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MODEL REFERENCE ADAPTIVE CONTROL OF WEB GUIDES By ARAVIND SESHADRI Bachelor of Engineering University of Madras Tamil Nadu, India 2003 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May 2007 MODEL REFERENCE ADAPTIVE CONTROL OF WEB GUIDES Thesis Approved: Dr. Prabhakar R. Pagilla Thesis Advisor Dr. Karl N. Reid Dean of the College of Engineering, Architecture & Technology Dr. Gary E. Young Committee Member Dr. A. Gordon Emslie Dean of the Graduate College ii ACKNOWLEDGMENTS I wish to express my deepest gratitude to my advisor, Dr. Prabhakar R. Pagilla, for his intelligent supervision, friendship and support throughout my graduate program. I am indebted to him for his motivation and encouragement, which kept me focused on my goals. I would like to thank him for supporting me as a research assistant, during which time I was fortunate to learn a lot about the latest advancements in web handling and control systems. I would like to extend my warmest thanks to my committee members: Dr. Karl N. Reid and Dr. Gary E. Young for their support, suggestions and time. I would like to thank my colleagues, at Oklahoma State University, Anil Abbaraju, Mauro Cimino, Ramamurthy Dwivedula, Reza Jafari, Seshadri Kuppuswamy, Pranav Kumar Peddi Ravi, Ryan Ratliff, Nilesh Siraskar, Diao Yu, and Yunfei Zou for their timely support and suggestions. Finally, I greatly appreciate the support, patience and motivation from my parents and my brother. Without them I would have never reached this level in my life. iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Lateral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.1 Remotely Pivoted Guide (Steering Guide) . . . . . . . . . . . . . . . . . . . 6 1.1.2 Offset Pivot Guide (Displacement Guide) . . . . . . . . . . . . . . . . . . . 8 1.2 Lateral Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Need for a Different Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Adaptive Control Design forWeb Guiding 13 2.1 Introduction to Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Online Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Parameter Estimation: An Example . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 Adaptive Regulation using a Reference Model: An Example . . . . . . . . 22 2.4 Simplified Guide Adaptive Controller Design . . . . . . . . . . . . . . . . . . . . . 24 2.4.1 Three parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . 26 2.4.2 Simplified GAC with an Estimator . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.3 Four Parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . . 31 2.5 Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5.1 Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5.2 Adaptive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 iv 3 Experimental Results 45 3.1 Experimental Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Process Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.2 Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Three Parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . 51 3.3.1 Experiments with opaque web . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.2 Experiments with transparent web . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 Four Parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . 58 3.5 Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6 Systematic Procedure for Adaptive Controller Implementation . . . . . . . . . . . 60 4 Friction Compensation inWeb Guides 65 4.1 Static Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Friction Compensation Based on Static Models . . . . . . . . . . . . . . . . . . . . 69 4.2.1 Friction Compensation forWeb Guides Based on a Static Model . . . . . . 70 4.2.2 Friction Parameters Identification . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Adaptive Static Friction Compensation forWeb Guides . . . . . . . . . . . . . . . 76 4.4 Adaptive Friction Compensation with Web Dynamics . . . . . . . . . . . . . . . . 85 4.5 Adaptive Friction Compensation using RLS Algorithm . . . . . . . . . . . . . . . 89 4.5.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5 Histogram : A New Performance Metric for Web Guiding 94 5.1 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.1.1 Normally Distributed Histograms . . . . . . . . . . . . . . . . . . . . . . . 96 5.1.2 Symmetric, NonNormal, ShortTailed Histograms . . . . . . . . . . . . . 101 5.1.3 Symmetric, NonNormal, LongTailed Histograms . . . . . . . . . . . . . 102 5.1.4 Symmetric Bimodal histogram . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.1.5 Skewed NonNormal histogram . . . . . . . . . . . . . . . . . . . . . . . . 105 5.1.6 Symmetric Histogram with Outliers . . . . . . . . . . . . . . . . . . . . . . 107 5.1.7 Ideal Error Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 v 6 Summary and Future Work 110 BIBLIOGRAPHY 113 A Offset Adaptation 117 B Mathematical Preliminaries 120 B.1 Continuous Functions and their Limits . . . . . . . . . . . . . . . . . . . . . . . . . 120 B.2 InputOutput Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 B.3 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 B.4 Positive Real and Strictly Positive Real Transfer Functions . . . . . . . . . . . . . 126 C Model Reference Adaptive Control: Supplement 128 C.1 MRAC for Relative Degree 1 System . . . . . . . . . . . . . . . . . . . . . . . . . . 128 C.2 Choice of parameter p0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 D Additional Experimental Results 132 D.1 Three Parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . 132 D.1.1 Experiments with opaque web . . . . . . . . . . . . . . . . . . . . . . . . . 132 D.1.2 Experiments with transparent web . . . . . . . . . . . . . . . . . . . . . . . 134 D.2 Four Parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . 136 D.2.1 Experiments with opaque web . . . . . . . . . . . . . . . . . . . . . . . . . 136 D.2.2 Experiments with transparent web . . . . . . . . . . . . . . . . . . . . . . . 141 D.3 Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 D.3.1 Experiments with opaque web . . . . . . . . . . . . . . . . . . . . . . . . . 145 D.3.2 Experiments with transparent web . . . . . . . . . . . . . . . . . . . . . . . 150 vi LIST OF TABLES Table Page 4.1 Kamberoller Guide Motor Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Coulomb and Viscous Friction Estimates . . . . . . . . . . . . . . . . . . . . . . . . 73 vii LIST OF FIGURES Figure Page 1.1 A Web Material used for Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Experimental Web Handling System . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 An Example of aWeb Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 End Pivoted Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Center Pivoted Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Offset Pivot Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7 Remotely Pivoted Guide or Steering Guide . . . . . . . . . . . . . . . . . . . . . . 5 1.8 A Schematic of a Remotely Pivoted Guide . . . . . . . . . . . . . . . . . . . . . . . 7 1.9 A Schematic of an Offset Pivot Guide . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 Adaptive Gain Scheduling Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Adaptive Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Model Reference Adaptive Control System . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Online Parameter Adjustment Mechanism . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Adaptive Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 Simulation Results for 3Parameter GAC . . . . . . . . . . . . . . . . . . . . . . . 30 2.7 Simulink Block Diagram for 3Parameter GAC . . . . . . . . . . . . . . . . . . . . 30 2.8 Simulation Results for 3Parameter GAC with an Estimator . . . . . . . . . . . . . 32 2.9 Simulink Block Diagram for 3Parameter GAC with an Estimator . . . . . . . . . 32 2.10 Simulation Results for 4Parameter GAC . . . . . . . . . . . . . . . . . . . . . . . 38 2.11 Simulation Results for 4Parameter GAC with a Pulse Disturbance . . . . . . . . 39 2.12 Simulink Block Diagram for 4Parameter GAC . . . . . . . . . . . . . . . . . . . . 40 2.13 Simulation Results for Guide Adaptie Controller . . . . . . . . . . . . . . . . . . . 43 2.14 Simulation Results for Guide Adaptie Controller with a Pulse Disturbance . . . . 43 2.15 Simulink Block Diagram for Guide Adaptie Controller . . . . . . . . . . . . . . . 44 viii 3.1 Experimental Web Handling Platform . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Line Schematic of the Experimental Web Handling Platform . . . . . . . . . . . . 46 3.3 offset pivot Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Remotely Pivoted Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.6 Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.7 The Effect of Opacity on Sensor Gain for an Infrared Sensor . . . . . . . . . . . . 50 3.8 Performance Comparison: 3 Parameter, Sine Disturbance, 300 fpm, Opaque Web 53 3.9 Adaptive Controller: 3Parameter, Sine Disturbance, 300 fpm, Opaque Web . . . 54 3.10 Performance Comparison: 3Parameter, Pulse Disturbance, 300 fpm, Opaque Web 54 3.11 Adaptive Controller: 3Parameter, 300 and 500 fpm, Step Reference Changes, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.12 Performance Comparison: 3Parameter, 500 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.13 Adaptive Controller: 3Parameters, 500fpm, Sine Disturbance, Transparent Web . 56 3.14 Performance Comparison: 3Parameter, 500 fpm, Pulse Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.15 Adaptive Controller: 4Paramater, 300 fpm, Sine Disturbance, Opaque Web . . . 58 3.16 Performance Comparison: 8Parameter, 300 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.17 Adaptive Controller: 8Parameter, 300 fpm, Sine Disturbance, Transparent Web . 61 3.18 Adaptive Controller: 8Parameter, 300 fpm, SineDisturbance, SteadyState, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.19 Performance Comparison: 8Parameter, 500 fpm, Pulse Disturbance, Effect of p0, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.1 Static Friction Model with Coulomb Friction Effect . . . . . . . . . . . . . . . . . . 66 4.2 Static Friction Model with Coulomb and Viscous Effects . . . . . . . . . . . . . . . 67 4.3 Static Friction Model with Coulomb, Viscous and Stiction Effects . . . . . . . . . 68 4.4 Static Friction Model with Stribeck Effect . . . . . . . . . . . . . . . . . . . . . . . 68 4.5 Friction Compensation using Estimated Friction . . . . . . . . . . . . . . . . . . . 69 ix 4.6 Velocity Output for a Sinusoidal Input Voltage . . . . . . . . . . . . . . . . . . . . 73 4.7 Zero Velocity Crossing Friction Effect . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.8 Estimation of Stiction Using a Ramp Input . . . . . . . . . . . . . . . . . . . . . . 74 4.9 Stribeck Velocity Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.10 Simulink Block Diagram: Adaptive Friction Compensation . . . . . . . . . . . . . 80 4.11 Initial Condition Response (Angular Position) . . . . . . . . . . . . . . . . . . . . 81 4.12 Parameter Estimated with Initial Condition Response (Angular Position) . . . . . 81 4.13 Initial Condition Response Angular Velocity . . . . . . . . . . . . . . . . . . . . . 82 4.14 Parameter Estimated with Initial Condition Response (Angular Velocity) . . . . . 82 4.15 Simulink Block Diagram: Position Regulation . . . . . . . . . . . . . . . . . . . . . 83 4.16 Adaptive Static Friction Compensationwith Position regulation. θdes = 2, θ(0) = 0 and θ(˙0) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.17 Parameter Estimates for Adaptive Static Friction Compensation with Position Regulation. θdes = 2, θ(0) = 0 and θ(˙0) = 0 . . . . . . . . . . . . . . . . . . . . . . 84 4.18 Model Reference Adaptive Control with Adaptive Friction Compensation . . . . 85 4.19 Performance Comparision: 3Parameter, with and without Friction Compensation 92 4.20 Performance Comparision: 4Parameter, with and without Friction Compensation 93 4.21 Performance Comparision: 8Parameter, with and without Friction Compensation 93 5.1 Performance Comparison Based on Regulation Error: Clear Distinction . . . . . . 