Simulation Analysis for the Application of Four-Way Servovalve Controlled Cylinder in Hydraulic AGC System of Rolling Mills

Abstract

Three-way servovalve controlled cylinder (TSCC) is the main method used in hydraulic automatic gauge control (HAGC) system of rolling mill, which generally only foucus on the screw-down speed of HAGC cylinder under load resistance. However, with the development of plan view pattern control and variable gauge rolling, the lifting-up speed of HAGC cylinder is required to be as high as the screw-down speed. At present, high frequency response and large flow servovalves or even double servovalves in parallel are usually used to achieve high lifting-up speed, resulting in a substantial increase in equipment cost. In this paper, the nonlinear mathematical models of the TSCC and four-way servovalve controlled cylinder (FSCC) are firstly established, and the steady-state speed equations of the two methods are derived. Then, the steady-state speed of the two methods with different HAGC cylinder sizes under different load conditions is compared and analyzed, and it is proved that FSCC has obvious advantages in lifting-up speed. Finally, simulation experiments of variable gauge rolling on a 1050 mm cold-rolling mill are carried out. In the rolling process of transition zone where the roll gap increases, the FSCC has higher control precision and wider dynamic adjustment ability, which is more conducive to improving rolling speed, production efficiency and shape quality.

1. Introduction

Thickness precision is one of the most important dimension and quality indexes of plate and strip, and automatic thickness control (AGC) is an indispensable and important part in rolling field. Hydraulic AGC (HAGC) system controls the roll gap and pressure by adjusting servovalves to ensure the accuracy requirement of the production thickness. It has been rapidly popularized and applied in modern rolling industry because of its high control precision and fast response speed.

The HAGC control of conventional equal thickness rolling usually only focuses on the screw-down speed of AGC cylinders under load, and the TSCC method is adopted in most rolling mills with the HAGC system.^1,2,3) In the TSCC method, one working port of the servo valve is connected to the piston side of the AGC cylinder, and the ring side of the AGC cylinder is controlled by a constant back pressure P_b. Due to the fact that there is only one hydraulic spring in the TSCC, its control process has good stability. However, the lifting-up speed of the TSCC is slow because of the low P_b (2–5 MPa). In recent years, with the development of plan view pattern control and variable gauge rolling, the lifting-up speed of cylinder is required to be as high as the screw-down speed.^4,5,6) In the actual production process, due to the original systems using the TSCC method, the dimensional accuracy of the transition zone can only be ensured by reducing the rolling speed. For variable gauge rolling, reducing rolling speed will directly lead to low production efficiency. For plan view pattern control, reducing rolling speed not only leads to low production efficiency, but also increases the temperature difference between the head and tail of hot rolled plate, and loses the uniformity of material microstructure and properties.

In equipment upgrading or new equipment design, designers generally choose high frequency response and large flow servovalve or even double servovalves in parallel to improve the lifting-up speed of the TSCC,⁷⁾ but these methods cause a substantial increase in equipment and maintenance costs. Therefore, it is of great theoretical significance and practical application value to study how to improve the lifting-up speed of HAGC cylinder by improving hydraulic control method.

The FSCC is widely used in modern industries, such as flight motion simulators,^8,9,10) electro-hydraulic load simulators,^11,12) electro-hydraulic manipulators,^13,14) and so on. However, there are few reports of using FSCC in the HAGC control of rolling mills.

In this paper, the nonlinear mathematical models of the TSCC and the FSCC are firstly established, which fully consider the real-time variation of fluid volume related to piston position in two chambers of cylinder, and have higher accuracy than traditional modeling methods. Then, the steady-state speed equations of the two methods are derived. On this basis, the steady-state velocities of TSCC and FSCC with different cylinder sizes under different loads, oil source pressures and back pressures are compared and analyzed. Finally, simulation experiments of variable gauge rolling on a 1050 mm cold-rolling mill are carried out.

2. Modeling

2.1. Modeling of the TSCC

The hydraulic circuit diagram of TSCC is shown in Fig. 1, where P_s is the oil source pressure, Pa; P₀ is the pressure of oil return pipeline and generally considered as 0, Pa; P₁ is the pressure in the piston side, Pa; Q₁₁ is the flow into the piston side of the cylinder through the servovalve, m³/s; Q₁₂ is the flow out of the piston side of the cylinder through the servovalve, m³/s; M_t is the total mass of the cylinder’s movable part, kg; B_p is the viscous damping coefficient of the cylinder and generally considered as 0, N∙(m/s)⁻¹; x is the piston position, m; x_v is the valve spool position, m; K is the load stiffness, N/m; F is the generalized load force combined with half of the rolling force, piston gravity, balance force and friction force, N; A is one of the working ports of the servovalve, and B is the other one. In HAGC systems, the friction force is far less than the rolling force, so is the resultant force of piston gravity and balance force. Therefore, F can be approximated as half of the rolling force.

Fig. 1. The hydraulic circuit diagram of the TSCC.

The actual flow of the servovalve is dependent upon the valve spool position and valve pressure drop. The flow for a given valve pressure drop can be calculated using the square root function for sharp edge orifices

  

$$Q=C_{\text{d}}\omega x_{\text{v}}\sqrt{\frac{2}{\rho }\mathrm{\Delta }P}$$(1)

where C_d is the flow coefficient; ω is the area gradient, m; ΔP is the valve pressure drop, Pa; ρ is the fluid density, kg/m³.

