Decoupling Strategy and Mechanism-intelligent Model of Non Square Flatness Control System

Abstract

Taking 1420 mm UCM six-high cold rolling mill as the research object, a non square flatness control system with five input and four output is decoupled into a square subsystem with two input and two output which controls the primary and cubic flatness and a non square subsystem with three input and two output which controls the quadratic and quartic flatness by using the relative gain theory. By decomposing the unstable poles of the generalized inverse matrix of the non square system, the method of the generalized inverse matrix decoupling control the quadratic and quartic flatness is proposed, which solves the unstable problem of decoupling of non-square system. According to the characteristics of intermediate roll shifting, the variable model of roll shifting influence coefficient and the control strategy of minimum roll shifting adjustment and threshold are proposed. The dynamic characteristics of the system are improved and the adjustment of intermediate roll shifting is reduced. In order to overcome the shortcomings of low accuracy and poor generalization ability of shallow neural network, a mechanism-intelligent influence matrix model based on big data and deep neural network is proposed. Simulation calculation and industrial application show that the control system runs stably, the adjustment speed is fast, the control precision is high, the change of intermediate roll shifting is small, and it is suitable for online control.

1. Introduction

In the production of cold rolled strip, flatness control is an important key technology. How to establish a practical flatness control strategy and model has always been a major scientific and technological problem.^1,2,3,4,5) At present, a large number of cold rolled strips are rolled by six-high rolling mills. There are five ways of online flatness adjustment means, such as roll tilting (RT), work roll bending (WRB, symmetrical, the same below), work roll asymmetric bending (WRAB), intermediate roll bending (IRB, symmetrical, the same below) and intermediate roll shifting (IRS, symmetrical, the same below). The flatness is usually characterized by four flatness components of primary, quadratic, cubic and quartic.⁶⁾ In theory, each adjustment mean has an effect on each flatness component, that is, the coupling problem of flatness control exists.⁷⁾ There is a certain distance between the Shapemeter and the roll gap, and it takes a period of time for the measured flatness to reach the Shapemeter, that is, there is a time-delay problem in flatness measurement.⁸⁾ Moreover, the influence coefficient of each adjustment mean on each flatness component changes from time to time in the rolling process, that is, there is a nonlinear problem.⁹⁾ It can be seen that the flatness control system is a coupled time-delay nonlinear system with five input and four output.

For the component flatness control method, the former proposed the component flatness decoupling control method,¹⁰⁾ and established the decoupling model by calculating the influence matrix to realize the flatness control. Over the years, many scholars have established the flatness decoupling control model based on mechanism or intelligent method,^{11,12,13,14,15,16,17)} and have achieved certain research results, but they are only limited to square control problems, and have not studied non square control problems (the number of adjustment means and flatness components are not equal). In the past, the intelligent models for calculating the influence matrix are based on shallow neural network, whose training samples are limited, and the ability to express complex functions is limited.¹⁸⁾ The rolling process parameters are many, the model is very complex, highly nonlinear, and the amount of industrial field data is big.^19,20) Using the shallow neural network model to calculate the influence matrix has large error and poor generalization ability, having an impact on the follow-up control strategy formulation and decoupling model design. Moreover, these studies are limited to the decoupling of the system, without considering the time-delay of the system and the control strategy of the IRS. In the rolling process, the time-delay will make the control system produce larger overshoot and longer adjustment time. The greater the online adjustment of IRS, the more frequently it is used, the more serious the roll wear and the lower the service life.

In view of the above problems, this paper studies the non square flatness control system five input four output coupling time-delay system in engineering. Through the calculation and analysis of the non square relative gain matrix,²¹⁾ the decoupling strategy and generalized inverse decoupling model are proposed; according to the change of the influence coefficient of the adjustment means, the variable model of the influence coefficient of the IRS is proposed; in order to reduce the adjustment of IRS, the control strategy of minimum roll shifting adjustment and threshold is proposed. The deep neural network (DNN) model of flatness adjustment influence matrix is established by using the calculation data of mechanism model and the measured data of rolling process, so as to improve the calculation speed, accuracy of influence matrix and the generalization ability of network, meeting the requirements of online control. Finally, combining internal model control (IMC) and Smith prediction model,²²⁾ the IMC-Smith controller is built, which solves the time-delay, and applies the control strategy and model proposed in this paper to the industrial site successfully.

