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composed of an optimization-based feedforward controller and a proportional feedback controller with Smith predictor is derived for the lateral motion and shape. The performance of the proposed control concept is demonstrated in simulation studies based on the validated model.


Introduction
Roughing mills are typically used for the first thickness-reduction steps of steel slabs after casting. Typically, 5 to 7 consecutive rolling passes are applied to reach the desired plate thickness. To counteract spreading and to control the width of the resulting plate, some roughing mills are equipped with edger rolls. Typically, they are mounted at the entry side of the mill stand and are active in forward rolling passes only. Fig. 1 shows a roughing mill equipped with edger rolls. Ideally, the plate centerline coincides with the lateral centerline of the rolling mill and the desired rectangular plate shape is perfectly realized (no wedge, no camber). In reality, however, plates tend to rotate in the roll gap and a camber is formed, due to lateral inhomogeneities like temperature or thickness gradients caused by the casting or the reheating process. These inhomogeneities may also entail an unwanted lateral thickness wedge. Imperfections of the plate shape not only reduce the product quality. In the worst case, they can cause excessive wear of side guides or even collisions with the rolling mill or equipment along the roller table. High repair costs and unproductive downtime of the plant may be the consequences. To prevent such undesirable scenario, active control of the plate motion and camber is usually required.
In the literature, the roughing process is often analyzed by the finite element method (FEM) [3][4][5]. These works focus on the question of how camber occurs and how it can be reduced. While [5] suggests reducing camber by tilting the roll during the rolling pass, [3,4] suggest influencing the camber by exerting lateral forces on the plate while it is clamped in the roll gap. Although the contribution of exerting forces on the plate to reduce camber is quite strong, these works neither consider the lateral plate motion nor do they propose a suitable control concept to use lateral forces in a feedback loop.
In [1], a mathematical model of the lateral plate motion in a roughing mill is proposed and used to analyze how the plate motion can be influenced by exerting lateral forces with edger rolls. It is further investigated how forces on the entry or exit side of the roll gap influence the stability of the plate motion. A Fig. 1. Reversing roughing mill: forward pass with active edger rolls [1,2]. review of models for the evolution of the plate contour is given in [6]. There, it is discussed how different effects like thickness inhomogeneities, roll gap alignment or lateral forces influence the resulting camber.
The shape of the plate as seen from the top view has to be captured to measure the camber, which, for instance, is required for feedback control. In the literature, different strategies for camber measurement are documented. In [7], three position measurement devices are used to obtain a polynomial representation of the product contour. The measured contour is then used for feedback control of the product curvature. More recent works like [8][9][10][11][12][13][14] use 2D-CCD or C-MOS cameras to record the plate from the top view. Afterwards, image processing techniques like edge detection (e.g. the Canny algorithm [15]) and image stitching are employed to get the product contour. In [16][17][18][19], the main focus is the measurement of the lateral strip position in a finishing mill. In these applications, the contour is of minor importance. In [17,18], the measured lateral strip position is used for strip steering control. In addition, [20] focuses on stabilizing the process in longitudinal direction.
Most of the existing solutions concentrate on measuring the plate contour (heavy plates in roughing mills) or the lateral position (strips in finishing mills). In contrast to the works presented so far, in [21], both the motion and the contour of the plate are of major interest. In [21], a dynamic model of a heavy-plate roughing mill is reported and both the lateral plate motion and the plate contour are estimated in real time using an optimizationbased approach. The measurement system used in the current paper is based on [21]. In [22,23], the camber evolution in roughing mills is controlled without edger rolls. In these works, the work rolls are hydraulically adjusted and the work roll tilt serves as a control input for camber reduction. This control input can also be adjusted while the plate moves through the roll gap.
In the considered industrial plant, the roll gap tilt is electromechanically adjusted by self-retaining screws. Thus, the roll gap tilt cannot be modified during a rolling pass but only between the passes. However, the plant is equipped with edger rolls that can be moved also during the rolling pass. In [2], a control concept is developed to reduce camber and also ensure a safe plate motion during the rolling pass with the lateral forces applied by edger rolls. For safety reasons, a cascade control structure with admittance control in the inner loop is proposed, see [2]. This subordinate admittance control loop limits the performance of the control concept.
The present work builds on and significantly extends the results of [2]. The main contributions of the present work are: • A new solution to omit the inner admittance control loop is developed.
• The model is expanded and validated by measurement data from the industrial plant. • The effectiveness of edger rolls as a control input for the lateral plate motion and camber is analyzed.
• A two-degrees-of-freedom control concept for the lateral plate motion and camber is developed and tested based on the validated model. It uses the lateral position of the edger rolls as control input.
The paper is organized as follows: Section 2 summarizes the models from [1,2,6] and shows how they can be reformulated from lateral edger forces to lateral edger positions as control inputs. In Section 3, the measurement system [21] used in the considered industrial plant is briefly discussed. It is also shown how the model from Section 2 can be validated based on these measurements. The validated model is then used to design a model-based control concept in Section 4. A two-degrees-offreedom control concept for the lateral plate motion and camber is proposed by commanding the lateral position of the edger rolls. The control concept is evaluated in simulation studies in Section 5. Section 6 gives a short conclusion and suggests further possible research in this field.