The rated flow rate Q_R of servovalve under rated valve pressure drop ΔP_N is obtained from Eq. (1)

  

$$Q_R=C_{\mathrm{d}}\omega x_{\mathrm{v}\text{max}}\sqrt{\frac{2\mathrm{\Delta }{P}_N}{\rho }}$$(2)

where x_vmax is the maximum spool displacement, m.

Equations (1) and (2) are combined to obtain

  

$$Q=Q_{\text{R}}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{\mathrm{\Delta }P}{\mathrm{\Delta }{P}_N}}$$(3)

In the piston screw-down process, the flow into the piston side of the cylinder through the servovalve is

  

$$Q_{11}=Q_{\text{R}}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{P_{\mathrm{s}}-P_{\hspace{0.17em}1}}{\mathrm{\Delta }{P}_N}}$$(4)

In the piston lifting-up process, the flow out of the piston side of the cylinder through the servovalve is

  

$$Q_{12}=Q_{\text{R}}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{P_{\hspace{0.17em}1}}{\mathrm{\Delta }{P}_N}}$$(5)

Fluid is compressible, and its bulk modulus is an important factor affecting the dynamic performance of hydraulic control systems. In this paper, a virtual transition state as shown in Fig. 2 is proposed to analyze the compression process of fluid during the movement of the cylinder. The virtual transition state does not exist in the actual movement process of the cylinder, but introducing this concept will make the physical meanings of various volume related variables in the compression process clearer. From time 0 to the virtual transition state, the volume of fluid flowing into or out of the piston side of the cylinder through the servo valve is equal to the change in volume of the piston side, so the fluid is not compressed, and the pressure inside the piston side remains unchanged. From the virtual transition state to time t, there is no fluid passing through the servo valve, and the cylinder piston moves by Δx under external loads, inertial forces, etc., to the actual position x. It can be seen that before compression occurs, the original volume of the fluid is V₁ at the virtual transition state, and the compression amount is the volume change of piston side ΔV₁ caused by Δx. Therefore, considering the fluid compressibility, the pressure in the piston side is

  

$$P_1=P_{\text{10}}+{\beta }_{\text{e}}\frac{\mathrm{\Delta }{V}_1}{V_1}$$(6)

where P₁₀ is the initial pressure in the piston side, Pa; β_e is the bulk modulus of the fluid, Pa.

Fig. 2. The decomposition diagram of fluid compression state.

Consider Q₁ as a unified variable for Q₁₁ and Q₁₂. When x_v≥0, Q₁=Q₁₁. When x_v<0, Q₁=Q₁₂. In this way, during the movement of the piston, the fluid volume in the piston side is

  

$$V_1=V_{10}+{\int }_0^t{Q}_1\text{d}t$$(7)

where V₁₀ is the sum of the initial fluid volume in the piston side at t=0 and the pipeline volume from the working port A of the servovalve to the joint of the piston side, m³.

The volume compression of the fluid in the piston side is

  

$$\mathrm{\Delta }{V}_1={\int }_0^t{Q}_1\text{d}t-x{A}_1$$(8)

where A₁ is the piston area of the cylinder, m². The value of x is 0 at t=0.

By combining Eqs. (6), (7), and (8), the pressure in the piston side during the piston motion can be obtained as

  

$$P_1=P_{\text{10}}+{\beta }_{\text{e}}\frac{{\int }_0^t{Q}_1\text{d}t-x{A}_1}{V_{10}+{\int }_0^t{Q}_1\text{d}t}$$(9)

The force balance equation of the piston is

  

$$P_1{A}_{\text{1}}-P_{\text{b}}{A}_{\text{2}}-F=M_{\text{t}}\frac{{\text{d}}^{2}x}{\text{d}{t}^2}+B_{\text{p}}\frac{\text{d}x}{\text{d}t}+Kx$$(10)

where A₂ is the ring area of the cylinder, m².

2.2. Modeling of the FSCC

The hydraulic circuit diagram of FSCC is shown in Fig. 3. where P₂ is the pressure in the ring side, Pa. Q₂₁ is the flow out of the ring side of the cylinder through the servovalve, m³/s; Q₂₂ is the flow into the ring side of the cylinder through the servovalve, m³/s; A is one of the working ports of the servovalve, and B is the other one. During the piston motion, the equation of the pressure in the piston side is the same as Eq. (9).

Fig. 3. The hydraulic circuit diagram of the FSCC.

