# Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System +
### Overview of the Design Options Considered  
The control design options considered are two types of process models in combination with two types of disturbance observers (Table 1). The parameters of the PI controller \(C(s)=K_{p}+K_{i}/s\) (Figure 4), which correspond to the ARCs and DOBs considered with (33), (47), (56), and (60) in the proportional zone of the control, are summarized in Table 2. The parameters are expressed in terms of a closed-loop time constant \(T_{c}\) corresponding to a stabilizing P controller \(K_{P}\) and of a filter time constant \(T_{f}\) used in DOB to reconstruct the input disturbance for the plant model (8). The parameters of the current local model are adjusted according to the setpoint \(w\) as the scheduling parameter \(\alpha\), which provides \(\Delta w=w-y_{p}=0\). **Remark 4** (The essence of integral action). : _Although none of the four solutions considered contains an explicit integrator, the integral action is generated implicitly by compensating for the estimate of the input disturbance, which is computed by comparing the reconstructed input to the process \(u_{a}\) and the output of the controller \(u\) (see Figure 2). In the case of ARC0, it is most easily reconstructed based on ultralocal process models (i.e., integral model with \(a=0\)) as the steady-state value of the controller output [29]. When \(a\
eq 0\), the steady-state output of the process must also be considered in GS-ARC1 to increase the accuracy of \(d_{i}\) reconstruction._  
**Remark 5** (Research on GS-ARCs and DOBs). : _ARCs have been used extensively in industry for almost a century. However, their various applications, which arise from the use of two types of linear models based on generally first-order non-linear processes and utilizing simplified disturbance observers (which reconstruct input disturbances just in steady states), are virtually unknown in the literature. The model-based features of ARCs were overshadowed by the heuristic PID control approaches and were also overlooked and misunderstood by the MCT of the 1960s [25; 26; 27]. Uncovering the specifics of the applied DOB would be even more difficult in MCT, because reconstructing and compensating for disturbances for the first-order processes based on ESO and an identical state and disturbance observer yields a characteristic second-order polynomial in (9) [31]._  
_This also hides the connection between ARC and ADRC [68; 32], the latter being essentially a state-space approach simplified by the formulation for ultralocal models with \(a=0\). For the postmodern IMC, the direct connection with ARCs is not obvious, as the output disturbances of ultralocal (integral) models are unobservable [31]. Reciprocal relationships were also not sufficiently clarified for DOBC when DOB was used as an extension of PI control [69]. Paradoxically, simultaneous application of integral action with DOB could lead to degradation of the resulting loop performance, \begin{table}
\begin{tabular}{c c c c} \hline ARC0 & Automatic Reset Controller based on ultralocal linear process model \\ \hline ARC1 & Automatic Reset Controller based on local linear process model \\ \hline DOB0 & Stabilizing P controller with Disturbance Observer based on ultralocal linear process model \\ \hline DOB1 & Stabilizing P controller with Disturbance Observer based on local linear process model \\ \hline \end{tabular}
\end{table} 
Table 1: Controller abbreviations. \begin{table}
\begin{tabular}{c c c c c} \hline \(-\) & **ARC0** & **DOB0** & **ARC1** & **DOB1** \\ \hline \(K_{p}K_{s}\) & \(\dfrac{1}{T_{c}}\) & \(\dfrac{1}{T_{c}}+\dfrac{1}{T_{f}}\) & \(\dfrac{1}{T_{c}}-a\) & \(\dfrac{1}{T_{c}}+\dfrac{1}{T_{f}}-a\) \\ \hline \(K_{i}K_{s}\) & \(\dfrac{1}{T_{c}T_{f}}\) & \(\dfrac{1}{T_{c}T_{f}}\) & \(\dfrac{1}{T_{c}T_{f}}\) & \(\dfrac{1}{T_{c}T_{f}}\) \\ \hline \(T_{p}\) & \(T_{f}\) & \(T_{c}+T_{f}\) & \(T_{f}(1-aT_{c})\) & \(T_{c}+T_{f}-aT_{c}T_{f}\) \\ \hline \end{tabular}
\end{table}