95 5.2 Performance Comparison Based on Regulation Error: Difficult to Compare . . . 95 5.3 Performance Comparison using Histograms: . . . . . . . . . . . . . . . . . . . . . 97 5.4 Performance Comparison using Histograms: . . . . . . . . . . . . . . . . . . . . . 97 5.5 Normally distributed Histogram, with Zero Mean . . . . . . . . . . . . . . . . . . 98 5.6 Mean of a Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.7 Variance of a Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.8 Experimental Data: Bottom Plot Shows Normal Distribution . . . . . . . . . . . . 100 5.9 ShortTailed Histogram Characteristic with “Fat” Body . . . . . . . . . . . . . . . 101 5.10 Experimental Data: Top Plot Indicates ShortTailed Distribution . . . . . . . . . . 102 5.11 LongTailed Histogram Characterized by “Lean” Body and Long Tails . . . . . . 103 5.12 Experimental Data: Bottom Plot Shows LongTailed Distribution . . . . . . . . . 103 x 5.13 Symmetric Bimodal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.14 Experimental Data: Top Plot Indicates Bimodal Distribution . . . . . . . . . . . . 105 5.15 A Right Skewed Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.16 Experimental Data: Skewed Histogram . . . . . . . . . . . . . . . . . . . . . . . . 106 5.17 A Histogram with Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.18 Experimental Data: Presence of Outliers . . . . . . . . . . . . . . . . . . . . . . . . 108 5.19 Experimental Data: The Ideal Distribution for Guiding Applications . . . . . . . 109 A.1 Output voltage range and typical curves for opaque and transparent webs . . . . 118 A.2 Offset adaptation on the opaque web . . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.3 Offset adaptation on the transparent web . . . . . . . . . . . . . . . . . . . . . . . 119 D.1 Performance Comparison: 3Parameter, 500 fpm, Sine Disturbance, Opaque Web 132 D.2 Adaptive Controller: 3Parameter, 500 fpm, Sine Disturbance, Opaque Web . . . 133 D.3 Performance Comparison: 3Parameter, 500 fpm, Pulse Disturbance, Opaque Web 133 D.4 Performance Comparison: 3Parameter, 300 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 D.5 Adaptive Controller: 3Parameter, 300 fpm, Sine Disturbance, Transparent . . . . 135 D.6 Performance Comparison: 3Parameter, 300 fpm, Pulse Disturbance, Transparent 135 D.7 Performance Comparison: 4Parameters, 300 fpm, Sine Disturbance, Opaque Web 136 D.8 Adaptive Controller: 4Parameters, 300 fpm, Sine Disturbance, Opaque Web . . . 137 D.9 Performance Comparison: 4Parameters, 500 fpm, Sine Disturbance, Opaque Web 137 D.10 Adaptive Controller: 4Parameters, 500 fpm, Sine Disturbance, Opaque Web . . . 138 D.11 Performance Comparison: 4Parameters, 300 fpm, Pulse Disturbance, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 D.12 Adaptive Controller : 4Parameters, 300 fpm, Pulse Disturbance, SteadyState, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 D.13 Performance Comparison: 4Parameters, 500 fpm, Pulse Disturbance, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 D.14 Performance Comparison: 4Parameters, 500 fpm, Pulse Disturbance, Steady State, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 xi D.15 Adaptive Controller: 4Parameter, 300 and 500 fpm, Step Reference Changes, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 D.16 Performance Comparison: 4Parameters, 300 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 D.17 Adaptive Controller: 4Parameters, 300 fpm, Sine Disturbance, Transparent Web 142 D.18 Performance Comparison: 4Parameters, 500 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 D.19 Adaptive Controller: 4Parameters, 500 fpm, Sine Disturbance, Transparent Web 143 D.20 Performance Comparison: 4Parameters, 300 fpm, Pulse Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 D.21 Performance Comparison: 4Parameters, 500 fpm, Pulse Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 D.22 Adaptive Controller: 4Parameters, 300 and 500 fpm, Step Reference Changes, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 D.23 Performance Comparison: 8Parameters, 300 fpm, Sine Disturbance, Opaque Web 145 D.24 Adaptive Controller: 8Parameters, 300 fpm, Sine Disturbance, Opaque Web . . . 146 D.25 Adaptive Controller: 8Parameters, 300 fpm, Sine Disturbance, SetadyState, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 D.26 Performance Comparison: 8Parameters, 500 fpm, Sine Disturbance, Opaque Web 147 D.27 Adaptive Control: 8Parameters, 500 fpm, Sine Disturbance, Opaque Web . . . . 147 D.28 Adaptive Control: 8Parameters, 500 fpm, SineDisturbance, SteadyState, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 D.29 Performance Comparison: 8Parameters, 300 fpm, Pulse Disturbance, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 D.30 Performance Comparison: 8Parameters, 500 fpm, Pulse Disturbance, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 D.31 Performance Comparison: 8Parameters, 500 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 D.32 Adaptive Controller: 8Parameters, 500 fpm, Sine Disturbance, Transparent Web 151 D.33 Adaptive Controller: 8Parameters, 500 fpm, Sine Disturbance, SteadyState, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 xii D.34 Performance Comparison: 8Parameters, 300 fpm, Pulse Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 D.35 Performance Comparison: 8Parameters, 500 fpm, Pulse Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 xiii NOMENCLATURE SYMBOLS Cm : transmission ratio e : error e1 : tracking error E : modulus of elastcity of web ǫ1 : estimation error F : friction force Fc : Coulomb friction coefficient Fs : static friction coefficient Fv : viscous friction coefficient γ : gain : gain matrix i : current I : moment of inertia J : rotor inertia km : motor parameter or high frequency gain for a reference model kp : high frequency gain for a plant model K = r T EI : web span parameter Ke : back e.m.f constant Kt : torque constant/sensitivity xiv SYMBOLS L : inductance or length of span L1 : distance from the guide roller to instant center L : Laplace operator L−1 : inverse Laplace operator μ : mean n∗ : relative degree ω : regresor vector ωn : natural frequency Wm(s) : reference model transfer function φ : filtered regressor vector r : reference command R : resistance Rm(s) : denominator polynomial of reference model Rp(s) : denominator polynomial of plant model R : set of all real numbers sgn(.) : signum function σ : standard deviation σ2 : variance T : torque or Tension τ : time constant θ : angular position or parameter vector θ∗ : true parameter vector θ0 : roller misalignment u,Up : input to a plant v : velocity vs : Stribeck velocity constant xv SYMBOLS x : state variable x1 : distance from the guide roller to the instant center y : output of a plant ˆy : estimator output Y0 : initial lateral position misalignment yL, YL : lateral edge position ym : output of a reference model ζ : damping ratio Z : guide position Zm(s) : numerator polynomial of reference model Zp(s) : numerator polynomial of plant model xvi ABBREVIATIONS BIBO : Boundedinput boundedoutput I/O : InputOutput LMS : Least Mean Square LTI : Linear Time Invariant MRAC : Model Reference Adaptive Control OPG : Offset Pivot Guide P : Proportional PD : ProportionalDerivative PE : Persistent Excitation PI : ProportionalIntegral PR : Positive Real RLS : Recursive Least Square RPG : Remotely Pivoted Guide SISO : Single Input Single Output SPR : Strictly Positive Real xvii Chapter 1 Introduction The term web is used to describe materials which have their length considerably larger than their width and width considerably larger than thickness. Webs are materials which are manufactured and processed in a continuous, flexible strip form. Webs consist of a broad spectrum of materials that are used extensively in every day life such as plastics, paper, textile, metals and composites. Typically web materials are manufactured into rolls since it is easy to transport and process the materials in the rolled form. An example of a web material is shown in Figure 1.1. Figure 1.1: A Web Material used for Packaging Web handling is a term that is used to refer to the study of the behavior of the web while it is transported and controlled through the processing machinery from an unwind roll to a 1 rewind roll. A typical operation involves transporting a web in rolled, unfinished form from an unwind roll to a rewind roll through processing machinery where the required processing operations are performed. An example of such a process is commonly seen in the metals industries. The web (metal strip) that is to be processed is transported on rollers to various sections where different operations like coating, painting, drying, slitting, etc., are performed. The process line generally has unwind and rewind rolls, many idle rollers and one or more intermediate driven rollers. An experimental web handling system with unwind and rewind sections is shown in Figure 1.2. REWIND SECTION UNWIND SECTION DRIVEN ROLLER IDLE ROLLERS WEB Figure 1.2: Experimental Web Handling System The longitudinal dynamics of theweb is the behavior of theweb in the direction of transport of the web. Web transport velocity and web tension are two key variables of interest that affect the longitudinal behavior of the web. The lateral dynamics of the web is the behavior of the web perpendicular to the direction of transport of the web and in the plane of web. Several parameters which affect the lateral web dynamics include web material, tension, transport velocity, and web geometry, etc. The quality of the finished web depends on how well the web is handled on the rollers during transport. The longitudinal and lateral control of the web on rollers play a critical role in the quality of the finished product. 2 The focus of thiswork is on control of lateral dynamics of aweb. Adaptive control strategies that are capable of providing the required performance in the presence of the variations in the process and web parameters are investigated. The suitability of these control strategies and their ability to provide the required performance are studied in detail, both from theoretical and experimental perspectives. Web guiding (also called as lateral control) involves controlling web fluctuations in the plane of the web and perpendicular to web travel. Web guiding is important because rollers in any web handling machinery tend to have inherent misalignment problems and this may cause the web to move laterally on the rollers. The lateral movement of the web on the rollers may produce wrinkles or slackness in the web, or the web may completely fall off the rollers. A number of web processes like printing, coating, winding may get affected severely due to the web lateral motion and it becomes important to maintain the lateral position of the web. Web guides are used to maintain the lateral position of the web on rollers during transport. EDGE SENSOR GUIDE ROLLER LATERAL LONGITUDINAL WEB GUIDE MECHANISM ACTUATOR Figure 1.3: An Example of aWeb Guide Aweb guidemechanismtypically consist of a roller sitting on a pivoted base, whosemotion is controlled to change the axis of rotation of the roller. The working principle of all web guides is based on awell understood fundamental principle inweb handling – a web approaching a roller 3 will always tend to orient itself perpendicular to the axis of rotation of the roller. The lateral motion of the web is controlled by changing the axis of rotation of the guide roller. The lateral position of the web is measured using an edge sensor. Based on this measurement as feedback the axis of rotation of the guide roller is controlled tomaintain the lateral position at the required location. Figure 1.3 shows a web guide mechanism with an electromechanical actuator and an infrared sensor. Web guides are positioned at different locations in an industrial process line where guiding is required. Guides located at either ends in a process line are usually called terminal guides [1]. An unwind