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ANALYSIS OF THE UNWIND SECTION OF AN INDUSTRIAL WEB PROCESSING LINE By BENJAMIN PACINI Bachelor of Science Mechanical and Aerospace Engineering University of Colorado at Colorado Springs Colorado Springs, CO, USA 2008 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE July, 2011 ANALYSIS OF THE UNWIND SECTION OF AN INDUSTRIAL WEB PROCESSING LINE Thesis Approved: Dr. Prabhakar R. Pagilla Thesis Advisor Dr. Gary E. Young Committe Member Dr. Lawrence L. Hoberock Committe Member Dr. Mark E. Payton Dean of the Graduate College ii ACKNOWLEDGMENTS There are several people I would like to thank for their contributions to my thesis. First and foremost I would like to express my greatest appreciation to my advisor Dr. Prabhakar R. Pagilla for his invaluable guidance, supervision, and friendship throughout my graduate studies. I am extremely grateful for his support, encouragement, and technical insights. I would like to extend my thanks to my master’s committee members: Dr. Gary E. Young for his friendship and guidance throughout my graduate studies and Dr. Lawrence L. Hoberock for his support and suggestions in completion of this work. Their guidance and understanding made the development of this thesis a positive learning experience. I would also like to thank my colleagues at Oklahoma State University: Pramod Raul, Muthappa PonjandaMadappa, Youwei Lu, Kadhim Jabbar, Shyam Konduri, Mauro Cimino, Carlo Branca, Aravind Seshadri, and Supraj Paleti. Their friendship and support (both technical and moral) have been an essential part of my graduate studies. I owe a special debt of gratitude to Jamie Lynch, Tim Gottlob, and the other employees at Armstrong World Industries for their invaluable assistance throughout the duration of this project. Additionally, I would like to thank the Oklahoma Center for the Advancement of Science and Technology (OCAST) and the National Science Foundation (NSF) for their funding throughout the project. iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Coating and Fusion Line . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Models, Control Strategies, and Evaluation of Parameters for the Unwind Section of a Coating and Fusion Line 10 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Simplification of the CFL . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Longitudinal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.1 Linearized Dynamics . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Parameter Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.1 Viscoelastic Parameter Evaluation . . . . . . . . . . . . . . . . 25 2.4.2 Friction Torque Evaluation . . . . . . . . . . . . . . . . . . . . 30 2.5 Current Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.1 RSLogix5000 Operation . . . . . . . . . . . . . . . . . . . . . 33 2.5.2 Unwind Roll Control Strategy . . . . . . . . . . . . . . . . . . 36 2.5.3 Pull Roll 1 and Unwind Accumulator Control . . . . . . . . . 40 2.5.4 Pull Roll 2 Control . . . . . . . . . . . . . . . . . . . . . . . . 43 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 iv 3 Analysis of Unwind Roll Control and Improvements 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.1 Parameter Values and Initial Conditions . . . . . . . . . . . . 48 3.2.2 Model Simulation Results . . . . . . . . . . . . . . . . . . . . 49 3.2.3 Measured Data From the CFL . . . . . . . . . . . . . . . . . . 55 3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3 Strategies For Improvement of the Existing Control Strategy . . . . . 64 3.3.1 Strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.2 Strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3.3 Strategy 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.4 Experimental Data Analysis . . . . . . . . . . . . . . . . . . . . . . . 84 3.4.1 Strategy 1 Experimental Results . . . . . . . . . . . . . . . . . 84 3.4.2 Strategy 2 Experimental Results . . . . . . . . . . . . . . . . . 93 3.4.3 Current Control Strategy Using Varying PID Gains . . . . . . 101 3.4.4 Strategy 3 Experiment Discussion . . . . . . . . . . . . . . . . 109 3.4.5 Summary of All Experiments . . . . . . . . . . . . . . . . . . 109 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4 Comparison of Torque and Velocity Control 115 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Stability Boundary Mapping . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.1 Stable to Unstable Transition . . . . . . . . . . . . . . . . . . 117 4.3 Controller Parameter Stability Regions for Velocity Control . . . . . . 119 4.3.1 Inner Velocity Loop Stability Boundary . . . . . . . . . . . . . 121 4.3.2 Outer Tension Loop Stability Boundary for Velocity Control . 124 4.4 Tension Loop Stability Boundary for Torque Control . . . . . . . . . 129 v 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5 Pull Roll 1 and Unwind Accumulator Analysis 134 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2 System Simulation Using the Current Control Strategy . . . . . . . . 135 5.2.1 Span Parameters and Initial Conditions . . . . . . . . . . . . . 138 5.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2.3 Model Verification Using Measured Data from the CFL . . . . 145 5.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.3 Improvement 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.3.1 Model Simulation Using Improvement 1 . . . . . . . . . . . . 151 5.3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6 Conclusions and Future Work 157 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 BIBLIOGRAPHY 162 vi LIST OF TABLES Table Page 2.1 Test 1 Parameters and Measurements . . . . . . . . . . . . . . . . . . 31 2.2 Test 2 Parameters and Measurements . . . . . . . . . . . . . . . . . . 32 2.3 Average Friction Torque . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Priorities and Periods of RSLogix Tasks . . . . . . . . . . . . . . . . 34 2.5 Parameter Values Used in RSLogix PIDs . . . . . . . . . . . . . . . . 35 3.1 Parameter Values Used in Simulation . . . . . . . . . . . . . . . . . . 49 3.2 Simulation Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . 50 4.1 Web Line Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.1 Parameters Used in the Simulation of the System Shown in Fig. 5.1 . 139 5.2 Controller Gains Used in the Simulation of the System Shown in Fig. 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.3 Initial Conditions Used in the Simulation of the System Shown in Fig. 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 vii LIST OF FIGURES Figure Page 1.1 Control system with outer tension loop and inner velocity loop . . . . 2 1.2 Control system with tension loop only . . . . . . . . . . . . . . . . . 2 1.3 Unwind section of the CFL . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Simplified model of Unwind Roll to Pull Roll 2 . . . . . . . . . . . . . 11 2.2 Maxwell element in parallel with a linear spring . . . . . . . . . . . . 14 2.3 Two span accumulator with control volume . . . . . . . . . . . . . . . 15 2.4 Displaced dancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Zero, vertical, and displaced positions of Dancer 1 . . . . . . . . . . . 20 2.6 Free body diagram of Dancer 1 . . . . . . . . . . . . . . . . . . . . . 21 2.7 Section of web line for linearized dynamics . . . . . . . . . . . . . . . 23 2.8 Typical stress versus time for material during tensile testing . . . . . 29 2.9 Stress versus time for measured data and bestfit model . . . . . . . . 30 2.10 Execution times of each task[5] . . . . . . . . . . . . . . . . . . . . . 34 2.11 Block diagram of RS Logix PID . . . . . . . . . . . . . . . . . . . . . 35 2.12 Control strategy for Unwind Roll . . . . . . . . . . . . . . . . . . . . 36 2.13 Control strategy for Pull Roll 1 and Unwind Accumulator under normal operating conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.14 Control strategy for Pull Roll 1 and Unwind Accumulator while emptying 44 2.15 Control strategy for Pull Roll 1 and Unwind Accumulator while filling 44 2.16 Control strategy for Pull Roll 2 . . . . . . . . . . . . . . . . . . . . . 44 3.1 Velocity profile of Pull Roll 1 . . . . . . . . . . . . . . . . . . . . . . 47 viii 3.2 Unwind Roll radius (Model Simulation) . . . . . . . . . . . . . . . . . 50 3.3 Control variable for Span 1 with tension PI (Model Simulation) . . . 51 3.4 PI controller output percentage (Model Simulation) . . . . . . . . . . 51 3.5 Unwind Roll peripheral velocity (Model Simulation) . . . . . . . . . . 51 3.6 Span 1 tension (Model Simulation) . . . . . . . . . . . . . . . . . . . 52 3.7 Unwind Roll velocity during EF portion of roll (Model Simulation) . 54 3.8 Pull Roll 1 velocity using Brake 1 (Measured data) . . . . . . . . . . 56 3.9 Control variable using Brake 1 (Measured data) . . . . . . . . . . . . 56 3.10 Tension in Span 1 using Brake 1 (Measured data) . . . . . . . . . . . 57 3.11 Span 1 tension using controller with increased Ki,unw (Model Simulation) 59 3.12 Pull Roll 1 velocity using Brake 2 (Measured data) . . . . . . . . . . 60 3.13 Control variable using Brake 2 (Measured data) . . . . . . . . . . . . 60 3.14 Tension in Span 1 using Brake 2 (Measured data) . . . . . . . . . . . 60 3.15 Unwind roll velocity using Strategy 1 (Model Simulation) . . . . . . . 67 3.16 Control variable using Strategy 1 (Model Simulation) . . . . . . . . . 68 3.17 PI Output percentage using Strategy 1 (Model Simulation) . . . . . . 68 3.18 Tension using Strategy 1 (Model Simulation) . . . . . . . . . . . . . . 69 3.19 Unwind Roll velocity during EF portion of roll using Strategy 1 (Model Simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.20 Tension oscillation amplitudes still increase as the material roll depletes using Strategy 1 (Model Simulation) . . . . . . . . . . . . . . . . . . 71 3.21 Proportional gain as a function of Unwind Roll radius . . . . . . . . . 74 3.22 Proportional gain using Strategy 2 (Model Simulation) . . . . . . . . 75 3.23 Integral gain using Strategy 2 (Model Simulation) . . . . . . . . . . . 75 3.24 Unwind roll velocity using Strategy 2 (Model Simulation) . . . . . . . 76 3.25 Control variable using Strategy 2 (Model Simulation) . . . . . . . . . 76 3.26 PI Output percentage using Strategy 2 (Model Simulation) . . . . . . 77 ix 3.27 Tension using Strategy 2 (Model Simulation) . . . . . . . . . . . . . . 77 3.28 Rescaled Span 1 tension using Strategy 2 (Model Simulation) . . . . 78 3.29 PI Output percentage using Strategy 3 (Model Simulation) . . . . . . 81 3.30 Controller output using Strategy 3 (Model Simulation) . . . . . . . . 81 3.31 Tension using Strategy 3 (Model Simulation) . . . . . . . . . . . . . . 82 3.32 Unwind roll velocity using Strategy 3 (Model Simulation) . . . . . . . 82 3.33 Rescaled Span 1 tension using Strategy 3 (Model Simulation) . . . . 82 3.34 Tension using the current control strategy with Brake 1 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . 86 3.35 Pull Roll 1 velocity using the current control strategy with Brake 1 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . 86 3.36 Control variable using the current control strategy with Brake 1 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . . . . 87 3.37 Tension using Strategy 1 with Brake 1 . . . . . . . . . . . . . . . . . 87 3.38 Pull Roll 1 velocity using Strategy 1 with Brake 1 . . . . . . . . . . . 88 3.39 Control variable using Strategy 1 with Brake 1 . . . . . . . . . . . . . 88 3.40 Control variable for a typical roll using Strategy 1 and the current control strategy with Brake 1 . . . . . . . . . . . . . . . . . . . . . . 89 3.41 Tension using the current control strategy with Brake 2 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . 90 3.42 Pull Roll 1 velocity using the current control strategy with Brake 2 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . 90 3.43 Control variable using the current control strategy with Brake 2 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . . . . 91 3.44 Tension using Strategy 1 with Brake 2 . . . . . . . . . . . . . . . . . 91 3.45 Pull Roll 1 velocity using Strategy 1 with Brake 2 . . . . . . . . . . . 92 3.46 Control variable using Strategy 1 with Brake 2 . . . . . . . . . . . . . 92 x 3.47 Control variable for the current control strategy using Brake 1 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . . 94 3.48 Pull Roll 1 speed for the current control strategy using Brake 1 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . 94 3.49 Tension for current control strategy using Brake 1 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.50 Control variable for Strategy 2 using Brake 1 . . . . . . . . . . . . . . 95 3.51 Pull Roll 1 Speed for Strategy 2 using Brake 1 . . . . . . . . . . . . . 96 3.52 Tension for Strategy 2 using Brake 1 . . . . . . . . . . . . . . . . . . 96 3.53 Control variable for current control strategy using Brake 2 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . . 98 3.54 Pull Roll 1 Speed for current control strategy using Brake 2 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . . 98 3.55 Tension for current control strategy using Brake 2 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.56 Control variable for Strategy 2 using Brake 2 . . . . . . . . . . . . . . 99 3.57 Pull Roll 1 speed for Strategy 2 using Brake 2 . . . . . . . . . . . . . 100 3.58 Tension for Strategy 2 using Brake 2 . . . . . . . . . . . . . . . . . . 100 3.59 Pull Roll 1 speed for the current control strategy using Brake 1 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . 102 3.60 Control variable for the current control strategy using Brake 1 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . . . 102 3.61 Tension for current control strategy using Brake 1 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.62 Pull Roll 1 Speed for Strategy 2a using Brake 1 . . . . . . . . . . . . 103 3.63 Control variable for Strategy 2a using Brake 1 . . . . . . . . . . . . . 104 3.64 Tension for Strategy 2a using Brake 1 . . . . . . . . . . . . . . . . . . 104 xi 3.65 Pull Roll 1 speed for the current control strategy using Brake 2 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . 105 3.66 Control variable for the current control strategy using Brake 2 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . . . 106 3.67 Tension for current control strategy using Brake 2 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.68 Pull Roll 1 Speed for Strategy 2a using Brake 2 . . . . . . . . . . . . 107 3.69 Control variable for Strategy 2a using Brake 2 . . . . . . . . . . . . . 107 3.70 Tension for Strategy 2a using Brake 2 . . . . . . . . . . . . . . . . . . 107 4.1 Block diagram of velocity controlled unwind roll with outer tension loop115 4.2 Block diagram of torque controlled unwind roll . . . . . . . . . . . . . 116 4.3 General block diagram of plant with controller . . . . . . . . . . . . . 116 4.4 Block diagram of inner velocity loop . . . . . . . . . . . . . . . . . . 121 4.5 Root invariant regions in KpvvKivv space for velocity loop (the number of unstable poles in each region is indicated) . . . . . . . . . . . . . . 124 4.6 Block Diagram of the Outer Tension Loop . . . . . . . . . . . . . . . 124 4.7 Root invariant regions in KptvKitv space for Kpvv = 1 and Kivv = ¯K ivv (the number of unstable poles in each region is indicated) . . . . . . . 128 4.8 Root invariant regions in KptvKitv space for Kpvv = 1 and Kivv > ¯K ivv (the number of unstable poles in each region is indicated) . . . . . . . 129 4.9 Root invariant regions in KptvKitv space for Kpvv = 1 and Kivv < ¯K ivv (the number of unstable poles in each region is indicated) . . . . . . . 130 4.10 Root invariant regions in KpttKitt space (the number of unstable poles in each region is indicated) . . . . . . . . . . . . . . . . . . . . . . . . 132 5.1 Simplified model of the CFL from Pull Roll 1 to Pull Roll 2 . . . . . 135 xii 5.2 Reference profiles for Pull Roll 1 speed and Unwind Accumulator carriage height and velocity (Model Simulation) . . . . . . . . . . . . . . 137 5.3 Pull Roll 1 speed and Unwind Accumulator carriage height and velocity (Model Simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.4 Dancer 1 position in percentage of maximum stroke (Model Simulation)142 5.5 Dancer 1 Trim scaling factor (Model Simulation) . . . . . . . . . . . . 142 5.6 Accumulator carriage speed during initial portion of the emptying process (Model Simulation) . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.7 Pull Roll 1 speed and Unwind Accumulator carriage height and velocity146 5.8 Dancer 1 position in percentage of maximum stroke . . . . . . . . . . 147 5.9 Dancer 1 Trim scaling factor . . . . . . . . . . . . . . . . . . . . . . . 147 5.10 Accumulator carriage speed during initial portion of the emptying process148 5.11 Reference profiles for Pull Roll 1 speed and Unwind Accumulator carriage height and velocity using Improvement 1 (Model Simulation) . . 152 5.12 Pull Roll 1 speed and Unwind Accumulator carriage height and velocity using Improvement 1 (Model Simulation) . . . . . . . . . . . . . . . . 153 5.13 Dancer 1 position in percentage of maximum stroke using Improvement 1 (Model Simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.14 Dancer 1 Trim scaling factor using Improvement 1 (Model Simulation) 154 xiii NOMENCLATURE Chapters 2 and 3 A Cross sectional area of the web b Damping constant in Maxwell element bw Lateral width of the web dx Dancer 1 trim E Spring constant in Maxwell element Ev Spring constant in Maxwell element F Force applied to keep Dancer 1 vertical Fc Interaction force between accumulator carriage and motor Fgc Gravitational force of accumulator carriage and rollers g Acceleration due to gravity Ji Inertia of element i Jc Equivalent inertia of accumulator motor, gears, carriage, and rollers Jmc Inertia of accumulator motor Ki,UW Integral PI gain Kp,UW Proportional PI gain Kd,D1 Derivative gain of Dancer 1 PID Ki,AC Integral gain of accumulator speed loop PI Ki,D1 Integral gain of Dancer 1 PID Ki,PR1 Integral gain of Pull Roll 1 speed loop PI Ki,PR2 Integral gain of Pull Roll 2 speed loop PI xiv Kp,AC Proportional gain of accumulator speed loop PI Kp,D1 Proportional gain of Dancer 1 PID Kp,PR1 Proportional gain of Pull Roll 1 speed loop PI Kp,PR2 Proportional gain of Pull Roll 2 speed loop PI Kpm Scale factor used during manual control for the Unwind Roll kn Unit conversion from Control Variable to brake pressure l Length from the dancer pivot to the dancer roller center lF Length from the dancer pivot to the point where F is applied Li Length of span i lL Length from the dancer pivot to the center of gravity of the dancer lever arm lx Length from the dancer pivot to the point where dancer linear displacement is measured MAXI Maximum process variable MINI Minimum process variable MAXS Maximum engineering unit scaling value MINS Minimum engineering unit scaling value MAXCV Maximum control variable MINCV Minimum control variable mc Mass of accumulator carriage n Number of stress measurements from tensile test N Number of stress measurements in random subset nc Gear ratio between accumulator motor angular velocity and carriage linear velocity NAc Number of spans in the Unwind Accumulator ni Gear ratio of the motor to roller i for driven rollers OPI,UW PI output percentage of the Unwind Roll controller xv OPID,D1 PID output percentage of the Dancer 1 PID r Ratio statistic used to determine steady state Rc0 Radius of spindle that Unwind Roll is would upon Rc Radius of roller in accumulator carriage i Ri Radius of Roller i RRMSi,RRMSfi ith root mean square of random subset and its filtered value RMSi ith root mean square RRMS Average root mean square of random subset Si Scale factors used in the PI controllers. i = t1, b ti Tension in span i t3i Tension of i th span in the accumulator tci Tension in span i of Unwind Accumulator tw Thickness of the web Ttix Torque due to x direction component of tension i about the dancer pivot Ttiy Torque due to y direction component of tension i about the dancer pivot tmeas Averaged measured tension used in the Unwind Roll control tr Reference tension ui Control input to roller i vi Velocity of roller i vc Velocity of the accumulator carriage vcr Reference velocity of the accumulator carriage viref , vir Velocity reference of roller i vls Line speed reference v1r Speed reference for Pull Roll 1 v1rr Ramped speed reference for Pull Roll 1 xvi v3i Velocity of i th roller in the accumulator xc Position of the accumulator carriage xt Measured linear displacement of the dancer roller xt,max Maximum linear displacement of Dancer 1 xtr Reference displacement of Dancer 1 i Angle from horizontal that tension i is applied on the dancer roller i Wrap angle of the web around the dancer from horizontal to the point where tension i is applied on the dancer roller i Filter factors 2 fi, 2 fi ith filtered numerator and denominator variance !i Angular velocity of Roller i !c Angular velocity of accumulator motor shaft w Density of the web ¯ Model value of stress during tensile test i Web stress of Span i mi ith Web stress measurement during tensile test 2n , 2 d Numerator and denominator variances mc Torque produced by the accumulator motor cmd,i Commanded torque of element i f Friction torque p Time constant for brake pneumatic device d Angular displacement of the dancer roll from vertical n Angular displacement of the dancer roll from dancer’s zero position to vertical "i Strain in span i "3i Strain of i th span in the accumulator xvii Chapter 4 A Cross sectional area of the web b Damping constant in Maxwell element bw Lateral width of the web dx Dancer 1 trim Cij Controller for loop i (tension/velocity) and control strategy j (velocity/torque) E Spring constant in Maxwell element Ev Spring constant in Maxwell element Evel Variational velocity error Ji Inertia of roller i Kijk Integral PI gain of loop j (tension/velocity) and control strategy k (velocity/torque) Kpjk Proportional PI gain of loop j (tension/velocity) and control strategy k (velocity/torque) L1 Length of Span 1 Ri Radius of roller i tr Reference tension vi Velocity of roller i vr Velocity reference rjk Real component of the characteristic equation of loop j (tension/velocity) and control strategy k (velocity/torque) ijk Imaginary component of the characteristic equation of loop j (tension/velocity) and control strategy k (velocity/torque) i ith time constant "i Strain in span i "r Reference strain xviii Chapter 5 dx Dancer 1 trim F Force applied to keep Dancer 1 vertical g Acceleration due to gravity Ji Inertia of element i Jc Inertia of accumulator motor, gears, carriage, and rollers Kd,D1 Derivative gain of Dancer 1 PID Ki,AC Integral gain of accumulator speed loop PI Ki,D1 Integral gain of Dancer 1 PID Ki,PR1 Integral gain of Pull Roll 1 speed loop PI Ki,PR2 Integral gain of Pull Roll 2 speed loop PI Kp,AC Proportional gain of accumulator speed loop PI Kp,D1 Proportional gain of Dancer 1 PID Kp,PR1 Proportional gain of Pull Roll 1 speed loop PI Kp,PR2 Proportional gain of Pull Roll 2 speed loop PI Li Length of span i mgc Mass of accumulator carriage and rollers nc Gear ratio between accumulator motor angular velocity and carriage linear velocity NAc Number of spans in the Unwind Accumulator ni Gear ratio of the motor to roller i for driven rollers Rc Radius of roller in accumulator carriage i Ri Radius of Roller i ti Tension in span i t3i Tension of i th span in the accumulator tr Reference tension ui Control input to roller i xix vi Velocity of roller i v3i Velocity of i th roller in the accumulator vc Velocity of the accumulator carriage vls Line speed reference vref , vr Velocity reference xc Position of the accumulator carriage xt Measured linear displacement of the dancer roller f Friction torque xx CHAPTER 1 Introduction 1.1 Background Any material that is produced in continuous flexible form in a rolltoroll fashion is known as web and the manufacturing of such materials is referred to as web handling. Many of the consumer products today, such as paper, diapers, textiles, and laminate flooring, are made in web form. In the manufacture of such products, control of the longitudinal dynamics is essential to ensure high quality goods. The most important facet of longitudinal control is maintaining web tension at appropriate values. If web tension is not well regulated, there is a high potential for damaging the product and the web handling machinery, resulting in extra costs for the manufacturing company and, consequently, higher prices for the consumer. A web line is the series of processes and components that are used in the production of the web material. The typical processes include printing, coating, heating, and cooling of the web, and these operations are essential in the manufacture of the product. The standard components of a web line include an unwinder, accumulators, pull rollers, idle rollers, dancers, loadcell rollers, a winder, and other machinery used in the processing of the web material such as ovens, printers, coaters, and heating/ cooling rollers. The unwind section of a web line consists of the elements that aid in dispensing web into the web line. These components include the unwinder, accumulator, pull rollers, idle rollers, dancers, and loadcell rollers and they are discussed below. The unwinder is an apparatus that contains a roll of web material (referred to as 1 Tension Controller Velocity Controller S S Velocity Dynamics Tension Dynamics +  + +  Reference Tension Reference Velocity Figure 1.1: Control system with outer tension loop and inner velocity loop Tension Controller S Velocity Dynamics Tension Dynamics +  Reference Tension Figure 1.2: Control system with tension loop only the unwind roll) and a corresponding control device. This is always the first element since it supplies the material for the entire web line. The control of this component can be achieved using either of the two methods presented below. 1. An outer loop that utilizes web tension feedback to provide a correction to the reference of an inner loop that controls the speed of the unwind roll (see Fig. 1.1) 2. A single tension loop that controls the torque applied to the unwind roll (see Fig. 1.2) The first approach uses a motor attached to the material roll whereas the latter utilizes only a brake. During the operation of the web line, the unwind roll will eventually be depleted and will need to be replaced. Additionally, the material from the new roll must be affixed to the previous web material in a process called splicing. There are two types of splicing: (1) a static method called the zerospeed splicing where both the previous and new webs are stationary when they are connected and (2) an “on 2 the fly” method where both webs are moving as they are attached to one another. This process provides the continuity in the manufacture of web product. However, during the changing of these rolls, the operation of the processing portion of the line must not be interrupted or else the web may be damaged. If the operation of the line is halted the web may overheat from being in an oven for too long or a coating may be too thick from being stopped at a printing station. To ensure the continuity of the web line, the accumulator is utilized. An accumulator is a structure that contains two sets of parallel idle rollers with one set fixed and the other on an extendable carriage. The carriage will move either vertically or horizontally depending on the size of the accumulator and the weight of the material. The web is alternately wound about a fixed and then a mobile roller so that the accumulator is able to either supply the rest of the line with web (in the case of an accumulator downstream of the unwinder) or receive web from the line (in the case of an accumulator upstream of the winder). In the former case, when the unwinder stops for a roll change, the accumulator carriage descends (contracts towards the fixed rollers) at a rate so that web is supplied at the correct speed. When the new roll is ready, the web upstream of the accumulator is driven faster than the web downstream which allows the carriage to move upwards (extend away from the fixed rollers), restoring the accumulator to its original height so that it is reset for the next roll change. The converse motions are seen for the case of the winder. Pull rolls are driven rollers that propel the web through the line. They are most often controlled in a similar fashion to the type (1) unwinder, with an outer tension loop that provides a correction to the reference for the inner speed loop. This speed reference modification alters the speed of the pull roll in order to correct a tension error. However, there is one type of pull roll that does not use a tension loop and is strictly under velocity control. This roller is called the master speed roller and it dictates the process speed for the entire line. Not every roller is driven, however. 3 Dancer 1 Master Speed Roller Pull Roll 1 Unwind Accumulator R30 Load Cell Unwind Rolls Figure 1.3: Unwind section of the CFL Those that are not driven are referred to as idle rollers, and their purpose is to support the web as it travels through the web line. There are two types of dancers, active and passive. Active dancers are rollers that displace (either linearly or rotationally) in order to vary the adjacent span lengths as a method to control tension. Passive dancers are components that use transducers to measure the displacement resulting from variations in tension, thus providing an indirect method for determining web tension. The motion can be either linear or rotational and for each type there is a normalizing force that is applied such that the passive dancer is at equilibrium in the nominal position when the tensions in the adjacent spans are at the reference value. Thus, if the tension changes, the balance of forces will be disrupted and the dancer will move. Load cells are elements that are also used for measuring web tension. They are attached to idle rollers and display the numerical value of the force the web is applying to the roller. 