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A Water Tank Level Control System with Time Lag Using CGSA and Nonlinear Switch Decoration

A Water Tank Level Control System with Time Lag Using CGSA and Nonlinear Switch Decoration Article A Water Tank Level Control System with Time Lag Using CGSA and Nonlinear Switch Decoration Weifeng Xu *, Xianku Zhang * and Haoze Wang Navigation College, Dalian Maritime University, Dalian 116026, China * Correspondence: weifeng_xu@dlmu.edu.cn (W.X.); zhangxk@dlmu.edu.cn (X.Z.) Abstract: Tank level control has some unavoidable factors such as disturbance, non-linearity, and time lag. This paper proposes a simple and robust control scheme with nice energy-saving effects and smooth output to improve the quality of the controller and meet real-world application requirements. A linear controller is first designed using a third-order closed-loop gain-shaping algorithm. We then use an arcsine function to modify the system with non-linear switching to reduce the effect of the non-linear modification on the dynamic performance of the control system. Furthermore, we use the Nyquist stability criterion to demonstrate the stability of the closed-loop system in the presence of time lag. The results of the final simulation experiment show that the controller not only has high control quality but also has the characteristics of energy saving and smooth output under the condition of lag and pump performance constraints. These features are necessary for extending the life of the pump and enhancing the applicability of the tank level controller. Keywords: level control; closed-loop gain-shaping; nonlinear switching modification; time-lag system 1. Introduction Level control is commonly applied in large industrial automation production. In the shipping industry, the level control of various ship compartments plays a vital role in coor- dinating the safe navigation of ships, such as oil tanks, ballast tanks, and freshwater tanks. However, tank-level control systems are often subject to uncertainties such as time-varying Citation: Xu, W.; Zhang, X.; Wang, H. state, non-linearity, and time lag. So many scholars have conducted a lot of research on this A Water Tank Level Control System problem. Li et al. [1] and Zhang et al. [2] designed the water tank level controller based on with Time Lag Using CGSA and the PID method, the simulation experiments showed that the PID controller has a better Nonlinear Switch Decoration. Appl. control performance. However, there is still the problem of complex parameter rectification Syst. Innov. 2023, 6, 12. https:// to some extent. Thus, the idea of fuzzy logic and PID control worked together to solve the doi.org/10.3390/asi6010012 problem of parameter tuning [3,4]. Olivas et al. [5] used the ant colony algorithm (ACO) to Academic Editor: Christos Douligeris further optimize the fuzzy controller for the water tank. However, this type of intelligent algorithm is more complex and computationally intensive. Young et al. [6] proposed a Received: 10 December 2022 sliding mode control theory. It is widely used in applications due to its robustness and Revised: 31 December 2022 simplicity. The water tank level controller designed based on the sliding mode control Accepted: 3 January 2023 theory [7–10] worked well. However, the discontinuous switching characteristics of the Published: 16 January 2023 variable structure can cause jitter and vibration in the system. It reduces the service life of the pump and is not conducive to practical engineering applications. Therefore, it is necessary to design a simple tank level control system with the following characteristics. Copyright: © 2023 by the authors. (1) Clear engineering implications and low calculation load. (2) Under the constraints Licensee MDPI, Basel, Switzerland. of pump performance constraints, external environmental interference, and time lag, the This article is an open access article controller should reduce energy loss and extend the service life of the pump while ensuring distributed under the terms and high control effects. conditions of the Creative Commons So, we use a closed-loop gain-shaping algorithm (CGSA) to design the controller. Attribution (CC BY) license (https:// The algorithm is simplified as an engineering application of robust control and has the creativecommons.org/licenses/by/ advantages of a simple design process and obvious physical significance. Zhang et al. [11] 4.0/). Appl. Syst. Innov. 2023, 6, 12. https://doi.org/10.3390/asi6010012 https://www.mdpi.com/journal/asi Appl. Syst. Innov. 2023, 6, 12 2 of 11 proposed a new control scheme for industrial multiple-input multiple-output (MIMO) sys- tems with a time lag using the CGSA method. Guan et al. [12] used CGSA to design a robust PID fin control system to achieve ship rocking reduction control. Jiang et al. [13] designed a wireless network control system based on CGSA and verified its role in a ship course keeping control through a simulation platform. The non-linear modification technique adds a non-linear function component between the control input and the system model. It is often combined with CGSA algorithms to solve the issue of high energy consumption. The technique is widely used in various fields such as ship heading control [14–17], track keeping [18–20], pressure control in insulated spaces [21], parameter identification of ship response models [22], tank level control [23–26], etc. However, for tank level control, CGSA was modified by different functions, such as the arctan [23], sinusoidal function [24], and S function [25]. The control system was modified by the Gaussian function and applied to the control of liquid tanks of LNG vessels [26]. The above research results were searching for the effect of different nonlinear functions on the modification of the CGSA algorithm and applying them to different fields. In this paper, we propose a nonlinear switching theory to reduce its impact on the dynamic performance of the controller while ensuring the original decoration effect. Moreover, we introduce new evaluation metrics to analyze the effectiveness of the control system. The main contributions are listed by (1) We design a concise linear controller using a third-order closed-loop gain-shaping algorithm and modify the linear control law by a nonlinear switching modification tech- nique based on the arcsine function. Meanwhile, a pure time lag component of 0.8 s is introduced to fit the realistic situation, and finally, we use the Nyquist stability criterion to demonstrate the stability of the control system. (2) When doing simulation experiments, we consider the performance constraints of the pump and little interference. Then, we introduce a new evaluation index system to analyze comparatively the controller. The experimental results show that this controller has significant advantages in terms of energy saving, safety, and smoothing. The remainder of the paper is organized as follows: In Section 2, we simplify the non-linear tank level model. In Section 3, the control system is designed. In Section 4, we perform stability analysis in the presence of time lag. In Section 5, We provide simulation examples, and Section 6 concludes. 2. Tank Level Model Assumption. An idealized single tank exists, where both input and output regulating valves are in action to achieve the set height. The key mathematical notations used in this model are listed in Table 1. Table 1. Notations and descriptions. Notations Descriptions Q Steady-state value of input water flow DQ Increment of input water flow Q Steady-state value of output water flow DQ Increment of output water flow h Liquid level height h Steady-state value of the liquid level Dh Increment of liquid level u Regulating the opening of valves A Cross-sectional area of the water tank R Resistance of load valves at the outflow end Du Change in opening of control valve The difference between the inflow and outflow is dv dh DQ DQ = = A (1) dt dt Appl. Syst. Innov. 2023, 6, 12 3 of 11 where V is the tank liquid storage volume. DQ = K D (2) i u u where K is the valve flow coefficient. Q = A 2gh (3) 0 0 where K is the cross-sectional area of the output tube. Equation (3) can be converted to linear at the equilibrium point (h , Q ), then the 0 0 liquid resistance R can be expressed as Dh R = (4) DQ Substituting Equations (2) and (4) into Equation (1) and converting them into transfer function form, we receive h(s) K G(s) = = (5) Q (s) s(T s + 1) i 0 where, K = K R, T = R A. The linear model (5) is used for the design of the controller, 0 u 0 while the non-linear model (1) (3) is used for system simulation to verify the robustness of the designed controller. Assuming that the height of the tank used for this paper is 2 m, the cross-sectional area of the tank is 1 m , the cross-sectional area of the output pipe is 2 3 0.05 m , the initial level is 0.5 m, and the maximum inlet volume of the tank is 0.5 m /s, then R = 2, K = 0.4, K = 0.8s, T = 2 [22]. 0 0 However, the simplified model responds nearly 40% faster than the actual situa- tion [27]. To improve the simulation accuracy of the linear tank control system, a first-order inertial system 1/5s + 1 was added to Equation (5), and it was verified that the regulation time was similar to the actual situation in [27]. Thus, this paper uses this model to improve the confidence of the system simulation, and the new model transfer function is h(s) 0.8 G(s) = = (6) Q (s) s(2s + 1)(5s + 1) 3. Control System Structure Design In this section, we first design the controller using the CGSA and then modify the Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW 4 of 12 above-level-control system using an arcsine function. We add a pure time lag component after the level model to fit the actual situation. The control system structure is shown in Figure 1. Figure 1. Configuration diagram of level keeping control based on nonlinear switch decoration. Figure 1. Configuration diagram of level keeping control based on nonlinear switch decoration. 