## 5 TCLab Results
### PID Controller Tuning
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method** & _IAE_ & _TV_ \\ \hline CC & 2300.9 & 2647.9 \\ AMIGO & 2953.5 & 1860.6 \\ PSO-I & **2235.8** & 3594.1 \\ PSO-II & 2259.9 & 2102.6 \\ PSO-III & 2351.1 & 2065.8 \\ PSO-IV & 2442.0 & **1788.8** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: TCLab physical results obtained for the several methods considering the time interval [95–620 s]. The best results are represented in bold. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & _Kp_ & _Ti (s)_ & _Td (s)_ & _IAE_ & _TV_ \\ \hline PSO-I & 13.83 & 39.54 & 7.14 & 4442.0 & **2843.8** \\ GWO-I & 13.82 & 39.49 & 7.13 & **4441.8** & **2843.8** \\ PSO-II & 9.64 & 49.29 & 5.70 & **3831.4** & 2444.7 \\ GWO-II & 10.40 & 47.55 & 6.08 & 3834.4 & **2429.0** \\ PSO-III & 10.36 & 47.70 & 6.05 & 4438.5 & **2921.4** \\ GWO-III & 10.23 & 47.97 & 5.97 & **4437.5** & 2924.8 \\ PSO-IV & 9.53 & 49.69 & 5.65 & **4430.7** & 2936.6 \\ GWO-IV & 10.17 & 48.14 & 5.93 & 4436.8 & **2927.0** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: TCLab simulation results obtained with the PSO and grey wolf optimization (GWO) algorithms. The best results are represented in bold. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & _Kp_ & _Ti (s)_ & _Td (s)_ & _IAE_ & _TV_ \\ \hline CC & 13.47 & 45.99 & 7.00 & 2389.4 & 2962.2 \\ AMIGO & 4.69 & 72.96 & 9.48 & 3004.9 & 1972.3 \\ PSO-I & 13.83 & 39.54 & 7.14 & **2273.6** & 3600.4 \\ PSO-II & 9.64 & 49.29 & 5.70 & 2298.5 & 2216.7 \\ PSO-III & 10.36 & 47.70 & 6.05 & 2392.5 & 2164.8 \\ PSO-IV & 9.53 & 49.69 & 5.65 & 2482.7 & **1888.4** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: TCLab physical results obtained for the several methods considering the entire test time. The best results are represented in bold. The results obtained for PSO with just the ITAE minimization criterion (PSO-I) are presented in Figure 13. As it can be seen from Tables 3 and 4, only minimizing the IAE results in the best possible IAE value and worse TV than with the CC and AMIGO methods. The results obtained by using cost functions (8) with \(\alpha=1\) and \(\beta=70\) (PSO-II), \(\alpha=1\) and \(\beta=80\) (PSO-III), \(\alpha=1\) and \(\beta=100\) (PSO-IV), are presented respectively in Figures 14-16  
Observing the values obtained with the four PSO cases, the IAE value increases as expected. With the increase of the TV factor, \(\beta\) in the cost function, the corresponding TV value decreases. Depending on the TV requirements, it is possible to obtain better results for both IAE and TV compared with the CC and AMIGO tuning rules, as expected. Figure 17 presents the overlap responses obtained with the TCLab regarding the following methods: CC, AMIGO, and PSO-IV. As it can be observed from the tracking response in the top plot, CC gains perform slightly better than the PSO-IV gains and these two are much better than AMIGO. However, regarding the control signals in the bottom plot, the optimized PSO gains result in a less aggressive and less irregular control signal variation than CC and AMIGO. The results also consider an output disturbance applied at instant \(t\) = 600 s, as presented in Figure 18. In this case, the disturbance was caused by blowing with a straw near the transistor 1 for 5 s, to decrease the temperature. As it can be observed, the response also tracks well from the output disturbance. Figure 16: Simulated and TCLab responses using PSO, case IV (J = IAE + 100 TV). Figure 15: Simulated and TCLab responses using PSO, case III (J = IAE + 80 TV).