4 1.2 Coating and Fusion Line The focus of this thesis is on the unwind section of a Coating and Fusion Line (CFL) of the Armstrong World Industries plant located in Stillwater, Oklahoma. The CFL is the final web line in the manufacture of several different brands of laminate flooring material. The main purpose of this line is to apply a coating that provides a protective layer for the printed laminate material. The unwind section of the CFL is shown in Fig. 1.3 and includes two Unwind Rolls, Pull Roll 1, the 18span Unwind Accumulator, Dancer 1, and the Master Speed Roller (also known as Pull Roll 2). The two Unwind Rolls are alternately used to supply web for the rest of the line. They are the type (2) unwinders from the above discussion and each are controlled by separate brakes that utilize the same algorithm. Since the Unwind Rolls are brake controlled, Pull Roll 1 is their sole means of rotation; the Unwind Rolls rotate as Pull Roll 1 draws the web. This driven roller is controlled in the typical fashion and uses position measurements from Dancer 1 as feedback for the outer tension loop. Similar to Pull Roll 1, the Unwind Accumulator has a controller with an outer loop that uses tension feedback from Dancer 1 to correct the reference for the inner speed loop for the carriage. Dancer 1 is of the passive pendulum type but provides translational displacement data. This is accomplished by a linear transducer that measures the movement of a point on the pivot lever. The final component of the unwind section is Pull Roll 2. This is the master speed roller for the entire CFL and as such is only under velocity control. Note that the control structure utilized in all of the controllers are the ProportionalIntegralDerivative (PID) type. 1.3 Thesis Outline The main purpose of this thesis is to analyze and improve upon the current control strategies of the unwind section of the CFL. This web line was developed in an ad 5 hoc manner so the analysis contained herein gives a greater understanding of how the system operates. Additionally, the improvements suggested in this thesis will prove useful in increasing the functionality of the CFL and the overall quality of the flooring products. Chapter 2 discusses the longitudinal web dynamics, the current control strategies employed on the CFL, and the parameter evaluations. References [1], [2], [4], and [3] are used in the development of the dynamic equations. Reference [1], which discussed a method for decentralized control of a web line, provided the equations for the velocity of an unwind roll and for the rate of change of the unwind roll radius. Additionally, the dynamics for driven and idle rollers were presented. The velocity of the rollers within the accumulator were given in Reference [2], which compared an industrial accumulator controller to one developed using Lyapunov’s second method. The web used in the CFL is assumed to be viscoelastic, so Hooke’s Law cannot be used to relate stress and strain for a span with fixed length, as was done in Reference [1]. To describe the viscoelastic behavior, Reference [4] is utilized. This article detailed the relationship between web tension and strain for a viscoelastic material. Reference [3] is used to derive the strain dynamics for spans of varying length (i.e., for spans within an accumulator and those immediately adjacent to a dancer). After the dynamics are derived, the control strategies for the Unwind Roll, Pull Rolls 1 and 2, and the Unwind Accumulator are discussed. The tension loop calculations are performed in the RSLogix5000 software, the operation of which is detailed in References [5] and [6]. The evaluation of the friction term and viscoelastic parameters are also included in this chapter. The friction is assumed to be a constant resistive torque and is determined via a test where an idler roller is accelerated to a predetermined velocity and then is allowed to slow to a stop using only friction. The time required for this deceleration is recorded and used in the friction torque calculation. The viscoelastic parameters are determined using the results of tensile 6 tests of the flooring material from Reference [12]. A heuristic optimization method developed in Reference [9] is used to fit a viscoelastic stress equation to the tensile test data. The model parameters that produce a stress curve that best matches the tensile test data are selected. This method is similar to that used in Reference [7] which developed a procedure for modeling biological tissues. Reference [10] is used to determine the stopping criteria for the optimizer. In Chapter 3, the control strategy of the Unwind Roll is analyzed. First, a simplified system model containing the Unwind Roll and Pull Roll 1 is constructed and then verified by comparing simulation results with data collected from the CFL. Three new strategies are proposed in an effort to improve certain disadvantageous aspects of the current control strategy. The first strategy decreases the web tension sampling time and increases the resolution of the tension measurements and controller output. The second strategy utilizes timevarying PID gains and each of the modifications from the first strategy. The third strategy uses feedforward control with corrections provided by a PID in addition to the modifications from the first strategy. These improvements are verified by simulations and subsequently employed on the CFL. The results of these experiments are compared against the performance of the current strategy, and based on this, a recommendation that will improve the tension performance is given. Chapter 4 discusses the comparison between the type (1) and (2) unwinder control strategies. The control structure for each scenario is developed as well as their corresponding closed loop characteristic equation for the tension dynamics. The basis for this analysis is the stability regions of their respective controller parameter spaces. The procedure outlined in Reference [11] is used to map the stability boundary in the root space to each of the controller parameter spaces using their closed loop characteristic equations of the tension dynamics. Chapter 5 describes the analysis and improvements of the control strategies for 7 Pull Roll 1 and the Unwind Accumulator. First a simplified model is developed that includes the major elements from Pull Roll 1 to Pull Roll 2. This is compared with data measured from the CFL for model verification. Subsequently, one improvement is suggested that will increase the tension regulation performance. A simulation is then used to demonstrate its effectiveness. 1.4 Contributions The contributions of the work presented in this thesis are summarized below: • The control strategies for the Unwind Roll, Pull Roll 1, and Unwind Accumulator were summarized into block diagram form. This will prove useful for the operators of the CFL to increase their understanding of the operation of these components. • The disparity in the performance of the two brakes used to control the Unwind Roll was discovered during the work on this thesis. When using the same control algorithm, the controller output and tension performance for each brake can differ significantly. • Three strategies for improving the control of the Unwind Roll are presented and supported by simulations. Additionally, the first two strategies were implemented onto the CFL and showed improvement over the current strategy. Based on the results of the experimentation, the strategy that most effectively increases the tension performance in this portion of the CFL is proposed as the recommended controller. • While implementing the new control strategies for the Unwind Roll, the tension measurement resolution was increased by four times. This change was made permanent after the experimental employment of the first new strategy as it provides more accurate feedback for their controller. 8 • An algorithm to calculate the radius of the Unwind Roll was created in the controller software and is available for use. Each of the measurements required to compute this value were already accessible, however, through the implementation of the experiments, the radius calculation algorithm was generated. • One improvement was suggested for Pull Roll 1 and the Unwind Accumulator to decrease the motion of Dancer 1. This modification requires altering the ramp rate of Pull Roll 1 and altering the deceleration profile of Pull Roll 1 during the initiation of the emptying procedure. Simulations show that this change results in decreased dancer motion and hence improved tension regulation. 9 CHAPTER 2 Models, Control Strategies, and Evaluation of Parameters for the Unwind Section of a Coating and Fusion Line 2.1 Introduction This chapter details the development of a model for the portion of the Coating and Fusion Line (CFL) that includes all elements from the Unwind Roll to the Master Speed Roll. This includes the derivation of mathematical models that describe the dynamics of the web and of the web line components. Additionally, this model involves the evaluation of web parameters as well as the description of the control strategies employed on the CFL. In subsequent chapters, this model will be used to analyze the existing control strategies and will also be utilized in the development and evaluation of improvements to the control of specific elements of the CFL. The simplification of the unwind section of the CFL is presented in Section 2.2. Subsequently, Section 2.3 discusses the derivation of the strain, tension, and velocity equations of the web and the dynamic equations of certain components of the line (such as the dancer and the accumulator). Following this discussion, a description of the procedures used to evaluate the viscoelastic parameters and the friction torque is presented in Section 2.4. The control strategy for maintaining the web tension and velocity is then given for each of the controlled components in Section 2.5. Section 2.6 concludes this chapter with a discussion of the applicability of the material presented herein to the subsequent chapters. 10 2.2 Simplification of the CFL For simulation and analysis purposes, the portion of the CFL shown in Fig. 1.3 was simplified in Fig. 2.1, which shows the section of the CFL under consideration following several modifications. The first alteration is that only the major components of the line will be considered, meaning that the simplified model will only contain the Unwind Roll, Pull Roll 1, the Unwind Accumulator, Dancer 1, and the Master Speed Roll. Additionally, the load cell roll is shown with dashed lines to signify that it will provide tension feedback for Span 1 but not contribute directly to the dynamics of the system. As can be seen in Fig. 2.1, other rolls are also included (such as Accumulator Entry/Exit Rolls) in an attempt to match the actual configuration as accurately as possible. It should be noted that both the span lengths between components and the wrap angle of the web around the rollers will be maintained even though this is not depicted. The last simplifying assumption is that the tensions within the wound material roll and the span after the Master Speed Roll are set to the reference tension value for their corresponding portions of the CFL. Span 7 tr v0(t) Master Speed Roller (Pull Roll #2) Dancer #1 R30 Accumulator Exit Roll Accumulator Entry Roll Unwind Roll Span 5 Span 6 Span 31 Span 2 Span 1 Pull Roll #1 (Load Cell) v1(t) v2(t) v31 (t) v4(t) v5(t) v6(t) v7(t) v317 (t) Span 318 Figure 2.1: Simplified model of Unwind Roll to Pull Roll 2 11 2.3 Longitudinal Dynamics This section describes the tension, velocity, and strain dynamic equations for the portion of the CFL shown in Fig. 2.1. Equation (2.1) given below describes the web velocity dynamics at the Unwind Roller with f0 as the friction term which can take several forms based on the friction model [1]. v˙0(t) = t1(t)R2 0(t) J0(t) − n0R0(t)u0(t) J0(t) − f0R0(t) J0(t) − twv2 0(t) 2 J0(t) J0(t) R2 0(t) − 2 wbwR2 0(t) (2.1) where v0 is the peripheral velocity of the Unwind Roll, R0 is the radius of the Unwind Roll, t1 is the web tension in Span 1, J0 is the inertia of the Unwind Roll, n0 is the conversion between controller output and applied braking torque, u0 is the control torque applied to the Unwind Roll, tw is the web thickness, w is the web density, and bw is the lateral web width. Notice in Equation (2.1) that the radius, R0, and the inertia, J0, are shown to vary with time. The reason is because the material roll radius (and hence the inertia) becomes smaller as material is released into the web line. This occurs at a rate given by the following equation [1]. ˙R 0(t) − twv0(t) 2 R0(t) (2.2) The following equation describes the velocity dynamics for Rollers i = 1, 2, 4, 6, and 7 [1]. Note that rollers 2, 4, and 6 are idle rollers thus for i = 2, 4, and 6, ui(t) = 0. Ji Ri v˙i(t) = (ti+1(t) − ti(t))Ri + niui(t) − f (2.3) where Ji is the inertia of Roller i, Ri is the radius of Roller i, vi is the peripheral velocity of Roller i, ti (ti+1) is the web tension in Span i (i + 1), ni is the gear ratio 12 between the motor shaft and roller shaft of Roller i, ui is the control input for Roller i, and f is the constant friction torque applied to the roller. Equation (2.4) describes the velocity of the web at the roller within the accumulator for i = 1, ..., 17 [2]. In actuality, the dynamics depend on the accumulator carriage velocity. However it is assumed that it does not significantly affect the value of v3i(t) (the peripheral speed of Roller i within the accumulator) since the carriage velocity is much slower than that of v3i(t). Note that t318(t) t4(t). J3i R3i v˙3i(t) = (t3i+1(t) − t3i(t))R3i − f (2.4) where J3i is the inertia of Roller i within the accumulator, R3i is the radius of Roller i within the accumulator, and t3i (t3i+1) is the web tension in Span i (i + 1) within the accumulator. The velocity v5(t) (the peripheral speed of Dancer 1 roller) is dependent on the motion of the dancer roller. However, as in the case of the accumulator, the dancer roller motion does not significantly affect the velocity of the web, so it is ignored. The equation describing the dynamics of the web velocity at this location is given below [1]. J5 R5 v˙5(t) = (t6(t) − t5(t))R5 − f (2.5) where J5 is the inertia of Dancer 1 roller, R5 is the radius of Dancer 1 roller, and t5 and t6 are the web tensions in the spans upstream and downstream of Dancer 1, respectively. The tension dynamics were derived by assuming that the web material exhibits viscoelastic behavior. The viscoelastic characteristics are captured using the model shown in Fig. 2.2 which shows a Maxwell model in parallel with a linear spring where Ev is the spring constant for the Maxwell component, b is the damping constant for the Maxwell component, and E is also a spring constant [4]. The stress and strain in 13 the model are related through Equation (2.6) [4]. E Ev b Figure 2.2: Maxwell element in parallel with a linear spring 1 b i(t) + 1 Ev ˙ i(t) = E b "i(t) + 1 + E Ev "˙i(t) (2.6) where i is the web stress in Span i and "i is the web strain in Span i. Since i = ti/A (A is the web crosssectional area), the tension in Span i, for i = 1, ..., 7, is related to the strain by Equation (2.7). ˙ ti(t) = − Ev b ti(t) + EEvA b "i(t) + (Ev + E)A"˙i(t) (2.7) By using the law of conservation of mass around a control volume containing the span between two fixed adjacent rollers along with the assumptions that the strain is small and uniform along the span, the following relationship between the peripheral velocity of upstream and downstream rollers and the strain can be derived [4]. "˙i(t) = vi(t) Li (1 − "i(t)) − vi−1(t) Li (1 − "i−1(t)) (2.8) where Li is the length of Span i. Equation (2.8) describes the strain dynamics for i = 1, 2, and 7. The strain dynamics for the spans in the accumulator can be derived from Reference [3]. Assuming that the density and viscoelastic parameters are constant over the web’s cross section and assuming that the strain is small and constant along the 14 span, the conservation of mass relationship for the control volume shown in Fig. 2.3 can be written as shown in Equation (2.9) [3]. 3i1 (t) xc(t) ! " Control Volume " v3i1 (t) v3i (t) v3i+1 (t) 3i+1 3 (t) i (t) Figure 2.3: Two span accumulator with control volume "Z xc(t) 0 dx # d dt (1 − "3i(t)) + (1 − "3i(t)) d dt "Z xc(t) 0 dx # (2.9) = v3i−1(t) 1 − "3i−1(t) − v3i(t) [1 − "3i(t)] where xc is the accumulator carriage height and "3i ("3i−1) is the strain of Span i (i − 1) within the accumulator. Using Leibnitz rule to perform the differentiation of the second term on the left hand side of Equation (2.9), the strain dynamics for spans within the accumulator can be derived [3]. "˙3i(t) = (1 − "3i (t)) ˙xc(t) xc(t) + v3i (t) (1 − "3i (t)) xc(t) − 1 − "3i−1 (t) v3i−1 (t) xc(t) (2.10) The linear velocity of the accumulator carriage, vc, is related to the angular velocity of the motor that raises and lowers the carriage, !c, by Equation (2.11). vc(t) = nc!c(t) (2.11) 15 where nc has units of length. The motor dynamics is given by Equation (2.12) and the carriage dynamics is presented in Equations (2.13) and (2.14). Jcm!˙ c(t) = mc(t) − ncFc(t) (2.12) mc¨xc(t) = Fc(t) − XNAc i=1 t3i(t) − Fgc (2.13) x˙ c(t) = vc(t) (2.14) where Jcm is the inertia of the accumulator carriage motor, mc is the control torque applied to the accumulator motor shaft, Fc(t) is the interaction force between the motor and the carriage, mc is the mass of the accumulator carriage, NAc is the number of accumulator spans, and Fgc is the gravitational force of the accumulator carriage. Combining Equations (2.11) through (2.14) results in Equation (2.15) which shows the accumulator carriage dynamics reflected to the motor side. Note that Jc = Jcm+n2c mc. Jc!˙ c(t) = mc(t) − nc XNAc i=1 t3i(t) − ncFgc (2.15) To derive the equations for the strain dynamics for the spans immediately upstream and downstream of the dancer (Spans 5 and 6), the dancer is approximated as a two span accumulator. The strain equation for a span within an accumulator derived in Reference [3] will be used with the accumulator carriage position and velocity terms replaced with Li(t) and ˙L i(t), respectively, where i = 5, 6 for Spans 5 and 6, respectively. Thus the strain in Span 5 is given by Equation (2.16) [3] and that of Span 6 is presented in Equation (2.17) [3]. "˙5(t) = (1 − "5 (t)) ˙L 5 (t) L5(t) + v5 (t) (1 − "5 (t)) L5(t) − (1 − "4 (t)) v4 (t) L5(t) (2.16) "˙6(t) = (1 − "6 (t)) ˙L 6 (t) L6(t) + v6 (t) (1 − "6 (t)) L6(t) − (1 − "5 (t)) v5 (t) L6(t) (2.17) 16 The lengths L5(t) and L6(t) can be represented as nominal lengths plus varying lengths. The nominal length of Span 5, L5n, is distance AB from Fig. 2.4 and that of Span 6, L6n, is CD. Although the dancer motion is purely rotational, it is assumed that the angular displacement of the dancer from vertical, d(t), is small. Therefore, its displacement can be approximated as being the arc length from E to E0 (see Fig. 2.4). This distance in terms of d(t) is shown in Equation (2.18) below. EE0 = l d(t) (2.18) where l is the distance OE in Fig. 2.4. Thus, the total length of Spans 5 and 6 can be approximated as shown in Equations (2.19) and (2.20). L5(t) = L5n + l d(t) (2.19) L6(t) = L6n + l d(t) (2.20) The first time derivative of Equations (2.19) and (2.20) are given below. ˙L 5(t) = l ˙d(t) (2.21) ˙L 6(t) = l ˙d(t) (2.22) However, Dancer 1 does not have its zero position at vertical nor does it measure angular displacement; its zero position is a constant n clockwise from vertical (see Fig. 2.5). The dancer displacement is measured with a linear transducer which is placed a distance lx (distance OA in Fig. 2.5) down the lever arm from the pivot. This measured displacement, xt(t), is related to EE0 and d(t) by the following equation. d(t) = EE0 l = xt(t) − lx sin( n) lx (2.23) 17 Thus d(t) is related to xt(t) by the following. d(t) = xt(t) − lx sin( n) lx (2.24) The time derivative of Equation (2.24) is given below. ˙d(t) = x˙ t(t) lx (2.25) Thus, combing Equations (2.19) through (2.22) with (2.24) and (2.25), the total lengths of Spans 5 and 6 and their first time derivatives in terms of the measured displacement, xt(t), can be calculated as shown in Equations (2.26) through (2.29). L5(t) = L5n + l lx (xt(t) − lx sin( n)) (2.26) L6(t) = L6n + l lx (xt(t) − lx sin( n)) (2.27) ˙L 5(t) = l lx x˙ t(t) (2.28) ˙L 6(t) = l lx x˙ t(t) (2.29) Substituting Equations (2.26) and (2.28) into Equation (2.16) and Equations (2.27) and (2.29) into Equation (2.17) gives the strain dynamics in Spans 5 and 6 in terms of the measured dancer displacement and are given in Equations (2.30) and (2.31), respectively. "˙5(t) = (1 − "5 (t)) lx˙ t(t) L5n lx + lxt (t) − llx sin ( n) + (1 − "5(t)) lxv5 (t) L5nlx + lxt (t) − llx sin ( n) (2.30) − (1 − "4 (t)) lxv4 (t) L5n lx + lxt (t) − llx sin ( n) 18 "˙6(t) = (1 − "6 (t)) lx˙ t(t) L6n lx + lxt (t) − llx sin ( n) + (1 − "6(t)) lxv6 (t) L6nlx + lxt (t) − llx sin ( n) (2.31) − (1 − "5 (t)) lxv5 (t) L6n lx + lxt (t) − llx sin ( n) (t) d C A D B E B‘ E‘ D‘ 5 6 5 6 O Figure 2.4: Displaced dancer The dancer velocity dynamics are determined from the dancer free body diagram which is shown in Fig. 2.6. The constant applied force F is applied to keep the dancer vertical when the tension in Spans 5 and 6 are equal to the reference tension. This force is applied with a piston air cylinder device which is on a pivot so that F is not always completely horizontal. However, assuming small variations in tension such that the dancer movement is not significant, the direction of F can be approximated as being horizontal. Note that the web wrap angle of the dancer is not 180 degrees. This means that t5(t) and t6(t) are not applied horizontally. Thus there is a vertical component associated with each tension. It is also assumed that due to the small dancer movement, 19 O A A Xt(t) n(t) d(t) Dancer Zero Poistion Displaced Dancer Position Figure 2.5: Zero, vertical, and displaced positions of Dancer 1 the angles with respect to horizontal at which t5(t) and t6(t) are applied at Dancer 1 remain constant. Using Fig. 2.6, the torques due to the horizontal and vertical components of t5(t) and t6(t) about the pivot point of Dancer 1 (point O from Fig. 2.6) are calculated to be as follows. Tt5x = −(l cos( d(t)) + Rd sin( 5))t5(t) cos( 5) (2.32) Tt5y = −(l sin( d(t)) + Rd cos( 5))t5(t) sin( 5) (2.33) Tt6x = −(l cos( d(t)) − Rd sin( 6))t6(t) cos( 6) (2.34) Tt6y = (l sin( d(t)) + Rd cos( 6))t6(t) sin( 6) (2.35) where Tt5x is the torque at the dancer pivot (point O form Fig. 2.6) due to the horizontal component of t5(t), Tt5y is the torque at the dancer pivot due to the vertical component of t5(t), Tt6x is the torque at the dancer pivot due to the horizontal 20 F d (t) m g R m g L t 6 (t) t 5 (t) !6 !5 "5 "6 O A B C Figure 2.6: Free body diagram of Dancer 1 component of t6(t), Tt6y is the torque at the dancer pivot due to the vertical component of t6(t), i is the angle from horizontal that the tension in Span i is applied on the dancer roller, i is the wrap angle of the web around the dancer from horizontal to the point where the tension in Span i is applied on the dancer roller, Rd is the radius of the Dancer 1 roller, and l is distance OC in Fig. 2.6. Using the above relations and Fig. 2.6, the equation of motion of Dancer 1 about O can be derived. The equation of motion in terms of d(t) is as follows. 21 Jdt ¨ d(t) = FlF cos( d(t)) − lLmLg sin( d(t)) − lmRg sin( d(t)) + (l sin( d(t)) + Rd cos( 6))t6(t) sin( 6) − (l cos( d(t)) − Rd sin( 6))t6(t) cos( 6) − (l sin( d(t)) + Rd cos( 5))t5(t) sin( 5) − (l cos( d(t)) + Rd sin( 5))t5(t) cos( 5) (2.36) where Jdt is the inertia of Dancer 1, mL is the combined mass of both dancer pivot arms, mR is the mass of the Dancer 1 roller, and lL and lF are distances OA and OB, respectively. Equation (2.25) gives the first time derivative of d(t) in terms of x˙ t(t). Differentiating this equation again will result in Equation (2.37), the second derivative of d(t) in terms of ¨xt(t). ¨ d(t) = ¨xt(t) lx (2.37) Substituting Equations (2.24) and (2.37) into Equation (2.36) and assuming d(t) is small so that cos( d(t)) 1 and sin( d(t)) d(t) yields the equation of motion of Dancer 1 in terms of its measured linear displacement. This is given below. Jdt lx ¨xt(t) = FlF + −lLmLg − lmRg + l sin( 6)t6(t) − lt5(t) sin( 5) lx xt(t) + (−l sin( n) sin( 6) + Rd cos( 6) sin( 6) − l cos( 6) + Rd sin( 6) cos( 6))t6(t) + (l sin( n) sin( 5) − Rd cos( 5) sin( 5) − l cos( 5) − Rd sin( 5) cos( 5))t5(t) + lLmLg sin( n) + lmRg sin( n) (2.38) 2.3.1 Linearized Dynamics This section describes the linearization of Equations (2.1), (2.7), and (2.8) for the simplified model shown in Fig. 2.7. 22 v0(t) Unwind Roll Span 1 (Load Cell) Pull Roll #1 t1(t), 1(t) v1(t) vref Figure 2.7: Section of web line for linearized dynamics The first step is to assume that the Unwind Roll radius R0(t) is slowly changing so that R0(t) constant. Additionally, it is assumed that the reference velocity for Pull Roll 1, vr, is constant and the wound in strain, "0, is equal to the reference value and is also constant. Thus Equations (2.1) and (2.8) reduce to Equations (2.39) and (2.40). v˙0(t) = t1(t)R2 0 J0 − n0R0u0(t) J0 − f0R0 J0 (2.39) "˙1(t) = vr L1 (1 − "1(t)) − v0(t) L1 (1 − "r) (2.40) The second step is to define the variational dynamics by using the following relationships: v0(t) = V0(t) + vr, t1(t) = T1(t) + tr, and "1(t) = 1(t) + "r, where V0(t), T1(t), and 1(t) are, respectively, the velocity, tension, and strain deviations from their corresponding reference values. Additionally, u0(t) = U0(t) + u0r, where u0r is the control input required to maintain equilibrium and U0(t) is the deviation of the control input from the equilibrium value. With these substitutions, Equations (2.7), (2.39), and (2.40) become Equations (2.41) through (2.43). 23 ˙V 0(t) = R2 0 J0 (T1(t) + tr) + n0R0 J0 (U0(t) + u0r) − R0 J0 f (2.41) ˙T 1(t) = − Ev b (T1(t) + tr) + EEvA b ( 1(t) + "r) + (Ev + E)A˙ 1(t) (2.42) ˙ 1(t) = vr L1 (1 − 1(t) − "r) − V0(t) + v0r L1 (1 − "r) (2.43) At equilibrium, V0(t), ˙V 0(t), T1(t), ˙T 1(t), 1(t), ˙ 1(t), and U0(t) are all zero. Thus, the equilibrium conditions are as follows. u0r = −R0tr + f n0 (2.44) "r = tr EA (2.45) vr0 = vr (2.46) Inserting the equilibrium conditions into Equations (2.41) through (2.43) yields the linearized variational dynamics given below. ˙V 0(t) = R2 0 J0 T1(t) + n0R0 J0 U0(t) (2.47) ˙T 1(t) = − Ev b T1(t) + EEvA b 1(t) + (Ev + E)A˙ 1(t) (2.48) ˙ 1(t) = − vr L1 1(t) − (1 − "r) L1 V0(t) (2.49) 2.4 Parameter Evaluation This section describes the evaluation of the viscoelastic parameters and the friction torque found in the dynamic equations. 24 2.4.1 Viscoelastic Parameter Evaluation The terms E, Ev, and b are obtained by using nonlinear regression analysis to match a stress model derived from Equation (2.6) to data from tensile test results of two web materials. The tensile tests were conducted in Reference [12]. This procedure is similar to one conducted in Reference [7]. The two materials tested were the Felt, Hot Melt Calendar, Gel (FHG) composite and the Royelle Felt and Gel (RFG) composite. These materials were chosen because they are indicative of the materials that are used in the unwind section of the CFL. The tensile tests were conducted per ASTM D638. Stress Model Derivation Before the regression analysis can be discussed, the model of the web stress during the tensile tests, ¯ (t), must be derived. A tensile test consists of loading a strip of material in tension such that the strain rate is constant until the specimen ruptures [8]. After each test, the strain data was plotted versus time. These plots showed that all of the strain data had the form of a straight line with zero intercept and a slope of 0.0024 in/in s . Hence, Equation (2.6) becomes a first order differential equation as given below. 1 b ¯ (t) + 1 Ev ˙¯ (t) = 0.0024E b t + 0.0024 1 + E Ev (2.50) The solution to Equation (2.50) gives the web stress as a function of time during the tensile tests and is given below. ¯ (t) = 0.0024Et + 0.0024b − 0.0024be−Ev b t (2.51) Regression Analysis Now that the stress model has been chosen, the regression analysis procedure can be discussed. The objective of the regression analysis is to find the viscoelastic parame 25 ters such that min {E,Ev,b} J = vuut Xn i=1 (¯ i − mi)2 n (2.52) where mi is the ith stress measurement from the tensile test, ¯ i is the corresponding model stress value, and n is the total number of measurements. J represents the rootmeansquare (RMS) value between the measured data and the model values. ¯ i is calculated using Equation (2.51) and the time stamp corresponding to mi. The method chosen to solve the optimization statement given in Equation (2.52) is the Cyclic Heuristic Search [9]. The procedure for this method is as follows. Initial values for E, Ev, and b are randomly selected and are used to calculate the n ¯ i values. These are used in Equation (2.52) to determine the base objective function value, J0. E is then incremented by dE, the n ¯ i values are calculated using Equation (2.51), and the trial objective function value, Jt, is determined using Equation (2.52). If Jt < J0, then Jt becomes the new base objective function value and dE is increased. Otherwise, J0 remains the base objective function value and dE decreases in magnitude and changes sign. This procedure is repeated with Ev and b, which completes one optimization cycle. The above procedure is repeated until the stopping criterion is satisfied, thus yielding the E, Ev, and b set that best matches the model described by Equation (2.51) to the tensile test data. The selected stopping criterion for the optimization procedure defined in Reference [10], will now be described. The RMS between the measured data and the model will generally asymptotically decrease with each optimizer iteration to the minimum. Likewise, the RMS of a random sampling (RRMS) of a random subset of the data will also show this trend and will have random perturbations. Viewing the RRMS over progressive iterations gives the optimization process the appearance of a noisy system transitioning from transient to a steady state value. If the steady state condition is detected, then the minimum defined by Equation (2.52) has been found and 26 optimization iterations should cease. The method for determining steady state is summarized below. Let r, the ratio statistic that determines steady state, be defined as follows. r = 2n 2 d (2.53) where 2n is a moving average and 2 d is a measure of variance along the data trend [9]. At steady state, 2 d is an unbiased estimate, assuming both the data and noise are independently distributed. The expressions for these two variances are given below. 2n = 1 N − 1 XN i=1 (RRMSi − RRMS)2 (2.54) 2 d = 1 2(N − 1) XN i=1 (RRMSi − RRMSi−1)2 ! (2.55) where RRMSi is the RRMS of the ith optimization iteration and RRMS is the average RRMS over the past N optimization iterations. Note that these variances are calculated at each optimizer iteration. The calculations associated with Equations (2.54) and (2.55) present a computational burden. Thus, a technique which requires less computational effort was used. This technique uses exponentiallyweighted moving averages (firstorder filtered values) in place of the variances defined by Equations (2.54) and (2.55). The expressions for these exponentiallyweighted moving averages, 2 fi and 2 fi , are given below. 