3.1. Introduction of the Closed-Loop Gain Shaping Algorithm The closed-loop 3.1. Introd gain-shaping uction of the Close algorithm d-Loop Gain is a pr Sh oposed aping Alsimple gorithmand robust control method for stable MIMO processes based on H robust control theory. The core of the The closed-loop gain-shaping algorithm is a proposed simple and robust control method is to determine the final expression of the closed-loop transfer function matrix of method for stable MIMO processes based on H robust control theory. The core of the the system and to design the robust controller using the main parameters of the closed- method is to determine the final expression of the closed-loop transfer function matrix of loop system, i.e., the maximum singular value, bandwidth of frequency, high frequency the system and to design the robust controller using the main parameters of the closed- asymptote slope, and spectrum peak of the closed-loop. loop system, i.e., the maximum singular value, bandwidth of frequency, high frequency asymptote slope, and spectrum peak of the closed-loop. A control algorithm based on closed-loop gain-shaping is given by observing the mixed sensitivity singular value curve of control S/T (see Figure 2) and the correlation SG =+ 1/ (1 K) between the sensitivity function S( , G is the controlled object and K is the TG=+ K/(1 GK) controller) and the complementary sensitivity function T( ). According to the four engineering parameters of the maximum singular value, the bandwidth of fre- quency, the high frequency asymptote slope, and the spectrum peak of the closed-loop, the result of the hybrid sensitivity control algorithm using H control is used to construct the complementary sensitivity function T. The correlation between T and the sensitivity function T is applied to indirectly construct the sensitivity function T, and finally, the con- troller K is inverted. Figure 2. Typical S&T singular value curve. 3.2. Controller Design We use the third-order CGSA to design the tank level controller and set the band- width frequency of the closed-loop system to 1 T . Then the complementary sensitivity function of the tank level control system at this time is also the closed-loop transfer func- tion of the system. Gs K s () ( ) (7) 1 +Gs K s () () Ts + 1 () By substituting Equation (6) into Equation (7), the final robust controller is obtained as Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW 4 of 12 Figure 1. Configuration diagram of level keeping control based on nonlinear switch decoration. 3.1. Introduction of the Closed-Loop Gain Shaping Algorithm The closed-loop gain-shaping algorithm is a proposed simple and robust control method for stable MIMO processes based on H robust control theory. The core of the method is to determine the final expression of the closed-loop transfer function matrix of the system and to design the robust controller using the main parameters of the closed- loop system, i.e., the maximum singular value, bandwidth of frequency, high frequency Appl. Syst. Innov. 2023, 6, 12 4 of 11 asymptote slope, and spectrum peak of the closed-loop. A control algorithm based on closed-loop gain-shaping is given by observing the mixed sensitivity singular value curve of control S/T (see Figure 2) and the correlation A control algorithm based on closed-loop gain-shaping is given by observing the mixed sensitivity singular value curve of H control S/T (see Figure 2) and the correlation SG =+ 1/ (1 K) K between the sensitivity function S( , G is the controlled object and is the between the sensitivity function S(S = 1/(1 + GK), G is the controlled object and K is the TG=+ K/(1 GK) controller) and the complementary sensitivity function T( ). According to controller) and the complementary sensitivity function T(T = GK/(1 + GK)). According the four engineering parameters of the maximum singular value, the bandwidth of fre- to the four engineering parameters of the maximum singular value, the bandwidth of quency, the high frequency asymptote slope, and the spectrum peak of the closed-loop, frequency, the high frequency asymptote slope, and the spectrum peak of the closed-loop, the the r result esult o offthe the hybrid sen hybrid sensitivity sitivitycontr contol rol algo algorithm rithm usin usingg H H contr control i ol issused used to to constr construct uct the complementary sensitivity function T. The correlation between T and the sensitivity the complementary sensitivity function T. The correlation between T and the sensitivity function T is applied to indirectly construct the sensitivity function T, and finally, the function T is applied to indirectly construct the sensitivity function T, and finally, the con- controller K is inverted. troller K is inverted. Figure Figure 2. 2. T Typical S&T singular value curve ypical S&T singular value curve. . 3.2. Controller Design 3.2. Controller Design We use the third-order CGSA to design the tank level controller and set the bandwidth We use the third-order CGSA to design the tank level controller and set the band- frequency of the closed-loop system to 1/T . Then the complementary sensitivity function width frequency of the closed-loop system to 1 T . Then the complementary sensitivity of the tank level control system at this time is also the closed-loop transfer function of function of the tank level control system at this time is also the closed-loop transfer func- the system. tion of the system. G(s)K(s) 1 = (7) 1 + G(s)K(s) (T s + 1) Gs K s () ( ) (7) By substituting Equation (6) into Equation (7), the final robust controller is obtained as 1 +Gs K s () () Ts + 1 () By substituting Equation (6) into Equation (7), the final robust controller is obtained K(s) = (8) GT s(T s + 3T s + 3) 1 1 as 1 As can be seen from Equation (8), the controller designed using the third-order closed- loop gain-shaping algorithm is in the form of a typical PD controller with an oscillating component in series. It is simple and easy to implement, solving the problem of complex parameterization and the unclear physical meaning of conventional PD controllers. 3.3. Improved Non-Linear Switch Modification The non-linear modifier switching technique is essentially a segmentation function. In this paper, the segmentation function is constructed using the arcsine function in the following form. aa sin(bu), u  2 f (u) = (9) u, u > 2 where ab < 1 and b 6= 0, a 6= 0. It indicates that the system output differs significantly from the set output when u is large. It is often the initial stage of the control process or a situation where a major distur- bance occurs. At this time, the arcsine non-linear modification does not work, maintaining the dynamic performance of the control law (8). It means that the difference between the system output and the set output is little when u is small. It is often the stable phase of the control process or a situation where small disturbances occur. Now the arcsine non-linear modification takes effect, making the control input smaller with little impact on Appl. Syst. Innov. 2023, 6, 12 5 of 11 the dynamic performance of the control law (8) and thus reducing energy consumption. Since the dynamic performance of the control law (8) remains unchanged for u > 2. So, we list the following analysis for the case where u  2. When u  2, we choose to retain the first-order Taylor expansion of the arcsine function. aa sin(bu)  abu (10) (1) Effect on the steady state of the system Assume that the input is a unit step signal and then analyze the output steady-state values using a modified model of the control object. According to the final value theorem of the Rasch transform, we obtain Equation (11). GKab C(¥) = lims 1+GKab s s!0 ab 2 2 T s T s +3T s+3 ( ) 1 1 = lim ab 1+ (11) s!0 T s T s +3T s+3 ( ) 1 1 1 ab = lim T s T s +3T s+3 +ab 1 ( 1 ) s!0 = 1 Therefore, the non-linear switching modification of the arcsine function does not affect the final steady state of the system. (2) Effect on the dynamic performance C(s) GKab = (12) R(s) 1 + GKab Equation (12) is the transfer function of the closed-loop system. According to the closed-loop gain-shaping theory, the open-loop transfer function GK of the system meets the requirements of high gain at low frequencies and low gain at high frequencies when ab < 1. Therefore, in the low-frequency range of Equation (12) compared with the standard feedback system GK/(1 + GK), adding ab has little effect on the dynamic performance of the system. (3) Effect on controller output u Kab = (13) u 1 + GKab Equation (13) is the transfer function from the input to the controller output. The numerator of Equation (13) decreases more significantly than the denominator. So ab will reduce the control output. Non-linear switching technology is the introduction of ab to reduce energy consumption during the stabilization phase of the control, but at the same time to reduce the output. 4. Stability Analysis In this section, we begin with an individual analysis of the control law (8) to explore its inherent stability performance. Afterward, the control system, which has the addition of non-linear switching modifications and time lag, is proved to be stable. The stability analysis of a tank level feedback controller designed based on a closed- loop gain-shaping algorithm is commonly used in the Lyapunov stability theory [22–24]. In this paper, the controller follows this scheme and finds that it is necessary to construct and solve a positive definite real symmetric matrix, which is computationally more complex and abstract. Therefore, we refer to the proof of [19] to analyze the stability of the third-order closed-loop gain-forming controller using the Nyquist stability criterion in the frequency domain approach. Appl. Syst. Innov. 