Looper-gauge Integrated Control in Hot Strip Finishing Mill Using Inverse Linear Quadratic Theory

Abstract

Hot strip rolling is a significant process in the fields of manufacturing and processing. In order to further improve the control accuracy of looper angle, strip tension and gauge in hot strip finishing mill, an innovative looper-gauge integrated control scheme is developed in this paper. Based on the inverse linear quadratic (ILQ) theory, a proposed control scheme is designed for the looper-gauge integrated system. First, considering the numerous interactions between looper angle, strip tension and gauge, and the disturbances from several sources, a dynamic 6th order state space model is established and validated. Then, the control scheme based on ILQ theory is imported into gauge-looper integrated system. The desired poles are placed according to the dynamic characteristics requirements. The state feedback optimal control law is determined by an improved ILQ design method. Then the multivariable looper-gauge integrated control system is constructed. The proposed control scheme has the explicit ability to achieve desired looper and gauge control performances, with less external disturbances and no sensitivity of strip dimension changing. The effectiveness of the proposed looper-gauge integrated control scheme compared with traditional control strategies is shown in the simulation results. The interactions between looper control and guage control are also minimized.

1. Introduction

Rolling is one of the most effective processes in metallic materials manufacturing. During hot strip rolling process the casting slabs are rolled into thin steel strips. A typical schematic diagram of a hot rolling plant is shown in the Fig. 1. In finishing rolling process, a 30 mm thickness transfer bar is gradually rolled by each stand, and its thickness is finally reduced to 0.8–12 mm.^1,2) The looper installed interstand controlled by strip tension and mass flow affects both the strip thickness and the operation stability. The speed of upstream stand is regulated to keep the looper angle at its setup reference, whilst the looper motor torque is controlled to maintain the strip tension at suitable range. The strip thickness accuracy is controlled by automatic gauge control (AGC) through adjusting roll gaps of mills.^3,4) Since there are strong interactions among the control processes of the strip gauge, looper angle and tension, the finishing mill control is a complex integrated problem. The conventional PI control strategy is widely used in the above control processes, but the dynamic relations between them are not effectively considered. With the increasingly demand to improve strip thickness and operation stability, the development of advanced control in hot finishing mill control is always a focus problem.^5,6)

Fig. 1.

Schematic drawing of a hot rolling plant. (Online version in color.)

Considering the interaction between looper angle and tension, various looper-tension control system techniques have been introduced. Noh et al. proposed a non-interactive looper angle and strip tension decoupling control method through nonlinear disturbance observer.⁷⁾ Based on feedback linearization, the looper angle and strip tension almost disturbance decoupling control scheme was proposed by Zhong and Wang.⁸⁾ Combining with a minimal-order disturbance observer and an on-line subspace identifier based on state space model for the looper system, Park and Hwang constructed the adaptive receding horizon controller.⁹⁾ Yin et al. investigated a dynamic matrix control algorithm that realized the optimal control of the looper angle and tension integrated system.¹⁰⁾ Based on the guaranteed cost sliding mode control, Wang et al. proposed a nonlinear control algorithm to further improve the control accuracy of looper uncertain system.¹¹⁾ However, these approaches are designed only for looper angle and strip tension system, that means looper-tension control and AGC have been implemented independently. As an important disturbance for the looper dynamics, if there is any changing the mass flow of the stand because of regulation of AGC, then looper angle and strip tension will be disturbed.¹²⁾ Meanwhile, the strip tension is an important parameter affecting strip thickness. Therefore, considering the interactions among strip thickness, tension and looper angle, the multivariable control looks more effective.

Recently, some control schemes on looper-gauge integrated control system have been proposed. To enhance system robustness, Hearns et al. studied the looper-gauge system controller with the multivariable H_∞ control strategy.¹³⁾ In order to balance gauge and looper angle control performance against each other, also they designed multivariable controller with both state feedback and estimation.¹⁴⁾ Okada et al. proposed a looper-gauge controller based on optimal servo theory and the model decoupling method, and the controller has already applied to the whole productions process.¹⁵⁾ Pittner and Simaan proposed looper-gauge system based on the state-dependent Riccati equation (SDRE) technique, and the proposed control scheme can effectively cope with the rapid changes during strip threading.^16,17) Such control schemes have showed good performance, however, they are not intuitive and hard to tune on site due to the complicated control structures.

Inverse Linear Quadratic (ILQ) is the inverse problem of Linear Quadratic (LQ). According to the desired dynamic and steady performance index of system, ILQ control is regarded as an efficient control strategy since it guarantees the optimum performance by the suitable control gains selecting.¹⁸⁾ The state feedback control was imported into system to obtain the best control effect. ILQ theory has been successfully applied to tandem cold rolling mill, strip gage and tension controller. Also mill balance controller is designed based on the ILQ design method and applied to an actual process.^19,20,21) Hwang and Kim proposed an ILQ controller on the basis of state feedback linearized model, high speed and high accuracy control loops are very effective in keeping both looper angle and strip tension within target values after the controller is applied in actual finishing mill control system.²²⁾ In this paper, looper-gauge integrated control system is designed based on ILQ theory to realize the cooperative optimization control for strip tension, looper angle and strip gauge with better anti-interference ability and better dynamic control performance

This paper is organized as follows: Section 2 gives a brief description of process model and its state space representation. In Section 3, the proposed control scheme based on ILQ theory is demonstrated. In Section 4, the control performances of ILQ control scheme are compared with that of traditional control strategies. Conclusions are presented in Section 5.

2. Process Model of Hot Strip Finishing Mill

In this section, an overview of the process model with strip tension, looper angle and gauge model is given to synthesize the integrated system. The looper, strip and interstand geometry for rolling stands are shown in Fig. 2.

Fig. 2.

Schematic diagram of the intermediate stands, looper and the strip. (Online version in color.)

