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Control of Curvature and Contact Force of a Metal Strip at the Strip-Roll Contact Point
Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th World Congress The International Federation of Automatic The International Federation of Congress Automatic Control Control Toulouse, France, July 9-14, 2017 Proceedings of the 20th World The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com Toulouse, France, July 9-14, 2017 The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFACof PapersOnLine 50-1 (2017) 11325–11330 Control Curvature and Contact Force of Control of Curvature and Contact Force of Control of Curvature and Contact Force of Control of Curvature and Contact Force of a Metal Strip at the Strip-Roll Contact Control of Curvature and Contact Force of a Metal Strip at the Strip-Roll Contact a the a Metal Metal Strip Strip at at Point the Strip-Roll Strip-Roll Contact Contact a Metal Strip at Point the Strip-Roll Contact Point Point ∗ G. Stadler ∗∗ A. Point Steinboeck ∗∗ ∗∗ A. Kugi ∗
∗ G. Stadler ∗∗ A. Steinboeck ∗∗ A. Kugi G. A. Kugi ∗ G. Stadler Stadler ∗ A. A. Steinboeck Steinboeck ∗∗ A. Kugi ∗∗ ∗ G. Stadler A. Steinboeck Kugi ∗Control in the Steel Laboratory for Model-BasedA.Process ∗ Christian Doppler ∗ Christian Doppler Laboratory for Model-Based Process Control in the Steel Doppler Laboratory for Model-Based Process Control in ∗ Christian Industry, Automation and Control Institute, TU Wien, Austria, Christian Doppler Laboratory for Model-Based Process Control in the the Steel Steel Industry, Automation and Control Institute, TU Wien, Austria, ∗ Industry, Automation and Control Institute, TU Wien, Austria, (e-mail: {stadler, kugi}@acin.tuwien.ac.at). Christian Doppler Laboratory for Model-Based Process Control in the Steel Industry, Automation and Control Institute, TU Wien, Austria, (e-mail: {stadler, kugi}@acin.tuwien.ac.at). ∗∗ (e-mail: kugi}@acin.tuwien.ac.at). Automation and {stadler, Control Institute, TU Wien, (e-mail: Industry, Automation and Control Institute, TUAustria Wien, Austria, ∗∗ (e-mail: {stadler, kugi}@acin.tuwien.ac.at). ∗∗ Automation and Control Institute, TU Wien, Austria (e-mail: and Control Institute, TU Wien, Austria (e-mail: ∗∗ Automation [email protected]) (e-mail: {stadler, kugi}@acin.tuwien.ac.at). Automation and Control Institute, TU Wien, Austria (e-mail: [email protected]) ∗∗ [email protected]) Automation and Control Institute, TU Wien, Austria (e-mail: [email protected]) [email protected]) Abstract: The control of the curvature and the contact force at the strip-roll contact point of Abstract: The The control control of of the curvature curvature and and the the contact force force at at the the strip-roll contact contact point point of of Abstract: a metal strip in an experimental device isand investigated. thisatpurpose, a material model Abstract: The control of the the curvature the contact contactFor force the strip-roll strip-roll contact pointand of a metal strip in an experimental device is investigated. For this purpose, a material model and a instrip an experimental device investigated. For this aafeasible material model steadystrip state deformation model is are derived. Using these models, combinations Abstract: The control of the curvature and the contact force the strip-roll contact pointand of aa metal metal an experimental is investigated. For thisatpurpose, purpose, material model and steadystrip stateinstrip strip deformation device model are derived. Using these models, feasible combinations a steady state deformation model are derived. Using these models, combinations curvature and force are computed. A control structure is proposed that comprises metal strip instrip ancontact experimental device is investigated. For this purpose, afeasible material model and aof steady state deformation model are derived. Using these models, feasible combinations of curvature curvature and and contact contact force force are are computed. computed. A A control structure structure is is proposed that that comprises comprises of feedforward and adeformation feedback Exponential stability of the models, closed-loop system is shown. a steady stateand strip model are derived. Usingstructure these feasible combinations of curvature contact force part. are computed. A control control is proposed proposed that comprises a feedforward and a feedback part. Exponential stability of the closed-loop system is shown. shown. a feedforward a feedback part. Exponential stability of the closed-loop system is from and aand numerical example the robustness of the controller against model-plant of curvature contact force are verify computed. A control structure is proposed that comprises aResults feedforward and a feedback part. Exponential stability of the closed-loop system is shown. Results from a numerical example verify the robustness of the controller against model-plant Results from a numerical example verify the against model-plant mismatches its asuitability for disturbance rejection. a feedforward feedback part. Exponential stabilityofofthe thecontroller closed-loop system is shown. Results fromand aand numerical example verify the robustness robustness mismatches and its suitability suitability for disturbance disturbance rejection. of the controller against model-plant mismatches its for rejection. Results fromand a numerical example verify the robustness of the controller against model-plant mismatches and its suitability for disturbance rejection. © 2017, IFACand (International Federationdisturbance of Automaticrejection. Control) Hosting by Elsevier Ltd. All rights reserved. mismatches its suitability Keywords: Mechanical system, for Mathematical models, Static models, Boundary value problem, Keywords: Mechanical system, Mathematical models, Static models, Boundary value problem, Keywords: Mechanical system, Mathematical models, Static models, models, Boundary value value problem, problem, Model inversion, Steel industry, Closed-loop control, Lyapunov stability Keywords: Mechanical system, Mathematical models,Lyapunov Static Boundary Model inversion, Steel industry, Closed-loop control, stability Model inversion, Steel industry, Closed-loop control, Lyapunov stability Keywords: Mechanical system, Mathematical models,Lyapunov Static models, Boundary value problem, Model inversion, Steel industry, Closed-loop control, stability Model1.inversion, Steel industry, Closed-loop control, Lyapunov stability INTRODUCTION amount of curvature based on the desired residual stress 1. INTRODUCTION INTRODUCTION amount of curvature based on the desired residual stress 1. amount of curvature based on state. Another example of a contact problemresidual betweenstress strip 1. INTRODUCTION amount of curvature based on the the desired desired residual stress state. Another example of aa contact problem between strip state. Another example of contact problem between strip 1. INTRODUCTION and roll is the looper in tandem rolling mills. Here, amount of curvature based on the desired residual stress state. Another example of a contact problem between stripa In a typical strip processing plant, cold-rolled steel is heat and roll is the looper in tandem rolling mills. Here, a In a typical strip processing plant, cold-rolled steel is heat and roll is the looper in tandem rolling mills. Here, controller ensures the correct contactproblem force between looper state.roll Another example of atandem contact between stripa In a typical strip processing plant, cold-rolled steel is heat and is the looper in rolling mills. Here, a treated in continuous annealing lines in order to improve controller ensures the correct contact force between looper In a typical strip processing plant, cold-rolled steel is heat treated in in continuous continuous annealing annealing lines lines in in order order to to improve improve roll controller ensures the correct contact force between looper and strip, e.g., et al., 2014; Choia and roll is the see, looper in (Steinboeck tandem rolling mills. Here, treated controller ensures the correct contact force between looper properties such as hardness, strength, and ductility of steel In a typical strip processing plant, cold-rolled steel is heat and strip, see, e.g., (Steinboeck et al., 2014; Choi treated in such continuous annealing lines and in order to improve properties as hardness, hardness, strength, ductility of steel steel roll roll and strip, see, (Steinboeck et al., Choi Furthermore, by means the2014; rotational controller ensures thee.g., correct contact force between looper properties such as strength, and ductility of rollal., and2007). strip, see, e.g., (Steinboeck etof al., 2014; Choi sheets (Matthews and annealing James, 2010). horizontal and et treated in such continuous lines Both in order to improve et al., 2007). Furthermore, by means of the rotational properties as hardness, strength, and ductility of steel sheets (Matthews and James, 2010). Both horizontal and et al., 2007). Furthermore, by means of the rotational speed difference between two consecutive mill stands, the roll and strip, see, e.g., (Steinboeck et al., 2014; Choi sheets (Matthews and James, 2010). Both horizontal and et al., 2007). Furthermore, by means of the rotational vertical(Matthews continuous annealing lines exist. The temperatures properties such as hardness, strength, and ductility of steel speed difference between two consecutive mill stands, the sheets and James, 2010). Both horizontal and vertical continuous continuous annealing annealing lines lines◦exist. exist. The The temperatures temperatures speed between two consecutive the ◦ length of the strip between these stands of ismill controlled. et al., difference 2007). Furthermore, by means thestands, rotational vertical speed difference between two consecutive mill stands, the in annealing lines range from 730 C up to 1200 C (Lanksheets (Matthews and James, 2010). Both horizontal and ◦exist. The temperatures ◦ length of the strip between these stands is controlled. vertical continuous annealing lines ◦C up to 1200 ◦C (Lankin annealing lines range from 730 length of the strip between these stands is controlled. speed difference between two consecutive mill stands, the in annealing lines range from 730 C up to 1200 C (Lank◦exist. ◦ length of the