\)(s) and inlet flow rate \(Q\)(s) is derived and given in (33). \[\frac{H_{2}(s)}{Q(s)}=\frac{R_{1}C_{1}R_{2}s+R_{2}}{R_{1}C_{1}R_{2}C_{2}s^{2}+( R_{1}C_{1}+R_{2}C_{2}+R_{2}C_{1})s+1}. \tag{33}\]  
In order to understand the effect of the dead time in the process output, five seconds in dead time was introduced in the derived transfer function. Let \(R_{1}=R_{2}=\)1; \(C_{1}=C_{2}=\) 1. Now the resulting transfer function is given in (34):  
\[\frac{H_{2}(s)}{Q(s)}=\frac{(s+1)e^{-5s}}{(s^{2}+3s+1)}. \tag{34}\]  
It is a minimum phase stable system. Let us consider that the level control system is time critical and it is required to control the second tank level \(h_{2}(t)\) by manipulating the inflow rate \(q(t)\). Since it is required to control \(h_{2}(t)\) using the RTD-A controller, the actual system in Equation (34) is reduced into the FOPDT system. The three methods in Tool-I were used to obtain the FOPDT model of the process. The result of each method is presented in Table 3. The accuracy of the reduced model in terms of the integral square error (ISE) performance measure was also compared. It was found that the fraction incomplete method based model was more accurate for this interacting liquid level system. Hence, in this section, the remaining analysis was carried out with the obtained accurate reduced model in (35). \[G(s)=\frac{e^{-4.82s}}{2.69s+1}. \tag{35}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Methods** & **Skogestad** & **Two Point** & **Fraction Incomplete** \\ \hline Reduced Model \(G\)(s) & \(\frac{e^{-4.19s}}{2.809s+1}\) & \(\frac{e^{-4.8s}}{2.68s+1}\) & \(\frac{e^{-4.8s}}{2.69s+1}\) \\ \hline ISE & 0.066 & 0.0518 & 0.0444 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance comparison of order reduction techniques. Figure 7: Two tank liquid-level system [4]. The open loop response of a reduced order model is displayed in Figure 8. The fraction incomplete method-based model response was very much closer to the actual second order process response than other methods [4]. The implementation of the RTD-A controller with a simulation tool on the liquid level control is shown in Figure 9. The performance of the RTD-A controller with two tuning algorithms and the PID controller was examined in controlling the \(h_{2}(t)\) without uncertainties and disturbance. The structure of the PID controller transfer function used is given in (36). The PID tuning parameters proportional gain (\(K_{P}\)), integral gain (\(K_{I}\)), derivative gain (\(K_{D}\)) and filter co-efficient (\(T_{f}\)) used in the simulation were determined using the auto tune function in MATLAB. \[G_{C}(s)=K_{P}+\frac{K_{I}}{s}+\frac{K_{D}s}{T_{f}s+1}. \tag{36}\]  
The tuning parameters for the process with uncertainty and disturbance using two tuning rules are given in Table 4. The sampling time of 0.1 s was considered in the controller implementation. The PID auto tuned values were \(K_{P}=0.496\), \(K_{I}=0.116\), \(K_{D}=-0.857\) and \(T_{f}=0.128\). The setpoint for \(h_{2}(t)\) was considered as one meter.