2. Flatness Decoupling Control System

2.1. Flatness Control Principle of UCM Six-high Cold Rolling Mill

In a 1420 mm UCM (universal crown control mill) six-high cold rolling mill, there are five ways of online flatness adjustment, including RT, WRB, WRAB, IRB and IRS. Let the bending force of work roll on operation side and driving side are F_wo and F_wd respectively, (F_wd + F_wo)/2 is defined as work roll symmetrical bending force, and F_wd − F_wo is defined as work roll asymmetric bending force. The principle of flatness control is shown in Fig. 1, in which, σ = {σ₁,σ₂,...,σ_m} is a vector of the flatness of each section along the strip width direction measured by the Shapemeter, and m is the number of measurement sections. A = {a₁,a₂,a₃,a₄} is the measured flatness component coefficient vector obtained through flatness pattern recognition,²³⁾ where a₁, a₂, a₃ and a₄ represent the measured primary, quadratic, cubic and quartic flatness component coefficients respectively. A^T = $\{a_1^T,a_2^T,a_3^T,a_4^T\}$ is the target flatness component coefficient vector, $a_1^T$, $a_2^T$, $a_3^T$ and $a_4^T$ represent the primary, quadratic, cubic and quartic target flatness component coefficient respectively. E = A^T – A = {e₁,e₂,e₃,e₄} is the vector of flatness deviation component coefficient, where e₁, e₂, e₃ and e₄ represent the primary, quadratic, cubic and quartic flatness deviation coefficient components respectively. ΔU = {Δu₁,Δu₂,Δu₃,Δu₄,Δu₅} is the regulator vector of a given flatness adjustment parameter, where Δu₁, Δu₂, Δu₃, Δu₄ and Δu₅ are the given RT displacement adjustment, WRB, WRAB, IRB bending force adjustment and IRS displacement adjustment, respectively. ΔU^* = $\{\mathrm{\Delta }{u}_1^{*},\mathrm{\Delta }{u}_2^{*},\mathrm{\Delta }{u}_3^{*},\mathrm{\Delta }{u}_4^{*},\mathrm{\Delta }{u}_5^{*}\}$ is the actual output flatness adjustment vector, $\mathrm{\Delta }{u}_1^{*}$, $\mathrm{\Delta }{u}_2^{*}$, $\mathrm{\Delta }{u}_3^{*}$, $\mathrm{\Delta }{u}_4^{*}$ and $\mathrm{\Delta }{u}_5^{*}$ are the actual adjustment of RT displacement adjustment, WRB, WRAB, IRB bending force adjustment and IRS displacement adjustment respectively. In each control cycle, the flatness control system makes the target flatness A^T minus the measured flatness A initially obtain the flatness deviation E. Then, it calculates the adjustment ΔU according to the flatness deviation E and sends it to the hydraulic system. Lastly, the hydraulic system of each actuator outputs the actual adjustment ΔU^* to the roll system according to its dynamic response. Meanwhile, the flatness of the roll gap is changed by changing its shape. The flatness after rolling reaches the Shapemeter after τ = l/v time (l is the distance between the roll gap and the Shapemeter, and v is the rolling speed).

Fig. 1.

Flatness control principle.

2.2. Open-loop of Non Square Flatness Control System

As shown in Fig. 1, the 1420 mm UCM six-high cold rolling mill is a non square coupling time-delay nonlinear system with five input and four output. The open-loop control system is shown in Fig. 2, in which G_TM(s) = G_T(s)G_M(s), G_WM(s) = G_W(s)G_M(s), G_IM(s) = G_I(s)G_M(s), G_SM(s) = G_S(s)G_M(s), and G_T(s), G_W(s), G_I(s), G_S(s), G_M(s) are the transfer functions of RT, WRB, IRB, IRS and Shapemeter respectively. Δa₁, Δa₂, Δa₃ and Δa₄ are the change of the flatness coefficients of primary, quadratic, cubic and quartic flatness respectively, $a_1^0$, $a_2^0$, $a_3^0$ and $a_4^0$ are the initial measured (last time) primary, quadratic, cubic and quartic flatness component respectively. Element c_ij is the open-loop gain of the flatness control system, whose physical meaning is the influence coefficient of the j-th adjustment mean on the i-th flatness component coefficient,¹²⁾ which constitutes the open-loop gain matrix of the flatness control called the flatness adjustment influence matrix C.   

$$C=\left[\begin{array}{lllll}{c}_{11}\hfill & c_{12}\hfill & c_{13}\hfill & c_{14}\hfill & c_{15}\hfill \\ c_{21}\hfill & c_{22}\hfill & c_{23}\hfill & c_{24}\hfill & c_{25}\hfill \\ c_{31}\hfill & c_{32}\hfill & c_{33}\hfill & c_{34}\hfill & c_{35}\hfill \\ c_{41}\hfill & c_{42}\hfill & c_{43}\hfill & c_{44}\hfill & c_{45}\hfill \end{array}\right],c_{ij}=\mathrm{\Delta }{a}_i/\mathrm{\Delta }{u}_j^{*}$$(1)

Fig. 2.

Open-loop of non square flatness control system.

The hydraulic system of RT, WRB, IRB and IRS of cold rolling mill and the Shapemeter can be approximated to a first-order model,⁸⁾ and their respective transfer functions are as Formula (2), in which, T_T, T_W, T_I, T_S and T_M is the time constant of the RT, WRB, IRB, IRS system and Shapemeter respectively. The open loop transfer function matrix of flatness control system is Formula (3).   

$$\begin{array}{l}{G}_T(s)={\displaystyle \frac{1}{T_{T}s+1}},G_W(s)={\displaystyle \frac{1}{T_{W}s+1}},G_I(s)={\displaystyle \frac{1}{T_{I}s+1}}\\ G_S(s)={\displaystyle \frac{1}{T_{S}s+1}},G_M(s)={\displaystyle \frac{1}{T_{M}s+1}}\end{array}$$(2)