2.1. Deformation Resistance

Since strip temperature is the primary determiner of the deformation resistance, it should be predicted accurately. Conduction of the rolls or water loss ΔT_C, and radiation in the inter-stand region ΔT_R are two main causes of heat loss. At the same time, there is an energy input ΔT_W due to the work done by rolls during the deformation.²³⁾   

$$\mathrm{\Delta }{T}_{\text{C}}=\frac{4\beta (T-T_{roll})}{0.3\mathrm{\hspace{0.17em} }{h}_{in}+0.7\mathrm{\hspace{0.17em} }{h}_{out}}\sqrt{\frac{\lambda }{\rho C\pi }\frac{l_c}{\omega R}}$$(1)

  

$$\begin{array}{l}\mathrm{\Delta }{T}_R=\\ {\left[{(T+273)}^{-3}+{\displaystyle \frac{6{\epsilon }_s{\sigma }_{s}l}{\rho C\omega R}}\left({\displaystyle \frac{1}{w}}+{\displaystyle \frac{1}{l}}+{\displaystyle \frac{1}{h_{out}}}\right)\right]}^{-1/3}-(T+273)\end{array}$$(2)

  

$$\mathrm{\Delta }{T}_W=\frac{P\eta }{w{l}_c\rho C}ln\frac{h_{in}}{h_{out}}$$(3)

It is very useful when the strips have a vivid skid mark. Here, this skid mark signal means a long-term yield stress variation in the rolling process caused by local slab temperature fluctuation in a reheating furnace. It is the major part of the yield stress variation.

The yield stress of steel increases with increasing strain, increasing strain rate and decreasing temperature. Though there are several plots for estimation of the yield stress k_f, an estimate of k_f made by using the comprehensive set of empirical relationships developed by Shida²⁴⁾ is more specific.

2.2. Strip Thickness

Sims’ rolling force model is presented widely in the literature as being useful for control development. He assumes that the product of the interfacial shear stress and the angular variable is negligible when compared to other terms and that sticking friction. The separating rolling force per unit width exerted on the rolls by the deforming metal sheet is obtained as:   

$$P=Q_P\sqrt{R^{\prime }(h_{in}-h_{out})}(K-0.7{\sigma }_{in}-0.3{\sigma }_{out})$$(4)

where K = 1.15k_f for finishing rolling process, $R^{\prime }=R$$\left[1+\frac{16(1-{\nu }^2)P}{\pi E_r(h_{in}-h_{out})}\right]$.

Q_P is a geometrical factor which is a non-linear function of flattened work roll radius R′, the inlet and exit gauges in various functional forms, a simply Q_P developed by Ford and Alexander²⁵⁾ is used in this study:   

$$Q_P=0.786+\frac{\sqrt{1-e}}{\sqrt{2(2-e)}}\frac{\sqrt{R^{\prime }(h_{in}-h_{out})}}{h_{in}+h_{out}}$$(5)

where $e=\frac{h_{in}-h_{out}}{h_{in}}$.

Exit thickness deviation is estimated as follows based on the gaugemeter principle with the roll gap deviation and the rolling force deviation. It can be expressed as:   

$$\mathrm{\Delta }{h}_{out}=\mathrm{\Delta }S+\frac{\mathrm{\Delta }P}{M_m}$$(6)

Substituting Eqs. (4) and (5) to Eq. (6) leads to:   

$$\mathrm{\Delta }{h}_{out,i+1}=f_{th}(h_{in,i+\text{1}},\mathrm{\hspace{0.17em} }{\sigma }_i,\mathrm{\hspace{0.17em} }{\sigma }_{i+\text{1}},\mathrm{\hspace{0.17em} }{T}_{i+\text{1}},\mathrm{\hspace{0.17em} }{V}_{i+\text{1}},\mathrm{\hspace{0.17em} }{S}_{i+\text{1}})$$(7)

2.3. Forward Slip

Slip is a difference between work roll circumferential speed and the strip speed. The relative velocities of the strip and the roll have been identified as having an effect on the rate of straining, lubrication, friction, scaling and the interfacial forces. The forward slip and backward slip, which are given in terms of the relative velocity, depends on many factors such as entry and exit gauges, back and front strip tension, and friction conditions. The forward slip equation derived by Sims²⁶⁾ is used in this study.   

$$f=\frac{R^{\prime }}{h_{out}}{\varphi }_n^2$$(8)

where   

$$\begin{array}{l}{\varphi }_n=\\ \sqrt{{\displaystyle \frac{h_{out}}{R^{\prime }}}}tan\\ \left[{\displaystyle \frac{1}{2}}arctan\sqrt{{\displaystyle \frac{e}{1-e}}}+{\displaystyle \frac{\pi }{8}}ln(1-e)\sqrt{{\displaystyle \frac{h_{out}}{R^{\prime }}}}+{\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{h_{out}}{R^{\prime }}}}{\displaystyle \frac{{\sigma }_{out}-{\sigma }_{in}}{K}}\right]\end{array}$$(9)

As the principle of metal flow equilibrium, h_in(1 − b) = h_out(1 + f), the backward slip can be expressed as:   

$$b=\text{1}-\frac{h_{out}}{h_{in}}(\text{1}+f)$$(10)

2.4. Strip Tension Dynamic Model

Interstand strip tension is defined by the Hook’s law, which is a function of Young’s modulus E:   

$${\sigma }_i(t)=E\left[\frac{L(\theta )-(L-\xi (t))}{L-\xi (t)}\right]$$(11)

where strip tension σ_i is the same as the ith stand exit tension σ_out,i and the i + 1th stand entry tension σ_in,i+1, L(θ) is the geometry looper length between the intermediate stands, and L – ξ(t) is uncertain accumulated strip length because of the mass flow difference between the intermediate stands.   

$$\begin{array}{l}L(\theta )=AB+AC\\ AB=\sqrt{{(L_{l}sin\theta -d+r)}^2+{(a+L_{l}cos\theta )}^2}\\ AC=\sqrt{{(L_{l}sin\theta -d+r)}^2+{(L-a-L_{l}cos\theta )}^2}\end{array}$$(12)

  

$$\begin{array}{l}\xi (t)={\int }_{t_0}^t(v_{in,i+1}-v_{out,i})\text{d}t\\ \dot{\xi }(t)=V_{i+1}(1-b_{i+1})-V_i(1+f_i)\end{array}$$(13)