2 fi = 2 RRMSi − RRMSfi−1 2 + (1 − 2) 2 fi−1 (2.56) RRMSfi = 1RRMSi + (1 − 1)RRMSfi−1 (2.57) 2 fi = 3 (RRMSi − RRMSi−1)2 + (1 − 3) 2 fi−1 (2.58) 27 where 1, 2, and 3 are filter factors and are comparable to the inverse of N. Equations (2.56) and (2.58) replace Equations (2.54) and (2.55), respectively. Additionally, the filtered value, RRMSfi , is used instead of the RRMS. The equivalent of Equation (2.53) can now be written as the following. ri = (2 − 1) 2 fi 2 fi (2.59) At steady state, the expected value of r is unity, otherwise it is much larger. However, due to the noise presented by the RRMS calculation, it is possible for the optimizer to not be at steady state and yet have an r value near unity. To minimize the probability of accepting this steady state condition when it is not true, Reference [9] recommends using 1 = 2 = 3 = 0.05 with a critical rvalue of 0.8. Thus, at some optimizer iteration when the rvalue is below 0.8, it is assumed that steady state has been reached and thus the minimum has been found. The entire regression procedure can now be described. After the completion of each optimizer iteration, the RRMS is calculated and used in Equations (2.57) through (2.58). The ratio statistic for that iteration is calculated using Equation (2.59). If this value is less than 0.8, the optimizer has found the best possible E, Ev, and b set and thus stops iterating. Otherwise, the optimizer iterates again and the above procedure repeats until the ratio statistic drops below 0.8. Regression Analysis Results Five separate tensile tests were conducted on both FHG and RFG materials. A sample of a typical stress versus time curve for the tensile tests is shown in Fig. 2.8. There are three distinct regions. During Region 1, there is calibration error in the tensile test machine that occurs for each test. Region 2 is the period from the correction of the calibration error to the rupture of the specimen. Region 3 is the portion of the test after the specimen ruptures. Only the data from Region 2 was 28 0 10 20 30 −5 0 5 10 15 x 104 Time (sec) Stress (lbf/ft2) Region 1 Region 2 Region 3 Figure 2.8: Typical stress versus time for material during tensile testing used in the regression analysis as it is the only reliable data. In order to be certain that the best set of viscoelastic parameters has indeed been found, the required number of independent random starts of the regression analysis optimizer must be determined. From Reference [9], in order to be c confident that at least one of the best f × 100% results have been found, M independent random starts are required. The value of M is determined using Equation (2.60). M = integer ln(1 − c) ln(1 − f) (2.60) Thus, in order to be 95% confident that at least one of the best 10% results are found, the number of independent random starts of the regression analysis optimizer must be 28. Therefore, to determine the viscoelastic parameters, the regression analysis was performed 28 times for each of the ten sets of data. The viscoelastic parameter set that corresponded to the best results were chosen to be the “true” E, Ev, and b values. Best was defined as the smallest RMS value. The data that yielded the best results came from the FHG material. From the regression analysis, the viscoelastic parameters were determined to be: E = 9.75×105 29 lbf/ft2, Ev = 9.75 × 106 lbf/ft2, and b = 5.01 × 107 lbf·s/ft2. Figure 2.9 shows the data and the corresponding bestfit curve for the given viscoelastic parameters. As seen from the plot, the model fits the measured data well and thus the above E, Ev, and b values are sufficient to use in simulations. 0 2 4 6 8 10 0 5 10 15 x 104 Time (sec) Stress (lbf/ft2) Measured Data Model Data Figure 2.9: Stress versus time for measured data and bestfit model 2.4.2 Friction Torque Evaluation The friction present in the bearings of rollers has two components: a viscous friction that is proportional to velocity and a constant friction torque. For modeling purposes, it is assumed that the viscous friction is negligible and so the majority of the bearing friction is assumed to be the constant friction torque. In order to determine an accurate value for this term, a test was performed on two typical idle rollers used in CFL. For this test, the roller was hand spun to a speed higher than the desired test speed. A hand held tachometer was used to track the speed of the roller as it slowed due only to friction. Once the speed of the roller reached the desired test speed, a timer was started and the tachometer was removed from the roller. The time for the roller to come to a complete stop was measured for ten trials. The first test was performed on a 10.5 inch diameter roller, and all ten trials 30 Trial Initial Velocity Stopping Time Friction Torque fpm sec (ft · lbf) 1 99.9 55.28 0.309 2 100.01 60.57 0.283 3 100.99 58.14 0.297 4 98.78 61.04 0.277 5 100.02 58.05 0.295 6 100.93 60.00 0.288 7 100.73 60.00 0.287 8 100.66 61.07 0.282 9 100.80 55.93 0.309 10 99.81 56.49 0.302 Table 2.1: Test 1 Parameters and Measurements were performed at a test speed of 100 feet per minute (fpm). The second test was performed on a 9.5 inch diameter roller with six trials performed at a test speed of 100 fpm and four trials at 120 fpm. The results of the tests are presented in Tables 2.1 and 2.2. To determine the friction torque, the free body diagram of the roller is used. Since there were no other forces besides friction acting on the rollers, the equation of motion for the test rollers is as given in the following. J!(˙t) = − f (2.61) This can be approximated as J 4! 4t = − f (2.62) 31 Trial Initial Velocity Stopping Time Friction Torque fpm sec (ft · lbf) 1 99.41 64.83 0.226 2 100.46 60.83 0.244 3 100.17 65.12 0.227 4 99.38 73.36 0.200 5 99.45 71.02 0.207 6 100.98 73.77 0.202 7 120.92 90.80 0.197 8 120.10 85.95 0.206 9 119.93 87.32 0.203 10 120.21 87.84 0.202 Table 2.2: Test 2 Parameters and Measurements where J is the test roller inertia, 4! is the change in velocity of the roll in 4t stopping time, and f is the friction torque. Using classical methods for determining inertia, the friction torque can be solved for assuming that it is a constant value throughout the duration of each trial. Tables 2.1 and 2.2 show the calculated friction torques for each trial using the aforementioned calculation. The average friction torque values from both tests at each target speed are displayed in Table 2.3. As can be seen from the results, the friction torques for the 9.5 in idler roller are approximately equal even though they were run at different speeds. 32 Test Target Velocity Friction Torque (fpm) (ft · lbf) 1 100 0.293 2 100 0.218 120 0.202 Table 2.3: Average Friction Torque 2.5 Current Control Strategies 2.5.1 RSLogix5000 Operation Before the control strategy used in this section of the CFL can be discussed, the exact functionality of the RSLogix ladder logic and its PID controller must be detailed first. Ladder logic is a programming method that uses routines containing a series of rungs that have commands which are executed in order from top to bottom. The entire control strategy of the CFL is composed of multiple tasks consisting of several routines. The tasks involve performing safety checks, computing variables, and several other duties. Only one task can be executed at a time, so each is assigned a different execution period and a priority value from 1 to 15, where 1 is the highest priority and 15 is the lowest. Each task is implemented every period, but it can be interrupted by a higher priority task that happens to occur at the same time. When this happens, the higher priority task executes completely first followed by continuation of the lower priority task from where it was interrupted [5]. On the CFL, there are three tasks that are pertinent to this report: a task that governs the rewind section of the CFL (called the STI task), one that calculates the various reference values for variables of the CFL (called the Line References task), and another that controls the status of certain line components (called the Main task). The Unwind Roll uses an RSLogix PID for tension control which resides in 33 = Task Executed = Task Interrupted 0 10 20 30 40 50 60 70 80 90 STI Line References Main Time (ms) Figure 2.10: Execution times of each task[5] the Main task whereas the PID that controls tension for both Pull Roll 1 and the Unwind Accumulator (Dancer 1 PID) resides in the Line References task. Table 2.4 lists the priorities and execution periods of each of these tasks. Table 2.4: Priorities and Periods of RSLogix Tasks Task Priority Period STI 1 30 ms Line References 5 10 ms Main 6 15 ms Figure 2.10 shows a graphic representation of the times when each of the three aforementioned tasks are implemented. Notice that every 30 ms when each task is scheduled to be performed, the STI task interrupts the other two due to its higher priority. Additionally, the scheduled execution time is constant regardless of any intrusions by a higher priority task. Figure 2.11 shows the PID procedure used by RSLogix in block diagram form that is utilized for automatic control[6]. Table 2.5 shows the values of the Output Bias percentage, maximum (MAXI) and minimum (MINI) process variable (PV), maximum (MAXS) and minimum (MINS) engineering unit scaling value, and maximum (MAXCV) and minimum (MINCV) control variable (CV) value for the Unwind Roll and Dancer 1 controllers. The process variable is the measured feedback and the 34 control variable is the value output by the PID that is sent to an actuator[6]. +  PV + + MAXSMINS MAXIMINI Kp,RSL s K + i,RSL+KSP d,RSLs + + +  MINI MINS 100 MAXSMINS Error Output % MAXCVMINCV 100 MINCV CV Figure 2.11: Block diagram of RS Logix PID Table 2.5: Parameter Values Used in RSLogix PIDs Variable Unwinder PI Dancer 1 PID Bias 0 0 MAXI 100 100 MINI 0 0 MAXS 100 100 MINS 0 0 MAXCV 100 1.1 MINCV 0 0.9 The PID equation first uses the error between the SP and PV to calculate the output. Note that both the error and the output are expressed as percentages of the engineering unit range. The last step performed by the PID function as used in RSLogix is to convert the PID output percentage into the units of the control variable [6]. 35 + – Roller Dynamics Tension & Strain Dynamics Manual Control Time < 10 s Time > 10 s Automatic Control Kpm tr PI Output t1 v0 Round Control Variable Average St1 St1 Round Round Unwind Tension PI knSb tps+1 Figure 2.12: Control strategy for Unwind Roll 2.5.2 Unwind Roll Control Strategy The Unwind Roll torque is controlled by a brake. Therefore, the rotation of the roll is due to the material being pulled by Pull Roll 1. This means that there is no direct velocity control of the Unwind Roll. Thus, there is only tension control via braking for the portion of the web line spanning from the material roll to Pull Roll 1. The tension control strategy presently being employed is shown in Fig. 2.12. This scheme has two parts: manual control and automatic control. The manual control is utilized only for the first 10 seconds after the start up of Pull Roll 1 whereafter the automatic control is employed. When the switch from manual to automatic control occurs, the RSLogix software calculates the accumulated error required to produce the same CV output that is generated by the manual control. This process results in a smooth transition from open loop to closed loop control without causing a sudden increase or decrease in the CV [6]. During manual control, a scale factor, Kpm in Fig. 2.12, input by an operator multiplies the tension set point to produce the PI output percentage. For the simulations 36 conducted in Section 3.2, Kpm has a value of 0.6. One reason for implementing the manual control is that it allows RSLogix to acquire 10 tension measurements so that when automatic control is initiated, the tension averaging calculation (to be discussed later) will utilize actual tension measurements as opposed to using the measurements that were recorded before the Unwind Roll was initiated. For automatic control, web tension measurements taken by a load cell are used as feedback to produce the controller output. The tension measurements are sent to the RSLogix software as integer values where they are sampled once every second and then averaged with the past 9 tension samples. This mean, set as an integer value, is subtracted from the tension set point. This difference is used in the PI equation to produce the PI output percentage that, when rounded to the nearest integer, is the control variable. During automatic control when the actual tension and the set point are different, the integral term of the PI accumulates this error between the two values. At the instant when the error becomes zero (thus rendering the proportional term of the PI to be zero), the integral part of the PI is able to supply the output required to maintain the desired tension because of the accumulated error. One important fact to note is that the update time for the PI is 0.2 seconds so the PI continues to integrate in between the tension samples. Thus for a large error, the brake would apply an extreme (either high or low depending on the sign of the error) and continuously increasing or decreasing amount of torque to the Unwind Roll until the next sample is measured. The PI equation is displayed in Equation (2.64) below. The PI output percentage, computed either using the manual or the automatic method, is limited to 1% to 80%. By using Fig. 2.11 with Table 2.5, it may appear that the PI output percentage and control variable have the same numerical value. However, the control variable is set as an integer value and thus is the integer equivalent of the PI output percentage. The control variable corresponds to an out 37 put voltage. This voltage is then transformed into pressure via a voltagetopressure (E/P) transducer. The pressure is then applied to brake pucks which generate the braking torque on the Unwind Roll. In an effort to increase efficiency, two separate unwind structures, and thus two different brakes (designated Brake 1 and Brake 2), are used so while one is supplying the CFL with material, the other can be reloaded. These brakes are individually configured and thus the configuration of the two brakes can be different (as they most often are) which, as will be seen in Section 3.4, results in dissimilar tension performances and controller outputs for a given PID controller. The dynamics of the pneumatic device used to supply the pressure for the brakes are assumed to be first order of the following form. pp˙(t) + p(t) = knCV (t) (2.63) where p is the time constant for the pneumatic device, p(t) is the pressure within this device, kn is the unit conversion constant from units of control variable to psi, and CV (t) is the control variable value sent from the controller. In order to determine p, a simple test was conducted. A step increase was made in the CV and the time required for the pressure in the pneumatic device to reach its steady state value was recorded. From elementary systems analysis, dividing this rise time by four will result in the time constant of the device given an increase in CV. This procedure was repeated for a step decrease in CV. The resulting time constants from these two tests are shown below. p = 8>< >: 1.625 for CV increasing 0.625 for CV decreasing The determination of kn is discussed later. Note that there are several scaling factors in Fig. 2.12. The factor of St1 accounts for the fact that in the RSLogix program, the tension values are represented as percentages of 400 lbf, the maximum tension that can be induced in the web by the brake when a full material roll is attached. This means that the tension PI equation 38 for the Unwind Roll is given by Equation (2.64). OPI,UW = Kp,UW(St1(tr − tmeas(t))) + Ki,UW Z St1(tr − tmeas(t)) (2.64) where OPI,UW is the PI Output, Kp,UW is the proportional gain, Ki,UW is the integral gain, tr is the reference tension, and tmeas is the tension feedback. Since the CV is just the integer equivalent of the PI output percentage, the transformation from the latter to the former is shown in Fig. 2.12 as a rounding block. In order to find values for kn and Sb (the conversion from braking pressure to applied torque), the following information gathered from the control program was used. The PI is calibrated so that 100% of PI output (CV value of MAXCV for the Unwinder from Table 2.5) corresponds to 10 volts and the voltage to pressure transducer (E/P device) is set so that 10 volts corresponds to 46 pounds per square inch (psi) of pressure. Thus kn = (46/10) × (10/MAXCV ) = 46/MAXCV . To determine the amount of braking torque per psi of applied pressure, the following reasoning is used. It is assumed from the RSLogix files that 100% of PI output corresponds to 400 lbf of induced tension. Since the web tension effected by the brake varies with roll radius, it is assumed that this relation was determined using a full material roll. This assumption leads to Equation (2.65) which shows the relation between brake pressure and the corresponding applied torque. Sb = R0i × tmax Pmax (2.65) where R0i is the initial material roll radius, tmax is 400 lbf, and Pmax is the applied braking pressure corresponding to 100% of PI output. Although Equation (2.65) was derived presuming a full material roll, it is assumed that Sb is constant throughout the entire unwinding of the roll. In the model verification section (3.2) of Chapter 3, this value is adjusted in order to better match data measured from the CFL. 39 2.5.3 Pull Roll 1 and Unwind Accumulator Control Pull Roll 1 uses tension feedback and velocity control. The tension control is used to produce a correction to the velocity reference value. In the CFL, the tension feedback is provided by Dancer 1 which uses the linear transducer mentioned in Section 2.3 to measure the linear displacement of the dancer. This displacement value is then passed to the ladder logics program as a percentage of the maximum transducer stroke where it is compared with the reference dancer position within a PID in the software. The PID equation is given in Equation (2.66). OPID,D1(t) = Kp,D1ex(t) + Ki,D1 Z ex(t)dt + Kd,D1 dex(t) dt (2.66) where OPID,D1 is the PI Output, Kp,D1 is the proportional gain, Ki,D1 is the integral gain, Kd,D1 is the derivative gain, and ex(t) = 100 (xtr − xt(t)) /xtmax is the dancer displacement error as a percentage of the maximum transducer stroke. This output is scaled from 0.9 to 1.1 which will provide ±10% trim to the Pull Roll 1 velocity. This calculation is shown in Equation (2.67) below. dx(t) = OPID,D1(t) MAXCVD1 −MINCVD1 100 +MINCVD1 (2.67) where dx(t), referred to as Dancer 1 Trim, is the output of the PID and MAXCVD1 and MINCVD1 are the values shown in Table 2.5 for the Dancer 1 PID. Dancer 1 Trim is not the velocity correction, but a scaling factor that, when multiplied by the reference line speed (Pull Roll 2 speed reference), gives the corrected speed reference for Pull Roll 1 when the accumulator carriage is stationary. Equation (2.68) shows the values for the Pull Roll 1 reference speed, v1r for the various Unwind Accumulator phases (see discussion below). v1r(t) = 8>>>>< >>>>: vlsdx(t) if carriage is stationary vlsdx(t) + 50fpm if filling 0 if emptying (2.68) 40 where vls is the line reference speed. The speed reference that is sent to the drive is the ramped equivalent of the value determined from Equation (2.68). This ramped value is calculated at every scan of the “Line References” routine and is shown in Equation (2.69) below. v1rr(t) = 8>>>>< >>>>: v1rr(t − T) + 0.2 if v1r(t) > v1rr(t − T) v1rr(t − T) − 0.2 if v1r(t) < v1rr(t − T) 0 if 0.5s after emptying intiation (2.69) where v1rr(t) is called the ramped velocity reference and T is the sampling period. The velocity control for Pull Roll 1 is performed in a PID internal to a Rockwell Powerflex 700 drive attached to the motor which powers Pull Roll 1. This drive bases its PID calculations on motor speed and not web line velocity. Thus v1r(t) needs to be converted to motor speed, which is accomplished using the gear ratio between the drive motor and the attached roller. The drive uses speed feedback from an encoder which is attached to the motor. This value is compared to the speed reference calculated above to produce a speed error which is sent to the PID inside the drive. The PID will then provide the appropriate amount of torque in order to drive Pull Roll 1 at the desired velocity. The control strategy of the Unwind Accumulator is as follows. When the current roll is near depletion and must be replaced, the accumulator must empty in order to supply the rest of the line with material. To begin this process, an operator presses the Unwinder Stop Push Button when the material roll is empty. In actuality, this button stops Pull Roll 1 (making its speed reference equal to zero) since it directly controls the speed of the material roll. To stop the emptying process, an operator presses the Unwinder Start Push Button (i.e., Pull Roll 1 is started) when a new material roll is ready. At this stage, the accumulator capacity is low and needs to be replenished before the next roll change. In order to begin filling the accumulator, the Unwinder Start Push Button must be pressed first and then the Accumulator 41 Fill Button must also be pressed. During the filling process, Pull Roll 1 is driven 50 feet per minute (fpm) faster than the line reference speed, but it is still controlled as described above. The accumulator stops filling when it reaches a limit switch located at the top of the accumulator structure. During the emptying phase and the initial and final portions of the filling procedure, the accumulator uses tension feedback and carriage velocity control. When the carriage is moving, dancer position feedback is used to create a reference speed for the carriage in order to maintain the reference tension. This feedback is provided by Dancer 1. The calculated Dancer 1 Trim from Equation (2.67) is multiplied by the speed reference for Pull Roll 2. This product is then subtracted from the ramped speed reference for Pull Roll 1 for the corresponding accumulator phase. This difference, when divided by the number of accumulator spans, gives the speed reference for the accumulator carriage. The calculation of the reference speed for the carriage, vcr, is shown in Equation (2.70). vcr(t) = 1 NAc (v1rr(t) − vlsdx(t)) (2.70) Note that at the initiation of the emptying process, v1r(t) will be set to zero but the ramp defined by Equation (2.69) is allowed to work for another 0.5 seconds. This allows the accumulator carriage to accelerate per Equation (2.70). Afterwards, v1rr(t) is set to zero. As will be seen in Chapter 5, this sudden step in the ramped reference causes a speed mismatch between Pull Roll 1 and the Unwind Accumulator. Once Pull Roll 1 has accelerated to the nominal fill speed, the carriage will have a constant speed reference of 50/NAc since the vlsdx(t) terms from Equations (2.68) and (2.70) will cancel. Therefore, during this time, the accumulator will be under speed control only. The velocity control of the accumulator carriage is performed using a Powerflex 700 drive. The drive is attached to a motor which, through a series of gears and turn 42 +  0 FPM Unwind Accumulator Control Pull Roll 1 Control Sd 1N Roller Dynamics Tension & Strain PR2 Dynamics Spd Ref Velocity PID Dancer Dynamics Carriage Motor Dynamics Carriage Velocity PID Dancer 1PID n1 Nc Sd dx vc xtr v1 xt 1 Nc n1 + 1  Figure 2.13: Control strategy for Pull Roll 1 and Unwind Accumulator under normal operating conditions screws, raises and lowers the carriage. The feedback used internally in the drive is the encoder measured motor speed, not the carriage velocity. Thus, the calculated carriage reference speed as given by Equation (2.70) above is converted into motor speed using the gear ratio that determines the amount of linear displacement of the carriage per revolution of the drive motor. The feedback measured from the encoder is subtracted from this reference value. This error is then used in the drive’s PID in order to actuate the motor to the speed that will produce the desired carriage velocity. See Figs. 2.13 to 2.15 for the control strategies of Pull Roll 1 and the Unwind Accumulator under normal operating conditions, while emptying, and while filling. 2.5.4 Pull Roll 2 Control Pull Roll 2 is the Master Speed Roll for the entire CFL and therefore does not use tension feedback to control its motor speed. The only control utilized for Pull Roll 2 is the speed control performed internally in the Powerflex 700 drive. This drive operates exactly like the drive for Pull Roll 1 except the speed reference is not provided by a dancer but is a constant value that is input by an operator. See Fig. 2.16 for the block diagram depicting the control strategy for Pull Roll 2. 43 – Roller Dynamics Tension & Strain Dynamics PR2 Spd Ref Velocity PID Dancer Dynamics Carriage Motor Dynamics Carriage Velocity PID Dancer 1 PID 1N Unwind Accumulator Control Pull Roll 1 Control Sd Sd 1 n1 Nc dx vc xtr vr1! 0 FPM v1 xt +  " + " + 1 Nc + " n1 Figure 2.14: Control strategy for Pull Roll 1 and Unwind Accumulator while emptying +  + +  Pull Roll 1 Control +  Unwind Accumulator Control ! +  + vfill Sd Sd n1 1N Roller Dynamics Tension & Strain Dynamics PR2 Spd Ref Velocity PID Dancer Dynamics Carriage Motor Dynamics Carriage Velocity PID Dancer 1 PID xtr dx vc v1 xt ! ! 1 1 Nc ! Nc ! n1 Figure 2.15: Control strategy for Pull Roll 1 and Unwind Accumulator while filling +  Roller Dynamics Velocity n7 PID v 7 vr,PR2 1 n7 Figure 2.16: Control strategy for Pull Roll 2 44 2.6 Conclusion The equations, parameters, and control strategies presented herein will be utilized in the subsequent chapters. The dynamics and control strategies associated with the Unwind Roll are used in Chapters 3 and 4. In Chapter 3, this data will be utilized to develop a simulation model and derive improvements for the current control strategy. In Chapter 4, the linearized dynamics are employed in a technique that compares the stability regions for systems with velocity and torque controlled Unwind Rolls. The control strategies and dynamics corresponding to Pull Roll 1, the Unwind Accumulator, Dancer 1, and Pull Roll 2 are used in Chapter 5 to generate a model simulation which is utilized to analyze and improve upon the current control strategies for Pull Roll 1 and Unwind Accumulator. 45 CHAPTER 3 Analysis of Unwind Roll Control and Improvements 3.1 Introduction This chapter discusses the modeling and analysis of the existing control strategy of the Unwind Roll. The analysis will be based on a model simulation using the control strategy described in Subsection 2.5.2. The credibility of the model will be based on comparison of the model output with data measured from the CFL. Based on the deficiencies of the current control strategy, three strategies will be derived with the goal of improving the tension performance. Each of these strategies are simulated and compared with the results obtained using the model of the existing strategy. Experiments are then performed on the CFL to test the effectiveness of these modifications. In this chapter, Section 3.2 discusses the verification of the system model. The improvements to the existing strategy are presented in Section 3.3, followed by the results of the experimental implementation in Section 3.4. Section 3.5 concludes this chapter with a discussion of the results obtained herein. 3.2 Model Verification This section discusses the model verification for the system presented in Fig. 2.7. A model simulation is conducted that employs the control strategy for the Unwind Roll as described in Subsection 2.5.2 as well as the mathematical models given by Equations (2.1), (2.7), (2.8), and (2.63), which describe the relevant dynamics. It is 46 assumed that Pull Roll 1 is running at the reference velocity profile shown in Fig. 3.1. This figure was constructed by assuming that the Pull Roll 1 velocity is the speed reference that would be commanded if Dancer 1 is at its reference position. This simplification allows for the analysis of this section of the CFL. An Scurve instead of step changes in speed was also implemented in order to facilitate realistic speed changes. 