Steering Control of Metal Strips Using a Pivoted Guide Roller

IFAC Workshop on Mining, Minerals and Metal Processing IFAC Workshop on Mining, Minerals and Metal Processing IFAC Workshop on Mining, Minerals and Metal Processing August 25-28, 2015. Oulu, Finland IFAC Workshop on Mining, Minerals and Metal Processing August 25-28, 2015. Oulu, Finland Available online at www.sciencedirect.com August 25-28, 2015. Oulu, Finland August 25-28, 2015. Oulu, Finland

ScienceDirect IFAC-PapersOnLine 48-17 (2015) 137–142

Steering Control of Metal Strips Steering Control of Metal Strips Steering Control of Metal Strips Steering Control of Metal Strips Using a Pivoted Guide Roller Using a Pivoted Guide Roller Using a Pivoted Guide Roller Using a Pivoted Guide Roller ∗ † A. Steinboeck ∗ A. Kugi † A. Steinboeck ∗ A. Kugi † A. A. Steinboeck Steinboeck ∗ A. A. Kugi Kugi † ∗ ∗ Automation and Control Institute, Vienna University of Technology, ∗ Automation and Control Institute, Vienna University of Technology, and Vienna University of ∗ AutomationGußhausstraße 27-29, 1040 Vienna, Austria AutomationGußhausstraße and Control Control Institute, Institute, Vienna University of Technology, Technology, 27-29, 1040 Vienna, Austria Gußhausstraße 27-29, 1040 Vienna, Austria (e-mail: [email protected]) Gußhausstraße 27-29, 1040 Vienna, Austria (e-mail: [email protected]) † (e-mail: [email protected]) for Model-Based Control in the Steel † Christian Doppler (e-mail:Laboratory [email protected]) Laboratory for Model-Based Control in the Steel † Christian Doppler Christian Doppler Laboratory for Model-Based Control in the † Industry, Automation and Control Institute, Vienna University of Christian Automation Doppler Laboratory for Model-Based Control in the Steel Steel Industry, and Control Institute, Vienna University of Industry, Automation and Control Institute, Vienna University of Technology, Gußhausstraße 27-29, 1040 Vienna, Austria Industry, Automation and Control Institute, Vienna University of Technology, Gußhausstraße 27-29, 1040 Vienna, Austria Technology, Gußhausstraße 27-29, 1040 Vienna, Austria (e-mail: [email protected]) Technology, (e-mail: Gußhausstraße 27-29, 1040 Vienna, Austria [email protected]) (e-mail: (e-mail: [email protected]) [email protected]) Abstract: Controlling the lateral position of moving metal strips is a frequent task in the Abstract: Controlling the lateral position of moving metal strips is a frequent task in the Abstract: Controlling the position moving metal strips is task in metal industry. industry. For production production lines with aaof guide roller at the the entry, dynamical model is Abstract: Controlling the lateral lateral position ofguide moving metal strips is aaaa frequent frequent task in the the metal For lines with roller at entry, dynamical model is metal industry. For production lines with a guide roller at the entry, a dynamical model is developed and validated by measurements from an industrial plant. A reduced-order state metal industry. For production lines with afrom guideanroller at the plant. entry, Aa dynamical model is developed and validated by measurements industrial reduced-order state developed validated by from an plant. reduced-order state observer is is and designed to estimate estimate unmeasurable states. As the the system system is open-loop open-loop unstable, developed and validated by measurements measurements fromstates. an industrial industrial plant. A A reduced-order state observer designed to unmeasurable As is unstable, observer is to unmeasurable states. is unstable, output-feedback controllers, state-feedback controllers, andthe constrained model predictive predictive observer is designed designed to estimate estimate unmeasurable states. As As theaa system system is open-loop open-loop unstable, output-feedback controllers, state-feedback controllers, and constrained model output-feedback controllers, state-feedback controllers, and a constrained model predictive controller are developed. Their performance is compared in a simulated example problem. output-feedback controllers, state-feedback controllers, anda simulated a constrained model predictive controller are developed. Their performance is compared in example problem. controller are developed. performance is in example problem. controller are(International developed. Their Their performance is compared compared in aa simulated simulated problem. © 2015, IFAC Federation of Automatic Control) Hosting by Elsevierexample Ltd. All rights reserved. Keywords: Strip steering, lateral dynamics of moving web, double integrator, reduced-order Keywords: Strip dynamics of web, double reduced-order Keywords: Strip steering, steering, lateral lateral dynamics of moving moving web,constrained double integrator, integrator, reduced-order observer, output-feedback control,dynamics state-feedback model predictive control control, Keywords: Strip steering, lateral of moving web, double integrator, reduced-order observer, output-feedback control, state-feedback control, constrained model predictive control observer, output-feedback control, state-feedback control, constrained model predictive control observer, output-feedback control, state-feedback control, constrained model predictive control Guide roller Guide rollers 1. INTRODUCTION Guide roller Guide rollers 1. INTRODUCTION Guide roller Guide rollers 1. INTRODUCTION Guide roller Guide rollers 1. INTRODUCTION In the metal