In the piston screw-down process, the flow out of the ring side of the cylinder through the servovalve is

  

$$Q_{21}=Q_{\text{R}}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{P_{\hspace{0.17em}2}}{\mathrm{\Delta }{P}_N}}$$(11)

In the piston lifting-up process, the flow into the ring side of the cylinder through the servovalve is

  

$$Q_{22}=Q_{\text{R}}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{P_{\mathrm{s}}-P_{\hspace{0.17em}2}}{\mathrm{\Delta }{P}_N}}$$(12)

Considering the fluid compressibility, the pressure in the ring side is

  

$$P_2=P_{\text{20}}+{\beta }_{\text{e}}\frac{\mathrm{\Delta }{V}_2}{V_2}$$(13)

Consider Q₂ as a unified variable for Q₂₁ and Q₂₂. When x_v≥0, Q₂=Q₂₁. When x_v<0, Q₂=Q₂₂. In this way, during the movement of the piston, the fluid volume in the ring side is

  

$$V_2=V_{20}-{\int }_0^t{Q}_2\text{d}t$$(14)

where V₂₀ is the sum of the initial fluid volume in the ring side at t=0 and the pipeline volume from the working port B of the servovalve to the joint of ring side, m³.

The volume compression of the fluid in the ring side is

  

$$\mathrm{\Delta }{V}_2=x{A}_2-{\int }_0^t{Q}_2\text{d}t$$(15)

By combining Eqs. (13), (14), and (15), the pressure in the ring side during the piston motion can be obtained as

  

$$P_2=P_{\text{20}}+{\beta }_{\text{e}}\frac{x{A}_2-{\int }_0^t{Q}_2\text{d}t}{V_{20}-{\int }_0^t{Q}_2\text{d}t}$$(16)

2.3. Discussion

The hydraulic servo system is a typical nonlinear system, and traditional modeling methods approximate linearization of the system. As a textbook, reference [15] provides detailed descriptions of the traditional linearization modeling methods for TSCC and FSCC, which differ from the modeling method proposed in this paper mainly in two aspects.¹⁵⁾ First, traditional methods perform Taylor series expansion on Eq. (1) at the steady-state operating point of the servovalve to obtain a local incremental linear model. Second, when considering fluid compression, assuming that the cylinder operates near the neutral position, V₁ and V₂ in Eqs. (6) and (13) are taken as constant values, and ΔV₁ and ΔV₂ are determined by the instantaneous flow relationship of the flow continuity equation. It can be seen that the model established by traditional methods is suitable for situations where the adjustment range of the servovalve working point and the working displacement of the cylinder are both small.

The hydraulic servo system model established by traditional methods is linear. Based on it, important concepts such as hydraulic natural frequency and hydraulic damping ratio are proposed by classical control theory. It is convenient to use time-domain or frequency-domain method to conduct in-depth research on the stability and dynamic performance of the system. The research results based on traditional models provide theoretical basis for the design of hydraulic servo systems and lay an important theoretical foundation for the development of hydraulic servo control.

For traditional equal thickness rolling, the role of hydraulic AGC is to eliminate the influence of rolling process parameters (such as raw material thickness, hardness, deformation resistance, etc.) fluctuations by adjusting the roll gap, so as to achieve the expected target thickness. The piston displacement of the HGC cylinders is very small. Moreover, the servovalves usually operate within a very small range near the zero spool position. Therefore, modeling and analysis through traditional methods can achieve sufficient accuracy.¹⁶⁾

However, for variable gauge rolling, especially for plan view pattern control, the maximum thickness difference in the rolling transition zone can even reach 10 mm. For AGC cylinders with a stroke generally within 50 mm, using traditional methods to take the denominator of the fraction in Eqs. (9) and (16) as a constant will result in an error of nearly 40% in calculating the working pressure inside the cylinder. In addition, the screw-down speed or lifting-up speed of the cylinder in the rolling transition zone is very high. At this time, the servovalve needs to work in a large positive or negative opening state, and this working state is normal, so it is not suitable to use traditional incremental linear model to describe the flow state of the servovalve.

In this paper, the square root function for sharp edge orifices is directly used to calculate the flow of the servovalve without linearization approximation, which can accurately reflect the actual flow characteristics of the servovalve at different spool positions and load states. What’s more, Eqs. (9) and (16) fully consider the real-time variation of fluid volume related to piston position, and accurately reflect the actual state of fluid compression. Overall, in the research of variable thickness rolling, the model proposed in this paper has higher accuracy compared to traditional model.

3. Comparison and Analysis of Steady-state Piston Speed of TSCC and FSCC

3.1. Calculation of Steady-state Piston Speed of TSCC

In steady state, the piston acceleration is 0. In addition, for HAGC system, the viscous damping force and spring load are far less than the rolling force, so they can be ignored in steady-state force analysis. Equation (10) can be simplified to

  

$$P_{1\text{w}}{A}_{\text{1}}-P_{\text{b}}{A}_{\text{2}}=F$$(17)

where P_1w is the pressure in the piston side in steady-state, Pa.

