# Performance Optimal PI controller Tuning Based on Integrating Plus Time Delay Models
## 5 Simulation Examples  
In the following simulations (if possible), we compare Algorithm 1, with the recommended MP parameter settings, vs. the SIMC tuning rule ([PERSON] et al., 2016). We continue with studying the reference example considered in Section 4. See also ([PERSON], 2016) for additional details on this example. **Example 2** (Reference Example). : _The same IPTD example as in ([PERSON], 2016) is used, i.e., a process model, \(H_{p}(s)=k\frac{\epsilon^{-\tau s}}{s}\), with gain velocity \(k=1\) and time delay \(\tau=1\) is considered._  
_The time-domain responses given a prescribed robustness, \(M_{s}=1.59\), are illustrated in Figure 5. The corresponding PI controller parameters, indices and margins are given in Table 6. The margins for the controllers are all acceptable, i.e., \(GM>2\) and \(PM>30\) as in ([PERSON] et al., 2016)._  
Figure 5: Example 2 (Reference example). Consider PI control of an IPTD process model, \(H_{p}(s)=\frac{\epsilon^{-\tau}}{s}\). 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 \(T_{c}=1.24\,\tau\) and Algorithm 1 where the MP parameter \(\varepsilon=2.5\) (proposed in Section 4) and RTDE \(\delta=1.79\). An output disturbance unit step is presented at time \(t=0\) and an input disturbance unit step at time \(t=50\). **Example 3** (Lag-dominant system). : _An air-heater was studied in [19] and it was found that a FOPTD model with process gain \(K=5.7\), time delay \(\tau=4\) and time constant \(T=60\), gives a sufficient model approximation. We approximate the FOPTD model as an IPTD process where the gain velocity (slope) \(k=\frac{K}{T}=0.095\) and time delay, \(\tau=4\)._  
_The Pareto performance objective \(\int\) vs. \(M_{s}\) trade-off curves are shown in Figure 6. In terms of the main performance objective \(V_{M}\) it can be seen in Table 7 that \(\bar{c}=2.5\) is \(\frac{V_{\bar{c}=2.7}}{V_{\bar{c}=2.5}}=1.9\) times better than \(\bar{c}=2.7\) and \(\frac{V_{\mathrm{SMC}}}{V_{\bar{c}=2.5}}=12.2\) times better than SIMC._  
_The time-domain responses, given a prescribed robustness, \(M_{s}=1.59\), are illustrated in Figure 7. The corresponding PI controller parameters, indices and margins are given in Table 8. The margins for the controllers are all acceptable, i.e., \(GM>2\) and \(PM>30\) as in [14]. Notice, that the prescribed MTDE, \(d\tau_{\text{max}}=\delta\tau=7.16\) is almost equal the exact \(DM=7.51\)._ \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Alg. 1** & **SIMC** & **PO PI** \\ \hline \(K_{p}\) & 0.41 & 0.45 & 0.41 \\ \(T_{i}\) & 6.14 & 8.96 & 6.28 \\ \(\mathrm{IAE}_{py}\) & 4.39 & 4.24 & 4.37 \\ \(\mathrm{IAE}_{uu}\) & 15.26 & 20.06 & 15.39 \\ \(J\) & 1.52 & 1.64 & 1.52 \\ TV & 3.33 & 3.12 & 3.31 \\ GM & 3.56 & 3.34 & 3.54 \\ PM & 44.57 & 50.02 & 44.94 \\ DM & 1.79 & 1.90 & 1.80 \\ \(M_{s}\) & 1.59 & 1.59 & 1.59 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Example 2. Consider PI control of the IPTD process model, \(H_{p}(s)=\frac{\bar{c}^{-s}}{s}\). The table shows the controller parameter, indices and margins are given for prescribed robustness \(M_{s}=1.59\) for the following methods: Alg. 1 (\(\bar{c}=2.5,\delta=1.79\)), SIMC (\(T_{c}=1.24\)\(\tau\)) and PO PI (\(M_{s}=1.59\)). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(\bar{c}\) **= 2.5** & \(\bar{c}\) **= 2.7** & **SIMC** \\ \hline \(V_{M}\)/e-2 & 0.57 & 1.08 & 6.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Example 3. The table shows the comparison of the settings for Algorithm 1 and SIMC using the main performance objective \(V_{M}\) (Equation (42)).