industry, lateral steering of metal strips is a In industry, lateral steering of strips In the the metal metal industry, lateral steering of metal metal strips is is aaa frequent control task. For instance, it occurs in continuous In the metal industry, lateral steering of metal strips is frequent control task. For instance, it occurs in continuous frequent control task. task. For instance, instance, it occurs occurs in continuous continuous Strip Strip strip processing lines and and strip transport transport facilities. Paper frequent control For it in Strip 1 Strip 1 strip lines facilities. Paper 2 2 Strip 1 Strip 1 strip processing processing lines and strip strip transport facilities. Paper 2 making, printing,lines coating, and the production of printed Strip Strip0 1 Pivot 2 Pivot strip processing and strip transport facilities. Paper 1 Pivot 2 3 2 3 0 making, printing, coating, and the production of printed a) b) making, printing, printing, coating, and theindustrial production of printed printed 0 1 Pivot 2 3 0 1 Pivot electronic devices are other and typical applications, Pivot 2 3 a) b) making, coating, the production of 0 3 0 3 Pivot Pivot electronic devices are other typical industrial applications, a) b) 0 3 0 3 electronic devices are other other typicalstrips, industrial applications, a) b) where guidance of moving often referred to lateral electronic devices are typical industrial applications, Fig. 1. Web steering systems, a) remotely-pivoted guide, where lateral guidance of moving strips, often referred to where lateral guidance of moving moving strips, strips, often often referred referred to to Fig. 1. Web steering systems, a) remotely-pivoted guide, as webs, is of central interest. where lateral guidance of Fig. 1. Web steering systems, a) remotely-pivoted b) offset-pivot guide. as webs, is of central interest. Fig. b) 1. offset-pivot Web steering systems, a) remotely-pivoted guide, guide, guide. as webs, webs, is is of of central central interest. interest. b) offset-pivot guide. as Shelton and Reid (1971a,b) provided a comprehensive b) offset-pivot guide. Shelton Reid provided aa comprehensive spans of the strip. Both systems are typically used Shelton and and Reid (1971a,b) (1971a,b) provided comprehensive analysis on modeling techniques for idealized idealized and real real vertical Shelton and Reid (1971a,b) provided a comprehensive vertical spans of the strip. Both systems are typically used analysis on modeling techniques for and vertical spans the Both systems are used analysis on modeling techniques for idealized and real as intermediate guides, i. e., in continuous processing moving webs. Here, the term idealized means and that real the vertical spans of of web the strip. strip. Both systems are typically typically used analysis on modeling techniques for idealized as intermediate web guides, i. e., in continuous processing moving webs. Here, the term idealized means that the as intermediate web guides, i. e., in continuous processing moving webs. Here, the term idealized means that the lines with uniform upand downstream strip tension. The material properties of the web are neglected and the web as intermediate web guides, i. e., in continuous processing moving webs. Here, the term idealized means that the lines with uniform upand downstream strip tension. The material properties of the web are neglected and the web lines with uniform upand downstream strip tension. The material properties of the web are neglected and the web different situation shown in Fig. 2 is considered in this is considered as an inelastic membrane with zero shear lines with uniform upand downstream strip tension. The material properties ofinelastic the webmembrane are neglected and theshear web different situation shown in Fig. 2 is considered in this is considered as an with zero different situation shown in Fig. 2 is considered in this paper. There is no tension the strip section upstream of is considered as an inelastic membrane with zero shear in stiffness and longitudinal fibres that are straight between different situation shown in Fig. 2 is considered in this is considered as an inelastic membrane with zero shear paper. There is no tension in the strip section upstream of stiffness and longitudinal fibres that are straight between paper. There is no tension in the strip section upstream of stiffness and longitudinal fibres that are straight between the pivoted guide (roll 1 in Fig. 2), which is the first roll the guideand rollers. Modeling a real web means that the paper. Thereguide is no (roll tension inFig. the 2), stripwhich section upstream of stiffness longitudinal fibres that web are straight between the pivoted 1 in is the first roll the guide rollers. Modeling a real means that the the pivoted guide (roll 1 in Fig. 2), which is the first roll the guide rollers. Modeling a real web means that the after a looping pit. Generally, this guide roller is connected elastic properties of the material are taken into account. the pivoted guide (roll 1 in Fig. 2), which is the first roll the guide rollers. Modeling a real web means that the after a looping pit. Generally, this guide roller is connected elastic of material are into aa electric looping pit. this roller elastic properties properties of the the material are taken taken into account. account. to an that operates in generator mode. A Hence, the web may undergo elastic deformation in the after after looping drive