  

$$\begin{array}{l}G(s)=\\ \left[\begin{array}{lllll}{c}_{11}{G}_{TM}(s)\hfill & c_{12}{G}_{WM}(s)\hfill & c_{13}{G}_{WM}(s)\hfill & c_{14}{G}_{IM}(s)\hfill & c_{15}{G}_{SM}(s)\hfill \\ c_{21}{G}_{TM}(s)\hfill & c_{22}{G}_{WM}(s)\hfill & c_{23}{G}_{WM}(s)\hfill & c_{24}{G}_{IM}(s)\hfill & c_{25}{G}_{SM}(s)\hfill \\ c_{31}{G}_{TM}(s)\hfill & c_{32}{G}_{WM}(s)\hfill & c_{33}{G}_{WM}(s)\hfill & c_{34}{G}_{IM}(s)\hfill & c_{35}{G}_{SM}(s)\hfill \\ c_{41}{G}_{TM}(s)\hfill & c_{42}{G}_{WM}(s)\hfill & c_{43}{G}_{WM}(s)\hfill & c_{44}{G}_{IM}(s)\hfill & c_{45}{G}_{SM}(s)\hfill \end{array}\right]e^{-\tau s}\end{array}$$(3)

Because the pure time-delay of each flatness control loop is the same, it will not cause coupling effect, so the coupling part and time-delay of flatness control system can be treated separately, the system decoupling is carried out first, and then the time-delay compensation is carried out.

3. Decoupling and IRS Control Strategy of Non Square Flatness Control System

3.1. Relative Gain and Decoupling Strategy

Decoupling is an effective method to solve the coupling of multivariable control loops and improve the control performance. The essence of decoupling is to design a computational network to counteract the correlation in the control process, so that each control loop becomes a relatively independent control loop, equivalent to a single loop control system. With the increase of coupling loops, the complexity of decoupling controller increases and the practicability becomes worse. Therefore, it is necessary to analyze the coupled system by using the relative gain theory, find out the loops that need to be decoupled or not, formulate a reasonable decoupling strategy, and simplify the design of decoupling controller.

Relative gain is a level used to measure the effect of a preselected adjustment on a particular adjusted quantity. The calculation method of non square relative gain matrix $\overline{C}$ of the above non square flatness control system is shown in Formula (4).   

$$\overline{C}=C\otimes {(C^{+})}^T$$(4)

Where (C⁺)^T is the transposition of the generalized inverse of the open-loop gain matrix C, operator ⊗ is Hadamard multiplication, that is, matrix element multiplication. The meaning of each element in the non square relative gain matrix $\overline{C}$ is the influence degree of the j-th adjustment parameter on the i-th flatness component. The greater the absolute value of relative gain, the greater the influence. The four rolling conditions are calculated by using the established flatness control deformation mechanism model(mechanism model).²⁴⁾ The calculation results of non square influence matrix and relative gain matrix are shown in Table 1. RT and WRAB have great influence on primary and cubic flatness (value of ${\overline{C}}_{11,13,31,33}$), but little influence on quadratic and quartic flatness (values of ${\overline{C}}_{21,23,41,43}$, almost zero). WRB, IRB and IRS have little effect on primary and cubic flatness (values of ${\overline{C}}_{12,14,15,32,34,35}$, almost zero), but great influence on quadratic and quartic flatness (values of ${\overline{C}}_{22,24,25,42,44,45}$).

Table 1. Non square influence matrix and relative gain matrix of flatness adjustment.

Strip width/ mm	Rolling force/ 10 kN	Entry thickness/ mm	Exit thickness/ mm	Front tension/ kN	Back tension/ kN	IRS/ mm	Influence matrix C					Relative gain matrix $\overline{\mathrm{C}}$
1264	1130	2.40	1.60	145	110	82	−0.1305	0.0000	−0.2965	0.0001	0.0270	0.5957	−0.0000	0.4042	0.0000	0.0001
							0.0005	−0.3137	0.0000	0.0249	−0.3539	−0.0001	1.2614	0.0000	0.0158	−0.2771
							−0.0602	−0.0007	0.2013	−0.0004	−0.0654	0.4044	−0.0000	0.5956	−0.0001	0.0000
							0.0007	0.2345	0.0001	0.0347	1.1718	−0.0002	−0.2820	0.0000	0.0085	1.2737
230	864	1.04	0.79	107	80	115	−0.1843	−0.0002	−0.3192	−0.0009	0.0076	0.6577	0.0000	0.3423	0.0000	0.0000
							0.0010	−0.4120	0.0020	0.0400	−0.3100	−0.0001	1.1932	0.0002	0.0189	−0.2121
							−0.0644	−0.0002	0.2142	0.0003	−0.0441	0.3425	0.0000	0.6576	0.0000	−0.0001
							0.0010	0.3110	−0.0015	0.0390	1.2900	−0.0002	−0.2141	−0.0002	0.0065	1.2080
1197	890	0.62	0.5	96	82	155	−0.2044	−0.0002	−0.4250	0.0000	0.0127	0.6525	0.0000	0.3475	0.0000	−0.0000
							0.0000	−0.4440	0.0000	0.0480	−0.3400	0.000	1.1512	0.0000	0.0198	−0.1710
							−0.0823	−0.0001	0.3214	−0.0001	−0.0347	0.3474	−0.0000	0.6525	−0.0000	0.0001
							0.0000	0.3500	0.0000	0.0420	1.7700	0.0000	−0.1728	0.0000	0.0046	1.1682
1012	880	1.24	0.85	95	75	124	−0.1306	−0.0002	−0.3151	0.0000	0.0175	0.5227	0.0000	0.4780	0.0000	−0.0007
							0.0003	−0.1917	0.0015	0.0332	0.0740	−0.0000	0.9287	0.0001	0.0519	0.0194
							−0.0830	0.0000	0.2202	−0.0002	−0.0535	0.4748	0.0000	0.5208	−0.0007	0.0052
							0.0004	0.0805	−0.0015	0.0244	1.3450	−0.0001	0.0476	−0.0002	0.0465	0.9061