In the steady-state screw-down process, by combining Eqs. (4) and (17) after replacing P₁ in Eq. (4) with P_1w, we can get

  

$$Q_{11}=Q_{\text{R}}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{P_{\mathrm{s}}-k{P}_{\hspace{0.17em}\mathrm{b}}-\frac{F}{A_1}}{\mathrm{\Delta }{P}_N}}$$(18)

where $k=\frac{A_2}{A_{\text{1}}}$. Therefore, the steady-state screw-down speed of the TSCC is

  

$$v_1=\frac{Q_{\text{R}}}{A_1}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{P_{\mathrm{s}}-k{P}_{\hspace{0.17em}\mathrm{b}}-\frac{F}{A_1}}{\mathrm{\Delta }{P}_N}}$$(19)

In the steady-state lifting-up process, by combining Eqs. (5) and (17) after replacing P_1w in Eq. (5) with P_1w, we can get

  

$$Q_{12}=Q_{\text{R}}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{k{P}_{\hspace{0.17em}\mathrm{b}}+\frac{F}{A_1}}{\mathrm{\Delta }{P}_N}}$$(20)

Therefore, the steady-state lifting-up speed of the TSCC is

  

$$v_2=\frac{Q_{\text{R}}}{A_1}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{k{P}_{\hspace{0.17em}\mathrm{b}}+\frac{F}{A_1}}{\mathrm{\Delta }{P}_N}}$$(21)

3.2. Calculation of Steady-state Piston Speed of FSCC

Referring to the modeling of the steady-state speed of the TSCC, the steady-state force balance equation of the FSCC can be obtained as follows

  

$$P_{1\text{w}}{A}_{\text{1}}-P_{2\text{w}}{A}_{\text{2}}=F$$(22)

where P_2w is the steady-state pressure in the ring side, Pa. At the moment when the piston motion reaches steady-state, P_1w can be obtained from Eq. (9) as follows

  

$$P_{1\text{w}}=P_{\text{10}}+{\beta }_{\text{e}}\frac{{\int }_0^{t_{\text{w}}}{Q}_1\text{d}t-x_{\text{w}}{A}_1}{V_{10}+{\int }_0^{t_{\text{w}}}{Q}_1\text{d}t}$$(23)

where t_w is the time when the piston motion reaches steady-state, s; x_w is the piston displacement at t=t_w, m. When t>t_w, the pressure in the piston side is

  

$$P_1=P_{\text{1w}}+{\beta }_{\text{e}}\frac{{\int }_{t_{\text{w}}}^t{Q}_1\text{d}t-(x-x_{\text{w}})A_1}{V_{1\text{w}}+{\int }_{t_{\text{w}}}^t{Q}_1\text{d}t}$$(24)

where V_1w is the sum of the fluid volume in the piston side at t= t_w and the pipeline volume from the working port A of the servovalve to the joint of the piston side, m³. After the piston motion reaches steady-state, the flow Q₁ through the servovalve is all used to compensate for the volume change in the piston side, and there is no further compression or rebound of the fluid, so

  

$${\int }_{t_{\text{w}}}^t{Q}_1\text{d}t=(x-x_{\text{w}})A_1$$(25)

Therefore, when t>t_w, P₁=P_1w. By taking the derivative of time on both sides of Eq. (25), we can get

  

$$Q_1=\frac{\text{d}x}{\text{d}t}{A}_1$$(26)

At the moment when the piston motion reaches steady-state, P_2w can be obtained from Eq. (16) as follows

  

$$P_{2\text{w}}=P_{\text{20}}+{\beta }_{\text{e}}\frac{x_{\text{w}}{A}_2-{\int }_0^{t_{\text{w}}}{Q}_2\text{d}t}{V_{20}-{\int }_0^{t_{\text{w}}}{Q}_2\text{d}t}$$(27)

When t>t_w, the pressure in the ring side is

  

$$P_2=P_{2\text{w}}+{\beta }_{\text{e}}\frac{(x-x_{\text{w}})A_2-{\int }_{t_{\text{w}}}^t{Q}_2\text{d}t}{V_{2\text{w}}-{\int }_{t_{\text{w}}}^t{Q}_2\text{d}t}$$(28)

where V_2w is the sum of the fluid volume in the ring side at t= t_w and the pipeline volume from the working port B of the servovalve to the joint of ring side, m³. After the piston motion reaches steady-state, the flow Q₂ through the servovalve is all used to compensate for the volume change in the ring side, and there is no further compression or rebound of the fluid, so

  

$$(x-x_{\text{w}})A_2={\int }_{t_{\text{w}}}^t{Q}_2\text{d}t$$(29)

Therefore, when t>t_w, P₂=P_2w. By taking the time derivative of both sides of Eq. (29), we can get

  

$$\frac{\text{d}x}{\text{d}t}{A}_2=Q_2$$(30)

After the piston motion reaches steady-state, Eqs. (26) and (30) are combined to obtain

  

$$\frac{Q_2}{A_2}=\frac{Q_1}{A_{\text{1}}}$$(31)

In the steady-state screw-down process, it can be obtained from Eq. (31) that

  

$$\frac{Q_{21}}{A_2}=\frac{Q_{11}}{A_{\text{1}}}$$(32)

In the steady-state lifting-up process, it can be obtained from Eq. (31) that

  

$$\frac{Q_{22}}{A_2}=\frac{Q_{12}}{A_{\text{1}}}$$(33)

By combining Eqs. (4), (11), (22) and (32), the steady-state screw-down speed of the FSCC can be obtained as follows

  

$$v_1^{\prime }=\frac{Q_{\mathrm{R}}}{A_1}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{1}{\mathrm{\Delta }{P}_N}\cdot \frac{P_{\mathrm{s}}-\frac{F}{A_1}}{k^3+1}}$$(34)

