# Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System +
## 3 Illustrative Example  
Variable dynamics of the system depending on the output and, respectively, the overall state of the system, is relatively common in practice. In order to avoid many important details of more demanding applications, such as they appear, e.g., in connection with attitude control [70], the included illustrative example considers a hydraulic one-tank system with a variable cross-section (see Figure 5), which is described by the differential equation  
\[\begin{array}{l}\dot{y}=\left[u-c\sqrt{\dot{y}}\right]/S(y);\\ y\in[0,Y];\;u\in[0,U];\;\;c=2;\\ S(y)=\pi r^{2}=\pi(r_{0}+ky)^{2};\;r_{0}=0.1;\;k=1.\end{array} \tag{63}\]  
The input flow rate \(q_{1}\) is considered a control signal (manipulated variable) and the output flow rate \(q_{2}\) depends on the liquid level \(h\). The parameters \(r_{0}\) and \(r=r_{0}+ky\) depend on the shape of the container, \(c\) on the resistance of the outlet tube. \(Y\) and \(U\) represent the limit values permissible for the given task.  
The results on the application of gain-scheduled PI controllers reported in [71] and supplemented by ARC0 in [5] are now extended by simulating the four options of the model-based control discussed above. Linearization around a steady state (14) with \(y_{p}=w\) results in a parametrization of the operating point by a single scheduling parameter \(\alpha\)  
\[y_{p}=\alpha=w;\;u_{p}=c\sqrt{\alpha}. \tag{64}\]  
The gain-scheduled model  
\[\begin{array}{l}\dot{\Delta y}=\dot{y}=f_{u}(\alpha)\Delta u-f_{y}(\alpha) \Delta y;\\ f_{u}(\alpha)=\left[\frac{\partial f}{\partial u}\right]_{y_{p}=\alpha;\;u_{ p}=c\sqrt{\alpha}}=\frac{1}{S(\alpha)}=\frac{1}{\pi(r_{0}+ka)^{2}};\\ f_{y}(\alpha)=-\left[\frac{\partial f}{\partial y}\right]_{y_{p}=\alpha;\;u_{ p}=c\sqrt{\alpha}}=\frac{c}{2S(\alpha)\sqrt{\alpha}};\end{array} \tag{65}\]  
can be specified by the process model \(K_{s}/(s+a)\) (17) as follows:  
\[\begin{array}{l}K_{s}=K_{s}(\alpha)=f_{u}(\alpha)=1/S(\alpha);\\ a=a(\alpha)=f_{y}(\alpha)=\frac{c}{2S(\alpha)\sqrt{\alpha}}.\end{array} \tag{66}\]  
Figure 5: One-tank hydro system with a variable cross-section.  
As mentioned above, there are four possible settings for the controller, depending on the applied model (ultralocal or local, denoted by indices 0 and 1) and the disturbance reconstruction by DOB (with the reconstruction in steady state denoted by 0 or already during transients with 1). The controller settings for the proportional band are calculated from (33), (47), (56), and (60), while the pre-filters are determined from (44), (51), (58), and (62). The main difference from [57] is that only one possible equilibrium state at zero was considered, while now there are infinitely many possible equilibrium states. The works of [6, 57, 58, 59, 60, 61, 62] focused on the control solutions for non-linear systems with multiple inputs and outputs by specifying the position of their poles. On the other hand, a single-input, single-output (SISO) system allows for easy comparison of previously unnoticed differences resulting from two types of linear model and determination of transient uniformity conditions with and without control signal saturation and measurement noise constraints.  
To manually adjust the historical analog ARCs, only a constant value \(y_{p}\) was considered when linearizing the system. When control algorithms are implemented with the currently available discrete-time processors, control performance can be significantly increased by adjusting the controller setting separately for each setpoint response based on the gain scheduling parameter \(\alpha=w\) (64).