strip between these stands is controlled. ford et al., 1985). While the strip is conveyed through a vertical continuous annealing lines The temperatures C up to 1200 ◦through C (Lank-a Figure 1 outlines the experimental device considered in in annealing lines range from 730 ford et al., 1985). While the strip conveyed ◦ is length of the strip between these stands is controlled. Figure 1 outlines the experimental device considered in ford et al., 1985). While the strip is conveyed through a horizontal line, it is730 bent (purely elastically or Cisup to 1200 C (Lankin annealing lines range from 11 outlines the experimental device considered in ford et al.,annealing 1985). While the strip conveyed through a Figure this paper. The strip, is typically rather thin (thickFigure outlines thewhich experimental device considered in horizontal annealing line, it is bent (purely elastically or this paper. The strip, which is typically rather thin (thickhorizontal annealing line, it is bent (purely elastically or elasto-plastically) theisstrip rolls to its dead load. ford et al.,annealing 1985). around While the isdue conveyed through a this paper. The strip, which is typically rather thin (thickhorizontal line, it bent (purely elastically or ness h ∈ [0.3 mm, 1.2 mm]), is tightened by a mechanical Figure 1 outlines the experimental device considered in this paper. The strip, which is typically rather thin (thickelasto-plastically) around the rolls due to its dead load. h ∈ [0.3 mm, 1.2 mm]), is tightened by aa mechanical elasto-plastically) around the to its dead load. The periodic strip-roll contact may due cause surface defects horizontal annealing line, it is rolls bent (purely elastically or ness ness h ∈ [0.3 mm, 1.2 mm]), is tightened by mechanical elasto-plastically) around the rolls due to its dead load. actuator which applies the tensional force n. The roll with this paper. The strip, which is typically rather thin (thickness h ∈ which [0.3 mm, 1.2 mm]), is tightened by aThe mechanical The periodic strip-roll contact may cause surface defects applies the tensional force n. roll with The periodic strip-roll contact may cause on rolls (Fukubayashi, 1995; Sawa and Oohori, 1995). elasto-plastically) around the rolls due to surface its deaddefects load. actuator actuator which applies the force n. roll with Thethe periodic strip-roll contact may cause surface defects aactuator radius has two1.2 actuated of freedom. That is, ness h ∈R [0.3 mm, mm]), isdegrees tightened by mechanical which applies the tensional tensional force n. aThe The roll with on the rolls (Fukubayashi, 1995; Sawa and Oohori, 1995). a radius R has two actuated degrees of freedom. That is, on the rolls (Fukubayashi, 1995; Sawa and Oohori, 1995). In literature, it is contact distinguished between mechanical Thethe periodic strip-roll may cause surface defects a radius R has two actuated degrees of freedom. That is, on rolls (Fukubayashi, 1995; Sawa and Oohori, 1995). its horizontal position x and the penetration depth ∆h actuator which applies the tensional force n. The roll with r a radius R has two actuated degrees of freedom. That is, In the literature, it is distinguished between mechanical its horizontal position x and the penetration depth ∆h In the literature, it is distinguished between mechanical r and reaction effectsitthat lead to the development of surface on the rolls (Fukubayashi, 1995; Sawa and Oohori, 1995). its horizontal position x and the penetration depth ∆h r In literature, is distinguished between mechanical can be controlled. The position of the strip-roll contact a radius R has two actuated degrees of freedom. That is, its horizontal position x and the penetration depth ∆h and reaction effects that lead to the development of surface r can be be controlled. controlled. The The position position of of the the strip-roll strip-roll contact contact and reaction lead to the development of surface defects. One effects way to reduce surface defects is tomechanical guarantee In the literature, itthat is distinguished between can and reaction effects that lead to the development of surface point is denoted by x and the length of the experimental its horizontal position x and the penetration depth ∆h 1 r can beis controlled. positionlength of the strip-roll contact defects. One way to reduce surface defects is to guarantee denoted by The x of the experimental defects. One way reduce defects is guarantee 1 and the a suitable of surface the rolldevelopment surface. be point and reaction effects lead to the ofmust surface point denoted x the of the defects. Onecomposition way to tothat reduce surface defects is to toIt guarantee device L. The by main purpose of of thethe experimental device can beis controlled. The position contact point isby denoted by x11 and and the length length of strip-roll the experimental experimental a suitable composition of the roll surface. It must be device by L. The main purpose of the experimental device a suitable composition of the roll surface. It must be wearand corrosion-resistant, it must prevent defects. Onecomposition way to reduce is to device L. The main purpose of the experimental a suitable of surface the and rolldefects surface. It guarantee must the be point is to analyze theby influence of the temperature, thedevice steel isby denoted x1 and the length of the experimental device by L. The main purpose of the experimental device wearand corrosion-resistant, and it must prevent the is to analyze the influence of the temperature, the steel wearand corrosion-resistant, and it must prevent the adhesion of foreign particles onto the roll surface (Hao, a suitable composition of the roll surface. It must be is to analyze the influence of the temperature, the wearand corrosion-resistant, and it must prevent the grade of the strip, the contact force, and the curvature on device by L. The main purpose of the experimental device is to analyze the influence of force, the temperature, the steel steel adhesion of foreign particles onto the roll surface (Hao, grade of the the strip, strip, the contact contact and the the curvature curvature on adhesion of foreign particles onto the roll surface (Hao, 2007; Fukubayashi, 1995, 2004). In this paper, mechanical wearand corrosion-resistant, and it must prevent the grade of the force, and on adhesion of foreign particles onto the roll surface (Hao, the evolution of surface defects. These experiments should is to analyze the influence of the temperature, the steel grade of the strip, the contact force, and the curvature on 2007; Fukubayashi, 1995, 2004). In this paper, mechanical the evolution of surface defects. These experiments should 2007; Fukubayashi, 1995, 2004). In this paper, mechanical effects like the contact force between the strip and the adhesion of foreign particles onto the roll surface (Hao, the evolution of surface defects. These experiments should 2007; Fukubayashi, 1995,force 2004). In this paper, mechanical help toofgain into the essential contact mechanisms grade the insight strip, the contact force, and the curvature on the evolution of surface defects. These experiments should effects like the contact between the strip and the to gain insight into the essential contact mechanisms effects like contact the strip and roll thethe curvature atforce the between strip-roll contact point are help 2007;and Fukubayashi, 1995, 2004). In this paper, mechanical help to gain insight the essential contact mechanisms effects like the contact force between the strip and the the between furnace rollsinto and strip. This paper focuses on the the evolution of surface defects. These experiments should help to gain insight into the essential contact mechanisms roll and the curvature at the strip-roll contact point are between furnace rolls and strip. This paper focuses on the roll and the curvature the strip-roll contact are examined. effects contactat strippoint and the furnace rolls and strip. This focuses on roll andlike thethe curvature atforce the between strip-roll the contact point are between mechanical aspects of thethe experimental device. Therefore, help to gain insight into essential contact mechanisms between furnace rolls This paper paper focuses on the the examined. mechanical aspects of and the strip. experimental device. Therefore, examined. roll and the curvature at the strip-roll contact point are mechanical aspects of the experimental device. Therefore, examined. model-based control mechanical parameters between furnace rolls strip. This paper focuses onassothe mechanical aspects of and the of experimental device. Therefore, A periodic strip-roll contact also occurs in the leveling the the model-based control of mechanical parameters assoexamined. A periodic strip-roll contact also occurs in the leveling the model-based mechanical parameters ciated with aspects such control contact problems is considered inassothis mechanical of the of experimental device. Therefore, A periodic strip-roll contact also occurs in the leveling the model-based control of mechanical parameters assoprocess, where the main objective is to reduce residual with such contact problems is considered in this A periodic strip-roll contact also occurs in the residual leveling ciated process, where the main objective is to reduce ciated with contact is this model-based mechanical parametersin process, where the main objective is to reduce ciated with such such control contact ofproblems problems is considered considered inassothis stresses and shape defects (residual curvatures) of the the A periodic strip-roll contact also occurs in the residual leveling process, where the main objective is to reduce residual stresses and and shape shape defects defects (residual (residual curvatures) curvatures) of of the the ciated with y such contact problems is considered in this stresses strip. However, in this process, the dead load process, where the main objective is to reduce residual y stressesHowever, and shape defects (residual curvatures) of the h y strip. in this process, the dead load of the ∆h h n strip. in process, the dead load the y w strip isHowever, not and the strip alternately stresses and sufficient shape defects (residual curvatures) h ∆h strip. However, in this this process, the is dead load of of bent the n w ∆h h strip is not sufficient and