As ξ(t) is rather small compared with L, in the denominator of Eq. (12), L – ξ(t) is approximated to L. The derivative σ_i(t) can be expressed as:   

$$\begin{array}{l}{\dot{\sigma }}_i(t)=f_{ten}(\theta ,\mathrm{\hspace{0.17em} }{V}_i,\mathrm{\hspace{0.17em} }{V}_{i+1},\mathrm{\hspace{0.17em} }{b}_{i+1},\mathrm{\hspace{0.17em} }{f}_i)\\ \hspace{1em}\hspace{1em}=\frac{E}{L}\left[\mathrm{\Delta }\dot{L}(\theta )+(v_{in,i+1}-v_{out,i})\right]\\ \hspace{1em}\hspace{1em}=\frac{E}{L}\{\left[-L_{l}sin(\theta -{\theta }_1)+L_{l}sin(\theta +{\theta }_2)\right]\\ \hspace{1em}\hspace{1em}\mathrm{\hspace{0.17em} }\dot{\theta }(t)+\left[V_{i+1}(1-b_{i+1})-V_i(1+f_i)\right]\}\end{array}$$(14)

where ${\theta }_1=\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(\frac{L_{l}sin\theta -d+r}{L_{l}cos\theta +a}\right)$, ${\theta }_2=\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(\frac{L_{l}sin\theta -d+r}{L-L_{l}cos\theta -a}\right)$.

2.5. Looper Dynamic Model

Several external forces such as interstand tension, strip gravity, strip bending force and inertia force acting on the looper. The total inertia is made up of the looper and strip respect to the pivoting point. The load torque on the looper is caused by the strip tension, strip weight, looper weight, and the force required to bend the strip. By the Newton’s second law, the looper dynamic behavior can be described by the following equation:   

$$J_{lp}\ddot{\theta }=M-M_T-M_{lp}$$(15)

  

$$\begin{array}{l}{M}_T=\sigma hw{L}_l[sin(\theta \text{+}{\theta }_2)-sin(\theta -{\theta }_1)]\\ \hspace{1em}\hspace{1em}+[16Ewc{(h/L)}^3+0.5W_{S}g]L_{l}cos\theta \\ \hspace{1em}\hspace{1em}+0.25W_S{L}_l^2\ddot{\theta }\left\{{[cos(\theta -{\theta }_1)]}^2+{[cos(\theta +{\theta }_2)]}^2\right\}\end{array}$$(16)

  

$$M_{lp}=(W_{R}g+0.5W_{L}g)L_{l}cos\theta$$(17)

Substituting Eqs. (16) and (17) to Eq. (15) leads to:   

$$\begin{array}{l}\ddot{\theta }=f_{lp}(M,\theta ,\sigma )\\ ={\displaystyle \frac{M-\left[16Ewc{(h/L)}^3+0.5W_{S}g+W_{R}g+0.5W_{L}g\right]L_{l}cos\theta }{J}}\\ -{\displaystyle \frac{\sigma hw{L}_l[sin(\theta +{\theta }_2)-sin(\theta -{\theta }_1)]}{J}}\end{array}$$(18)

where   

$$\begin{array}{l}J=J_{lp}+0.25W_S{L}_l^2\\ \mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\left\{{[cos(\theta -{\theta }_1)]}^2+{[cos(\theta +{\theta }_2)]}^2\right\}\end{array}$$(19)

2.6. Controlling Actuators

In hot strip finishing mill control process, the main controlling actuators are the main drive (work roll speed V_i), the looper drive (looper torque M_i) and the hydraulic cylinder (roll gap S_i+1), including automatic speed control (ASC), automatic torque control (ATC) and hydraulic gap control (HGC), respectively. All of the above actuators are modeled as first-order systems. Hence, they are assumed that:   

$${\dot{V}}_i=-\frac{V_i}{T_{spd}}+\frac{V_i^{ref}}{T_{spd}}$$(20)

  

$${\dot{M}}_i=-\frac{M_i}{T_{trq}}+\frac{M_i^{ref}}{T_{trq}}$$(21)

  

$${\dot{S}}_{i+1}=-\frac{S_{i+1}}{T_{hgc}}+\frac{S_{i+1}^{ref}}{T_{hgc}}$$(22)

where $V_i^{ref}$, $M_i^{ref}$, $S_{i+1}^{ref}$ stand for the rolling speed of the i th stand, the looper actuator torque, the roll gap of the i + 1 th stand, respectively. T_spd, T_trq, T_hgc are the time constants of the ASC, ATC and HGC, respectively.

2.7. Transfer Delay

The transport delay that is related to the travel of the strip thickness and the temperature between stands i and i+1, the transport delay of strip thickness can be expressed as first order lags:   

$$\mathrm{\Delta }{h}_{in,i+1}=\mathrm{\Delta }{h}_{out,i}{e}^{-T_{d,i}s}$$(23)

Unlike the thickness, the temperature is not constant for a strip segment, but decreases with time due to radiation and increases with strain rate due to rolling speed. The transport delay of temperature can be simply expressed as:   

$$\mathrm{\Delta }{T}_{i+1}=\mathrm{\Delta }{T}_i{e}^{-T_{d,i}s}+t_{{\omega }_i}\mathrm{\Delta }{\omega }_i+t_{T_i}\mathrm{\Delta }{T}_i$$(24)

2.8. State Space Model

The overall nonlinear mathematic model for looper-gauge system can be obtained by the Eqs. (7), (14), (18) and (20), (21), (22), (23), (24). The system is linearized about the setup value with Taylor’s series expansion. As a result, the state space representation of the system could be expressed by:   

$$\{\begin{array}{l}{\dot{\mathit{x}}}_i={\mathit{A}}_i{\mathit{x}}_i+{\mathit{B}}_i{\mathit{u}}_i+{\mathit{D}}_{1i}{\mathit{d}}_i\\ {\mathit{y}}_i={\mathit{C}}_i{\mathit{x}}_i+{\mathit{D}}_{2i}{\mathit{d}}_i\end{array}$$(25)

where x_i = [Δθ Δω_l Δσ_i ΔV_i ΔM_i ΔS_i+1]^T, ${\mathit{u}}_i={\left[\begin{array}{ccc}\mathrm{\Delta }{V}_i^{ref}& M_i^{ref}& S_{i+1}^{ref}\end{array}\right]}^T$, d_i = [Δh_in,i ΔT_in,_i Δh_in,i+1 ΔT_i+1 ΔV_i+1 ΔS_i]^T, y_i = [Δθ Δσ_i Δh_out,i+1]^T.   