0 500 1000 1500 0 50 100 150 O A B C D E F Time (sec) Speed Reference(fpm) Figure 3.1: Velocity profile of Pull Roll 1 This profile represents a scenario that shows all of the speed changes from the initiation of a new roll to roll depletion. The reference speed has six phases: initial startup to line speed (OA), holding constant at line speed (AB), ramping up to 50 feet per minute (fpm) greater than the line speed (BC), holding constant at 50 fpm plus line speed (CD), decelerating to line speed (DE), and holding constant at line speed until roll depletion (EF). The first two segments are used to advance the web splice that connects the new roll to the previous one through the accumulator. The line speed is maintained in AB in order to keep the accumulator carriage stationary since it will not move when the speed reference for Pull Roll 1 is equal to the line speed. This procedure is used because the web splice may not be able to endure the 47 increased stresses associated with the web passing through an accumulator when it is filling. Segments BC and CD fill the accumulator. This is possible since the web upstream is moving faster than the web downstream and thus the carriage must rise in order to maintain the web tension and velocity. At D, the accumulator is nearly full and thus Pull Roll 1 and the accumulator carriage must decelerate, a process that happens during DE. Segment EF shows the Pull Roll 1 speed reference when the accumulator carriage is at its maximum height and stationary. Since the carriage is not moving, the speed reference for Pull Roll 1 will be the line speed [2]. The time span during the constant velocity phases were approximated from data collected from the CFL. 3.2.1 Parameter Values and Initial Conditions Tables 3.1 and 3.2 show the parameter values and the initial conditions that were employed in the simulation. The constant friction torque acting on the Unwind Roll was determined from the friction torque test discussed in Subsection 2.4.2. This is an estimate since the friction test was conducted on idle rollers and a similar test was not performed on the Unwind Rolls. The controller gains currently used on the CFL were selected for Kp,unw and Ki,unw. The web thickness, tw, and density, w, were measured from CFL web samples. The length of Span 1, L1, and the lateral web width, bw, were obtained from a drawing of the physical line. Since the model shown in Fig. 2.7 ignored the idle rollers in this section of the CFL, the resonant frequencies introduced by these idle rollers are also ignored and an average tension model is used. Thus, L1 is selected to be the average length of the spans between the first Unwind Roll and Pull Roll 1. The web crosssectional area, A, was calculated as the product of tw and bw. E, Ev, and b are selected to be the values determined in Subsection 2.4.1. The initial Unwind Roll radius value was calculated by the following procedure. The length of material for one roll was determined by integrating the velocity profile 48 curve. As the volume of the material laying flat is the same as that of the wound material, this length was related to the initial material roll radius. This relationship leads to Equation (3.1), which gives the initial radius of the material roll. The initial tension of 12 lbf was selected because that is the same value as the measured data at the beginning of a new material roll. R0i = r Lwtw + R2 c0 (3.1) Table 3.1: Parameter Values Used in Simulation Variable Value Used in Simulation Units Kp,unw 0.5 None Ki,unw 0.005 sec−1 tw 0.003833 ft w 3.182 slug/ft3 bw 12.25 ft A 0.04696 ft2 E 9.75 × 105 lbf/ft2 Ev 9.75 × 106 lbf/ft2 b 5.01 × 107 lbf · s/ft2 L1 4.5 ft f0 0.293 ft · lbf 3.2.2 Model Simulation Results The results of the conducted model simulation are shown in Figs. 3.2 through 3.6. Note that three repetitions of the same simulation are shown for clarity when comparing these results with the measured data from the CFL. For simplicity, results of 49 Table 3.2: Simulation Initial Conditions Variable Initial Condition Unwind Roll Radius, R0 22 inches Unwind Roll Velocity, v0 0 fpm Span 1 Tension, t1 12 lbf 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 5 10 15 20 25 Unwind Radius, R 0 (in) Time (sec) Figure 3.2: Unwind Roll radius (Model Simulation) the model simulation are referred to as “simulated” data (i.e., the resulting tension data from the model simulation is referred to as the “simulated tension”). Figure 3.2 shows the material roll radius. The radius of the core that the material is wound upon is 6 inches and thus when R0(t) is equal to this value, the material roll will be completely depleted. The material roll is seen to be completely empty at the end of the simulation, as desired. Figure 3.3 shows the CV of the Span 1 tension PI controller. As expected, it is constant during the manual control phase. After an initial drop, the CV is seen to oscillate between values of 13 and 15 for approximately the first 300 seconds. Beyond this time, the CV begins to decrease since R0 is decreasing. A smaller Unwind Roll radius conveys that the material roll has a lower inertia and thus requires less braking torque to slow its rotation. This implies that it becomes easier to produce tension 50 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 50 Control Variable Time (sec) Figure 3.3: Control variable for Span 1 with tension PI (Model Simulation) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 50 PI Output (%) Time (sec) Figure 3.4: PI controller output percentage (Model Simulation) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 50 100 150 200 Unwind Roll Speed, v 0 (fpm) Time (sec) Figure 3.5: Unwind Roll peripheral velocity (Model Simulation) 51 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 Tension, t 1 (lbf) Time (sec) Actual Set Point Figure 3.6: Span 1 tension (Model Simulation) in Span 1 since the change in tension is a function of the upstream and downstream roller velocities. Moreover, braking torque induces tension in Span 1 by Equation (3.2) and hence as the material roll radius wanes, the braking torque required to effect the same amount of tension in Span 1 also reduces. braking = R0tind (3.2) where braking is the applied braking torque and tind is the corresponding induced tension. During the simulation, the CV oscillates at times of transition between subsequent values. This occurs since the CV can only take on integer values. In situations where less braking is required, the PI output has to decrease by an entire percentage before a change in the CV will occur. While the PI output is decreasing, the brake is applying approximately the same amount of resistive torque which means the tension will continue to rise. Once the PI Output reaches a value such that the CV changes, the change in braking torque may decrease too much, depending on the Unwind Roll radius. If this situation occurs, the tension will drop below the reference value, causing a rise in the PI output percentage, which, once it becomes large enough, will increase the CV to a value equal to or higher than it was initially. This cycle would repeat until the braking torque corresponding to the lower CV value is sufficient to keep the 52 tension at the desired value. As the radius decreases, these oscillations will occur more frequently since the drop in braking torque induces more tension as previously discussed. Hence it can be seen that these oscillations are due to the inability of the CV to take on the exact value calculated by the PI equation. The oscillations are also due to the slow reaction of the brake due to the dynamics of the pneumatic device. Once the CV changes, the pressure will also change but at a slower rate. This will cause errors in tension since the brake is not applying the amount of torque that is commanded by the controller. The controller will then alter the CV in order to correct for these tension errors, resulting in more CV oscillations. Comparing Figs. 3.1 and 3.5 shows that the Unwind Roll velocity generally follows the reference velocity relatively well. However, there are relevant differences during transitions from ramped to constant velocity and during the CV oscillations described above. Since the Unwind Roll rotation comes solely from Pull Roll 1, there is velocity error in the transition from constant velocity to acceleration of Pull Roll 1. At times when Pull Roll 1 decelerates (transitioning from BC to CD and from CD to DE from Fig. 3.1), the large inertia of the Unwind Roll causes it to overshoot the Pull Roll 1 velocity. This occurrence also contributes to the velocity error. Additionally, when the CV oscillates, the applied braking torque also oscillates causing deviations of the Unwind Roll velocity from the reference. As the roll radius decreases, the applied braking torque during the CV oscillations has a greater effect on the Unwind Roll velocity due to the smaller inertia. This causes the velocity oscillation amplitudes to grow as the simulation progresses as witnessed in Fig. 3.7 which shows the Unwind Roll velocity during the EF portion. The tension in Span 1 is displayed in Fig. 3.6. During the manual control phase, the tension starts at 12 lbf then increases to 193 lbf before returning to approximately 75 lbf. The reason for this trend is that at the initial start up, a large amount of tension is required to accelerate the stationary Unwind Roll. The tension drop after 53 400 500 600 700 800 900 1000 1100 1200 1300 1400 99 99.5 100 100.5 101 Unwind Roll Speed, v 0 (fpm) Time (sec) Actual Set Point Figure 3.7: Unwind Roll velocity during EF portion of roll (Model Simulation) the spike is a result of the Unwind Roll overshooting the velocity of Pull Roll 1 as explained above. Since the peripheral speed of the upstream roll is greater than that of the downstream roll, the tension in Span 1 decreases. The velocity error is also significant during the other Pull Roll 1 speed changes and transitions which causes the tension variation to be large during these times, as is seen in Fig. 3.6. The CV oscillations also cause fluctuations in tension. Since the pneumatic device is slow, the braking pressure is not able to match the changes in CV. Meanwhile, this changing pressure is causing the brake to vary the velocity of the Unwind Roll, as discussed above, which induces fluctuations in the tension. As was the trend with the Unwind Roll velocity, the tension oscillations become larger as the material roll radius decreases, a fact that is shown in Equation (3.2). Since changes in the CV correspond to similar changes in braking torque for both a larger roll and smaller roll, the induced tension will increase as the material roll decreases. The final trend seen in the simulated tension is that the tension drifts and does not oscillate about the reference value of 92 lbf but rather 100 lbf. In summary, the model simulation displayed characteristics expected from the system shown in Fig. 2.7 under the given conditions. To further verify this model, the above results are compared with measured data from the CFL in the subsequent 54 section. 3.2.3 Measured Data From the CFL As discussed in Subsection 2.5.2, two separate brakes are alternately used in the control of the Unwind Roll. Thus the measured data analysis is segregated into two sections, each comparing the measured data to the model simulated data. There are three sets of data for each brake. For Brake 1, the first data set was collected on a separate day than the data sets for the latter two sets. For Brake 2, each data set was collected from a different day. Thus the differences in the data for a given brake are attributed to several factors, namely, the Pull Roll 1 velocity profile, the web material, and the particular configuration of the brake. Brake 1 Figures 3.8 through 3.10 show the measured data for Brake 1. The beginning of each run is the instant that the Pull Roll 1 velocity increases from 0 fpm. Since the Unwind Roll velocity is not measured, Pull Roll 1 velocity is displayed so that the tension and control variable data can be correlated to the action of Pull Roll 1. The first roll operated at the same speeds that the simulations were conducted; however, it does have additional velocity changes during the portion where the Pull Roll 1 velocity is supposed to be near the line speed reference. The latter rolls have a profile shape similar to that of Fig. 3.1, but are ran at slower speeds. Despite these differences between these measured velocities and the velocity profile used in the model simulation, the comparison of the tension and control variable results are still valid. As expected, the control variable is constant for the first 10 seconds of each roll and then changes based upon the tension. Note that the constant portions at approximately 1500 seconds and 3400 seconds are observed because the controller provides 55 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 50 100 150 Speed (fpm) Time (sec) Figure 3.8: Pull Roll 1 velocity using Brake 1 (Measured data) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 50 Control Variable Time (sec) Figure 3.9: Control variable using Brake 1 (Measured data) a constant braking torque at both the beginning and at the end of material rolls. The large spike seen at the end of the first and third runs is because the brake pressure was very low (in the case of the former, the controller actually saturates), thus causing the tension to drop. However, due to the sampling of tension values and the slow sampling time, the controller cannot immediately react to these changes. Eventually, as the tension drops lower and lower, the current and accumulated errors becomes large, resulting in a drastic increase in the control variable. Except for the first data set, the control variable is seen to gradually rise to its maximum value and then continually and gradually decrease throughout the entire roll. These same trends are observed in 56 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 Tension (lbf) Time (sec) Actual Set Point Figure 3.10: Tension in Span 1 using Brake 1 (Measured data) the first data set, excluding the initial increase in control variable. For the particular system configuration, the initial CV was large enough to quickly increase the tension, eliminating the need to increase the CV once automatic control was initiated. Comparing Figs. 3.3 with 3.9 shows that the simulated controller follows the same trends as those seen in the first set of measured data; they both use similar magnitudes during the automatic control phase and they both continually decrease throughout the depletion of the Unwind Roll. Additionally, the CV jumps seen in the simulation data are also present in the first set of measured data. On the other hand, the model simulation did not accurately portray the CV trends seen in the second and third measured data sets. However, the difference is not significant since disparities in magnitude can be seen even between different sets of measured data. Thus for the simulated controller, the assumptions that related the controller output to the applied torque were for only a particular brake configuration. The measured tension data is shown in Fig. 3.10. Each data set begins with a large tension spike as Pull Roll 1 accelerates from 0 fpm, another spike as Pull Roll 1 accelerates from line speed, and then a drop in tension as Pull Roll 1 decelerates back to line speed after the accumulator has been filled. For the second and third data sets, the tension does not reach the reference value until approximately 130 57 and 160 seconds after initiation, respectively. The reason is that the slow increase in their respective control variables gradually increases the braking torque, causing the tension to slowly increase. This trend is not seen in the first data set because, as discussed previously, the initial CV was large enough to produce tension values near the set point. Near the depletion of their respective rolls, the first and third data sets are seen to decrease and the second data set begins oscillate with increasing amplitude. The reason for the former trend has been discussed already, however, note that although both control variables spiked at the end of their respective rolls, the tension did not respond. The reason is that the web had already released from the core and thus any changes in the brake would not effect the tension. For the second data set, the cause of the increasing oscillation amplitudes can be attributed to two factors: (1) the gains are not appropriate for the smaller roll size and (2) the CV fluctuations are inducing tension oscillations as discussed in Section 3.2.2. Comparing Figs. 3.6 and 3.10 illustrates that the simulated tension follows the general trends of the measured tension with a few exceptions. The simulation sufficiently modeled the large spikes during Pull Roll 1 speed transitions, but did not show the tension drop at the end of the roll that was present in the first and third data sets. However, the simulated model did display the increasing oscillation amplitudes seen in the second data set. The simulated data also has similar tension magnitudes seen in the first and second measured data sets. On the other hand, the measured data sets each oscillated about their corresponding reference values, a trend that the simulation was unable to accurately predict. However, consider Fig. 3.11 which shows the tension in Span 1 using the same model except with the integral gain five times larger. Note that the tension drift has been reduced and the tension oscillates about the set point. This indicates that the model is lacking some unknown scaling or gain that is present within the hardware. Additionally, the model did not accurately predict the oscillation frequency that was seen 58 in the measure data. One possible reason for this can attributed to the model relating the braking pressure to applied braking torque. The braking pressure inflates the brake pucks which apply a force on the Unwind Roll, causing the frictional torque. These dynamics were modeled using a constant gain, but the actual dynamics are more complicated and may exhibit stickslip phenomena due to the intermittent force applied by the brake pucks on the brake cylinder. Other possible factors contributing to the difference in oscillation frequency is the span length that was used in the model and that all of the idle rollers were ignored. The various spans between the Unwind Roll and load cell in the actual CFL contribute to the tension dynamics and can therefore influence the tension oscillation frequency observed in the load cell data. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 Tension, t 1 (lbf) Time (sec) Actual Set Point Figure 3.11: Span 1 tension using controller with increased Ki,unw (Model Simulation) Brake 2 Figures 3.12 through 3.14 show the measured data using Brake 2. The first and third data sets are operated at similar speeds that are seen in Fig. 3.1, while the second data set has lower speeds. All of the data sets have speed profiles that are similar to the simulated data. Thus, this data is comparable to the scenario presented in the simulation and can be used for model verification. The trends seen in the control variable for Brake 2 are very similar to Brake 1 59 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 50 100 150 Speed (fpm) Time (sec) Figure 3.12: Pull Roll 1 velocity using Brake 2 (Measured data) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 Control Variable Time (sec) Figure 3.13: Control variable using Brake 2 (Measured data) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 Tension (lbf) Time (sec) Actual Set Point Figure 3.14: Tension in Span 1 using Brake 2 (Measured data) 60 except for one major aspect. While using Brake 1, the CV slowly increased to a maximum and then gradually decreased throughout the roll. The CV for Brake 2, however, quickly rises to the maximum and does not have a gradual decrease; there are large durations where the control variable is nearly constant. This is explained by the brake being configured such that a small decrease in CV results in a large decrease in braking torque. This causes the CV fluctuations as the controller is attempting to provide the correct amount of torque. The simulated control variable shown in Fig. 3.3 is comparable to the first two measured data sets in Fig. 3.13. The first similarity is that the magnitudes shown in the simulation are relatively close to those seen in both of the measured data sets. Another trend that illustrates the correspondence is the presence of the CV oscillations. However, the simulated data gradually decreases throughout the entire roll and does not stay constant for large portions of time, a trend seen in the measured data. Additionally, the model simulation was not able to match the magnitudes observed in the third data set. However, as mentioned above, this is not a significant difference since the CV magnitude is seen to vary between the various measured data sets for the same brake. The tension data produced using Brake 2 followed similar patterns to those seen with Brake 1. There are large tension spikes during Pull Roll 1 acceleration and a drop in tension whenever Pull Roll 1 decelerates back to line speed after filling the accumulator. However, the tension reached the reference value much faster with Brake 2, and this can be attributed to the faster rise in the CV that was seen in the Brake 2 data. For the first data set, the tension begins to oscillate with increased amplitude as the material roll depletes. This trend is also slightly present in the second and third data sets, but is not as pronounced. The simulated tension data in Fig. 3.6 displays the general trends found in the 61 measured data in Fig. 3.14. As was the case with Brake 1, the simulated data matches the tension spikes at the Pull Roll 1 speed changes. Additionally, the oscillations of increasing amplitudes found in the measured tension were also present in the simulation. The time required for the simulated tension to reach the reference value was also similar to that observed in the measured data. However, as with Brake 1, the measured tension for Brake 2 was also able to oscillate about the reference value, a feat that was not accomplished in the simulated data. Additionally, the oscillation frequency observed in the measured tension was not accurately predicted by the model simulation. The reasons for these differences are the same as those described in the Brake 1 discussion. 3.2.4 Summary This analysis showed that the dynamic models, though unable to match the measured data perfectly, are able to predict several trends seen in the data collected from the CFL. The simulation was able to mimic the tension effects during the speed changes of Pull Roll 1 for both brakes. The model also produced tension values similar to those seen in the data. Another aspect present in the tension data of both brakes that was captured by the simulation were the oscillations of increasing amplitudes. Additionally, the time required for the simulated tension to rise to the reference value was similar to that observed in the measured data for both brakes. The control variable signal from the model simulations displayed the general attributes that were present in certain data sets of Brakes 1 and 2. The simulated CV showed continual decrease that was observed in each of the three data sets for Brake 1, but was only able to match the magnitudes of the first data set. Conversely, the model was able to match the CV magnitudes of the first two data sets of Brake 2, but did not show the large durations of nearly constant CV present in the Brake 2 data. Control variable fluctuations were also present in both brakes, a trend that was observed in 62 the simulated CV as well. However, there were aspects that were not modeled well by the simulation. One such characteristic is the steady state error observed in the tension of the model simulation. The measured tension for both brakes oscillated about the set point value, but the simulated tension did not. As described in the previous discussions, by increasing the integral gain in the model, the steady state error is reduced. This indicates that the model is missing a scaling or a gain factor. The measured tension oscillation frequencies were also not perfectly matched by the model. This can be attributed to three possible causes: the model relating braking pressure to friction torque, the span length used in the model simulation, and the absence of the idle rollers in the model. Another aspect that was not well predicted by the simulation was that the CV values for the second and third data sets for Brake 1 showed a large initial increase after the switch from manual control. Additionally, the CV magnitudes of these two data sets were not matched by the model simulation. For Brake 2, the measured CV showed large portions that were nearly constant, a trend not observed in the model simulation. The modeled CV also did not match the CV magnitude that was achieved in the third data set for Brake 2. Despite the shortcomings of the simulation, the major trends of the tension data were adequately predicted. The CV values between the two brakes are different as are the CV values for the three data sets for the same brake. Thus, it would not be possible for the simulation to match the CV trend of each data set for both brakes. However, the simulated CV did have certain attributes from each brake. When it did not accurately predict an aspect of one of the brakes, it matched this same characteristic aptly with the other. Therefore, the developed model sufficiently represents the portion of the CFL shown in Fig. 2.7. 63 3.3 Strategies For Improvement of the Existing Control Strategy This section contains a simulation based case study of three suggested improvements to the current control strategy employed to control the Unwind Roll. The first strategy is to decrease the tension measurement sampling time and increase the resolution of certain variables. The second strategy is similar to the first except that the PI gains will vary as a function of the radius. Lastly, the third strategy implements feedforward control action as well as the improvements employed in the second strategy. As discussed in Section 2.5, the current control algorithm for the Unwind Roll has the tension feedback and the control variable set as integers. Moreover, due to the scaling within the input modules, the resolution of the tension measurements is 0.25 values per lbf (vpl), where resolution is defined as given below. r = Nv R (3.3) where r is the resolution and Nv is the number of values that the parameter takes over the range of values R. For example, the tension as measured by the controller takes only one value for tension values that lie in 98 lbf and 102 lbf (98 lbf t1(t) < 102 lbf), and thus the resolution is r = 1/(102− 98) = 0.25 vpl. These factors dilute the tension data so that the controller is not utilizing accurate data. Since the control variable (CV) is an integer, the controller does not output the required amount with suitable precision, causing large fluctuations in tension. Moreover, the resolution of the CV is one, restricting the number of specific torque values that are able to be applied. The voltage sent from the controller to the brake pressure device is directly related to the CV. Therefore, if the CV can only assume a fixed number of values, the voltage, and hence the pressure, can also only maintain certain values. Additionally, the sampling period of the tension measurements is one second with the PI controller update time as 0.2 seconds. This means that the controller output is updated every 0.2 seconds but the error only updates every second. Thus, the 64 controller is just an integrator 0.8 out of every one second rendering the system to be open loop the majority of the time. This results in inaccurate control since the applied torque is not reacting to current errors in tension. Lastly, the controller used constant PI gains throughout the entire roll. For systems that are time invariant, this would acceptable. However, the inertia of the Unwind Roll changes as the web is continuously released from the roll and the radius decreases. This will cause a set of PI gains to be sufficient near the beginning of the new roll (full roll) while causing the tension performance to be degraded later. There are two solutions to this problem. The first is to define fixed gains such that the system remains stable throughout and allow for decreased performance. The second is to vary the gains as a function of the radius so that the system remains stable and satisfactory performance is achieved for all radii of the material roll. Each of the simulations conducted in Subsections 3.3.1 through 3.3.3 employed the model shown in Fig. 2.7 and the velocity profile from Fig. 3.1. Unless otherwise specified, the web properties from Table 3.1 and the initial conditions shown in Table 3.2 were utilized as well. As in Subsection 3.2.2, for each simulation, three repetitions of the same data are shown for clarity of comparison. 3.3.1 Strategy 1 As discussed above, the controller output and feedback measurements should be updated as often as is practical and that they be as accurate as possible. Moreover, to avoid the complications of multirate control systems, the controller update and measurement rates should be the same. Thus the current control strategy for the Unwind Roll is not ideal since it has different rates for updating the controller and measuring the feedback data in addition to using integer values. Strategy 1 is aimed at rectifying these shortcomings. The first change is reducing the tension measurement sampling time from one 65 second to 0.2 seconds. This will allow the controller to react to the most recent data, increasing the relevancy of the control action. Another benefit is that the controller will update at the same rate as the tension measurement, eliminating the need for analysis of the system as a multirate system. The second improvement seen in Strategy 1 is changing the scaling on the CV and the measured tension as well as setting the average tension as a floating point number. As mentioned above, the current scaling results in a tension resolution of only 0.25 vpl. This causes the feedback to be degraded and decreases the effectiveness of the controller. For Strategy 1, the scaling was changed so that the tension resolution is 1 vpl (i.e., the controller takes on one value for 99 lbf t1(t) <100 lbf). Although not as accurate as a floating point, this is a substantial improvement. The control variable is the integer equivalent of the PI Output percentage (the percentage of the maximum output the controller is able to provide). Thus, in order to change the controller output, the tension error must become large enough to vary the PI Output by 0.5 so that the CV changes as opposed to the CV adapting more precisely with the tension error. Moreover, slight corrections in the braking torque are not possible since the adjustment of the braking torque is directly related to the alteration of the CV. Using Equation (3.3), the current PI Output to CV resolution is one CV per percent (cvp). Under the new scaling for Strategy 1, the resolution is 10 cvp. Thus, the control variable is able to achieve 10 different values as the PI Output changes by 1%. The CV is now better suited to attain the value specified by the PI equation. This means that the CV jumps seen in Section 2.5 will be decreased, and thus improved tension control performance is expected. Since the sampling time was changed, the PI gains need to be retuned. Using a model simulation with Strategy 1, the PI gains were tuned until the best performance was achieved. This processes resulted in Kp = 0.01 and Ki = 0.15. The decrease in the proportional gain indicates that the Unwind Roll is sensitive to large changes 66 in the input and that the most effective method for controlling the tension is to apply a smooth braking torque. Increasing the proportional action will increase the portion of the controller output that is directly related to the tension error. This will cause abrupt changes in the controller output since the tension error is continuously fluctuating. These actions will produce spikes in tension due to the large adjustments in the braking torque
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Title  Analysis of the Unwind Section of an Industrial Web Processing Line 
Date  20110701 
Author  Pacini, Benjamin Robert 
Keywords  torque control, Web handling 
Document Type  
Full Text Type  Open Access 
Abstract  The purpose of this thesis is to analyze the control strategies and tension performance of the unwind section of an industrial web processing line that manufactures composite flooring materials. The main elements of the unwind section of the web line include a torque controlled unwind roll, two driven rollers, an accumulator, and a dancer. Dynamic models describing the interaction and control of these web line elements are derived and used to generate improvements to the existing control strategies. The controller structures used are the ProportionalIntegralDerivative type. Experiments are conducted on the improved strategies for the unwind roll to verify their performance. A comparison between a torque controlled unwind roll and a velocity controlled unwind roll is performed. The basis of comparison is the stability regions in their corresponding controller parameter spaces. Comparing model simulation results and measured data verifies that the model adequately predicts trends measured in the unwind section. Model simulations and experimentation using the improved unwind roll strategies show improved tension regulation performance. However, it is seen that the variability and inconsistency in the braking mechanisms may limit the effectiveness of these improvements. Simulation results indicate that the modified control strategy for the first pull roll and accumulator also increased tension regulation performance. The analysis of the stability regions for the torque controlled unwind roll and velocity controlled unwind roll show that there is a greater flexibility in selecting the controller gains for the velocity controlled unwind. 
