## 5 TCLab Results
### PID Controller Tuning
In this case, the gains converged to the following values: \(K_{p}=13.83\), \(T_{i}=39.54\) s, and \(T_{d}=7.14\) s. Figure 9 presents the SPT and LDR responses in the top plot and the control signal in the lower plot. As it can be observed, the system tracks well with no overshoot and with an acceptable load disturbance rejection. Figure 10 shows the best PID gains evolution over 70 iterations. It is possible to observe from Figure 10 that the parameters converged to values are very close to the final ones by iteration 40. Figure 11 presents the evolution obtained with the best swarm cost value, IAE, and the average of the entire swarm particles cost function values. As observed, there is a convergence between the average  
Figure 8: PSO evolution of the best integral absolute error (IAE) values and average swarm IAE values corresponding to four different inertia weight variation settings. Figure 9: TCLab set-point tracking (SPT) and disturbance rejection (DR) simulated responses with a PID controller optimized with the PSO algorithm (J = IAE). and the best values in between iterations 40 and 50. These results clearly indicate that 70 iterations are an adequate stopping criterion, in this case. After comparing the relative simulated values obtained for IAE and TV, the following cases were considered:  
* I: \(\alpha=1\) and \(\beta=0\). * II: \(\alpha=1\) and \(\beta=70\). * III: \(\alpha=1\) and \(\beta=80\). * IV: \(\alpha=1\) and \(\beta=100\). Notice that this selection was done based on several tests performed on the TCLab. For cases with an additive compromise function, more relevance was given to TV over IAE. Before presenting the TCLab PSO results, a performance comparison with another swarm optimization algorithm, the original Grey Wolf Optimization (GWO) [18], is presented. The GWO was compared with the classical PSO in a benchmark test suit when it was proposed [18]. The GWO swarm size, number of iterations, and PID controller gains interval are the same as the ones used for the PSO. The GWO algorithm uses a heuristic parameter, represented by _a_[18] responsible for the tradeoff swarm exploration and exploitation. The GWO parameter \(a\) is a linearly decreasing vector from \(a_{max}=2\) to \(a_{min}=0\) through the evolutionary process. The simulation results are presented in Table 2 for cases I and II. The best controller gains from an 11 run set converged to very similar IAE and TV values. This indicates that  
Figure 11: PSO evolution of the best IAE value and average swarm IAE values corresponding to the test in Figures 9 and 10. Figure 10: PSO evolution of the PID controller gains resulting in the final values used in the system response of Figure 9.  
with the appropriate settings the classical PSO performs, as well as the classical GWO for this control case study. More information regarding a comparison between PSO, GWO, and GSA for the PID control is reported in [19]. A compilation of the results with the classic tuning rules and PSO optimization is presented in Table 3. The best PSO gains presented in Table 2 are replicated in Table 3 for comparison. Note also that as the TCLab tests were performed in different day periods, the initial starting ambient temperature varies slightly from test to test. To see if this influenced the results analysis, the IAE and TV values considered only the period [95-620 s] (see Table 4). This interval starting point corresponds to 5 s prior to the second step change at t = 100 s where all the temperature steady-state values are approximately equal to 25 \({}^{\circ}\)C.  
The results obtained with the AMIGO PID settings: _Kp_ = 4.69, _Ti_ = 72.94 \(s\), and _Td_ = 9.48 s are presented in Figure 12. As it can be observed, the response presents a pronounced overshoot and worse load rejection disturbance compared with the CC response. The TV value obtained with the AMIGO settings is clearly better than the one obtained with CC (see Tables 3 and 4).