According to the above calculation and analysis, the RT and WRAB are used to decoupling control the primary and cubic flatness components; WRB, IRB and IRS are used to decoupling control the quadratic and quartic flatness components. The decoupling strategy of five input and four output non square flatness control system is shown in Fig. 3, in which one is a square coupling system with two input and two output, another is a non square coupling system with three input and two output. Without considering the time-delay, the transfer function matrix is shown in Formula (5).   

$$\begin{array}{l}{G}_{13}(s)=\left[\begin{array}{cc}{c}_{11}{G}_{TM}(s)& c_{13}{G}_{WM}(s)\\ c_{31}{G}_{TM}(s)& c_{33}{G}_{WM}(s)\end{array}\right],\\ G_{24}(s)=\left[\begin{array}{lll}{c}_{22}{G}_{WM}(s)\hfill & c_{24}{G}_{IM}(s)\hfill & c_{25}{G}_{SM}(s)\hfill \\ c_{42}{G}_{WM}(s)\hfill & c_{44}{G}_{IM}(s)\hfill & c_{45}{G}_{SM}(s)\hfill \end{array}\right]\end{array}$$(5)

Where G₁₃(s) is the square open-loop transfer function matrix of the primary and cubic flatness control subsystem, and G₂₄(s) is the non square open-loop transfer function matrix of the quadratic and quartic flatness control subsystem.

Fig. 3.

Decoupling strategy of flatness control system—2 independent subsystem.

3.2. Primary and Cubic Flatness Control Square Decoupling Model

For square systems, the common decoupling methods are feed forward decoupling, feedback decoupling, diagonal matrix decoupling, etc.^25,26) The diagonal matrix decoupling is widely used.²⁷⁾ It adds a dynamic decoupling matrix to the control system, so that the product of the matrix and the characteristic matrix (transfer function matrix) of the controlled object is diagonal matrix to realize decoupling. According to the diagonal matrix decoupling theory, the dynamic decoupling matrix G_D(s) should satisfy Formula (6), where G_D11(s), G_D31(s), G_D13(s) and G_D33(s) are the four elements of the dynamic decoupling matrix G_D(s).   

$$\begin{array}{l}\left[\begin{array}{cc}{c}_{11}{G}_{TM}(s)& c_{31}{G}_{WM}(s)\\ c_{31}{G}_{TM}(s)& c_{33}{G}_{WM}(s)\end{array}\right]\times \left[\begin{array}{cc}{G}_{D11}(s)& G_{D31}(s)\\ G_{D13}(s)& G_{D33}(s)\end{array}\right]\\ =\left[\begin{array}{cc}{G}_{TM}(s)& 0\\ 0& G_{WM}(s)\end{array}\right]\end{array}$$(6)

3.3. Quadratic and Quartic Flatness Control Non Square Decoupling Model

For the decoupling control problem of non square system, the conventional decoupling method is no longer applicable. At present, the generalized inverse matrix method is commonly used to decouple the non square system and realize the decoupling control. However, the generalized inverse matrix of the non square system has right half plane poles, which makes the control system unstable.²⁸⁾ Therefore, the unstable poles of the generalized inverse matrix are decomposed to eliminate the unstable poles of the decoupling controller, so as to ensure the stability of the control system in the paper.

For quadratic and quartic flatness control subsystem, G₂₄(s) is a row full rank matrix of 2 × 3, and its generalized inverse matrix $G_{24}^{+}(s)$ is as Formula (7).   

$$G_{24}^{+}(s)=G_{24}^T(-s){\left[G_{24}(s)G_{24}^T(-s)\right]}^{-1}$$(7)

G₂₄(s) can be decomposed into a diagonal matrix G_24N(s) with stable poles and a non square matrix G_24M(s) with unstable poles.   

$$G_{24}(s)=G_{24N}(s)G_{24M}(s)$$(8)

Where:   

$$G_{24N}(s)=\left[\begin{array}{cc}\prod _{i=1}^k{\left(\frac{-s+p_i}{s+p_i}\right)}^{l_i}& 0\\ 0& \prod _{i=1}^k{\left(\frac{-s+p_i}{s+p_i}\right)}^{l_i}\end{array}\right]$$(9)

p_i is the pole of generalized inverse matrix $G_{24}^{+}(s)$ in the right half plane, and l_i is the multiplicity of p_i.

According to Formula (8) and (9), G_24M(s) = G_24N(s)⁻¹G₂₄(s) is a 2 × 3 matrix, and its generalized inverse matrix is as Formula (10).   

$$\begin{array}{l}{G}_{24M}^{+}(s)={\left[G_{24N}^{-1}(s)G_{24}(s)\right]}^{+}\\ ={\left[G_{24N}^{-1}(s)G_{24}(s)\right]}^H{\left\{\left[G_{24N}^{-1}(s)G_{24}(s)\right]{\left[G_{24N}^{-1}(s)G_{24}(s)\right]}^H\right\}}^{-1}\\ =\left[G_{24}{(s)}^H{G}_{24N}^{-1}(s)\right]{\left[G_{24N}^{-1}(s)G_{24}(s)G_{24}{(s)}^H{G}_{24N}^{-1}(s)\right]}^{-1}\\ =G_{24}{(s)}^H{G}_{24N}^{-1}(s)G_{24}(s){\left[G_{24N}(s)G_{24}{(s)}^H\right]}^{-1}{G}_{24N}(s)\\ =G_{24}{(s)}^H{\left[G_{24N}(s)G_{24}{(s)}^H\right]}^{-1}{G}_{24N}(s)\\ =G_{24}^{+}(s)G_{24N}(s)\end{array}$$(10)

