# Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System +
## 3 Illustrative Example
### Constrained Control
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & \multicolumn{8}{c}{**ARC0-1, \(T_{c}=0.1\), \(T_{f}=0.1\)**} \\ \hline \(w\) & **0.1** & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** & **0.7** & **0.8** & B & **1** \\ \hline \(IAE\) & 0.011935 & 0.013234 & 0.014865 & 0.015264 & 0.015883 & 0.016313 & 0.016681 & 0.016897 & 0.016894 & 0.017535 \\ \hline \(TV_{0}(y)\) & 0.17787 & 0.18706 & 0.19206 & 0.1947 & 0.19134 & 0.19496 & 0.1966 & 0.19815 & 0.19652 & 0.20149 \\ \hline \(TV_{1}(u)\) & 32.616486 & 76.042108 & 135.58941 & 208.66087 & 297.53243 & 404.31701 & 531.64265 & 668.94032 & 825.90369 & 1011.4014 \\ \hline \multicolumn{8}{c}{DOB-1, \(T_{c}=0.1\), \(T_{f}=0.1\)} \\ \hline \(w\) & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ \hline \(IAE\) & 0.021189 & 0.021393 & 0.021954 & 0.021669 & 0.02128 & 0.021412 & 0.021776 & 0.021572 & 0.021252 & 0.021844 \\ \hline \(TV_{0}(y)\) & 0.35134 & 0.35656 & 0.35706 & 0.35482 & 0.349 & 0.35141 & 0.34833 & 0.34973 & 0.34757 & 0.35408 \\ \hline \(TV_{1}(u)\) & 66.142893 & 153.40691 & 273.39205 & 420.57951 & 599.70507 & 816.62546 & 1074.0832 & 1351.1266 & 1669.2539 & 2043.106 \\ \hline \multicolumn{8}{c}{ARCO-5, \(T_{c}=0.1\), \(T_{f}=0.5\)} \\ \hline \(w\) & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ \hline \(IAE\) & 0.050488 & 0.050709 & 0.050805 & 0.05057 & 0.050332 & 0.050285 & 0.050795 & 0.050626 & 0.050296 & 0.050531 \\ \hline \(TV_{0}(y)\) & 0.1594 & 0.1537 & 0.1507 & 0.149 & 0.1474 & 0.1459 & 0.1453 & 0.1469 & 0.1443 & 0.1480 \\ \hline \(TV_{1}(u)\) & 32.671235 & 76.303127 & 13  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & \multicolumn{6}{c}{**ARC0-J, \(T_{c}=0.5\,T_{f}=0.5\)**} \\ \hline \(w\) & **0.1** & **0.2** & **0.3** & **0.4** & **0.5** & **0.6** & **0.7** & **0.8** & **0.9** & **1** \\ \hline \(IAE\) & 0.051801 & 0.056311 & 0.059723 & 0.063522 & 0.067422 & 0.070043 & 0.07228 & 0.074821 & 0.076368 & 0.077478 \\ \hline \(TV_{0}(y)\) & 0.1009 & 0.1001 & 0.1015 & 0.1066 & 0.1087 & 0.1112 & 0.1136 & 0.1149 & 0.1170 & 0.1162 \\ \hline \(TV_{1}(u)\) & 19.7342 & 44.5925 & 79.5527 & 125.406 & 179.2605 & 248.6903 & 319.8459 & 407.0494 & 503.2121 & 606.7853 \\ \hline \multicolumn{6}{c}{DOB-1, \(T_{c}=0.5\,T_{f}=0.5\)} \\ \hline \(w\) & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ \hline \(IAE\) & 0.10119 & 0.10221 & 0.10244 & 0.10206 & 0.10195 & 0.10208 & 0.10179 & 0.10218 & 0.10135 & 0.10153 \\ \hline \(TV_{0}(y)\) & 0.2014 & 0.1935 & 0.1909 & 0.1949 & 0.1942 & 0.194 & 0.1949 & 0.1937 & 0.1965 & 0.1921 \\ \hline \(TV_{1}(u)\) & 39.874588 & 89.572495 & 159.52872 & 251.36843 & 359.19898 & 498.2467 & 640.76059 & 815.39492 & 1008.0409 & 1215.4798 \\ \hline \multicolumn{6}{c}{ARC0-5, \(T_{c}=0.5\,T_{f}=2.5\)} \\ \hline \(w\) & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 & 1 \\ \hline \(IAE\) & 0.2493 & 0.25727 & 0.25237 & 0.25132 & 0.25136 & 0.2507 & 0.25026 & 0.25041 & 0.2503 & 0.25063 \\ \hline \(TV_{0}(y)\) & 0.0842 & 0.0727 & 0.0685 & 0.0687 & 0.0682 & 0.0678 & 0.0685 & 0.0674 & 0.0684 & 0.0668 \\ \hline \(TV_{1}(u)\) & 19.6364 & 44.6244 & 79.6069 & 125.5128 & 179.3979 & 248.8087 & 320.0882 & 407.3516 &The above analysis focused on the compensation of process non-linearities using the local and ultralocal model approximations. The controller parameters were chosen so that homogeneous closed-loop dynamics (in the sense of homogeneous values _iae_) is achieved over the entire range of permissible operating points. In this case, the controller parameters are also selected according to the limitations of the control signal. Due to the saturation constraints, almost constant values of _iae_ are no longer possible. Instead, we will focus on obtaining homogeneous shapes of the setpoint and disturbance responses at different amplitudes of the setpoint and disturbance steps. From the responses in Figure 14 and from the corresponding performance measures in Figure 15, it can be seen that the upper limit of the controller output \(U=2.05\) specifies the saturation limits.