pit. Generally, Generally, this guide guide roller is is connected connected elastic properties of the material are taken into account. to an electric drive that operates in generator mode. A Hence, the web may undergo elastic deformation in the to an electric drive that operates in generator mode. A pressure roll on top of the guide roller avoids or minimizes Hence, the web may undergo elastic deformation in the form of lateral bending (non-uniform stretch and bending to an electric drive that operates in generator mode. A Hence, the webbending may undergo elastic stretch deformation in the pressure roll on top of the guide roller avoids or minimizes form of lateral (non-uniform and bending pressure roll on top of the guide roller avoids or minimizes form of lateral bending (non-uniform stretch and bending slip between the strip and the roll. of longitudinal fibres). Shelton and Reid (1971a) showed pressure roll on top of the guide roller avoids or minimizes form of lateral bending (non-uniform stretch and bending slip between strip and roll. of fibres). Shelton and (1971a) showed between the the and the the of longitudinal longitudinal fibres). Sheltonproblem and Reid Reid (1971a)considershowed slip that the lateral fibres). beam bending (without slip the strip strip the roll. roll. with and without disof longitudinal Shelton and Reid (1971a) showed Shinbetween et al. (2004) usedand PI-control that the lateral beam bending problem (without considerthat the the lateraldeflection) beam bending bending problem (without consideret al. (2004) used PI-control with and without disation of shear can be analytically solved. The Shin that lateral beam problem (without considerShin et al. (2004) used PI-control with without disturbance feedforward for the second-order system of a real ation of shear deflection) can be analytically solved. The Shin et al. (2004) used PI-control with and and without disation of shear deflection) can be analytically solved. The turbance feedforward for the second-order system of a real dynamical models of both idealized and real moving webs ation of shear deflection) can be analytically solved.webs The turbance feedforward for the second-order system of a real dynamical models of both idealized and real moving turbance feedforward for the second-order system of a real dynamical models of both idealized and real moving webs are linear and can be expressed by firstor second-order dynamical models ofbeboth idealized and real moving webs l ku are and ku l are linear linearfunctions. and can can be be expressed expressed by by firstfirst- or or second-order second-order transfer ku h l are linear and can expressed by firstor second-order transfer functions. Guide roller ku h l transfer functions. h Guide roller transfer h Guide roller Figure 1 functions. shows the two most commonly used web steering X Z Guide roller Figure the used web steering Figure 1 1 shows shows the two two most most commonly commonly used web steering X systems, i. e., a remotely-pivoted guide and anweb offset-pivot Y Z X Figure 1 shows the two most commonly used steering x1 systems, i. e., aa remotely-pivoted guide and an offset-pivot Y Z x2 X r systems, i. e., remotely-pivoted guide and an offset-pivot x1 Y Z guide (Seshadri and Pagilla, 2011). In the former case, x2 x1 r systems, i. e., a remotely-pivoted guideInand anformer offset-pivot Y x2 guide (Seshadri and Pagilla, 2011). the case, r x Measurement 1 Pressure roll guide (Seshadri and Pagilla, 2011). In the former case, x2 only one guide roller (roll 2 in 2011). Fig. 1a) isthe pivoted. r Elastic Measurement Pressure roll guide (Seshadri and Pagilla, In former case, of strip position only one guide roller (roll 2 in Fig. 1a) is pivoted. Elastic Pivot Measurement Pressure roll of strip position only one guide roller (roll 2 in Fig. 1a) is pivoted. Elastic Y Measurement deformation of the strip occurs mainly between the two Pressure roll Pivot of strip position only one guide roller (rolloccurs 2 in Fig. 1a) isbetween pivoted.the Elastic Pivot Y Z deformation of the strip mainly two X of strip position Z Y Z Pivot deformation of the the strip occursguide mainly between the1 and two 0 1 2 Strip rollers adjacent to the pivoted (between rolls X Y Z deformation of strip occurs mainly between the two 0 1 2 Strip X Strip rollers adjacent to the pivoted guide (between rolls 1 and 0 1 2 3 X adjacent to the the pivoted guide(Fig. (between rolls and 3rollers in Fig. 1a). In the pivoted latter case 1b), rolls two 11guide Looping pit 0 1 2 3Strip rollers adjacent to guide (between and 33 in 1a). In latter case (Fig. 1b), two guide Looping pit 3 in Fig. Fig. 1a). In 2the the latter case (Fig. 1b), two guide Looping pit rollers (rolls 1 and in latter Fig. 1b) are (Fig. carried by two a pivoting 3 3 in Fig. 1a). In the case 1b), guide Looping pit rollers (rolls 11 and 22 in Fig. 1b) are carried by aa pivoting rollers (rolls and in Fig. 1b) are carried by pivoting Fig. 2. Web steering systems after looping pit. frame. Elastic deformation of the strip occurs mainly in the rollers (rolls 1 and 2 in Fig. 1b) are carried by a pivoting Fig. 2. Web steering systems after looping pit. frame. Elastic deformation of the strip occurs mainly in the Fig. 2. Web steering systems after looping pit. frame. Elastic deformation of the strip occurs mainly in the Fig. 2. Web steering systems after looping pit. frame. Elastic deformation of the strip occurs mainly in the