It can be seen that by multiplying $G_{24}^{+}(s)$ and G_24N(s), the pole of the right half plane of $G_{24}^{+}(s)$ is eliminated by the zero point of the right half plane of G_24N(s), so that $G_{24M}^{+}(s)$ does not contain unstable poles. Therefore, the decoupling matrix can be taken as $G_{24M}^{+}(s)$ and the open-loop transfer function after decoupling is as Formula (11). Because G_24N(s) is a diagonal matrix, the control system can be decoupled.   

$$G_{24}(s)G_{24M}^{+}=G_{24}(s)G_{24}^{+}(s)G_{24N}(s)=G_{24N}(s)$$(11)

The decoupling method of quadratic and quartic flatness closed-loop control subsystem is shown in Fig. 4, in which the dotted line is the generalized inverse decoupling matrix, $G_{24M11}^{+}(s)$, $G_{24M12}^{+}(s)$, $G_{24M21}^{+}(s)$, $G_{24M22}^{+}(s)$, $G_{24M31}^{+}(s)$, $G_{24M32}^{+}(s)$ are the six elements of generalized inverse decoupling matrix $G_{24M}^{+}(s)$, G_c2(s) and G_c4(s) are the quadratic and quartic flatness feedback controllers respectively, $e_2^{*}$ and $e_4^{*}$ are the quadratic and quartic flatness deviation coefficients output by the feedback controller. After decoupling, the quadratic and quartic flatness control subsystems are equivalent to two independent and uncoupled generalized single loop systems. The feedback controller can be designed according to the single loop system.

Fig. 4.

Closed-loop decoupling control system of quadratic and quartic flatness.

3.4. Change of Influence Coefficient of Flatness Adjustment

The above two decoupling control models are closely related to the influence coefficient of flatness adjustment. In order to design the controller accurately and efficiently, it is necessary to determine the influence coefficient accurately and reasonably. Taking the strip width of 1264 mm as an example, the mechanism model calculates the influence coefficient of RT, WRAB on primary and cubic flatness component, and WRB, IRB and IRS on quadratic and quartic flatness component, as shown in Fig. 5. In the execution domain, with the change of adjustment parameters, the influence coefficients of RT, WRAB on primary and cubic flatness, WRB and IRB on quadratic and quartic flatness have almost no change, while the influence coefficient of IRS on quadratic and quartic flatness changes greatly, which is basically linear distribution. After a large number of calculations, it is found that all specifications of strip have this rule. Therefore, a method to deal with the influence coefficient is proposed in this paper, that is, c₁₁, c₃₁, c₁₃, c₃₃, c₂₂, c₄₂, c₂₄ and c₄₄ remain unchanged, c₂₅ and c₄₅ change with the position of IRS.

Fig. 5.

Calculation results of influence coefficient change. (Online version in color.)

3.5. Control Strategy of IRS

For the quadratic and quartic flatness control system, the generalized inverse decoupling model directly uses the generalized inverse matrix to realize the system decoupling without considering the adjustment allocation problem. In the rolling process, the greater the adjustment of IRS and the higher the use frequency, the more serious the wear of the roller and the lower the service life. For the quadratic and quartic flatness control loop, there are infinite groups of solutions in the executable domain because two flatness components are controlled by three adjustment means. The adjustment of IRS calculated by generalized inverse decoupling is not the minimum in all solutions. In this paper, the control strategy of minimum roll shifting adjustment and threshold value of IRS as shown in Formula (12) and Fig. 6 is proposed. When the minimum roll shifting adjustment $\mathrm{\Delta }{u}_5^{min}$ obtained from Formula (12) is greater than the threshold value ε (0.5–1 mm), the flatness can be adjusted by three adjustment means of two roll bending and IRS. If the $\mathrm{\Delta }{u}_5^{min}$ is no more than the threshold ε, the position of IRS remains unchanged. Only two bending rolls are used to adjust the flatness and the diagonal matrix is used to decouple. The c₂₅, c₄₅ can be fitted by Formula (13).   

$$min(\mathrm{\Delta }{u}_5)\{\begin{array}{c}{c}_{22}\mathrm{\Delta }{u}_2+c_{24}\mathrm{\Delta }{u}_4+{\int }_{u_5^0}^{u_5^0+\mathrm{\Delta }{u}_5}{c}_{25}(u_5)d{u}_5+a_2^0=0\\ c_{42}\mathrm{\Delta }{u}_2+c_{44}\mathrm{\Delta }{u}_4+{\int }_{u_5^0}^{u_5^0+\mathrm{\Delta }{u}_5}{c}_{45}(u_5)d{u}_5+a_4^0=0\end{array}$$(12)

  

$$c_{25}(u_5)=k_{25}{u}_5+b_{25},c_{45}(u_5)=k_{45}{u}_5+b_{45}$$(13)

Where k₂₅, b₂₅, k₄₅ and b₄₅ are the fitting coefficients and $u_5^0$ is the initial setting value of IRS. When the absolute value of Δu₅ is the minimum, one of Δu₂ and Δu₄ must reach the limit value. Therefore, we only need to calculate the value of Δu₅ when Δu₂ or Δu₄ take the limit value, and select the smallest absolute value of Δu₅ as the minimum adjustment $\mathrm{\Delta }{u}_5^{min}$ of IRS. Because of the error in the calculation of the influence matrix, and the flatness may fluctuate in a small range during the rolling process, the total amount of WRB or IRB should reach 80%–90% of the maximum value when Δu₂ or Δu₄ is taken as the limit value.

