The flatness of the lead cathode plate affects the electrolytic efficiency of lead and the production efficiency of the whole lead electrolytic industry. However, the dynamic response of the force control system of the leveler is slow, and the anti-interference and robustness are poor. By comparing proportion integration differentiation (PID) control and feedback linearized synovial control two control strategies, MATLAB software was used for modeling and simulation analysis. The results show that the system with feedback linearized sliding mode control has faster response, higher precision and better robustness

Cai, Y. Q.

Wang, Y. F.

Yin, Q. Q.

English

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117METALURGIJA 63 (2024) 1, 117-120Y. F. WANG, Q. Q. YIN, Y. Q. CAIRESEARCH ON FORCE CONTROL SYSTEM  OF LEAD CATHODE LEVELERReceived – Primljeno: 2023-05-26Accepted – Prihvaćeno: 2023-08-25Original Scientific Paper – Izvorni znanstveni radThe flatness of the lead cathode plate affects the electrolytic efficiency of lead and the production efficiency of the whole lead electrolytic industry. However, the dynamic response of the force control system of the leveler is slow, and the anti-interference and robustness are poor. By comparing proportion integration differentiation (PID)  control and feedback linearized synovial control two control strategies, MATLAB software was used for modeling and simu-lation analysis. The results show that the system with feedback linearized sliding mode control has faster response, higher precision and better robustness.Keywords: lead, cathode sheet, levelling machine, force control, sliding mode control algorithmISSN 0543-5846METABK 63(1) 117-120 (2024)UDC – UDK 669.4:621.41/621.35:621.7:531.78.004.5=111INTRODUCTIONNon-ferrous lead electrolysis industry has brought great promotion to the development of economy. In the process of industrial production of electric lead, crude lead and fine lead are cast into a certain shape of anode plate and cathode plate respectively. The process dia-gram of lead electrolysis is shown in Figure 1. Y.F. Wang, Q.Q. Yin (E-mail: wyf@ncst.edu.cn), Y.Q. Cai, College of Mechanical Engineering, North China University of Science and Tech-nology, Hebei, Tangshan, China.ness of lead sheet even and the surface smooth, so as to improve the quality and electrolytic efficiency of lead sheet.LEVELING MACHINE HYDRAULIC CONTROL SYSTEM The leveling machine needs to squeeze the lead sheet, so the valve control cylinder system is required to adjust the liquid pressure. The hydraulic cylinder of the cathode leveling machine pushes the pressure plate in a cyclic process, which consists of a series of slopes and holding cycles. The working process is shown in Figure 3. In the process of closing the die, the hydraulic oil pushes the hydraulic cylinder out, the convex die press-ing plate and the concave die pressing plate move for-ward, the opening die retracted after closing the die, the punch and the concave die retracted back to the desig-nated position under the push of the piston bit of the hydraulic cylinder[1]. Figure 1  Schematic diagram of lead electrolysis production processLead cathode sheet is the lead sheet made by the lead cathode preparation unit after the electrolysis of crude lead, with high purity but uneven thickness and folds. The flatness of lead cathode sheet will affect the electrolytic efficiency of lead, so it is necessary to im-prove the flatness of lead sheet by leveling and pressing machine, as shown in Figure 2, so as to make the thick-Figure 2 Lead sheet diagramFigure 3 Work process of levelerIn the process of pressure preservation, the force on the lead sheet is maximum, and at this time, the grain is printed on the lead sheet to increase the stiffness of the lead sheet. The principle of a typical valve-controlled asymmetric cylinder system is shown in Figure 4.In the Figure, Xp represents the displacement of the piston rod, m represents the equivalent load mass, B 118  METALURGIJA 63 (2024) 1, 117-120Y. F. WANG et al.: RESEARCH ON FORCE CONTROL SYSTEM OF LEAD CATHODE LEVELERrepresents the equivalent load damping, K represents the equivalent load stiffness, F represents the equivalent external load force, Xv represents the spool displace-ment, A1 and A2 respectively represent the effective area of the rodless and roded cavities of the hydraulic cylin-der, p1 and p2 respectively represent the pressure of the two cavities of the hydraulic cylinder. q1 and q2 respec-tively represent the flow into and out of the two cham-bers of the hydraulic cylinder.The three equations are as follows: 1) Servo valve inlet and outlet oil flow   (1)Where  (Kd is the conversion factor), ps is the system operating pressure, p1 is the servo cylinder inlet chamber pressure, p2 is the servo cylinder return chamber pressure, and p0 is the system return pressure.2) Hydraulic cylinder flow continuity equation  (2)  (3)Where Cip is the internal leakage coefficient, Cep is the external leakage coefficient, L is the total stroke of the servo cylinder, L0 is the initial position of the servo cyl-inder piston, A1is the area of the rodless chamber of the servo cylinder, A2 is the area of the rodged chamber of the servo cylinder, xp is the displacement of the servo cylinder, βe is the effective volume elastic modulus of the hydraulic oil, Va1 is the volume of the piping con-necting the servo valve and the servo cylinder inlet, Va2 is the volume of the piping connecting the servo valve and the servo cylinder return.3) Hydraulic cylinder and load balance equation  (4)In addition, the electro-hydraulic servo valve is a double nozzle flapper force feedback electro-hydraulic servo valve, and its dynamic model can be described by the proportional link as follows:  (5)Where, Kaxv is the