2023, 6, 12 6 of 11 4.1. Improved Non-Linear Switch Modification Theorem 1. The closed-loop system of the controller designed using the third-order closed-loop gain- shaping algorithm is stable with amplitude margin h = 9 and phase angle margin g = 71.2528 under the condition T 6= 0. Proof: The open-loop transfer function of the system is obtained from Equation (8): H(s) = K(s)G(s) = 2 2 T s T s +3T s+3 ( ) 1 1 (14) p  p 3 3 3 3 T s T s+ + j T s+ j 1 1 1 2 2 2 2 Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW 7 of 12 Taking jw into the Equation (14). G(jw) = h p ih p i (15) Gjω = 1 () 3 3 3 3 T jw j(T w + ) + j(T w ) + 1 1 1 2 2 2 2  (16) ∠= Gjω −180  () Assume that the cut-off frequency is w and the phase angle junction frequency is w . c g The solution is jG(jw )j = 1 (16)  0.3273 \G jw = 180 ω =  c  1  (17) The solution is  0.3273 w ω= =  1 (17)  1 w = Further solving for the phase angle margin γ = 71.2528 , amplitude margin h = 9 . Further solving for the phase angle margin g = 71.2528 , amplitude margin h = 9. When wω= =0 0 , the real part of G(jw) is When , the real part of Gjω is () 3 1 −31 Re[G(jw)] = = (18)  Re Gjω==− () 4 2 (18) 4 2  44 22 T w + 3T w + 9 TT ωω++ 39 3 1 1 Apparently, Aωϕ =+∞ , /2 ω = −π . As ω changes from 0 to infinity, the ampli- () () Apparently, A(w) = +¥, j(w) = p/2. As w changes from 0 to infinity, the tude and phase angle decreases. When tends to infinity, the phase angle of all three amplitude and phase angle decreases. When w tends to infinity, the phase angle of all three complex complex vectors is vectors is 90 , and , and the the ampl amplitude itudeis is infinity infinity . . S Substituting ubstituting in into to E Equation quation (7) (7) w we e obtain A(w) = 0, j(w) = 3p/2. We make the magnitude-phase characteristic curve of obtain Aωϕ== 0, 3 ω −π/2 . We make the magnitude-phase characteristic curve of () () the system as shown in Figure 3. the system as shown in Figure 3. Figure 3. Amplitude phase characteristic curve of controller. Figure 3. Amplitude phase characteristic curve of controller. From the Nyquist stability criterion, we obtain ZP =−20 N= −2×0=0 (19) P is the number of poles in the right half-plane of the open-loop transfer function. N is the number of turns of the Nyquist curve enclosing −1, j0 and Z is the number of () poles in the right half-plane of the closed-loop transfer function. In summary, this closed- γ = 71.2528 loop system is stable. Its phase angle margin and amplitude margin h = 9 . 4.2. Control System Stability Analysis We add a pure time lag component of 0.8 s to the simulation system to fit the actual situation. However, this results in a degradation of the quality of the control and a reduc- tion in the stability of the system. For this reason, we plot the amplitude-phase character- istic curve for the combined effect of the non-linear maximum modifier and the 0.8 s pure Appl. Syst. Innov. 2023, 6, 12 7 of 11 From the Nyquist stability criterion, we obtain Z = P 2N = 0 2 0 = 0 (19) P is the number of poles in the right half-plane of the open-loop transfer function. N is the number of turns of the Nyquist curve enclosing (1, j0) and Z is the number of poles in the right half-plane of the closed-loop transfer function. In summary, this closed-loop system is stable. Its phase angle margin g = 71.2528 and amplitude margin h = 9. Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW 8 of 12 4.2. Control System Stability Analysis We add a pure time lag component of 0.8 s to the simulation system to fit the actual situation. However, this results in a degradation of the quality of the control and a reduction in the stability of the system. For this reason, we plot the amplitude-phase characteristic curve for the combined effect of the non-linear maximum modifier and the 0.8 s pure time time lag component as shown in Figure 4 and use the Nyquist stability criterion to deter- lag component as shown in Figure 4 and use the Nyquist stability criterion to determine mine the stability of the system. the stability of the system. Figure 4. Amplitude phase characteristic curve of system. Figure 4. Amplitude phase characteristic curve of system. The blue line shows the magnitude-phase characteristic curve of the original system. The black line is the amplitude-phase characteristic curve of the above two links acting The blue line shows the magnitude-phase characteristic curve of the original system. together. The black line is the amplitude-phase characteristic curve of the above two links acting Z = P 2N = 0 2 0 = 0 (20) 1 1 1 together. As a result, the system remains stable. However, the non-linear switching modification selectively changes the open-loop transfer function gain, affecting the amplitude margin of ZP=−20 N = −2×0=0 (20) the system. Additionally, the introduction of a pure time lag component reduces the phase 11 1 angle junction frequency of the system, thus reducing the phase angle margin. As a result, the system remains stable. However, the non-linear switching modifica- 5. Simulation Experiments tion selectively changes the open-loop transfer function gain, affecting the amplitude mar- In this section, simulations are used to verify the effectiveness of the proposed con- troller. The second-order CGSA controller modified by the Gaussian function has good gin of the system. Additionally, the introduction of a pure time lag component reduces energy efficiency and strong robustness in the literature [26]. We select it as a reference and the phase angle junction frequency of the system, thus reducing the phase angle margin. compare it with the proposed controller. 5. Simulation Experiments In this section, simulations are used to verify the effectiveness of the proposed con- troller. The second-order CGSA controller modified by the Gaussian function has good energy efficiency and strong robustness in the literature [26]. We select it as a reference and compare it with the proposed controller. 5.1. Design of Evaluation Indexes Traditional water tank control systems are evaluated through a comprehensive per- formance evaluation index [23–26]. It cannot analyze all aspects of the control system. Moreover, as research progresses, energy efficiency, safety, and economy become new goals to be pursued. For this reason, we introduce the following metrics [19] to evaluate the performance of the controller proposed in this paper in the presence of disturbances and time lag. The MTV is used to determine the degree of variation in pumping speed, which is necessary to study the life of the pump. The MAE is used to measure the response performance of the system output. Additionally, MIA is used to measure the energy con- sumption of the corresponding control algorithm, see Equations (21)–(23). MTV=− u u dt (21)  tt −1 () ( ) tt − ∞ 0 MIA = u dt (22)  t () tt − ∞ 0 1 ∞ MAE=− h h dt (23)  r t () tt − 0 ∞ 0 Appl. Syst. Innov. 2023, 6, 12 8 of 11 5.1. Design of Evaluation Indexes Traditional water tank control systems are evaluated through a comprehensive per- formance evaluation index [23–26]. It cannot analyze all aspects of the control system. Moreover, as research progresses, energy efficiency, safety, and economy become new goals to be pursued. For this reason, we introduce the following metrics [19] to evaluate the performance of the controller proposed in this paper in the presence of disturbances and time lag. The MTV is used to determine the degree of variation in pumping speed, which is necessary to study the life of the pump. The MAE is used to measure the response perfor- mance of the system output. Additionally, MIA is used to measure the energy consumption of the corresponding control algorithm, see Equations (21)–(23). MTV = u u dt (21) (t) (t1) t t ¥ t 0 0 Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW M I A = u dt 9 of(22) 12 (t) t t ¥ 0 M AE = h h dt (23) (t) t t ¥ 0 5.2. Comparative Experiments The set input signal for the level is a square wave that varies from 0.5 m to 1.5 m. We 5.2. Comparative Experiments study the actual tank level control situation and find that there are limitations to both the The set input signal for the level is a square wave that varies from 0.5 m to 1.5 m. We tank inlet flow rate and the inlet acceleration. We introduce a servo system for the inlet study the actual tank level control situation and find that there are limitations to both the water rate and acceleration to make the experiment fit the actual situation. We adopt the tank inlet flow rate and the inlet acceleration. We introduce a servo system for the inlet control variable method throughout the experiment. The conditions are as follows: (1) the water rate and acceleration to make the experiment fit the actual situation. We adopt the objects are all hypothetical water tanks; (2) the simulation time is 500 s and the step size control variable method throughout the experiment. The conditions are as follows: (1) the T = 1 is set to 0.1 s; (3) the control parameter is ; (4) the non-linear switching modification objects are all hypothetical water tanks; (2) 1 the simulation time is 500 s and the step size is set to 0.1 s; (3) the control parameter is T = 1; (4) the non-linear switching modification parameters are α = 1.03 , β = 0.065 ; (5) the interference range simulated with white noise parameters are a = 1.03, b = 0.065; (5) the interference range simulated 3 with white noise is −− 33   0.1m s is −× 410 ,41 × 0 ; (6) the maximum inlet flow rate is and the maximum inlet  3 3 3 4 10 , 4 10 ; (6) the maximum inlet flow rate is 0.1 m /s and the maximum inlet 3 2 0.01m s acceleration is . The control effects of small pump is shown in Figure 5. acceleration is 0.01 m /s . The control effects of small pump is shown in Figure 5. Figure 5. The control effects of small pump. Figure 5. The control effects of small pump. From the above results(see Table 2), we can obtain that for the same non-linear modi- From the above results(see Table 2), we can obtain that for the same non-linear mod- fication function, the third-order CGSA has a reduction of more than 65% in each metric ification function, the third-order CGSA has a reduction of more than 65% in each metric compared to the second-order CGSA. Therefore, the third-order CGSA has a more obvious compared to the second-order CGSA. Therefore, the third-order CGSA has a more obvi- advantage. Furthermore, the non-linear switch modification technique further reduces the ous advantage. Furthermore, the non-linear switch modification technique further re- three main indicators by 62%, 81%, and 67% compared to the Gaussian function. Then, to duces the three main indicators by 62%, 81%, and 67% compared to the Gaussian function. investigate the advantages of the improved model, we varied the maximum inlet flow rate Then, to investigate the advantages of the improved model, we varied the maximum inlet and maximum inlet acceleration to simulate different types of pumps while maintaining flow rate and maximum inlet acceleration to simulate different types of pumps while the other experimental conditions. The maximum inlet flow rate is set to 0.5 m /s, and the 3 2 maintaining the other experimental conditions.. The maximum inlet flow rate is set to maximum inlet acceleration is 0.05 m /s for next experiment. The control effects of large 3 32 0. pump 5m sis , a shown nd the ma in Figur ximum inlet e 6. acceleration is 0.05m s for next experiment. The con- trol effects of large pump is shown in Figure 6. Table 2. Controller performance of small map. Control Method MAE MIA MTV Second order CGSA + Gaussian function trim 0.7347 0.0566 0.0009 Third order CGSA + Gaussian function trim 0.3107 0.0260 0.0009 Third order CGSA + Switching nonlinear trim 0.1167 0.0049 0.0003 Appl. Syst. Innov. 