Note  Thesis 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  ANALYSIS OF THE UNWIND SECTION OF AN INDUSTRIAL WEB PROCESSING LINE By BENJAMIN PACINI Bachelor of Science Mechanical and Aerospace Engineering University of Colorado at Colorado Springs Colorado Springs, CO, USA 2008 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE July, 2011 ANALYSIS OF THE UNWIND SECTION OF AN INDUSTRIAL WEB PROCESSING LINE Thesis Approved: Dr. Prabhakar R. Pagilla Thesis Advisor Dr. Gary E. Young Committe Member Dr. Lawrence L. Hoberock Committe Member Dr. Mark E. Payton Dean of the Graduate College ii ACKNOWLEDGMENTS There are several people I would like to thank for their contributions to my thesis. First and foremost I would like to express my greatest appreciation to my advisor Dr. Prabhakar R. Pagilla for his invaluable guidance, supervision, and friendship throughout my graduate studies. I am extremely grateful for his support, encouragement, and technical insights. I would like to extend my thanks to my master’s committee members: Dr. Gary E. Young for his friendship and guidance throughout my graduate studies and Dr. Lawrence L. Hoberock for his support and suggestions in completion of this work. Their guidance and understanding made the development of this thesis a positive learning experience. I would also like to thank my colleagues at Oklahoma State University: Pramod Raul, Muthappa PonjandaMadappa, Youwei Lu, Kadhim Jabbar, Shyam Konduri, Mauro Cimino, Carlo Branca, Aravind Seshadri, and Supraj Paleti. Their friendship and support (both technical and moral) have been an essential part of my graduate studies. I owe a special debt of gratitude to Jamie Lynch, Tim Gottlob, and the other employees at Armstrong World Industries for their invaluable assistance throughout the duration of this project. Additionally, I would like to thank the Oklahoma Center for the Advancement of Science and Technology (OCAST) and the National Science Foundation (NSF) for their funding throughout the project. iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Coating and Fusion Line . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Models, Control Strategies, and Evaluation of Parameters for the Unwind Section of a Coating and Fusion Line 10 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Simplification of the CFL . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Longitudinal Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.1 Linearized Dynamics . . . . . . . . . . . . . . . . . . . . . . . 22 2.4 Parameter Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.1 Viscoelastic Parameter Evaluation . . . . . . . . . . . . . . . . 25 2.4.2 Friction Torque Evaluation . . . . . . . . . . . . . . . . . . . . 30 2.5 Current Control Strategies . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.1 RSLogix5000 Operation . . . . . . . . . . . . . . . . . . . . . 33 2.5.2 Unwind Roll Control Strategy . . . . . . . . . . . . . . . . . . 36 2.5.3 Pull Roll 1 and Unwind Accumulator Control . . . . . . . . . 40 2.5.4 Pull Roll 2 Control . . . . . . . . . . . . . . . . . . . . . . . . 43 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 iv 3 Analysis of Unwind Roll Control and Improvements 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.1 Parameter Values and Initial Conditions . . . . . . . . . . . . 48 3.2.2 Model Simulation Results . . . . . . . . . . . . . . . . . . . . 49 3.2.3 Measured Data From the CFL . . . . . . . . . . . . . . . . . . 55 3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3 Strategies For Improvement of the Existing Control Strategy . . . . . 64 3.3.1 Strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3.2 Strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3.3 Strategy 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.4 Experimental Data Analysis . . . . . . . . . . . . . . . . . . . . . . . 84 3.4.1 Strategy 1 Experimental Results . . . . . . . . . . . . . . . . . 84 3.4.2 Strategy 2 Experimental Results . . . . . . . . . . . . . . . . . 93 3.4.3 Current Control Strategy Using Varying PID Gains . . . . . . 101 3.4.4 Strategy 3 Experiment Discussion . . . . . . . . . . . . . . . . 109 3.4.5 Summary of All Experiments . . . . . . . . . . . . . . . . . . 109 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4 Comparison of Torque and Velocity Control 115 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2 Stability Boundary Mapping . . . . . . . . . . . . . . . . . . . . . . . 116 4.2.1 Stable to Unstable Transition . . . . . . . . . . . . . . . . . . 117 4.3 Controller Parameter Stability Regions for Velocity Control . . . . . . 119 4.3.1 Inner Velocity Loop Stability Boundary . . . . . . . . . . . . . 121 4.3.2 Outer Tension Loop Stability Boundary for Velocity Control . 124 4.4 Tension Loop Stability Boundary for Torque Control . . . . . . . . . 129 v 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5 Pull Roll 1 and Unwind Accumulator Analysis 134 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2 System Simulation Using the Current Control Strategy . . . . . . . . 135 5.2.1 Span Parameters and Initial Conditions . . . . . . . . . . . . . 138 5.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2.3 Model Verification Using Measured Data from the CFL . . . . 145 5.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.3 Improvement 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.3.1 Model Simulation Using Improvement 1 . . . . . . . . . . . . 151 5.3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6 Conclusions and Future Work 157 6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 BIBLIOGRAPHY 162 vi LIST OF TABLES Table Page 2.1 Test 1 Parameters and Measurements . . . . . . . . . . . . . . . . . . 31 2.2 Test 2 Parameters and Measurements . . . . . . . . . . . . . . . . . . 32 2.3 Average Friction Torque . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Priorities and Periods of RSLogix Tasks . . . . . . . . . . . . . . . . 34 2.5 Parameter Values Used in RSLogix PIDs . . . . . . . . . . . . . . . . 35 3.1 Parameter Values Used in Simulation . . . . . . . . . . . . . . . . . . 49 3.2 Simulation Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . 50 4.1 Web Line Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.1 Parameters Used in the Simulation of the System Shown in Fig. 5.1 . 139 5.2 Controller Gains Used in the Simulation of the System Shown in Fig. 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 5.3 Initial Conditions Used in the Simulation of the System Shown in Fig. 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 vii LIST OF FIGURES Figure Page 1.1 Control system with outer tension loop and inner velocity loop . . . . 2 1.2 Control system with tension loop only . . . . . . . . . . . . . . . . . 2 1.3 Unwind section of the CFL . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Simplified model of Unwind Roll to Pull Roll 2 . . . . . . . . . . . . . 11 2.2 Maxwell element in parallel with a linear spring . . . . . . . . . . . . 14 2.3 Two span accumulator with control volume . . . . . . . . . . . . . . . 15 2.4 Displaced dancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.5 Zero, vertical, and displaced positions of Dancer 1 . . . . . . . . . . . 20 2.6 Free body diagram of Dancer 1 . . . . . . . . . . . . . . . . . . . . . 21 2.7 Section of web line for linearized dynamics . . . . . . . . . . . . . . . 23 2.8 Typical stress versus time for material during tensile testing . . . . . 29 2.9 Stress versus time for measured data and bestfit model . . . . . . . . 30 2.10 Execution times of each task[5] . . . . . . . . . . . . . . . . . . . . . 34 2.11 Block diagram of RS Logix PID . . . . . . . . . . . . . . . . . . . . . 35 2.12 Control strategy for Unwind Roll . . . . . . . . . . . . . . . . . . . . 36 2.13 Control strategy for Pull Roll 1 and Unwind Accumulator under normal operating conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.14 Control strategy for Pull Roll 1 and Unwind Accumulator while emptying 44 2.15 Control strategy for Pull Roll 1 and Unwind Accumulator while filling 44 2.16 Control strategy for Pull Roll 2 . . . . . . . . . . . . . . . . . . . . . 44 3.1 Velocity profile of Pull Roll 1 . . . . . . . . . . . . . . . . . . . . . . 47 viii 3.2 Unwind Roll radius (Model Simulation) . . . . . . . . . . . . . . . . . 50 3.3 Control variable for Span 1 with tension PI (Model Simulation) . . . 51 3.4 PI controller output percentage (Model Simulation) . . . . . . . . . . 51 3.5 Unwind Roll peripheral velocity (Model Simulation) . . . . . . . . . . 51 3.6 Span 1 tension (Model Simulation) . . . . . . . . . . . . . . . . . . . 52 3.7 Unwind Roll velocity during EF portion of roll (Model Simulation) . 54 3.8 Pull Roll 1 velocity using Brake 1 (Measured data) . . . . . . . . . . 56 3.9 Control variable using Brake 1 (Measured data) . . . . . . . . . . . . 56 3.10 Tension in Span 1 using Brake 1 (Measured data) . . . . . . . . . . . 57 3.11 Span 1 tension using controller with increased Ki,unw (Model Simulation) 59 3.12 Pull Roll 1 velocity using Brake 2 (Measured data) . . . . . . . . . . 60 3.13 Control variable using Brake 2 (Measured data) . . . . . . . . . . . . 60 3.14 Tension in Span 1 using Brake 2 (Measured data) . . . . . . . . . . . 60 3.15 Unwind roll velocity using Strategy 1 (Model Simulation) . . . . . . . 67 3.16 Control variable using Strategy 1 (Model Simulation) . . . . . . . . . 68 3.17 PI Output percentage using Strategy 1 (Model Simulation) . . . . . . 68 3.18 Tension using Strategy 1 (Model Simulation) . . . . . . . . . . . . . . 69 3.19 Unwind Roll velocity during EF portion of roll using Strategy 1 (Model Simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.20 Tension oscillation amplitudes still increase as the material roll depletes using Strategy 1 (Model Simulation) . . . . . . . . . . . . . . . . . . 71 3.21 Proportional gain as a function of Unwind Roll radius . . . . . . . . . 74 3.22 Proportional gain using Strategy 2 (Model Simulation) . . . . . . . . 75 3.23 Integral gain using Strategy 2 (Model Simulation) . . . . . . . . . . . 75 3.24 Unwind roll velocity using Strategy 2 (Model Simulation) . . . . . . . 76 3.25 Control variable using Strategy 2 (Model Simulation) . . . . . . . . . 76 3.26 PI Output percentage using Strategy 2 (Model Simulation) . . . . . . 77 ix 3.27 Tension using Strategy 2 (Model Simulation) . . . . . . . . . . . . . . 77 3.28 Rescaled Span 1 tension using Strategy 2 (Model Simulation) . . . . 78 3.29 PI Output percentage using Strategy 3 (Model Simulation) . . . . . . 81 3.30 Controller output using Strategy 3 (Model Simulation) . . . . . . . . 81 3.31 Tension using Strategy 3 (Model Simulation) . . . . . . . . . . . . . . 82 3.32 Unwind roll velocity using Strategy 3 (Model Simulation) . . . . . . . 82 3.33 Rescaled Span 1 tension using Strategy 3 (Model Simulation) . . . . 82 3.34 Tension using the current control strategy with Brake 1 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . 86 3.35 Pull Roll 1 velocity using the current control strategy with Brake 1 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . 86 3.36 Control variable using the current control strategy with Brake 1 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . . . . 87 3.37 Tension using Strategy 1 with Brake 1 . . . . . . . . . . . . . . . . . 87 3.38 Pull Roll 1 velocity using Strategy 1 with Brake 1 . . . . . . . . . . . 88 3.39 Control variable using Strategy 1 with Brake 1 . . . . . . . . . . . . . 88 3.40 Control variable for a typical roll using Strategy 1 and the current control strategy with Brake 1 . . . . . . . . . . . . . . . . . . . . . . 89 3.41 Tension using the current control strategy with Brake 2 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . . . . . . . 90 3.42 Pull Roll 1 velocity using the current control strategy with Brake 2 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . 90 3.43 Control variable using the current control strategy with Brake 2 prior to implementation of Strategy 1 . . . . . . . . . . . . . . . . . . . . . 91 3.44 Tension using Strategy 1 with Brake 2 . . . . . . . . . . . . . . . . . 91 3.45 Pull Roll 1 velocity using Strategy 1 with Brake 2 . . . . . . . . . . . 92 3.46 Control variable using Strategy 1 with Brake 2 . . . . . . . . . . . . . 92 x 3.47 Control variable for the current control strategy using Brake 1 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . . 94 3.48 Pull Roll 1 speed for the current control strategy using Brake 1 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . 94 3.49 Tension for current control strategy using Brake 1 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.50 Control variable for Strategy 2 using Brake 1 . . . . . . . . . . . . . . 95 3.51 Pull Roll 1 Speed for Strategy 2 using Brake 1 . . . . . . . . . . . . . 96 3.52 Tension for Strategy 2 using Brake 1 . . . . . . . . . . . . . . . . . . 96 3.53 Control variable for current control strategy using Brake 2 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . . 98 3.54 Pull Roll 1 Speed for current control strategy using Brake 2 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . . 98 3.55 Tension for current control strategy using Brake 2 prior to implementation of Strategy 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 3.56 Control variable for Strategy 2 using Brake 2 . . . . . . . . . . . . . . 99 3.57 Pull Roll 1 speed for Strategy 2 using Brake 2 . . . . . . . . . . . . . 100 3.58 Tension for Strategy 2 using Brake 2 . . . . . . . . . . . . . . . . . . 100 3.59 Pull Roll 1 speed for the current control strategy using Brake 1 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . 102 3.60 Control variable for the current control strategy using Brake 1 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . . . 102 3.61 Tension for current control strategy using Brake 1 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.62 Pull Roll 1 Speed for Strategy 2a using Brake 1 . . . . . . . . . . . . 103 3.63 Control variable for Strategy 2a using Brake 1 . . . . . . . . . . . . . 104 3.64 Tension for Strategy 2a using Brake 1 . . . . . . . . . . . . . . . . . . 104 xi 3.65 Pull Roll 1 speed for the current control strategy using Brake 2 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . 105 3.66 Control variable for the current control strategy using Brake 2 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . . . 106 3.67 Tension for current control strategy using Brake 2 prior to implementation of Strategy 2a . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.68 Pull Roll 1 Speed for Strategy 2a using Brake 2 . . . . . . . . . . . . 107 3.69 Control variable for Strategy 2a using Brake 2 . . . . . . . . . . . . . 107 3.70 Tension for Strategy 2a using Brake 2 . . . . . . . . . . . . . . . . . . 107 4.1 Block diagram of velocity controlled unwind roll with outer tension loop115 4.2 Block diagram of torque controlled unwind roll . . . . . . . . . . . . . 116 4.3 General block diagram of plant with controller . . . . . . . . . . . . . 116 4.4 Block diagram of inner velocity loop . . . . . . . . . . . . . . . . . . 121 4.5 Root invariant regions in KpvvKivv space for velocity loop (the number of unstable poles in each region is indicated) . . . . . . . . . . . . . . 124 4.6 Block Diagram of the Outer Tension Loop . . . . . . . . . . . . . . . 124 4.7 Root invariant regions in KptvKitv space for Kpvv = 1 and Kivv = ¯K ivv (the number of unstable poles in each region is indicated) . . . . . . . 128 4.8 Root invariant regions in KptvKitv space for Kpvv = 1 and Kivv > ¯K ivv (the number of unstable poles in each region is indicated) . . . . . . . 129 4.9 Root invariant regions in KptvKitv space for Kpvv = 1 and Kivv < ¯K ivv (the number of unstable poles in each region is indicated) . . . . . . . 130 4.10 Root invariant regions in KpttKitt space (the number of unstable poles in each region is indicated) . . . . . . . . . . . . . . . . . . . . . . . . 132 5.1 Simplified model of the CFL from Pull Roll 1 to Pull Roll 2 . . . . . 135 xii 5.2 Reference profiles for Pull Roll 1 speed and Unwind Accumulator carriage height and velocity (Model Simulation) . . . . . . . . . . . . . . 137 5.3 Pull Roll 1 speed and Unwind Accumulator carriage height and velocity (Model Simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.4 Dancer 1 position in percentage of maximum stroke (Model Simulation)142 5.5 Dancer 1 Trim scaling factor (Model Simulation) . . . . . . . . . . . . 142 5.6 Accumulator carriage speed during initial portion of the emptying process (Model Simulation) . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.7 Pull Roll 1 speed and Unwind Accumulator carriage height and velocity146 5.8 Dancer 1 position in percentage of maximum stroke . . . . . . . . . . 147 5.9 Dancer 1 Trim scaling factor . . . . . . . . . . . . . . . . . . . . . . . 147 5.10 Accumulator carriage speed during initial portion of the emptying process148 5.11 Reference profiles for Pull Roll 1 speed and Unwind Accumulator carriage height and velocity using Improvement 1 (Model Simulation) . . 152 5.12 Pull Roll 1 speed and Unwind Accumulator carriage height and velocity using Improvement 1 (Model Simulation) . . . . . . . . . . . . . . . . 153 5.13 Dancer 1 position in percentage of maximum stroke using Improvement 1 (Model Simulation) . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.14 Dancer 1 Trim scaling factor using Improvement 1 (Model Simulation) 154 xiii NOMENCLATURE Chapters 2 and 3 A Cross sectional area of the web b Damping constant in Maxwell element bw Lateral width of the web dx Dancer 1 trim E Spring constant in Maxwell element Ev Spring constant in Maxwell element F Force applied to keep Dancer 1 vertical Fc Interaction force between accumulator carriage and motor Fgc Gravitational force of accumulator carriage and rollers g Acceleration due to gravity Ji Inertia of element i Jc Equivalent inertia of accumulator motor, gears, carriage, and rollers Jmc Inertia of accumulator motor Ki,UW Integral PI gain Kp,UW Proportional PI gain Kd,D1 Derivative gain of Dancer 1 PID Ki,AC Integral gain of accumulator speed loop PI Ki,D1 Integral gain of Dancer 1 PID Ki,PR1 Integral gain of Pull Roll 1 speed loop PI Ki,PR2 Integral gain of Pull Roll 2 speed loop PI xiv Kp,AC Proportional gain of accumulator speed loop PI Kp,D1 Proportional gain of Dancer 1 PID Kp,PR1 Proportional gain of Pull Roll 1 speed loop PI Kp,PR2 Proportional gain of Pull Roll 2 speed loop PI Kpm Scale factor used during manual control for the Unwind Roll kn Unit conversion from Control Variable to brake pressure l Length from the dancer pivot to the dancer roller center lF Length from the dancer pivot to the point where F is applied Li Length of span i lL Length from the dancer pivot to the center of gravity of the dancer lever arm lx Length from the dancer pivot to the point where dancer linear displacement is measured MAXI Maximum process variable MINI Minimum process variable MAXS Maximum engineering unit scaling value MINS Minimum engineering unit scaling value MAXCV Maximum control variable MINCV Minimum control variable mc Mass of accumulator carriage n Number of stress measurements from tensile test N Number of stress measurements in random subset nc Gear ratio between accumulator motor angular velocity and carriage linear velocity NAc Number of spans in the Unwind Accumulator ni Gear ratio of the motor to roller i for driven rollers OPI,UW PI output percentage of the Unwind Roll controller xv OPID,D1 PID output percentage of the Dancer 1 PID r Ratio statistic used to determine steady state Rc0 Radius of spindle that Unwind Roll is would upon Rc Radius of roller in accumulator carriage i Ri Radius of Roller i RRMSi,RRMSfi ith root mean square of random subset and its filtered value RMSi ith root mean square RRMS Average root mean square of random subset Si Scale factors used in the PI controllers. i = t1, b ti Tension in span i t3i Tension of i th span in the accumulator tci Tension in span i of Unwind Accumulator tw Thickness of the web Ttix Torque due to x direction component of tension i about the dancer pivot Ttiy Torque due to y direction component of tension i about the dancer pivot tmeas Averaged measured tension used in the Unwind Roll control tr Reference tension ui Control input to roller i vi Velocity of roller i vc Velocity of the accumulator carriage vcr Reference velocity of the accumulator carriage viref , vir Velocity reference of roller i vls Line speed reference v1r Speed reference for Pull Roll 1 v1rr Ramped speed reference for Pull Roll 1 xvi v3i Velocity of i th roller in the accumulator xc Position of the accumulator carriage xt Measured linear displacement of the dancer roller xt,max Maximum linear displacement of Dancer 1 xtr Reference displacement of Dancer 1 i Angle from horizontal that tension i is applied on the dancer roller i Wrap angle of the web around the dancer from horizontal to the point where tension i is applied on the dancer roller i Filter factors 2 fi, 2 fi ith filtered numerator and denominator variance !i Angular velocity of Roller i !c Angular velocity of accumulator motor shaft w Density of the web ¯ Model value of stress during tensile test i Web stress of Span i mi ith Web stress measurement during tensile test 2n , 2 d Numerator and denominator variances mc Torque produced by the accumulator motor cmd,i Commanded torque of element i f Friction torque p Time constant for brake pneumatic device d Angular displacement of the dancer roll from vertical n Angular displacement of the dancer roll from dancer’s zero position to vertical "i Strain in span i "3i Strain of i th span in the accumulator xvii Chapter 4 A Cross sectional area of the web b Damping constant in Maxwell element bw Lateral width of the web dx Dancer 1 trim Cij Controller for loop i (tension/velocity) and control strategy j (velocity/torque) E Spring constant in Maxwell element Ev Spring constant in Maxwell element Evel Variational velocity error Ji Inertia of roller i Kijk Integral PI gain of loop j (tension/velocity) and control strategy k (velocity/torque) Kpjk Proportional PI gain of loop j (tension/velocity) and control strategy k (velocity/torque) L1 Length of Span 1 Ri Radius of roller i tr Reference tension vi Velocity of roller i vr Velocity reference rjk Real component of the characteristic equation of loop j (tension/velocity) and control strategy k (velocity/torque) ijk Imaginary component of the characteristic equation of loop j (tension/velocity) and control strategy k (velocity/torque) i ith time constant "i Strain in span i "r Reference strain xviii Chapter 5 dx Dancer 1 trim F Force applied to keep Dancer 1 vertical g Acceleration due to gravity Ji Inertia of element i Jc Inertia of accumulator motor, gears, carriage, and rollers Kd,D1 Derivative gain of Dancer 1 PID Ki,AC Integral gain of accumulator speed loop PI Ki,D1 Integral gain of Dancer 1 PID Ki,PR1 Integral gain of Pull Roll 1 speed loop PI Ki,PR2 Integral gain of Pull Roll 2 speed loop PI Kp,AC Proportional gain of accumulator speed loop PI Kp,D1 Proportional gain of Dancer 1 PID Kp,PR1 Proportional gain of Pull Roll 1 speed loop PI Kp,PR2 Proportional gain of Pull Roll 2 speed loop PI Li Length of span i mgc Mass of accumulator carriage and rollers nc Gear ratio between accumulator motor angular velocity and carriage linear velocity NAc Number of spans in the Unwind Accumulator ni Gear ratio of the motor to roller i for driven rollers Rc Radius of roller in accumulator carriage i Ri Radius of Roller i ti Tension in span i t3i Tension of i th span in the accumulator tr Reference tension ui Control input to roller i xix vi Velocity of roller i v3i Velocity of i th roller in the accumulator vc Velocity of the accumulator carriage vls Line speed reference vref , vr Velocity reference xc Position of the accumulator carriage xt Measured linear displacement of the dancer roller f Friction torque xx CHAPTER 1 Introduction 1.1 Background Any material that is produced in continuous flexible form in a rolltoroll fashion is known as web and the manufacturing of such materials is referred to as web handling. Many of the consumer products today, such as paper, diapers, textiles, and laminate flooring, are made in web form. In the manufacture of such products, control of the longitudinal dynamics is essential to ensure high quality goods. The most important facet of longitudinal control is maintaining web tension at appropriate values. If web tension is not well regulated, there is a high potential for damaging the product and the web handling machinery, resulting in extra costs for the manufacturing company and, consequently, higher prices for the consumer. A web line is the series of processes and components that are used in the production of the web material. The typical processes include printing, coating, heating, and cooling of the web, and these operations are essential in the manufacture of the product. The standard components of a web line include an unwinder, accumulators, pull rollers, idle rollers, dancers, loadcell rollers, a winder, and other machinery used in the processing of the web material such as ovens, printers, coaters, and heating/ cooling rollers. The unwind section of a web line consists of the elements that aid in dispensing web into the web line. These components include the unwinder, accumulator, pull rollers, idle rollers, dancers, and loadcell rollers and they are discussed below. The unwinder is an apparatus that contains a roll of web material (referred to as 1 Tension Controller Velocity Controller S S Velocity Dynamics Tension Dynamics +  + +  Reference Tension Reference Velocity Figure 1.1: Control system with outer tension loop and inner velocity loop Tension Controller S Velocity Dynamics Tension Dynamics +  Reference Tension Figure 1.2: Control system with tension loop only the unwind roll) and a corresponding control device. This is always the first element since it supplies the material for the entire web line. The control of this component can be achieved using either of the two methods presented below. 1. An outer loop that utilizes web tension feedback to provide a correction to the reference of an inner loop that controls the speed of the unwind roll (see Fig. 1.1) 2. A single tension loop that controls the torque applied to the unwind roll (see Fig. 1.2) The first approach uses a motor attached to the material roll whereas the latter utilizes only a brake. During the operation of the web line, the unwind roll will eventually be depleted and will need to be replaced. Additionally, the material from the new roll must be affixed to the previous web material in a process called splicing. There are two types of splicing: (1) a static method called the zerospeed splicing where both the previous and new webs are stationary when they are connected and (2) an “on 2 the fly” method where both webs are moving as they are attached to one another. This process provides the continuity in the manufacture of web product. However, during the changing of these rolls, the operation of the processing portion of the line must not be interrupted or else the web may be damaged. If the operation of the line is halted the web may overheat from being in an oven for too long or a coating may be too thick from being stopped at a printing station. To ensure the continuity of the web line, the accumulator is utilized. An accumulator is a structure that contains two sets of parallel idle rollers with one set fixed and the other on an extendable carriage. The carriage will move either vertically or horizontally depending on the size of the accumulator and the weight of the material. The web is alternately wound about a fixed and then a mobile roller so that the accumulator is able to either supply the rest of the line with web (in the case of an accumulator downstream of the unwinder) or receive web from the line (in the case of an accumulator upstream of the winder). In the former case, when the unwinder stops for a roll change, the accumulator carriage descends (contracts towards the fixed rollers) at a rate so that web is supplied at the correct speed. When the new roll is ready, the web upstream of the accumulator is driven faster than the web downstream which allows the carriage to move upwards (extend away from the fixed rollers), restoring the accumulator to its original height so that it is reset for the next roll change. The converse motions are seen for the case of the winder. Pull rolls are driven rollers that propel the web through the line. They are most often controlled in a similar fashion to the type (1) unwinder, with an outer tension loop that provides a correction to the reference for the inner speed loop. This speed reference modification alters the speed of the pull roll in order to correct a tension error. However, there is one type of pull roll that does not use a tension loop and is strictly under velocity control. This roller is called the master speed roller and it dictates the process speed for the entire line. Not every roller is driven, however. 3 Dancer 1 Master Speed Roller Pull Roll 1 Unwind Accumulator R30 Load Cell Unwind Rolls Figure 1.3: Unwind section of the CFL Those that are not driven are referred to as idle rollers, and their purpose is to support the web as it travels through the web line. There are two types of dancers, active and passive. Active dancers are rollers that displace (either linearly or rotationally) in order to vary the adjacent span lengths as a method to control tension. Passive dancers are components that use transducers to measure the displacement resulting from variations in tension, thus providing an indirect method for determining web tension. The motion can be either linear or rotational and for each type there is a normalizing force that is applied such that the passive dancer is at equilibrium in the nominal position when the tensions in the adjacent spans are at the reference value. Thus, if the tension changes, the balance of forces will be disrupted and the dancer will move. Load cells are elements that are also used for measuring web tension. They are attached to idle rollers and display the numerical value of the force the web is applying to the roller. 