Mathematical model
The first part of this section summarizes the results from [1,2,6]. Based on these state-of-the-art models for the lateral plate motion and camber, some extensions are proposed with the objective to replace the lateral edger force by the lateral edger position as a control input. In addition to the lateral force/position of the edgers, the curvature and thickness profile of the incoming plate and the roll gap tilt are considered as model inputs. For brevity, the argument t for the time is omitted throughout the paper. frame (x, y, z) is located at the center of the mill stand. For quantities at the entry side, the superscript -(e. g. z = 0 − ) and on the exit side, the superscript + (e. g. z = 0 + ) will be used.

State-of-the-art model for lateral plate motion
According to [1], the mathematical model of the plate motion follows from the material derivative D Dt (·) = ∂ ∂t (·) + v en ∂ ∂z (1) of the angular displacement ϕ (z) and the lateral position w (z) of the plate. The variable v en denotes the mean entry velocity of the plate. These material derivatives yield where ω − denotes the rotational speed of the plate with respect to the axis y. In (2b), zero lateral material flow is assumed in the roll gap. The spatial derivatives in (2) can be computed from Timoshenko's beam theory [24] in the form In (3), M (z) denotes the local bending moment in the beam, E (z) is Young's modulus, and I − y defines the area moment of inertia of the beam with respect to the axis y. The initial curvature of the beam is denoted by κ (z), Q − is the shear force, G (z) the shear modulus and A − s is the shear area of the plate.
The rotational speed ω − in (2) is computed in the form based on the material velocity profile V − (X ) at the entry of the roll gap, with the plate width w 0 and X being the lateral distance measured from the plate centerline. Thus, ω − also depends on the conditions in the roll gap. As in [1], the roll gap model of Sims [25] with the local roll force q roll per unit width, the work rolls circumferential speed u r , the mean yield stress k fm , the entry thickness h − of the plate and the roll gap height H (which is assumed to be the exit thickness h + of the plate) is used and extended to capture the influence of tensile stresses Σ − and Σ + [26]. With the linear approximations α (X) =ᾱ + ∆α , the velocity profile follows as with the sensitivities The The processed length z in of the incoming plate serves as independent variable. In the time-free formulation, variations of v en do not affect the dynamic behavior.

State-of-the-art model for camber
The analytical relation for the outgoing camber according to [6] is expanded by the last term in (11) to cover elastic deformation outside the roll gap which is rolled in and influences the resulting camber κ + . Here, λ =h − /H is the ratio of the mean input and mean output thicknessh − andH, respectively. With (4) and analogous considerations for ω + , the algebraic relation can be deduced. The sensitivities K + Σ − , K + h , and K + H determine how the tensile stress ∆Σ − , the thickness wedge ∆h − of the incoming plate, and the tilt of the roll gap ∆H influence the output velocity profile and thus also the resulting camber κ + . The sensitivities satisfy [6] K −