guidemaintains the lateral position of thewebwhich is fed into the processing line, whereas a rewind guide maintains the lateral position of the processed web which is wound onto a roll in the rewind section. Apart fromterminal guiding, web guides are extensively used in the intermediate process sections and they are referred to as intermediate guides. The intermediate web guides are classified based on the way in which the axis of rotation of the guide roller is changed. Figure 1.4 shows an end pivoted guide where the change in the axis of rotation of the roller is about a pivot point which is at one end of the roller. Similarly the center pivoted guide shown in Figure 1.5 has its pivot point in the center of the guide roller. An offsetpivot guide (shown in Figure 1.6) utilizes a pair of rollers to change the axis of rotation while in a remotely pivoted guide (shown in Figure 1.7) the guide roller moves along a curved path to change its axis. These are some commonly used intermediate web guides in the web handling industry. WEB PIVOT POINT EDGE SENSOR GUIDE ROLLER AXES OF ROTATION Figure 1.4: End Pivoted Guide WEB PIVOT POINT EDGE SENSOR GUIDE ROLLER AXES OF ROTATION Figure 1.5: Center Pivoted Guide 4 PIVOT CARRIER MOUNTING BASE ENTERING SPAN EXITING SPAN FIXED ENTERING ROLLER FIXED EXITING ROLLER WEB PIVOT POINT EDGE SENSOR AXES OF ROTATION WEB Figure 1.6: Offset Pivot Guide WEB GUIDE ROLLER FIXED ENTERING IDLER ROLLER CENTERLINE OF MACHINE AND GUIDE ASSEMBLY CENTER OF ROTATION OF GUIDE ROLLER (INSTANT CENTER) AXIS OF ROTATION GUIDE ROLLER WEB EDGE SENSOR ENTERING SPAN PREENTERING SPAN EXITING SPAN Figure 1.7: Remotely Pivoted Guide or Steering Guide 5 1.1. LATERAL DYNAMICS 1.1 Lateral Dynamics In order to control aweb guide it is important to understand the physics behind the transport of web on rollers. Lateral and longitudinal dynamics of a moving web are dependent on various process parameters like transport velocity, web tension, web material, and the geometry of the web material, etc. Understanding the lateral dynamics of the web is important for designing an effective lateral controller for the web guide mechanism. Lateral dynamics of the web was first modeled in [2], based on the assumption that the web behaves like a string. A major improvement to the mathematical model for the lateral dynamics of the web was described by J. J. Shelton [3]. A first order model of an ideal web was presented in [4] followed by a secondorder model [5] by considering the web as an Euler beam. A model for the web lateral dynamics for a multiroll system was developed in [6] based on the Timoshenko beam theory. A stochastic modeling formulation for the lateral dynamics was developed in [7]. An overview of the lateral and longitudinal dynamics along with a historic perspective of modeling and control of moving webs was presented in [8]. A detailed modeling of the lateral dynamics of the web for different types of intermediate guides was presented in [9, 10]. The two types of intermediate guides which are considered in this work are a remotely pivoted guide (also called as a steering guide) and an offsetpivot guide (also called as a displacement guide). The web span lateral dynamics for the two guides are similar and hence the same controller design can be implemented on both the guides. Even though the focus of this research is on these two intermediate guides, the theory developed can be adapted to other guides as well. 1.1.1 Remotely Pivoted Guide (Steering Guide) The action of the remotely pivoted guide on the lateral position of the web is given by YL(s) = G1s(s)Z(s) + G2s(s)θ0(s) + G3s(s)Y0(s) (1.1) where YL(s) is the Laplace transform of the web lateral position, Z(s) is the input to the guide in the lateral direction, θ0(s) denotes the entering span roller misalignment and Y0(s) is the initial lateral position misalignment (see Figure 1.8). The transfer functions in equation (1.1) are given by G1s(s) = s2 + β2s + β1 s2 + β2s + β0 , G2s(s) = β′3 s2 + β2s + β0 , and G3s(s) = −β3s + β0 s2 + β2s + β0 (1.2) 6 1.1. LATERAL DYNAMICS Y0 X Y L v Z YL 0 L x1 INSTANT CENTER ORIGINAL CENTER q q Figure 1.8: A Schematic of a Remotely Pivoted Guide where β0 = 1 τ 2 (KL)2(coshKL − 1) KLsinhKL − 2(coshKL − 1) , 1 τ 2 f1(KL) β1 = L τ 2x1 KL(KLcoshKL − sinhKL) KLsinhKL − 2(coshKL − 1) , L τ 2x1 f2(KL) β2 = 1 τ KL(KLcoshKL − sinhKL) KLsinhKL − 2(coshKL − 1) , 1 τ f2(KL) β3 = 1 τ KL(sinhKL − KL) KLsinhKL − 2(coshKL − 1) , 1 τ f3(KL) β′3 = L τ 2 KL(sinhKL − KL) KLsinhKL − 2(coshKL − 1) , L τ 2 f3(KL) where the web span parameter K is defined as K2 = T EI , E is the modulus of elasticity of web, I is the moment of inertia of the web, T is the web tension, L is the length of the entering span, τ = L/v is the time constant, v is the web transport velocity, and x1 is the distance from the guide roller to its instant center of rotation. The variables θ0(s) and Y0(s) are considered as the disturbances, and the objective of the web guide is to reject these disturbances to maintain the lateral position downstream of the web guide. Thus the effect of the input guide displacement, Z(s), to the lateral position of the web, YL(s), is given by YL(s) = s2 + β2s + β1 s2 + β2s + β0 Z(s) (1.4) 7 1.1. LATERAL DYNAMICS 1.1.2 Offset Pivot Guide (Displacement Guide) Y 3 3 YL Y L B X3 3 A v L A B C D L1 0 Y 1 YL X Y L v L 3 Z L B PIVOT AXIS (ROLLER POSITION) L C PIVOT AXIS Figure 1.9: A Schematic of an Offset Pivot Guide The lateral dynamics of the web with a displacement guide is influenced by the input to the guide mechanism, Z(s), and the initial lateral position misalignment, Y0(s), and is given by YL(s) = G1dZ(s) + G2dY0(s) (1.5) where G1d(s) = − f3(KL) τ s + f1(KL) τ 2 s2 + f2(KL3) τ3 s L1 − L L1 s2 + f2(KL3) τ3 s + f1(KL3) τ 2 3 s2 + f2(KL) τ s + f1(KL) τ 2 + s2 + f2(KL) τ s + f2(KL)L τ 2L1 s2 + f2(KL) τ s + f1(KL) τ 2 G2d(s) = − f3(KL3) τ3 s + f1(KL3) τ 2 3 − f3(KL) τ s + f1(KL) τ 2 s2 + f2(KL3) τ3 s + f1(KL3) τ 2 3 s2 + f2(KL) τ s + f1(KL) τ 2 with L3 as the span length as shown in Figure 1.9 and τ3 = L3/v. Displacement guides are typically installed such that the length of the guide span (denoted by L) is very close to the distance from the pivot axis to the second guide roller (denoted by L1) 8 1.2. LATERAL CONTROL (see Figure 1.9). When L ≈ L1, the transfer function G1d(s) is simplified [9], and the structure is same as the transfer function G1s(s) in the dynamics of the remotely pivoted guide given by equation (1.2). With this simplification, the dynamics from the guide input to the web lateral position is given by YL(s) = s2 + β2s + β′1 s2 + β2s + β0 Z(s) (1.6) where β′1 = f2(KL)L τ 2L1 . 1.2 Lateral Control Lateral control involves the design of a closedloop control system for regulating the lateral position of the web in a process line using a web guide mechanism. The guide mechanism includes an actuator which provides the input to the system and a feedback sensor which is used to measure the lateral position of the web. Current industrial controllers for web guiding are simple and do not consider the web lateral dynamics in the controller design. A typical industrial controller consists of three loops. A very fast current loop, a velocity loop and an outer position loop. The two inner loops are designed based on the actuator dynamics, and they regulate the current and velocity of the actuator. The actuator dynamics is assumed to be completely known. The outer position loop is designed to regulate the lateral position of the web, and the compensation is usually based on a Proportional (P) controller or a ProportionalIntegral (PI) controller. The error in the lateral position of the web drives the two inner loops. Hence all the three loops work in unison to regulate the lateral position of the web. Since the inner two loops are driven by the outer position loop, the lateral positionmeasurement becomes critical. Depending on the web handling application, the type of sensor used to measure the lateral position of the web varies. Applications which handle opaque webs commonly use infrared sensors while transparent film applications employ ultrasonic sensors. The position loop is dependent on this sensor measurement and the sensor gain affects the position loop. Both analog and digital industrial controllers commonly have a manual variable gain, which can be used for tuning the controller gains based on the type of the web. Based on the sensor gain, the controller gain is adjusted appropriately. A number of strategies have been presented in the literature to control the lateral position of the web. The strategies include Proportional (P) control [11] and [7], ProportionalDerivative 9 1.3. NEED FOR A DIFFERENT CONTROL STRATEGY (PD) control [11], a state estimationproportional gain strategy [12], a state variable feedback control strategy [13], estimated velocity feedback control [14] and [15], controller based on frequency domain design [16], etc. In most of the control strategies cited above, it is assumed that the parameters that affect the lateral dynamics of the web are known. Some of the key parameters that affect the lateral dynamics are the web material properties, web geometry, transport velocity, web tension, etc. These parameters may vary due to process condition variations or due to the processing of different web materials in the same process line. Unless these parameters are measured regularly it is difficult to know the exact dynamic model of the web for each situation. 1.3 Need for a Different Control Strategy Some potential shortcomings of the existing lateral control strategies are given in the following. • Existing industrial strategies do not consider the web lateral dynamics in the analysis and design of the lateral controllers. Inclusion of the lateral web dynamics in the control design process has the potential to significantly improve the guiding performance in the presence of process variations as well as many machine induced lateral disturbances. • Sensor gain changes as a result of web material variations cause poor guiding with existing fixed gain controllers. • Parameters of the dynamic model are not known. In a model based controller design for lateral guiding, the knowledge of the parameters like tension, web transport velocity, web material geometry, etc., are important to achieve better tracking performance. But in most industrial applications these process parameters are not known to the guide controller. Additionally, most industrial process lines are designed to process different web materials under different operating conditions. The goal is to find a controller that is capable of providing specified guiding performance which has the ability to overcome many of the limitations of the existing fixed gain controllers, including the ones given above. To achieve this goal, adaptive control strategies are investigated in this thesis. 10 1.4. CONTRIBUTIONS 1.4 Contributions Two main contributions of this work are summarized below: 1. Model reference adaptive control designs that are applicable to web guiding are developed. A systematic approach for industrial implementation of these new adaptive strategies are developed. Additionally, practical industrial implementation guidelines are proposed. 2. A new performance metric that clearly highlights the web guiding performance is developed. The newmetric is based on histograms. Profiles of commonly observed histograms are studied and their occurrence in guiding situations are analyzed. This novel performance metric can be used as a metric for tuning controllers as well as a diagnostic tool for lateral web guiding applications. In addition to the primary contributions, several other secondary contributions are summarized below: 1. Simplified adaptive controllers are developed based on the approximation of the lateral dynamics by reduced order models. 2. Extensive experimentation of the adaptive strategies on an experimental platform containing various intermediate web guides is carried out. Further, an often used industrial control strategy is simultaneously implemented for all situations, and the results are compared. 