Gain-Scheduled Disturbance Observer-Based Saturated Controllers for Non-Linear First-Order System

Almost a century ago, the first industrial controllers were introduced to the market, labeled as automatic reset and later generalized to hyper-reset or pre-act. Recently, it has been shown that such control solutions can be characterized as model-based solutions with a simplified disturbance observer developed for an integrating model. The aforementioned controllers, albeit under the name of proportional–integral–derivative (PID) controllers, are still the most commonly used control solutions in practice. With the help of a new interpretation, however, it can be shown that PID controllers are also very well suited for controlling processes with complex non-linear dynamics. This paper investigates the design and feasibility of a family of gain-scheduling controllers for saturated non-linear systems described by a first-order differential equation. It is shown that the process can be linearized either by using locally applicable linear models or by using more narrowly applicable ultralocal models. By combining both approaches, an innovative linearization method around the steady states of the process input and output is proposed. This novel approach emphasizes that the entire process input signal has to be constructed by adding the control increment calculated by the linearization to the value of the considered operating point. Thus, it avoids the uncertainties of those methods, which are based on achieving the actual controller output by integrating the calculated differential values. Another advantage of model-based design is that the saturation of the control signal is included in the design from the outset. Therefore, the undesired integration (windup), which is typical for controllers with explicit integral action, is prevented. The proposed design is illustrated using the control of a liquid tank with variable cross-section as a function of the liquid level. The model-based approach is also used in the evaluation of the transients, where homogeneous responses were obtained over the whole range of process output values. Responses were more homogeneous when simple ultralocal models were used, regardless of controller saturation constraints. Finally, all important innovative aspects of the design are highlighted by a comparison with gain-scheduled PI controller design based on velocity implementation.