Robust Controller Decorated by Nonlinear S Function and Its Application to Water Tank

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.


Introduction
The level control of water tanks is a benchmark problem in the control field that can be used to assess the performance of a new control algorithm [1]. It is extensively used in many industrial fields. In particular, ocean vessels have a large number of liquid tanks, such as ballast tanks, oil tanks, fresh water tanks, and sludge tanks. They perform different services to guarantee safe navigation. Automatic control first originated from the needs of industrial processes. Since then, this technology has been developed for more than 70 years and has penetrated into every aspect of human life. However, in recent decades, the research on improving controller design has entered a bottleneck stage; there are hardly any new breakthroughs.
The development of control theory can be roughly divided into three stages: manual control, open-loop control and closed-loop control, and the closed-loop controls have transferred from process control to state control where the essence is still linear feedback [2]. If we use formulas to denote the closed-loop control then u = f (e) · e (where u represents the controller output, e represents the system error). The effectiveness of control law f (e) is judged by whether the output will reach the set value when the system error occurs for a variety of reasons. Moreover, the conventional error feedback is linear feedback, which means e is feedback to the input of the controller either intact or filtered. Therefore, sometimes the control input is unnecessarily large. For this reason, several research studies attempted to use nonlinear feedback to replace linear feedback [3,4], that is u = f (e) · g(e) (g is a nonlinear function). Along the same lines, one can also adjust the control law output, which is called nonlinear decoration with h(u) = f (e) · e (where h is a nonlinear function) [5,6]. The nonlinear-decorated control can improve system performance; therefore, the relevant research is worth to be conducted.

Mathematical Model
Reference [15] puts forward one special case of concise robust control which is known as PID control. Undoubtedly, simplicity, reliability and obvious physical meaning are the typical features of PID control with extensive application in industrial control engineering. As a result, in order to reach better self-adaption, robustness, accuracy, and easier to regulate parameters, the typical PID control has been improved and modified into different modes gradually, e.g., self-adaptive PID control, self-tuning PID control, gain scheduling PID control, robustness PID control, etc., [15][16][17].
According to references [18,19], designing a controller using robust control theory has received more attention to, and it will be better to use the closed-loop gain shaping method to maintain the robustness of controllers [19][20][21][22].

Mathematic Model and Calculation of Parameters
If the controlled plant G can be shown as the second-order strictly appropriate plant of (1): Then, the robust controller K can be designed according to the closed-loop gain shaping algorithm as (2) [2].
where 1/T 1 is the frequency of system bandwidth.  Figure 1 shows a simplified physical model; the switch sign indicates the water pump. In a single tank system (Figure 1), Q i indicates the steady-state value of input water flux, and ∆Q i means the increment of input water flux. Q o indicates the steady-state value of output water flux, and ∆Q o means the increment of output water flux. h means the height of water level, and h o means the steady-state value of water level, ∆h is the increment of the water level. u is the opening value of the adjustable input valve. A is the cross sectional area, R is the water resistance at the output valve, and V is the water volume in the tank. According to the correlation of material balance, the initial tank balance is displayed as: When the adjustable input valve has an increment ∆u, the actual water level will change correspondingly. Thus, the output water flux will change, which is mainly caused by the variety of water levels while keeping the other output valves unchanged.
where 1 1/ T is the frequency of system bandwidth. Figure 1 shows a simplified physical model; the switch sign indicates the water pump. In a single tank system (Figure 1) is the increment of the water level. u is the opening value of the adjustable input valve. A is the cross sectional area, R is the water resistance at the output valve, and V is the water volume in the tank. According to the correlation of material balance, the initial tank balance is displayed as: ,

Mathematic Model of Water tank
When the adjustable input valve has an increment u Δ , the actual water level will change correspondingly. Thus, the output water flux will change, which is mainly caused by the variety of water levels while keeping the other output valves unchanged. In this manuscript, the tank model is a first-order inertial system, and the tank inlet control valve model is equivalent to an integral, as shown in (Figures 2 and 3)  In this manuscript, the tank model is a first-order inertial system, and the tank inlet control valve model is equivalent to an integral, as shown in (Figures 2 and 3) where 1 1/ T is the frequency of system bandwidth. is the increment of the water level. u is the opening value of the adjustable input valve. A is the cross sectional area, R is the water resistance at the output valve, and V is the water volume in the tank. According to the correlation of material balance, the initial tank balance is displayed as: ,