3. Commonly used friction models are investigated and an adaptive friction compensation scheme based on static friction model is proposed for web guiding applications. A simplified adaptive controller with friction compensation is proposed and an indirect friction compensation scheme based on recursive least squares is implemented. 4. The output voltage of an optical sensor depends on the opacity of theweb. A newmethod to determine the range of the optical sensor is developed (see Appendix A). The rest of the document is organized as follows. In Chapter 2, a detailed description of the design of model reference adaptive control strategies suitable for web guiding is given. The results of the experiments carried out on an experimental web handling platform with the 11 1.4. CONTRIBUTIONS proposed adaptive control strategies are presented in Chapter 3. A detailed discussion of the results along with practical industrial implementation guidelines based on the experimental observations are also presented in Chapter 3. Chapter 4 gives friction compensation techniques that are applicable to web guiding. In Chapter 5, a new performance metric for web guiding based on histograms is discussed. Chapter 6 summarizes the thesis and provides suggestions for future work. 12 Chapter 2 Adaptive Control Design for Web Guiding 2.1 Introduction to Adaptive Control To adapt means to change to meet requirements or adjust to new circumstances. An adaptive controller is a scheme that adapts to changes in process dynamics and disturbances. Adaptive controllers are commonly used when there is a considerable change in the process dynamics and disturbances, which may not be compensated by using fixed gain controllers. One practical application of an adaptive control scheme is the autopilot guidance system in aircrafts. The process dynamics of an aircraft depends on various parameters such as speed, altitude, head wind, tail wind, etc., and the aircraft also experiences various external disturbances. It is difficult to design a linear fixed gain feedback controller which can perform well under different operating conditions and in the presence of various disturbances. In the early autopilot design, the adaptive scheme that was commonly used was gain scheduling. Various parameters that affect the process dynamics of an aircraft have a direct relationship with process outputs (or measured variables), and hence these outputs can be used to change the controller parameters directly. Hence it is possible to determine a suitable constant gain linear feedback control strategy for each operating condition. The model is thus linearized around those operating conditions. This is called gain scheduling because the control scheme determines the operating condition based on the process outputs and consequently an appropriate controller is chosen to compensate for the process dynamics (or process gain). The system has basically two loops as shown in Figure 2.1. The inner loop is a standard feedback loop while the outer loop adjusts the controller in the inner loop based on the operating conditions. Gain scheduling can be regarded as a mapping from process parameters to controller parameters [17]. 13 2.1. INTRODUCTION TO ADAPTIVE CONTROL CONTROLLER PLANT CONTROL SIGNAL GAIN SCHEDULE REFERENCE COMMAND OUTPUT CONTROLLER PARAMETERS OPERATING CONDITIONS Figure 2.1: Adaptive Gain Scheduling Scheme Gain scheduling is possible only when the variations (or the process dynamics) are directly correlated to the measured variables. In other words, gain scheduling is possible only when the operating conditions could be characterized based on these measured variables. Variation in disturbance characteristics cannot be compensated using gain scheduling and hence adaptive controllers are used. The modern autopilot for aircrafts employs adaptive controllers to compensate for various disturbances that act on the plane. The modern autopilot system is not limited to level flight. Current systems include different autopilot schemes for taxi, takeoff, ascent, level flight, descent, landing and taxi back to the terminal. Since the disturbance changes frequently, it is reasonable to adjust the controller parameters to cope with the disturbance characteristics. A common adaptive control scheme is illustrated in Figure 2.2. Similar to gain scheduling, there are two loops. The inner loop is a standard feedback loop with a controller. The outer loop has a parameter adjustment mechanism which modifies the controller based on variations. Direct and Indirect Adaptive Control An adaptive controller consists of two main subsystems. A parameter estimator and a controller based on the parameter estimator. The parameter adjustment block in Figure 2.2 is the parameter estimator which estimates the unknown parameters, based on the output of the plant, reference command, control signal, etc. The estimated parameters are then used by the controller to compute the control signal to the plant. The way in which the parameters are estimated, also referred to as the adaptive law, along with the way in which the control signal is calculated, gives rise to two main kinds of adaptive control schemes, indirect and direct. Indirect 14 2.1. INTRODUCTION TO ADAPTIVE CONTROL CONTROLLER PLANT CONTROL SIGNAL PARAMETER ADJUSTMENT REFERENCE COMMAND OUTPUT CONTROLLER PARAMETERS Figure 2.2: Adaptive Control System adaptive control involves the estimation of actual plant parameters which are used to compute the controller parameters. The controller parameters are computed indirectly and hence the name indirect adaptive control. In direct adaptive control, the plant model is parametrized in terms of the controller parameters which are estimated directly without intermediate calculations involving plant parameter estimates [18]. Indirect adaptive control is also referred to as explicit adaptive control while direct adaptive control is also referred as implicit adaptive control because the control design is based on explicit or implicit plantmodel estimation. Both types of control schemes can be used for minimumphase plants, but difficulty arises when designing an indirect adaptive control scheme for nonminimumphase plants [18]. In direct adaptive control scheme, since the plant parameters are parametrized in terms of the controller parameters, the convergence of parameters to their true values is not of utmost importance. The parameter estimation is usually driven by the output error. Although the parameters do not converge to their true values, the controller is capable of meeting the performance requirement on the output error convergence. On the contrary, convergence of parameters to their true values is of utmost importance while using indirect adaptive control. The convergence of parameters and its importance will be discussed later in this chapter. Model Reference Adaptive Control In a Model Reference Adaptive Control (MRAC) scheme the control law and adaptive law are designed such that the closedloop Input/Output (I/O) properties of the plant exactly match a reference model. The reference model is any dynamic model which meets a desired closedloop 15 2.2. ONLINE PARAMETER ESTIMATION performance requirements for the plant. Hence the control law for the plant is designed such that the closedloop dynamics matches the dynamics of the reference model. There are two kinds ofMRAC strategies, direct and indirectMRAC. A schematic of directMRAC is shown in Figure 2.3. CONTROLLER PLANT CONTROL SIGNAL PARAMETER ADJUSTMENT REFERENCE COMMAND OUTPUT CONTROLLER PARAMETERS REFERENCE MODEL ERROR _ + Figure 2.3: Model Reference Adaptive Control System In this thesis a direct model reference adaptive control scheme is investigated for web guides. In the following sections, the design and analysis of common adaptive control schemes will be presented along with their application to web guides. Online parameter estimation, a key component in adaptive control design, is discussed in section 2.2. A simple adaptive control example based on online parameter estimation is discussed in section 2.3. Model reference adaptive control schemes suitable for web guiding are developed in sections 2.4 and 2.5. Model reference adaptive control schemes, based on a simplified models for the web dynamics, are proposed in section 2.4. Amodel reference adaptive scheme based on the complete web dynamic model is presented in 2.5. 2.2 Online Parameter Estimation Online parameter estimation is required for any adaptive controller design. Both in direct and indirect adaptive control schemes some form of parameter estimation is carried out. In 16 2.2. ONLINE PARAMETER ESTIMATION indirect adaptive control the plant parameters are estimated. But in direct adaptive control the controller parameters are estimated. In both the cases parameters are estimated in realtime and the controller is adjusted appropriately. PARAMETER ESTIMATOR _ + PLANT OUTPUT ESTIMATED OUTPUT ESTIMATION ERROR CONTROL r(t) ^ y(t) REFERENCE y(t) u(t) Figure 2.4: Online Parameter Adjustment Mechanism In an online parameter estimation problem a parameter adjustment mechanism is utilized to estimate the unknown plant parameters, as shown in Figure 2.4. The parameter estimator is designed such that its structure is similar to the plant structure. The estimator is driven by the input to the plant u(t), reference command r(t) and the output of the plant y(t). The output of the estimator ˆy(t) is constantly compared with the output of the plant. The difference between the estimator output and the actual plant output is called the estimation error. The parameters in the estimator are constantly adjusted so that the estimation error is minimized. Standard optimization techniques like the gradientdescent, leastsquares, Newton’s method, etc., can be used to minimize the estimation error in realtime (see Appendix B in [18]). CONTROLLER PLANT CONTROL SIGNAL PARAMETER ADJUSTMENT REFERENCE COMMAND OUTPUT CONTROLLER PARAMETERS Figure 2.5: Adaptive Control System 17 2.2. ONLINE PARAMETER ESTIMATION An online estimation problem involve three key steps. First the parameter estimator has to be parametrized appropriately in terms of the unknown parameters. The second step involves the design of the adaptive law which defines the way in which the parameters are updated. The adaptive law is designed such that the optimization techniques used to minimize the estimation error results in a stable system. The final step involves the design of the plant inputs such that the estimates converge to their true values. The final step is important only for identification problems. For direct adaptive control problems the minimization of the output error is important rather than the convergence of parameters. In the following section a simple scalar plant of unknown parameters will be considered and the procedure for designing an online parameter estimation scheme will be discussed. The example is based on the procedure described in [18]. Additionally, various considerations for designing the input and the stability properties of the adaptive system will be discussed. 