# Nonlinear Tank-Level Control Using Dahlin Algorithm Design and PID Control
## 5 Conclusions  
In this paper, the Dahlin algorithm was used for the level control in single tank and coupled tank systems for the first time. Both systems are nonlinear but were effectively linearized with the proposed procedures. A simple modification of the Dahlin controller resulted in a \"non-ringing\" control signal and a stable, error-free, aperiodic response of the desired dynamics. Additionally, the lower limit of the decrease in the desired time constant, which ensures the desired behavior of the system, is determined. A comparison of the proposed controller with the PID controllers, which are usually applied, are represented through transient responses and performance indexes. The provided MATLAB Simulink simulations indicate that the proposed modified Dahlin controller has more advantages than the conventional PID controllers.  
In the future, special attention will be paid to the application of the experimental validation of the obtained results and the application of the metaheuristic algorithms to estimate optimal PID controller parameters. Furthermore, attention will be paid to the testing of the control system robustness in terms of tank dimension.  
Conceptualization, T.D., S.A. and M.C.; methodology, T.D., S.A., M.C. and A.M.; software, S.A. and A.M.; validation, S.A., M.C. and A.M.; formal analysis, T.D. and S.A., investigation, T.D., S.A. and N.M.; resources, S.A., A.M. and N.M.; data curation, T.D., S.A. and N.M.; writing--original draft preparation, T.D., S.A. and N.M.; writing--review and editing, M.C. and A.M.; visualization, T.D., S.A. and N.M.; supervision, M.C. and A.M.; project administration, T.D., S.A. and A.M. All authors have read and agreed to the published version of the manuscript.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Trial and Error** & **Auto-Tuning** & **Ziegler–Nichols** & **Cohen–Coon** & **Modified Dahlin** \\  & [7] & [7] & [7] & [7] & **1/\(\Lambda\) = 4 s** \\ \hline Rise time & 24 s & 9.14 s & 3.29 s & 2.81 s & 5 s \\ \hline Overshoot & 6.86\% & 1.81\% & 38.5\% & 33.7\% & 0\% \\ \hline Setting time & 84.4 s & 53.3 s & 32.1 s & 23.59 s & 7 s \\ \hline Steady-state error & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 16: Performance of the transient response for the CT system reported in [7].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Methods** & **Trial and Error** & **Auto–Tuning** & **Ziegler–Nichols** & **Cohen–Coon** & **Modified Dahlin** \\  & **and Error** & **Tuning** & **Nichols** & **Coon** & **1/\(\Lambda\) = 4 s** \\ \hline IE & 14.24 & 9.246 & 0.4068 & 0.4198 & 5.009 \\ \hline IAE & 16.95 & 9.272 & 6.036 & 4.28 & 5.009 \\ \hline ISE & 10.21 & 4.783 & 2.371 & 1/836 & 2.787 \\ \hline ITAE & 259.5 & 188.4 & 58.77 & 26.58 & 22.46 \\ \hline ITASE & 81.45 & 188.4 & \(-7.552\) & \(-6.886\) & 22.46 \\ \hline \hline \end{tabular}
\end{table}
Table 17: Performance evaluation determined from [7].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Methods** & **Ziegler–Nichols** & **Ciancone** & **Pole** & **Modified Dahlin** \\  & **Nichols** & **Correlation** & **Placement** & **1/\(\Lambda\) = 4 s** \\ \hline IE & 0.4115 & 1.027 & 0.4274 & 5.017 \\ \hline IAE & 32.87 & 78.39 & 153.6 & 5.017 \\ \hline ISE & 13.69 & 31.27 & 60.7 & 2.789 \\ \hline ITAE & 1597 & 9507 & \(3.538\times 10^{4}\) & 22.69 \\ \hline ITASE & \(-27.94\) & \(-71.95\) & 38.93 & 22.69 \\ \hline \hline \end{tabular}
\end{table}
Table 18: Performance evaluation determined from [6].