Application of Robust Control in the Level
When the adjustable input valve has an increment u Δ , the actual water level will change correspondingly. Thus, the output water flux will change, which is mainly caused by the variety of water levels while keeping the other output valves unchanged. In this manuscript, the tank model is a first-order inertial system, and the tank inlet control valve model is equivalent to an integral, as shown in (Figures 2 and 3)  The difference between the input and output water flux is In Formula (3), ∆Q i is caused by ∆u, then where K u is the constant of water flow of valve. The output water flow is relevant with the height of the water level, A 0 is the crosssectional area of the output pipe, shown as The above formula can be further linearized at the balance point (h o , Q o ), then We can substitute the Formulas (4) and (6) into (3), and adopt Laplace transformation to obtain the transfer function for a single water tank where K 0 = K u R, T 0 = RA. The linear model of (7) is used to design the controller and the nonlinear model of (3)~ (5)   The difference between the input and output water flux is In formula (3), where u K is the constant of water flow of valve.
The output water flow is relevant with the height of the water level, 0 A is the crosssectional area of the output pipe, shown as The above formula can be further linearized at the balance point ( ) We can substitute the Formulas (4) and (6) into (3), and adopt Laplace transformation to obtain the transfer function for a single water tank If we assume the parameters of a water tank are: the height of the tank is 2 m, the area of the tank base is 1 m 2 , the sectional area of the pipe is 0.05 m 2 , the initial water level is 0.5 m, and the max inlet ratio of water intake is 0.5 m 3 /s. Then, the formula of water level and inlet ratio is where H represents the water level in the tank, and Q i represents the input flux ratio. Formula (7) can also be shown as If taking the disturbance term into consideration (interference is used to prove the stability of the system.), then Formula (9) can be improved as where the w is limited disturbance term, and also w ∞ ≤ ρ.

Controller Design Based on the Closed-Loop-Gain Algorithm
According to Formula (7), the closed-loop gain algorithm was used in the controller design (as Figure 4). If 1/T 1 = 1 rad/s, all parameters of robust controller can be obtained as k p = 1 0.8 , k i = 0, k d = 2 0.8 . This study emphasizes on the application of the nonlinear S function 1−e −2u 1+e −2u in modifying the output of the controller mathematically and experimentally. The new PD controller, improved by the S function, which is proved by its result in improving the controller performance.
where the w is limited disturbance term, and also w ρ ∞ ≤ .

Controller design based on the closed-loop-gain algorithm
According to Formula (7), the closed-loop gain algorithm was used in the controller design (as Figure 4). If , all parameters of robust controller can be obtained This study emphasizes on the application of the nonlinear S function

Controller Design
In this study, the Lyapunov stability theory was applied to launch the stability analysis of the water tank feedback controller, which is designed based on the closed-loopgain algorithm.
First, the state variable of the control system is interpreted as Based on Formula (11), the inlet ratio of the water tank control system can be expressed as (12). At the same time, the parameters' values are 1 By combining Formulas (8) and (12), we obtain:

Controller Design
In this study, the Lyapunov stability theory was applied to launch the stability analysis of the water tank feedback controller, which is designed based on the closed-loop-gain algorithm.
First, the state variable of the control system is interpreted as X = x 1 x 2 T , Based on Formula (11), the inlet ratio of the water tank control system can be expressed as (12). At the same time, the parameters' values are a 1 = 1, a 0 = 0, a 2 = 2, b = 0.8. Combine with (2).
By combining Formulas (8) and (12), we obtain: Then, by rearranging Formulas (11) and (13), the state equation of the controlled system can be written as . X = AX + bw (14) For Formula (14), the Lyapunov function of the system is interpreted as where P represents a positive definite real symmetric matrix, and P = P T . Ignoring the external disturbance term w, then According to the Lyapunov stability theory, the following function is needed to ensure the stability of the system at origin, for any interpreted positive definite real symmetric matrix.
Define positive definite real symmetric matrix, and make positive definite real symmetric matrix P as P = P 11 P 12 P 12 P 22 Putting Formulas (14) and (18) into Formula (17), to obtain To expand Formula (19) and obtain the solutions as The following functions have to be met for maintaining the positive definiteness of matrix P P 11 > 0 By solving Function (21), we obtain Thus, in the case of T 0 > √ 0.8 = 0.9, if T 1 > 0 is ensured, then the controller will keep the balanced state at the origin.
For the purpose of verifying the robustness of the controller proposed by this study against external distraction, the external distraction term w is considered in Formula (9), and then the formula of Lyapunov function as shown in Formula (10), is [14] .
To define a = max(T 1 , k), then Formula (26), indicated that when state variable of the system X ∞ > ρ/ 2(1 − a 2 ), then . V(x) < 0. For further discussing the robustness for the external distraction, L 2 which is the performance index of gain robust is interpreted as where µ 1 and µ 2 are positive and small, and the following theorem is obtained. Theorem: in the controller design for the water tank level control system, as shown in Formula (14) by applying the closed-loop gain algorithm, and the parameter T 1 of controller satisfying the condition of 0 < T 1 ≤ 1, the state variable of the system X ∞ > ρ/ 2(1 − a 2 ) ensures the whole controller uniform ultimate boundness and also achieves the performance index of gain robust L 2 , which is related to the controller parameter T 1 .
This theorem can be proved by integrating Formula (26) from t = 0 to t = t 0 , to obtain On the basis of the definition of, Formula (28) can be derived as After discussing the bounds of a as shown in Formula (29), a can be stated as [22] a = T 1 , The formula 1 − a 2 > 0 is a requisite, for the purpose of keeping the negative definite of Formula (26), and then the range of controller parameters is derived according to Formula (30) To summarize, based on the result of the Formula (27) about the performance index of gain robust and Formula (29), when the state variable of water tank control system satisfies the requirement of X ∞ > ρ/ 2(1 − a 2 ), if the parameter T 1 of the controller of closed-loop gain control law can meet 0 < T 1 ≤ 1, then the water tank control system shown in Formula (14) will obtain the performance index of gain robust L 2 , as shown in Formula (27), which is 1/2(1 − a 2 ) related to the parameter of controller parameter T 1 .
If the S function is applied to decorate the output (u) of the existing controller, then Based on the Taylor series expansion, the Formula (33) can be obtain When u is not too large, then When γ = 1.5, the above derivation is still valid. The above proof is based on the reference [14] using the L 2 gain robustness performance index. In the actual simulation, it is found that the requirement 0 < T 1 < 1 is conservative and the H ∞ robust theorem is used to re-prove.
According to Formulas (11) and (14), if the output matrix C = 0 1 is added, the sensitivity function S = C(sI − A) −1 B of the system can be obtained, followed by the complementary sensitivity function T of the system.
According to the system robustness theorem, Formula (37) can be obtained as According to Formula (36), if you want to satisfy Formula (38), the damping coefficient of a second-order oscillation system requiring a complementary sensitivity function is greater than or equal to 1. Therefore, According to Young's inequality, it can be obtained that Therefore, Formula (39) is automatically satisfied; hence under the condition that the state variables of water tank level control system satisfy X ∞ > ρ/ 2(1 − a 2 ), as long as T 1 > 0 and T 0 > 0, the robustness of the designed controller can be guaranteed, which is more relaxed than the original condition, and also consistent with the simulation results.