the strip is alternately bent n y w R x strip is not sufficient and the strip is alternately bent ∆h between the rollers of the leveler. In this way, a specific strip. However, in this process, the dead load of the n w strip is not sufficient and the strip is alternately bent h R x between the rollers of the leveler. In this way, a specific R x ∆h n between the rollers of the leveler. In this way, a specific w R x amount of curvature is imposed onto the strip at each roll. strip is not sufficient and the strip is alternately bent x1 between rollers of the leveler. this way, specific amount of ofthe curvature is imposed imposed ontoInthe the strip at aeach each roll. x R x 1 amount curvature is onto strip at roll. Batty and Lawson the force power between rollers(1965) of theinvestigated leveler. this way, aeach specific xr1 amount ofthe curvature is imposed ontoInthe strip atand roll. xr1 g Batty and Lawson (1965) investigated the force and power L x Batty and investigated the force power g requirements of the(1965) leveling process. they did not amount of Lawson curvature is imposed ontoHowever, the strip atand each roll. xrr1 Batty and Lawson (1965) investigated the force and power g L requirements of the leveling process. However, they did not L g requirements of the leveling process. However, they did not x consider the evolution of the curvature. Kaiser et al. (2014) Batty and Lawson (1965) investigated the force and power L r requirements of the leveling process. However, they did not g consider the evolution of the curvature. Kaiser et al. (2014) L Fig. 1. Geometry of the experimental device. consider the evolution of the curvature. Kaiser et al. (2014) proposed a method to determine the required sequence and requirements of the leveling process. However, they did not consider the evolution of the curvature. Kaisersequence et al. (2014) Fig. proposed a method to determine the required and Fig. 1. 1. Geometry Geometry of of the the experimental experimental device. device. proposed a to the and consider the evolution of the curvature. Kaisersequence et al. (2014) proposed a method method to determine determine the required required sequence and Fig. 1. Geometry of the experimental device. Fig. 1. Geometry of the experimental device. proposed©a 2017 method Copyright IFACto determine the required sequence and 11817
Copyright © 2017 11817 Copyright 2017 IFAC IFAC 11817Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017, 2017 IFAC 11817 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2017 IFAC 11817 10.1016/j.ifacol.2017.08.1665
Proceedings of the 20th IFAC World Congress 11326 G. Stadler et al. / IFAC PapersOnLine 50-1 (2017) 11325–11330 Toulouse, France, July 9-14, 2017
paper. In fact, the contact force between strip and roll ∆q and the curvature κ at the strip-roll contact point have to be controlled. The paper is structured as follows: First, suitable models for material and strip deformation are derived in Section 2. Examination of the resulting boundary value problem leads to feasible combinations of curvature and contact force for the considered simulation device. Section 3 is concerned with the design of the proposed control structure. Furthermore, the stability of the closed-loop system is investigated, where exponential stability is shown. Finally, a numerical example in Section 4 analyzes the performance of the controller against model-plant mismatches and its suitability for disturbance rejection. 2. MODELING 2.1 Material model In the following, derivatives with respect to the length coordinate x are denoted by (·) = d(·)/dx and derivatives ˙ = d(·)/dt. with respect to the time t are denoted by (·) Furthermore, w = w(x) and κ(x) = κ are the deflection and the curvature of the strip, respectively. The following material model was originally proposed by Stadler et al. (2015). However, in this paper, only the linear elastic case is considered, whereas the ideal-elastic-ideal plastic case was considered in (Stadler et al., 2015). An initially plain strip with zero residual stresses and a plane state of stress without shear stresses is assumed. Because the mean tensile stress caused by strip tension is small compared to the yield stress and also small compared to the bending stress, zero mean stress is a valid assumption. In the geometrically linear case (small deflections), the curvature is approximated by w κ= (1) 3 ≈ w . 1 + w 2 2 Since the width b of the strip satisfies b h and due to the fact that bending strains in lateral direction vanish, the plate stiffness effect has to be considered. This can be done by replacing Young’s modulus E by an effective Young’s modulus E ˜= , (2) E (1 − ν 2 ) where E may be temperature dependent (Varshni, 1970). The bending moment mb normalized with respect to the width is thus defined by ˜ 3κ Eh . (3) mb = − 12
at x = 0 and x = L. As in (Steinboeck et al., 2015), it is assumed that the tensional force n in the strip is uniform in longitudinal direction. Furthermore, small deflections w are assumed, linear geometric relations are considered, and because the considered beam is slender, the EulerBernoulli hypothesis is applied. In case of thick beams, the previous assumptions are rendered invalid and the Timoshenko beam theory may be used (Pilkey, 2002). Based on these assumptions, the quasi-static deflection w of the strip can be computed from the differential equation gρh − nw − w = 0 (4) supplemented by appropriate boundary conditions. With m b = q and (3), this differential equation can be reformulated as a nonlinear boundary value problem (BVP) with known boundaries x = 0, x = L and a generally unknown boundary x = x1 w w ˜ 3) d w −12mb /(Eh , x ∈ (0, L) (5) m = q b dx ˜ 3) q gρh + 12nmb /(Eh with the boundary conditions w(0) = w0 , w(L) = w0 (6a) w (L) = 0 (6b) w (0) = 0, + + w(x− w (x− (6c) 1 ) = w(x1 ), 1 ) = w (x1 ) − + mb (x1 ) = mb (x1 ) (6d) w(x1 ) = R sin(arccos((x1 − xr )/R)) − R + ∆h . x− 1
(6e)
x+ 1
2.2 Steady-state strip deformation model
Here, and denote the left- and the right-hand limit of the strip-roll contact point x1 , respectively. The boundary condition (6e) ensures that the strip-roll contact point x1 is on the roll. Because x1 represents an unknown boundary point in (6), an additional equation has to be stated for the computation of x1 . That is, the slopes of the strip and the roll have to be consistent at this point. With w(x1 ) = R2 − (xr − x1 )2 (7a) x r − x1 (7b) w (x1 ) = R2 − (xr − x1 )2 it follows that (8) (w (x1 ))2 R2 = (xr − x1 )2 (1 + (w (x1 ))2 ) and therefore Rw (x1 ) x 1 = xr − . (9) 1 + (w (x1 ))2 has to hold. An analytical solution of the differential equation (5) itself does exist. However, the resulting constants of integration can only be numerically computed based on the boundary conditions (6) and (9). Thus, a BVP-solver (multiple shooting method) was used to solve (5), (6), and (9).
For similar mechanical structures, a systematic derivation of the governing differential equation and its boundary conditions has been carried out in (Pilkey, 2002). Due to the fact that only slow roll movements are considered, acceleration forces can be neglected. The dead weight gρh with the gravitational acceleration g and the mass density of steel ρ represents a uniformly distributed load on the strip. Another external load is the tensional force applied
For the following analysis, parameters from Tab. 1 in Section 4 were used. The upper part of Fig. 2 shows the inputs nb and ∆h required for certain curvature values κ and contact forces ∆q for the horizontal roll position xr = x1 = L/2 = 0.6 m. Analogous plots exist for any other point xr ∈ [R, L − R]. Gray isolines represent constant contact forces ∆q and black isolines represent constant curvatures κ. The intersection of two such lines gives the required control inputs for the respective values
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Fig. 3. Block diagram of the proposed 2-DOF control structure. position xr as a user defined input trajectory. Its time derivatives are limited to ensure that the assumption of a quasi-static bending process is tenable. The manipulated variables set by the controller are the tensional force n and T the penetration depth ∆h of the roll, i.e., u = [n ∆h] . As usual in cascade control structures, actuator dynamics are neglected for control design and stability proof of the outer control loop. Simulation studies in Section 4 justify this approach.
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Fig. 2. Contour plot (top) of curvature and contact force at xr = x1 = L/2 = 0.6 m. Feasible regions in the curvature/contact force plain (bottom) for various horizontal roll positions xr . ∆q and κ. The input box-constraints N = nb ∈ [0, 100] N and ∆h ∈ [0, 0.16] m are shown as a gray dash-dotted rectangle. The edges of this rectangle can be mapped to the curvature/contact force plain shown in the bottom part of Fig. 2 for various horizontal roll positions xr . Markers highlight the correspondence between corner points of the feasible regions in both parts of Fig. 2. That is, all feasible curvature/contact force combinations for a specific horizontal roll position xr are given by the respective enclosed area in the lower part of Fig. 2. 3. CONTROLLER DESIGN The considered control task is to establish a certain curvature κ and contact force ∆q at the strip-roll contact point x1 . For this control task, a two-degrees-of-freedom control structure (2-DOF) comprising a feedforward and a feedback part is outlined in Fig. 3, see, e.g., (Goodwin et al., 2001). In a 2-DOF control structure, the feedforward controller contributes the main part of the control input, whereas the purpose of the feedback controller is to reject unknown disturbances and to compensate for model-plant mismatches. The controller considers the horizontal roll
The solution of (5), (6) and (9) with (3) can be formally written as a nonlinear static input-output relation f (mb (x1 )) κ(x1 ) = Υ(u, xr ) . (10) = y= + ∆q(x1 ) q(x− 1 ) − q(x1 ) In terms of the input u and the output y, Υ is a bijective mapping. For given values xr , κd , and ∆q d , the feedforward controller uses (10) and numerically computes the open-loop control input uff , so that d κ nff yd = , x ), with u = (11) = Υ(u ff r ff ∆hff ∆q d