By combining Eqs. (5), (12), (22) and (33), the steady-state lifting-up speed of the FSCC can be obtained as follows

  

$$v_2^{\prime }=\frac{Q_{\mathrm{R}}}{A_1}\frac{x_{\text{v}}}{x_{\text{vmax}}}\sqrt{\frac{1}{\mathrm{\Delta }{P}_N}\cdot \frac{k{P}_{\mathrm{s}}+\frac{F}{A_1}}{k^3+1}}$$(35)

3.3. Comparison and Analysis of Steady-state Speed

Under the same conditions of servovalve, x_v, D, d and F, the ratio n₁ of steady-state screw-down speeds of FSCC and TSCC can be obtained by combining Eq. (19) with Eq. (34) as follows

  

$$n_1=\frac{v_1^{\prime }}{v_1}=\sqrt{\frac{1}{k^3+1}\cdot \frac{P_s-\frac{F}{A_1}}{P_s-\frac{F}{A_1}-k{P}_{\hspace{0.17em}\mathrm{b}}}}$$(36)

Under the same conditions of servovalve, x_v, D, d and F, the ratio n₁ of steady-state lifting-up speeds of FSCC and TSCC can be obtained by combining Eqs. (21) and (35) as follows

  

$$n_2=\frac{v_2^{\prime }}{v_2}=\sqrt{\frac{1}{k^3+1}\cdot \frac{k{P}_{\mathrm{s}}+\frac{F}{A_1}}{k{P}_{\hspace{0.17em}\mathrm{b}}+\frac{F}{A_1}}}$$(37)

It can be seen that n₁ and n₂ are influenced by k, A₁, P_s, F, and P_b. Among them, A₁ is determined by the piston diameter D. For a HAGC cylinder, its piston rod diameter d is generally calculated by empirical methods based on D. Equations (36) and (37) can be used to calculate the n₁ and n₂ of commonly used series of HAGC cylinders under different oil source pressures, load forces, and back pressures. The calculation results are shown in Tables 1, 2, 3. It should be noted that F_max in the tables is not a constant, but corresponds to the maximum output force of different cylinder sizes under their rated working pressures. For example, when D=320 mm, d=220 mm, and the rated working pressure of the cylinder is 25 MPa, F_max=2010 kN. When D=1450 mm, d=1300 mm, and the rated working pressure of the cylinder is 25 MPa, F_max=41280 kN.

Table 1. The values of n₁ and n₂ when P_s=21 MPa, P_b=3 MPa.

cylinder sizes	n1					n2
	0	0.25Fmax	0.5Fmax	0.75Fmax	Fmax	0	0.25Fmax	0.5Fmax	0.75Fmax	Fmax
D=320 mm	0.97	0.98	1.00	1.03	1.13	2.47	1.53	1.32	1.22	1.16
d=220 mm
D=550 mm	1.01	1.01	1.02	1.04	1.10	2.60	1.45	1.27	1.19	1.14
d=450 mm
D=900 mm	1.01	1.01	1.02	1.03	1.06	2.63	1.34	1.19	1.13	1.10
d=800 mm
D=1100 mm	1.01	1.01	1.02	1.03	1.05	2.64	1.30	1.17	1.11	1.09
d=1000 mm
D=1250 mm	1.01	1.01	1.02	1.03	1.07	2.63	1.36	1.20	1.14	1.11
d=1100 mm
D=1450 mm	1.01	1.01	1.02	1.03	1.06	2.64	1.33	1.18	1.13	1.10
d=1300 mm

Table 2. The values of n₁ and n₂ when P_s=21 MPa, P_b=6 MPa.

cylinder sizes	n1					n2
	0	0.25Fmax	0.5Fmax	0.75Fmax	Fmax	0	0.25Fmax	0.5Fmax	0.75Fmax	Fmax
D=320 mm	1.01	1.04	1.07	1.16	1.54	1.75	1.35	1.22	1.15	1.11
d=220 mm
D=550 mm	1.03	1.05	1.07	1.11	1.26	1.84	1.33	1.20	1.14	1.11
d=450 mm
D=900 mm	1.03	1.03	1.05	1.07	1.15	1.86	1.26	1.15	1.11	1.08
d=800 mm
D=1100 mm	1.02	1.03	1.04	1.06	1.12	1.87	1.23	1.13	1.09	1.07
d=1000 mm
D=1250 mm	1.03	1.04	1.05	1.08	1.16	1.86	1.27	1.16	1.11	1.09
d=1100 mm
D=1450 mm	1.03	1.03	1.04	1.07	1.14	1.86	1.25	1.14	1.10	1.08
d=1300 mm

Table 3. The values of n₁ and n₂ when P_s=28 MPa, P_b=3 MPa.

cylinder sizes	n1					n2
	0	0.25Fmax	0.5Fmax	0.75Fmax	Fmax	0	0.25Fmax	0.5Fmax	0.75Fmax	Fmax
D=320 mm	0.96	0.97	0.98	1.01	1.13	2.85	1.56	1.32	1.22	1.16
d=220 mm
D=550 mm	1.00	1.00	1.01	1.03	1.10	3.00	1.47	1.27	1.18	1.14
d=450 mm
D=900 mm	1.07	1.01	1.02	1.03	1.06	3.04	1.34	1.19	1.13	1.10
d=800 mm
D=1100 mm	1.07	1.01	1.01	1.02	1.05	3.05	1.30	1.16	1.11	1.09
d=1000 mm
D=1250 mm	1.01	1.01	1.02	1.03	1.07	3.04	1.36	1.20	1.14	1.11
d=1100 mm
D=1450 mm	1.01	1.01	1.01	1.02	1.06	3.04	1.33	1.18	1.12	1.09
d=1300 mm