Model with edger position as control input
The equations of the plate dynamics (9) and the camber evolution (12) have the lateral edger force F − as control input. To prevent undesirable large edger positions, a formulation with the edger position w (−l e ) as control input is desirable for feedback control. In [2], admittance control was proposed to overcome the safety issue associated with the control input F − . In this section, (9) and (12) are reformulated so that the lateral position w (−l e ) replaces F − as control input.
As in [2] and outlined in Fig. 3, a reduced Young's modulus and shear modulus in the roll gap is assumed to capture the reduced deformation resistance in the roll gap i. e.
with the Poisson's ratio ν. The roll gap has the contact length L [27]. This simplified model does not cover plastic deformation in the roll gap in a physically correct way, however, the important benefit of the proposed approach is the resulting simple linear system structure.
On the entry side of the roll gap, two Timoshenko beams are coupled at z = −L (entry part of the roll gap). Integration of (3) with a coupling condition for the continuity of ϕ (−L) and w (−L) with the abbreviation In the roll gap (z = 0), the conditions ϕ [1,6]. For typical values of κ − , it can be shown that the double integral in (15a) is of minor importance and thus can be neglected The downstream part of the plate is assumed as a rigid body. In the following, an asymmetric mill stretch due to a nonzero lateral plate position or due to a thickness wedge [28] is neglected. Thus, inserting the lateral force F − from (16) into (9) and (12) yields the mathematical model with the abbreviations for the input gains, based on (9)

Model for the plate centerline
To compute the plate's centerline, (18b) is rewritten in the form the incoming plate curvature κ − and The relation with the independent variable ζ ∈ [ 0, L + ] and the outgoing plate length L + yields the centerline of the outgoing plate in the form where the terms cl x,u , cl − and cl ∆hH can be calculated separately. For better comparison, the centerlines cl + are rotated and shifted (rigid-body motion) according to the procedure of Appendix A.1 yieldingcl + .
The centerline cl − of the incoming plate is measured by the camera system before every rolling pass. Thuscl − can be calculated in advance. Because of the self-retaining character of the roll gap adjustment, the roll gap tilt ∆H = ∆h + and also the thickness wedge ∆h − are assumed as constant throughout the rolling pass. This yields a constant camber κ ∆hH (cf. (19c)) and thus the centerline which can also be calculated in advance. The relation can be systematically considered in the dynamic system (18a). This yields the linear system the states u ] T and the model input u = w (−l e ). After the rigid-body motion of cl x,u , the centerline of the outgoing plate can be written as

Model validation
In this section, the model (24) and (25) is validated based on the parameters and measurements of the industrial plant at voestalpine Stahl GmbH. For this purpose, experiments were conducted where edger rolls move in a lateral direction while the plate is rolled. During the experiments, the edger rolls were active in the third (17 plates) and the fifth rolling pass (38 plates).
Therefore, the curvature κ ∆hH is summarized as one uncertain parameter. In the following, the parameters E l and K Σ and κ ∆hH are estimated based on measurements. For an overview of known parameters and parameters to be estimated, see Table 1.

Measurement system at the industrial plant
At the considered industrial plant, two CMOS cameras are installed at the entry and exit side of the mill stand, above the roller table. Fig. 1 illustrates the setup. With the algorithm discussed in [21], it is possible to capture both the lateral position in the field of view (FOV) and the shape of the plate.
As usual in the roughing process, the work rolls are cooled with water sprays. This brings along that cooling water accumulates at the upstream upper plate surface and thereby reduces the contrast between the hot edge and the surrounding. In most of the rolling passes, neither the upstream lateral position nor the upstream shape of the plate can be accurately measured with the entry-side camera. The exit-side camera image is usually not deteriorated by cooling water and thus provides reliable measurement data.
However, at the time the plate passes the FOV of the exit-side camera, the lateral position and shape of the plate between the roll gap and the FOV is unknown. As a consequence, a reliable calculation of the lateral position of the plate in the roll gap is only possible after the rolling pass but not in real time. This measurement delay does not matter for validation purposes but is a significant obstacle for feedback control.