holds. Because the desired reference trajectories are known in advance (chosen by the operators), the required openloop control input can be calculated off-line. 3.2 Feedback controller
To reduce the effects of model-plant mismatches and disturbances, the feedforward controller is supplemented by a feedback controller. It is assumed that the contact force ∆q is measured by a force sensor and that the curvature κ is optically measured. For compact notation, consider that Υred (u, x1 ) is the solution function of the reduced model (5) and (6), where the value of x1 is assumed to be constant and given by the solution of (11). As shown in (Lunze, 1989), local linearization of (10) at the point uff , yd = Υred (uff , x1 ), and x1 for the undisturbed case udis = 0 gives u = uff + uf b (12a) ∂Υred (u, x1 ) uf b = yd + Juf b . (12b) y = yd + ∂u u=uff yf b J
Here, it is assumed that the change of x1 due to the feedback controller uf b is negligibly small. In the following, the arguments uff , yd and x1 are omitted to keep the notation compact. It is worth noting that all quantities,
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×10−3
including the following regular state transformation, depend on these arguments. To simplify the controller design, the multi-input/multi-output (MIMO) system is decoupled into two single-input-single-output (SISO) relations. Therefor, the Jordan normal form ˜ = (D(x))−1 JD(x) (13) J is calculated, where D(x) denotes the transformation matrix consisting of the eigenvectors of J. For the sake of clarity, the argument of D is omitted in the following. ˜f b = ˜ f b = D−1 uf b and y The linearized relation between u −1 D yf b is then given by ˜ uf b ˜f b = J˜ (14) y ˜ It is now possible to design with the diagonal matrix J. two decoupled SISO controllers, which reduce the control error ˜ = D−1 (yd − y) = −˜ e yf b . (15) In this paper, PI control laws ˜ I + KP D−1 (yd − y) ˜fb = u (16a) u −1 d ˙u ˜ I = KI D (y − y) (16b)
with the diagonal matrices KI and KP are used for this purpose. Finally, the feedback law transformed back to the original space reads as nf b ˜ I + KP D−1 (yd − y) . uf b = =D u (17) ∆hf b
3.3 Stability proof
In this section, the stability of the closed-loop system and its robustness against model-plant mismatches is analyzed. Because the plant is represented by the nonlinear static function (10), which gives bounded outputs for bounded inputs, only the stability of the integrator state (16b) has to be analyzed. To analyze robustness, the strip thickness used by the controller will be reduced by 5 %. From the ˜˙ I = 0 of (16b), it follows that steady state condition u y = yd holds at any steady state of the system. For such a steady state, the feedback output based on (17) ˜ I,0 denotes the steady state of is uf b,0 = D˜ uI,0 , where u the integrator. The values of uf b,0 can be computed by solving (10), (11), and (12a). If the same parameters are used in (10) and (11), uf b,0 = 0 is obtained. If different parameters are used in (10) and (11), e.g., to simulate a model-plant mismatch, uf b,0 = 0. This can be converted into the steady state value ˜ I,0 = D−1 uf b,0 (18) u ˜I − u ˜ I,0 , the of the integrator. With the error state xI = u positive definite Lyapunov function 1 V = xT xI (19) 2 I can be defined. Its time derivative is given by −1 ˙ I = xT V˙ = xT (yd − y) , (20) Ix I KI D where y is the solution of the nonlinear implicit equation ˜ I,0 + KP D−1 (yd − y))) . (21) y = Υ(uff + D(xI + u For exponential stability, 2 V˙ ≤ −α xI , α > 0 (22) 2
has to hold in a region around the origin, (cf. Khalil, 2002). Based on (19), (22) reduces to V˙ ≤ −2αV . (23)
0
V˙
−2 −4 −6 −0.02
xI
0 ,2
−0.02
0 0.02
0.02
xI ,1
Fig. 4. Time derivate V˙ of the Lyapunov function V with T xI = [xI,1 xI,2 ] for a strip thickness deviation of −5 %, κd = −3/m, and ∆q d = 450 N/m. V˙ depends on xI , yd , and the strip-roll contact point x1 . Because an analytical proof of inequality (23) is rather difficult, its validity is graphically analyzed for the representative case κd = −3/m and ∆q d = 450 N/m. For this purpose, (20) is numerically evaluated for the range xr ∈ [0.2, 1.0] m. The worst case value maxxr ∈[0.2,1.0]m (V˙ ) is shown in Fig. 4. The illustrated range for xI includes the intervals κ ∈ [0.79κd , 1.20κd ] and ∆q ∈ [0.83∆q d , 1.14∆q d ]. Clearly, V˙ is locally negative definite and the upper value α = 1.1/s in (23) can be identified. Consequently, the error state decreases at least according to (24) xI (t) 2 ≤ xI (t0 ) 2 e−α(t−t0 ) , t ≥ t0 ≥ 0 . 4. NUMERICAL RESULTS For the following simulation analysis, the actuator dynamics of the tensional force n and the penetration depth ∆h are approximated by means of second-order delay systems with time constant T = 50 ms and damping ratio 1.0. The sampling time of the controller is set to Ts = 10 ms. Figure 5 shows the simulation results for the proposed 2-DOF control structure. A strip thickness deviation of −5 % is used. Furthermore, the influence of different disturbances acting on the output and the control variables is investigated. On the left-hand side of Fig. 5, the deT sired output is yd = [−2/m 300 N/m] , whereas yd = T [−3/m 450 N/m] on the right-hand side of Fig. 5. In both scenarios, the horizontal roll position xr follows the same ramp trajectory ranging from 0.2 m to L/2 = 0.6 m. For symmetry reasons, it is sufficient to only simulate the case xr ≤ L/2. Due to the simulated model-plant mismatch, the control error is relatively high at the beginning of the simulation, where a constant horizontal roll position xr is considered. However, the feedback controller ensures that the control error is nearly zero after one second. At
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t = 2 s and t = 14 s, a 3 s long output disturbance, e.g., an output error of ydis = [0 ∆qdis ] = [0 ±30 N/m], occurs, cf. Fig 3. Again, the controller quickly compensates for these disturbances. Next, the influence of disturbances udis = [ndis ∆hdis ] acting on the control variables ∆h and n is investigated. The constant disturbances ∆hdis = −40 mm and ndis = 60 N/m start at t = 8 s and t = 11 s, respectively. Note that solid black lines in the second and third row of Fig. 5 refer to the disturbed outputs n and ∆h of the respective actuator. Once again, about one second is needed until the control error is nearly zero. To sum up, throughout the entire trajectory, a good control performance is achieved. These results also justify the assumption of negligible actuator dynamics when designing the outer control loop. Table 1. Parameters of the considered experimental device. Description
Parameter
Value
Unit
Mass density of steel Young’s modulus of steel Gravitational acceleration Thickness of strip Roll radius Poisson’s ratio
ρ E g h R ν
7850 210 9.81 0.65 60 0.3
kg/m3 kN/mm2 m/s2 mm mm -
5. CONCLUSION AND OUTLOOK This paper deals with the MIMO control of the curvature and the contact force at the strip-roll contact point of a metal strip in an experimental device. A 2-DOF control structure comprising a feedforward and a feedback part in the form of a PI controller is developed. Feasible regions in the curvature/contact force plain are presented for various roll positions considering the input box-constraints. The robustness of the control structure against model-plant mismatches is demonstrated by means of a Lyapunovbased stability proof. Furthermore, a simulation study with various disturbance scenarios shows a good disturbance rejection behavior of the controller. Future work will concern the transfer of the proposed control strategy to the laboratory experiment. ACKNOWLEDGEMENTS Financial support by the Austrian Federal Ministry of Science, Research and Economy and the National Foundation for Research, Technology and Development, and voestalpine Stahl GmbH is gratefully acknowledged.
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Fukubayashi, H.H. (2004). Present furnace and pot roll coatings and future development. In Proceedings of the International Thermal Spray Conference 2004, 125–131. ASM International, Osaka, Japan. Goodwin, G.C., Graebe, S.F., and Salgado, M.E. (2001). Control System Design. Prentice Hall, Upper Saddle River, NJ, USA, 1st edition. Hao, R. (2007). Thermal spraying technology and its applications in the iron & steel industry in China. In Proceedings of the International Thermal Spray Conference, Thermal Spray 2007: Global Coating Solutions, 291–296. Beijing, China. Kaiser, R., Hatzenbichler, T., Buchmayr, B., and Antretter, T. (2014). Simulation of the roller straightening process with respect to residual stresses and the curvature trend. In International Conference on Residual Stresses 9 (ICRS 9), volume 768 of Materials Science Forum, 456–463. Khalil, H.K. (2002). Nonlinear Systems. Prentice Hall, New Jersey, 3rd edition. Lankford, W., Samways, N., Craven, R., and McGannon, H. (1985). The Making, Shaping, and Treating of Steel. United States Steel, 10th edition. Lunze, J. (1989). Robust Multivariable Feedback Control. Prentice-Hall, New Jersey, 1st edition. Matthews, S. and James, B. (2010). Review of thermal spray coating applications in the steel industry: Part 1 – Hardware in steel making to the continuous annealing process. Journal of Thermal Spray Technology, 19(6), 1267–1276. Pilkey, W. (2002). Analysis and Design of Elastic Beams: Computational Methods. John Wiley & Sons, New York, 1st edition. Sawa, M. and Oohori, J. (1995). Application of thermal spraying technology at steelworks, Thermal spraying: Current status and future trends. In Proceedings of the 14th International Thermal Spray Conference, 37– 42. High Temperature Society of Japan, Kobe, Japan. Stadler, G., Steinboeck, A., Baumgart, M., and Kugi, A. (2015). Modellierung des Umschlingungswinkels eines auf Rollen gef¨ uhrten Metallbandes. at – Automatisierungstechnik, 63(8), 646–655. Steinboeck, A., Baumgart, M., Stadler, G., Saxinger, M., and Kugi, A. (2015). Dynamical models of axially moving rods with tensile and bending stiffness. IFACPapersOnLine, 48(1), 598–603. Steinboeck, A., M¨ uhlberger, G., and Kugi, A. (2014). Control of strip tension in a rolling mill based on loopers and impedance control. IFAC Proceedings Volumes, 47(3), 10646–10651. 19th IFAC World Congress. Varshni, Y.P. (1970). Temperature dependence of the elastic constants. Physical Review B, 2, 3952–3958.
