# Nonlinear Tank-Level Control Using Dahlin Algorithm Design and PID Control
## 4 Discussion  
This paper presents the level control of the ST and CT. For this purpose, it was necessary to design different controllers. In particular, the Dahlin controller and standard PID controllers were used. Both systems are nonlinear. In order to design the controllers, it was necessary to linearize them. As a result, different approaches to model linearization were applied. In the case of a ST system, the authors suggest that linearization should be performed based on the obtained system response curve. The gain factor, the time constant, and the transport delay can be easily determined from the recorded curve. In the case of higher-order systems, we propose linearization by development into the Taylor series. The Dahlin algorithm applied for both tank systems control provides very good results. The application of the Dahlin controller enables the aperiodic response of the system with \begin{table}
\begin{tabular}{c c c} \hline \hline  & Ziegler–Nichols Method \\ \hline \(K_{P}\) & \(K_{I}\) & \(K_{D}\) \\ \hline
121 & 0 & 0 \\
108.9 & 3.4389 & 0 \\
145.2 & 7.6421 & 689.7 \\ \hline \multicolumn{3}{c}{Takahashi tuning method} \\ \hline \(K_{P}\) & \(K_{I}\) & \(K_{D}\) \\ \hline
121 & 0 & 0 \\
107.1805 & 3.4389 & \\
141.3789 & 7.6421 & 689.7 \\ \hline \multicolumn{3}{c}{Auto-tuning method} \\ \hline \(K_{P}\) & \(K_{I}\) & \(K_{D}\) \\ \hline C & 110.452 & 1.479 & 1896.5737 \\ D & 35.2523 & 0.30011 & 804.0765 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: PID controller parameter selection for the CT system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Controller**}} & \multicolumn{1}{c}{**Rise Time**} & \multicolumn{1}{c}{**Overshoot**} & \multicolumn{1}{c}{**Setting Time**} & \multicolumn{1}{c}{**Steady-State Error**} \\ \hline \multirow{2}{*}{ Ziegler–Nichols method} & P & 9.34 s & 83.2\% & 567 s & 0 \\ \cline{2-5}  & PID & 7.27 s & 66.5\% & 233 s & 0 \\ \hline \multirow{2}{*}{Takahashi method} & P & 9 s & 91.3\% & 1140 s & 0 \\ \cline{2-5}  & PID & 8 s & 71.7\% & 347 s & 0 \\ \hline \multirow{2}{*}{Auto-tuning method} & C & 5.78 s & 9.69\% & 59.1 s & 0 \\ \cline{2-5}  & D & 16 s & 8.7\% & 107 s & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The transient response parameters with the PID controllers. the desired time constant. The problem with using this controller can be the sudden change in the control signal, which is difficult to follow for most actuators. According to [PERSON], this problem can be effectively overcome by eliminating three poles, namely, a pair of complex conjugate poles and one real negative pole within the unit circle. In this way, a modified Dahlin controller is designed, which provides a very similar response as the Dahlin controller, but a control signal with \"non-ringing\" behavior. In the first-order system, the elimination of the mentioned three poles is relatively simple if the system has a transport delay of about 2 s, i.e., \(N=2\). In this case, the modified Dahlin controller becomes a Dahlin PI controller. However, in the case of a system with the transport delay greater than 2, i.e., \(N\) greater than 2, the elimination of these three poles becomes more difficult. In this case, [PERSON] suggests the design of a Dahlin PI controller [30]. This controller, in the example of a single tank, when \(N=4\), did not provide satisfactory results. The desired aperiodic response was not achieved, but a response with an overshoot as shown in Figure 13, although the sudden changes in the control signal were eliminated as presented in Figure 12. Therefore, the authors propose the elimination of only one pole, namely, the real negative pole, by including \(z=1\) in the factor \((C_{1}+C_{2}z^{-1})\), (19), (31). The proposed modified Dahlin controller provided very good results for a ST system. The response almost matched the Dahlin controller, and the ringing of the control signal was eliminated as shown in Figures 11b and 13.