2.2.1 Parameter Estimation: An Example Consider a firstorder system described by x˙ = ax + bu, x(0) = x0 (2.1) where x is the output, u is the input and a, b are unknown constants. We also assume that the system is stable, i.e., a < 0 and the input u is bounded. The objective is to develop an online estimation algorithmfor estimating the unknown parameters using themeasured signals x and u. For estimation and adaptive control problems, an effective parametrization of estimator in terms of unknown plant parameters is extremely important. A simple parametrization would be x(s) = [ a b ] x(s) s u(s) s (2.2) where x(s) and u(s) are the Laplace transforms of x(t) and u(t). This parametrization is not desirable due to the presence of the integrator. A similar parametrization which is implementable can be obtained by using a low pass filter for the measured signals x and u. Let the plant be parametrized by adding and subtracting amx, where am > 0, am ∈ R1. The 18 2.2. ONLINE PARAMETER ESTIMATION parameter am is the low pass filter parameter whose value will be chosen later. x˙ = −amx + (a + am)x + bu (2.3a) x = 1 s + am [(a + am)x + bu] (2.3b) = θ∗⊤φ (2.3c) where θ∗⊤ = [b (a + am)] is the unknown parameter vector and φ = [ 1 s+am u 1 s+am x]⊤ is filtered measurement. Define the estimate of x as ˆx = θ⊤φ (2.4) where θ is the estimate of the unknown parameter vector in equation (2.3c), i.e., θ = [ˆb (ˆa + am)]. Define the estimation error as ǫ1 = x − ˆx (2.5) The estimation error dynamics is given by ˙ ǫ1 = −amǫ1 − ˜ax −˜ bu (2.6) where ˜a , ˆa − a and ˜b , ˆb − b. Equation (2.6) describes the behavior of the estimator which is influenced by the parameters and their estimates. If the parameter estimates converge to the true parameter values, i.e., ˆa = a,ˆb = b, then ǫ1 → 0 exponentially1, since am > 0. The adaptive law for updating the parameter estimates, ˆa and ˆb , can be derived using the estimation error ǫ1 as ˙ˆ a = f1(ǫ1, x, ˆx, u), ˙ˆb = f2(ǫ1, x, ˆx, u) (2.7) Notice that f1 and f2 are functions of the measured signals which are chosen such that the equilibrium state ˆae = a, ˆb e = b, ǫ1e = 0 (2.8) of equations (2.6) and (2.7) is uniformly stable or asymptotically stable or exponentially stable. Let us consider a Lyapunov function candidate as a function of the three states ǫ1, ˜a,˜b V (ǫ1, ˜a,˜b ) = 1 2 (ǫ21 + ˜a2 +˜b 2) (2.9) 1The choice of am determines the rate of convergence of ǫ1, since the decay is as per e−amt. 19 2.2. ONLINE PARAMETER ESTIMATION The function V is positive definite, decrescent and radially unbounded in R3. The time derivative of V along the trajectories of equations (2.6) and (2.7) is ˙V (ǫ1, ˜a,˜b ) = ǫ1(−amǫ1 − ˜ax −˜ bu) + ˜af1 +˜ bf2 (2.10a) = −amǫ21 − ˜axǫ1 −˜ buǫ1 + ˜af1 +˜ bf2 (2.10b) If f1 = ǫ1x, f2 = ǫ1u, we have ˙V = −amǫ21 ≤ 0. (2.11) The function ˙V is negative semidefinite since for all ˜a,˜b ∈ R and ǫ1 = 0, ˙V = 0. The parameter update law is given by ˙ˆ a = ǫ1x, ˙ˆb = ǫ1u (2.12) where ˆx is generated using equation (2.4). 2.2.1.1 Analysis Applying Theorem B.2, we conclude that V is a Lyapunov function and the equilibrium given by equation (2.8) is uniformly stable. Also V > 0, ˙V ≤ 0 implies that V ∈ L∞ which further implies that ǫ1, ˜a, ˜b ∈ L∞. Since ǫ1 = x − ˆx ∈ L∞ and x is bounded, ˆx ∈ L∞. Hence all the signals in the system described by equation (2.6) are bounded. Additionally, ǫ122 = Z ∞ 0 ǫ21 (τ )dτ = − 1 am Z ∞ 0 ˙Vdτ = − 1 am (V∞ − V0), V ∈ L∞ ⇒ ∃V∞ lim t→∞ V = V∞ ⇒ ǫ122 exists ⇒ ǫ1 ∈ L2 From equation (2.6), since ǫ1, ˜a, x,˜b, u ∈ L∞, ˙ ǫ1 ∈ L∞. Since ǫ1 ∈ L2 ∩ L∞ and ˙ ǫ1 ∈ L∞, using Lemma B.2, ǫ1 → 0 as t → 0. Fromequation (2.12), ˙ˆ a, ˙ˆb → 0 as t → ∞. Nowwe have established that ǫ1, ˙ˆ a, ˙ˆb → 0, as t → ∞ but this does not imply that ˜a,˜b → 0 as t → ∞. From equation (2.9) it is clear that limt→∞ V = V∞ = 1 2 (˜a2 +˜b 2) but this does not mean that ˜a,˜b have a limit. Notice that there is no restriction on the input u apart from being bounded. To guarantee parameter convergence additional conditions have to be imposed on the input u. 2.2.1.2 Sufficiently Rich Signals In this section the properties of the input signal for parameter convergence are discussed in detail. The notion of sufficiently rich signal and persistent excitation are introduced. 20 2.2. ONLINE PARAMETER ESTIMATION The general solution for the differential equation (2.1) is x(t) = eatx0 + Z t 0 ea(t− )bu(τ )dτ (2.14) The first part of equation (2.14) decays to zero as t → ∞ since a < 0 . Taking Laplace transform on both sides for equation (2.1) with zero initial condition we get X(s) = b s − a U(s) = G(s)U(s) (2.15) If u = 0, then the solution carries information about the parameter a alone, and if x0 = 0, information about both the parameters a, b cannot be extracted from the input/output signals. Similarly if u = c, c 6= 0, a constant, then u(t) = c ⇒ U(s) = c s ⇒ X(s) = b c s(s − a) = −b c/a s + b c/a s − a ⇒ x(t) = −b c a + b c a eat x(t) → −bc a as t → ∞. Hence, the input/output signals only have the information about the ratio b a , and not about the individual values of a and b. If u(t) = sin ω0t, then X(s) = ω0 b (s2 + ω2 0)(s − a) ⇒ x(t) = k1eat + L−1{ k2s + k3 s2 + ω2 0 } where k1, k2, k3 are constants which depend on a and b. Notice that as a < 0 and t → ∞, x(t) = L−1{ k2s + k3 s2 + ω2 0 } = G(jω0) sin(ω0t + ̺), ̺ = ∠G(jω0) G(jω0) = p b ω2 0 + a2 , ∠G(jω0) = tan−1 ImgG(jω0) ReG(jω0) By observing the magnitude and the phase of the output signal at steadystate, the unknown constants a and b can be determined. Hence, with a sinusoidal input the unknown parameters can be estimated. Such a signal is a sufficiently rich signal for this system. The property of the signal u to be sufficiently rich is called persistent excitation (PE). Persistent excitation is very important in identification problems where parameter convergence is of utmost importance. 21 2.3. ADAPTIVE CONTROL For adaptive control problems where output error convergence to zero is desired, it is not critical. For more information on persistent excitation refer to [17] (pages 63 – 73), [18] (pages 177 – 180) and [22] (Chapter 6). 2.3 Adaptive Control In this section the design of model reference adaptive control is introduced by considering a scalar tracking problem. The stability analysis and convergence properties use the mathematical preliminaries given in Appendix B. 2.3.1 Adaptive Regulation using a ReferenceModel: An Example Consider a scalar plant with dynamics described by x˙ = ax + u, x(0) = x0 (2.19) where a is the unknown plant parameter. It is desired to apply a bounded control input, u, such that the closed system is stable and x → xm as t → ∞, where xm is the output of the reference model given by x˙m = −amxm, xm(0) = xm0, am > 0 (2.20) 2.3.1.1 Control Law Let u = −k∗x be the control law such that a − k∗ = −am. In other words if the parameter a is known, then one can choose k∗ = a+am so that the closedloop dynamics of equation (2.19) is same as the reference model (2.20). If x0 = xm0, then x(t) = xm(t) for all t ≥ 0. And if x0 6= xm0 then the transient response of the closedloop system will differ from the reference model, at steady state x(t) = xm(t). Because the parameter a is unknown, the feedback gain k∗ cannot be calculated and hence the control law cannot be implemented. A control law based on the estimate of the unknown parameter is chosen as u = −k(t)x (2.21) where k(t) is the estimate of k∗. 22 2.3. ADAPTIVE CONTROL 2.3.1.2 Adaptive Law An adaptive law is now developed to estimate the unknown parameter by posing the problem as an online estimation problem. Let us define the tracking error as e1 = x − xm. Therefore, e˙1 = x˙ − x˙m = (a − k)x + amxm (2.22a) = ax + amxm − kx = ax + amxm − kx + k∗x − k∗x = ax −{zk∗x}+amxm + k∗x{−z kx} = −amx {+z amxm}−˜kx, ˜k = k − k∗ e˙1 = −ame1 − ˜kx (2.22b) ⇒ e1 = 1 s + am (−˜kx) = Gc(s)(−˜kx) The transfer function Gc(s) is SPR from Theorem B.6. Hence, SPRLyapunov2 design scheme is considered. All the signals in equation (2.22b) can bemeasured except ˜k which is a function of time. The error dynamics in equation (2.22b) is in a suitable form to choose the adaptive law for k(t) such that e1 → 0 as t → ∞. Let ˙˜ k(t) = f1(e1, x, u) where f1 is a function to be chosen. Let V (e1, ˜k) = e21 2 + ˜k2 2γ (2.23) be a Lyapunov function candidate with γ > 0 for the system described by ˙ e1 and ˙k . Choosing f1 = γe1x, we get ˙V = −ame21 ≤ 0 and ˙k = γe1x, k(0) = k0 (2.24) 2.3.1.3 Analysis Because V > 0 and ˙V ≤ 0, V is bounded i.e., V ∈ L∞ ⇒ e1 ∈ L∞, ˜k ∈ L∞ (from (2.23)). e1 = x − xm ⇒ x ∈ L∞ (since xm ∈ L∞). Additionally, Z ∞ 0 e21 (τ )dτ = Z ∞ 0 ˙V (τ )dτ = V (0) − V (∞) ⇒ ˙V ∈ L1 ⇒ e1 ∈ L2 (2.25) 2In this design the estimation error is related to the parameter by a SPR transfer function. Once in this form, the KYP ( Lemma B.5) or MKY (Lemma B.6) Lemmas can be invoked to choose an appropriate Lyapunov function V such that ˙V 0. Refer to chapter 4 (OnLine Parameter Estimation) in [18]. 23 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN ˜k ∈ L∞ ⇒ k ∈ L∞ since, k∗ is constant. From equation (2.22b) we get e˙1 ∈ L∞. Therefore, all the signal in the closedloop system are bounded. Because e˙1 ∈ L∞ and e1 ∈ L2 ∩ L∞, from lemma B.2, e1(t) → 0 as t → ∞. u = −kx ⇒ u ∈ L∞. Hence, all the signals in the closedloop system are bounded and the tracking error, e1, converges to zero. It has to be noted that, even though k ∈ L∞, ˙k → 0 as t → ∞, ˜k may not converge to zero. Convergence of ˜k to zero can be gauranteed only when ˙k ∈ L∞, k ∈ Lp ∩ L∞ where p ∈ [1,∞] (using lemma B.2). 2.4 Simplified Guide Adaptive Controller Design In the rest of this chapter model reference adaptive controllers called Guide Adaptive Controller (GAC) for web guiding applications are developed. In this section a simplified approximation of the GAC is designed to regulate the position of the web by considering a simple lateral web dynamic model. Recall that the transfer functions for both the steering guide and the displacement guide are of relative degree n∗ = 0 and of the order n = 2. By approximating the system with a reduced order model, the number of estimated parameters in the adaptive control design is reduced and the controller implementation is simpler. The transfer function for a steering guide (between the guide position and the lateral web position) is given by yL(s) = s2 + β2s + β1 s2 + β2s + β0 Z(s) (2.26) The transfer function between the voltage input and the lateral web position is given by yL(s) = kmCm(s2 + β2s + β1) s(s + a)(s2 + β2s + β0) up(s) (2.27) where km and a are motor parameters and Cm is the transmission ratio between the actuator position and the guide position. Ignoring the dynamics of the web and considering only the static gain of the lateral dynamics, we get yL(s) = kmCmβ1 s(s + a)β0 = K s(s + a) up(s) , kp Zp Rp up(s). (2.28) The transfer function kp Zp(s) Rp(s) represents a general transfer function where Zp(s), Rp(s) are monic polynomials and kp is a constant. The complexity of the adaptive controller depends on the degree of the polynomials Zp(s) and Rp(s). The knowledge of the sign of the constant kp simplifies the adaptive controller design. 24 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN The dynamics in equation (2.28) in time domain is given by y¨L + ay˙L = Kup. (2.29) Define state variables y1 = yL and y2 = y˙L. The dynamics in the state space form is y˙1 y˙2 = 0 1 0 −a y1 y2 + 0 K up (2.30a) = Ay + Bup (2.30b) yL = [1 0]y = C⊤y (2.30c) Notice that since K > 0, a > 0, the system is controllable. Since the system is controllable, a suitable reference model can be chosen such that the closedloop plant dynamics matches the referencemodel dynamics. A referencemodel with the same relative degree as the plantmodel is chosen to be ym r = ω2n s2 + 2ζωns + ω2n = km Zm Rm = Wm(s) (2.31) The reference model in state space form is x˙ 1 x˙ 2 = 0 1 −ω2n −2ζωn x1 x2 + 0 ω2n r (2.32a) x˙ = Acx + Bcr (2.32b) ym = [1 0]x = C⊤ c x (2.32c) Two different types of adaptive controllers can be designed for the simplified system given in equation (2.28), each of which has a different number of estimated parameters. By reducing the number of estimated parameters, the controller is simple. But with very few parameters, it may not be possible to sufficiently capture the dynamic behavior of the system. Theoretical and simulation results are presented for the two adaptive controllers designed based on the reduced order lateral dynamicmodels. The results indicate that for lateral guiding applications the simplified adaptive controllers are able to provide good tracking performance in the presence of process variations and disturbances. Experimental results that illustrate the effect of the number of the estimated parameters on guiding performance will be discussed in the next chapter. 