$${\mathit{A}}_i=\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 0\\ \frac{\partial f_{lp}}{\partial \theta }& \frac{\partial f_{lp}}{\partial {\omega }_l}& \frac{\partial f_{lp}}{\partial {\sigma }_i}& 0& \frac{\partial f_{lp}}{\partial M_i}& 0\\ 0& \frac{\partial f_{ten}}{\partial {\omega }_l}& \frac{\partial f_{ten}}{\partial {\sigma }_i}& \frac{\partial f_{ten}}{\partial V_i}& 0& \frac{\partial f_{ten}}{\partial S_{i+1}}\\ 0& 0& 0& \frac{-1}{T_{spd}}& 0& 0\\ 0& 0& 0& 0& \frac{-1}{T_{trq}}& 0\\ 0& 0& 0& 0& 0& \frac{-1}{T_{hgc}}\end{array}\right]$$

  

$$\begin{array}{cc}{\mathit{B}}_i=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ \frac{1}{T_{spd}}& 0& 0\\ 0& \frac{1}{T_{trq}}& 0\\ 0& 0& \frac{1}{T_{hgc}}\end{array}\right]& {\mathit{C}}_i=\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& \frac{\partial f_{th}}{\partial {\sigma }_i}& 0& 0& \frac{\partial f_{th}}{\partial S_{i+1}}\end{array}\right]\end{array}$$

  

$${\mathit{D}}_{1i}=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ \frac{\partial f_{ten}}{\partial h_{in,i}}& \frac{\partial f_{ten}}{\partial T_i}& \frac{\partial f_{ten}}{\partial h_{in,i+1}}& \frac{\partial f_{ten}}{\partial T_{i+1}}& \frac{\partial f_{ten}}{\partial V_{i+1}}& \frac{\partial f_{ten}}{\partial S_i}\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]$$

  

$${\mathit{D}}_{2i}=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& \frac{\partial f_{th}}{\partial h_{in,i+1}}& \frac{\partial f_{th}}{\partial T_{i+1}}& \frac{\partial f_{th}}{\partial V_{i+1}}& 0\end{array}\right]$$

2.9. Model Validation

This paper takes the 6th stand, the 7th stand, and the 6th looper of 1700 mm hot finishing mills as control objects. For the sake of completeness, all the related mechanical and rolling parameters obtained from the plant are listed in Table 1. A comprehensive simulation has been implemented with rolling operation. The Validation is used to compare the plant and simulation outputs for the main process variables of stands and looper.

Table 1. Mechanical and rolling parameters of the plant.

Variable	Value	Variable	Value	Variable	Value
L	5.5 m	θ	0.297 rad	Ttrq	0.02 s
d	0.155 m	hout,6	2.97 mm	Thgc	0.03 s
r	0.15 m	hout,7	2.527 mm	V7	8.5 m/s
Ws	189.9 kg	f6	0.0552	S7	3.12 mm
WL	260.0 kg	σ5	6.64 MPa	T7	908.9°C
R6	323.5 mm	a	1.5 m	hin,6	3.764 mm
Mm,7	5400 kN/mm	Ll	0.76 m	hin,7	2.97 mm
Tspd	0.12 s	Jlp	480 kg·m2	E	150 GPa
V6	7.13 m/s	WR	370.0 kg	f7	0.0384
S6	3.53 mm	w	1265 mm	σ6	7.03 MPa
T6	930.6°C	R7	324.3 mm

As described above, the main disturbances are the strip temperature and gauge. According to the entry temperature of the mill, strip temperature of finishing mills can be calculated stand by stand. The temperature of transfer bar measured at mill entry, the entry temperature of stand 6, and the entry thickness of stand 6 are shown in Fig. 3, respectively.

Fig. 3.

Main disturbances of the integrated system.

Since the roll gap of stand 6 is a constant value, and the speed of last stand is accelerating, the process is named speed-up rolling. There has been no tension meters installed between stands, tension is controlled by varying the torque supplied to the looper using torque reference calculation block. The torque of the looper motor is adjusted according to looper angle and strip weight to maintain strip tension to the reference value. The strip temperature, looper angle and strip thickness can be obtained from the comprehensive models. The comparison of real and simulated profiles are provided in Fig. 4. All profiles show that the simulated results well agree with the real measured ones. Although the magnitudes do not always match, the drifts are slow and not likely to have any effect on process dynamics.

Fig. 4.

Comparison of actual and simulated profiles. (Online version in color.)

3. Controller Design Based on ILQ Theory

3.1. ILQ Design Theory

For the optimal control problem of a linear control system, if the performance index is described by a quadratic cost function of both state variables and control variables, the optimal control problem is called the Linear Quadratic (LQ) regulator problem, ILQ is the inverse problem of LQ.¹⁸⁾ Consider the following linear system:   

$$\{\begin{array}{l}\dot{\mathit{x}}=\mathit{A}\mathit{x}+\mathit{B}\mathit{u}\\ \mathit{y}=\mathit{C}\mathit{x}\end{array}$$(26)

Lemma 1: A control law K is both optimal and stable if and only if there exist P > 0 and R > 0 satisfying:   

$$\mathit{P}\mathit{H}+{\mathit{H}}^T\mathit{P}>0$$(27)

  

$${\mathit{B}}^T\mathit{P}=\mathit{R}\mathit{K}$$(28)

where H = BK/2 − A.

Considering that A and B are of the form:   

$$\mathit{A}=\left[\begin{array}{cc}{\mathit{A}}_{11}& {\mathit{A}}_{12}\\ {\mathit{A}}_{21}& {\mathit{A}}_{22}\end{array}\right]\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathit{B}=\left[\begin{array}{c}\mathit{0}\\ \mathit{I}\end{array}\right]$$(29)

where $\mathit{A}\in {\mathit{R}}^{n\times n}$, ${\mathit{A}}_{11}\in {\mathit{R}}^{(n-m)\times (n-m)}$, ${\mathit{A}}_{22}\in {\mathit{R}}^{m\times m}$, $\mathit{I}\in {\mathit{R}}^{m\times m}$ is the identity matrix.