2023, 6, 12 9 of 11 Table 2. Controller performance of small map. Control Method MAE MIA MTV Second order CGSA + Gaussian function trim 0.7347 0.0566 0.0009 Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW 10 of 12 Third order CGSA + Gaussian function trim 0.3107 0.0260 0.0009 Third order CGSA + Switching nonlinear trim 0.1167 0.0049 0.0003 Figure 6. The control effects of large pump. Figure 6. The control effects of large pump. From the above results(see Table 3), we can obtain that the MAE, MIA, and MTV of the From the above results(see Table 3), we can obtain that the MAE, MIA, and MTV of third-order CGSA controller decreased by 84%, 91%, and 69%, compared to the controller the third-order CGSA controller decreased by 84%, 91%, and 69%, compared to the con- with second-order CGSA in the case of the same non-linear modification function. In the troller with second-order CGSA in the case of the same non-linear modification function. case of the same order CGSA, the non-linear switching modification technique proposed In the case of the same order CGSA, the non-linear switching modification technique pro- in this paper results in a 70% and 74% decrease in the MIA and MTV, respectively, but a posed in this paper results in a 70% and 74% decrease in the MIA and MTV, respectively, 20% increase in the MAE value compared to the Gaussian function. Notably, this is due but a 20% increase in the MAE value compared to the Gaussian function. Notably, this is to a decrease in response speed in comparison. However, the overshoot of the algorithm due to a decrease in response speed in comparison. However, the overshoot of the algo- proposed in this paper decreases obviously, which is more significant for the actual control rithm proposed in this paper decreases obviously, which is more significant for the actual of the tank level. In summary, in the presence of time lag, a small amount of interference, control of the tank level. In summary, in the presence of time lag, a small amount of inter- pump speed limitations, etc., the third-order CGSA and non-linear switching modification ference, pump speed limitations, etc., the third-order CGSA and non-linear switching techniques proposed in this paper have significant advantages in terms of control quality, modification techniques proposed in this paper have significant advantages in terms of energy savings, and algorithmic smoothness. The non-linear modified switching technique control quality, energy savings, and algorithmic smoothness. The non-linear modified proposed in this paper performed well in both experiments. It shows that it works well for switching technique proposed in this paper performed well in both experiments. It shows different types of water tanks. Therefore it has great application potential. that it works well for different types of water tanks. Therefore it has great application Table 3. Controller performance of large pump. potential. Control Method MAE MIA MTV Table 3. Controller performance of large pump. Second order CGSA + Gaussian function trim 0.5121 0.1452 0.0049 Control Method MAE MIA MTV Third order CGSA + Gaussian function trim 0.0795 0.0129 0.0015 Second Thiror d or der CGSA + der CGSA +Gaus Switching sian funct nonlinearion ttrimrim 0.5121 0 0.0956 .10.0039452 0.0.00040049 Third order CGSA + Gaussian function trim 0.0795 0.0129 0.0015 Third order CGSA + Switching nonlinear trim 0.0956 0.0039 0.0004 We listed two reasons for the energy-saving effect and smoothness of the controller proposed in this paper. We listed two reasons for the energy-saving effect and smoothness of the controller (1) The controller using second-order CGSA is equivalent to adding a first-order proposed in this paper. filtering component to PD control. The controller using third-order CGSA is equivalent (1) The controller using second-order CGSA is equivalent to adding a first-order fil- to adding a second-order low-pass filter to the PD control. PD control will introduce tering component to PD control. The controller using third-order CGSA is equivalent to high-frequency noise. High-frequency noise can be filtered by both first and second- adding a second-order low-pass filter to the PD control. PD control will introduce high- order low-pass filters. However, the transition band of the second-order low-pass filter is frequency noise. High-frequency noise can be filtered by both first and second-order low- narrower. Then unwanted interference signals will decay faster, and the noise is filtered pass filters. out more cleanly However, the transi . It can be verified tion bby and the of the secon better performance d-order low of -pass filter is the third-order narrower CGSA . Then unwanted interference signals will decay faster, and the noise is filtered out more controller for the same non-linear modifier function. clean(2) ly. IUnlike t can be ve non-linear rified by modification the better petechniques rformance o that f the sacrifice third-or dynamic der CGSA con performance troller for to the reduce same ener non gy -lin consumption, ear modifier function. the non-linear switching modification technique proposed in this paper can selectively change the system gain. The dynamic performance of the system (2) Unlike non-linear modification techniques that sacrifice dynamic performance to reduce energy consumption, the non-linear switching modification technique proposed in this paper can selectively change the system gain. The dynamic performance of the system remains in the initial stage of the control process, and the control input decreases after reaching a steady state to reduce energy consumption. Appl. Syst. Innov. 2023, 6, 12 10 of 11 remains in the initial stage of the control process, and the control input decreases after reaching a steady state to reduce energy consumption. 6. Conclusions This paper designs a concise and robust control scheme for water tank levels that can further improve the control quality. We first design a linear controller using a third-order closed-loop gain-shaping algorithm. The control law is then modified using a nonlinear switching modification technique based on an arcsine function. Additionally, the stability of the closed-loop system with time lag is demonstrated using the Nyquist stability criterion. We use a hypothetical tank model with a small disturbance, a pure lag of 0.8 s, and a pump servo system to verify the controller. Take one of the experiments as an example, compared to the second-order CGSA controller modified by the Gaussian function, the third-order CGSA controller modified by the nonlinear switch in this paper decreases by 81%, 97%, and 92% in MAE, MIA, and MIV. In the case of the same order CGSA, the MIA and MTV of the system with the non-linear switching modification technique reduce by 70% and 74%, respectively. Therefore, the controller reduces energy consumption and helps to extend the service life of the pump while maintaining a better control effect, which is of great significance to the actual large-size tank level control system. Finally, we give a theoretical analysis of the reasons for the excellent energy saving and smoothness of the controller in this paper. Compared with the second-order CGSA simple nonlinear technique, the control algorithm in this paper is more effective in filtering out high-frequency disturbances, which improves the energy-saving effect while further increasing the smoothness of the algorithm. In the future, we will automatically optimize the control parameters using energy saving and other indicators as constraints. Author Contributions: Conceptualization, W.X. and H.W.; methodology, W.X.; software, W.X.; validation, X.Z. and W.X.; writing—original draft preparation, W.X.; writing—review and editing, W.X. and H.W.; supervision, X.Z. and W.X. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Science Foundation of China (Grant No. 51679024), Dalian Innovation Team Support Plan in the Key Research Field (Grant No. 2020RT08), and the Fundamental Research Funds for the Central University (Grant No. 3132021139). Data Availability Statement: Not applicable. Acknowledgments: Much appreciation to each reviewer for their valuable comments and sugges- tions to improve the quality of this note. The authors would like to thank anonymous reviewers for their valuable comments to improve the quality of this article. Conflicts of Interest: The authors declare no conflict of interest. References 1. Li, L.; Li, J.; Gu, J.; Hua, L. Research on PID Control of Double Tank Based on QPSO Algorithm. Control. Eng. China 2021, 28, 1553–1558. [CrossRef] 2. Zhang, X.; Jia, X. Robust PID Algorithm Based on Closed-Loop Gain Shaping and Its Application in Liquid level Control. Shipbuild. China 2000, 41, 37–41. [CrossRef] 3. Zhao, K. Self-adaptive Fuzzy PID Control for Three-tank Water. In Proceedings of the 5th International Conference on Machine Vision (ICMV)-Algorithms, Pattern Recognition and Basic Technologies, Wuhan, China, 20–21 October 2012; p. 87841. [CrossRef] 4. 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Using Sine Function-Based Nonlinear Feedback to Control Water Tank Level. Energies 2021, 14, 7602. [CrossRef] 25. Zhang, X.K.; Song, C.Y. Robust Controller Decorated by Nonlinear S Function and Its Application to Water Tank. Appl. Syst. Innov. 2021, 4, 64. [CrossRef] 26. Zhao, Z.J.; Zhang, X.K.; Li, Z. Tank-Level Control of Liquefied Natural Gas Carrier Based on Gaussian Function Nonlinear Decoration. J. Mar. Sci. Eng. 2020, 8, 695. [CrossRef] 27. Zhang, X.K.; Zhang, G.Q. Nonlinear Feedback Theory and Its Application to Ship Motion Control; Dalian Maritime University Press: Dalian, China, 2000; pp. 135–136. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). 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A Water Tank Level Control System with Time Lag Using CGSA and Nonlinear Switch Decoration