25 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN 2.4.1 Three parameter Guide Adaptive Controller 2.4.1.1 Control Law Our objective is to design an adaptive control law such that the closedloop plant matches the reference model. Consider a control law with a feedback term and a feed forward term. up = L∗y + P∗r = [L∗1 L∗2 P∗] y1 y2 r = θ∗⊤ω (2.33a) where L∗1 ,L∗2 and P∗ are the parameter values which make the closedloop system match the reference model. Substituting up in equation (2.30) y˙ = 0 1 0 −a y + 0 K [L∗1 L∗2 P∗] y1 y2 r = 0 1 0 −a y + 0 0 KL∗1 KL∗2 y + 0 KP∗ r = 0 1 KL∗1 KL∗2 − a y + 0 KP∗ r (2.34) Comparing the closedloop system in equation (2.34) with the desired reference model given by equation (2.32a) , the desired control parameters that match the closedloop system with the reference model are L∗1 = − ω2n K L∗2 = a − 2ζωn K P∗ = ω2n K 2.4.1.2 Adaptive Law Since the parameters K and a are unknown, it is not possible to find the exact values of the control parameters. The adaptive control problem is posed as an estimation problem with an objective of minimizing the estimation error. Let us estimate L∗1 ,L∗2 , P∗ and use the estimates (θ⊤ = [L1 L2 P]) in the control input. 26 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN Consider the openloop system y˙ = 0 1 0 −a y + 0 K up (2.35) Adding and subtracting the desired control input we get y˙ = 0 1 0 −a y + 0 K h up + θ∗⊤ω − θ∗⊤ω i (2.36) where θ∗⊤ = [L∗1 L∗2 P∗] , ω⊤ = [y1 y2 r] y˙ = 0 1 0 −a y + 0 K [up + L∗1 y1 + L∗2 y2 + P∗r − L∗1 y1 − L∗2 y2 − P∗r] (2.37) = 0 1 0 −a y + 0 K [L∗1 y1 + L∗2 y2 + P∗r]  {z } + 0 K [up − L∗1 y1 − L∗2 y2 − P∗r]  {z } (2.38) = 0 1 −ω2n −2ζωn y + 0 ω2n r  {z } + 0 K h up − θ∗⊤ω i  {z } (2.39) = A cy {+zBcr}+ 0 K h up − θ∗⊤ω i  {z } (2.40) Denote the state estimation error by ǫ , y−x and the tracking error by e1 , yL −ym. Then, ˙ ǫ = Acǫ + B h up − θ∗⊤ω i = Acǫ + B˜θ⊤ω, ˜θ = θ − θ∗ (2.41) e1 = [1 0]ǫ = C⊤ c ǫ (2.42) Therefore, e1 = C⊤ c [sI − Ac]−1 0 K h up − θ∗⊤ω i (2.43) Since Wm(s) = C⊤ c [sI − Ac]−1 0 KP∗ , we have (2.44) e1 = Wm(s)ρ∗ ˜θ⊤ω (2.45) 27 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN where ρ∗ = 1 P∗ . One can think of an adaptive law based on the SPRLyapunov approach and hence could take advantage of the KYP (B.5) and MKY lemmas (B.6). Since the relative degree of the reference model is 2, Wm(s) is not SPR and hence a modification to the system has to be made so that the error e1 relates to the parameters through a SPR transfer function. With this modification the adaptive controller design procedure is similar to a relative degree one system (controller design for a relative degree one system is presented in Appendix C.1). If the controller can have differentiators, one can choose a transfer function L(s) = (s + p0) such that Wm(s)L(s) is SPR and follow the same procedure as in Appendix C.1. Narendra and Valavani [23] introduced a new control structure which was able to make Wm(s)L(s) SPR without using differentiators in the controller. Since Wm(s) has a relative degree 2, let us use the operator PL(θ) , L(s)θL(s)−1 in equation (2.45) and follow the same modification to the control structure and the error equation as in [23] to find the adaptive control law. The error equation changes to ˙ ǫ = Acǫ + Bρ∗ ˜θ⊤φ, B = (s + p0)BP∗ (2.46a) e1 = Wm(s)(s + p0)  {z } ρ∗ ˜θ⊤φ, φ = 1 s + p0 ω (2.46b) = Wm1(s)ρ∗ ˜θ⊤φ, where Wm1(s) is SPR (2.46c) = Wm1(s)ρ∗(θ⊤φ − θ∗⊤φ) (2.46d) = Wm1(s)ρ∗(L(s)−1θ⊤ω − θ∗⊤φ) (2.46e) Let the estimate of tracking error be ˆe1 = Wm1(s)ρ(L(s)−1up − θ⊤φ) (2.47) Notice that equation (2.47) is similar to equation (C.3). In order to use the procedure as in Appendix C.1, the estimation error ǫ1 = e1 − ˆe1 should be e1, hence the controller has to be modified such that the term (L(s)−1up − θ⊤φ) is zero. up = L(s)θ⊤φ = (s + p0)θ⊤φ = ˙ θ⊤φ + θ⊤ ˙φ + p0θ⊤φ = ˙ θ⊤φ + θ⊤ (˙φ + p0φ)  {z } = θ˙⊤φ + θ⊤ω (2.48a) 28 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN With the appropriate modifications to the estimation error dynamics and the controller, the dynamics of the system resemblems a relative degree one system. The procedure in Appendix C.1 is followed to find the adaptive law that results in a stable system with limt→∞ e1(t) = 0. The adaptive law is θ˙ = −e1φsgn(ρ∗) = −e1φ ∵ ρ∗ > 0 (2.49) Notice that the controller has a differentiator, but θ˙ can be obtained from the adaptive law given in equation (2.49). This modified structure was first introduced in [23]. For a detailed analysis of this problem (relative degree = 2) refer to chapter 5, pages 199  226, [22] and the article by Narendra and Valavani [23]. 2.4.1.3 Simulation In order to understand the validity of the simplified model, the GAC based on the simplified model was simulated in Simulink using the complete plant model. The lateral web dynamics block in Figure 2.7 represents the second order model for the web dynamics. The adaptive law and the control law are based on the simplifiedmodel. Figure 2.6 shows the performance of the controller with a sinusoidal reference trajectory. The top plot shows the tracking performance, the middle shows the controller output while the bottom plot shows the time history of the estimated parameters. Notice that the estimated parameters reach a steady value but these parameter values are not necessarily their true values. The parameter update stops as soon as the error reaches zero. The Simulink block diagram for this simulation is shown in Figure 2.7. The simulation results indicate that the adaptive control scheme based on the approximate model is capable of regulating the position of the web. A sinusoidal reference with a bias was used as the output of the reference model in order to observe the transient, steadystate and tracking performance of the controller. 2.4.2 Simplified GAC with an Estimator In consideration of the implementation of the controller in practice note that it is not possible to obtain the measurement of the lateral velocity y˙L (Notice that the control input depends on ω which depends on y˙L). One possible way to obtain the lateral velocity y˙L would be to use the finite difference approximation from the lateral position measurement yL. One can also setup an estimator for the measurement y˙L based on the previous assumption that the dynamics of 29 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 −10 −5 0 5 10 15 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −1 −0.5 0 0.5 1 1.5 Time history of estimated parameters Time in seconds Parameter values y m y p L1 L2 P Figure 2.6: Simulation Results for 3Parameter GAC Web and Actuator Dynamics Sensor Gain Adaptive Controller Adaptive Controller Regressor Adaptive Law ym r error lateral position yp control up 1 yp ym u e e omega up omega r yp 1 Reference Model lateral position yp reference r yp omega 1 yp1 du/dt 2 1 Regressor Adaptive Law −Gamma error omega theta dot theta control up phi 1 1 th s 1 s K*u 2 K*u 1 Figure 2.7: Simulink Block Diagram for 3Parameter GAC 30 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN the web is a constant and if the motor velocity measurement is available. The relationship between the motor velocity and the lateral velocity can be approximated as y˙L = Kw θ˙ = Kwvm (2.50) where y˙L is the lateral web velocity, Kw is a constant which is the approximated lateral web dynamics and θ˙ = vm is the motor velocity. Filtering both left and right hand side of equation (2.50) by a low pass filter F(s) = 1 s+1 and taking the Laplace transform we get sF(s)yL = KwF(s)vm (2.51) Let an intermediate system be defined as Z = ϕ∗⊤φ (2.52) where Z , F(s)vm, φ = sF(s)yL and ϕ∗ = 1 Kw . Hence φ is obtained by filtering the lateral position measurement using the filter s s+1 and Z is obtained by filtering the motor velocity vm by using the filter 1 s+1 . Let e be the estimation error y − ϕ⊤φ with ϕ being the estimate of ϕ∗. Notice that this is a linear estimation problem and hence the minimum can be reached in a single step if the update is in the direction of negative gradient of the cost function. It is common to choose the cost function as J(ϕ) = e2 2 and hence the gradient is▽J(ϕ) = ee˙ = −eφ. Hence the update law for the ϕ is ϕ˙ = γeφ, where γ > 0 and ϕ(0) = ϕ0. From the estimate of ϕ = 1 Kw , we can use the equation (2.50) to find the estimated lateral velocity of the web. 2.4.2.1 Simulation Figure 2.8 shows the simulation result for the simplified guide adaptive controller with the estimated lateral velocity. The top plot shows the tracking performance of the controller while the middle plot shows the control effort and the bottom plot shows the estimated parameters. The simulation results indicate a similar performance as observed with the previous controller. The Simulink model is presented in shown in Figure 2.9. 2.4.3 Four Parameter Guide Adaptive Controller Notice that the adaptive controllers derived in the previous section require two measurements, the lateral position yL and the velocity y˙L. It is possible to derive a model reference adaptive 31 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 0 5 10 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −1 0 1 2 Time history of estimated parameters Time in seconds Parameter values y m y p L1 L2 P Figure 2.8: Simulation Results for 3Parameter GAC with an Estimator Web and Actuator Dynamics Sensor Gain Adaptive Controller Regressor ym r error Lateral Position yp control up Tachometer Feedback yp dot estimator 1 yp ym u e e omega up omega r yp yp dot 2 1 Adaptive Controller Adaptive Law −Gamma error omega theta dot theta control up phi 1 th 1 s 1 s K*u 2 K*u 1 Regressor Adaptive Law lateral position yp reference r yp r omega estimated lateral velocity 3 1 2 1 Figure 2.9: Simulink Block Diagram for 3Parameter GAC with an Estimator 32 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN control law for the system described by equation (2.30) using one measurement, the lateral position measurement yL. The following section describes the design process. 2.4.3.1 Control Law Let us define the following control law for the system described by equation (2.30) so that the dynamics of the closedloop plant matches the dynamics of the reference model given in equation (2.32a) . up = θ∗ 1 1 (s) up + θ∗ 2 1 (s) yL + θ∗ 3yL + c∗0 r (2.53) where (s) = s + a0 be a Hurwitz polynomial and θ∗ i ’s, c∗0 are the true parameters which make the closedloop system dynamics match the reference model dynamics. The usefulness of parametrizing the control law in this manner will be evident later. Rewriting the control law in terms of the feedback and feedforward terms we get up = θ∗ 3s + (θ∗ 2 + a0θ∗ 3) s + (a0 − θ∗ 1) yL + (s + a0)c∗0 s + (a0 − θ∗ 1) r (2.54) Substituting the control law given by equation (2.54) in the system described by equation (2.28) we get s2yL + asyL = K θ∗ 3s s + (a0 − θ∗ 1) yL + θ∗ 2 + a0θ∗ 3 s + (a0 − θ∗ 1) yL + (s + a0)c∗0 s + (a0 − θ∗ 1) r (2.55) (s2 + as)(s + (a0 − θ∗ 1)) − Kθ∗ 3s − K(θ∗ 2 + a0θ∗ 3) yL = K(s + a0)c∗0 r (2.56) yL r = K(s + a0)c∗0 s3 + (a0 − θ∗ 1 + a)s2 + [a(a0 − θ∗ 1) − Kθ∗ 3] s − K(θ∗ 2 + a0θ∗ 3) = w2n s2 + 2ζωns + ω2n (2.57) Notice that in order to have the closedloop system dynamics match the reference model dynamics, there has to be a polezero cancellation. The cancellation will occur at −a0 and hence the filter (s) = s+a0 has to be Hurwitz. If (s) is not Hurwitz, then the polezero cancellation will occur in the righthalf plane which is not desirable. If Kc∗0 = ω2n then, (s + a0)(s2 + 2ζωns + ω2n ) = s3 + (a0 − θ∗ 1 + a)s2 + [a(a0 − θ∗ 1) − Kθ∗ 3] s − K(θ∗ 2 + a0θ∗ 3) (2.58a) 33 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN Equating both we obtain the true parameters to be θ∗ 1 = a − 2ζωn (2.59a) θ∗ 3 = 1 K (aa0 − a2 + 2ζωna − ω2n − 2ζωna0) (2.59b) θ∗ 2 = − a0ω2n K − a0θ∗ 3 (2.59c) c∗0 = ω2n K (2.59d) It is not always possible to find the desired control law which can make the closedloop system match the reference model. A few conditions on the type of the plant model and the reference model have to be made [18]. 1. The polynomial Zp(s) should be Hurwitz and monic. 2. The order, n, of the plant should be known or at least an upper bound should be known. 3. The relative degree of the plant (n∗) should be known. 4. The sign of kp should be known. 5. Zm(s),Rm(s) should be monic and Hurwitz and the order of the reference model should not be greater than the order of the plant. 