Any optimal control law K satisfied Eq. (28) can be expressed as:   

$$\mathit{K}={\mathit{V}}^{-1}\mathrm{\Sigma }\mathit{V}\left[{\mathit{F}}_1,\mathrm{\hspace{0.17em} }\mathit{I}\right]$$(30)

by proper choice of a real nonsingular matrix V, a real diagonal matrix Σ =$diag\mathrm{\hspace{0.17em} }({{\sigma }^{\prime }}_1,\mathrm{\hspace{0.17em} }{{\sigma }^{\prime }}_2,\mathrm{\hspace{0.17em} }\cdots {{\sigma }^{\prime }}_m)$ or Σ = σ′I, and a real matrix F₁.

Let S = diag (s₁, s₂, …s_n−m) be the real Jordan form of A₁₁–A₁₂F, and T₁ be a real nonsingular matrix such that:   

$$({\mathit{A}}_{11}-{\mathit{A}}_{12}{\mathit{F}}_1){\mathit{T}}_1={\mathit{T}}_1\mathit{S}$$(31)

Define G = −F₁T₁, set T₁ = [t₁, t₂, …, t_n−m], G = [g₁, g₂, …, g_n−m], Eq. (31) can be written as:   

$${\mathit{t}}_i={({\mathit{s}}_i\mathit{I}-{\mathit{A}}_{11})}^{-1}{\mathit{A}}_{12}{\mathit{g}}_i$$(32)

F₁ can be uniquely determined by:   

$${\mathit{F}}_1=-\mathit{G}{\mathit{T}}_1^{-1}$$(33)

if we choose a G, and then obtain the nonsingular T₁, to satisfy Eqs. (31) and (32) for some suitable chosen S.

Set $\mathrm{\Pi }=diag({\pi }_1,\mathrm{\hspace{0.17em} }{\pi }_2,\mathrm{\hspace{0.17em} }\cdots {\pi }_{n-m})$, ${\pi }_i\ne 0$ can be chosen arbitrarily, $\overline{\mathit{T}}=\left[\begin{array}{cc}{\mathit{T}}_1\mathrm{\Pi }& \mathbf{0}\\ \mathit{G}\mathrm{\Pi }& {\mathit{V}}^{-1}\end{array}\right]$

And then   

$$\overline{\mathit{H}}={\mathit{T}}^{-1}\mathit{H}\mathit{T}=\left[\begin{array}{cc}-\mathit{S}& -{\mathrm{\Pi }}^{-1}{\overline{\mathit{A}}}_{12}\\ -{\overline{\mathit{A}}}_{21}\mathrm{\Pi }& -{\overline{\mathit{A}}}_{22}+\mathrm{\Sigma }/2\end{array}\right]$$(34)

where ${\overline{\mathit{A}}}_{12}={\mathit{T}}^{-1}{\mathit{A}}_{12}{\mathit{V}}^{-1}$, ${\overline{\mathit{A}}}_{21}=\mathit{V}({\mathit{A}}_{21}{\mathit{T}}_1+{\mathit{A}}_{22}\mathit{G}-\mathit{G}\mathit{S})$, ${\overline{\mathit{A}}}_{22}=\mathit{V}({\mathit{A}}_{22}+{\mathit{F}}_1{\mathit{A}}_{21}){\mathit{V}}^{-1}$.

Lemma 2: The matrix $\overline{\mathit{H}}$, with which Eq. (27) is satisfied, is copositive, if   

$${\sigma }_i^{\prime }>{\lambda }_{max}(\overline{\mathit{E}})\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }1\le i\le m$$(35)

where $\overline{\mathit{E}}=({\overline{\mathit{A}}}_{22}+{\overline{\mathit{A}}}_{22}^{\text{T}})$$-\left[{({\mathrm{\Pi }}^{-1}{\overline{\mathit{A}}}_{12})}^{\text{T}}+{\overline{\mathit{A}}}_{21}\mathrm{\Pi }\right]$${(\mathit{S}+{\mathit{S}}^{\text{T}})}^{-1}$${\left[{({\mathrm{\Pi }}^{-1}{\overline{\mathit{A}}}_{12})}^{\text{T}}+{\overline{\mathit{A}}}_{21}\mathrm{\Pi }\right]}^{\text{T}}$, λ_max(●) denotes the maximum eigenvalue of a matrix.

As shown above, F₁ and Σ are obtained, and V can be chosen arbitrarily, the optimal control law K can be determined in the form of Eq. (30).

3.2. Controller Design

With the ILQ control design theory, the control law for the process represented by Eq. (25) is tried to be designed. With the theory presented, a very important prerequisite is that the control matrix has the same form as matrix B in Eq. (29). In traditional ILQ design process, the dominant poles {S_i} and vectors {g_i} are selected respectively, the parameters in {g_i} are determined by trial and error method.

Define the following matrices:   

$${\mathit{A}}_A=\left[\begin{array}{cc}{\mathit{A}}_i& {\mathit{B}}_i\\ 0& 0\end{array}\right]\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{\mathit{B}}_A=\left[\begin{array}{c}0\\ \mathit{I}\end{array}\right]$$(36)

  

$${\mathit{A}}_{cl}=\left[\begin{array}{cc}{\mathit{A}}_i& 0\\ {\mathit{C}}_i& 0\end{array}\right]\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{\mathit{B}}_{cl}=\left[\begin{array}{c}{\mathit{B}}_i\\ 0\end{array}\right]$$(37)

There are following relationships between them:   

$${\mathit{A}}_A={\mathbf{\Gamma }}^{-1}{\mathit{A}}_{cl}\mathbf{\Gamma }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }{\mathit{B}}_A={\mathbf{\Gamma }}^{-1}{\mathit{B}}_{cl}$$(38)

where $\mathbf{\Gamma }=\left[\begin{array}{cc}{\mathit{A}}_i& {\mathit{B}}_i\\ {\mathit{C}}_i& 0\end{array}\right]$ is a nonsingular matrix.