Fig. 6.

Control strategy of IRS.

4. Mechanism-intelligent Model of Flatness Adjustment Influence Matrix

4.1. Mechanism-intelligent Combination Method

There are two calculation methods of flatness adjustment influence matrix, one is mechanism model based on the basic theory of rolling, another is artificial intelligence model based on measured data. The mechanism model has a solid theoretical foundation and strong analysis and prediction ability. A large number of studies have been carried out in recent decades,^24,29,30) but the model is still huge and complex, the calculation time is long, and the calculation results have certain deviations, so it is difficult to adapt to the online control. Artificial intelligence model, which simulates the thinking mode of human brain, summarizes the relationship from experience and data, has strong self-learning ability, simple model, fast calculation speed, and the calculation results are close to the actual situation, adapting to the online control. Each of the two models has its own advantages. By combining the two models an accurate and reliable online flatness control model can be established, which is shown in Fig. 7, in which α is the weighting coefficient and can be taken as 0–1.

Fig. 7.

Mechanism-intelligent calculation of influence matrix.

4.2. DBN-DNN Structure and Training Methods

DNN network is large in scale, complex in structure and easy to fall into local optimal solution. Hinton et al.³¹⁾ proposed the DBN-DNN model in 2006, which uses the weights trained by Deep Belief Networks (DBN) to initialize the DNN, and greatly overcomes the problem of local optimal solution. Take DBN-DNN with 3-layer hidden layer structure as an example. The structure is shown in Fig. 8, which is composed of three Restricted Boltzmann Machine (RBM) units. RBM has two layers, the upper layer is hidden layer and the lower layer is visible layer. When a DBN is stacked, the output layer (hidden layer) of the previous RBM is used as the input layer (visible layer) of the next RBM unit, and then the basic DBN structure is formed. Finally, by adding another output layer, the final DBN-DNN structure is obtained.

Fig. 8.

DBN-DNN neural network structure.

DBN-DNN training process:

(1) The normalization data vector is input and Gaussian distribution is used to initialize all layer weights of DBN-DNN.

(2) The unsupervised layer by layer greedy training method is used to train DBN layer by layer, that is, RBM1 (weight is updated to contrast divergence algorithm) is trained by using input data. After the training is completed, the output of RBM1 is used as the input of RBM2, and then RBM2 is trained, and so on until RBM3 (the last RBM) training is completed.

(3) Input the output data vector in the output layer, and gradient back propagation algorithm is used to adjust the network.

4.3. DBN-DNN Calculation of Influence Matrix

The strip rolling process is affected by many factors, such as rolling mill parameters, strip parameters, process parameters and control parameters. In order to avoid too large network structure and make the model simple and practical, only some main parameters affecting the rolling process are selected such as strip width B, entry thickness h₀, exit thickness h₁, rolling force P, elastic modulus E_s, front tension σ₁, back tension σ₀, total execution amount u₁, u₂, u₃, u₄, u₅ of RT, WRB, WRAB, IRB, IRS are taken as input of the network. The primary, quadratic, cubic and quartic flatness component are the network output. The structure is shown in Fig. 9, and the calculation method of influence matrix is Formula (1).

Fig. 9.

Calculation model of DBN-DNN influence matrix.

The calculation data of mechanism model are generated offline according to rolling process parameters and flatness adjustment mechanism model, in which 50000 groups are selected as training data and 10000 groups are test data. The measured data of rolling process are generated from a large number of accumulated data in production, in which 50000 groups are selected as training data and 10000 groups are taken as test data. The number of network layers and hidden layer neurons are determined by many simulation experiments.³²⁾ DBN-DNN1 and DBN-DNN2 are selected as 8 layers, and the network structure is 12-200-150-100-60-30-10-4. The purpose of the large number of neurons in the first hidden layer is to make the input data features fully mixed, and the number of neurons in each hidden layer decreases layer by layer, which is to conducive to network regression. After simulation, the network batch size is 100, the training times of DBN is 200, the gradient back-propagation algorithm is Adam, and the learning rate is 0.001. Taking DBN-DNN2 as an example, the mean square error of each RBM network is shown in Fig. 10(a). After training, the mean square error of each hidden layer is significantly reduced. In order to illustrate the function of DBN initialization and the advantages of DNN, taking DBN-DNN2 as an example, the training times of DBN are set to 0 and 200 respectively, and a single hidden layer BP neural network with the same number of hidden layer neurons is built. The mean square error decrease comparison of the three cases is shown in Fig. 10(b). It can be seen that the mean square error of the DNN network after initialization by DBN rapidly drops to near 10⁻⁷, while the DNN network not initialized by DBN falls into the local optimal solution, and the mean square error only drops to around 345, which indicates that the weights initialized by DBN are closer to the global optimal solution than those by not initialized by DBN; the mean square error of BP neural network with the same number of hidden neurons only drops to around 207, which indicates that the calculation accuracy is higher when the number of hidden layers is deepened.

Fig. 10.

Comparison of training process.