6. The relative degree of the reference model should be same as that of the plant model. From equation (2.57) it can be seen that if the zeros of the reference model are not the same as the plant zeros, then the plant zeros will be cancelled by the control in order to match the closedloop system dynamics to the reference model dynamics. If Zp(s) is not Hurwitz, then polezero cancellation would occur in the right half plane which is not desirable. If Zp(s) is not monic, then the constant termof the leading coefficient can be pulled out and augmented to the gain kp. The requirement for the sign of kp to be known will be evident from the adaptive law derivation. Without the knowledge of the relative degree of the plant, it would not be possible to design an adaptive control law. The order of the plant is important because it is not always possible to find a parametrized control law such that the closedloop systemdynamicsmatches the reference model. But we can use the Bezout Identity to overcome the problem (refer to [22] sections 5.4.1 and 5.4.2). In order to obtain proper parametrization of the controller parameters θ∗ i ’s, both the numerator and denominator polynomials of the referencemodel transfer function should be Hurwitz. 34 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN In order to have the minimum number of controller parameters, it is necessary that both polynomials of the transfer function of the reference model be monic. Since the parameters a and K are unknown it is not possible to use the control law in equation (2.54) and hence the adaptive control problem is posed as an estimation problem with the objective of minimizing the parameter estimation error. We can follow the same procedure of adding and subtracting the desired control effort, but first we need to represent the control law in the statespace form in order to simplify the analysis. We shall also define an augmented system which is useful for our analysis both in this section as well as in the next section. up = θ∗ 1 1 s + a0 up + θ∗ 2 1 s + a0 yL + θ∗ 3yL + c∗0 r up = θ∗⊤ω where θ∗⊤ = [θ∗ 1, θ∗ 2, θ∗ 3, c∗0 ] and ω⊤ = 1 s + a0 up, 1 s + a0 yL, yL, r . ω˙ 1 = Fω1 + gup, ω1(0) = 0, F = −a0 (2.60a) ω˙ 2 = Fω2 + gyp, ω2(0) = 0, g = 1 (2.60b) The statespace representation of the closedloop system with the desired control effort can be obtained by augmenting the states y of the system described by equation (2.30) with the states of the controller in equation (2.60). ˙Y = AsY + Bsc∗0 r (2.61a) yL = C⊤ s Y (2.61b) where Y = y⊤, ω1, ω2 ⊤ and y˙ = Ay + Bup = Ay + B [θ∗ 1ω1 + θ∗ 2ω2 + θ∗ 3yL + c∗0 r] (2.62a) ω˙ 1 = Fω1 + gup = Fω1 + g [θ∗ 1ω1 + θ∗ 2ω2 + θ∗ 3yL + c∗0 r] (2.62b) ω˙ 2 = Fω2 + gyp = Fω2 + gC⊤y (2.62c) ∴ ˙Y = A + Bθ∗ 3C⊤ Bθ∗ 1 Bθ∗ 2 gθ∗ 3C⊤ F + gθ∗ 1 gθ∗ 2 gC⊤ 0 F Y + c∗0 B g 0 r (2.63a) yL = [C⊤ 0 0]Y (2.63b) 35 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN From equation (2.57) yL(s) r(s) = K(s + a0)c∗0 s3 + (a0 − θ∗ 1 + a)s2 + [a(a0 − θ∗ 1) − Kθ∗ 3] s − K(θ∗ 2 + a0θ∗ 3) = Wm(s) (2.64) and from equation (2.61) yL(s) r(s) = C⊤ s (sI − As)−1Bsc∗0 = Wm(s) = ym r (2.65) The reference model can be described by the following state space representation. ˙Y m = AsYm + Bsc∗0 r (2.66a) ym = C⊤ s Ym (2.66b) Similarly, the augmented closedloop system with the control up can be obtained as ˙Y c = A0Yc + Bsup (2.67a) yL = C⊤ s Yc (2.67b) where Yc = y⊤, ω1, ω2 ⊤ . y˙ = Ay + Bup (2.68a) ω˙ 1 = Fω1 + gup (2.68b) ω˙ 2 = Fω2 + gyp = Fω2 + gC⊤y (2.68c) Therefore, ˙Y c = A 0 0 0 F 0 gC⊤ 0 F Yc + B g 0 up (2.69a) yL = [C⊤ 0 0]Yc (2.69b) 2.4.3.2 Adaptive Law Our objective now is to obtain an estimate of θ∗ so that the closedloop system described in equation (2.69) matches the reference model in equation (2.66). Adding and subtracting the 36 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN desired control effort θ∗⊤ω we get ˙Y c = A0Yc + Bsup + Bsθ∗⊤ω − Bsθ∗⊤ω (2.70a) = A0Yc + Bsθ∗⊤  {z ω}+Bs(up − θ∗⊤ω) (2.70b) = AsYc + Bsc∗0 r + Bs(up − θ∗⊤ω) (2.70c) yL = C⊤ s Yc (2.70d) Let ǫ = Yc − Ym be the state estimation error and e1 = yL − ym be the tracking error. Hence the error equations can be obtained from equations (2.70) and (2.66) as ˙ ǫ = Asǫ + Bs(up − θ∗⊤ω), ǫ(0) = ǫ0 (2.71a) e1 = C⊤ s ǫ (2.71b) Notice that e1 up − θ∗⊤ω = C⊤ s (sI − As)−1Bs (2.72) and Wm(s) = C⊤ s (sI − As)−1Bsc∗0 (2.73) ⇒ e1 = Wm(s)ρ∗(up − θ∗⊤ω), ρ∗ = 1 c∗0 (2.74) The error equation is similar to the form seen in equation (C.5) and the same procedure can be continued if Wm(s) is SPR. Since the relative degree of the reference model is 2, Wm(s) is not SPR and hence we shall use the operator PL(θ) as before and the modified error equation is ˙ ǫ = Asǫ + Bcρ∗ ˜θ⊤φ (2.75a) e1 = C⊤ s ǫ (2.75b) which is similar to the error equation (2.46). The same procedure is followed and the adaptive law and control law which result in a stable system with e1(t) → 0 as t → ∞ are given by θ˙ = −e1φsgn(ρ∗) = −e1φ (2.76a) up = θ⊤ω + θ˙⊤φ (2.76b) 2.4.3.3 Simulation Figure 2.10 shows the performance of the adaptive controller. The plant used in this simulation is the complete plantmodel instead of the simplified plant. The top plot shows the tracking performance, the middle plot shows the control input, while the bottom plot shows the estimated 37 2.5. GUIDE ADAPTIVE CONTROLLER parameters. The Simulink block diagram for the simulation is shown in Figure 2.12. Similar to the previous controller the adaptive controller with four parameters is capable of regulating the lateral position of the web. By observing the control effort for the two controllers one can notice that the controller with four parameters exhibits a better transient response behavior. This can be attributed to the fact that the controller with three parameters has no knowledge of the control effort generated. The estimated parameters settle to a steady state value after the plant starts tracking the sinusoidal reference. A common disturbance which helps in characterizing the transient response is a pulse disturbance. The performance of the controller with a pulse disturbance is shown in Figure 2.11 and the plots indicate that the controller is capable of rejecting the pulse disturbances. 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 0 5 10 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −1 0 1 2 3 Time history of estimated parameters Time in seconds Parameter Values y m y p q 1 q 2 q 3 c 0 Figure 2.10: Simulation Results for 4Parameter GAC 2.5 Guide Adaptive Controller In the previous section a simplified guide adaptive controller was designed based on the assumption that the dynamics of the web is a constant. Here in this section we shall consider the complete dynamics of the web and develop a guide adaptive controller. The transfer function for a steering guide is given by equation (2.27) and its state space form is y˙ = Ay + Bup, yL = C⊤y (2.77) 38 2.5. GUIDE ADAPTIVE CONTROLLER 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 −10 −5 0 5 10 15 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −2 −1 0 1 2 3 Time history of estimated parameters Time in seconds Parameter values y m y p q 1 q 2 q 3 c 0 Figure 2.11: Simulation Results for 4Parameter GAC with a Pulse Disturbance Consider the reference model in equation (2.32a) and the objective is to design a control law such that the closedloop system matches the reference model. So far our first step in designing an adaptive controller has been to find the desired control input which can ensure that the closedloop system matches the reference model. Then we would pose the problem as an estimation problem and use the estimated parameters in the control law. In a general model reference adaptive control problems with relative degree n∗ = 1, it is simple to choose a parametrized control law which can algebraically render the closedloop system to have the same roots as the reference model. But in a general adaptive control problem with relative degree n∗ ≥ 2, finding the desired control is not simple. In the previous section, we were able to choose a desired control (θ∗⊤ω) which resulted in a closedloop system that matched the reference model. As discussed in [22] (section 5.4.1 and 5.4.2), Bezout Identity can be used to show that it is always possible to find a parameter vector θ∗ so that the following control law [18] will result in a closedloop system whose dynamics is same as the reference model dynamics: up = θ∗ 1 ⊤ α(s) (s) up + θ∗ 2 ⊤ α(s) (s) yL + θ∗ 3yL + c∗0 r (2.78) where α(s) = [s2 s 1]⊤, (s) is an arbitrary monic Hurwitz polynomial of degree 3 which has to be chosen. Similar to the procedure in the previous section (section (2.60) on page 35) we can 39 2.5. GUIDE ADAPTIVE CONTROLLER Web and Actuator Dynamics Sensor Gain Adaptive Controller Adaptive Controller Regressor Adaptive Law ym r error lateral position yp control up 1 yp ym u e e omega up omega r yp 1 Reference Model Regressor Adaptive Law −Gamma error omega theta dot theta control up phi 1 1 th s 1 s K*u 2 K*u 1 Control up lateral position yp reference r omega 1 omega 2 yp r omega 1 1 s 1 s g g K*u K*u 3 2 1 Figure 2.12: Simulink Block Diagram for 4Parameter GAC 40 2.5. GUIDE ADAPTIVE CONTROLLER represent the control law in the statespace form as ω˙ 1 = Fω1 + gup, ω1(0) = 0 (2.79a) ω˙ 2 = Fω2 + gyp, ω2(0) = 0 (2.79b) up = θ∗⊤ω (2.79c) where ω1, ω2, θ1, θ2 ∈ R3, θ∗ = [θ∗ 1 ⊤ θ∗ 2 ⊤ θ∗ 3 c∗0 ] and ω = [ω⊤ 1 ω⊤ 2 yL r]. The matrix F and g are realized as F = −λ2 −λ1 −λ0 1 0 0 0 1 0 , g = 1 0 0 (2.80) where (s) = s3 + λ2s2 + λ1s + λ0. The adaptive control design is similar to the design in the previous section but with more estimation parameters which are a result of considering the complete web lateral dynamic model. Notice that in equation (2.59) we have shown the existence of the true parameters with the desired control given by equation (2.53). Similarly, it is possible to show the existence of the true parameters for the adaptive controller with the desired control given in equation (2.78). Same procedure as in the previous section can be followed to obtain the adaptive law and control law (refer to chapter 6 in [18] for a comprehensive formulation). 2.5.1 Control Law up = θ⊤ω + θ˙⊤φ (2.81a) φ = 1 s + p0 ω (2.81b) ω = [ω⊤ 1 ω⊤ 2 yL r] (2.81c) ω˙ 1 = Fω1 + gup, ω1(0) = 0 (2.81d) ω˙ 2 = Fω2 + gyp, ω2(0) = 0 (2.81e) 2.5.2 Adaptive Law θ˙ = −e1φ, e1 = yL − ym (2.82) 41 2.5. GUIDE ADAPTIVE CONTROLLER 2.5.3 Analysis It has been shown in [18] that the above adaptive law along with the control law results in a stable closedloop system with e1(t) → 0 as t → ∞. The proof is similar to the one described in section C.1.0.1. The chosen Lyapunovlike function is V (˜θ, ǫ) = ǫ⊤Pcǫ 2 + ˜θ⊤−1 ˜θρ∗ 2 and with the above control law and adaptive law the time derivative of V is given by ˙V = − ǫ⊤qq⊤ǫ 2 − ǫ⊤νcL⊤c ǫ 2 where q is a vector and νc > 0 is a scalar and Lc is a positive definite matrix. Because V > 0 and ˙V ≤ 0, V is bounded or V ∈ L∞ ⇒ ǫ, θ, ˜θ ∈ L∞. With ǫ ∈ L∞ we get Yc, Ym ∈ L∞. In addition to proving that u = θ⊤ω ∈ L∞, we need to prove that u = θ⊤ω + θ˙⊤φ ∈ L∞. Since φ is a filtered version of ω and F being a stable filter, it is clear that u ∈ L∞. 2.5.4 Simulation Figures 2.13 and 2.14 show the performance of guide adaptive controller. The top plot in each figure shows the tracking performance, the middle plot shows the controller output while the bottom plot shows the estimated parameters. In Figure 2.14 in addition to the sinusoidal reference, a pulse disturbance of 0.25 V amplitude is introduced at two time instants, 4 and 12 seconds. The controller is capable of rejecting the pulse disturbance. The Simulink block diagram for the GAC is shown in Figure 2.15. 