Copyright IFAC 2015 137 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright © IFAC 2015 137 Copyright IFAC responsibility 2015 137Control. Peer review©under of International Federation of Automatic Copyright © IFAC 2015 137 10.1016/j.ifacol.2015.10.092

IFAC MMM 2015 138 August 25-28, 2015. Oulu, Finland

A. Steinboeck et al. / IFAC-PapersOnLine 48-17 (2015) 137–142

moving web steered by an offset-pivot guide. For the same system, Ho et al. (2008) simulated the use of PID-control. Wang et al. (2005) used PI-control and a Smith predictor with PI-control for the second-order model with time delay of a real moving web steered by a pivoted roller. The time delay accounts for the spatial offset of a downstream position measurement. Seshadri and Pagilla (2011) argued that fixed gain controllers are not suitable for systems with unknown or changing model parameters. Hence, they proposed an adaptive controller for web guiding systems based on a generic second-order reference model. This controller can be used for both offset-pivot guides and remotelypivoted guides but requires several design parameters to be tuned. Seshadri and Pagilla (2011) conducted laboratory experiments for this parameter tuning process. All models and control solutions mentioned so far are parameterized in terms of the time. If the longitudinal velocity of the web varies, these models are thus timevariant, which implies that control parameters have to be updated, e. g., by gain scheduling (Wang et al., 2005) or adaptive control (Seshadri and Pagilla, 2011). Schulmeister and Kozek (2009) presented a dynamical model of the lateral position of an endless steel strip, i. e., a process belt, where the strip traveling distance rather than the time is used as an independent variable. The advantage of such a time-free formulation is that it is invariant with respect to changing strip velocities. Therefore, this timefree formulation is also used in this paper. In most published models, the strip tension is considered to be uniform along both the up- and downstream direction and the lateral position at upstream rollers (e. g., roll 0 in Fig. 1) is assumed to be a known system input. The configuration considered in this paper (cf. Fig. 2) is different: The lateral position at roll 0 is unknown and irrelevant. The strip tension varies from zero in the looping pit to full tension after roll 3. Another fundamental assumption of most published models of moving webs is zero slip between the web and the guide rollers. However, it is not clear whether the zero-slip assumption is tenable for the longitudinal direction if the shear and the lateral bending stiffness of the moving web is high, which is characteristic for wide metal strips. In view of the existing strip steering solutions and the special requirements of strip steering after a looping pit (cf. Fig. 2), the objective of the current paper is to develop a mathematical model and controllers for the lateral strip position of the considered system. The paper is organized as follows: In Section 2, a dynamical model of the system is derived, parameterized, and validated. A state observer and various controllers are designed in Sections 3 and 4, respectively. The control performance is demonstrated in an example scenario in Section 4. 2. IDEALIZED LATERAL STRIP MOTION Consider the web steering system outlined in Fig. 2. The angular position ku of the pivoted entry roller is controlled by a hydraulic cylinder, which has the position u. Here, k is a geometric constant. Subordinate control loops are considered as ideal and u is a system input. The steering system should ensure that the strip enters the 138

downstream section of the processing line at a laterally centered position. 2.1 Mathematical Model A dynamical model of the lateral strip position is developed based on the following assumptions for the strip section between the rollers 1 and 2 in Fig. 2: A1) The strip has a flat and wrinkle-free shape. A2) The centerline of the strip (dash-dotted line in Fig. 2) rolls over the guides 1 and 2 without slip at the upper vertices. All other points of the strip may experience slip along the direction Z as they pass these vertices. A3) The centerline of the strip is straight. A4) Acceleration forces and lateral vibrations are negligible, i. e., a purely kinematic model is considered. A5) For the angular position of the guide roller, |ku| ≪ 1 holds. The lateral displacement of the strip is small compared to min{r, l}. Hence, geometrically linear relations can be used and displacements of the center point of the guide roller along the direction Z can be neglected. In particular, assumption A2) is different compared to existing models of moving webs (cf. Section 1). Assumption A2) seems reasonable because of the tensile force in the strip, the effect of the pressure roll, and the high stiffness of the strip against shear and lateral bending. The distance between the center point of the guide roller and the pivot axis is r = 2.06 m. The rolls 1 and 2 have the longitudinal distance l = 4.14 m. The lateral strip position y at a distance h = 1.68 m from the guide roller is the system output and is optically measured. Let w be the accumulated strip length, which serves as the independent variable of the dynamical model (instead of the time t). Derivatives with respect to w are abbreviated in the form ( · )′ = d( · )/dw. The lateral distances between the centerline of the strip and the center points of the rolls 1 and 2 serve as system states x1 (w) and x2 (w), respectively, and are assembled in the vector x = [x1 , x2 ]T . Whenever confusion is ruled out, the argument w is omitted. The centerline of the strip between the rolls 1 and 2 and the axis Z span the angle (x2 − (x1 + rku))/l, which is equivalent to −x′2 , i. e., the negative change of x2 per unit accumulated strip length. Similarly, the rate of change of x1 is equivalent to the angle ku + (x1 + rku − x2 )/l. Hence, the state-space representation of the dynamical model follows in the form   k r+l 1 1 −1 x+ u ∀w > 0 (1a) x′ = r l 1 −1 l with the initial state x(0) = x0 and the output equation rk 1 (1b) y = [l − h h] x + (l − h)u. l l This system is linear, independent of the strip velocity dw/dt, and invariant with respect to w. The system features direct feedthrough and its transfer function ((l − h)¯ s + 1)(r¯ s + 1)k y˜(¯ s) = (2) u ˜(¯ s) l¯ s2 shows that it is a double integrator, i. e., it is unstable. In (2),˜ labels a signal in the continuous Laplace domain and s¯ ∈ C is the Laplace variable with the unit 1/m.