Simulation Experiments and Results
The Simulink toolbox of MATLAB (2016b) was used for the simulation experiment, and its diagrammatic drawing is shown in Figure 5. Compared with the fuzzy controller proposed in reference [18], the robust control proposed in reference [17] has better robust stability and simplicity. However, the improved PD controller proposed in this study is a kind of S function nonlinear decorated control, aiming to achieve energy-saving performance based on the two controllers mentioned above.
The input signal is a square wave with amplitude changing from 0.5 m to 1.5 m. The Formula (2) in this study was applied to design a PD controller. When the max inlet ratio of the tank is ensured at 0.5 m 3 /s due to the effect of nonlinear decorated robust PD control, the mean control input is 0.0273 m 3 /s. The simulation results are displayed in Figure 6. There are no overshoot and quick tracking achieved. Considering the complicated nonlinear model that is used in the simulation experiments while the controller is designed according to the simplified linear model, the system robustness is also guaranteed. (In Figure 6b, the blue line represents the set level and the red line represents the simulated level).
As displayed in Figure 7, which shows the effect of nonlinear feedback, its mean control input is 0.0302 m 3 /s. The control input of the improved algorithm is reduced by 9.6%, compared to the nonlinear-decorated control and the nonlinear feedback control. The purpose of saving energy by applying the S-function-decorated control is then obtained.
The mean control input of nonlinear decorated robust control is 0.0349 m 3 /s if the inlet ratio is decreased from 0.5 m 3 /s to 0.4 m 3 /s, as shown in Figure 8. The control output of the level is almost unchanged with satisfactory robustness.  The input signal is a square wave with amplitude changing from 0.5 m to 1.5 m . The Formula (2) in this study was applied to design a PD controller. When the max inlet ratio of the tank is ensured at 3 0.5m / s due to the effect of nonlinear decorated robust PD control, the mean control input is 3 0.0273 m / s . The simulation results are displayed in Figure 6. There are no overshoot and quick tracking achieved. Considering the complicated nonlinear model that is used in the simulation experiments while the controller is designed according to the simplified linear model, the system robustness is also guaranteed. (In Figure 6b, the blue line represents the set level and the red line represents the simulated level) (a) (b) Figure 6. Simulation results of nonlinear decorated robust controller: (a) represents the control input curve (b) represents the water level curve.
As displayed in Figure 7, which shows the effect of nonlinear feedback, its mean control input is 3 0.0302 m / s . The control input of the improved algorithm is reduced by 9.6%, compared to the nonlinear-decorated control and the nonlinear feedback control. The purpose of saving energy by applying the S-function-decorated control is then obtained.  The input signal is a square wave with amplitude changing from 0.5 m to 1.5 m . The Formula (2) in this study was applied to design a PD controller. When the max inlet ratio of the tank is ensured at 3 0.5m / s due to the effect of nonlinear decorated robust PD control, the mean control input is 3 0.0273 m / s . The simulation results are displayed in Figure 6. There are no overshoot and quick tracking achieved. Considering the complicated nonlinear model that is used in the simulation experiments while the controller is designed according to the simplified linear model, the system robustness is also guaranteed. (In Figure 6b, the blue line represents the set level and the red line represents the simulated level) (a) (b) Figure 6. Simulation results of nonlinear decorated robust controller: (a) represents the control input curve (b) represents the water level curve.