42 2.5. GUIDE ADAPTIVE CONTROLLER 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 −1 0 1 2 3 4 5 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −0.5 0 0.5 1 1.5 Time history of estimated parameters Time in seconds Parameter values y m y p Figure 2.13: Simulation Results for Guide Adaptie Controller 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 −10 −5 0 5 10 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −1 −0.5 0 0.5 1 1.5 Time history of estimated parameters Time in seconds Parameter values y m y p Figure 2.14: Simulation Results for Guide Adaptie Controller with a Pulse Disturbance 43 2.5. GUIDE ADAPTIVE CONTROLLER Web and Actuator Dynamics Sensor Gain Adaptive Controller Adaptive Controller Regressor Adaptive Law ym r error lateral position yp control up 1 yp ym u e e omega up omega r yp 1 Reference Model Regressor Adaptive Law −Gamma error omega theta dot theta control up phi 1 1 th s 1 s K*u 2 K*u 1 Control up lateral position yp reference r omega 1 omega 2 yp r omega 1 1 s 1 s g g F F 3 2 1 Figure 2.15: Simulink Block Diagram for Guide Adaptie Controller 44 Chapter 3 Experimental Results In the previous chapter the design and analysis of model reference adaptive control strategies suitable for web guiding were presented. Three different guide adaptive controllers were developed based on the number of estimated parameters in the control law. Two adaptive controllers were designed based on a simplified model for the lateral web dynamics and one was based on the complete model. In this chapter the controller designs are implemented on an experimental web platform containing two intermediate web guides and their performance under different operating conditions are evaluated. Guidelines for implementation of the adaptive algorithms are given and discussed. 3.1 Experimental Platform The experimental web handling platform used for implementing the designed controllers is shown in Figure 3.1. A line schematic of the platform is shown in Figure 3.2. The platform is an endless web line with several idle rollers and one driven roller (Master Speed Roller). The platform does not have an unwind or a rewind section; the web runs in a loop around the rollers. The transport velocity of the web is set by the speed of the driven roller. In order to provide traction between the driven roller and the web, a nip roller is used on the driven roller. The nip roller applies pressure at the contact surface to maintain web traction on the driven roller during the startup of the line. The platform is equipped with a passive dancer and an active dancer which can be used to control the tension of the web in the platform [24, 25]. In an active dancer, the dancer roller position is controller by an actuator. The tension in the web line can be increased or decreased by varying the position of the dancer roller. In a passive dancer mechanism the dancer roller is 45 3.1. EXPERIMENTAL PLATFORM Remotely Pivoted Guide OffsetPivot Guide Figure 3.1: Experimental Web Handling Platform MOTOR LOAD CELL LOAD CELL REMOTELY PIVOTED GUIDE NIP ROLLER ACTIVE DANCER SYSTEM PASSIVE DANCER AIR PISTON AIR PISTON MASTER SPEED ROLLER OFFSETPIVOT GUIDE EDGE SENSOR INFRARED SENSOR Figure 3.2: Line Schematic of the Experimental Web Handling Platform 46 3.1. EXPERIMENTAL PLATFORM Ultrasonic Sensor Web Figure 3.3: offset pivot Guide Infrared Sensor Web Figure 3.4: Remotely Pivoted Guide free to move about a pivot or on a linear slide [25]. The dancer roller in a passive dancer is not actuated but floats due to the pressure from an air cylinder. There are two web guides in the platform; an offset pivot guide and a remotely pivoted guide. The offset pivot guide is a Fife Narrow web guide and the remotely pivoted guide is a Fife Kamberoller guide. The offset pivot guide is equipped with an ultrasonic sensor and the remotely pivoted guide is equipped with an infrared sensor. The actuators (servo motors) and the sensors of each guide mechanism are connected to a dSPACE DS1103 realtime board through Fife controller hardware. The Kamberoller guide and its infrared sensor is connected to the realtime hardware through a Fife A9 analog controller. Electrical provisions are made such that the guide can be controlled either with the A9 controller or with the dSPACE hardware. The offset pivot guide is connected to the dSPACE hardware through a Fife DP01 controller. The offset pivot guide can only be controlled using the dSPACE hardware. The feedback signals from the sensors and the control signals to the actuators are processed by the dSPACE hardware. The DS1103 hardware has eight A/D converters for a total of 20 A/D channels, of which 16 channels are multiplexed by four converters. The digital output from the realtime hardware is converted into analog output by 8 D/A converters. The input and output range of the dSPACE board is ± 10 V. The dSPACE board is driven by a PowerPC CPU running at 400 MHz. The realtime control software is written in C using dSPACE RTLib realtime libraries. The control software is driven by interrupts which occur every sampling period. At each sampling time the following operations are performed by the realtime software: 1. the sensor feedback signals are read; 47 3.2. EXPERIMENTAL PROCEDURE 2. the control algorithm is executed and the required control effort is calculated; and 3. the control effort is then supplied to the actuators. The dSPACE board is installed inside a host computer running Microsoft Windows 2000 operating system. The dSPACE ControlDesk software is utilized to communicate between the realtime hardware and the host computer. The realtime hardware is capable of buffering the data using its internal memory. Trace variables can be setup so that the buffered data in the realtime control hardware can be stored/displayed in the host computer. These trace variables can also be used to change the variable values in the realtime control software while executing in realtime. 3.2 Experimental Procedure Experiments were carried out to observe the guiding performance of the designed adaptive controllers for process variations and disturbances. The experimental results presented in this thesis cover only the Kamberoller guide. Experiments were also conducted on the offset pivot guide and similar results were observed. The Kamberoller is located four spans downstream of the offset pivot guide. Any arbitrary disturbance can be created using the offset pivot guide and the Kamberoller can be used to reject the disturbance. On the contrary, since the web has to travel 13 spans fromthe Kamberoller to the offset pivot guide, the lateral disturbances generated at the Kamberoller do not propagate to the offset pivot guide. The Kamberoller guide is connected to the dSPACE hardware through the Fife A9 controller. Electrical modifications to the A9 controller were made so that either the A9 controller or the controller implemented on the dSPACE hardware can be used to control the Kamberoller guide. This setup provides a convenient way to compare the two controllers. Such an arrangement is not available with the offset pivot guide. The adaptive control schemes were compared with an A9 analog controller (PI controller). The A9 controller is the only industrial controller in the experimental platform which can be directly compared with the adaptive control schemes. Modifications to the existing setup are required in order to incorporate other industrial controllers in the experimental platform. The following sections present the process variations and the disturbances thatwere created for controller evaluation. 48 3.2. EXPERIMENTAL PROCEDURE 3.2.1 Process Variations The guiding performance of the developed adaptive control schemes were evaluated with different process parameters. Two main process parameters that affect the coefficients of the web lateral dynamic model are the web span parameter K and the longitudinal velocity of the web v. The parameter K is affected by the properties of the web material. Two different web materials with distinct physical characteristics were used; an optically opaque magnetic film like material (Figure 3.5) and an optically transparent polyethylene polymer web (Figure 3.6). The webs used were of different dimensions. The experiments were conducted with different web transport velocities and with different tensions in the web line. Figure 3.5: Opaque Web Figure 3.6: Transparent Web Whenever the web material is changed from opaque to transparent web, the gain of the infrared sensor changes. The full scale voltage range changes from 0  6 V (opaque) to approximately 5  6 V (transparent) as illustrated in Figure 3.7. One of the drawbacks of the existing controllers is their inability to cope with sensor gain variations. Due to sensor gain variations the guidepoint also changes. The guidepoint is the physical reference position of the web inside the sensing window. Notice from Figure 3.7, the minimum voltage output from the sensor 49 3.2. EXPERIMENTAL PROCEDURE is 5 V for a transparent web. This is due to the fact that the maximum signal attenuation with the transparent web is only 1 V. ACTUAL WEB POSITION SENSOR OUTPUT 0 V 0 in 0.7 in 6 V 5 V Transparent Web Opaque Web Figure 3.7: The Effect of Opacity on Sensor Gain for an Infrared Sensor Experimentswere conductedwith the twowebmaterials to evaluate the performance of the controllers for sensor gain variations. All the parameters of the controllers including the gain matrix and the filter parameters remained constant with both webs. Similarly the PI controller gains remained constant with both webs. Since the guidepoint changes with the opacity of the web material, different guidepoints were used with the two webs. For example, the guidepoint for the opaque was 3 V while the guidepoint with the transparent web was 5.4 V. Both guidepoints correspond to the same physical position of the web. A new technique for automatic guidepoint detection called offsetadaptation is presented in Appendix A. 3.2.2 Disturbances Some common lateral disturbances in industrial lines are the result of misaligned rollers, telescoped unwind rolls, splicing, improper web edges, wrinkled web, etc. To mimic these disturbances in the experimental web platform, sinusoidal, step and pulse disturbances were created by the offset pivot guide and with changes to the web edge. A common disturbance observed in an industrial web line is due to misaligned rollers. 50 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER It is seldom possible to have exact alignment for all the rollers in a web line. The angular misalignment of a series of rollers may produce lateral disturbances which can be periodic. Telescoping of rolls may produce periodic disturbances when unwound. In order to simulate a periodic disturbance, the offset pivot guide in the web line is used. The offset pivot guide, which is located before the Kamberoller in the web line (refer Figure 3.2), was made to follow a sinusoidal reference. As the sinusoid propagates, a periodic disturbance is created at the Kamberoller. In order to characterize the transient response, the performance of the controller with a pulse disturbance was evaluated in the experiments. The pulse disturbance was generated by adding a small strip of web material about one feet long and 0.2 inches wide to the edge of the web. Since the strip is of constant length, the duration of the pulse is dependent on the transport velocity of the web. As the web transport velocity increases, the pulse width reduces and tends to an impulse. In order to observe the performance characteristics such as the percentage overshoot and settling time, stepreferencechange experiments were conducted. The guidepoint or the lateral position reference was changed and the performance of the controllers for this change was evaluated. The sensor signals from both the infrared sensor and the ultrasonic sensor were noisy. In industrial controllers, analog or digital signal conditioning are carried out on the raw signal to filter the noise. The error signal for the adaptive controllers were unfiltered in order to observe the performance of the controllers with noisy measurements. 3.3 Three Parameter Guide Adaptive Controller The performance of the three parameter guide adaptive controller under different operating conditions and disturbances are presented in this section. The three parameter GAC requires an additional measurement, i.e., the lateral web velocity. While implementing the controller using the realtime hardware, the lateral velocity measurement was approximated by the finite difference of the lateral position. