# Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System +
## 3 Illustrative Example
### Comparison with Gain-Scheduled PI Control
This difference alone indicates that significant differences in the achieved transients can be expected from both solutions._  
Figure 16: Setpoint step responses of one-tank hydro system with a variable cross-section (63) with 2 DoF gain-scheduled ARC0 and DOB0 controllers, pre-filters (44) and (51), \(T_{c}=0.1\), \(T_{f}=T_{c}\), gain-scheduled 1 DOF PI1 controller according to [6] with \(T_{f}=T_{c}/10\) and its 2 DOF PI0 modification according to (80), (18) (left) and the corresponding performance measures (right). However, the comparison with the gain-scheduled PI controller according to [6] also opens new topics for further research. [PERSON] et al. did not consider the possibility of using two types of linear model--their approach was formulated only using local models. However, considering \(a=0\), this approach can also be reformulated for ultralocal models. They neglected the fact that the linear controller (76) should only be used to calculate small increments of the control signal \(\Delta u\) according to  
\[\Delta U(s)=\frac{1}{s}(K_{i}E(s)-K_{p}\frac{s}{T_{f}s+1}Y(s)), \tag{80}\]  
which are added to the working point \(u_{p}\) (18) (see Figure 17). [PERSON] et al. did not consider the possible homogenization of the output dynamics achieved by removing the zero of the transfer function (77) using a pre-filter. This can be easily implemented again only for ultralocal models (Figure 17), when the pre-filter time constant is independent of the operating point, whereby it holds  
\[F_{p}(s)=1/(1+T_{i}s);\ T_{i}=K_{p}/K_{i}=2T_{c} \tag{81}\]  
The curves corresponding to the 2 DoF PI0 controller are shown in Figure 16 (magenta). The effect of adding \(u_{p}\) to the PI controller output is similar to the changes discussed in Section 3.4 Offset impact. Even this modification of the GS-PI controller does not ensure the high homogeneity of the waveforms achieved with the DOB0-1 controller. For the \(IAE\) values, even ARC0-1 gives smaller and more homogeneous values than 1 DOF-PI1 or 2 DOF-PI0. The comparison of ARC0 and DOB0 controllers with GS-PI alternatives does not change even when choosing larger values of \(T_{c}=0.5\) in Figure 18. None of the modifications considered for GS-PI controllers ensures the homogeneity of \(IAE\) values in the entire range of \(w\) changes achieved by DOB0-1 and ARC0-1 controllers. The advantage of both 1 DOF-PI1 and 2 DOF-PI0 is smooth controller output waveforms, which approach ideal shapes. 1 DoF-PI1 and 2 DoF-PI0 complement each other. The former is suitable for larger values of \(w\), the latter for lower \(w\). However, this only highlights the long-term unrecognized and unused advantages of the solution, whose neglect is closely related to the prevailing interpretations of PID control and process linearization. It would also be interesting to quantitatively compare the thus modified GS-PI controller with the aforementioned ARC and DOB-based controllers, also from the point of view of measurement noise attenuation and saturated control. It may also be interesting to prove and compare the stability and domains of attraction of possible equilibria in such modified gain-scheduled PI controllers and in the above-designed ARCs and DOBs using appropriate methods; for example, Lyapunov functions. However, it is certainly not possible to solve all the open questions that have accumulated over the 9 decades of existence of automatic reset controllers in a single publication. \[U(s)=\frac{1}{s}(K_{i}E(s)+K_{p}sE(s))=\frac{1}{s}(K_{i}E(s)-K_{p}sY(s)). \tag{76}\]0 2 DoF velocity implementation of gain-scheduled PI controller with a pre-filter \(F_{p}(s)\) and inverse transformation from \(\Delta u\) to \(u\) according to (18).