# Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System +
## 3 Illustrative Example
### Responses with Unconstrained 1 DoF Controllers  
The transients of all four GS controller options, each of them corresponding to ten smaller setpoint steps with \(\Delta w=0.1\), are shown in Figure 8 above. Gradually increasing the setpoint variable through such a sequence of steps \(\Delta w\) allows the gain-scheduled transients with \(\alpha=w\) to be checked over a wider range of output values. Each of the closed-loop time constants \(T_{c}\) (67) is associated with two filter time constants \(T_{f}\). The simplified ARCs are expected to be more effective for increased values of \(T_{f}\). Consequently, more complex DOBs are also expected to be efficient at smaller \(T_{f}\) values. Obviously, the transients follow the gradually increasing set point quickly for all four controller options. In detail, they can be analyzed using appropriate performance measures in Figure 9 above. Normalized \(iae\) values (69) deviate from the ideal value \(iae=1\) in virtually all simulations.  
Figure 8: Step responses of one-tank hydro system with a variable cross-section (63) and increasing setpoint value with gain-scheduling controllers according to Figure 3 with \(u_{p}\) (26), (64) (above) and \(u_{p}=0\) (below); \(T_{c}=0.1\) (left); and \(T_{c}=0.5\) (right); \(T_{f}=T_{c}\) and \(T_{f}=5T_{c}\).  
Figure 9: Performance measures of the setpoint step responses of one-tank hydro system with a variable cross-section (63) from Figure 8 for \(T_{c}=0.1\) (left) and \(T_{c}=0.5\) (right), \(T_{f}=T_{c}\) and \(T_{f}=5T_{c}\): Above—normed _iae_ values (69), middle—\(TV_{1}(u)\); below—normed \(TV_{0}(y)\).  
For small values of \(w\), closed-loop responses are faster than should correspond to the value of \(T_{c}\) used for the calculation of \(K_{P}\). In contrast, closed-loop responses are slower for larger values of \(w\). The values \(TV_{0}>0\) indicate that all the response steps can exhibit a certain overshoot in the output. The evaluation with fast and slow settings of the stabilizing controller \(K_{P}\), each in combination with two \(T_{f}\) values, thus reveals some interesting facts:  
* The \(TV_{0}\) corresponding to the local ARC1 models is larger than in the case of the simpler ultralocal ARC0 models.
* If you choose a smaller filter time constant \(T_{f}=T_{c}\), the overshoots are relatively large when using ARCs, while for \(T_{f}=5T_{c}\) they are about the same as when using more complex DOBs.
* Smaller values of _iae_ for \(T_{f}=T_{c}\) can apparently be achieved with DOBs. But even here, the DOB0s based on the ultralocal model result in less deviations from monotonicity (\(TV_{0}\)) than the DOB1s.
* With the exception of ARCs, which correspond to \(T_{f}=T_{c}\), none of the options considered leads to excessive control effort with \(TV_{1}(u)>0\).  
In this evaluation carried out without measurement noise, Figure 9 middle shows excessive controller effort only for ARC0 and ARC1 with fast setting \(T_{c}=0.1\) and \(T_{f}=T_{c}\). For \(T_{c}=0.5\) and \(T_{f}=T_{c}\), only ARC1-1 based on more complex local models gives more excessive controller effort.