From Tables 1, 2, 3, it can be seen that: (1) there is only one situation where n₁ is slightly less than 1 when the cylinder is small-sized with small rolling forces. However, there is a large margin between the maximum screw-down speed and the actual process requirements under small rolling forces, so this situation will not affect the HAGC screw-down capacity at all. In other cases, n₁ is larger than 1, indicating that the FSCC has an advantage in screw-down speed compared to the TSCC. (2) under all load conditions, the FSCC has a significant improvement in its lifting-up speed, especially under medium or small load conditions.

Comparing Tables 1 and 2, it can be seen that as the back pressure increases, the lifting-up speed of the TSCC increases, but n₁ significantly increases, indicating a loss of its screw-down speed. Moreover, excessive back pressure can also cause a loss of the maximum rolling capacity of the rolling mill.

Comparing Tables 1 and 3, it can be seen that with the increase of oil source pressure, there is a significant increase in n₂ within the medium and small rolling force ranges, indicating that increasing oil source pressure is beneficial for improving the lifting-up speed of the FSCC.

4. Simulation Application of Variable Gauge Rolling

The research object is the HAGC system of a 1050 mm single stand cold rolling mill. Maximum rolling force is required to be 10000 kN. The oil source pressure is 28 MPa. The piston diameter D of the AGC cylinder is 550 mm, and the piston rod diameter d is 450 mm. MOOG-D661-G35 servovalve is used, and its rated flow Q_R corresponding to the rated valve pressure drop ΔP_N=3.5 MPa is 90 L/min.

4.1. Dynamic Modeling of Servovalve

In this study, the servovalve is treated as a second-order system, and the transfer function between the valve spool position x_v(t) and the command signal s_v(t) can be described as

  

$$\frac{X_{\mathrm{v}}(s)}{S_{\mathrm{v}}(s)}=\frac{1}{\frac{s^2}{{\omega }_{\mathrm{n}}{}^2}+\frac{2\zeta }{{\omega }_{\text{n}}}s+1}\cdot \frac{x_{\mathrm{v}\text{max}}}{s_{\mathrm{v}\text{max}}}$$(38)

where X_v(s)and S_v(s) are the image functions of x_v(t) and s_v(t), respectively; s_vmax is the maximum command signal of the servovalve, A; ζ is the damping ratio of the servovalve; ω_n is the undamped natural frequency of the servovalve, rad/s.

The step response and frequency response curves of the servovalve can be obtained from the valve user manual, as shown in Fig. 4. It can be approximately calculated from the control theory of typical second-order systems that ζ=0.83 and ω_n=376 rad/s.

Fig. 4. Step response and frequency response curves of the servovalve.

4.2. Modeling of Rolling Force

The cold rolling process adopts process lubrication and has a small friction coefficient, so it is suitable to use Stone formula to calculate rolling force. The average unit pressure is calculated by

  

$$\overline{P}=\left(K_0-\frac{{\sigma }_{\text{f}}+{\sigma }_{\text{b}}}{2}\right)\cdot \frac{{\text{e}}^m-1}{m}$$(39)

where K₀ is deformation resistance, MPa; σ_f and σ_b are the forward and backward tensile stress, respectly, MPa.

  

$$m=\frac{fl}{\overline{h}}$$(40)

where f is the friction coefficient; l is the contact art length and $l=\sqrt{R\mathrm{\Delta }h}$, m; R is the work roll radius, m; h=0.5(h+H), m; h is the exit thickness of the strip, m; H is the entry thickness of the strip, m. The rolling force is

  

$$P=\overline{P}lB$$(41)

In the variable gauge rolling, H is initial thickness of the strip and considered as a constant. In the transition zone, h is a variable, which leads to the real-time change of rolling force. The rolling force is shared by two HAGC cylinders, so when calculating the rolling force by using Eq. (10), it should be taken

  

$$F=0.5P$$(42)

In the actual production process of rolling, constant control is carried out on the forward tension and backward tension T_b of the strip. Therefore, σ_f is calculated by

  

$${\sigma }_{\text{f}}=\frac{T_{\text{f}}}{Bh}$$(43)

where T_f is the forward tension, N.

σ_b is calculated by

  

$${\sigma }_{\text{b}}=\frac{T_{\text{b}}}{BH}$$(44)

where T_b is the backward tension, N.

Under zero rolling force condition, when the roll gap is H, take the piston position x as 0. Considering the elastic deformation of the rolling mill, the relationship between the exit thickness h and the piston position x satisfies

  

$$h=H-x+\frac{P}{K_{\text{m}}}$$(45)

where K_m is the mill modulus, N/m.

