# Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System +
### GS-ARC1 Generalized for Local Model  
When considering local (linear) models in which \(a\
eq 0\) is taken into account both in the process description and in the controller design, the simple stabilizing controller yields the closed-loop pole \(s_{1}=-K_{P1}K_{s}-a\) and the time constant  
\[T_{c}=T_{1}=-1/s_{1}=1/(K_{P1}K_{s}+a). \tag{53}\]  
This relationship can in turn be used to adjust the gain of the stabilizing controller \(K_{P1}\) according to the following formula:  
\[K_{P1}K_{s}=1/T_{1}-a. \tag{54}\]  
**Remark 2** (Model-based robustness measure).: _In the nominal case, the local model parameters used for the controller design are equal to the local model parameters that describe the process. Thus, this setting ensures a stability estimate at any value of \(T_{1}>0\) according to (54). However, if there is a significant difference between \(K_{P0}\) and \(K_{P1}\) for some \(T_{0}=T_{1}>0\), the controller depends heavily on the process linearization and the obtained controller robustness may be reduced (see, e.g., [67]). On the other hand, if \(K_{P0}\approx K_{P1}\), it is likely that the non-linear process can be described with sufficient precision by both linear models. This can be considered as an indication of a sufficient agreement between the expected and the real control responses even further away from the equilibrium points._  
If the choice of \(T_{f}\) satisfies \((T\!s+1)/(T_{f}\!s+1)\approx 1/(T_{f}\!s+1)\) at \(\Delta w=0\) (motivated by the achievement of smooth transients even with a significant amount of measurement noise), the parameter \(T\) of the process model can be neglected. This leads to the following transfer function of the equivalent stabilizing controller:  
\[R^{10}(s)=K_{P10}+\frac{a}{K_{s}}\frac{1}{T_{f10}\!s+1}. \tag{55}\]  
By considering the positive controller feedback, it yields an automatic reset series PI controller, whose gain is defined by the time constant \(T_{10}\) and the \"integral\" (i.e., automatic reset) time constant \(T_{f10}\)  
\[\begin{split}& C^{10}(s)=R^{10}\frac{1+T_{f10}\delta}{T_{f10} \!s}=K_{P10}\frac{1+T_{f10}\delta}{T_{f10}\delta}+\frac{a}{K_{s}T_{f10}\delta} \\ &=\frac{1-aT_{10}}{T_{10}K_{s}}\Bigg{[}1+\frac{1}{T_{f10}(1-aT_{ 10})s}\Bigg{]}=\frac{1-aT_{10}}{T_{10}K_{s}}+\frac{1}{T_{10}T_{f10}K_{s}}. \end{split} \tag{56}\]  
In the proportional band of the controller, the closed loop dynamics with ARC1 (56) becomes  
\[F_{\text{avg}}^{10}(s)=\frac{C^{10}(s)S(s)}{1+C^{10}(s)S(s)}=\frac{T_{f10}(1- aT_{10})s+1}{T_{f10}T_{10}\delta^{2}+T_{f10}s+1}=\frac{T_{f10}(1-T_{10}/T)s+1}{T_{ f10}T_{10}s^{2}+T_{f10}s+1}. \tag{57}\]  
Due to the numerator \(F_{\text{avg}}^{10}\), overshooting of the setpoint step responses can occur. It can be removed by using a pre-filter  
\[F_{p}^{10}(s)=1/[T_{f10}(1-T_{10}/T)s+1]. \tag{58}\]  
**Remark 3** (GS-ARCs with local and ultralocal models).: _The design of GS-ARCs using local and ultralocal models leads to formally identical controller Equations (33) and (56). The differences lie in the calculation of the stabilizing controller gains (31) and (54), in their relation to the closed-looptime constants, and in the design of the pre-filters (44) and (58) to eliminate overshoots. When designing the controller for unstable processes, the ARC0 controller using ultralocal models does not automatically guarantee the closed-loop stability._