# Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models
## 5 Simulation Examples
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Alg. 1** & **SIMC** & **PO PI** \\ \hline \(K_{p}\) & 0.78 & 0.91 & 0.85 \\ \(T_{i}\) & 5.35 & 7.04 & 6.06 \\ \(\text{IAE}_{pv}\) & 3.62 & 3.35 & 3.48 \\ \(\text{IAE}_{eu}\) & 6.83 & 7.74 & 7.14 \\ \(J\) & 1.41 & 1.41 & 1.39 \\ TV & 3.77 & 3.77 & 3.77 \\ GM & 6.74 & 6.13 & 6.39 \\ PM & 43.63 & 46.74 & 45.19 \\ DM & 1.54 & 1.49 & 1.51 \\ \(M_{s}\) & 1.59 & 1.59 & 1.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Example 4. The table shows the PI controller parameters, indices and margins are given for prescribed robustness \(M_{s}=1.59\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(\bar{c}=2.5\) & \(\bar{c}=2.7\) & **SIMC** \\ \hline \(V_{M}\)/e-3 & 0.7 & 0.3 & 6.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Example 4. The table shows the comparison of the different settings for Algorithm 1 and SIMC using the main performance \(V_{M}\) (Equation (42)). Figure 8: Example 4. Consider PI control of the higher order process model (Equation (47)). The figure illustrates the trade-off curves with the Pareto performance objective \(J\) (Equation (41)) and robustness \(M_{\text{s}}\) (Equation (10)). It shows the MP parameter settings \(\tilde{c}=2.5\) and \(\tilde{c}=2.7\) for Algorithm 1 proposed in Section 4. SIMC is added for comparison. Figure 9: Example 4. Consider PI control of the higher order process model (Equation (47)). The figure illustrates the time-domain responses, given a prescribed robustness \(M_{\text{s}}=1.59\), of the following methods: the PO PI controller vs. SIMC with closed loop time constant \(T_{c}=1.33\,\tau\), and PRC + Algorithm 1 where the MP parameter setting \(\varepsilon=2.7\) (proposed in Section 4) and RTDE \(\delta=1.63\). An output disturbance unit step is presented at time \(t=0\) and an input disturbance unit step at time \(t=35\). Note that the half-rule technique in SIMC is not compatible with process models containing complex poles/underdamped dynamics, hence, in such cases, we consider arguably the same algorithm as SIMC, i.e., Algorithm 1, where the MP parameter, \(\bar{c}=4\). An example of this is given in the following. **Example 5** (Underdamped system). : _An unmanned submersible vehicle studied in [21] is described partly by_  
\[H_{p}(s) = \frac{-2.6158(2.299s+1)}{(0.8131s+1)(0.5s+1)((7.692s)^{2}+1.738(7.6 92s)+1)}, \tag{48}\]  
_i.e., from commanded elevator deflection \(u\) to the pitch angle of the vehicle \(y\). We approximate Equation (48) by an IPTD model with gain velocity \(k=R_{1}=-0.145\) and time delay \(\tau=L=1.729\)._  
_The Pareto performance objective \(J\) vs. \(M_{s}\) trade-off curves are illustrated in Figure 10. In terms of the main performance objective \(V_{M}\), we show in Table 11 that \(\bar{c}=2.5\) is \(\frac{V_{s,\bar{c}}}{V_{s,\bar{c}-2.5}}=7.3\) times better than \(\bar{c}=4\) and \(\frac{V_{s,\bar{c}-2.7}}{V_{s,\bar{c}-2.5}}=1.3\) times better than \(\bar{c}=2.7\). Notice that \(\bar{c}=2.5\) is closest to optimal on the most robust part of the \(M_{s}\)-interval. Furthermore, \(\bar{c}=4\) is seen crossing both \(\bar{c}=2.5\) and \(\bar{c}=2.7\) around \(M_{s}=1.55\) and is closest to optimal on the less robust part._  
_The time-domain responses, given a prescribed robustness \(M_{s}=1.59\), are illustrated in Figure 11. The corresponding PI controller parameters, indices and margins are given in Table 12. The margins for the controllers are all acceptable, i.e., GM \(>2\) and PM \(>30\) as in [14]._  
Figure 10: Example 5. Consider PI control of the higher order underdamped process model (Equation (48)). The figure shows the trade-off curves with the Pareto performance objective \(J\) (Equation (41)) and robustness \(M_{s}\) (Equation (10)). It illustrates the PO PI controllers and PRC + Algorithm 1 variants with MP parameter settings \(\bar{c}=4.0\), \(\bar{c}=2.5\) and \(\bar{c}=2.7\).