4.3. Simulation of Variable Gauge Rolling

The models of TSCC and FSCC are combined with the models of servovalve and rolling force, and the closed-loop control systems can be founded through the controllers and position feedback. The structural block diagrams of closed-loop position control systems based on TSCC and FSCC for variable gauge rolling are shown in Figs. 5 and 6.

Fig. 5. Structural block diagram of closed-loop position control system based on the TSCC.

Fig. 6. Structural block diagram of closed-loop position control system based on the FSCC.

where h* is the ideal exit thickness of the strip, m. In the simulation process, when x_v≥0, the related flow is calculated by using Eqs. (4) and (11); When x_v<0, the related flow is calculated by Eqs. (5) and (12).

Figure 7 shows the target longitudinal section dimensions of a typical tailor rolled blank (TRB) with initial thickness H=2 mm. The maximum thickness of the TRB is 1.9 mm, the minimum thickness is 1.0 mm, and the horizontal length of the transition zone is 100 mm. h* can be easily calculated based on Fig. 7 and the rolling speed. The thickness accuracy of TRB in transition zone is required to be ±0.04 mm. The AB, CD, and EF sections belong to equal thickness rolling. The BC and DE sections belong to variable gauge rolling. In the BC section, the roll gap gradually decreases and the rolling force continuously increases. In the DE section, the roll gap gradually increases and the rolling force continuously decreases. Some key parameters required for simulation are shown in Table 4, where K_p is the proportional gain of the controller; K_i is the integral gain of the controller; T_f1 and T_b1 are the forward tension and backward tension at B=900 mm, respectively; T_f2 and T_b2 are the forward tension and backward tension at B=200 mm, respectively.

Fig. 7. Longitudinal section dimensions of a TRB.

Table 4. List of some key simulation parameters.

parameters	value	parameters	value
Ps	28 MPa	Pb	3 MPa
Mt	300 kg	βe	800 MPa
K	0 N/m	Bp	0 N∙(m/s)−1
V10	0.006 m3	V20	0.002 m3
Km	5.6 MN/mm	K0	265 MPa
Kp	20000	Ki	105
Tf1	18 kN	Tb1	36 kN
Tf2	4 kN	Tb2	8 kN
R	145 mm	f	0.17

It can be seen from Eq. (41) that the rolling force is proportional to the strip width B. The simulation experiments of variable gauge rolling are carried out under the conditions of B=900 mm and B=200 mm respectively.

4.3.1. Simulation Experiment at B=900 mm

PI controller is adopted. When the rolling speed is 0.2 m/s, the control effects of the TSCC and FSCC are shown in Fig. 8,¹⁷⁾ where S_r is the percentage of servovalve command s_v signal relative to maximum command signal s_vmax, %. When the rolling speed reaches 0.5 m/s, the control effects of the two methods are shown in Fig. 9. Under two rolling speeds, the comparison of key performance indexes of the two methods is shown in Table 5.

Fig. 8. Comparison of control effects under rolling speed 0.2 m/s at B=900 mm.

Fig. 9. Comparison of control effects under rolling speed 0.5 m/s at B=900 mm.

Table 5. Comparison of key performance indexes of the two methods under different rolling speeds at B=900 mm.

control method	When rolling speed is 0.2 m/s				When rolling speed is 0.5 m/s
	error_max/mm	error_min/mm	Sr_max/%	Sr_min/%	error_max/mm	error_min/mm	Sr_max/%	Sr_min/%
TSCC	0.0270	−0.0117	23.35	−54.04	0.0747	−0.0291	58.21	−100
FSCC	0.0149	−0.0113	22.69	−29.75	0.0369	−0.0280	55.94	−73.68

4.3.2. Simulation Experiment at B=200 mm

When the rolling speed is 0.2 m/s, the control effects of the TSCC and FSCC are shown in Fig. 10. When the rolling speed reaches 0.5 m/s, the control effects of the two methods are shown in Fig. 11. Under two rolling speeds, the comparison of key performance indexes of the two methods is shown in Table 6.

Fig. 10. Comparison of control effects under rolling speed 0.2 m/s at B=200 mm.

Fig. 11. Comparison of control effects under rolling speed 0.5 m/s at B=200 mm.

Table 6. Comparison of key performance indexes of the two methods under different rolling speeds at B=200 mm.

control method	When rolling speed is 0.2 m/s				When rolling speed is 0.5 m/s
	error_max/mm	error_min/mm	Sr_max/%	Sr_min/%	error_max/mm	error_min/mm	Sr_max/%	Sr_min/%
TSCC	0.0260	−0.0094	18.78	−52.12	0.1065	−0.0235	46.91	−100
FSCC	0.0152	−0.0093	18.55	−30.45	0.0370	−0.0232	46.36	−74.05

4.3.3. Stability Analysis

The open-loop frequency response characteristics of the system can be efficiently analyzed using the Model Linearizer tool in Simulink, which enables straightforward generation of Bode diagram for stability margin evaluation. Figure 12 presents the open-loop Bode diagram of the TSCC system under a rolling force of 800 kN, while Fig. 13 displays the corresponding plot for the FSCC system under identical operating conditions. Notably, the Bode characteristics of the HAGC system demonstrate significant dependence on rolling force variations. As evidenced by the process data in Figs. 8 and 9, the maximum rolling force reaches approximately 3895 kN at B=900 mm. To comprehensively assess system stability, phase and gain margins were calculated across varying rolling force conditions up to 4000 kN, with detailed comparative results presented in Table 7.

