# 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 \(K_{p}\) & 1.17 & 1.25 & 1.12 \\ \(T_{i}\) & 22.55 & 33.60 & 19.47 \\ \(\mathrm{IAE}_{py}\) & 15.13 & 13.53 & 15.70 \\ \(\mathrm{IAE}_{uu}\) & 17.73 & 25.16 & 15.89 \\ \(J\) & 1.39 & 1.52 & 1.37 \\ TV & 3.94 & 3.70 & 4.05 \\ GM & 3.36 & 3.22 & 3.46 \\ PM & 50.49 & 56.21 & 47.83 \\ DM & 7.51 & 8.08 & 7.26 \\ \(M_{s}\) & 1.59 & 1.59 & 1.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Example 3. The table shows the PI controller parameter, indices and margins are given for prescribed robustness \(M_{s}=1.59\). Figure 6: Example 3. Consider PI control of the FOPTD process model, \(H_{p}(s)=K\frac{\varepsilon^{-n}}{Ts+1}\), where \(K=5.7\), \(\tau=4\) and \(T=60\). The figure shows the trade-off curves with the Pareto performance objective \(J\) (Equation (41)) and robustness \(M_{s}\) (Equation (10)). It illustrates the MP parameters \(\tilde{c}=2.5\) and \(\tilde{c}=2.7\) for Algorithm 1 (proposed in Section 4). SIMC with set-point time constant \(T_{c}\) is added for comparison. Figure 7: Example 3. Consider PI control of the FOPTD process model, \(H_{p}(s)=K\frac{\varepsilon^{-n}}{Ts+1}\), where \(K=5.7\), \(\tau=4\) and \(T=60\). The figure illustrates the time-domain responses, given a prescribed robustness \(M_{s}=1.59\), of the following methods: the PO PI, SIMC with prescribed closed loop time constant \(T_{c}=1.10\,\tau\), and Algorithm 1 where the MP parameter \(\tilde{c}=2.5\) (proposed in Section 4) and RTDE \(\delta=1.56\). An output disturbance unit step is presented at time \(t=0\) and an input disturbance unit step at time \(t=140\). Some results regarding a couple of motivated higher order processes are presented in the following examples. Notice that SIMC offers the half-rule model reduction technique. However, for our case, we approximate the higher order systems by identifying two parameters, the unit reaction rate \(R_{1}\) and the lag \(L\) from a step response, i.e., the Process-Reaction Curve (PRC) as presented in the work of [PERSON] [9; 10; 11]. We denote the variant as follows: PRC + Algorithm 1. **Example 4** (Higher order process). : _A distillation column studied in [20] (p. 591) is partly described by the following process model,_  
\[H_{p}(s)=\frac{34}{(54s+1)(0.5s+1)^{2}}. \tag{47}\]  
_By identifying the lag \(L\) and the maximum slope (unit reaction rate) \(R_{1}\) from the PRC method we may approximate the process model as an IPTD model with gain velocity \(k=R_{1}=0.597\) and time delay \(\tau=L=0.923\)._  
_Using the half-rule technique in SIMC, we approximate a FOPTD model where the gain \(K=34\), time constant \(T=54+\frac{0.5}{2}=54.25\), and time delay \(\tau=0.5+\frac{0.5}{2}=0.75\)._  
_The Pareto Performance objective \(J\) vs. robustness \(M_{s}\) trade-off curves are illustrated in Figure 8. Notice, that \(\bar{c}=2.7\) is the closest to optimal on the most robust part of the \(M_{s}\)-interval. SIMC is crossing \(\bar{c}=2.7\) around \(M_{s}=1.64\) and is the closest to optimal on the less robust part. In terms of the main performance objective \(V_{M}\), we show in Table 9 that \(\bar{c}=2.7\) is \(\frac{V_{\bar{c}=2.5}}{V_{\bar{c}=2.7}}=2.3\) times better than \(\bar{c}=2.5\), and \(\frac{V_{\text{SIMC}}}{V_{\bar{c}=2.7}}=20.7\) times better than SIMC._  
_The time-domain responses, for a prescribed robustness \(M_{s}=1.59\), are illustrated in Figure 9. The corresponding PI controller parameters, indices and margins are given in Table 10. The margins for the controllers are all acceptable, i.e., GM \(>2\) and PM \(>30\), as in [14]. Notice, that the prescribed MTDE, \(d\tau_{max}=\delta\tau=1.50\) is almost equal to the exact \(DM=1.54\)._