# Optimal Robust PID Control for First- and Second-Order Plus Dead-Time Processes
## 3 Unified Optimal PID Tuning with Robust Stability
### Servo Optimization  
The optimized normalized PID parameters for the servo tracking optimization in the given finite plants are plotted as \(\circ\) symbols in Figures 1-3.  
Figure 1: Calculated and approximated optimal \(\kappa_{p}\) in the servo mode.  
To obtain the unified design method, the obtained normalized PID parameters are approximated by the following equations:  
Figure 3: Calculated and approximated optimal \(\tau_{d}\) in the servo mode.  
Figure 2: Calculated and approximated optimal \(\tau_{i}\) in the servo mode.  
\[\kappa_{p}(a,\tau_{0},M_{\rm s}^{d}) = a_{0}(a,M_{\rm s}^{d})+a_{1}(a,M_{\rm s}^{d})\tau_{0}^{a_{2}(a,M_{ \rm s}^{d})}, \tag{21}\] \[a_{0}(a,M_{\rm s}^{d}) = \frac{\alpha_{0}(M_{\rm s}^{d})+\alpha_{1}(M_{\rm s}^{d})a+\alpha_ {2}(M_{\rm s}^{d})a^{2}}{\alpha_{3}(M_{\rm s}^{d})+a},\] \[a_{1}(a,M_{\rm s}^{d}) = \frac{\alpha_{4}(M_{\rm s}^{d})+\alpha_{5}(M_{\rm s}^{d})a+\alpha _{6}(M_{\rm s}^{d})a^{2}}{\alpha_{7}(M_{\rm s}^{d})+a},\] \[a_{3}(a,M_{\rm s}^{d}) = \alpha_{8}(M_{\rm s}^{d})a^{5}+\alpha_{9}(M_{\rm s}^{d})a^{4}+ \alpha_{10}(M_{\rm s}^{d})a^{3}\] \[+ \alpha_{11}(M_{\rm s}^{d})a^{2}+\alpha_{12}a+\alpha_{13},\]  
\[\tau_{i}(a,\tau_{0},M_{\rm s}^{d}) = b_{0}(\tau_{0},M_{\rm s}^{d})+b_{1}(\tau_{0},M_{\rm s}^{d})\tau_ {0}, \tag{22}\] \[+ b_{2}(\tau_{0},M_{\rm s}^{d})\tau_{0}^{2}+b_{3}(\tau_{0},M_{\rm s }^{d})\tau_{0}^{3}\,\] \[b_{0}(\tau_{0},M_{\rm s}^{d}) = \beta_{0}(M_{\rm s}^{d})+\beta_{1}(M_{\rm s}^{d})\tau_{0},\] \[+ \beta_{2}(M_{\rm s}^{d})\tau_{0}^{2}+\beta_{3}(M_{\rm s}^{d})\tau _{0}^{3}\,\] \[b_{1}(\tau_{0},M_{\rm s}^{d}) = \beta_{4}(M_{\rm s}^{d})+\beta_{5}(M_{\rm s}^{d})\tau_{0},\] \[+ \beta_{6}(M_{\rm s}^{d})\tau_{0}^{2}+\beta_{7}(M_{\rm s}^{d})\tau _{0}^{3}\,\] \[b_{2}(\tau_{0},M_{\rm s}^{d}) = \beta_{8}(M_{\rm s}^{d})+\beta_{9}(M_{\rm s}^{d})\tau_{0},\] \[+ \beta_{10}(M_{\rm s}^{d})\tau_{0}^{2}+\beta_{11}(M_{\rm s}^{d}) \tau_{0}^{3}\,\] \[b_{3}(\tau_{0},M_{\rm s}^{d}) = \beta_{12}(M_{\rm s}^{d})+\beta_{13}(M_{\rm s}^{d})\tau_{0},\] \[+ \beta_{14}(M_{\rm s}^{d})\tau_{0}^{2}+\beta_{15}(M_{\rm s}^{d}) \tau_{0}^{3}\,\]  
Figure 7: Calculated and approximated optimal \(\tau_{d}\) in the regulation mode.  
The values of \(M_{\rm s}\) for \(M_{\rm s}^{d}=1.4\), 1.6, 1.8, and 2.0 are plotted in Figure 8, and the obtained values of \(M_{\rm s}\) are listed in Table 8. Since the errors for the prescribed maximum sensitivity are within \(\pm 3\%\), the effectiveness of the proposed method is confirmed.  
From Tables 4-8, the prescribed robust stability is sufficiently achieved, even though the optimal trade-off PID parameters are easily implemented using the calculated look-up tables.  
\begin{table}
\begin{tabular}{c c c c c} \hline \(M_{\rm s}^{d}\) & **1.4** & **1.6** & **1.8** & **2.0** \\ \hline \(\gamma_{0}\) & \(-0.01709\) & \(-0.01810\) & \(-0.007789\) & \(0.02155\) \\ \(\gamma_{1}\) & \(0.9843\) & \(0.4629\) & \(1.402\) & \(0.6965\) \\ \(\gamma_{2}\) & \(-0.1369\) & \(-0.004881\) & \(-0.5348\) & \(-0.1997\) \\ \(\gamma_{3}\) & \(1.448\) & \(0.4799\) & \(2.351\) & \(1.073\) \\ \(\gamma_{4}\) & \(0.1451\) & \(0.06998\) & \(0.1886\) & \(0.05419\) \\ \(\gamma_{5}\) & \(0.2027\) & \(0.1237\) & \(0.06031\) & \(0.1531\) \\ \(\gamma_{6}\) & \(0.02381\) & \(0.1178\) & \(0.2215\) & \(0.1676\) \\ \(\gamma_{7}\) & \(0.4183\) & \(0.2145\) & \(0.6713\) & \(0.2098\) \\ \(\gamma_{8}\) & \(0.3704\) & \(0.1160\) & \(0.1664\) & \(0.07437\) \\ \(\gamma_{9}\) & \(1.271\) & \(1.191\) & \(1.026\) & \(1.235\) \\ \(\gamma_{10}\) & \(-0.01542\) & \(-0.3177\) & \(-0.3190\) & \(-0.4418\) \\ \(\gamma_{11}\) & \(0.3743\) & \(0.1438\) & \(0.1948\) & \(0.07446\) \\ \hline \end{tabular}
\end{table}
Table 7: Coefficient parameters \(\gamma_{i}(M_{\rm s}^{d})\) for \(\tau_{d}\) in the regulation mode.  
Figure 8: \(M_{\rm s}\) in the regulation mode.