IFAC MMM 2015 August 25-28, 2015. Oulu, Finland

A. Steinboeck et al. / IFAC-PapersOnLine 48-17 (2015) 137–142

In contrast to most published models of web steering systems (cf. Section 1), the lateral strip position x1 at the entry roller (roll 1 in Fig. 2) is not a given input but a state variable. This noteworthy feature of the considered system is rooted in the absence of strip tension and lateral stiffness against shear and bending upstream of roll 1.

0.02 y (m) 0

measured simulated 0

50

100 w (m)

150

100 w (m)

150

200

0.02 u (m) 0 −0.02 b)

0

50

50

100

150 w (m)

200

250

300

0

50

100

150 w (m)

200

250

300

u (m) 0 −0.01 b)

Fig. 4. Signals used for model validation. also known as Luenberger observer, is designed. The zeroorder-hold method with the sampling period ws yields the discrete state-space model     kws 2(r + l) + ws 1 l + ws −ws x+ ui (3) xi+1 = ws l − ws i 2l 2r + ws l

0.05

a)

0 a) 0.01

Because the system is open-loop unstable, parameter identification and model validation are delicate tasks. The only currently available measurement signals from the real plant were recorded during (unsuccessful) attempts of stabilizing the system by black-box PI feedback control.

−0.05

measured simulated

−0.02

2.2 Parameter Identification and Model Validation

y (m) 0

139

200

Fig. 3. Signals used for model identification. The measured signals shown in Fig. 3 (black lines) were used to estimate the unknown input gain k, the initial state x0 , and the unknown gains of the PI controller. First, k and x0 were identified by simple least-squares estimation using the plant model and the measured in- and output signals. The estimated value for k is 0.1625 rad/m. Then, the PI-controller gains were computed based on the frequency and damping ratio of the measured output signal y and the model of the closed-loop system. The frequency and damping ratio can, for instance, be estimated using the discrete Hilbert transform (Shin and Hammond, 2008). Fig. 3 shows simulation results based on these identified parameters as gray lines. The mean absolute deviation between measured and simulated signals is 0.0025 m for y and 0.0030 m for u. The small phase shift between the measured and the simulated signal u could be eliminated if in addition to k more parameters of (1) were identified. This was not done to ensure both a transparent identification process and simple transferability of the model. The estimated value k was validated using data from a second measurement with different initial values and a differently tuned PI controller. The corresponding simulation results are shown in Fig. 4 and confirm that k has been accurately estimated. Here, the mean absolute deviation between measured and simulated signals is 0.0012 m for y and 0.0011 m for u. 3. OBSERVER DESIGN State-feedback controllers require knowledge of the system state x, which is not directly measured. Therefore, a discrete reduced-order observer (Franklin et al., 2015), 139

∀ i ≥ 0. Here, i is the discrete index that defines the grid points wi = iws and xi is the corresponding system state, i. e., xi = x(wi ). The output equation (1b) is also used for the discrete system. Based on the observability matrix, it can be easily shown that the system is fully observable. Transformation to the sensor coordinates x1 and y−r(1−h/l)ku and design of a reduced-order observer with the state x ¯i yields  ws  ws x¯i − (1 − kO )2 yi x ¯i+1 = 1 + (1 − kO ) h h  kw s ((2(r + l) + ws )(1 − kO ) + 2hkO ) + 2l  kws r (l − h)(1 − kO )2 ui + lh with x ¯0 = 0 and the output equation     1 1 h kO h ˆi = x ¯i + yi x h h−l h kO h + (1 − kO )l   kr kO h (l − h)ui − hl kO h + (1 − kO )l