REFERENCES Batty, F.A. and Lawson, K. (1965). Heavy plate levellers. Journal of the Iron and Steel Institute, 203, 1115–1128. Choi, I., Rossiter, J., and Fleming, P. (2007). Looper and tension control in hot rolling mills: A survey. Journal of Process Control, 17(6), 509–521. Fukubayashi, H.H. (1995). Coating for high temperature pickup and wear resistant applications, Thermal spraying: Current status and future trends. In Proceedings of the 14th International Thermal Spray Conference, 47– 52. High Temperature Society of Japan, Kobe, Japan. 11821
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0.6
0.4
0.4
0.2 1500
0.2 1500
curvature κ (1/m)
penetration depth (m)
tensional force (N/m)
xr (m)
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nff ndis
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∆hff ∆hdis
0.06 0.04
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0
−1.8
−2.8 −2.9
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340 ∆qdis (N/m) contact force ∆q (N/m)
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8
10 12 t (s)
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T
20
Fig. 5. Simulation results with 2-DOF controller for yd = [−2/m 300 N/m] (left) and yd = [−3/m 450 N/m] (right) for a strip thickness deviation of −5 %. Different disturbances acting on the output and on the control variables are examined. 11822
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IFACof PapersOnLine 50-1 (2017) 11325–11330 Control Curvature and Contact Force of Control of Curvature and Contact Force of Control of Curvature and Contact Force of Control of Curvature and Contact Force of a Metal Strip at the Strip-Roll Contact Control of Curvature and Contact Force of a Metal Strip at the Strip-Roll Contact a the a Metal Metal Strip Strip at at Point the Strip-Roll Strip-Roll Contact Contact a Metal Strip at Point the Strip-Roll Contact Point Point ∗ G. Stadler ∗∗ A. Point Steinboeck ∗∗ ∗∗ A. Kugi ∗
∗ G. Stadler ∗∗ A. Steinboeck ∗∗ A. Kugi G. A. Kugi ∗ G. Stadler Stadler ∗ A. A. Steinboeck Steinboeck ∗∗ A. Kugi ∗∗ ∗ G. Stadler A. Steinboeck Kugi ∗Control in the Steel Laboratory for Model-BasedA.Process ∗ Christian Doppler ∗ Christian Doppler Laboratory for Model-Based Process Control in the Steel Doppler Laboratory for Model-Based Process Control in ∗ Christian Industry, Automation and Control Institute, TU Wien, Austria, Christian Doppler Laboratory for Model-Based Process Control in the the Steel Steel Industry, Automation and Control Institute, TU Wien, Austria, ∗ Industry, Automation and Control Institute, TU Wien, Austria, (e-mail: {stadler, kugi}@acin.tuwien.ac.at). Christian Doppler Laboratory for Model-Based Process Control in the Steel Industry, Automation and Control Institute, TU Wien, Austria, (e-mail: {stadler, kugi}@acin.tuwien.ac.at). ∗∗ (e-mail: kugi}@acin.tuwien.ac.at). Automation and {stadler, Control Institute, TU Wien, (e-mail: Industry, Automation and Control Institute, TUAustria Wien, Austria, ∗∗ (e-mail: {stadler, kugi}@acin.tuwien.ac.at). ∗∗ Automation and Control Institute, TU Wien, Austria (e-mail: and Control Institute, TU Wien, Austria (e-mail: ∗∗ Automation [email protected]) (e-mail: {stadler, kugi}@acin.tuwien.ac.at). Automation and Control Institute, TU Wien, Austria (e-mail: [email protected]) ∗∗ [email protected]) Automation and Control Institute, TU Wien, Austria (e-mail: [email protected]) [email protected]) Abstract: The control of the curvature and the contact force at the strip-roll contact point of Abstract: The The control control of of the curvature curvature and and the the contact force force at at the the strip-roll contact contact point point of of Abstract: a metal strip in an experimental device isand investigated. thisatpurpose, a material model Abstract: The control of the the curvature the contact contactFor force the strip-roll strip-roll contact pointand of a metal strip in an experimental device is investigated. For this purpose, a material model and a instrip an experimental device investigated. For this aafeasible material model steadystrip state deformation model is are derived. Using these models, combinations Abstract: The control of the curvature and the contact force the strip-roll contact pointand of aa metal metal an experimental is investigated. For thisatpurpose, purpose, material model and steadystrip stateinstrip strip deformation device model are derived. Using these models, feasible combinations a steady state deformation model are derived. Using these models, combinations curvature and force are computed. A control structure is proposed that comprises metal strip instrip ancontact experimental device is investigated. For this purpose, afeasible material model and aof steady state deformation model are derived. Using these models, feasible combinations of curvature curvature and and contact contact force force are are computed. computed. A A control structure structure is is proposed that that comprises comprises of feedforward and adeformation feedback Exponential stability of the models, closed-loop system is shown. a steady stateand strip model are derived. Usingstructure these feasible combinations of curvature contact force part. are computed. A control control is proposed proposed that comprises a feedforward and a feedback part. Exponential stability of the closed-loop system is shown. shown. a feedforward a feedback part. Exponential stability of the closed-loop system is from and aand numerical example the robustness of the controller against model-plant of curvature contact force are verify computed. A control structure is proposed that comprises aResults feedforward and a feedback part. Exponential stability of the closed-loop system is shown. Results from a numerical example verify the robustness of the controller against model-plant Results from a numerical example verify the against model-plant mismatches its asuitability for disturbance rejection. a feedforward feedback part. Exponential stabilityofofthe thecontroller closed-loop system is shown. Results fromand aand numerical example verify the robustness robustness mismatches and its suitability suitability for disturbance disturbance rejection. of the controller against model-plant mismatches its for rejection. Results fromand a numerical example verify the robustness of the controller against model-plant mismatches and its suitability for disturbance rejection. © 2017, IFACand (International Federationdisturbance of Automaticrejection. Control) Hosting by Elsevier Ltd. All rights reserved. mismatches its suitability Keywords: Mechanical system, for Mathematical models, Static models, Boundary value problem, Keywords: Mechanical system, Mathematical models, Static models, Boundary value problem, Keywords: Mechanical system, Mathematical models, Static models, models, Boundary value value problem, problem, Model inversion, Steel industry, Closed-loop control, Lyapunov stability Keywords: Mechanical system, Mathematical models,Lyapunov Static Boundary Model inversion, Steel industry, Closed-loop control, stability Model inversion, Steel industry, Closed-loop control, Lyapunov stability Keywords: Mechanical system, Mathematical models,Lyapunov Static models, Boundary value problem, Model inversion, Steel industry, Closed-loop control, stability Model1.inversion, Steel industry, Closed-loop control, Lyapunov stability INTRODUCTION amount of curvature based on the desired residual stress 1. INTRODUCTION INTRODUCTION amount of curvature based on the desired residual stress 1. amount of curvature based on state. Another example of a contact problemresidual betweenstress strip 1. INTRODUCTION amount of curvature based on the the desired desired residual stress state. Another example of aa contact problem between strip state. Another example of contact problem between strip 1. INTRODUCTION and roll is the looper in tandem rolling mills. Here, amount of curvature based on the desired residual stress state. Another example of a contact problem between stripa In a typical strip processing plant, cold-rolled steel is heat and roll is the looper in tandem rolling mills. Here, a In a typical strip processing plant, cold-rolled steel is heat and roll is the looper in tandem rolling mills. Here, controller ensures the correct contactproblem force between looper state.roll Another example of atandem contact between stripa In a typical strip processing plant, cold-rolled steel is heat and is the looper in rolling mills. Here, a treated in continuous annealing lines in order to improve controller ensures the correct contact force between looper In a typical strip processing plant, cold-rolled steel is heat treated in in continuous continuous annealing annealing lines lines in in order order to to improve improve roll controller ensures the correct contact force between looper and strip, e.g., et al., 2014; Choia and roll is the see, looper in (Steinboeck tandem rolling mills. Here, treated controller ensures the correct contact force between looper properties such as hardness, strength, and ductility of steel In a typical strip processing plant, cold-rolled steel is heat and strip, see, e.g., (Steinboeck et al., 2014; Choi treated in such continuous annealing lines and in order to improve properties as hardness, hardness, strength, ductility of steel steel roll roll and strip, see, (Steinboeck et al., Choi Furthermore, by means the2014; rotational controller ensures thee.g., correct contact force between looper properties such as strength, and ductility of rollal., and2007). strip, see, e.g., (Steinboeck etof al., 2014; Choi sheets (Matthews and annealing James, 2010). horizontal and et treated in such continuous lines Both in order to improve et al., 2007). Furthermore, by means of the rotational properties as hardness, strength, and ductility of steel sheets (Matthews and James, 2010). Both horizontal and et al., 2007). Furthermore, by means of the rotational speed difference between two consecutive mill stands, the roll and strip, see, e.g., (Steinboeck et al., 2014; Choi sheets (Matthews and James, 2010). Both horizontal and et al., 2007). Furthermore, by means of the rotational vertical(Matthews continuous annealing lines exist. The temperatures properties such as hardness, strength, and ductility of steel speed difference between two consecutive mill stands, the sheets and James, 2010). Both horizontal and vertical continuous continuous annealing annealing lines lines◦exist. exist. The The temperatures temperatures speed between two consecutive the ◦ length of the strip between these stands of ismill controlled. et al., difference 2007). Furthermore, by means thestands, rotational vertical speed difference between two consecutive mill stands, the in annealing lines range from 