4 1.2 Coating and Fusion Line The focus of this thesis is on the unwind section of a Coating and Fusion Line (CFL) of the Armstrong World Industries plant located in Stillwater, Oklahoma. The CFL is the final web line in the manufacture of several different brands of laminate flooring material. The main purpose of this line is to apply a coating that provides a protective layer for the printed laminate material. The unwind section of the CFL is shown in Fig. 1.3 and includes two Unwind Rolls, Pull Roll 1, the 18span Unwind Accumulator, Dancer 1, and the Master Speed Roller (also known as Pull Roll 2). The two Unwind Rolls are alternately used to supply web for the rest of the line. They are the type (2) unwinders from the above discussion and each are controlled by separate brakes that utilize the same algorithm. Since the Unwind Rolls are brake controlled, Pull Roll 1 is their sole means of rotation; the Unwind Rolls rotate as Pull Roll 1 draws the web. This driven roller is controlled in the typical fashion and uses position measurements from Dancer 1 as feedback for the outer tension loop. Similar to Pull Roll 1, the Unwind Accumulator has a controller with an outer loop that uses tension feedback from Dancer 1 to correct the reference for the inner speed loop for the carriage. Dancer 1 is of the passive pendulum type but provides translational displacement data. This is accomplished by a linear transducer that measures the movement of a point on the pivot lever. The final component of the unwind section is Pull Roll 2. This is the master speed roller for the entire CFL and as such is only under velocity control. Note that the control structure utilized in all of the controllers are the ProportionalIntegralDerivative (PID) type. 1.3 Thesis Outline The main purpose of this thesis is to analyze and improve upon the current control strategies of the unwind section of the CFL. This web line was developed in an ad 5 hoc manner so the analysis contained herein gives a greater understanding of how the system operates. Additionally, the improvements suggested in this thesis will prove useful in increasing the functionality of the CFL and the overall quality of the flooring products. Chapter 2 discusses the longitudinal web dynamics, the current control strategies employed on the CFL, and the parameter evaluations. References [1], [2], [4], and [3] are used in the development of the dynamic equations. Reference [1], which discussed a method for decentralized control of a web line, provided the equations for the velocity of an unwind roll and for the rate of change of the unwind roll radius. Additionally, the dynamics for driven and idle rollers were presented. The velocity of the rollers within the accumulator were given in Reference [2], which compared an industrial accumulator controller to one developed using Lyapunov’s second method. The web used in the CFL is assumed to be viscoelastic, so Hooke’s Law cannot be used to relate stress and strain for a span with fixed length, as was done in Reference [1]. To describe the viscoelastic behavior, Reference [4] is utilized. This article detailed the relationship between web tension and strain for a viscoelastic material. Reference [3] is used to derive the strain dynamics for spans of varying length (i.e., for spans within an accumulator and those immediately adjacent to a dancer). After the dynamics are derived, the control strategies for the Unwind Roll, Pull Rolls 1 and 2, and the Unwind Accumulator are discussed. The tension loop calculations are performed in the RSLogix5000 software, the operation of which is detailed in References [5] and [6]. The evaluation of the friction term and viscoelastic parameters are also included in this chapter. The friction is assumed to be a constant resistive torque and is determined via a test where an idler roller is accelerated to a predetermined velocity and then is allowed to slow to a stop using only friction. The time required for this deceleration is recorded and used in the friction torque calculation. The viscoelastic parameters are determined using the results of tensile 6 tests of the flooring material from Reference [12]. A heuristic optimization method developed in Reference [9] is used to fit a viscoelastic stress equation to the tensile test data. The model parameters that produce a stress curve that best matches the tensile test data are selected. This method is similar to that used in Reference [7] which developed a procedure for modeling biological tissues. Reference [10] is used to determine the stopping criteria for the optimizer. In Chapter 3, the control strategy of the Unwind Roll is analyzed. First, a simplified system model containing the Unwind Roll and Pull Roll 1 is constructed and then verified by comparing simulation results with data collected from the CFL. Three new strategies are proposed in an effort to improve certain disadvantageous aspects of the current control strategy. The first strategy decreases the web tension sampling time and increases the resolution of the tension measurements and controller output. The second strategy utilizes timevarying PID gains and each of the modifications from the first strategy. The third strategy uses feedforward control with corrections provided by a PID in addition to the modifications from the first strategy. These improvements are verified by simulations and subsequently employed on the CFL. The results of these experiments are compared against the performance of the current strategy, and based on this, a recommendation that will improve the tension performance is given. Chapter 4 discusses the comparison between the type (1) and (2) unwinder control strategies. The control structure for each scenario is developed as well as their corresponding closed loop characteristic equation for the tension dynamics. The basis for this analysis is the stability regions of their respective controller parameter spaces. The procedure outlined in Reference [11] is used to map the stability boundary in the root space to each of the controller parameter spaces using their closed loop characteristic equations of the tension dynamics. Chapter 5 describes the analysis and improvements of the control strategies for 7 Pull Roll 1 and the Unwind Accumulator. First a simplified model is developed that includes the major elements from Pull Roll 1 to Pull Roll 2. This is compared with data measured from the CFL for model verification. Subsequently, one improvement is suggested that will increase the tension regulation performance. A simulation is then used to demonstrate its effectiveness. 1.4 Contributions The contributions of the work presented in this thesis are summarized below: • The control strategies for the Unwind Roll, Pull Roll 1, and Unwind Accumulator were summarized into block diagram form. This will prove useful for the operators of the CFL to increase their understanding of the operation of these components. • The disparity in the performance of the two brakes used to control the Unwind Roll was discovered during the work on this thesis. When using the same control algorithm, the controller output and tension performance for each brake can differ significantly. • Three strategies for improving the control of the Unwind Roll are presented and supported by simulations. Additionally, the first two strategies were implemented onto the CFL and showed improvement over the current strategy. Based on the results of the experimentation, the strategy that most effectively increases the tension performance in this portion of the CFL is proposed as the recommended controller. • While implementing the new control strategies for the Unwind Roll, the tension measurement resolution was increased by four times. This change was made permanent after the experimental employment of the first new strategy as it provides more accurate feedback for their controller. 8 • An algorithm to calculate the radius of the Unwind Roll was created in the controller software and is available for use. Each of the measurements required to compute this value were already accessible, however, through the implementation of the experiments, the radius calculation algorithm was generated. • One improvement was suggested for Pull Roll 1 and the Unwind Accumulator to decrease the motion of Dancer 1. This modification requires altering the ramp rate of Pull Roll 1 and altering the deceleration profile of Pull Roll 1 during the initiation of the emptying procedure. Simulations show that this change results in decreased dancer motion and hence improved tension regulation. 9 CHAPTER 2 Models, Control Strategies, and Evaluation of Parameters for the Unwind Section of a Coating and Fusion Line 2.1 Introduction This chapter details the development of a model for the portion of the Coating and Fusion Line (CFL) that includes all elements from the Unwind Roll to the Master Speed Roll. This includes the derivation of mathematical models that describe the dynamics of the web and of the web line components. Additionally, this model involves the evaluation of web parameters as well as the description of the control strategies employed on the CFL. In subsequent chapters, this model will be used to analyze the existing control strategies and will also be utilized in the development and evaluation of improvements to the control of specific elements of the CFL. The simplification of the unwind section of the CFL is presented in Section 2.2. Subsequently, Section 2.3 discusses the derivation of the strain, tension, and velocity equations of the web and the dynamic equations of certain components of the line (such as the dancer and the accumulator). Following this discussion, a description of the procedures used to evaluate the viscoelastic parameters and the friction torque is presented in Section 2.4. The control strategy for maintaining the web tension and velocity is then given for each of the controlled components in Section 2.5. Section 2.6 concludes this chapter with a discussion of the applicability of the material presented herein to the subsequent chapters. 10 2.2 Simplification of the CFL For simulation and analysis purposes, the portion of the CFL shown in Fig. 1.3 was simplified in Fig. 2.1, which shows the section of the CFL under consideration following several modifications. The first alteration is that only the major components of the line will be considered, meaning that the simplified model will only contain the Unwind Roll, Pull Roll 1, the Unwind Accumulator, Dancer 1, and the Master Speed Roll. Additionally, the load cell roll is shown with dashed lines to signify that it will provide tension feedback for Span 1 but not contribute directly to the dynamics of the system. As can be seen in Fig. 2.1, other rolls are also included (such as Accumulator Entry/Exit Rolls) in an attempt to match the actual configuration as accurately as possible. It should be noted that both the span lengths between components and the wrap angle of the web around the rollers will be maintained even though this is not depicted. The last simplifying assumption is that the tensions within the wound material roll and the span after the Master Speed Roll are set to the reference tension value for their corresponding portions of the CFL. Span 7 tr v0(t) Master Speed Roller (Pull Roll #2) Dancer #1 R30 Accumulator Exit Roll Accumulator Entry Roll Unwind Roll Span 5 Span 6 Span 31 Span 2 Span 1 Pull Roll #1 (Load Cell) v1(t) v2(t) v31 (t) v4(t) v5(t) v6(t) v7(t) v317 (t) Span 318 Figure 2.1: Simplified model of Unwind Roll to Pull Roll 2 11 2.3 Longitudinal Dynamics This section describes the tension, velocity, and strain dynamic equations for the portion of the CFL shown in Fig. 2.1. Equation (2.1) given below describes the web velocity dynamics at the Unwind Roller with f0 as the friction term which can take several forms based on the friction model [1]. v˙0(t) = t1(t)R2 0(t) J0(t) − n0R0(t)u0(t) J0(t) − f0R0(t) J0(t) − twv2 0(t) 2 J0(t) J0(t) R2 0(t) − 2 wbwR2 0(t) (2.1) where v0 is the peripheral velocity of the Unwind Roll, R0 is the radius of the Unwind Roll, t1 is the web tension in Span 1, J0 is the inertia of the Unwind Roll, n0 is the conversion between controller output and applied braking torque, u0 is the control torque applied to the Unwind Roll, tw is the web thickness, w is the web density, and bw is the lateral web width. Notice in Equation (2.1) that the radius, R0, and the inertia, J0, are shown to vary with time. The reason is because the material roll radius (and hence the inertia) becomes smaller as material is released into the web line. This occurs at a rate given by the following equation [1]. ˙R 0(t) − twv0(t) 2 R0(t) (2.2) The following equation describes the velocity dynamics for Rollers i = 1, 2, 4, 6, and 7 [1]. Note that rollers 2, 4, and 6 are idle rollers thus for i = 2, 4, and 6, ui(t) = 0. Ji Ri v˙i(t) = (ti+1(t) − ti(t))Ri + niui(t) − f (2.3) where Ji is the inertia of Roller i, Ri is the radius of Roller i, vi is the peripheral velocity of Roller i, ti (ti+1) is the web tension in Span i (i + 1), ni is the gear ratio 12 between the motor shaft and roller shaft of Roller i, ui is the control input for Roller i, and f is the constant friction torque applied to the roller. Equation (2.4) describes the velocity of the web at the roller within the accumulator for i = 1, ..., 17 [2]. In actuality, the dynamics depend on the accumulator carriage velocity. However it is assumed that it does not significantly affect the value of v3i(t) (the peripheral speed of Roller i within the accumulator) since the carriage velocity is much slower than that of v3i(t). Note that t318(t) t4(t). J3i R3i v˙3i(t) = (t3i+1(t) − t3i(t))R3i − f (2.4) where J3i is the inertia of Roller i within the accumulator, R3i is the radius of Roller i within the accumulator, and t3i (t3i+1) is the web tension in Span i (i + 1) within the accumulator. The velocity v5(t) (the peripheral speed of Dancer 1 roller) is dependent on the motion of the dancer roller. However, as in the case of the accumulator, the dancer roller motion does not significantly affect the velocity of the web, so it is ignored. The equation describing the dynamics of the web velocity at this location is given below [1]. J5 R5 v˙5(t) = (t6(t) − t5(t))R5 − f (2.5) where J5 is the inertia of Dancer 1 roller, R5 is the radius of Dancer 1 roller, and t5 and t6 are the web tensions in the spans upstream and downstream of Dancer 1, respectively. The tension dynamics were derived by assuming that the web material exhibits viscoelastic behavior. The viscoelastic characteristics are captured using the model shown in Fig. 2.2 which shows a Maxwell model in parallel with a linear spring where Ev is the spring constant for the Maxwell component, b is the damping constant for the Maxwell component, and E is also a spring constant [4]. The stress and strain in 13 the model are related through Equation (2.6) [4]. E Ev b Figure 2.2: Maxwell element in parallel with a linear spring 1 b i(t) + 1 Ev ˙ i(t) = E b "i(t) + 1 + E Ev "˙i(t) (2.6) where i is the web stress in Span i and "i is the web strain in Span i. Since i = ti/A (A is the web crosssectional area), the tension in Span i, for i = 1, ..., 7, is related to the strain by Equation (2.7). ˙ ti(t) = − Ev b ti(t) + EEvA b "i(t) + (Ev + E)A"˙i(t) (2.7) By using the law of conservation of mass around a control volume containing the span between two fixed adjacent rollers along with the assumptions that the strain is small and uniform along the span, the following relationship between the peripheral velocity of upstream and downstream rollers and the strain can be derived [4]. "˙i(t) = vi(t) Li (1 − "i(t)) − vi−1(t) Li (1 − "i−1(t)) (2.8) where Li is the length of Span i. Equation (2.8) describes the strain dynamics for i = 1, 2, and 7. The strain dynamics for the spans in the accumulator can be derived from Reference [3]. Assuming that the density and viscoelastic parameters are constant over the web’s cross section and assuming that the strain is small and constant along the 14 span, the conservation of mass relationship for the control volume shown in Fig. 2.3 can be written as shown in Equation (2.9) [3]. 3i1 (t) xc(t) ! " Control Volume " v3i1 (t) v3i (t) v3i+1 (t) 3i+1 3 (t) i (t) Figure 2.3: Two span accumulator with control volume "Z xc(t) 0 dx # d dt (1 − "3i(t)) + (1 − "3i(t)) d dt "Z xc(t) 0 dx # (2.9) = v3i−1(t) 1 − "3i−1(t) − v3i(t) [1 − "3i(t)] where xc is the accumulator carriage height and "3i ("3i−1) is the strain of Span i (i − 1) within the accumulator. Using Leibnitz rule to perform the differentiation of the second term on the left hand side of Equation (2.9), the strain dynamics for spans within the accumulator can be derived [3]. "˙3i(t) = (1 − "3i (t)) ˙xc(t) xc(t) + v3i (t) (1 − "3i (t)) xc(t) − 1 − "3i−1 (t) v3i−1 (t) xc(t) (2.10) The linear velocity of the accumulator carriage, vc, is related to the angular velocity of the motor that raises and lowers the carriage, !c, by Equation (2.11). vc(t) = nc!c(t) (2.11) 15 where nc has units of length. The motor dynamics is given by Equation (2.12) and the carriage dynamics is presented in Equations (2.13) and (2.14). Jcm!˙ c(t) = mc(t) − ncFc(t) (2.12) mc¨xc(t) = Fc(t) − XNAc i=1 t3i(t) − Fgc (2.13) x˙ c(t) = vc(t) (2.14) where Jcm is the inertia of the accumulator carriage motor, mc is the control torque applied to the accumulator motor shaft, Fc(t) is the interaction force between the motor and the carriage, mc is the mass of the accumulator carriage, NAc is the number of accumulator spans, and Fgc is the gravitational force of the accumulator carriage. Combining Equations (2.11) through (2.14) results in Equation (2.15) which shows the accumulator carriage dynamics reflected to the motor side. Note that Jc = Jcm+n2c mc. Jc!˙ c(t) = mc(t) − nc XNAc i=1 t3i(t) − ncFgc (2.15) To derive the equations for the strain dynamics for the spans immediately upstream and downstream of the dancer (Spans 5 and 6), the dancer is approximated as a two span accumulator. The strain equation for a span within an accumulator derived in Reference [3] will be used with the accumulator carriage position and velocity terms replaced with Li(t) and ˙L i(t), respectively, where i = 5, 6 for Spans 5 and 6, respectively. Thus the strain in Span 5 is given by Equation (2.16) [3] and that of Span 6 is presented in Equation (2.17) [3]. "˙5(t) = (1 − "5 (t)) ˙L 5 (t) L5(t) + v5 (t) (1 − "5 (t)) L5(t) − (1 − "4 (t)) v4 (t) L5(t) (2.16) "˙6(t) = (1 − "6 (t)) ˙L 6 (t) L6(t) + v6 (t) (1 − "6 (t)) L6(t) − (1 − "5 (t)) v5 (t) L6(t) (2.17) 16 The lengths L5(t) and L6(t) can be represented as nominal lengths plus varying lengths. The nominal length of Span 5, L5n, is distance AB from Fig. 2.4 and that of Span 6, L6n, is CD. Although the dancer motion is purely rotational, it is assumed that the angular displacement of the dancer from vertical, d(t), is small. Therefore, its displacement can be approximated as being the arc length from E to E0 (see Fig. 2.4). This distance in terms of d(t) is shown in Equation (2.18) below. EE0 = l d(t) (2.18) where l is the distance OE in Fig. 2.4. Thus, the total length of Spans 5 and 6 can be approximated as shown in Equations (2.19) and (2.20). L5(t) = L5n + l d(t) (2.19) L6(t) = L6n + l d(t) (2.20) The first time derivative of Equations (2.19) and (2.20) are given below. ˙L 5(t) = l ˙d(t) (2.21) ˙L 6(t) = l ˙d(t) (2.22) However, Dancer 1 does not have its zero position at vertical nor does it measure angular displacement; its zero position is a constant n clockwise from vertical (see Fig. 2.5). The dancer displacement is measured with a linear transducer which is placed a distance lx (distance OA in Fig. 2.5) down the lever arm from the pivot. This measured displacement, xt(t), is related to EE0 and d(t) by the following equation. d(t) = EE0 l = xt(t) − lx sin( n) lx (2.23) 17 Thus d(t) is related to xt(t) by the following. d(t) = xt(t) − lx sin( n) lx (2.24) The time derivative of Equation (2.24) is given below. ˙d(t) = x˙ t(t) lx (2.25) Thus, combing Equations (2.19) through (2.22) with (2.24) and (2.25), the total lengths of Spans 5 and 6 and their first time derivatives in terms of the measured displacement, xt(t), can be calculated as shown in Equations (2.26) through (2.29). L5(t) = L5n + l lx (xt(t) − lx sin( n)) (2.26) L6(t) = L6n + l lx (xt(t) − lx sin( n)) (2.27) ˙L 5(t) = l lx x˙ t(t) (2.28) ˙L 6(t) = l lx x˙ t(t) (2.29) Substituting Equations (2.26) and (2.28) into Equation (2.16) and Equations (2.27) and (2.29) into Equation (2.17) gives the strain dynamics in Spans 5 and 6 in terms of the measured dancer displacement and are given in Equations (2.30) and (2.31), respectively. "˙5(t) = (1 − "5 (t)) lx˙ t(t) L5n lx + lxt (t) − llx sin ( n) + (1 − "5(t)) lxv5 (t) L5nlx + lxt (t) − llx sin ( n) (2.30) − (1 − "4 (t)) lxv4 (t) L5n lx + lxt (t) − llx sin ( n) 18 "˙6(t) = (1 − "6 (t)) lx˙ t(t) L6n lx + lxt (t) − llx sin ( n) + (1 − "6(t)) lxv6 (t) L6nlx + lxt (t) − llx sin ( n) (2.31) − (1 − "5 (t)) lxv5 (t) L6n lx + lxt (t) − llx sin ( n) (t) d C A D B E B‘ E‘ D‘ 5 6 5 6 O Figure 2.4: Displaced dancer The dancer velocity dynamics are determined from the dancer free body diagram which is shown in Fig. 2.6. The constant applied force F is applied to keep the dancer vertical when the tension in Spans 5 and 6 are equal to the reference tension. This force is applied with a piston air cylinder device which is on a pivot so that F is not always completely horizontal. However, assuming small variations in tension such that the dancer movement is not significant, the direction of F can be approximated as being horizontal. Note that the web wrap angle of the dancer is not 180 degrees. This means that t5(t) and t6(t) are not applied horizontally. Thus there is a vertical component associated with each tension. It is also assumed that due to the small dancer movement, 19 O A A Xt(t) n(t) d(t) Dancer Zero Poistion Displaced Dancer Position Figure 2.5: Zero, vertical, and displaced positions of Dancer 1 the angles with respect to horizontal at which t5(t) and t6(t) are applied at Dancer 1 remain constant. Using Fig. 2.6, the torques due to the horizontal and vertical components of t5(t) and t6(t) about the pivot point of Dancer 1 (point O from Fig. 2.6) are calculated to be as follows. Tt5x = −(l cos( d(t)) + Rd sin( 5))t5(t) cos( 5) (2.32) Tt5y = −(l sin( d(t)) + Rd cos( 5))t5(t) sin( 5) (2.33) Tt6x = −(l cos( d(t)) − Rd sin( 6))t6(t) cos( 6) (2.34) Tt6y = (l sin( d(t)) + Rd cos( 6))t6(t) sin( 6) (2.35) where Tt5x is the torque at the dancer pivot (point O form Fig. 2.6) due to the horizontal component of t5(t), Tt5y is the torque at the dancer pivot due to the vertical component of t5(t), Tt6x is the torque at the dancer pivot due to the horizontal 20 F d (t) m g R m g L t 6 (t) t 5 (t) !6 !5 "5 "6 O A B C Figure 2.6: Free body diagram of Dancer 1 component of t6(t), Tt6y is the torque at the dancer pivot due to the vertical component of t6(t), i is the angle from horizontal that the tension in Span i is applied on the dancer roller, i is the wrap angle of the web around the dancer from horizontal to the point where the tension in Span i is applied on the dancer roller, Rd is the radius of the Dancer 1 roller, and l is distance OC in Fig. 2.6. Using the above relations and Fig. 2.6, the equation of motion of Dancer 1 about O can be derived. The equation of motion in terms of d(t) is as follows. 21 Jdt ¨ d(t) = FlF cos( d(t)) − lLmLg sin( d(t)) − lmRg sin( d(t)) + (l sin( d(t)) + Rd cos( 6))t6(t) sin( 6) − (l cos( d(t)) − Rd sin( 6))t6(t) cos( 6) − (l sin( d(t)) + Rd cos( 5))t5(t) sin( 5) − (l cos( d(t)) + Rd sin( 5))t5(t) cos( 5) (2.36) where Jdt is the inertia of Dancer 1, mL is the combined mass of both dancer pivot arms, mR is the mass of the Dancer 1 roller, and lL and lF are distances OA and OB, respectively. Equation (2.25) gives the first time derivative of d(t) in terms of x˙ t(t). Differentiating this equation again will result in Equation (2.37), the second derivative of d(t) in terms of ¨xt(t). ¨ d(t) = ¨xt(t) lx (2.37) Substituting Equations (2.24) and (2.37) into Equation (2.36) and assuming d(t) is small so that cos( d(t)) 1 and sin( d(t)) d(t) yields the equation of motion of Dancer 1 in terms of its measured linear displacement. This is given below. Jdt lx ¨xt(t) = FlF + −lLmLg − lmRg + l sin( 6)t6(t) − lt5(t) sin( 5) lx xt(t) + (−l sin( n) sin( 6) + Rd cos( 6) sin( 6) − l cos( 6) + Rd sin( 6) cos( 6))t6(t) + (l sin( n) sin( 5) − Rd cos( 5) sin( 5) − l cos( 5) − Rd sin( 5) cos( 5))t5(t) + lLmLg sin( n) + lmRg sin( n) (2.38) 2.3.1 Linearized Dynamics This section describes the linearization of Equations (2.1), (2.7), and (2.8) for the simplified model shown in Fig. 2.7. 22 v0(t) Unwind Roll Span 1 (Load Cell) Pull Roll #1 t1(t), 1(t) v1(t) vref Figure 2.7: Section of web line for linearized dynamics The first step is to assume that the Unwind Roll radius R0(t) is slowly changing so that R0(t) constant. Additionally, it is assumed that the reference velocity for Pull Roll 1, vr, is constant and the wound in strain, "0, is equal to the reference value and is also constant. Thus Equations (2.1) and (2.8) reduce to Equations (2.39) and (2.40). v˙0(t) = t1(t)R2 0 J0 − n0R0u0(t) J0 − f0R0 J0 (2.39) "˙1(t) = vr L1 (1 − "1(t)) − v0(t) L1 (1 − "r) (2.40) The second step is to define the variational dynamics by using the following relationships: v0(t) = V0(t) + vr, t1(t) = T1(t) + tr, and "1(t) = 1(t) + "r, where V0(t), T1(t), and 1(t) are, respectively, the velocity, tension, and strain deviations from their corresponding reference values. Additionally, u0(t) = U0(t) + u0r, where u0r is the control input required to maintain equilibrium and U0(t) is the deviation of the control input from the equilibrium value. With these substitutions, Equations (2.7), (2.39), and (2.40) become Equations (2.41) through (2.43). 23 ˙V 0(t) = R2 0 J0 (T1(t) + tr) + n0R0 J0 (U0(t) + u0r) − R0 J0 f (2.41) ˙T 1(t) = − Ev b (T1(t) + tr) + EEvA b ( 1(t) + "r) + (Ev + E)A˙ 1(t) (2.42) ˙ 1(t) = vr L1 (1 − 1(t) − "r) − V0(t) + v0r L1 (1 − "r) (2.43) At equilibrium, V0(t), ˙V 0(t), T1(t), ˙T 1(t), 1(t), ˙ 1(t), and U0(t) are all zero. Thus, the equilibrium conditions are as follows. u0r = −R0tr + f n0 (2.44) "r = tr EA (2.45) vr0 = vr (2.46) Inserting the equilibrium conditions into Equations (2.41) through (2.43) yields the linearized variational dynamics given below. ˙V 0(t) = R2 0 J0 T1(t) + n0R0 J0 U0(t) (2.47) ˙T 1(t) = − Ev b T1(t) + EEvA b 1(t) + (Ev + E)A˙ 1(t) (2.48) ˙ 1(t) = − vr L1 1(t) − (1 − "r) L1 V0(t) (2.49) 2.4 Parameter Evaluation This section describes the evaluation of the viscoelastic parameters and the friction torque found in the dynamic equations. 24 2.4.1 Viscoelastic Parameter Evaluation The terms E, Ev, and b are obtained by using nonlinear regression analysis to match a stress model derived from Equation (2.6) to data from tensile test results of two web materials. The tensile tests were conducted in Reference [12]. This procedure is similar to one conducted in Reference [7]. The two materials tested were the Felt, Hot Melt Calendar, Gel (FHG) composite and the Royelle Felt and Gel (RFG) composite. These materials were chosen because they are indicative of the materials that are used in the unwind section of the CFL. The tensile tests were conducted per ASTM D638. Stress Model Derivation Before the regression analysis can be discussed, the model of the web stress during the tensile tests, ¯ (t), must be derived. A tensile test consists of loading a strip of material in tension such that the strain rate is constant until the specimen ruptures [8]. After each test, the strain data was plotted versus time. These plots showed that all of the strain data had the form of a straight line with zero intercept and a slope of 0.0024 in/in s . Hence, Equation (2.6) becomes a first order differential equation as given below. 1 b ¯ (t) + 1 Ev ˙¯ (t) = 0.0024E b t + 0.0024 1 + E Ev (2.50) The solution to Equation (2.50) gives the web stress as a function of time during the tensile tests and is given below. ¯ (t) = 0.0024Et + 0.0024b − 0.0024be−Ev b t (2.51) Regression Analysis Now that the stress model has been chosen, the regression analysis procedure can be discussed. The objective of the regression analysis is to find the viscoelastic parame 25 ters such that min {E,Ev,b} J = vuut Xn i=1 (¯ i − mi)2 n (2.52) where mi is the ith stress measurement from the tensile test, ¯ i is the corresponding model stress value, and n is the total number of measurements. J represents the rootmeansquare (RMS) value between the measured data and the model values. ¯ i is calculated using Equation (2.51) and the time stamp corresponding to mi. The method chosen to solve the optimization statement given in Equation (2.52) is the Cyclic Heuristic Search [9]. The procedure for this method is as follows. Initial values for E, Ev, and b are randomly selected and are used to calculate the n ¯ i values. These are used in Equation (2.52) to determine the base objective function value, J0. E is then incremented by dE, the n ¯ i values are calculated using Equation (2.51), and the trial objective function value, Jt, is determined using Equation (2.52). If Jt < J0, then Jt becomes the new base objective function value and dE is increased. Otherwise, J0 remains the base objective function value and dE decreases in magnitude and changes sign. This procedure is repeated with Ev and b, which completes one optimization cycle. The above procedure is repeated until the stopping criterion is satisfied, thus yielding the E, Ev, and b set that best matches the model described by Equation (2.51) to the tensile test data. The selected stopping criterion for the optimization procedure defined in Reference [10], will now be described. The RMS between the measured data and the model will generally asymptotically decrease with each optimizer iteration to the minimum. Likewise, the RMS of a random sampling (RRMS) of a random subset of the data will also show this trend and will have random perturbations. Viewing the RRMS over progressive iterations gives the optimization process the appearance of a noisy system transitioning from transient to a steady state value. If the steady state condition is detected, then the minimum defined by Equation (2.52) has been found and 26 optimization iterations should cease. The method for determining steady state is summarized below. Let r, the ratio statistic that determines steady state, be defined as follows. r = 2n 2 d (2.53) where 2n is a moving average and 2 d is a measure of variance along the data trend [9]. At steady state, 2 d is an unbiased estimate, assuming both the data and noise are independently distributed. The expressions for these two variances are given below. 