# Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System +
## 4 Conclusions  
The work took advantage of the recent revelation that official interpretations of PID control do not explain the essence of historical industrial controllers based on automatic reset. A new interpretation of automatic reset controllers considers them as model-based solutions with a simplified disturbance observer designed on the basis of integrative models with first-order dominant dynamics. The work therefore focused on additional study of their use for control of non-linear systems with first-order dominant dynamics. The choice of the objective appropriate to the essence of automatic reset controllers together with a consistent solution of the linearization problem led to the fact that the proposed controllers enable achieving higher performance and higher independence from the choice of the operating point than PID control approaches. These results are interpreted as a consequence of more accurate respect for essential aspects of controlled processes, while PID control is based only on secondary equivalent system properties valid only in a narrow range of transient responses in the proportional control band. This paper also showed that the proposal, which includes four different types of controllers based on two types of linear process models and two types of disturbance observers, also includes other well-known approaches including disturbance observer control (DOBC), modern control theory (MCT), and active disturbance rejection control (ADRC). It also pointed out frequent inconsistencies regarding the linearization of non-linear processes arising from the transition from the calculated increments of the control signal to its total value. In this way, it not only shows the ways to better use different approaches to controlling non-linear processes, but also to better understand their mutual connections. A detailed analysis of four approaches to process linearization and two types of disturbance observers applied to the control design of the considered first-order non-linear system showed the clear advantages of automatic reset-based solutions based on ultralocal models compared to gain-scheduled PI controller design. Their results can be further improved by using full DOB with inversion of the ultralocal model. Figure 18: 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), \(T_{c}=0.5\), \(T_{f}=T_{c}\), 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). In future research, several types of linear models will be developed for more complex non-linear systems with time delays. Controllers with higher-order derivatives are expected to be the most suitable for such models [19; 82; 83; 84; 85; 86; 87; 88; 89]. Writing--original draft preparation, M.H. ; writing--review and editing, M.H., P.B. and D.V. ; simulations, M.H. ; project administration, P.B. All authors have read and agreed to the published version of the manuscript. This research was supported in part by the following grants: Grant No. 1/0821/25 financed by the Scientific Grant Agency of the Ministry of Education, Research, Development and Youth of the Slovak Republic (VEGA); Grant No. APVV-21-0125 financed by the Slovak Research and Development Agency; Research Program P2-0001 (Systems and Control) and research project L2-3166 (Supervisory control system for plant-wide optimization of wastewater treatment plant operation) financed by the Slovenian Research and Innovation Agency; Grant Agreement No 875047 (project RUBY) financed by Clean Hydrogen Partnership (EU Horizon 2020). Not applicable. The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author. Supported by E-Academia Slovaca, a non-profit organization, Sadmelijska 1, 831 06 Bratislava, Slovakia.