The system matrices A_A and B_A match the form of Eq. (29), by placing the desired poles based on ILQ design theory, optimal control law K_A can be obtained as:   

$${\mathit{K}}_A={\mathit{V}}^{-1}\mathrm{\Sigma }\mathit{V}\left[{\mathit{F}}_1,\mathrm{\hspace{0.17em} }\mathit{I}\right]$$(39)

Considering the similarity transformation in Eq. (38), the optimal control law K_cl of system (A_cl, B_cl) is:   

$${\mathit{K}}_{cl}=\left[{\mathit{K}}_F,\mathrm{\hspace{0.17em} }{\mathit{K}}_I\right]={\mathit{K}}_A{\mathbf{\Gamma }}^{-1}$$(40)

Substituting the control laws given by Eqs. (39) and (40) into the systems (A_A, B_A) and (A_cl, B_cl), it can be shown that:   

$$\begin{array}{l}{\overline{\mathit{A}}}_A={\mathit{A}}_A-{\mathit{B}}_A{\mathit{K}}_A={\mathrm{\Gamma }}^{-1}({\mathit{A}}_{cl}-{\mathit{B}}_{cl}{\mathit{K}}_{cl})\mathrm{\Gamma }\\ \hspace{1em}\hspace{1em}={\mathrm{\Gamma }}^{-1}{\overline{\mathit{A}}}_{cl}\mathrm{\Gamma }\end{array}$$(41)

  

$${\overline{\mathit{A}}}_{cl}={\mathit{A}}_{cl}-{\mathit{B}}_{cl}{\mathit{K}}_{cl}=\left[\begin{array}{cc}{\mathit{A}}_i-{\mathit{B}}_i{\mathit{K}}_F& -{\mathit{B}}_i{\mathit{K}}_I\\ {\mathit{C}}_i& 0\end{array}\right]$$(42)

Supposing that {f_cl,i} is the corresponding eigenvector of eigenvalues {s_i} for ${\overline{\mathit{A}}}_{cl}$, and {f_A,i} is the corresponding eigenvector of eigenvalues {s_i} for ${\overline{\mathit{A}}}_A$. Then the following relationship can be established:   

$${\mathit{f}}_{A,i}={\mathrm{\Gamma }}^{-1}{\mathit{f}}_{cl,i}=\left[\begin{array}{c}{\mathit{t}}_i\\ {\mathit{g}}_i\end{array}\right]$$(43)

Taking the dominant poles {s_i} as desired poles, we can place poles for (A_A, B_A) based on ILQ theory, the vectors {g_i} are determined without trials, and the vectors {t_i} are the same with achieved by Eq. (32).

Assuming y_ref is the reference of looper angle, strip tension and strip thickness. Based on $\dot{q}=y-y_{ref}$, the close loop system with feedback control is:   

$$\{\begin{array}{l}\left[\begin{array}{c}\dot{\mathit{x}}\\ \dot{\mathit{q}}\end{array}\right]=\left[\begin{array}{cc}{\mathit{A}}_i-{\mathit{B}}_i{\mathit{K}}_F& -{\mathit{B}}_i{\mathit{K}}_I\\ {\mathit{C}}_i& 0\end{array}\right]\left[\begin{array}{c}\mathit{x}\\ \mathit{q}\end{array}\right]+\left[\begin{array}{c}0\\ -\mathit{I}\end{array}\right]{\mathit{y}}_{ref}\\ \mathit{y}=\left[\begin{array}{cc}{\mathit{C}}_i& 0\end{array}\right]\left[\begin{array}{c}\mathit{x}\\ \mathit{q}\end{array}\right]\end{array}$$(44)

Set V as an identity matrix, the structure of looper-gauge integrated control system based on ILQ control theory is show as Fig. 5, [K_F0, K_I0] can be deduced by Eqs. (39) and (40):   

$$\left[{\mathit{K}}_{F0},{\mathit{K}}_{I0}\right]={\mathrm{\Sigma }}^{-1}\left[{\mathit{K}}_F,{\mathit{K}}_I\right]=\left[{\mathit{F}}_1,\mathit{I}\right]{\mathrm{\Gamma }}^{-1}$$(45)

Fig. 5.

Structure of ILQ control algorithm.

To construct the proposed controller, the proposed algorithm will involve the following steps.

Step 1: According to the desired response of all outputs in looper-gauge integrated system, the desired dominant poles are chosen, and the matrix S can be formed as:   

$$\mathit{S}=diag\mathrm{\hspace{0.17em} }(-35,\mathrm{\hspace{0.17em} }-30,\mathrm{\hspace{0.17em} }-25,\mathrm{\hspace{0.17em} }-20,\mathrm{\hspace{0.17em} }-2.09+2.0i,\mathrm{\hspace{0.17em} }-2.09-2.0i)$$

Step 2: The desired poles are placed for (A_cl B_cl) system, the corresponding eigenvectors {f_cl,i} of dominant poles {s_i} are determined.

Step 3: The corresponding eigenvectors {f_A,i} are obtained from Eq. (43), and the matrices G and T₁ are constructed:   

$$\mathit{G}=\left[\begin{array}{rrrrrr}\hfill -3.16e-5& \hfill 1.22e-4& \hfill 4.44e-4& \hfill -3.09e-4& \hfill 0.0086& \hfill 0.0086\\ \hfill -0.0013& \hfill 0.0124& \hfill -0.0041& \hfill -0.0078& \hfill -0.0040& \hfill -0.0040\\ \hfill 3.50e-4& \hfill -0.0012& \hfill -0.0060& \hfill -0.0126& \hfill 0.0064& \hfill 0.0064\end{array}\right]$$

  

$${\mathit{T}}_1=\left[\begin{array}{rrrrrr}\hfill -1.82e-4& \hfill -5.94e-6& \hfill 3.86e-4& \hfill -6.92e-4& \hfill 0.0039& \hfill -0.0885\\ \hfill 0.0064& \hfill 1.78e-04& \hfill -0.0097& \hfill 0.0138& \hfill -0.1852& \hfill 0.1722\\ \hfill 0.0266& \hfill 0.0013& \hfill -0.0294& \hfill 0.0337& \hfill -0.0947& \hfill 0.1640\\ \hfill 9.88e-6& \hfill -4.71e-5& \hfill -2.22e-4& \hfill 2.21e-4& \hfill -0.0103& \hfill 0.0082\\ \hfill -0.0042& \hfill 0.0310& \hfill -0.0081& \hfill -0.0130& \hfill 0.0020& \hfill -0.0041\\ \hfill -0.0070& \hfill -0.0122& \hfill -0.0240& \hfill -0.0316& \hfill -0.0096& \hfill 0.0062\end{array}\right]$$