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Article A Water Tank Level Control System with Time Lag Using CGSA and Nonlinear Switch Decoration Weifeng Xu *, Xianku Zhang * and Haoze Wang Navigation College, Dalian Maritime University, Dalian 116026, China * Correspondence: weifeng_xu@dlmu.edu.cn (W.X.); zhangxk@dlmu.edu.cn (X.Z.) Abstract: Tank level control has some unavoidable factors such as disturbance, non-linearity, and time lag. This paper proposes a simple and robust control scheme with nice energy-saving effects and smooth output to improve the quality of the controller and meet real-world application requirements. A linear controller is first designed using a third-order closed-loop gain-shaping algorithm. We then use an arcsine function to modify the system with non-linear switching to reduce the effect of the non-linear modification on the dynamic performance of the control system. Furthermore, we use the Nyquist stability criterion to demonstrate the stability of the closed-loop system in the presence of time lag. The results of the final simulation experiment show that the controller not only has high control quality but also has the characteristics of energy saving and smooth output under the condition of lag and pump performance constraints. These features are necessary for extending the life of the pump and enhancing the applicability of the tank level controller. Keywords: level control; closed-loop gain-shaping; nonlinear switching modification; time-lag system 1. Introduction Level control is commonly applied in large industrial automation production. In the shipping industry, the level control of various ship compartments plays a vital role in coor- dinating the safe navigation of ships, such as oil tanks, ballast tanks, and freshwater tanks. However, tank-level control systems are often subject to uncertainties such as time-varying Citation: Xu, W.; Zhang, X.; Wang, H. state, non-linearity, and time lag. So many scholars have conducted a lot of research on this A Water Tank Level Control System problem. Li et al. [1] and Zhang et al. [2] designed the water tank level controller based on with Time Lag Using CGSA and the PID method, the simulation experiments showed that the PID controller has a better Nonlinear Switch Decoration. Appl. control performance. However, there is still the problem of complex parameter rectification Syst. Innov. 2023, 6, 12. https:// to some extent. Thus, the idea of fuzzy logic and PID control worked together to solve the doi.org/10.3390/asi6010012 problem of parameter tuning [3,4]. Olivas et al. [5] used the ant colony algorithm (ACO) to Academic Editor: Christos Douligeris further optimize the fuzzy controller for the water tank. However, this type of intelligent algorithm is more complex and computationally intensive. Young et al. [6] proposed a Received: 10 December 2022 sliding mode control theory. It is widely used in applications due to its robustness and Revised: 31 December 2022 simplicity. The water tank level controller designed based on the sliding mode control Accepted: 3 January 2023 theory [7–10] worked well. However, the discontinuous switching characteristics of the Published: 16 January 2023 variable structure can cause jitter and vibration in the system. It reduces the service life of the pump and is not conducive to practical engineering applications. Therefore, it is necessary to design a simple tank level control system with the following characteristics. Copyright: © 2023 by the authors. (1) Clear engineering implications and low calculation load. (2) Under the constraints Licensee MDPI, Basel, Switzerland. of pump performance constraints, external environmental interference, and time lag, the This article is an open access article controller should reduce energy loss and extend the service life of the pump while ensuring distributed under the terms and high control effects. conditions of the Creative Commons So, we use a closed-loop gain-shaping algorithm (CGSA) to design the controller. Attribution (CC BY) license (https:// The algorithm is simplified as an engineering application of robust control and has the creativecommons.org/licenses/by/ advantages of a simple design process and obvious physical significance. Zhang et al. [11] 4.0/). Appl. Syst. Innov. 2023, 6, 12. https://doi.org/10.3390/asi6010012 https://www.mdpi.com/journal/asi Appl. Syst. Innov. 2023, 6, 12 2 of 11 proposed a new control scheme for industrial multiple-input multiple-output (MIMO) sys- tems with a time lag using the CGSA method. Guan et al. [12] used CGSA to design a robust PID fin control system to achieve ship rocking reduction control. Jiang et al. [13] designed a wireless network control system based on CGSA and verified its role in a ship course keeping control through a simulation platform. The non-linear modification technique adds a non-linear function component between the control input and the system model. It is often combined with CGSA algorithms to solve the issue of high energy consumption. The technique is widely used in various fields such as ship heading control [14–17], track keeping [18–20], pressure control in insulated spaces [21], parameter identification of ship response models [22], tank level control [23–26], etc. However, for tank level control, CGSA was modified by different functions, such as the arctan [23], sinusoidal function [24], and S function [25]. The control system was modified by the Gaussian function and applied to the control of liquid tanks of LNG vessels [26]. The above research results were searching for the effect of different nonlinear functions on the modification of the CGSA algorithm and applying them to different fields. In this paper, we propose a nonlinear switching theory to reduce its impact on the dynamic performance of the controller while ensuring the original decoration effect. Moreover, we introduce new evaluation metrics to analyze the effectiveness of the control system. The main contributions are listed by (1) We design a concise linear controller using a third-order closed-loop gain-shaping algorithm and modify the linear control law by a nonlinear switching modification tech- nique based on the arcsine function. Meanwhile, a pure time lag component of 0.8 s is introduced to fit the realistic situation, and finally, we use the Nyquist stability criterion to demonstrate the stability of the control system. (2) When doing simulation experiments, we consider the performance constraints of the pump and little interference. Then, we introduce a new evaluation index system to analyze comparatively the controller. The experimental results show that this controller has significant advantages in terms of energy saving, safety, and smoothing. The remainder of the paper is organized as follows: In Section 2, we simplify the non-linear tank level model. In Section 3, the control system is designed. In Section 4, we perform stability analysis in the presence of time lag. In Section 5, We provide simulation examples, and Section 6 concludes. 2. Tank Level Model Assumption. An idealized single tank exists, where both input and output regulating valves are in action to achieve the set height. The key mathematical notations used in this model are listed in Table 1. Table 1. Notations and descriptions. Notations Descriptions Q Steady-state value of input water flow DQ Increment of input water flow Q Steady-state value of output water flow DQ Increment of output water flow h Liquid level height h Steady-state value of the liquid level Dh Increment of liquid level u Regulating the opening of valves A Cross-sectional area of the water tank R Resistance of load valves at the outflow end Du Change in opening of control valve The difference between the inflow and outflow is dv dh DQ DQ = = A (1) dt dt Appl. Syst. Innov. 2023, 6, 12 3 of 11 where V is the tank liquid storage volume. DQ = K D (2) i u u where K is the valve flow coefficient. Q = A 2gh (3) 0 0 where K is the cross-sectional area of the output tube. Equation (3) can be converted to linear at the equilibrium point (h , Q ), then the 0 0 liquid resistance R can be expressed as Dh R = (4) DQ Substituting Equations (2) and (4) into Equation (1) and converting them into transfer function form, we receive h(s) K G(s) = = (5) Q (s) s(T s + 1) i 0 where, K = K R, T = R A. The linear model (5) is used for the design of the controller, 0 u 0 while the non-linear model (1) (3) is used for system simulation to verify the robustness of the designed controller. Assuming that the height of the tank used for this paper is 2 m, the cross-sectional area of the tank is 1 m , the cross-sectional area of the output pipe is 2 3 0.05 m , the initial level is 0.5 m, and the maximum inlet volume of the tank is 0.5 m /s, then R = 2, K = 0.4, K = 0.8s, T = 2 [22]. 