ˆ i . For asymptotic stability, the for the estimated state x observer gain kO ∈ R has to be chosen so that the eigenvalue zO = 1 + (1 − kO )ws /h is inside the unit circle. Throughout this paper, the eigenvalue zO = 0.95 and the sampling period ws = 0.1 m are used. Simulation results using this observer are shown in Sections 4.2 and 4.3, where state-feedback controllers are used for stabilizing the system. 4. CONTROL DESIGN

In this section, four different discrete feedback controllers are designed and compared based on a simulation scenario. The control objective is to keep the system output y close to its desired value yd = 0. The input u is constrained by the inequalities |u| ≤ 0.152 35 m (4a) |u′ | ≤ 0.010 61. (4b) In essence, the velocity du/dt of the hydraulic cylinder is limited. This limit can be converted into a constraint on u′ using the relation u′ dw/dt = du/dt. The constraint (4b)

IFAC MMM 2015 140 August 25-28, 2015. Oulu, Finland

A. Steinboeck et al. / IFAC-PapersOnLine 48-17 (2015) 137–142

has been computed for a maximum strip velocity dw/dt = 2 m/s. In the same way, a less conservative constraint for variable strip velocities can be computed as a function of w, which should then be systematically considered in the control design. Such a variable constraint on u′ is a slight disadvantage of the time-free model formulation chosen in this paper. However, the controller presented in Section 4.3 can perfectly cope with variable constraints. The constraint (4b) is formulated in a continuous manner. In this continuous format, it is also implemented as an add-on to the following discrete controllers. The zeroorder-hold output of the respective control law is simply restricted to the continuous constraints (4). In Section 4.3, these constraints are additionally implemented in the discrete control law itself.

10 7.5 unstable ws (m) 5 2.5 stable 0

20

0

and converted to a discrete transfer function (z-transfer function) using the bilinear transform with the sampling period ws = 0.1 m. The first form of C(s) given in (5) uses the parameters VP (proportional gain), VD (differential gain), and Ω > 0 and proved useful for parameter tuning. The second form of C(s) given in (5) is often referred to as lead compensator (η ∈ (0, 1)) or lag compensator (η > 1) and facilitates a simple stability analysis. In this work, only the case η ∈ (0, 1] is considered. The identities VD = VP T (1 − η) and Ω = 1/(ηT ) can be used for conversion between the two forms. P-control is obtained if VD = 0 or equivalently η = 1, PD-control otherwise. u

Controller

G(s) Plant

y + n +

Fig. 5. Basic output-feedback control loop. Assume that G(s) is the bilinear transform of the ztransfer function corresponding to (3). Note that, G(s) �= y˜(¯ s)/˜ u(¯ s) according to (2). The closed-loop system is stable if and only if the transfer functions C(s)/(1 + C(s)G(s)) and G(s)/(1 + C(s)G(s)) are BIBO stable (Franklin et al., 2015). For PD-control (η ∈ (0, 1)), T > 0 and Routh’s test yield that the closed-loop system is stable if the conditions VP > 0 (6a) (6b) ws < 2(T + r + l − h) ws < 2 ws < 2

ηl VP k

+ r(l − h)

r+l−h T (r + l − h) + r(l − h) +

(6c) l VP k

T +r+l−h √ b − b2 − 4ac if b2 − 4ac ≥ 0 ws < 2a

20

0.5

1 VP (-)

1.5

2

Fig. 6. Stability regions for P- and PD-control. with the abbreviations a=T +r+l−h  l  b = 2 (T + r + l − h)2 + r(l − h) + VP k  c = 4 (T + r) (T r + (l − h)(T + r + l − h))  l (T (1 − η) + r + l − h) + VP k are satisfied. For P-control (η = 1), (6) must hold for arbitrary values of T . In fact, satisfaction of (6a)–(6c) with T = 0 is sufficient. The stability regions according to (6) are shown in Fig. 6. Fig. 7 shows the simulated plant outputs and control inputs of the closed-loop system for various controller parameterizations. In the considered scenario, the initial state is x0 = [0.05 m, 0.02 m]T and the reference signal yd jumps at w = 100 m from 0 m to 0.05 m to demonstrate the performance of the controllers. All PD-controllers use 1/Ω = 2.45 m. The P-controller tends to cause overshoots of y. The results of the PD-controller show that the differential gain reduces the overshoots of y. For all controllers, 0.06 0.04 y (m) 0.02

VP = 1.6, VP = 0.4, VP = 0.8, VP = 1.6, Reference

0

VD = 0 m VD = 2 m VD = 4 m VD = 8 m value yd

0

50

100 w (m)

150

200

0

50

100 w (m)

150

200

a) 0.04 0.02 u (m) 0 −0.02 −0.04 b)

140

15

stable 0

b)

(6d) (6e)

10 VP (-)

unstable

15 ws (m) 10 5

Basic output-feedback controllers (cf. Fig. 5) are simple to design and only require a measurement of the output y. In particular, they do not need the system state x to be known. In this work, discrete feedback controllers with proportional (P) and differential (D) gain are considered. The controller transfer function in the frequency domain for the discrete-time case can be formulated as s 1 + sT C(s) = VP + VD , (5) s = VP 1+ Ω 1 + sηT

C(s)

5

= 0.20, T = 2 m = 0.20, T = 4 m = 0.20, T = 6 m = 0.20, T = 8 m = 0.47, T = 2 m = 0.47, T = 4 m = 0.47, T = 6 m = 0.47, T = 8 m = 0.73, T = 2 m = 0.73, T = 4 m = 0.73, T = 6 m = 0.73, T = 8 m = 1, P-control

25

4.1 P- and PD-Control

yd e + −

0

a)

η η η η η η η η η η η η η

Fig. 7. Results for P- and PD-control without noise.