Approximation of uncertain parameters
The uncertain parameters E l , K Σ and κ ∆hH can be estimated based on (16) and (25). This estimation requires a measurement of F − and cl + .
When measuring the lateral edger force F − , friction forces in the adjustment system of the edgers have to be considered. To determine these friction forces, edgers were moved without a plate in the roll gap and a Stribeck friction curve [29] was identified (cf. Appendix A.2). The identified friction force is then compensated for an accurate measurement of F − .
For their estimation, the parameters E l , K Σ and κ ∆hH are normalized with respect to their nominal values yielding f E l , f K Σ , and f ∆hH . The optimization problem reads as  subject to (16) and (25), with the scalar weighting parameters χ F > 0 and χ cl > 0. The parameters have to be identified for every rolling pass and every product class separately. Fig. 4 shows the relative frequency distributions of the identified parameters. The left and center frequency distribution show the optimized parameters f E l and f K Σ for all strips where edgers were moved during the fifth rolling pass (38 plates). Because the peaks in the distributions are quite distinct, the use of constant parameters f E l and f K Σ is a reasonable approximation. Thus, all further investigations are performed with the parameters f E l and f K Σ identified by the optimization problem (26). The right frequency distribution in Fig. 4 refers to the parameter f ∆hH for all strips where measurement data are available in the fifth rolling pass (irrespective of a lateral motion of the edgers; 144 plates). Because of uncertainties like temperature gradient, offset, and mill stretch, the parameter f ∆hH is subject to more fluctuations.
It turned out that the factor f ∆hH calculated in the fourth rolling pass correlates with the factor determined in the fifth rolling pass (cf. Fig. 5). This brings along that during the production process (26) only needs to be calculated in the fourth rolling pass. The factor f ∆hH for the fifth rolling pass can then be determined in advance and without optimization.
During the production process, the roll gap tilt ∆H is only rarely changed and thus the influence of ∆h − and ∆H on cl + (cf. (19c)) cannot be clearly separated. It would be desirable to use ∆H as control input. However, this is not possible at the considered plant, i. e., ∆H is considered as an externally defined system input not available for control purpose.

Validation of the lateral plate position
In this section, the lateral plate position W is compared to the measurement data from the considered industrial plant. This comparison is only made for forward passes, where the edger

Validation of the shape model
This section is dedicated to the validation of the outgoing centerline shapecl + of the plate according to (25). A reliable measurement of this shape is available after the plate has passed the FOV of the exit-side camera system [21]. After the final rolling pass, the shape of the plate is measured once again [9,10]. Fig. 7 shows a representative measured (red [21]/orange [9,10]) and calculated (blue) centerline after the fifth rolling pass. The lateral position of the edger rolls (green) is transformed so that the figure shows which part of the plate experiences which lateral position of the edger rolls. The measured and calculated centerlines match quite well. Fig. 7 shows that a significant deformation appears when the edger rolls start or stop moving. This behavior can be explained if the force F − is examined for the same plate in Fig. 8. The force peaks evoke changes of the camber of the plate according to (12).

Controller design
The mathematical model from Section 2, which was validated in Section 3, serves as a basis for the controller design. The control objectives are twofold: • Safe plate motion: In the best case, the plate does not touch any side guides or other parts along the line.
• Correct shape: The final shape of the plate should be as straight as possible.
The measurement of the outgoing shape cl + with the measurement system described in Section 3.1 starts as soon as the head end of the plate reaches the FOV of the exit-side camera system. Because camber essentially develops in the roll gap, but can only be measured in the FOV of the downstream camera system, the measurement exhibits a delay. This lack of real-time measurement is also the reason why the head end of the plate cannot be straightened by simple feedback control. Therefore, to achieve the stated control objectives, an optimal feedforward controller is designed in a first step. It is then supplemented by a feedback controller to further reduce the remaining camber. Hence, a two-degrees-of-freedom (2-DOF) control concept is proposed (cf. Fig. 9). Note that the feedforward controller uses the centerline cl + and the feedback controller the camber κ + of the outgoing plate as control variable.

Optimal feedforward control
In the following, an optimal feedforward control law for the lateral edger position w (−l e ) = u FF is derived based on (24) and (25) and inspired by [30]. In contrast to [30] where the input u and the states x are used with an embedded integrator, in this work only the input variable u is used with an embedded integrator. Furthermore, the predictive control is extended by a term to compensate for the incoming centerline cl − and for the contribution cl ∆hH to the centerline.
Using the spatial discretization ∆z − and the zero-order-hold method, (24) and (25) can be written in discretized form with the matrices ] (29) and the output y = [ W cl + ] T . Here, z in = k∆z − is the processed plate length.  It is considered that the initial position of the edger rolls matches the initial position of the plate in the roll gap at the beginning of the rolling pass The objective is to get a zero output Y = [ This objective is reflected in the optimization problem (for the derivation, see Appendix A.4) subject to (27) and (30), with the user-defined positive definite weighting matrices χ S,Y and χ ∆u . This yields the optimal input trajectory and the optimal initial lateral position of the edger rolls. Optimizing this initial lateral position is strongly recommended as it helps to significantly reduce the mean lateral position of the plate during the roll pass. In practice, the choice k = 0 and l = N P being the plate length where the edger rolls are active proved useful. Finally, it is emphasized that (24), (27) and (31) are only applicable while the plate is clamped in the roll gap and between the edger rolls. After this period, i. e., when the plate has left the edger rolls, the model changes to (9) with F − = 0. Because then the edger rolls do not have any influence, their lateral positions are simply held constant.