Step 4: The matrix F₁ is determined by Eq. (33):   

$${\mathit{F}}_1=\left[\begin{array}{rrrrrr}\hfill 0.2080& \hfill 0.1959& \hfill -0.0402& \hfill -2.2482& \hfill -0.0014& \hfill 0.0135\\ \hfill 0.0926& \hfill 1.5898& \hfill -0.3804& \hfill -25.524& \hfill -0.4174& \hfill 0.0358\\ \hfill 0.6129& \hfill 4.6606& \hfill -1.1227& \hfill -72.499& \hfill -0.0675& \hfill -0.0467\end{array}\right]$$

Step 5: The matrix $\mathrm{\Pi }=diag$ (934.45, 391.30, 628.42, 636.56, 350.62, 350.62) is selected, the ${\lambda }_{max}(\overline{\mathit{E}})$ given by Eq. (35) is 1393.4. In this work, set σ′ = 1500 > 1393.4, and weighting matrix Σ = diag (1500, 1500, 1500).

Step 6: The optimal control law is given by Eq. (45):   

$${\mathit{K}}_{F0}=\left[\begin{array}{rrrrrr}\hfill 0.1134& \hfill -0.0073& \hfill 2.53e-4& \hfill 0.12& \hfill -8.47e-22& \hfill -6.34e-21\\ \hfill -0.0967& \hfill 3.0895& \hfill 0.0052& \hfill 0.00& \hfill 0.020& \hfill -3.24e-17\\ \hfill -0.1298& \hfill -0.3579& \hfill 0.0147& \hfill 0.00& \hfill 2.18e-19& \hfill 0.030\end{array}\right]$$

  

$${\mathit{K}}_{I0}=\left[\begin{array}{rrr}\hfill 0.0930& \hfill -0.0336& \hfill 0.0398\\ \hfill 48.624& \hfill 26.161& \hfill -0.1785\\ \hfill -5.0093& \hfill 0.5482& \hfill 3.0001\end{array}\right]$$

4. Simulation Results

For the purpose of evaluating the performance of the ILQ controller designed in section 3.2, various dynamic simulations are carried out. In order to verify the effectiveness of the proposed controller, simulation results of the traditional independently PI algorithm and ILQ + PI control (the looper angle and strip tension are controlled by ILQ controller, and AGC is controlled by PI controller) technique are simultaneously reported. The block diagram of PI, ILQ + PI and ILQ controllers are shown in Fig. 6, and LQ has the same structure as ILQ. The general control performances were illustrated by strip gauge, looper angle and tension step testing response, and several disturbances are considered for anti-interference ability verification. Furthermore, model mismatching simulations with various strip dimensions are conducted. In general terms, the disturbances and strip dimensions are sufficient to envelope most abnormal conditions encountered in actual operating process. The simulations are carried out to verify the control performance under various abnormal conditions.

Fig. 6.

Block diagram of system controllers.

4.1. Step Testing Response

In hot strip finishing mill, the control performance of looper and gauge system will directly affect the stability of strip tension and dimensional precision of strip. Step testing signals are added to the references of strip gauge, looper angle and strip tension, respectively. The simulation results are shown in Figs. 7, 8, 9.

Fig. 7.

The outputs of system in presence of strip gauge step testing. (Online version in color.)

Fig. 8.

The outputs of system in presence of looper angle step testing. (Online version in color.)

Fig. 9.

The outputs of system in presence of strip tension step testing. (Online version in color.)

The traditional PI controller results in long response time for gauge and tension control, and a little short response time for looper angle control but with large overshoot. Moreover, the fluctuations of other parameters are seriously greater than ILQ controller. For the ILQ + PI controller, since the interaction between looper angle and strip tension is concerned, good performance for looper system is realized. However, with gauge step signal, there will be an extreme fluctuation in strip tension. As for LQ and ILQ controllers, there are better performances in the response time and overshoot with ILQ controller. The fluctuations of other parameters are greatly reduced compare with the above controllers. To summarize, ILQ controller has a better set point tracking performance. There are more advantages than PI, ILQ + PI and LQ controllers in maintaining the stability of system.

4.2. Disturbance Rejection

In the hot strip rolling process, the whole rolling state is complex and variable. The integrated system in hot strip finishing mill can be affected by many disturbances from several sources. The main disturbances come from temperature variations, strip speed variations and roll eccentricity. Strip temperature mainly varies with skid marks, which are occur at low frequencies. Roll eccentricity is caused by irregularities in the mill rolls and roll bearings, it is smaller in size but at much higher frequencies. The speed disturbance comes from the setup deviations at the mills, and the forward and backward slips, which are influenced by the forward and backward tension. The system outputs in presence of incoming temperature sine disturbance of 20°C and 0.2 Hz, rolling speed step disturbance of 0.01 m/s, and roll eccentricity sine disturbance of 0.025 mm and 1.0 Hz are shown in Figs. 10, 11, 12.

Fig. 10.

The outputs of system in presence of skid marks disturbance. (Online version in color.)

Fig. 11.

The outputs of system in presence of rolling speed disturbance. (Online version in color.)

Fig. 12.

The outputs of system in presence of backup roll eccentricity disturbance. (Online version in color.)

For PI controller, all of the disturbances bring obvious fluctuations in strip gauge, looper angle and tension. ILQ, ILQ + PI and LQ controllers have the same performances in several situations, and the control effects of ILQ controller are better than that of ILQ + PI and LQ controllers in general situations. From the above results, it is clear that ILQ controller has good disturbance rejection performance for looper-gauge integrated system.

4.3. Various Strip Dimensions

In real hot strip production, every strip rolled in the mill may have different dimensions (cross sectional areas). Thus, the looper-gauge integrated system must be robust to ensure strip quality and mill operating stability. Here, under the conditions of constant controller parameters, the control effects of different cross sectional areas (CSA) were investigated by PI, ILQ + PI, LQ and ILQ controllers. The strip dimensions (represented by thickness×width) used for simulations are as follows: 1.5 mm × 965 mm, 2.0 mm × 1115 mm, 2.5 mm × 1265 mm, 3.5 mm × 1415 mm and 5.0 mm × 1565 mm. The same as section 4.1, step testing signals were added to the references, the simulation results of all strip dimensions are shown in Figs. 13, 14, 15, which are exported in the form of nephograms.

