# Robust Controller Decorated by Nonlinear S Function and Its Application to Water Tank
###### Abstract  
In this manuscript, a concept of modifying the results of the existing robust controller decorated by a nonlinear S function is presented to improve the system performance. A case-based study of level control of water tanks illustrates the effectiveness of nonlinear decoration in improving robustness and controlling energy-saving performance with an S-function-decorated robust controller. The performance of the controlled system was analyzed through Lyapunov stability theorem and robust control theory, and was evaluated with a performance index. By demonstrating three comparing simulations of different scenes, it testifies to the fact that the nonlinear decorated robust controller meets the requirement of improving the system performance index. Compared with the nonlinear feedback and the fuzzy control, the performance index of the system using a nonlinear decorated controller is reduced by more than 10% with satisfactory robustness. This nonlinear decorated robust controller is proven to be energy efficient, simple and clear and easy to use, valuable for extensive application.  
S function; nonlinear decoration; robust control; water tank; level control +
Footnote †: journal: Applied System  
**Citation:** [PERSON]; [PERSON] Robust Controller Decorated by Nonlinear S Function and Its Application to Water Tank. _Appl. Syst. Innov._**2021**, \(4\), 64.  
Reference [7] explained the four typical working conditions of water level fluctuations in surge tanks, and a cascade load regulation method for optimal attenuation of water level fluctuation under superposition conditions was proposed. Reference [8] discussed the design of PLC-based fractional-order controller for volume control of industry water tank; the purpose was to verify the stability and robustness of fractional-order discrete PID-feedback-loop for different approximation methods and approximation orders. Reference [9] introduced the improved smith predictor and the improved inverse calculation method to design the controller of the conical tank level process and compared the energy utilization and performance of the actuator in a nonlinear process with traditional PI controller. In order to stabilize the well water level, [PERSON] also designed another PI controller [10]. The principle and algorithm of dynamic matrix predictive control are described in [11]. Through the experimental analysis of different models of single, double and triple tanks, a classical double tank model was proposed, which was shown to have great efficiency improvement in the control process compared to the conventional model.  
In addition, Reference [12], using nonlinear ordinary differential equations described two mathematical models in order to research the adaptive control of nonlinear systems. Nonlinear robust controllers have also been widely used in other fields. However, all of the references mentioned above were in-depth studies based on different opinions and achieved some results about water tank control on the basis of distinctive control methods. Yet, there is little research based on the orientation of energy efficiency, leading to some prospective potential in this aspect.  
On the basis of the references [13; 14], the contributions of this paper are mainly twofold: (1) the condition of robust controller solution in reference [14] is reproved by using H\({}_{\infty}\) theory, and the condition of robust performance is relaxed from \(0\leq T_{1}\leq 1\) to \(T_{1}>0\); (2) the nonlinear feedback based on the arctangent function in [13] is changed to the nonlinear decoration based on S function and the control performance is further improved.