0 0 However, the simplified model responds nearly 40% faster than the actual situa- tion [27]. To improve the simulation accuracy of the linear tank control system, a first-order inertial system 1/5s + 1 was added to Equation (5), and it was verified that the regulation time was similar to the actual situation in [27]. Thus, this paper uses this model to improve the confidence of the system simulation, and the new model transfer function is h(s) 0.8 G(s) = = (6) Q (s) s(2s + 1)(5s + 1) 3. Control System Structure Design In this section, we first design the controller using the CGSA and then modify the Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW 4 of 12 above-level-control system using an arcsine function. We add a pure time lag component after the level model to fit the actual situation. The control system structure is shown in Figure 1. Figure 1. Configuration diagram of level keeping control based on nonlinear switch decoration. Figure 1. Configuration diagram of level keeping control based on nonlinear switch decoration. 3.1. Introduction of the Closed-Loop Gain Shaping Algorithm The closed-loop 3.1. Introd gain-shaping uction of the Close algorithm d-Loop Gain is a pr Sh oposed aping Alsimple gorithmand robust control method for stable MIMO processes based on H robust control theory. The core of the The closed-loop gain-shaping algorithm is a proposed simple and robust control method is to determine the final expression of the closed-loop transfer function matrix of method for stable MIMO processes based on H robust control theory. The core of the the system and to design the robust controller using the main parameters of the closed- method is to determine the final expression of the closed-loop transfer function matrix of loop system, i.e., the maximum singular value, bandwidth of frequency, high frequency the system and to design the robust controller using the main parameters of the closed- asymptote slope, and spectrum peak of the closed-loop. loop system, i.e., the maximum singular value, bandwidth of frequency, high frequency asymptote slope, and spectrum peak of the closed-loop. A control algorithm based on closed-loop gain-shaping is given by observing the mixed sensitivity singular value curve of control S/T (see Figure 2) and the correlation SG =+ 1/ (1 K) between the sensitivity function S( , G is the controlled object and K is the TG=+ K/(1 GK) controller) and the complementary sensitivity function T( ). According to the four engineering parameters of the maximum singular value, the bandwidth of fre- quency, the high frequency asymptote slope, and the spectrum peak of the closed-loop, the result of the hybrid sensitivity control algorithm using H control is used to construct the complementary sensitivity function T. The correlation between T and the sensitivity function T is applied to indirectly construct the sensitivity function T, and finally, the con- troller K is inverted. Figure 2. Typical S&T singular value curve. 3.2. Controller Design We use the third-order CGSA to design the tank level controller and set the band- width frequency of the closed-loop system to 1 T . Then the complementary sensitivity function of the tank level control system at this time is also the closed-loop transfer func- tion of the system. Gs K s () ( ) (7) 1 +Gs K s () () Ts + 1 () By substituting Equation (6) into Equation (7), the final robust controller is obtained as Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW 4 of 12 Figure 1. Configuration diagram of level keeping control based on nonlinear switch decoration. 3.1. Introduction of the Closed-Loop Gain Shaping Algorithm The closed-loop gain-shaping algorithm is a proposed simple and robust control method for stable MIMO processes based on H robust control theory. The core of the method is to determine the final expression of the closed-loop transfer function matrix of the system and to design the robust controller using the main parameters of the closed- loop system, i.e., the maximum singular value, bandwidth of frequency, high frequency Appl. Syst. Innov. 2023, 6, 12 4 of 11 asymptote slope, and spectrum peak of the closed-loop. A control algorithm based on closed-loop gain-shaping is given by observing the mixed sensitivity singular value curve of control S/T (see Figure 2) and the correlation A control algorithm based on closed-loop gain-shaping is given by observing the mixed sensitivity singular value curve of H control S/T (see Figure 2) and the correlation SG =+ 1/ (1 K) K between the sensitivity function S( , G is the controlled object and is the between the sensitivity function S(S = 1/(1 + GK), G is the controlled object and K is the TG=+ K/(1 GK) controller) and the complementary sensitivity function T( ). According to controller) and the complementary sensitivity function T(T = GK/(1 + GK)). According the four engineering parameters of the maximum singular value, the bandwidth of fre- to the four engineering parameters of the maximum singular value, the bandwidth of quency, the high frequency asymptote slope, and the spectrum peak of the closed-loop, frequency, the high frequency asymptote slope, and the spectrum peak of the closed-loop, the the r result esult o offthe the hybrid sen hybrid sensitivity sitivitycontr contol rol algo algorithm rithm usin usingg H H contr control i ol issused used to to constr construct uct the complementary sensitivity function T. The correlation between T and the sensitivity the complementary sensitivity function T. The correlation between T and the sensitivity function T is applied to indirectly construct the sensitivity function T, and finally, the function T is applied to indirectly construct the sensitivity function T, and finally, the con- controller K is inverted. troller K is inverted. Figure Figure 2. 2. T Typical S&T singular value curve ypical S&T singular value curve. . 3.2. Controller Design 3.2. Controller Design We use the third-order CGSA to design the tank level controller and set the bandwidth We use the third-order CGSA to design the tank level controller and set the band- frequency of the closed-loop system to 1/T . Then the complementary sensitivity function width frequency of the closed-loop system to 1 T . Then the complementary sensitivity of the tank level control system at this time is also the closed-loop transfer function of function of the tank level control system at this time is also the closed-loop transfer func- the system. tion of the system. G(s)K(s) 1 = (7) 1 + G(s)K(s) (T s + 1) Gs K s () ( ) (7) By substituting Equation (6) into Equation (7), the final robust controller is obtained as 1 +Gs K s () () Ts + 1 () By substituting Equation (6) into Equation (7), the final robust controller is obtained K(s) = (8) GT s(T s + 3T s + 3) 1 1 as 1 As can be seen from Equation (8), the controller designed using the third-order closed- loop gain-shaping algorithm is in the form of a typical PD controller with an oscillating component in series. It is simple and easy to implement, solving the problem of complex parameterization and the unclear physical meaning of conventional PD controllers. 3.3. Improved Non-Linear Switch Modification The non-linear modifier switching technique is essentially a segmentation function. In this paper, the segmentation function is constructed using the arcsine function in the following form. aa sin(bu), u  2 f (u) = (9) u, u > 2 where ab < 1 and b 6= 0, a 6= 0. It indicates that the system output differs significantly from the set output when u is large. It is often the initial stage of the control process or a situation where a major distur- bance occurs. At this time, the arcsine non-linear modification does not work, maintaining the dynamic performance of the control law (8). It means that the difference between the system output and the set output is little when u is small. It is often the stable phase of the control process or a situation where small disturbances occur. Now the arcsine non-linear modification takes effect, making the control input smaller with little impact on Appl. Syst. Innov. 2023, 6, 12 5 of 11 the dynamic performance of the control law (8) and thus reducing energy consumption. Since the dynamic performance of the control law (8) remains unchanged for u > 2. So, we list the following analysis for the case where u  2. When u  2, we choose to retain the first-order Taylor expansion of the arcsine function. aa sin(bu)  abu (10) (1) Effect on the steady state of the system Assume that the input is a unit step signal and then analyze the output steady-state values using a modified model of the control object. According to the final value theorem of the Rasch transform, we obtain Equation (11). GKab C(¥) = lims 1+GKab s s!0 ab 2 2 T s T s +3T s+3 ( ) 1 1 = lim ab 1+ (11) s!0 T s T s +3T s+3 ( ) 1 1 1 ab = lim T s T s +3T s+3 +ab 1 ( 1 ) s!0 = 1 Therefore, the non-linear switching modification of the arcsine function does not affect the final steady state of the system. (2) Effect on the dynamic performance C(s) GKab = (12) R(s) 1 + GKab Equation (12) is the transfer function of the closed-loop system. According to the closed-loop gain-shaping theory, the open-loop transfer function GK of the system meets the requirements of high gain at low frequencies and low gain at high frequencies when ab < 1. Therefore, in the low-frequency range of Equation (12) compared with the standard feedback system GK/(1 + GK), adding ab has little effect on the dynamic performance of the system. (3) Effect on controller output u Kab = (13) u 1 + GKab Equation (13) is the transfer function from the input to the controller output. The numerator of Equation (13) decreases more significantly than the denominator. So ab will reduce the control output. Non-linear switching technology is the introduction of ab to reduce energy consumption during the stabilization phase of the control, but at the same time to reduce the output. 4. Stability Analysis In this section, we begin with an individual analysis of the control law (8) to explore its inherent stability performance. Afterward, the control system, which has the addition of non-linear switching modifications and time lag, is proved to be stable. The stability analysis of a tank level feedback controller designed based on a closed- loop gain-shaping algorithm is commonly used in the Lyapunov stability theory [22–24]. In this paper, the controller follows this scheme and finds that it is necessary to construct and solve a positive definite real symmetric matrix, which is computationally more complex and abstract. Therefore, we refer to the proof of [19] to analyze the stability of the third-order closed-loop gain-forming controller using the Nyquist stability criterion in the frequency domain approach. Appl. Syst. Innov. 