2n = 1 N − 1 XN i=1 (RRMSi − RRMS)2 (2.54) 2 d = 1 2(N − 1) XN i=1 (RRMSi − RRMSi−1)2 ! (2.55) where RRMSi is the RRMS of the ith optimization iteration and RRMS is the average RRMS over the past N optimization iterations. Note that these variances are calculated at each optimizer iteration. The calculations associated with Equations (2.54) and (2.55) present a computational burden. Thus, a technique which requires less computational effort was used. This technique uses exponentiallyweighted moving averages (firstorder filtered values) in place of the variances defined by Equations (2.54) and (2.55). The expressions for these exponentiallyweighted moving averages, 2 fi and 2 fi , are given below. 2 fi = 2 RRMSi − RRMSfi−1 2 + (1 − 2) 2 fi−1 (2.56) RRMSfi = 1RRMSi + (1 − 1)RRMSfi−1 (2.57) 2 fi = 3 (RRMSi − RRMSi−1)2 + (1 − 3) 2 fi−1 (2.58) 27 where 1, 2, and 3 are filter factors and are comparable to the inverse of N. Equations (2.56) and (2.58) replace Equations (2.54) and (2.55), respectively. Additionally, the filtered value, RRMSfi , is used instead of the RRMS. The equivalent of Equation (2.53) can now be written as the following. ri = (2 − 1) 2 fi 2 fi (2.59) At steady state, the expected value of r is unity, otherwise it is much larger. However, due to the noise presented by the RRMS calculation, it is possible for the optimizer to not be at steady state and yet have an r value near unity. To minimize the probability of accepting this steady state condition when it is not true, Reference [9] recommends using 1 = 2 = 3 = 0.05 with a critical rvalue of 0.8. Thus, at some optimizer iteration when the rvalue is below 0.8, it is assumed that steady state has been reached and thus the minimum has been found. The entire regression procedure can now be described. After the completion of each optimizer iteration, the RRMS is calculated and used in Equations (2.57) through (2.58). The ratio statistic for that iteration is calculated using Equation (2.59). If this value is less than 0.8, the optimizer has found the best possible E, Ev, and b set and thus stops iterating. Otherwise, the optimizer iterates again and the above procedure repeats until the ratio statistic drops below 0.8. Regression Analysis Results Five separate tensile tests were conducted on both FHG and RFG materials. A sample of a typical stress versus time curve for the tensile tests is shown in Fig. 2.8. There are three distinct regions. During Region 1, there is calibration error in the tensile test machine that occurs for each test. Region 2 is the period from the correction of the calibration error to the rupture of the specimen. Region 3 is the portion of the test after the specimen ruptures. Only the data from Region 2 was 28 0 10 20 30 −5 0 5 10 15 x 104 Time (sec) Stress (lbf/ft2) Region 1 Region 2 Region 3 Figure 2.8: Typical stress versus time for material during tensile testing used in the regression analysis as it is the only reliable data. In order to be certain that the best set of viscoelastic parameters has indeed been found, the required number of independent random starts of the regression analysis optimizer must be determined. From Reference [9], in order to be c confident that at least one of the best f × 100% results have been found, M independent random starts are required. The value of M is determined using Equation (2.60). M = integer ln(1 − c) ln(1 − f) (2.60) Thus, in order to be 95% confident that at least one of the best 10% results are found, the number of independent random starts of the regression analysis optimizer must be 28. Therefore, to determine the viscoelastic parameters, the regression analysis was performed 28 times for each of the ten sets of data. The viscoelastic parameter set that corresponded to the best results were chosen to be the “true” E, Ev, and b values. Best was defined as the smallest RMS value. The data that yielded the best results came from the FHG material. From the regression analysis, the viscoelastic parameters were determined to be: E = 9.75×105 29 lbf/ft2, Ev = 9.75 × 106 lbf/ft2, and b = 5.01 × 107 lbf·s/ft2. Figure 2.9 shows the data and the corresponding bestfit curve for the given viscoelastic parameters. As seen from the plot, the model fits the measured data well and thus the above E, Ev, and b values are sufficient to use in simulations. 0 2 4 6 8 10 0 5 10 15 x 104 Time (sec) Stress (lbf/ft2) Measured Data Model Data Figure 2.9: Stress versus time for measured data and bestfit model 2.4.2 Friction Torque Evaluation The friction present in the bearings of rollers has two components: a viscous friction that is proportional to velocity and a constant friction torque. For modeling purposes, it is assumed that the viscous friction is negligible and so the majority of the bearing friction is assumed to be the constant friction torque. In order to determine an accurate value for this term, a test was performed on two typical idle rollers used in CFL. For this test, the roller was hand spun to a speed higher than the desired test speed. A hand held tachometer was used to track the speed of the roller as it slowed due only to friction. Once the speed of the roller reached the desired test speed, a timer was started and the tachometer was removed from the roller. The time for the roller to come to a complete stop was measured for ten trials. The first test was performed on a 10.5 inch diameter roller, and all ten trials 30 Trial Initial Velocity Stopping Time Friction Torque fpm sec (ft · lbf) 1 99.9 55.28 0.309 2 100.01 60.57 0.283 3 100.99 58.14 0.297 4 98.78 61.04 0.277 5 100.02 58.05 0.295 6 100.93 60.00 0.288 7 100.73 60.00 0.287 8 100.66 61.07 0.282 9 100.80 55.93 0.309 10 99.81 56.49 0.302 Table 2.1: Test 1 Parameters and Measurements were performed at a test speed of 100 feet per minute (fpm). The second test was performed on a 9.5 inch diameter roller with six trials performed at a test speed of 100 fpm and four trials at 120 fpm. The results of the tests are presented in Tables 2.1 and 2.2. To determine the friction torque, the free body diagram of the roller is used. Since there were no other forces besides friction acting on the rollers, the equation of motion for the test rollers is as given in the following. J!(˙t) = − f (2.61) This can be approximated as J 4! 4t = − f (2.62) 31 Trial Initial Velocity Stopping Time Friction Torque fpm sec (ft · lbf) 1 99.41 64.83 0.226 2 100.46 60.83 0.244 3 100.17 65.12 0.227 4 99.38 73.36 0.200 5 99.45 71.02 0.207 6 100.98 73.77 0.202 7 120.92 90.80 0.197 8 120.10 85.95 0.206 9 119.93 87.32 0.203 10 120.21 87.84 0.202 Table 2.2: Test 2 Parameters and Measurements where J is the test roller inertia, 4! is the change in velocity of the roll in 4t stopping time, and f is the friction torque. Using classical methods for determining inertia, the friction torque can be solved for assuming that it is a constant value throughout the duration of each trial. Tables 2.1 and 2.2 show the calculated friction torques for each trial using the aforementioned calculation. The average friction torque values from both tests at each target speed are displayed in Table 2.3. As can be seen from the results, the friction torques for the 9.5 in idler roller are approximately equal even though they were run at different speeds. 32 Test Target Velocity Friction Torque (fpm) (ft · lbf) 1 100 0.293 2 100 0.218 120 0.202 Table 2.3: Average Friction Torque 2.5 Current Control Strategies 2.5.1 RSLogix5000 Operation Before the control strategy used in this section of the CFL can be discussed, the exact functionality of the RSLogix ladder logic and its PID controller must be detailed first. Ladder logic is a programming method that uses routines containing a series of rungs that have commands which are executed in order from top to bottom. The entire control strategy of the CFL is composed of multiple tasks consisting of several routines. The tasks involve performing safety checks, computing variables, and several other duties. Only one task can be executed at a time, so each is assigned a different execution period and a priority value from 1 to 15, where 1 is the highest priority and 15 is the lowest. Each task is implemented every period, but it can be interrupted by a higher priority task that happens to occur at the same time. When this happens, the higher priority task executes completely first followed by continuation of the lower priority task from where it was interrupted [5]. On the CFL, there are three tasks that are pertinent to this report: a task that governs the rewind section of the CFL (called the STI task), one that calculates the various reference values for variables of the CFL (called the Line References task), and another that controls the status of certain line components (called the Main task). The Unwind Roll uses an RSLogix PID for tension control which resides in 33 = Task Executed = Task Interrupted 0 10 20 30 40 50 60 70 80 90 STI Line References Main Time (ms) Figure 2.10: Execution times of each task[5] the Main task whereas the PID that controls tension for both Pull Roll 1 and the Unwind Accumulator (Dancer 1 PID) resides in the Line References task. Table 2.4 lists the priorities and execution periods of each of these tasks. Table 2.4: Priorities and Periods of RSLogix Tasks Task Priority Period STI 1 30 ms Line References 5 10 ms Main 6 15 ms Figure 2.10 shows a graphic representation of the times when each of the three aforementioned tasks are implemented. Notice that every 30 ms when each task is scheduled to be performed, the STI task interrupts the other two due to its higher priority. Additionally, the scheduled execution time is constant regardless of any intrusions by a higher priority task. Figure 2.11 shows the PID procedure used by RSLogix in block diagram form that is utilized for automatic control[6]. Table 2.5 shows the values of the Output Bias percentage, maximum (MAXI) and minimum (MINI) process variable (PV), maximum (MAXS) and minimum (MINS) engineering unit scaling value, and maximum (MAXCV) and minimum (MINCV) control variable (CV) value for the Unwind Roll and Dancer 1 controllers. The process variable is the measured feedback and the 34 control variable is the value output by the PID that is sent to an actuator[6]. +  PV + + MAXSMINS MAXIMINI Kp,RSL s K + i,RSL+KSP d,RSLs + + +  MINI MINS 100 MAXSMINS Error Output % MAXCVMINCV 100 MINCV CV Figure 2.11: Block diagram of RS Logix PID Table 2.5: Parameter Values Used in RSLogix PIDs Variable Unwinder PI Dancer 1 PID Bias 0 0 MAXI 100 100 MINI 0 0 MAXS 100 100 MINS 0 0 MAXCV 100 1.1 MINCV 0 0.9 The PID equation first uses the error between the SP and PV to calculate the output. Note that both the error and the output are expressed as percentages of the engineering unit range. The last step performed by the PID function as used in RSLogix is to convert the PID output percentage into the units of the control variable [6]. 35 + – Roller Dynamics Tension & Strain Dynamics Manual Control Time < 10 s Time > 10 s Automatic Control Kpm tr PI Output t1 v0 Round Control Variable Average St1 St1 Round Round Unwind Tension PI knSb tps+1 Figure 2.12: Control strategy for Unwind Roll 2.5.2 Unwind Roll Control Strategy The Unwind Roll torque is controlled by a brake. Therefore, the rotation of the roll is due to the material being pulled by Pull Roll 1. This means that there is no direct velocity control of the Unwind Roll. Thus, there is only tension control via braking for the portion of the web line spanning from the material roll to Pull Roll 1. The tension control strategy presently being employed is shown in Fig. 2.12. This scheme has two parts: manual control and automatic control. The manual control is utilized only for the first 10 seconds after the start up of Pull Roll 1 whereafter the automatic control is employed. When the switch from manual to automatic control occurs, the RSLogix software calculates the accumulated error required to produce the same CV output that is generated by the manual control. This process results in a smooth transition from open loop to closed loop control without causing a sudden increase or decrease in the CV [6]. During manual control, a scale factor, Kpm in Fig. 2.12, input by an operator multiplies the tension set point to produce the PI output percentage. For the simulations 36 conducted in Section 3.2, Kpm has a value of 0.6. One reason for implementing the manual control is that it allows RSLogix to acquire 10 tension measurements so that when automatic control is initiated, the tension averaging calculation (to be discussed later) will utilize actual tension measurements as opposed to using the measurements that were recorded before the Unwind Roll was initiated. For automatic control, web tension measurements taken by a load cell are used as feedback to produce the controller output. The tension measurements are sent to the RSLogix software as integer values where they are sampled once every second and then averaged with the past 9 tension samples. This mean, set as an integer value, is subtracted from the tension set point. This difference is used in the PI equation to produce the PI output percentage that, when rounded to the nearest integer, is the control variable. During automatic control when the actual tension and the set point are different, the integral term of the PI accumulates this error between the two values. At the instant when the error becomes zero (thus rendering the proportional term of the PI to be zero), the integral part of the PI is able to supply the output required to maintain the desired tension because of the accumulated error. One important fact to note is that the update time for the PI is 0.2 seconds so the PI continues to integrate in between the tension samples. Thus for a large error, the brake would apply an extreme (either high or low depending on the sign of the error) and continuously increasing or decreasing amount of torque to the Unwind Roll until the next sample is measured. The PI equation is displayed in Equation (2.64) below. The PI output percentage, computed either using the manual or the automatic method, is limited to 1% to 80%. By using Fig. 2.11 with Table 2.5, it may appear that the PI output percentage and control variable have the same numerical value. However, the control variable is set as an integer value and thus is the integer equivalent of the PI output percentage. The control variable corresponds to an out 37 put voltage. This voltage is then transformed into pressure via a voltagetopressure (E/P) transducer. The pressure is then applied to brake pucks which generate the braking torque on the Unwind Roll. In an effort to increase efficiency, two separate unwind structures, and thus two different brakes (designated Brake 1 and Brake 2), are used so while one is supplying the CFL with material, the other can be reloaded. These brakes are individually configured and thus the configuration of the two brakes can be different (as they most often are) which, as will be seen in Section 3.4, results in dissimilar tension performances and controller outputs for a given PID controller. The dynamics of the pneumatic device used to supply the pressure for the brakes are assumed to be first order of the following form. pp˙(t) + p(t) = knCV (t) (2.63) where p is the time constant for the pneumatic device, p(t) is the pressure within this device, kn is the unit conversion constant from units of control variable to psi, and CV (t) is the control variable value sent from the controller. In order to determine p, a simple test was conducted. A step increase was made in the CV and the time required for the pressure in the pneumatic device to reach its steady state value was recorded. From elementary systems analysis, dividing this rise time by four will result in the time constant of the device given an increase in CV. This procedure was repeated for a step decrease in CV. The resulting time constants from these two tests are shown below. p = 8>< >: 1.625 for CV increasing 0.625 for CV decreasing The determination of kn is discussed later. Note that there are several scaling factors in Fig. 2.12. The factor of St1 accounts for the fact that in the RSLogix program, the tension values are represented as percentages of 400 lbf, the maximum tension that can be induced in the web by the brake when a full material roll is attached. This means that the tension PI equation 38 for the Unwind Roll is given by Equation (2.64). OPI,UW = Kp,UW(St1(tr − tmeas(t))) + Ki,UW Z St1(tr − tmeas(t)) (2.64) where OPI,UW is the PI Output, Kp,UW is the proportional gain, Ki,UW is the integral gain, tr is the reference tension, and tmeas is the tension feedback. Since the CV is just the integer equivalent of the PI output percentage, the transformation from the latter to the former is shown in Fig. 2.12 as a rounding block. In order to find values for kn and Sb (the conversion from braking pressure to applied torque), the following information gathered from the control program was used. The PI is calibrated so that 100% of PI output (CV value of MAXCV for the Unwinder from Table 2.5) corresponds to 10 volts and the voltage to pressure transducer (E/P device) is set so that 10 volts corresponds to 46 pounds per square inch (psi) of pressure. Thus kn = (46/10) × (10/MAXCV ) = 46/MAXCV . To determine the amount of braking torque per psi of applied pressure, the following reasoning is used. It is assumed from the RSLogix files that 100% of PI output corresponds to 400 lbf of induced tension. Since the web tension effected by the brake varies with roll radius, it is assumed that this relation was determined using a full material roll. This assumption leads to Equation (2.65) which shows the relation between brake pressure and the corresponding applied torque. Sb = R0i × tmax Pmax (2.65) where R0i is the initial material roll radius, tmax is 400 lbf, and Pmax is the applied braking pressure corresponding to 100% of PI output. Although Equation (2.65) was derived presuming a full material roll, it is assumed that Sb is constant throughout the entire unwinding of the roll. In the model verification section (3.2) of Chapter 3, this value is adjusted in order to better match data measured from the CFL. 39 2.5.3 Pull Roll 1 and Unwind Accumulator Control Pull Roll 1 uses tension feedback and velocity control. The tension control is used to produce a correction to the velocity reference value. In the CFL, the tension feedback is provided by Dancer 1 which uses the linear transducer mentioned in Section 2.3 to measure the linear displacement of the dancer. This displacement value is then passed to the ladder logics program as a percentage of the maximum transducer stroke where it is compared with the reference dancer position within a PID in the software. The PID equation is given in Equation (2.66). OPID,D1(t) = Kp,D1ex(t) + Ki,D1 Z ex(t)dt + Kd,D1 dex(t) dt (2.66) where OPID,D1 is the PI Output, Kp,D1 is the proportional gain, Ki,D1 is the integral gain, Kd,D1 is the derivative gain, and ex(t) = 100 (xtr − xt(t)) /xtmax is the dancer displacement error as a percentage of the maximum transducer stroke. This output is scaled from 0.9 to 1.1 which will provide ±10% trim to the Pull Roll 1 velocity. This calculation is shown in Equation (2.67) below. dx(t) = OPID,D1(t) MAXCVD1 −MINCVD1 100 +MINCVD1 (2.67) where dx(t), referred to as Dancer 1 Trim, is the output of the PID and MAXCVD1 and MINCVD1 are the values shown in Table 2.5 for the Dancer 1 PID. Dancer 1 Trim is not the velocity correction, but a scaling factor that, when multiplied by the reference line speed (Pull Roll 2 speed reference), gives the corrected speed reference for Pull Roll 1 when the accumulator carriage is stationary. Equation (2.68) shows the values for the Pull Roll 1 reference speed, v1r for the various Unwind Accumulator phases (see discussion below). v1r(t) = 8>>>>< >>>>: vlsdx(t) if carriage is stationary vlsdx(t) + 50fpm if filling 0 if emptying (2.68) 40 where vls is the line reference speed. The speed reference that is sent to the drive is the ramped equivalent of the value determined from Equation (2.68). This ramped value is calculated at every scan of the “Line References” routine and is shown in Equation (2.69) below. v1rr(t) = 8>>>>< >>>>: v1rr(t − T) + 0.2 if v1r(t) > v1rr(t − T) v1rr(t − T) − 0.2 if v1r(t) < v1rr(t − T) 0 if 0.5s after emptying intiation (2.69) where v1rr(t) is called the ramped velocity reference and T is the sampling period. The velocity control for Pull Roll 1 is performed in a PID internal to a Rockwell Powerflex 700 drive attached to the motor which powers Pull Roll 1. This drive bases its PID calculations on motor speed and not web line velocity. Thus v1r(t) needs to be converted to motor speed, which is accomplished using the gear ratio between the drive motor and the attached roller. The drive uses speed feedback from an encoder which is attached to the motor. This value is compared to the speed reference calculated above to produce a speed error which is sent to the PID inside the drive. The PID will then provide the appropriate amount of torque in order to drive Pull Roll 1 at the desired velocity. The control strategy of the Unwind Accumulator is as follows. When the current roll is near depletion and must be replaced, the accumulator must empty in order to supply the rest of the line with material. To begin this process, an operator presses the Unwinder Stop Push Button when the material roll is empty. In actuality, this button stops Pull Roll 1 (making its speed reference equal to zero) since it directly controls the speed of the material roll. To stop the emptying process, an operator presses the Unwinder Start Push Button (i.e., Pull Roll 1 is started) when a new material roll is ready. At this stage, the accumulator capacity is low and needs to be replenished before the next roll change. In order to begin filling the accumulator, the Unwinder Start Push Button must be pressed first and then the Accumulator 41 Fill Button must also be pressed. During the filling process, Pull Roll 1 is driven 50 feet per minute (fpm) faster than the line reference speed, but it is still controlled as described above. The accumulator stops filling when it reaches a limit switch located at the top of the accumulator structure. During the emptying phase and the initial and final portions of the filling procedure, the accumulator uses tension feedback and carriage velocity control. When the carriage is moving, dancer position feedback is used to create a reference speed for the carriage in order to maintain the reference tension. This feedback is provided by Dancer 1. The calculated Dancer 1 Trim from Equation (2.67) is multiplied by the speed reference for Pull Roll 2. This product is then subtracted from the ramped speed reference for Pull Roll 1 for the corresponding accumulator phase. This difference, when divided by the number of accumulator spans, gives the speed reference for the accumulator carriage. The calculation of the reference speed for the carriage, vcr, is shown in Equation (2.70). vcr(t) = 1 NAc (v1rr(t) − vlsdx(t)) (2.70) Note that at the initiation of the emptying process, v1r(t) will be set to zero but the ramp defined by Equation (2.69) is allowed to work for another 0.5 seconds. This allows the accumulator carriage to accelerate per Equation (2.70). Afterwards, v1rr(t) is set to zero. As will be seen in Chapter 5, this sudden step in the ramped reference causes a speed mismatch between Pull Roll 1 and the Unwind Accumulator. Once Pull Roll 1 has accelerated to the nominal fill speed, the carriage will have a constant speed reference of 50/NAc since the vlsdx(t) terms from Equations (2.68) and (2.70) will cancel. Therefore, during this time, the accumulator will be under speed control only. The velocity control of the accumulator carriage is performed using a Powerflex 700 drive. The drive is attached to a motor which, through a series of gears and turn 42 +  0 FPM Unwind Accumulator Control Pull Roll 1 Control Sd 1N Roller Dynamics Tension & Strain PR2 Dynamics Spd Ref Velocity PID Dancer Dynamics Carriage Motor Dynamics Carriage Velocity PID Dancer 1PID n1 Nc Sd dx vc xtr v1 xt 1 Nc n1 + 1  Figure 2.13: Control strategy for Pull Roll 1 and Unwind Accumulator under normal operating conditions screws, raises and lowers the carriage. The feedback used internally in the drive is the encoder measured motor speed, not the carriage velocity. Thus, the calculated carriage reference speed as given by Equation (2.70) above is converted into motor speed using the gear ratio that determines the amount of linear displacement of the carriage per revolution of the drive motor. The feedback measured from the encoder is subtracted from this reference value. This error is then used in the drive’s PID in order to actuate the motor to the speed that will produce the desired carriage velocity. See Figs. 2.13 to 2.15 for the control strategies of Pull Roll 1 and the Unwind Accumulator under normal operating conditions, while emptying, and while filling. 2.5.4 Pull Roll 2 Control Pull Roll 2 is the Master Speed Roll for the entire CFL and therefore does not use tension feedback to control its motor speed. The only control utilized for Pull Roll 2 is the speed control performed internally in the Powerflex 700 drive. This drive operates exactly like the drive for Pull Roll 1 except the speed reference is not provided by a dancer but is a constant value that is input by an operator. See Fig. 2.16 for the block diagram depicting the control strategy for Pull Roll 2. 43 – Roller Dynamics Tension & Strain Dynamics PR2 Spd Ref Velocity PID Dancer Dynamics Carriage Motor Dynamics Carriage Velocity PID Dancer 1 PID 1N Unwind Accumulator Control Pull Roll 1 Control Sd Sd 1 n1 Nc dx vc xtr vr1! 0 FPM v1 xt +  " + " + 1 Nc + " n1 Figure 2.14: Control strategy for Pull Roll 1 and Unwind Accumulator while emptying +  + +  Pull Roll 1 Control +  Unwind Accumulator Control ! +  + vfill Sd Sd n1 1N Roller Dynamics Tension & Strain Dynamics PR2 Spd Ref Velocity PID Dancer Dynamics Carriage Motor Dynamics Carriage Velocity PID Dancer 1 PID xtr dx vc v1 xt ! ! 1 1 Nc ! Nc ! n1 Figure 2.15: Control strategy for Pull Roll 1 and Unwind Accumulator while filling +  Roller Dynamics Velocity n7 PID v 7 vr,PR2 1 n7 Figure 2.16: Control strategy for Pull Roll 2 44 2.6 Conclusion The equations, parameters, and control strategies presented herein will be utilized in the subsequent chapters. The dynamics and control strategies associated with the Unwind Roll are used in Chapters 3 and 4. In Chapter 3, this data will be utilized to develop a simulation model and derive improvements for the current control strategy. In Chapter 4, the linearized dynamics are employed in a technique that compares the stability regions for systems with velocity and torque controlled Unwind Rolls. The control strategies and dynamics corresponding to Pull Roll 1, the Unwind Accumulator, Dancer 1, and Pull Roll 2 are used in Chapter 5 to generate a model simulation which is utilized to analyze and improve upon the current control strategies for Pull Roll 1 and Unwind Accumulator. 45 CHAPTER 3 Analysis of Unwind Roll Control and Improvements 3.1 Introduction This chapter discusses the modeling and analysis of the existing control strategy of the Unwind Roll. The analysis will be based on a model simulation using the control strategy described in Subsection 2.5.2. The credibility of the model will be based on comparison of the model output with data measured from the CFL. Based on the deficiencies of the current control strategy, three strategies will be derived with the goal of improving the tension performance. Each of these strategies are simulated and compared with the results obtained using the model of the existing strategy. Experiments are then performed on the CFL to test the effectiveness of these modifications. In this chapter, Section 3.2 discusses the verification of the system model. The improvements to the existing strategy are presented in Section 3.3, followed by the results of the experimental implementation in Section 3.4. Section 3.5 concludes this chapter with a discussion of the results obtained herein. 3.2 Model Verification This section discusses the model verification for the system presented in Fig. 2.7. A model simulation is conducted that employs the control strategy for the Unwind Roll as described in Subsection 2.5.2 as well as the mathematical models given by Equations (2.1), (2.7), (2.8), and (2.63), which describe the relevant dynamics. It is 46 assumed that Pull Roll 1 is running at the reference velocity profile shown in Fig. 3.1. This figure was constructed by assuming that the Pull Roll 1 velocity is the speed reference that would be commanded if Dancer 1 is at its reference position. This simplification allows for the analysis of this section of the CFL. An Scurve instead of step changes in speed was also implemented in order to facilitate realistic speed changes. 