730 C up to 1200 C (Lanksheets (Matthews and James, 2010). Both horizontal and ◦exist. The temperatures ◦ length of the strip between these stands is controlled. vertical continuous annealing lines ◦C up to 1200 ◦C (Lankin annealing lines range from 730 length of the strip between these stands is controlled. speed difference between two consecutive mill stands, the in annealing lines range from 730 C up to 1200 C (Lank◦exist. ◦ length of the strip between these stands is controlled. ford et al., 1985). While the strip is conveyed through a vertical continuous annealing lines The temperatures C up to 1200 ◦through C (Lank-a Figure 1 outlines the experimental device considered in in annealing lines range from 730 ford et al., 1985). While the strip conveyed ◦ is length of the strip between these stands is controlled. Figure 1 outlines the experimental device considered in ford et al., 1985). While the strip is conveyed through a horizontal line, it is730 bent (purely elastically or Cisup to 1200 C (Lankin annealing lines range from 11 outlines the experimental device considered in ford et al.,annealing 1985). While the strip conveyed through a Figure this paper. The strip, is typically rather thin (thickFigure outlines thewhich experimental device considered in horizontal annealing line, it is bent (purely elastically or this paper. The strip, which is typically rather thin (thickhorizontal annealing line, it is bent (purely elastically or elasto-plastically) theisstrip rolls to its dead load. ford et al.,annealing 1985). around While the isdue conveyed through a this paper. The strip, which is typically rather thin (thickhorizontal line, it bent (purely elastically or ness h ∈ [0.3 mm, 1.2 mm]), is tightened by a mechanical Figure 1 outlines the experimental device considered in this paper. The strip, which is typically rather thin (thickelasto-plastically) around the rolls due to its dead load. h ∈ [0.3 mm, 1.2 mm]), is tightened by aa mechanical elasto-plastically) around the to its dead load. The periodic strip-roll contact may due cause surface defects horizontal annealing line, it is rolls bent (purely elastically or ness ness h ∈ [0.3 mm, 1.2 mm]), is tightened by mechanical elasto-plastically) around the rolls due to its dead load. actuator which applies the tensional force n. The roll with this paper. The strip, which is typically rather thin (thickness h ∈ which [0.3 mm, 1.2 mm]), is tightened by aThe mechanical The periodic strip-roll contact may cause surface defects applies the tensional force n. roll with The periodic strip-roll contact may cause on rolls (Fukubayashi, 1995; Sawa and Oohori, 1995). elasto-plastically) around the rolls due to surface its deaddefects load. actuator actuator which applies the force n. roll with Thethe periodic strip-roll contact may cause surface defects aactuator radius has two1.2 actuated of freedom. That is, ness h ∈R [0.3 mm, mm]), isdegrees tightened by mechanical which applies the tensional tensional force n. aThe The roll with on the rolls (Fukubayashi, 1995; Sawa and Oohori, 1995). a radius R has two actuated degrees of freedom. That is, on the rolls (Fukubayashi, 1995; Sawa and Oohori, 1995). In literature, it is contact distinguished between mechanical Thethe periodic strip-roll may cause surface defects a radius R has two actuated degrees of freedom. That is, on rolls (Fukubayashi, 1995; Sawa and Oohori, 1995). its horizontal position x and the penetration depth ∆h actuator which applies the tensional force n. The roll with r a radius R has two actuated degrees of freedom. That is, In the literature, it is distinguished between mechanical its horizontal position x and the penetration depth ∆h In the literature, it is distinguished between mechanical r and reaction effectsitthat lead to the development of surface on the rolls (Fukubayashi, 1995; Sawa and Oohori, 1995). its horizontal position x and the penetration depth ∆h r In literature, is distinguished between mechanical can be controlled. The position of the strip-roll contact a radius R has two actuated degrees of freedom. That is, its horizontal position x and the penetration depth ∆h and reaction effects that lead to the development of surface r can be be controlled. controlled. The The position position of of the the strip-roll strip-roll contact contact and reaction lead to the development of surface defects. One effects way to reduce surface defects is tomechanical guarantee In the literature, itthat is distinguished between can and reaction effects that lead to the development of surface point is denoted by x and the length of the experimental its horizontal position x and the penetration depth ∆h 1 r can beis controlled. positionlength of the strip-roll contact defects. One way to reduce surface defects is to guarantee denoted by The x of the experimental defects. One way reduce defects is guarantee 1 and the a suitable of surface the rolldevelopment surface. be point and reaction effects lead to the ofmust surface point denoted x the of the defects. Onecomposition way to tothat reduce surface defects is to toIt guarantee device L. The by main purpose of of thethe experimental device can beis controlled. The position contact point isby denoted by x11 and and the length length of strip-roll the experimental experimental a suitable composition of the roll surface. It must be device by L. The main purpose of the experimental device a suitable composition of the roll surface. It must be wearand corrosion-resistant, it must prevent defects. Onecomposition way to reduce is to device L. The main purpose of the experimental a suitable of surface the and rolldefects surface. It guarantee must the be point is to analyze theby influence of the temperature, thedevice steel isby denoted x1 and the length of the experimental device by L. The main purpose of the experimental device wearand corrosion-resistant, and it must prevent the is to analyze the influence of the temperature, the steel wearand corrosion-resistant, and it must prevent the adhesion of foreign particles onto the roll surface (Hao, a suitable composition of the roll surface. It must be is to analyze the influence of the temperature, the wearand corrosion-resistant, and it must prevent the grade of the strip, the contact force, and the curvature on device by L. The main purpose of the experimental device is to analyze the influence of force, the temperature, the steel steel adhesion of foreign particles onto the roll surface (Hao, grade of the the strip, strip, the contact contact and the the curvature curvature on adhesion of foreign particles onto the roll surface (Hao, 2007; Fukubayashi, 1995, 2004). In this paper, mechanical wearand corrosion-resistant, and it must prevent the grade of the force, and on adhesion of foreign particles onto the roll surface (Hao, the evolution of surface defects. These experiments should is to analyze the influence of the temperature, the steel grade of the strip, the contact force, and the curvature on 2007; Fukubayashi, 1995, 2004). In this paper, mechanical the evolution of surface defects. These experiments should 2007; Fukubayashi, 1995, 2004). In this paper, mechanical effects like the contact force between the strip and the adhesion of foreign particles onto the roll surface (Hao, the evolution of surface defects. These experiments should 2007; Fukubayashi, 1995,force 2004). In this paper, mechanical help toofgain into the essential contact mechanisms grade the insight strip, the contact force, and the curvature on the evolution of surface defects. These experiments should effects like the contact between the strip and the to gain insight into the essential contact mechanisms effects like contact the strip and roll thethe curvature atforce the between strip-roll contact point are help 2007;and Fukubayashi, 1995, 2004). In this paper, mechanical help to gain insight the essential contact mechanisms effects like the contact force between the strip and the the between furnace rollsinto and strip. This paper focuses on the the evolution of surface defects. These experiments should help to gain insight into the essential contact mechanisms roll and the curvature at the strip-roll contact point are between furnace rolls and strip. This paper focuses on the roll and the curvature the strip-roll contact are examined. effects contactat strippoint and the furnace rolls and strip. This focuses on roll andlike thethe curvature atforce the between strip-roll the contact point are between mechanical aspects of thethe experimental device. Therefore, help to gain insight into essential contact mechanisms between furnace rolls This paper paper focuses on the the examined. mechanical aspects of and the strip. experimental device. Therefore, examined. roll and the curvature at the strip-roll contact point are mechanical aspects of the experimental device. Therefore, examined. model-based control mechanical parameters between furnace rolls strip. This paper focuses onassothe mechanical aspects of and the of experimental device. Therefore, A periodic strip-roll contact also occurs in the leveling the the model-based control of mechanical parameters assoexamined. A periodic strip-roll contact also occurs in the leveling the model-based mechanical parameters ciated with aspects such control contact problems is considered inassothis mechanical of the of experimental device. Therefore, A periodic strip-roll contact also occurs in the leveling the model-based control of mechanical parameters assoprocess, where the main objective is to reduce residual with such contact problems is considered in this A periodic strip-roll contact also occurs in the residual leveling ciated process, where the main objective is to reduce ciated with contact is this model-based mechanical parametersin process, where the main objective is to reduce ciated with such such control contact ofproblems problems is considered considered inassothis stresses and shape defects (residual curvatures) of the the A periodic strip-roll contact also occurs in the residual leveling process, where the main objective is to reduce residual stresses and and shape shape defects defects (residual (residual curvatures) curvatures) of of the the ciated with y such contact problems is considered in this stresses strip. However, in this process, the dead load process, where the main objective is to reduce residual y stressesHowever, and shape defects (residual curvatures) of the h y strip. in this process, the dead load of the ∆h h n strip. in process, the dead load the y w strip isHowever, not and the strip alternately stresses and sufficient shape defects (residual curvatures) h ∆h strip. However, in this this process, the is dead load of of bent the n w ∆h h strip is not sufficient