As displayed in Figure 7, which shows the effect of nonlinear feedback, its mean control input is 3 0.0302 m / s . The control input of the improved algorithm is reduced by 9.6%, compared to the nonlinear-decorated control and the nonlinear feedback control. The purpose of saving energy by applying the S-function-decorated control is then obtained.   The effect of nonlinear feedback is shown in Figure 9, its mean control input is  The effect of nonlinear feedback is shown in Figure 9, its mean control input is 0.0365 m 3 /s when the inlet ratio is reduced from 0.5 m 3 /s to 0.4 m 3 /s. Compared with the nonlinear-decorated control and the nonlinear feedback control with a limited inlet ratio, the control input of the improved algorithm is reduced by 4.4%. The performance of linear feedback fuzzy control is displayed in Figure 10, the mean control input is 3 0.0369 m / s , and the effect of linear feedback fuzzy control is displayed in Figure 11 when the limit of inlet ratio is 3 0.4m / s , with the mean control input as 3 0.0421m / s . The mean control input of nonlinear-decorated robust control is reduced by 26.1%, compared with the linear feedback fuzzy control. The purpose of energy-saving is met. Further, the fuzzy control has a slight overshoot. The performance of linear feedback fuzzy control is displayed in Figure 10, the mean control input is 0.0369 m 3 /s, and the effect of linear feedback fuzzy control is displayed in Figure 11 when the limit of inlet ratio is 0.4 m 3 /s, with the mean control input as 0.0421 m 3 /s. The mean control input of nonlinear-decorated robust control is reduced by 26.1%, compared with the linear feedback fuzzy control. The purpose of energy-saving is met. Further, the fuzzy control has a slight overshoot.
The comprehensive control performance index J of (42) is used to evaluate the above three kinds of control algorithms.
J is a function that controls input energy and output error, and the smaller the value is, the better. Table 1 presents a quantitative comparison of the control simulation results mentioned above, and studies the efficiency of the proposed algorithm. It is worth nothing that the response performance specification J of the nonlinear-decorated robust method is decreased by 13.0% and 26.3%, respectively, in contrast to the nonlinear feedback and the fuzzy controller. When the model has some perturbation (the max inlet ratio is changed), the control performance is also satisfactory with good robustness. The performance of linear feedback fuzzy control is displayed in Figure 10, the mean control input is 3 0.0369 m / s , and the effect of linear feedback fuzzy control is displayed in Figure 11 when the limit of inlet ratio is 3 0.4m / s , with the mean control input as 3 0.0421m / s . The mean control input of nonlinear-decorated robust control is reduced by 26.1%, compared with the linear feedback fuzzy control. The purpose of energy-saving is met. Further, the fuzzy control has a slight overshoot.  The comprehensive control performance index J of (42) is used to evaluate the above three kinds of control algorithms. J is a function that controls input energy and output error, and the smaller the value is, the better. Table 1 presents a quantitative comparison of the control simulation results mentioned above, and studies the efficiency of the proposed algorithm. It is worth nothing that the response performance specification J of the nonlinear-decorated robust method is decreased by 13.0% and 26.3%, respectively, in contrast to the nonlinear feedback and the fuzzy controller. When the model has some perturbation (the max inlet ratio is changed), the control performance is also satisfactory with good robustness.    [13] 474.6 608.9 Fuzzy control algorithm 560.1 648.9 Variable rate to the nonlinear feedback (%) [13] 13.0% 12.3% Variable rate to the fuzzy control algorithm (%) 26.3% 17.7%

Conclusions
Through comparing experiments, the control algorithm resulting in the nonlinear decorated robust control has the best system performance index, which can reduce the system performance index up to 13.0% and 26.3%, respectively, compared with the nonlinear feedback and the fuzzy control. The improved controller presented in this study is based on the robust controller proposed in reference [15], when properly preserving its notable robust stability and concision. The most important advantage of the improved edition is its remarkable energy-saving competence. This achievement results from the introduction of the S function to nonlinear-decorated-control to achieve the energy-saving target. The modification method analyzed in this study can be applied to other cases in the engineering industry on a larger scale.