Fig. 12. The open-loop Bode diagram of TSCC when rolling force is 800 kN. (Online version in color.)

Fig. 13. The open-loop Bode diagram of FSCC when rolling force is 800 kN. (Online version in color.)

Table 7. Comparison of stability margin between TSCC and FSCC under different rolling force.

rolling force/kN	TSCC				FSCC
	gain crossover frequency/rad/s	phase margin/°	phase crossover frequency/rad/s	gain margin/dB	gain crossover frequency/rad/s	phase margin/°	phase crossover frequency/rad/s	gain margin/dB
0	67.2	75.4	376	18	87.5	70.8	376	15.7
800	140	58.7	376	11.5	157	54.4	376	10.4
1600	183	48	376	9.04	198	44.1	376	8.25
2400	214	40	376	7.47	227	36.6	376	6.84
3200	238	33.7	376	6.32	250	30.7	376	5.78
4000	258	28.6	376	5.41	268	26	376	4.94

The analysis reveals two critical observations: First, both control strategies exhibit decreasing stability margins with increasing rolling force magnitudes. Second, while TSCC consistently demonstrates superior gain and phase margin performance compared to FSCC across all tested conditions, the FSCC system maintains adequate stability margins throughout the operational range.

4.3.4. Discussion

Comparing Tables 5 with 6, it can be seen that: (1) For the screw-down process with the same rolling speed and control method, both the maximum control error and the maximum servovalve command signal at B=900 mm are greater than those at B=200 mm; The reason is that the rolling force is proportional to the strip width, and the rolling force is the resistance of the piston movement in the screw-down process, so the larger rolling force is unfavorable to the dynamic performance of the control system. (2) For the lifting-up process with the same rolling speed and control method, both the maximum control error and the servovalve command signal at B=900 mm are less than the maximum control error at B=200 mm. The reason is that the rolling force is the driving force of the piston movement in the lifting-up process, and the larger rolling force is beneficial to the dynamic performance of the control system.

As can be seen from Figs. 8, 9, 10, 11, with the increase of rolling speed, the control errors in the screw-down process and the lifting-up process increase obviously, and the servo command signal also increases significantly. The reason is that with the increase of rolling speed, in order to ensure the same size of transition zone, it is necessary to increase the screw-down speed and lifting-up speed of the piston in equal proportion, which makes the control more difficult. When the rolling speed is 0.2 m/s, whether B=900 mm or B=200 mm, the control error can meet the control requirement. When the rolling speed is 0.5 m/s, whether B=900 mm or B=200 mm, the control error cannot meet the control requirement, and the control error is greater when B=200 mm.

The control effects of the TSCC and the FSCC are compared and analyzed by taking the variable gauge rolling of the strip with B=200 mm as an example. The details are as follows:

(1) In the transition zone with reduced roll gap, the control accuracy and servovalve command signal of the two methods are basically the same, because the screw-down ability of the two methods is equivalent under different load conditions as shown in Table 3.

(2) In the transition zone with increased roll gap, when the rolling speed is 0.2 m/s, the control accuracy of both methods meets the control requirements, but the precision of the FSCC is obviously higher than that of the TSCC. The reason is that the FSCC has better lifting-up ability than the TSCC under different loads. After reaching −30.45%, the servovalve command signal of the FSCC is even adjusted back a little to achieve high tracking accuracy. However, the servovalve command signal of the TSCC is significantly bigger than that of the FSCC, and it continues to increase until the transition zone ends.

(3) In the transition section where the roll gap increases, when the rolling speed is 0.5 m/s, the control accuracy advantage of the FSCC is more obvious. The servovalve command signal of the TSCC has reached −100%, but it still cannot prevent the trend of increasing control error, indicating that the lifting-up speed demand for the transition zone with roll gap increased has exceeded the lifting-up capacity limit of the TSCC. However, the maximum command of the FSCC is −74.05%, and there is still a certain adjustment margin.

5. Conclusions

(1) In the lifting-up process of the HAGC cylinder, the rolling force is the driving force. In the transition zone where the roll gap increases, the rolling force rapidly decreases, which is not conducive to achieving high lifting-up speed of the AGC cylinder. The back pressure of the TSCC is small, and high lifting-up speed under the medium and small rolling force cannot be guaranteed. The accuracy of transition section dimensions can only be ensured by reducing the rolling speed, which seriously affects production efficiency and product performance.

(2) Compared with the TSCC, the FSCC has similar screw-down ability, but has significant advantages in lifting-up speed, especially under medium and small rolling forces. It is more suitable for the process characteristics of plan view pattern control and variable gauge rolling.

(3) The TSCC only has one hydraulic spring in the ring side, which is easier to achieve high stability and it is suitable for HAGC systems for equal thickness plate and strip rolling. For the plan view pattern control and variable gauge rolling, the FSCC has higher control accuracy and dynamic adjustment ability in the transition zone with increased roll gap, which is more conducive to improving rolling speed and production efficiency.

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