2023, 6, 12 6 of 11 4.1. Improved Non-Linear Switch Modification Theorem 1. The closed-loop system of the controller designed using the third-order closed-loop gain- shaping algorithm is stable with amplitude margin h = 9 and phase angle margin g = 71.2528 under the condition T 6= 0. Proof: The open-loop transfer function of the system is obtained from Equation (8): H(s) = K(s)G(s) = 2 2 T s T s +3T s+3 ( ) 1 1 (14) p  p 3 3 3 3 T s T s+ + j T s+ j 1 1 1 2 2 2 2 Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW 7 of 12 Taking jw into the Equation (14). G(jw) = h p ih p i (15) Gjω = 1 () 3 3 3 3 T jw j(T w + ) + j(T w ) + 1 1 1 2 2 2 2  (16) ∠= Gjω −180  () Assume that the cut-off frequency is w and the phase angle junction frequency is w . c g The solution is jG(jw )j = 1 (16)  0.3273 \G jw = 180 ω =  c  1  (17) The solution is  0.3273 w ω= =  1 (17)  1 w = Further solving for the phase angle margin γ = 71.2528 , amplitude margin h = 9 . Further solving for the phase angle margin g = 71.2528 , amplitude margin h = 9. When wω= =0 0 , the real part of G(jw) is When , the real part of Gjω is () 3 1 −31 Re[G(jw)] = = (18)  Re Gjω==− () 4 2 (18) 4 2  44 22 T w + 3T w + 9 TT ωω++ 39 3 1 1 Apparently, Aωϕ =+∞ , /2 ω = −π . As ω changes from 0 to infinity, the ampli- () () Apparently, A(w) = +¥, j(w) = p/2. As w changes from 0 to infinity, the tude and phase angle decreases. When tends to infinity, the phase angle of all three amplitude and phase angle decreases. When w tends to infinity, the phase angle of all three complex complex vectors is vectors is 90 , and , and the the ampl amplitude itudeis is infinity infinity . . S Substituting ubstituting in into to E Equation quation (7) (7) w we e obtain A(w) = 0, j(w) = 3p/2. We make the magnitude-phase characteristic curve of obtain Aωϕ== 0, 3 ω −π/2 . We make the magnitude-phase characteristic curve of () () the system as shown in Figure 3. the system as shown in Figure 3. Figure 3. Amplitude phase characteristic curve of controller. Figure 3. Amplitude phase characteristic curve of controller. From the Nyquist stability criterion, we obtain ZP =−20 N= −2×0=0 (19) P is the number of poles in the right half-plane of the open-loop transfer function. N is the number of turns of the Nyquist curve enclosing −1, j0 and Z is the number of () poles in the right half-plane of the closed-loop transfer function. In summary, this closed- γ = 71.2528 loop system is stable. Its phase angle margin and amplitude margin h = 9 . 4.2. Control System Stability Analysis We add a pure time lag component of 0.8 s to the simulation system to fit the actual situation. However, this results in a degradation of the quality of the control and a reduc- tion in the stability of the system. For this reason, we plot the amplitude-phase character- istic curve for the combined effect of the non-linear maximum modifier and the 0.8 s pure Appl. Syst. Innov. 2023, 6, 12 7 of 11 From the Nyquist stability criterion, we obtain Z = P 2N = 0 2 0 = 0 (19) P is the number of poles in the right half-plane of the open-loop transfer function. N is the number of turns of the Nyquist curve enclosing (1, j0) and Z is the number of poles in the right half-plane of the closed-loop transfer function. In summary, this closed-loop system is stable. Its phase angle margin g = 71.2528 and amplitude margin h = 9. Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW 8 of 12 4.2. Control System Stability Analysis We add a pure time lag component of 0.8 s to the simulation system to fit the actual situation. However, this results in a degradation of the quality of the control and a reduction in the stability of the system. For this reason, we plot the amplitude-phase characteristic curve for the combined effect of the non-linear maximum modifier and the 0.8 s pure time time lag component as shown in Figure 4 and use the Nyquist stability criterion to deter- lag component as shown in Figure 4 and use the Nyquist stability criterion to determine mine the stability of the system. the stability of the system. Figure 4. Amplitude phase characteristic curve of system. Figure 4. Amplitude phase characteristic curve of system. The blue line shows the magnitude-phase characteristic curve of the original system. The black line is the amplitude-phase characteristic curve of the above two links acting The blue line shows the magnitude-phase characteristic curve of the original system. together. The black line is the amplitude-phase characteristic curve of the above two links acting Z = P 2N = 0 2 0 = 0 (20) 1 1 1 together. As a result, the system remains stable. However, the non-linear switching modification selectively changes the open-loop transfer function gain, affecting the amplitude margin of ZP=−20 N = −2×0=0 (20) the system. Additionally, the introduction of a pure time lag component reduces the phase 11 1 angle junction frequency of the system, thus reducing the phase angle margin. As a result, the system remains stable. However, the non-linear switching modifica- 5. Simulation Experiments tion selectively changes the open-loop transfer function gain, affecting the amplitude mar- In this section, simulations are used to verify the effectiveness of the proposed con- troller. The second-order CGSA controller modified by the Gaussian function has good gin of the system. Additionally, the introduction of a pure time lag component reduces energy efficiency and strong robustness in the literature [26]. We select it as a reference and the phase angle junction frequency of the system, thus reducing the phase angle margin. compare it with the proposed controller. 5. Simulation Experiments In this section, simulations are used to verify the effectiveness of the proposed con- troller. The second-order CGSA controller modified by the Gaussian function has good energy efficiency and strong robustness in the literature [26]. We select it as a reference and compare it with the proposed controller. 5.1. Design of Evaluation Indexes Traditional water tank control systems are evaluated through a comprehensive per- formance evaluation index [23–26]. It cannot analyze all aspects of the control system. Moreover, as research progresses, energy efficiency, safety, and economy become new goals to be pursued. For this reason, we introduce the following metrics [19] to evaluate the performance of the controller proposed in this paper in the presence of disturbances and time lag. The MTV is used to determine the degree of variation in pumping speed, which is necessary to study the life of the pump. The MAE is used to measure the response performance of the system output. Additionally, MIA is used to measure the energy con- sumption of the corresponding control algorithm, see Equations (21)–(23). MTV=− u u dt (21)  tt −1 () ( ) tt − ∞ 0 MIA = u dt (22)  t () tt − ∞ 0 1 ∞ MAE=− h h dt (23)  r t () tt − 0 ∞ 0 Appl. Syst. Innov. 2023, 6, 12 8 of 11 5.1. Design of Evaluation Indexes Traditional water tank control systems are evaluated through a comprehensive per- formance evaluation index [23–26]. It cannot analyze all aspects of the control system. Moreover, as research progresses, energy efficiency, safety, and economy become new goals to be pursued. For this reason, we introduce the following metrics [19] to evaluate the performance of the controller proposed in this paper in the presence of disturbances and time lag. The MTV is used to determine the degree of variation in pumping speed, which is necessary to study the life of the pump. The MAE is used to measure the response perfor- mance of the system output. Additionally, MIA is used to measure the energy consumption of the corresponding control algorithm, see Equations (21)–(23). MTV = u u dt (21) (t) (t1) t t ¥ t 0 0 Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW M I A = u dt 9 of(22) 12 (t) t t ¥ 0 M AE = h h dt (23) (t) t t ¥ 0 5.2. Comparative Experiments The set input signal for the level is a square wave that varies from 0.5 m to 1.5 m. We 5.2. Comparative Experiments study the actual tank level control situation and find that there are limitations to both the The set input signal for the level is a square wave that varies from 0.5 m to 1.5 m. We tank inlet flow rate and the inlet acceleration. We introduce a servo system for the inlet study the actual tank level control situation and find that there are limitations to both the water rate and acceleration to make the experiment fit the actual situation. We adopt the tank inlet flow rate and the inlet acceleration. We introduce a servo system for the inlet control variable method throughout the experiment. The conditions are as follows: (1) the water rate and acceleration to make the experiment fit the actual situation. We adopt the objects are all hypothetical water tanks; (2) the simulation time is 500 s and the step size control variable method throughout the experiment. The conditions are as follows: (1) the T = 1 is set to 0.1 s; (3) the control parameter is ; (4) the non-linear switching modification objects are all hypothetical water tanks; (2) 1 the simulation time is 500 s and the step size is set to 0.1 s; (3) the control parameter is T = 1; (4) the non-linear switching modification parameters are α = 1.03 , β = 0.065 ; (5) the interference range simulated with white noise parameters are a = 1.03, b = 0.065; (5) the interference range simulated 3 with white noise is −− 33   0.1m s is −× 410 ,41 × 0 ; (6) the maximum inlet flow rate is and the maximum inlet  3 3 3 4 10 , 4 10 ; (6) the maximum inlet flow rate is 0.1 m /s and the maximum inlet 3 2 0.01m s acceleration is . The control effects of small pump is shown in Figure 5. acceleration is 0.01 m /s . The control effects of small pump is shown in Figure 5. Figure 5. The control effects of small pump. Figure 5. The control effects of small pump. From the above results(see Table 2), we can obtain that for the same non-linear modi- From the above results(see Table 2), we can obtain that for the same