In order to test the training effect of the network, taking DBN-DNN2 as an example, 500 training samples and 500 test samples are randomly selected. The calculation results are shown in Fig. 11. It can be seen that the calculation mean square error of training samples and test samples, is all less than 10⁻⁶, which shows that the network has high calculation accuracy and good generalization ability. Taking the first rolling condition in Table 1 as an example, the influence matrix calculated according to the mechanism model, DBN-DNN1 and DBN-DNN2 models is shown in Formula (14), and the calculation time is 10.531 s, 0.13 ms and 0.11 ms respectively. It can be seen that the calculation results of DBN-DNN1 are basically the same as those of mechanism model, but the calculation speed is greatly improved, which meets the needs of online control. The calculated results of the DBN-DNN2 model are quite different from the mechanism model, which indicates that there is a certain deviation between the calculation results of the mechanism model and the actual situation, so it is necessary to introduce the DBN-DNN2 model.   

$$\begin{array}{l}{C}_{\mathrm{M}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{s}\mathrm{m}}=\left[\begin{array}{rrrrr}\hfill -0.1305& \hfill 0.0000& \hfill -0.2965& \hfill -0.0001& \hfill 0.0270\\ \hfill 0.0005& \hfill -0.3137& \hfill 0.0000& \hfill 0.0249& \hfill -0.3539\\ \hfill -0.0602& \hfill -0.0007& \hfill 0.2013& \hfill -0.0004& \hfill -0.0654\\ \hfill 0.0007& \hfill 0.2345& \hfill 0.0001& \hfill 0.0347& \hfill 1.1718\end{array}\right]\\ \hspace{1em}\hspace{0.5em}{C}_{\mathrm{D}\mathrm{N}\mathrm{N}1}=\left[\begin{array}{rrrrr}\hfill -0.1325& \hfill 0.0024& \hfill -0.2935& \hfill -0.0013& \hfill 0.0293\\ \hfill -0.0005& \hfill -0.3121& \hfill 0.0010& \hfill 0.0252& \hfill -0.3545\\ \hfill -0.0603& \hfill -0.0002& \hfill 0.2012& \hfill 0.0002& \hfill -0.0649\\ \hfill 0.0002& \hfill 0.2342& \hfill 0.0000& \hfill 0.0341& \hfill 1.1732\end{array}\right]\\ \hspace{1em}\hspace{0.5em}{C}_{\mathrm{D}\mathrm{N}\mathrm{N}2}=\left[\begin{array}{rrrrr}\hfill -0.1842& \hfill 0.0001& \hfill -0.3993& \hfill 0.0008& \hfill 0.0127\\ \hfill 0.001\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }& \hfill -0.4262& \hfill 0.0001& \hfill 0.0329& \hfill -0.2218\\ \hfill -0.0351& \hfill 0.0004& \hfill 0.2897& \hfill 0.0001& \hfill -0.0657\\ \hfill 0.0002& \hfill 0.3610& \hfill -0.0003& \hfill 0.0315& \hfill 1.3813\end{array}\right]\end{array}$$(14)

Fig. 11.

Training model test.

5. Simulation and Comparison

5.1. Flatness Control Simulation

In a 1420 mm UCM six-high cold rolling mill, the variation range of RT, WRB, WRAB, IRB and IRS are −100 μm–100 μm, −200 kN–500 kN, −150 kN–150 kN, −500 kN–500 kN, −200 mm–200 mm respectively. And the time constant of T_T, T_W, T_I, T_S, T_M are 0.05, 0.15 s, 0.25 s, 0.40 s, 0.003 s respectively. The normal rolling speed is 300 m/min–800 m/min, and the distance between Shapemeter and roll gap is 2.0 m.

In order to verify the correctness of decoupling strategy and generalized inverse decoupling model, the closed-loop simulation model as shown in Fig. 12 is established. To solve the time-delay of the system, the IMC-Smith algorithm proposed in paper (20) is adopted, and the filter time constant of each controller is taken as 0.4. Taking the first rolling condition in Table 1 as an example, the initial setting value of RT, WRB, WRAB and IRB are 0 μm, 130 kN, 0 kN, 160 kN, the influence coefficient of IRS is taken as the variable value, the influence coefficient of other actuators is taken fixed value. And the input speed of flatness control is 60 m/min, and the flatness measured delay time τ = 2 s, the initial flatness is $a_1^0$ = 22.54, $a_2^0$ = 55.28, $a_3^0$ = −18.158, $a_4^0$ = −85.35, and the target flatness is set as A^T = {0,0,0,0}, the change process of flatness component and adjustment is shown in Fig. 13. It can be seen that each adjustment can reach the ideal value quickly without overshoot, and the each flatness component can reach the target value quickly without overshoot, and the control process is stable. The simulation results show that the separately decoupling control strategy of the first, cubic and quadratic and quartic flatness is effective, and the process of the quadratic and quartic flatness which is controlled by generalized inverse decoupling is stable.

Fig. 12.

Flatness closed-loop decoupling control system.

Fig. 13.

Simulation results of closed-loop control of flatness.

To illustrate the advantages of the variable model of the influence coefficient of IRS, taking the quadratic and quartic flatness control loop as examples, the influence coefficient of the IRS is taken as the fixed value and the variable value respectively. The change process of the flatness component is shown in Fig. 14. It can be seen that when the influence coefficient of the IRS is variable value, the adjustment time of the quadratic flatness and quartic flatness are 0.5 s and 1.2 s faster than the fixed value respectively. It fully explains the necessity of changing the influence coefficient of IRS with the position of it.