and the strip is alternately bent n y w R x strip is not sufficient and the strip is alternately bent ∆h between the rollers of the leveler. In this way, a specific strip. However, in this process, the dead load of the n w strip is not sufficient and the strip is alternately bent h R x between the rollers of the leveler. In this way, a specific R x ∆h n between the rollers of the leveler. In this way, a specific w R x amount of curvature is imposed onto the strip at each roll. strip is not sufficient and the strip is alternately bent x1 between rollers of the leveler. this way, specific amount of ofthe curvature is imposed imposed ontoInthe the strip at aeach each roll. x R x 1 amount curvature is onto strip at roll. Batty and Lawson the force power between rollers(1965) of theinvestigated leveler. this way, aeach specific xr1 amount ofthe curvature is imposed ontoInthe strip atand roll. xr1 g Batty and Lawson (1965) investigated the force and power L x Batty and investigated the force power g requirements of the(1965) leveling process. they did not amount of Lawson curvature is imposed ontoHowever, the strip atand each roll. xrr1 Batty and Lawson (1965) investigated the force and power g L requirements of the leveling process. However, they did not L g requirements of the leveling process. However, they did not x consider the evolution of the curvature. Kaiser et al. (2014) Batty and Lawson (1965) investigated the force and power L r requirements of the leveling process. However, they did not g consider the evolution of the curvature. Kaiser et al. (2014) L Fig. 1. Geometry of the experimental device. consider the evolution of the curvature. Kaiser et al. (2014) proposed a method to determine the required sequence and requirements of the leveling process. However, they did not consider the evolution of the curvature. Kaisersequence et al. (2014) Fig. proposed a method to determine the required and Fig. 1. 1. Geometry Geometry of of the the experimental experimental device. device. proposed a to the and consider the evolution of the curvature. Kaisersequence et al. (2014) proposed a method method to determine determine the required required sequence and Fig. 1. Geometry of the experimental device. Fig. 1. Geometry of the experimental device. proposed©a 2017 method Copyright IFACto determine the required sequence and 11817
Copyright © 2017 11817 Copyright 2017 IFAC IFAC 11817Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017, 2017 IFAC 11817 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2017 IFAC 11817 10.1016/j.ifacol.2017.08.1665
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paper. In fact, the contact force between strip and roll ∆q and the curvature κ at the strip-roll contact point have to be controlled. The paper is structured as follows: First, suitable models for material and strip deformation are derived in Section 2. Examination of the resulting boundary value problem leads to feasible combinations of curvature and contact force for the considered simulation device. Section 3 is concerned with the design of the proposed control structure. Furthermore, the stability of the closed-loop system is investigated, where exponential stability is shown. Finally, a numerical example in Section 4 analyzes the performance of the controller against model-plant mismatches and its suitability for disturbance rejection. 2. MODELING 2.1 Material model In the following, derivatives with respect to the length coordinate x are denoted by (·) = d(·)/dx and derivatives ˙ = d(·)/dt. with respect to the time t are denoted by (·) Furthermore, w = w(x) and κ(x) = κ are the deflection and the curvature of the strip, respectively. The following material model was originally proposed by Stadler et al. (2015). However, in this paper, only the linear elastic case is considered, whereas the ideal-elastic-ideal plastic case was considered in (Stadler et al., 2015). An initially plain strip with zero residual stresses and a plane state of stress without shear stresses is assumed. Because the mean tensile stress caused by strip tension is small compared to the yield stress and also small compared to the bending stress, zero mean stress is a valid assumption. In the geometrically linear case (small deflections), the curvature is approximated by w κ= (1) 3 ≈ w . 1 + w 2 2 Since the width b of the strip satisfies b h and due to the fact that bending strains in lateral direction vanish, the plate stiffness effect has to be considered. This can be done by replacing Young’s modulus E by an effective Young’s modulus E ˜= , (2) E (1 − ν 2 ) where E may be temperature dependent (Varshni, 1970). The bending moment mb normalized with respect to the width is thus defined by ˜ 3κ Eh . (3) mb = − 12
at x = 0 and x = L. As in (Steinboeck et al., 2015), it is assumed that the tensional force n in the strip is uniform in longitudinal direction. Furthermore, small deflections w are assumed, linear geometric relations are considered, and because the considered beam is slender, the EulerBernoulli hypothesis is applied. In case of thick beams, the previous assumptions are rendered invalid and the Timoshenko beam theory may be used (Pilkey, 2002). Based on these assumptions, the quasi-static deflection w of the strip can be computed from the differential equation gρh − nw − w = 0 (4) supplemented by appropriate boundary conditions. With m b = q and (3), this differential equation can be reformulated as a nonlinear boundary value problem (BVP) with known boundaries x = 0, x = L and a generally unknown boundary x = x1 w w ˜ 3) d w −12mb /(Eh , x ∈ (0, L) (5) m = q b dx ˜ 3) q gρh + 12nmb /(Eh with the boundary conditions w(0) = w0 , w(L) = w0 (6a) w (L) = 0 (6b) w (0) = 0, + + w(x− w (x− (6c) 1 ) = w(x1 ), 1 ) = w (x1 ) − + mb (x1 ) = mb (x1 ) (6d) w(x1 ) = R sin(arccos((x1 − xr )/R)) − R + ∆h . x− 1
(6e)
x+ 1
2.2 Steady-state strip deformation model
Here, and denote the left- and the right-hand limit of the strip-roll contact point x1 , respectively. The boundary condition (6e) ensures that the strip-roll contact point x1 is on the roll. Because x1 represents an unknown boundary point in (6), an additional equation has to be stated for the computation of x1 . That is, the slopes of the strip and the roll have to be consistent at this point. With w(x1 ) = R2 − (xr − x1 )2 (7a) x r − x1 (7b) w (x1 ) = R2 − (xr − x1 )2 it follows that (8) (w (x1 ))2 R2 = (xr − x1 )2 (1 + (w (x1 ))2 ) and therefore Rw (x1 ) x 1 = xr − . (9) 1 + (w (x1 ))2 has to hold. An analytical solution of the differential equation (5) itself does exist. However, the resulting constants of integration can only be numerically computed based on the boundary conditions (6) and (9). Thus, a BVP-solver (multiple shooting method) was used to solve (5), (6), and (9).
For similar mechanical structures, a systematic derivation of the governing differential equation and its boundary conditions has been carried out in (Pilkey, 2002). Due to the fact that only slow roll movements are considered, acceleration forces can be neglected. The dead weight gρh with the gravitational acceleration g and the mass density of steel ρ represents a uniformly distributed load on the strip. Another external load is the tensional force applied
For the following analysis, parameters from Tab. 1 in Section 4 were used. The upper part of Fig. 2 shows the inputs nb and ∆h required for certain curvature values κ and contact forces ∆q for the horizontal roll position xr = x1 = L/2 = 0.6 m. Analogous plots exist for any other point xr ∈ [R, L − R]. Gray isolines represent constant contact forces ∆q and black isolines represent constant curvatures κ. The intersection of two such lines gives the required control inputs for the respective values
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−
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Fig. 3. Block diagram of the proposed 2-DOF control structure. position xr as a user defined input trajectory. Its time derivatives are limited to ensure that the assumption of a quasi-static bending process is tenable. The manipulated variables set by the controller are the tensional force n and T the penetration depth ∆h of the roll, i.e., u = [n ∆h] . As usual in cascade control structures, actuator dynamics are neglected for control design and stability proof of the outer control loop. Simulation studies in Section 4 justify this approach.
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900 600 300 0 −10
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Fig. 2. Contour plot (top) of curvature and contact force at xr = x1 = L/2 = 0.6 m. Feasible regions in the curvature/contact force plain (bottom) for various horizontal roll positions xr . ∆q and κ. The input box-constraints N = nb ∈ [0, 100] N and ∆h ∈ [0, 0.16] m are shown as a gray dash-dotted rectangle. The edges of this rectangle can be mapped to the curvature/contact force plain shown in the bottom part of Fig. 2 for various horizontal roll positions xr . Markers highlight the correspondence between corner points of the feasible regions in both parts of Fig. 2. That is, all feasible curvature/contact force combinations for a specific horizontal roll position xr are given by the respective enclosed area in the lower part of Fig. 2. 3. CONTROLLER DESIGN The considered control task is to establish a certain curvature κ and contact force ∆q at the strip-roll contact point x1 . For this control task, a two-degrees-of-freedom control structure (2-DOF) comprising a feedforward and a feedback part is outlined in Fig. 3, see, e.g., (Goodwin et al., 2001). In a 2-DOF control structure, the feedforward controller contributes the main part of the control input, whereas the purpose of the feedback controller is to reject unknown disturbances and to compensate for model-plant mismatches. The controller considers the horizontal roll
The solution of (5), (6) and (9) with (3) can be formally written as a nonlinear static input-output relation f (mb (x1 )) κ(x1 ) = Υ(u, xr ) . (10) = y= + ∆q(x1 ) q(x− 1 ) − q(x1 ) In terms of the input u and the output y, Υ is a bijective mapping. For given values xr , κd , and ∆q d , the feedforward controller uses (10) and numerically computes the open-loop control input uff , so that d κ nff yd = , x ), with u = (11) = Υ(u ff r ff ∆hff ∆q d
holds. Because the desired reference trajectories are known in advance (chosen by the operators), the required openloop control input can be calculated off-line. 3.2 Feedback controller
