# 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 \(\bar{\epsilon}\) & **2.5** & **2.7** & **4** \\ \hline \(V_{M}\)/e-2 & 0.84 & 1.05 & 6.13 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Example 5. The table shows the different MP parameter settings for the PRC + Algorithm 1 variant with corresponding main performance \(V_{M}\) (Equation (42)). Figure 11: Example 5. Consider PI control of the higher order underdamped process model (Equation (48)). The figure illustrates the time-domain responses, given a prescribed robustness \(M_{s}=1.59\), of following methods: the PO PI and the PRC + Algorithm 1 where the MP parameter setting \(\bar{\epsilon}=2.5\) (proposed in Section 4) and RTDE \(\delta=2.20\). An output disturbance unit step is presented at time \(t=0\) and an input disturbance unit step at time \(t=80\). \begin{table}
\begin{tabular}{c c c} \hline \hline  & **PRC + Alg. 1** & **PO PI** \\ \hline \(K_{p}\) & \(-1.42\) & \(-1.70\) \\ \(T_{i}\) & 12.18 & 14.90 \\ \(\text{IAE}_{\text{py}}\) & 6.41 & 5.88 \\ \(\text{IAE}_{\text{xu}}\) & 8.59 & 8.72 \\ \(J\) & 1.04 & 1.00 \\ TV & 4.62 & 5.06 \\ GM & 13.80 & 11.84 \\ PM & 43.90 & 44.20 \\ DM & 3.03 & 2.74 \\ \(M_{s}\) & 1.59 & 1.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Example 5. The corresponding controller parameter, indices and margins are given for prescribed robustness \(M_{s}=1.59\). Last, we propose a tuning variant based on the PRC and Algorithm 1, as above. However, now, the gain velocity in the IPTD model is, instead, varying proportionally, i.e., \(k=R_{1}\zeta\), where \(\zeta\) is considered as a tuning parameter. TO simplify the tuning, we propose to set the RTDE \(\delta=\bar{c}\) equal constant (i.e., an ad hoc suggestion). This means that the only tuning parameter is \(\zeta\). We denote this variant as follows: \(\zeta\)-PRC + Algorithm 1. **Example 6** (\(\zeta\)-PRC variant). : _Consider the same process model as studied in Example 5. The model is approximated by an IPTD model, where the gain velocity is varied, \(k=R_{1}\zeta=-0.145\zeta\), and time delay, \(\tau=L=1.729\). In this example, we set the RTDE \(\delta=\bar{c}=2.7\)._  
_It can be seen in Figure 12 that the PO PI curve and the \(\zeta\)-PRC curve are indistinguishable. This is quite a surprising result. In terms of the main performance objective \(V_{M}\), we show in Table 13 that \(\zeta\)-PRC is \(\frac{V_{\zeta}-\text{PRC}}{\text{PRC}}=4e+3\) times better than PRC variant._  
_The time-domain responses, for a prescribed robustness \(M_{\text{s}}=1.59\), are illustrated in Figure 13. As a consequence of the above, these responses are also indistinguishable. The corresponding PI controller parameters, indices and margins are given in Table 14._  
Figure 12: Example 6. PI control of the higher order underdamped process model (Equation (48)). 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 PO PI controllers with robustness \(M_{\text{s}}\) and the \(\zeta\)-PRC + Algorithm 1 variant where the RTDE \(\delta=\bar{c}=2.7\) is fixed and the main tuning parameter is \(\zeta\). \begin{table}
\begin{tabular}{l c c} \hline \hline
**Variant** & **PRC** & \(\mathbf{\zeta}\)**-PRC** \\ \hline \(V_{M}\)/e-4 & 83.6 & 0.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 13: Example 6. Comparing the following variants, \(\zeta\)-PRC + Algorithm 1 with MP parameter and MTDE settings \(\bar{c}=\delta=2.7\) and \(\zeta=0.74\), and the PRC + Algorithm 1 with MP parameter setting \(\bar{c}=2.5\), using the main performance \(V_{M}\) defined in Equation (42). Figure 13: Example 6. PI control of the higher order underdamped process model (Equation (48)). The figure illustrates the time-domain responses, given a prescribed robustness \(M_{s}=1.59\), for the following methods: the PO PI and the \(\zeta\)-PRC + Algorithm 1 variant with MP parameter and MTDE settings \(\bar{c}=\delta=2.7\), and tuning parameter \(\zeta=0.74\).