IFAC MMM 2015 August 25-28, 2015. Oulu, Finland

A. Steinboeck et al. / IFAC-PapersOnLine 48-17 (2015) 137–142

0.06

0.04

0.04 y (m) 0.02

VP = 1.6, VP = 0.4, VP = 0.8, VP = 1.6, Reference

0 −0.02

141

0

100 w (m)

50

a)

0.02 y (m) 0

VD = 0 m VD = 2 m VD = 4 m VD = 8 m value yd

150

zC = 0.975 zC = 0.980 zC = 0.985 zC = 0.990 Reference value yd

−0.02

200

0

50

100 w (m)

150

200

0

50

100 w (m)

150

200

a)

0.04 0.02 u (m) 0 −0.02 −0.04

0.04 0.02 u (m) 0 −0.02 −0.04 0

50

100 w (m)

b)

150

200 b)

Fig. 10. Results for state-feedback control without noise.

Fig. 8. Results for P- and PD-control with noise. the constraint (4b) is active at the beginning of the scenario (non-zero initial state) and at the jump of yd . The same scenario was simulated a second time with a random disturbance input n (cf. Fig. 5), e. g., measurement noise, to analyze the robustness of the controller. The results are shown in Fig. 8. The disturbance n was chosen to be uniformly distributed in the range [−0.01 m, 0.01 m]. Despite this unrealistically large disturbance, the controllers still achieve a good performance. 4.2 State-Space-Pole Placement In this section, a linear state-feedback controller is designed for the discrete system (3) with the output equation (1b) by means of pole placement. Based on the reachability matrix, it can be easily shown that the (unconstrained) system is fully reachable. A state-feedback control loop ˆ i from an observer is shown that uses the estimated state x in Fig. 9. In this paper, the observer from Section 3 with the eigenvalue zO = 0.95 is used. The control law uses the sampling period ws = 0.1 m and reads as T ˆ i + gC yd,i . ui = −kC x (7) The feedback gain kC is computed according to Ackermann’s formula (Franklin et al., 2015) setting both eigenvalues to the same value zC taken from the set {0.975, 0.98, 0.985, 0.99}. The scalar input gain gC is computed to have a unity steady-state gain from the reference input yd to the output y. yd Controller ˆ x

u

Plant

Observer

y + n +

0.06 0.04 zC = 0.975 zC = 0.980 zC = 0.985 zC = 0.990 Reference value yd

0.02 y (m) 0 −0.02 0

50

100 w (m)

150

200

0

50

100 w (m)

150

200

a) 0.04 0.02 u (m) 0 −0.02 −0.04 b)

Fig. 11. Results for state-feedback control with noise. It turns out that the observer has little influence on the achieved control performance. Its eigenvalue zO = 0.95 is smaller (faster) than zC ∈ {0.975, 0.98, 0.985, 0.99} and the estimation error is generally small. Due to this benign behavior of the estimation error, the observer does not have a negative effect on the overall control performance. 4.3 Model Predictive Control In contrast to all controllers discussed so far, model predictive control (MPC) can systematically incorporate constraints. In this section, the discrete system (3) with the output equation (1b) and the input constraints (4) is controlled by linear MPC using the optimization problem

Fig. 9. State-feedback control loop. Figs. 10 and 11 show the simulated plant outputs and control inputs of the closed-loop system according to Fig. 9 with the control law (7) for various controller parameterizations and the same scenarios (initial state, reference signal, disturbance input) that were also considered in Section 4.1. If the eigenvalue zC is chosen too small, the constraint (4b) is frequently active, which entails overshoots of y. Clearly, without such constraints, the chosen eigenvalues zC ∈ {0.975, 0.98, 0.985, 0.99} would not cause any overshoots. 141

min J(xi , ui ) =

ui ∈RN

N  j=1

qy,j (yi+j − yd,i+j )2 + qu u2i+j

(8a)

s. t. (3), (1b), (4). (8b) Here, the integer N is the length of the prediction horizon, ui = [ui+1 , . . . , ui+N ]T is the vector of inputs during this horizon, and qy,j ≥ 0 and qu,j ≥ 0 are weighting factors. The constraint (4b) is implemented in a discrete manner using the approximation u′i = (ui − ui−1 )/ws . The problem (8) can be easily rewritten as a standard