Fig. 14.

Effect of variable model for influence coefficient of IRS.

In order to verify the feasibility of the control strategy of minimum roll shifting adjustment and threshold, ignoring the slight influence of RT and WRAB on the quadratic and quartic flatness components. The generalized inverse decoupling model and diagonal matrix decoupling model proposed in this paper are used for simulation calculation. The influence coefficient of IRS on quadratic and quartic flatness is fitted as Formula (15).   

$$\begin{array}{l}{c}_{25}(u_5)=0.0007u_5-0.4149\\ c_{45}(u_5)=-0.0052u_5+1.5968\end{array}$$(15)

The total amount of WRB and IRB is 85% of design extreme value. Through calculating, the minimum adjustment of intermediate roll shifting is $\mathrm{\Delta }{u}_5^{min}$ ≈ 36mm. The change process of flatness component and adjustment is shown in Fig. 15. Compared with the simulation results in Fig. 14, it can be seen that the change process of quadratic and quartic flatness is the same, the change of WRB increases from 114.53 kN to 158.55 kN, IRB changes in the direction during the adjustment process, the adjustment increases from −13.56 kN to 268.72 kN, and the adjustment of IRS decreases from 57.58 mm to 36 mm. The simulation results show that the control strategy of minimum roll shifting adjustment and threshold setting is effective, and the adjustment of IRS is greatly reduced.

Fig. 15.

Simulation results of control strategy of IRS.

5.2. Industrial Application

The flatness control method proposed in this paper has been applied to 1420 mm UCM six-high cold rolling mill. DBN-DNN1 and DBN-DNN2 which have been trained complete are packaged into Dynamic Link Library (DLL) files for the main program to call. The weighted coefficient α is taken as 0.5, and the minimum adjustment of IRS is $\mathrm{\Delta }{u}_5^{min}$ ≈ 32mm. LabVIEW^TM is a computer software name registered trademark of National Instruments. The control program is written with LabVIEW as the development environment, and its application principle is shown in Fig. 16. Ni OPC server is the bridge between LabVIEW and S7-400 to establish data connection, and plays the function of communication protocol conversion. The flatness control program first reads the parameters needed for flatness control in S7-400 through Ni OPC server, then reads the measured flatness of the flatness measuring host through the switch, calculates the flatness control adjustment and writes it into S7-400 through Ni OPC server, and finally S7-400 gives the control adjustment to the corresponding actuator through its input and output (I/O) equipment. The factory application effect of the first rolling condition in Table 1 as shown in Fig. 17. When the rolling speed is greater than 30 m/min, the sampling starts and the period is 50 ms. It can be seen that the WRB does not overshoot, while the RT, WRAB and IRB produce small overshoot and the adjustment time is 4.5 s. When the IRS reaches 114 mm, it does not change. The mean square error of flatness decreases rapidly and stably. And it is stable at about 1.5I after flatness control. The engineering examples show that the decoupling strategy, decoupling model, influence coefficient variable model, minimum roll shifting adjustment and threshold control strategy of IRS meet the needs of online flatness control, the overshoot of each adjustment is small, and the influence matrix calculated by mechanism-intelligent is close to the reality. After long-term application, the strip flatness can be controlled within 5I for the flatness of incoming material less than 45I, less than 1.5I for strip thickness above 1.5 mm, 2.5I for 1.0–1.5 mm, 4I for 0.3–1.0 mm, and 5I for less 0.3 mm strip thickness. Moreover, the overshoot of each adjustment mean is small and the adjustment time is within 6 s. At present, the performance of the flatness control system is reliable, and it can meet the requirements of online rolling and the requirements of subsequent process, and the flatness control effect is remarkable.

Fig. 16.

Application principle of flatness control system.

Fig. 17.

Application example of flatness control.

6. Conclusion

(1) In this paper, a control strategy is proposed to adjust the primary and cubic flatness by RT and WRAB, and quadratic and quartic flatness adjusted by WRB, IRB and IRS. The complex five input and four output decoupling control problem is transformed into a square system with two input and two output and a non square system with three input and two output, which simplifies the design of control system. By decomposing the unstable poles of generalized inverse matrix of non square system, a new decoupling method of generalized inverse matrix for quadratic and quartic flatness control loop is proposed, which ensures the stability of the control system.

(2) In view of the large change of the IRS influence coefficient with the roll shifting position, a variable model of the roll shifting influence coefficient is proposed, which improves the dynamic characteristics and reduces the adjustment time of the system. In order to increase the service life of rolls, the control strategy of minimum roll shifting adjustment and threshold is proposed to reduce the use frequency and adjustment of IRS.

(3) A mechanism-intelligent influence matrix calculation model based on big data and DNN optimized by DBN is proposed, which improves the calculation speed, accuracy and generalization ability of the influence matrix, adapts to the requirements of online control, and provides more accurate calculation results for the formulation of control strategy and decoupling model.

(4) The industrial application shows that the thicker the strip is, the easier the flatness control is. When the strip thickness is greater than 1.5 mm, the flatness mean square error can be controlled within 2I. And the thinner the strip thickness is, the more difficult the flatness control is. When the strip thickness is less than 0.3 mm, the flatness mean square error can be controlled within 5I.

Acknowledgements

This work is supported by the National Science and Technology Support Program (Grant No. 2011BAF15B00) and the Natural Science Foundation Project of Hebei province (Grant No. E2016203482).

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