0 500 1000 1500 0 50 100 150 O A B C D E F Time (sec) Speed Reference(fpm) Figure 3.1: Velocity profile of Pull Roll 1 This profile represents a scenario that shows all of the speed changes from the initiation of a new roll to roll depletion. The reference speed has six phases: initial startup to line speed (OA), holding constant at line speed (AB), ramping up to 50 feet per minute (fpm) greater than the line speed (BC), holding constant at 50 fpm plus line speed (CD), decelerating to line speed (DE), and holding constant at line speed until roll depletion (EF). The first two segments are used to advance the web splice that connects the new roll to the previous one through the accumulator. The line speed is maintained in AB in order to keep the accumulator carriage stationary since it will not move when the speed reference for Pull Roll 1 is equal to the line speed. This procedure is used because the web splice may not be able to endure the 47 increased stresses associated with the web passing through an accumulator when it is filling. Segments BC and CD fill the accumulator. This is possible since the web upstream is moving faster than the web downstream and thus the carriage must rise in order to maintain the web tension and velocity. At D, the accumulator is nearly full and thus Pull Roll 1 and the accumulator carriage must decelerate, a process that happens during DE. Segment EF shows the Pull Roll 1 speed reference when the accumulator carriage is at its maximum height and stationary. Since the carriage is not moving, the speed reference for Pull Roll 1 will be the line speed [2]. The time span during the constant velocity phases were approximated from data collected from the CFL. 3.2.1 Parameter Values and Initial Conditions Tables 3.1 and 3.2 show the parameter values and the initial conditions that were employed in the simulation. The constant friction torque acting on the Unwind Roll was determined from the friction torque test discussed in Subsection 2.4.2. This is an estimate since the friction test was conducted on idle rollers and a similar test was not performed on the Unwind Rolls. The controller gains currently used on the CFL were selected for Kp,unw and Ki,unw. The web thickness, tw, and density, w, were measured from CFL web samples. The length of Span 1, L1, and the lateral web width, bw, were obtained from a drawing of the physical line. Since the model shown in Fig. 2.7 ignored the idle rollers in this section of the CFL, the resonant frequencies introduced by these idle rollers are also ignored and an average tension model is used. Thus, L1 is selected to be the average length of the spans between the first Unwind Roll and Pull Roll 1. The web crosssectional area, A, was calculated as the product of tw and bw. E, Ev, and b are selected to be the values determined in Subsection 2.4.1. The initial Unwind Roll radius value was calculated by the following procedure. The length of material for one roll was determined by integrating the velocity profile 48 curve. As the volume of the material laying flat is the same as that of the wound material, this length was related to the initial material roll radius. This relationship leads to Equation (3.1), which gives the initial radius of the material roll. The initial tension of 12 lbf was selected because that is the same value as the measured data at the beginning of a new material roll. R0i = r Lwtw + R2 c0 (3.1) Table 3.1: Parameter Values Used in Simulation Variable Value Used in Simulation Units Kp,unw 0.5 None Ki,unw 0.005 sec−1 tw 0.003833 ft w 3.182 slug/ft3 bw 12.25 ft A 0.04696 ft2 E 9.75 × 105 lbf/ft2 Ev 9.75 × 106 lbf/ft2 b 5.01 × 107 lbf · s/ft2 L1 4.5 ft f0 0.293 ft · lbf 3.2.2 Model Simulation Results The results of the conducted model simulation are shown in Figs. 3.2 through 3.6. Note that three repetitions of the same simulation are shown for clarity when comparing these results with the measured data from the CFL. For simplicity, results of 49 Table 3.2: Simulation Initial Conditions Variable Initial Condition Unwind Roll Radius, R0 22 inches Unwind Roll Velocity, v0 0 fpm Span 1 Tension, t1 12 lbf 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 5 10 15 20 25 Unwind Radius, R 0 (in) Time (sec) Figure 3.2: Unwind Roll radius (Model Simulation) the model simulation are referred to as “simulated” data (i.e., the resulting tension data from the model simulation is referred to as the “simulated tension”). Figure 3.2 shows the material roll radius. The radius of the core that the material is wound upon is 6 inches and thus when R0(t) is equal to this value, the material roll will be completely depleted. The material roll is seen to be completely empty at the end of the simulation, as desired. Figure 3.3 shows the CV of the Span 1 tension PI controller. As expected, it is constant during the manual control phase. After an initial drop, the CV is seen to oscillate between values of 13 and 15 for approximately the first 300 seconds. Beyond this time, the CV begins to decrease since R0 is decreasing. A smaller Unwind Roll radius conveys that the material roll has a lower inertia and thus requires less braking torque to slow its rotation. This implies that it becomes easier to produce tension 50 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 50 Control Variable Time (sec) Figure 3.3: Control variable for Span 1 with tension PI (Model Simulation) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 50 PI Output (%) Time (sec) Figure 3.4: PI controller output percentage (Model Simulation) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 50 100 150 200 Unwind Roll Speed, v 0 (fpm) Time (sec) Figure 3.5: Unwind Roll peripheral velocity (Model Simulation) 51 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 Tension, t 1 (lbf) Time (sec) Actual Set Point Figure 3.6: Span 1 tension (Model Simulation) in Span 1 since the change in tension is a function of the upstream and downstream roller velocities. Moreover, braking torque induces tension in Span 1 by Equation (3.2) and hence as the material roll radius wanes, the braking torque required to effect the same amount of tension in Span 1 also reduces. braking = R0tind (3.2) where braking is the applied braking torque and tind is the corresponding induced tension. During the simulation, the CV oscillates at times of transition between subsequent values. This occurs since the CV can only take on integer values. In situations where less braking is required, the PI output has to decrease by an entire percentage before a change in the CV will occur. While the PI output is decreasing, the brake is applying approximately the same amount of resistive torque which means the tension will continue to rise. Once the PI Output reaches a value such that the CV changes, the change in braking torque may decrease too much, depending on the Unwind Roll radius. If this situation occurs, the tension will drop below the reference value, causing a rise in the PI output percentage, which, once it becomes large enough, will increase the CV to a value equal to or higher than it was initially. This cycle would repeat until the braking torque corresponding to the lower CV value is sufficient to keep the 52 tension at the desired value. As the radius decreases, these oscillations will occur more frequently since the drop in braking torque induces more tension as previously discussed. Hence it can be seen that these oscillations are due to the inability of the CV to take on the exact value calculated by the PI equation. The oscillations are also due to the slow reaction of the brake due to the dynamics of the pneumatic device. Once the CV changes, the pressure will also change but at a slower rate. This will cause errors in tension since the brake is not applying the amount of torque that is commanded by the controller. The controller will then alter the CV in order to correct for these tension errors, resulting in more CV oscillations. Comparing Figs. 3.1 and 3.5 shows that the Unwind Roll velocity generally follows the reference velocity relatively well. However, there are relevant differences during transitions from ramped to constant velocity and during the CV oscillations described above. Since the Unwind Roll rotation comes solely from Pull Roll 1, there is velocity error in the transition from constant velocity to acceleration of Pull Roll 1. At times when Pull Roll 1 decelerates (transitioning from BC to CD and from CD to DE from Fig. 3.1), the large inertia of the Unwind Roll causes it to overshoot the Pull Roll 1 velocity. This occurrence also contributes to the velocity error. Additionally, when the CV oscillates, the applied braking torque also oscillates causing deviations of the Unwind Roll velocity from the reference. As the roll radius decreases, the applied braking torque during the CV oscillations has a greater effect on the Unwind Roll velocity due to the smaller inertia. This causes the velocity oscillation amplitudes to grow as the simulation progresses as witnessed in Fig. 3.7 which shows the Unwind Roll velocity during the EF portion. The tension in Span 1 is displayed in Fig. 3.6. During the manual control phase, the tension starts at 12 lbf then increases to 193 lbf before returning to approximately 75 lbf. The reason for this trend is that at the initial start up, a large amount of tension is required to accelerate the stationary Unwind Roll. The tension drop after 53 400 500 600 700 800 900 1000 1100 1200 1300 1400 99 99.5 100 100.5 101 Unwind Roll Speed, v 0 (fpm) Time (sec) Actual Set Point Figure 3.7: Unwind Roll velocity during EF portion of roll (Model Simulation) the spike is a result of the Unwind Roll overshooting the velocity of Pull Roll 1 as explained above. Since the peripheral speed of the upstream roll is greater than that of the downstream roll, the tension in Span 1 decreases. The velocity error is also significant during the other Pull Roll 1 speed changes and transitions which causes the tension variation to be large during these times, as is seen in Fig. 3.6. The CV oscillations also cause fluctuations in tension. Since the pneumatic device is slow, the braking pressure is not able to match the changes in CV. Meanwhile, this changing pressure is causing the brake to vary the velocity of the Unwind Roll, as discussed above, which induces fluctuations in the tension. As was the trend with the Unwind Roll velocity, the tension oscillations become larger as the material roll radius decreases, a fact that is shown in Equation (3.2). Since changes in the CV correspond to similar changes in braking torque for both a larger roll and smaller roll, the induced tension will increase as the material roll decreases. The final trend seen in the simulated tension is that the tension drifts and does not oscillate about the reference value of 92 lbf but rather 100 lbf. In summary, the model simulation displayed characteristics expected from the system shown in Fig. 2.7 under the given conditions. To further verify this model, the above results are compared with measured data from the CFL in the subsequent 54 section. 3.2.3 Measured Data From the CFL As discussed in Subsection 2.5.2, two separate brakes are alternately used in the control of the Unwind Roll. Thus the measured data analysis is segregated into two sections, each comparing the measured data to the model simulated data. There are three sets of data for each brake. For Brake 1, the first data set was collected on a separate day than the data sets for the latter two sets. For Brake 2, each data set was collected from a different day. Thus the differences in the data for a given brake are attributed to several factors, namely, the Pull Roll 1 velocity profile, the web material, and the particular configuration of the brake. Brake 1 Figures 3.8 through 3.10 show the measured data for Brake 1. The beginning of each run is the instant that the Pull Roll 1 velocity increases from 0 fpm. Since the Unwind Roll velocity is not measured, Pull Roll 1 velocity is displayed so that the tension and control variable data can be correlated to the action of Pull Roll 1. The first roll operated at the same speeds that the simulations were conducted; however, it does have additional velocity changes during the portion where the Pull Roll 1 velocity is supposed to be near the line speed reference. The latter rolls have a profile shape similar to that of Fig. 3.1, but are ran at slower speeds. Despite these differences between these measured velocities and the velocity profile used in the model simulation, the comparison of the tension and control variable results are still valid. As expected, the control variable is constant for the first 10 seconds of each roll and then changes based upon the tension. Note that the constant portions at approximately 1500 seconds and 3400 seconds are observed because the controller provides 55 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 50 100 150 Speed (fpm) Time (sec) Figure 3.8: Pull Roll 1 velocity using Brake 1 (Measured data) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 50 Control Variable Time (sec) Figure 3.9: Control variable using Brake 1 (Measured data) a constant braking torque at both the beginning and at the end of material rolls. The large spike seen at the end of the first and third runs is because the brake pressure was very low (in the case of the former, the controller actually saturates), thus causing the tension to drop. However, due to the sampling of tension values and the slow sampling time, the controller cannot immediately react to these changes. Eventually, as the tension drops lower and lower, the current and accumulated errors becomes large, resulting in a drastic increase in the control variable. Except for the first data set, the control variable is seen to gradually rise to its maximum value and then continually and gradually decrease throughout the entire roll. These same trends are observed in 56 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 Tension (lbf) Time (sec) Actual Set Point Figure 3.10: Tension in Span 1 using Brake 1 (Measured data) the first data set, excluding the initial increase in control variable. For the particular system configuration, the initial CV was large enough to quickly increase the tension, eliminating the need to increase the CV once automatic control was initiated. Comparing Figs. 3.3 with 3.9 shows that the simulated controller follows the same trends as those seen in the first set of measured data; they both use similar magnitudes during the automatic control phase and they both continually decrease throughout the depletion of the Unwind Roll. Additionally, the CV jumps seen in the simulation data are also present in the first set of measured data. On the other hand, the model simulation did not accurately portray the CV trends seen in the second and third measured data sets. However, the difference is not significant since disparities in magnitude can be seen even between different sets of measured data. Thus for the simulated controller, the assumptions that related the controller output to the applied torque were for only a particular brake configuration. The measured tension data is shown in Fig. 3.10. Each data set begins with a large tension spike as Pull Roll 1 accelerates from 0 fpm, another spike as Pull Roll 1 accelerates from line speed, and then a drop in tension as Pull Roll 1 decelerates back to line speed after the accumulator has been filled. For the second and third data sets, the tension does not reach the reference value until approximately 130 57 and 160 seconds after initiation, respectively. The reason is that the slow increase in their respective control variables gradually increases the braking torque, causing the tension to slowly increase. This trend is not seen in the first data set because, as discussed previously, the initial CV was large enough to produce tension values near the set point. Near the depletion of their respective rolls, the first and third data sets are seen to decrease and the second data set begins oscillate with increasing amplitude. The reason for the former trend has been discussed already, however, note that although both control variables spiked at the end of their respective rolls, the tension did not respond. The reason is that the web had already released from the core and thus any changes in the brake would not effect the tension. For the second data set, the cause of the increasing oscillation amplitudes can be attributed to two factors: (1) the gains are not appropriate for the smaller roll size and (2) the CV fluctuations are inducing tension oscillations as discussed in Section 3.2.2. Comparing Figs. 3.6 and 3.10 illustrates that the simulated tension follows the general trends of the measured tension with a few exceptions. The simulation sufficiently modeled the large spikes during Pull Roll 1 speed transitions, but did not show the tension drop at the end of the roll that was present in the first and third data sets. However, the simulated model did display the increasing oscillation amplitudes seen in the second data set. The simulated data also has similar tension magnitudes seen in the first and second measured data sets. On the other hand, the measured data sets each oscillated about their corresponding reference values, a trend that the simulation was unable to accurately predict. However, consider Fig. 3.11 which shows the tension in Span 1 using the same model except with the integral gain five times larger. Note that the tension drift has been reduced and the tension oscillates about the set point. This indicates that the model is lacking some unknown scaling or gain that is present within the hardware. Additionally, the model did not accurately predict the oscillation frequency that was seen 58 in the measure data. One possible reason for this can attributed to the model relating the braking pressure to applied braking torque. The braking pressure inflates the brake pucks which apply a force on the Unwind Roll, causing the frictional torque. These dynamics were modeled using a constant gain, but the actual dynamics are more complicated and may exhibit stickslip phenomena due to the intermittent force applied by the brake pucks on the brake cylinder. Other possible factors contributing to the difference in oscillation frequency is the span length that was used in the model and that all of the idle rollers were ignored. The various spans between the Unwind Roll and load cell in the actual CFL contribute to the tension dynamics and can therefore influence the tension oscillation frequency observed in the load cell data. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 Tension, t 1 (lbf) Time (sec) Actual Set Point Figure 3.11: Span 1 tension using controller with increased Ki,unw (Model Simulation) Brake 2 Figures 3.12 through 3.14 show the measured data using Brake 2. The first and third data sets are operated at similar speeds that are seen in Fig. 3.1, while the second data set has lower speeds. All of the data sets have speed profiles that are similar to the simulated data. Thus, this data is comparable to the scenario presented in the simulation and can be used for model verification. The trends seen in the control variable for Brake 2 are very similar to Brake 1 59 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 50 100 150 Speed (fpm) Time (sec) Figure 3.12: Pull Roll 1 velocity using Brake 2 (Measured data) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 10 20 30 40 Control Variable Time (sec) Figure 3.13: Control variable using Brake 2 (Measured data) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 100 200 300 Tension (lbf) Time (sec) Actual Set Point Figure 3.14: Tension in Span 1 using Brake 2 (Measured data) 60 except for one major aspect. While using Brake 1, the CV slowly increased to a maximum and then gradually decreased throughout the roll. The CV for Brake 2, however, quickly rises to the maximum and does not have a gradual decrease; there are large durations where the control variable is nearly constant. This is explained by the brake being configured such that a small decrease in CV results in a large decrease in braking torque. This causes the CV fluctuations as the controller is attempting to provide the correct amount of torque. The simulated control variable shown in Fig. 3.3 is comparable to the first two measured data sets in Fig. 3.13. The first similarity is that the magnitudes shown in the simulation are relatively close to those seen in both of the measured data sets. Another trend that illustrates the correspondence is the presence of the CV oscillations. However, the simulated data gradually decreases throughout the entire roll and does not stay constant for large portions of time, a trend seen in the measured data. Additionally, the model simulation was not able to match the magnitudes observed in the third data set. However, as mentioned above, this is not a significant difference since the CV magnitude is seen to vary between the various measured data sets for the same brake. The tension data produced using Brake 2 followed similar patterns to those seen with Brake 1. There are large tension spikes during Pull Roll 1 acceleration and a drop in tension whenever Pull Roll 1 decelerates back to line speed after filling the accumulator. However, the tension reached the reference value much faster with Brake 2, and this can be attributed to the faster rise in the CV that was seen in the Brake 2 data. For the first data set, the tension begins to oscillate with increased amplitude as the material roll depletes. This trend is also slightly present in the second and third data sets, but is not as pronounced. The simulated tension data in Fig. 3.6 displays the general trends found in the 61 measured data in Fig. 3.14. As was the case with Brake 1, the simulated data matches the tension spikes at the Pull Roll 1 speed changes. Additionally, the oscillations of increasing amplitudes found in the measured tension were also present in the simulation. The time required for the simulated tension to reach the reference value was also similar to that observed in the measured data. However, as with Brake 1, the measured tension for Brake 2 was also able to oscillate about the reference value, a feat that was not accomplished in the simulated data. Additionally, the oscillation frequency observed in the measured tension was not accurately predicted by the model simulation. The reasons for these differences are the same as those described in the Brake 1 discussion. 3.2.4 Summary This analysis showed that the dynamic models, though unable to match the measured data perfectly, are able to predict several trends seen in the data collected from the CFL. The simulation was able to mimic the tension effects during the speed changes of Pull Roll 1 for both brakes. The model also produced tension values similar to those seen in the data. Another aspect present in the tension data of both brakes that was captured by the simulation were the oscillations of increasing amplitudes. Additionally, the time required for the simulated tension to rise to the reference value was similar to that observed in the measured data for both brakes. The control variable signal from the model simulations displayed the general attributes that were present in certain data sets of Brakes 1 and 2. The simulated CV showed continual decrease that was observed in each of the three data sets for Brake 1, but was only able to match the magnitudes of the first data set. Conversely, the model was able to match the CV magnitudes of the first two data sets of Brake 2, but did not show the large durations of nearly constant CV present in the Brake 2 data. Control variable fluctuations were also present in both brakes, a trend that was observed in 62 the simulated CV as well. However, there were aspects that were not modeled well by the simulation. One such characteristic is the steady state error observed in the tension of the model simulation. The measured tension for both brakes oscillated about the set point value, but the simulated tension did not. As described in the previous discussions, by increasing the integral gain in the model, the steady state error is reduced. This indicates that the model is missing a scaling or a gain factor. The measured tension oscillation frequencies were also not perfectly matched by the model. This can be attributed to three possible causes: the model relating braking pressure to friction torque, the span length used in the model simulation, and the absence of the idle rollers in the model. Another aspect that was not well predicted by the simulation was that the CV values for the second and third data sets for Brake 1 showed a large initial increase after the switch from manual control. Additionally, the CV magnitudes of these two data sets were not matched by the model simulation. For Brake 2, the measured CV showed large portions that were nearly constant, a trend not observed in the model simulation. The modeled CV also did not match the CV magnitude that was achieved in the third data set for Brake 2. Despite the shortcomings of the simulation, the major trends of the tension data were adequately predicted. The CV values between the two brakes are different as are the CV values for the three data sets for the same brake. Thus, it would not be possible for the simulation to match the CV trend of each data set for both brakes. However, the simulated CV did have certain attributes from each brake. When it did not accurately predict an aspect of one of the brakes, it matched this same characteristic aptly with the other. Therefore, the developed model sufficiently represents the portion of the CFL shown in Fig. 2.7. 63 3.3 Strategies For Improvement of the Existing Control Strategy This section contains a simulation based case study of three suggested improvements to the current control strategy employed to control the Unwind Roll. The first strategy is to decrease the tension measurement sampling time and increase the resolution of certain variables. The second strategy is similar to the first except that the PI gains will vary as a function of the radius. Lastly, the third strategy implements feedforward control action as well as the improvements employed in the second strategy. As discussed in Section 2.5, the current control algorithm for the Unwind Roll has the tension feedback and the control variable set as integers. Moreover, due to the scaling within the input modules, the resolution of the tension measurements is 0.25 values per lbf (vpl), where resolution is defined as given below. r = Nv R (3.3) where r is the resolution and Nv is the number of values that the parameter takes over the range of values R. For example, the tension as measured by the controller takes only one value for tension values that lie in 98 lbf and 102 lbf (98 lbf t1(t) < 102 lbf), and thus the resolution is r = 1/(102− 98) = 0.25 vpl. These factors dilute the tension data so that the controller is not utilizing accurate data. Since the control variable (CV) is an integer, the controller does not output the required amount with suitable precision, causing large fluctuations in tension. Moreover, the resolution of the CV is one, restricting the number of specific torque values that are able to be applied. The voltage sent from the controller to the brake pressure device is directly related to the CV. Therefore, if the CV can only assume a fixed number of values, the voltage, and hence the pressure, can also only maintain certain values. Additionally, the sampling period of the tension measurements is one second with the PI controller update time as 0.2 seconds. This means that the controller output is updated every 0.2 seconds but the error only updates every second. Thus, the 64 controller is just an integrator 0.8 out of every one second rendering the system to be open loop the majority of the time. This results in inaccurate control since the applied torque is not reacting to current errors in tension. Lastly, the controller used constant PI gains throughout the entire roll. For systems that are time invariant, this would acceptable. However, the inertia of the Unwind Roll changes as the web is continuously released from the roll and the radius decreases. This will cause a set of PI gains to be sufficient near the beginning of the new roll (full roll) while causing the tension performance to be degraded later. There are two solutions to this problem. The first is to define fixed gains such that the system remains stable throughout and allow for decreased performance. The second is to vary the gains as a function of the radius so that the system remains stable and satisfactory performance is achieved for all radii of the material roll. Each of the simulations conducted in Subsections 3.3.1 through 3.3.3 employed the model shown in Fig. 2.7 and the velocity profile from Fig. 3.1. Unless otherwise specified, the web properties from Table 3.1 and the initial conditions shown in Table 3.2 were utilized as well. As in Subsection 3.2.2, for each simulation, three repetitions of the same data are shown for clarity of comparison. 3.3.1 Strategy 1 As discussed above, the controller output and feedback measurements should be updated as often as is practical and that they be as accurate as possible. Moreover, to avoid the complications of multirate control systems, the controller update and measurement rates should be the same. Thus the current control strategy for the Unwind Roll is not ideal since it has different rates for updating the controller and measuring the feedback data in addition to using integer values. Strategy 1 is aimed at rectifying these shortcomings. The first change is reducing the tension measurement sampling time from one 65 second to 0.2 seconds. This will allow the controller to react to the most recent data, increasing the relevancy of the control action. Another benefit is that the controller will update at the same rate as the tension measurement, eliminating the need for analysis of the system as a multirate system. The second improvement seen in Strategy 1 is changing the scaling on the CV and the measured tension as well as setting the average tension as a floating point number. As mentioned above, the current scaling results in a tension resolution of only 0.25 vpl. This causes the feedback to be degraded and decreases the effectiveness of the controller. For Strategy 1, the scaling was changed so that the tension resolution is 1 vpl (i.e., the controller takes on one value for 99 lbf t1(t) <100 lbf). Although not as accurate as a floating point, this is a substantial improvement. The control variable is the integer equivalent of the PI Output percentage (the percentage of the maximum output the controller is able to provide). Thus, in order to change the controller output, the tension error must become large enough to vary the PI Output by 0.5 so that the CV changes as opposed to the CV adapting more precisely with the tension error. Moreover, slight corrections in the braking torque are not possible since the adjustment of the braking torque is directly related to the alteration of the CV. Using Equation (3.3), the current PI Output to CV resolution is one CV per percent (cvp). Under the new scaling for Strategy 1, the resolution is 10 cvp. Thus, the control variable is able to achieve 10 different values as the PI Output changes by 1%. The CV is now better suited to attain the value specified by the PI equation. This means that the CV jumps seen in Section 2.5 will be decreased, and thus improved tension control performance is expected. Since the sampling time was changed, the PI gains need to be retuned. Using a model simulation with Strategy 1, the PI gains were tuned until the best performance was achieved. This processes resulted in Kp = 0.01 and Ki = 0.15. The decrease in the proportional gain indicates that the Unwind Roll is sensitive to large changes 66 in the input and that the most effective method for controlling the tension is to apply a smooth braking torque. Increasing the proportional action will increase the portion of the controller output that is directly related to the tension error. This will cause abrupt changes in the controller output since the tension error is continuously fluctuating. These actions will produce spikes in tension due to the large adjustments in the braking torque 



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