non-linear mod- fication function, the third-order CGSA has a reduction of more than 65% in each metric ification function, the third-order CGSA has a reduction of more than 65% in each metric compared to the second-order CGSA. Therefore, the third-order CGSA has a more obvious compared to the second-order CGSA. Therefore, the third-order CGSA has a more obvi- advantage. Furthermore, the non-linear switch modification technique further reduces the ous advantage. Furthermore, the non-linear switch modification technique further re- three main indicators by 62%, 81%, and 67% compared to the Gaussian function. Then, to duces the three main indicators by 62%, 81%, and 67% compared to the Gaussian function. investigate the advantages of the improved model, we varied the maximum inlet flow rate Then, to investigate the advantages of the improved model, we varied the maximum inlet and maximum inlet acceleration to simulate different types of pumps while maintaining flow rate and maximum inlet acceleration to simulate different types of pumps while the other experimental conditions. The maximum inlet flow rate is set to 0.5 m /s, and the 3 2 maintaining the other experimental conditions.. The maximum inlet flow rate is set to maximum inlet acceleration is 0.05 m /s for next experiment. The control effects of large 3 32 0. pump 5m sis , a shown nd the ma in Figur ximum inlet e 6. acceleration is 0.05m s for next experiment. The con- trol effects of large pump is shown in Figure 6. Table 2. Controller performance of small map. Control Method MAE MIA MTV Second order CGSA + Gaussian function trim 0.7347 0.0566 0.0009 Third order CGSA + Gaussian function trim 0.3107 0.0260 0.0009 Third order CGSA + Switching nonlinear trim 0.1167 0.0049 0.0003 Appl. Syst. Innov. 2023, 6, 12 9 of 11 Table 2. Controller performance of small map. Control Method MAE MIA MTV Second order CGSA + Gaussian function trim 0.7347 0.0566 0.0009 Appl. Syst. Innov. 2023, 6, x FOR PEER REVIEW 10 of 12 Third order CGSA + Gaussian function trim 0.3107 0.0260 0.0009 Third order CGSA + Switching nonlinear trim 0.1167 0.0049 0.0003 Figure 6. The control effects of large pump. Figure 6. The control effects of large pump. From the above results(see Table 3), we can obtain that the MAE, MIA, and MTV of the From the above results(see Table 3), we can obtain that the MAE, MIA, and MTV of third-order CGSA controller decreased by 84%, 91%, and 69%, compared to the controller the third-order CGSA controller decreased by 84%, 91%, and 69%, compared to the con- with second-order CGSA in the case of the same non-linear modification function. In the troller with second-order CGSA in the case of the same non-linear modification function. case of the same order CGSA, the non-linear switching modification technique proposed In the case of the same order CGSA, the non-linear switching modification technique pro- in this paper results in a 70% and 74% decrease in the MIA and MTV, respectively, but a posed in this paper results in a 70% and 74% decrease in the MIA and MTV, respectively, 20% increase in the MAE value compared to the Gaussian function. Notably, this is due but a 20% increase in the MAE value compared to the Gaussian function. Notably, this is to a decrease in response speed in comparison. However, the overshoot of the algorithm due to a decrease in response speed in comparison. However, the overshoot of the algo- proposed in this paper decreases obviously, which is more significant for the actual control rithm proposed in this paper decreases obviously, which is more significant for the actual of the tank level. In summary, in the presence of time lag, a small amount of interference, control of the tank level. In summary, in the presence of time lag, a small amount of inter- pump speed limitations, etc., the third-order CGSA and non-linear switching modification ference, pump speed limitations, etc., the third-order CGSA and non-linear switching techniques proposed in this paper have significant advantages in terms of control quality, modification techniques proposed in this paper have significant advantages in terms of energy savings, and algorithmic smoothness. The non-linear modified switching technique control quality, energy savings, and algorithmic smoothness. The non-linear modified proposed in this paper performed well in both experiments. It shows that it works well for switching technique proposed in this paper performed well in both experiments. It shows different types of water tanks. Therefore it has great application potential. that it works well for different types of water tanks. Therefore it has great application Table 3. Controller performance of large pump. potential. Control Method MAE MIA MTV Table 3. Controller performance of large pump. Second order CGSA + Gaussian function trim 0.5121 0.1452 0.0049 Control Method MAE MIA MTV Third order CGSA + Gaussian function trim 0.0795 0.0129 0.0015 Second Thiror d or der CGSA + der CGSA +Gaus Switching sian funct nonlinearion ttrimrim 0.5121 0 0.0956 .10.0039452 0.0.00040049 Third order CGSA + Gaussian function trim 0.0795 0.0129 0.0015 Third order CGSA + Switching nonlinear trim 0.0956 0.0039 0.0004 We listed two reasons for the energy-saving effect and smoothness of the controller proposed in this paper. We listed two reasons for the energy-saving effect and smoothness of the controller (1) The controller using second-order CGSA is equivalent to adding a first-order proposed in this paper. filtering component to PD control. The controller using third-order CGSA is equivalent (1) The controller using second-order CGSA is equivalent to adding a first-order fil- to adding a second-order low-pass filter to the PD control. PD control will introduce tering component to PD control. The controller using third-order CGSA is equivalent to high-frequency noise. High-frequency noise can be filtered by both first and second- adding a second-order low-pass filter to the PD control. PD control will introduce high- order low-pass filters. However, the transition band of the second-order low-pass filter is frequency noise. High-frequency noise can be filtered by both first and second-order low- narrower. Then unwanted interference signals will decay faster, and the noise is filtered pass filters. out more cleanly However, the transi . It can be verified tion bby and the of the secon better performance d-order low of -pass filter is the third-order narrower CGSA . Then unwanted interference signals will decay faster, and the noise is filtered out more controller for the same non-linear modifier function. clean(2) ly. IUnlike t can be ve non-linear rified by modification the better petechniques rformance o that f the sacrifice third-or dynamic der CGSA con performance troller for to the reduce same ener non gy -lin consumption, ear modifier function. the non-linear switching modification technique proposed in this paper can selectively change the system gain. The dynamic performance of the system (2) Unlike non-linear modification techniques that sacrifice dynamic performance to reduce energy consumption, the non-linear switching modification technique proposed in this paper can selectively change the system gain. The dynamic performance of the system remains in the initial stage of the control process, and the control input decreases after reaching a steady state to reduce energy consumption. Appl. Syst. Innov. 2023, 6, 12 10 of 11 remains in the initial stage of the control process, and the control input decreases after reaching a steady state to reduce energy consumption. 6. Conclusions This paper designs a concise and robust control scheme for water tank levels that can further improve the control quality. We first design a linear controller using a third-order closed-loop gain-shaping algorithm. The control law is then modified using a nonlinear switching modification technique based on an arcsine function. Additionally, the stability of the closed-loop system with time lag is demonstrated using the Nyquist stability criterion. We use a hypothetical tank model with a small disturbance, a pure lag of 0.8 s, and a pump servo system to verify the controller. Take one of the experiments as an example, compared to the second-order CGSA controller modified by the Gaussian function, the third-order CGSA controller modified by the nonlinear switch in this paper decreases by 81%, 97%, and 92% in MAE, MIA, and MIV. In the case of the same order CGSA, the MIA and MTV of the system with the non-linear switching modification technique reduce by 70% and 74%, respectively. Therefore, the controller reduces energy consumption and helps to extend the service life of the pump while maintaining a better control effect, which is of great significance to the actual large-size tank level control system. Finally, we give a theoretical analysis of the reasons for the excellent energy saving and smoothness of the controller in this paper. Compared with the second-order CGSA simple nonlinear technique, the control algorithm in this paper is more effective in filtering out high-frequency disturbances, which improves the energy-saving effect while further increasing the smoothness of the algorithm. In the future, we will automatically optimize the control parameters using energy saving and other indicators as constraints. Author Contributions: Conceptualization, W.X. and H.W.; methodology, W.X.; software, W.X.; validation, X.Z. and W.X.; writing—original draft preparation, W.X.; writing—review and editing, W.X. and H.W.; supervision, X.Z. and W.X. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Science Foundation of China (Grant No. 51679024), Dalian Innovation Team Support Plan in the Key Research Field (Grant No. 2020RT08), and the Fundamental Research Funds for the Central University (Grant No. 3132021139). 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Journal

Applied System Innovation – Multidisciplinary Digital Publishing Institute

Published: Jan 16, 2023

Keywords: level control; closed-loop gain-shaping; nonlinear switching modification; time-lag system

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