Fig. 13.

Response curves of strip gauge with gauge step signals. (Online version in color.)

Fig. 14.

Output curves of looper angle with gauge step signals. (Online version in color.)

Fig. 15.

Output curves of strip tension with gauge step signals. (Online version in color.)

For strip gauge step signal disturbances, the response time of all controllers are not much changed. However, with PI or ILQ + PI controller, the fluctuations of looper angle and tension are extremely larger. The ILQ controller is not sensitive to the changing of strip dimension. The fluctuations of looper angle and strip tension are much smaller than that in PI, LQ and ILQ + PI controllers.

The similar tests about disturbances of looper angel and tension also have been done. For looper angle step signal disturbances, control effects of all controllers are close to section 4.1. About 13% angle overshoot is obtained with PI controller, which is much bigger than with other controller. The gauge fluctuations of ILQ, LQ and ILQ + PI controllers are smaller than that of PI controller, while the tension fluctuation of PI controller is smaller than that of ILQ, LQ and ILQ + PI controllers. Fortunately, all these fluctuations are acceptable for hot strip finishing mill. For strip tension step signal disturbances, the PI, ILQ + PI and LQ controllers give large tension overshoot with different strip specifications, and the maximum overshoot of proposed ILQ controller is 5.57%. The fluctuations of strip gauge and looper angle with all controllers are acceptable.

The ILQ method is one of the optimal control strategies for the cross coupling phenomenon in multi-target tracking system. Its response model is directly determined based on dominant pole assignment with fast calculations and dynamic decoupling. Hence, ILQ controller can make the system obtain a relatively small overshoot and disturbance as fast response time. For the looper-gauge integrated system, balancing the coupling among looper angel, tension and gauge synthetically, improving response speed of looper angel, tension and gauge, and reducing the disturbance from looper torque, roll speed and roll gap can be realized by ILQ controller. Facing complex rolling conditions, ILQ controller has a strong ability to regulate on-line and the asymptotic behavior of ILQ method make the control system have strong robustness. From all of the above results, it is clear that ILQ controller is very effective for complication looper-gauge integrated system. The proposed ILQ controller obtains better control effect of strip gauge, looper angle and strip tension than that of PI, ILQ + PI and LQ controllers, and it is robust to model mismatching.

5. Conclusions

Taking the interactions among the looper angle, strip tension and gauge into account, a novel control scheme based on ILQ theory for looper-gauge integrated system has been investigated in this paper. The control scheme combines simplicity and effectiveness, according to the desired dynamic and steady performance, the strip gauge, looper angle and strip tension are cooperative controlled through desired closed-loop poles assignment and the weighting matrices tuning.

The control performances of the proposed ILQ controller are compared with that of PI controller, ILQ + PI controller and LQ controller, the simulation results are summarized follows. For ILQ controller, the response time of strip gauge, looper angle and strip tension are 92.0 ms, 1042.5 ms and 75.0 ms, respectively. With maximum 5.32% overshoot the fluctuations of other two parameters are smaller than PI, ILQ + PI and LQ controllers. The interrelated relations of the looper and gauge are integrated concerned, interaction between them is no longer a problem. The gauge control response time is extremely reduced, and the precision and stability are improved. With the various external rolling disturbances and different strip dimensions, good control performances, such as fast response, strong disturbance rejection and perfect robust stability can be obtained in looper-gauge integrated system by ILQ controller. Simulation results show the effectiveness of the proposed looper-gauge integrated control scheme compared with traditional control strategies. With the help of the powerful computing ability of programmable logic controller and the experience on ILQ controller for looper system, the ILQ controller for looper-gauge integrated system is prospective to use on site.

Acknowledgments

This work was financially supported by National Natural Science Foundation of China (No.: 51774084; 51704067; 51634002), National Key R&D Program of China (No.: 2017YFB0304100) and the Fundamental Research Funds for the Central Universities (No.: N160704004; N170708020).

Nomenclature

symbols

a: distance between upstream stand and looper, m

b: backward slip

β: thermal conductivity efficiency

C: specific heat capacity of rolled material, J/kg/°C

c: distance between passline and looper roll, m

d: distance between looper pivot and passline, m

ΔT_C: temperature conduction loss, °C

ΔT_R: temperature radiation loss, °C

ΔT_W: temperature deformation gain, °C

E: strip modulus of elasticity, GPa

e: reduction

E_r: roll modulus of elasticity, GPa

ε_s: strip surface emissivity

η: fraction of mechanical work transformed into heat

f: forward slip

f_lp: function of looper dynamic

f_ten: function of strip tension

f_th: function of strip thickness

g: gravitational constant, m/s²

h: strip thickness, mm

J: total inertia of the looper and strip, kg·m²

J_lp: looper inertia respect to the pivoting point, kg·m²

K: mean constrained yield stress, MPa

k_f: yield stress, MPa

L: distance between adjacent stands, m

L_l: looper arm length, m

l_c: contact length of rolled material with work rolls, m

λ: thermal conductivity of rolled material, W/m/°C

M: looper motor torque, N·m

M_lp: torque caused by looper weight, N·m

M_m: mill modulus, kN/mm

M_T: torque caused by strip, N·m

ν: Poisson’s ratio of work roll

ω: work roll angular speed, rad/s

ω_l: looper angular speed, rad/s

P: rolling force, kN

Q_P: deformation geometrical factor

R: work roll radius, m

r: radius of the looper roll, m

R′: flattened work roll radius, m

ρ: density of rolled material, kg/m³

S: roll gap, mm

σ: strip tension between adjacent stands, MPa

σ_s: Stefan-Boltzmann constant, W/m²/K⁴

T: strip temperature, °C

T_roll: work roll temperature, °C

θ: looper angle, rad

V: work roll speed, m/s

v: exit strip speed, m/s

w: strip width, mm

W_L: mass of looper arm, kg

W_R: mass of looper roll, kg

W_S: mass of interstand strip, kg

subscripts

i: i th stand or looper between i th and i + 1 th stand

i + 1: i + 1 th stand

in: entry of stand

out: exit of stand

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