To reduce the effects of model-plant mismatches and disturbances, the feedforward controller is supplemented by a feedback controller. It is assumed that the contact force ∆q is measured by a force sensor and that the curvature κ is optically measured. For compact notation, consider that Υred (u, x1 ) is the solution function of the reduced model (5) and (6), where the value of x1 is assumed to be constant and given by the solution of (11). As shown in (Lunze, 1989), local linearization of (10) at the point uff , yd = Υred (uff , x1 ), and x1 for the undisturbed case udis = 0 gives u = uff + uf b (12a) ∂Υred (u, x1 ) uf b = yd + Juf b . (12b) y = yd + ∂u u=uff yf b J
Here, it is assumed that the change of x1 due to the feedback controller uf b is negligibly small. In the following, the arguments uff , yd and x1 are omitted to keep the notation compact. It is worth noting that all quantities,
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×10−3
including the following regular state transformation, depend on these arguments. To simplify the controller design, the multi-input/multi-output (MIMO) system is decoupled into two single-input-single-output (SISO) relations. Therefor, the Jordan normal form ˜ = (D(x))−1 JD(x) (13) J is calculated, where D(x) denotes the transformation matrix consisting of the eigenvectors of J. For the sake of clarity, the argument of D is omitted in the following. ˜f b = ˜ f b = D−1 uf b and y The linearized relation between u −1 D yf b is then given by ˜ uf b ˜f b = J˜ (14) y ˜ It is now possible to design with the diagonal matrix J. two decoupled SISO controllers, which reduce the control error ˜ = D−1 (yd − y) = −˜ e yf b . (15) In this paper, PI control laws ˜ I + KP D−1 (yd − y) ˜fb = u (16a) u −1 d ˙u ˜ I = KI D (y − y) (16b)
with the diagonal matrices KI and KP are used for this purpose. Finally, the feedback law transformed back to the original space reads as nf b ˜ I + KP D−1 (yd − y) . uf b = =D u (17) ∆hf b
3.3 Stability proof
In this section, the stability of the closed-loop system and its robustness against model-plant mismatches is analyzed. Because the plant is represented by the nonlinear static function (10), which gives bounded outputs for bounded inputs, only the stability of the integrator state (16b) has to be analyzed. To analyze robustness, the strip thickness used by the controller will be reduced by 5 %. From the ˜˙ I = 0 of (16b), it follows that steady state condition u y = yd holds at any steady state of the system. For such a steady state, the feedback output based on (17) ˜ I,0 denotes the steady state of is uf b,0 = D˜ uI,0 , where u the integrator. The values of uf b,0 can be computed by solving (10), (11), and (12a). If the same parameters are used in (10) and (11), uf b,0 = 0 is obtained. If different parameters are used in (10) and (11), e.g., to simulate a model-plant mismatch, uf b,0 = 0. This can be converted into the steady state value ˜ I,0 = D−1 uf b,0 (18) u ˜I − u ˜ I,0 , the of the integrator. With the error state xI = u positive definite Lyapunov function 1 V = xT xI (19) 2 I can be defined. Its time derivative is given by −1 ˙ I = xT V˙ = xT (yd − y) , (20) Ix I KI D where y is the solution of the nonlinear implicit equation ˜ I,0 + KP D−1 (yd − y))) . (21) y = Υ(uff + D(xI + u For exponential stability, 2 V˙ ≤ −α xI , α > 0 (22) 2
has to hold in a region around the origin, (cf. Khalil, 2002). Based on (19), (22) reduces to V˙ ≤ −2αV . (23)
0
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Fig. 4. Time derivate V˙ of the Lyapunov function V with T xI = [xI,1 xI,2 ] for a strip thickness deviation of −5 %, κd = −3/m, and ∆q d = 450 N/m. V˙ depends on xI , yd , and the strip-roll contact point x1 . Because an analytical proof of inequality (23) is rather difficult, its validity is graphically analyzed for the representative case κd = −3/m and ∆q d = 450 N/m. For this purpose, (20) is numerically evaluated for the range xr ∈ [0.2, 1.0] m. The worst case value maxxr ∈[0.2,1.0]m (V˙ ) is shown in Fig. 4. The illustrated range for xI includes the intervals κ ∈ [0.79κd , 1.20κd ] and ∆q ∈ [0.83∆q d , 1.14∆q d ]. Clearly, V˙ is locally negative definite and the upper value α = 1.1/s in (23) can be identified. Consequently, the error state decreases at least according to (24) xI (t) 2 ≤ xI (t0 ) 2 e−α(t−t0 ) , t ≥ t0 ≥ 0 . 4. NUMERICAL RESULTS For the following simulation analysis, the actuator dynamics of the tensional force n and the penetration depth ∆h are approximated by means of second-order delay systems with time constant T = 50 ms and damping ratio 1.0. The sampling time of the controller is set to Ts = 10 ms. Figure 5 shows the simulation results for the proposed 2-DOF control structure. A strip thickness deviation of −5 % is used. Furthermore, the influence of different disturbances acting on the output and the control variables is investigated. On the left-hand side of Fig. 5, the deT sired output is yd = [−2/m 300 N/m] , whereas yd = T [−3/m 450 N/m] on the right-hand side of Fig. 5. In both scenarios, the horizontal roll position xr follows the same ramp trajectory ranging from 0.2 m to L/2 = 0.6 m. For symmetry reasons, it is sufficient to only simulate the case xr ≤ L/2. Due to the simulated model-plant mismatch, the control error is relatively high at the beginning of the simulation, where a constant horizontal roll position xr is considered. However, the feedback controller ensures that the control error is nearly zero after one second. At
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t = 2 s and t = 14 s, a 3 s long output disturbance, e.g., an output error of ydis = [0 ∆qdis ] = [0 ±30 N/m], occurs, cf. Fig 3. Again, the controller quickly compensates for these disturbances. Next, the influence of disturbances udis = [ndis ∆hdis ] acting on the control variables ∆h and n is investigated. The constant disturbances ∆hdis = −40 mm and ndis = 60 N/m start at t = 8 s and t = 11 s, respectively. Note that solid black lines in the second and third row of Fig. 5 refer to the disturbed outputs n and ∆h of the respective actuator. Once again, about one second is needed until the control error is nearly zero. To sum up, throughout the entire trajectory, a good control performance is achieved. These results also justify the assumption of negligible actuator dynamics when designing the outer control loop. Table 1. Parameters of the considered experimental device. Description
Parameter
Value
Unit
Mass density of steel Young’s modulus of steel Gravitational acceleration Thickness of strip Roll radius Poisson’s ratio
ρ E g h R ν
7850 210 9.81 0.65 60 0.3
kg/m3 kN/mm2 m/s2 mm mm -
5. CONCLUSION AND OUTLOOK This paper deals with the MIMO control of the curvature and the contact force at the strip-roll contact point of a metal strip in an experimental device. A 2-DOF control structure comprising a feedforward and a feedback part in the form of a PI controller is developed. Feasible regions in the curvature/contact force plain are presented for various roll positions considering the input box-constraints. The robustness of the control structure against model-plant mismatches is demonstrated by means of a Lyapunovbased stability proof. Furthermore, a simulation study with various disturbance scenarios shows a good disturbance rejection behavior of the controller. Future work will concern the transfer of the proposed control strategy to the laboratory experiment. ACKNOWLEDGEMENTS Financial support by the Austrian Federal Ministry of Science, Research and Economy and the National Foundation for Research, Technology and Development, and voestalpine Stahl GmbH is gratefully acknowledged.
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Fukubayashi, H.H. (2004). Present furnace and pot roll coatings and future development. In Proceedings of the International Thermal Spray Conference 2004, 125–131. ASM International, Osaka, Japan. Goodwin, G.C., Graebe, S.F., and Salgado, M.E. (2001). Control System Design. Prentice Hall, Upper Saddle River, NJ, USA, 1st edition. Hao, R. (2007). Thermal spraying technology and its applications in the iron & steel industry in China. In Proceedings of the International Thermal Spray Conference, Thermal Spray 2007: Global Coating Solutions, 291–296. Beijing, China. Kaiser, R., Hatzenbichler, T., Buchmayr, B., and Antretter, T. (2014). Simulation of the roller straightening process with respect to residual stresses and the curvature trend. In International Conference on Residual Stresses 9 (ICRS 9), volume 768 of Materials Science Forum, 456–463. Khalil, H.K. (2002). Nonlinear Systems. Prentice Hall, New Jersey, 3rd edition. Lankford, W., Samways, N., Craven, R., and McGannon, H. (1985). The Making, Shaping, and Treating of Steel. United States Steel, 10th edition. Lunze, J. (1989). Robust Multivariable Feedback Control. Prentice-Hall, New Jersey, 1st edition. Matthews, S. and James, B. (2010). Review of thermal spray coating applications in the steel industry: Part 1 – Hardware in steel making to the continuous annealing process. Journal of Thermal Spray Technology, 19(6), 1267–1276. Pilkey, W. (2002). Analysis and Design of Elastic Beams: Computational Methods. John Wiley & Sons, New York, 1st edition. Sawa, M. and Oohori, J. (1995). Application of thermal spraying technology at steelworks, Thermal spraying: Current status and future trends. In Proceedings of the 14th International Thermal Spray Conference, 37– 42. High Temperature Society of Japan, Kobe, Japan. Stadler, G., Steinboeck, A., Baumgart, M., and Kugi, A. (2015). Modellierung des Umschlingungswinkels eines auf Rollen gef¨ uhrten Metallbandes. at – Automatisierungstechnik, 63(8), 646–655. Steinboeck, A., Baumgart, M., Stadler, G., Saxinger, M., and Kugi, A. (2015). Dynamical models of axially moving rods with tensile and bending stiffness. IFACPapersOnLine, 48(1), 598–603. Steinboeck, A., M¨ uhlberger, G., and Kugi, A. (2014). Control of strip tension in a rolling mill based on loopers and impedance control. IFAC Proceedings Volumes, 47(3), 10646–10651. 19th IFAC World Congress. Varshni, Y.P. (1970). Temperature dependence of the elastic constants. Physical Review B, 2, 3952–3958.
REFERENCES Batty, F.A. and Lawson, K. (1965). Heavy plate levellers. Journal of the Iron and Steel Institute, 203, 1115–1128. Choi, I., Rossiter, J., and Fleming, P. (2007). Looper and tension control in hot rolling mills: A survey. Journal of Process Control, 17(6), 509–521. Fukubayashi, H.H. (1995). Coating for high temperature pickup and wear resistant applications, Thermal spraying: Current status and future trends. In Proceedings of the 14th International Thermal Spray Conference, 47– 52. High Temperature Society of Japan, Kobe, Japan. 11821
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Fig. 5. Simulation results with 2-DOF controller for yd = [−2/m 300 N/m] (left) and yd = [−3/m 450 N/m] (right) for a strip thickness deviation of −5 %. Different disturbances acting on the output and on the control variables are examined. 11822