servo valve gain, xr is the expected displacement, and Kx is the displacement sensor gain [2,3].LEVELING MACHINE FORCE CONTROL STRATEGY The state vector of the system is selected as piston rod displacement xp, piston rod speed , pressure with-out rod cavity p1 and pressure with rod cavity p2, then the state vector can be written as   (6)The input vector u is u = xr.According to the definition of relative order, the relative order of the system is obtained as (3). Since the order (4 ) of the system is greater than its relative order (3), the system has internal dynamic subsystem. Ac-cording to the feedback linearization method, the trans-formation relationship between the new state variable and the original state variable is constructed as  (7)Then the original nonlinear system equation of state can be transformed into the equation of state in linear space  (8)The control quantity u of the nonlinear system under the original coordinates can be obtained by the inverse coordinate transformation of the control quantity v of the linear system.  (9)When feedback linearization is completed, the order of equation of state (8) of the linear system is (3), which is less than the order of the original system (4), so there is a nonlinear internal dynamic subsystem. In order to achieve accurate stability control of the original system, the internal dynamic subsystem is also required to be stable, that is, the following conditions must be met  (10)Then , the dynamic subsystem is stable [4,5].Figure 4 Operating principle of valve controlled cylinder119Y. F. WANG et al.: RESEARCH ON FORCE CONTROL SYSTEM OF LEAD CATHODE LEVELERMETALURGIJA 63 (2024) 1, 117-120The system model obtained after feedback lineariza-tion is a third-order linear system, so the sliding mode surface can be designed as  (11)Where c1, c2 is the constant of the sliding mode sur-face.First determine the equivalent control:  (12)In order to ensure the generation of sliding mode,  is required, so the switching control quantity is selected  (13)Where, k is the switch control gain.The output of sliding mode control can be expressed as follows:  (14)Stability proof: Definition Lyapunov function ,For the third-order linear system after feedback lin-earization in this paper, there are:   (15)Then the sliding mode control system in the new co-ordinate system is stable. Since the existence of sign function will cause system flutter, boundary layer func-tion is used instead of sign function to reduce system flutter.  (16)Where,  is the boundary layer thickness.The improved feedback linearized sliding mode control rate can be obtained[6]:  (17)SIMULATION EXPERIMENT AND ANALYSISThe traditional PID control algorithm and feedback linearized sliding mode control algorithm were mod-eled on the MATLAB-Simulink platform, as shown in Figure5,6.The input is a sinusoidal force with a amplitude of 2 000 N and a step force with 5 000 N. The simulation parameters of the linearized sliding mode controller are k = –25 000, Ø = 10, and the PID control has the simula-tion parameters of kp = 8 000, ki = 0, kd = 20. The force tracking curves obtained by using different control al-gorithms are shown in Figure 7,8.It can be seen from Figure 7,8 that the feedback line-arized sliding mode control algorithm can effectively shorten the adjustment time of the system and the track-ing effect of the feedback linearized sliding mode control algorithm is better than that of PID control. This algo-rithm not only improves the response speed of the sys-tem, but also improves the tracking effect of the system.Figure 5 PID control modelFigure 6 FLSM control modelFigure 8 Step signalFigure 7 Sinusoidal signalCONCLUSIONIn this paper, the structure of lead cathode leveling electro-hydraulic servo system is analyzed, and a feed-back linearized sliding mode control method is designed and applied to the research of force control of valve cyl-inder. Through Simulink platform modeling, compared with the simulation experiment, we can see that the 120  METALURGIJA 63 (2024) 1, 117-120Y. F. WANG et al.: RESEARCH ON FORCE CONTROL SYSTEM OF LEAD CATHODE LEVELERfeedback linearized sliding mode controller designed in the respectively tracking sine and step force, the over-shoot is small and can reach a stable state quickly, the tracking force control process is relatively stable, no chatter, is beneficial to improve the quality of the lead cathode leveler. It has important practical significance to promote the development and progress of nonferrous metal lead electrolysis industry.REFERENCE[1] R.P.Xue. The Application Research of Electro-Hydraulic Proportional Force Control System of the Embossing Leveler[D]. Kunming University of Science and Techno-logy, (2014),1-8.[2] Samakwong T, Assawinchaichote W. PID Controller de-sign for electro-hydraulic servo valve system with genetic algorithm [J]. Procedia Computer Science, (2016)86, 91-94.[3] Yu J, Zhuang J, Yu D. Parameter optimization of PID con-troller based on complex system genetic algorithm in elec-tro-hydraulic servo control system [C]// Proceedings of the World Congress on Intelligent Control & Automation, Jul.6-8, 2012, Beijing, China.IEEE, (2012),30-35.[4] Chen H M, Renn J C, Su J P. Sliding mode control with varying boundary layers for an electro-hydraulic position servo system [J]. The International Journal of Advanced Manufacturing Technology, (2005)26,117-123.[5] Bessa W M, Dutra M S, Kreuzer E. Sliding mode control with adaptive fuzzy dead-zone compensation of an electro-hydraulic servo-system [J]. Journal of Intelligent and Ro-botic Systems, (2010)58,3-16.Note:  The responsible translator for English language is T.X. Zhang -Hebei Normal University, China.

Research on force control system of lead cathode leveler

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