IFAC MMM 2015 142 August 25-28, 2015. Oulu, Finland

A. Steinboeck et al. / IFAC-PapersOnLine 48-17 (2015) 137–142

0.04 y (m) 0.02

N = 50, qu = 0.01 N = 100, qu = 0.01 N = 50, qu = 0.10 N = 100, qu = 0.10 Reference value yd

0 0

50

a)

150

100 w (m)

200

0.04 0.02 u (m) 0 −0.02 −0.04

ACKNOWLEDGEMENTS 0

50

b)

100 w (m)

150

200

Fig. 12. Results for MPC without noise. 0.06 0.04 y (m) 0.02

N = 50, qu = 0.01 N = 100, qu = 0.01 N = 50, qu = 0.10 N = 100, qu = 0.10 Reference value yd

0 −0.02 0

50

a)

100 w (m)

150

200

0.04 0.02 u (m) 0 −0.02 −0.04 0 b)

were developed. The model features three novelties: It is a double integrator. It does not rely on zero-slip assumptions, which are especially questionable for high strip tension and high lateral shear and bending stiffness (wide strip). Its independent variable is the accumulated strip length instead of the time. To steer the strip, outputfeedback controllers, state-feedback controllers, and constrained MPC were designed. A simulated example scenario demonstrated that MPC gives a slightly better control performance than the other controllers. One reason for this superiority is the systematic consideration of constraints in MPC.

50

100 w (m)

150

200

Fig. 13. Results for MPC with noise. constrained quadratic program, for which highly efficient solution algorithms exist (Nocedal and Wright, 2006). The controller is also used in the control loop shown in Fig. 9. At the discrete index i, the first element of the optimized solution ui is applied to the plant. Instead of the ˆ i provided by the reducedinitial state xi , the estimate x order observer from Section 3 is used. Figs. 12 and 13 show the simulated plant outputs and control inputs of the closed-loop system according to Fig. 9 with MPC for various controller parameterizations and the same scenarios (initial state, reference signal, disturbance input) that were also considered in Section 4.1. The values N and qu are given in the figures. The values qy,j = 0.1 + j0.9/N , which were used in all cases, ensure prescient trajectory planning and a well-shaped step response. A longer prediction horizon (N = 100) reduces or avoids overshoots of y. Compared to the controllers designed in the previous sections, MPC reduces overshoots of y and it can better cope with constraints. It is the only controller that uses the idea of feedforward (output y raised before the jump of yd ). 5. CONCLUSIONS For a strip steering system with a guide roller at the entry, a dynamical model, an observer, and several controllers 142

The first author gratefully acknowledges financial support provided by the Austrian Academy of Sciences in the form of an APART-fellowship at the Automation and Control Institute of Vienna University of Technology. The second author gratefully acknowledges financial support provided by the Austrian Federal Ministry of Science, Research and Economy, the National Foundation for Research, Technology and Development. Thanks are also addressed to Andritz AG, Metals Division for posing the problem and providing measurement data. REFERENCES Franklin, G., Powell, J., and Emami-Naeini, A. (2015). Feedback Control of Dynamic Systems. Pearson, Upper Saddle River, NJ, 7th edition. Ho, T., Shin, H., and Lee, S. (2008). Development of a lateral control simulation software for roll-to-roll systems. In Proceedings of the 17th World Congress of the International Federation of Automatic Control (IFAC), Seoul, Korea, 11871–11876. Nocedal, J. and Wright, S. (2006). Numerical Optimization. Springer Series in Operations Research. Springer, New York, 2nd edition. Schulmeister, K. and Kozek, M. (2009). Modeling of lateral dynamics for an endless metal process belt. In Proceedings of the 6th Vienna Conference on Mathematical Modelling, Vienna, Austria, 1965–1973. Seshadri, A. and Pagilla, P. (2011). Adaptive control of web guides. In Preprints of the 18th World Congress of the International Federation of Automatic Control (IFAC), Milan, Italy, 8633–8638. Shelton, J. and Reid, K. (1971a). Lateral dynamics of a real moving web. Journal of Dynamic Systems, Measurement, and Control, 93(3), 180–186. Shelton, J. and Reid, K. (1971b). Lateral dynamics of an idealized moving web. Journal of Dynamic Systems, Measurement, and Control, 93(3), 187–192. Shin, K. and Hammond, J. (2008). Fundamentals of Signal Processing for Sound and Vibration Engineers. John Wiley & Sons, Chichester. Shin, K.H., Kwon, S.O., Kim, S.H., and Song, S.H. (2004). Feedforward control of the lateral position of a moving web using system identification. IEEE Transactions on Industry Applications, 40(6), 1637–1643. Wang, H., Logghe, D., and Miskin, D. (2005). Physical modelling and control of lateral web position for wallpaper making processes. Control Engineering Practice, 13(4), 401–412.