Feedback control
In the previous section, a feedforward controller was designed to calculate the optimal trajectory u FF of the lateral position of the edgers during the rolling pass. In the following, a feedback controller for the lateral edger position w (−l e ) = u FB is proposed to further reduce the remaining curvature of the plate. Hence, the controller uses the remaining camber κ + measured in the FOV of the exit-side camera as feedback.
This measurement exhibits a transport delay associated with the distance l FOV between the camera FOV and the roll gap. A well-known controller for such a delay system is the Smith predictor [31,32], see Fig. 10.
The first step of the Smith predictor design concerns a controller for the plant without delay. For this purpose, consider the plant transfer function (cf. (18)) with the Laplace variable s [33]. Here,( ·) is the Laplace transformation of (·) (z) and −s 1,2 are the real eigenvalues of A (cf. Appendix A.3). The camera system features a sampling rate of 20 frames/s, which is too low for the fast eigenvalue λ 2 . Therefore, the fast dynamics will be neglected in (32) and the lateral acceleration u ′′ of the edger is used as a new control input yielding the new design model To prevent a non-zero steady-state control error of the camber κ + , an integrating controller has to be employed for (33). In general, such a controller would demand a non-zero steady-state lateral acceleration of the edger rolls, which is undesirable. For instance, the elimination of a constant camber would require a constant acceleration of the edger rolls. As a reasonable alternative solution, a non-zero constant roll gap tilt would be suitable to eliminate a constant camber, see, e. g., [22,23]. This demonstrates that edger rolls are not the right actuators to compensate for a constant camber. They can, however, be advantageously used to eliminate zero-mean deviations from the desired straight plate profile. To still improve the plate centerline cl + by means of feedback control, a simple proportional controller with gain P is designed for (33). For the closed-loop system to be stable, the condition P > −1/V 1 must be satisfied.
In the next design step, the spatial distance l FOV between the roll gap and the exit-side camera is considered. To do so (cf. [31,32]), an internal model in the form of (18) is simulated (cf.   11. Comparison of plate centerlines and corresponding control inputs, measured without moving the edger rolls (red [21]/blue [9,10]), simulated with feedforward control only (orange) and simulated with both feedforward and feedback control (green).

Simulation studies
In the following, the performance of the developed control concept is analyzed in simulations. In Fig. 11, the following three scenarios are compared based on the centerline of the plate.
• Edgers are not moved during the rolling pass (measurement from the industrial plant, u = 0, red [21]/blue [9,10]). • Edgers are moved according to the feedforward controller only (simulation, u = u FF , orange). • Edgers are moved according to both feedforward and feedback control (simulation, u = u FF + u FB , green).
The simulation studies show that if feedforward control or the two-degrees-of-freedom control structure is used, the plates' centerline is significantly improved. In the objective function (31), the lateral position of the plate and in this way also the lateral position of the edger rolls are considered. Thanks to the predicting behavior of the feedforward controller, an acceptable lateral shift of the edger rolls and thus an acceptable lateral position of the plate in the roll gap can be achieved. The simulation results indicate that camber can be compensated by the proposed control concepts. In this case, local variations of the camber are mainly compensated by feedforward control whereas uncertainties of κ ∆hH (which cause a constant camber) are mainly reduced by feedback control. Fig. 11 also shows the corresponding input trajectories. As discussed in Section 3.3, the lateral position of the plate in the roll gap is more or less limited by the edgers' lateral position. Both, the centerline shape cl − of the incoming plate and the change cl ∆hH of this shape due to a thickness wedge and a roll gap tilt mainly cause a constant camber. As discussed in Section 4.2, constant camber requires edger rolls to accelerate throughout the rolling pass (cf. bottom part of Fig. 11). In view of these findings, the best control strategy would be to compensate a constant camber by tilting the roll gap and to reduce the local variations of the camber by a lateral motion of the edger rolls. Using the roll gap tilt as a control input is not feasible in the considered industrial plant and thus not part of this contribution.
In the following, the measurement results of 195 plates (not the plates from the parameter identification in Section 3.2) where the edger rolls were not moved during the production process, i. e.cl real x,u = 0, are superposed withcl x,u resulting from the feedforward (u = u FF ) or the two-degrees-of-freedom control law (u = u FF + u FB ) (cf. (24)). This superposition is valid because the influence of the control input u on the final shape of the plate cl + is reproduced quite well in simulation (cf. Section 3.4). In this way, the model-plant-mismatch e. g. caused by deviations in κ ∆hH is also considered in the simulation studies. Fig. 12 depicts the relative frequency distribution of the shape error for these 195 plates. The results in Fig. 12 show that the centerlines of the plates are improved with the proposed control concept. The mean error e rms is reduced from approximately 10 mm to approximately 8 mm with feedforward control only and to approximately 5 mm using the two-degrees-of-freedom control structure.

Conclusions
In heavy-plate rolling, there are many roughing mills where neither the roll gap height nor its tilt can be adjusted during the rolling pass. For such roughing mills, the question arises whether the lateral position of edger rolls can serve as an alternative control input.
To answer this question, the mathematical models from [1,2,6] were reformulated to use the edgers' lateral position as a control input. The resulting model was validated based on experimental data from an industrial plant. An optimization-based feedforward controller and a Smith predictor feedback controller were developed based on the validated mathematical model. The two control concepts can be combined in a two-degreesof-freedom control structure. The performance of the developed controllers was analyzed in simulations. The proposed control concept achieves a significant improvement of the resulting plate centerlines compared to the case where the edger rolls are not moved during the roll pass.
To eliminate a constant camber, constant lateral acceleration of the edgers is required. Hence, edger rolls are clearly not the right actuator to suppress a constant camber. However, they are capable of eliminating local variations of the camber. In particular the combination of an adjustable roll gap tilt and an adjustable lateral position of the edger rolls is expected to have a high potential. This configuration will be addressed in future research. Currently, the implementation of the proposed concept at the industrial plant of voestalpine Stahl GmbH is on the way. , The rotated and shifted centerline can thus be calculated as with the identity matrix I and the transformation matrix S.

A.2. Friction forces during lateral adjustment of the edgers
To investigate the friction forces in the adjustment mechanism of the edgers, force measurements were conducted with laterally moving edger rolls while no plate was clamped between them. Results of such a measurement are shown in Fig. 13. Since there are no other loads in this scenario, the measured forces are Based on these measurements, friction is modeled by the Stribeck curve [29] with the edgers' lateral velocity v e and the unknown model parameters r v , r C , r H , v 0 , and r off . They are optimized based on the cost function min rv ,r C ,r H ,v 0 ,r off subject to (39). The optimization problem (40) is individually solved for positive and negative velocities v e and for the drive and operator side. This leads to four parameter sets for r v , r C , r H , v 0 and r off . The resulting curves are also shown in Fig. 13.

A.3. Eigenvalues of A
In the following, the eigenvalues of the dynamic matrix A of (24) are calculated. The characteristic equation reads as Solving the quadratic part of the equation yields two real eigenvalues in the form with the residual term which is much smaller than the quadratic term in the square root. The eigenvalue with the smaller absolute value, which is dominant for the dynamics of (24), follows as This shows that the dominant dynamics of the lateral plate motion is (almost) independent of the uncertain parameters E l and K Σ .

A.4. Formulation of the quadratic problem
The solutions x k+l and y k+l from the model (27) at an arbitrary spatial point k + l directly follow as with ∆u k = u k − u k−1 (45c) With this formulation, the predicted future solution can be written as Note that S i,j denotes the entry of the matrix S in the ith row and jth column. The control objective yields the optimization problem min u k−1 ,∆UỸ subject to (46) and (50), with the user-defined positive definite weighting matrices χ Y and χ ∆u . Using χ S,Y =S T χ YS , this is equal to (31).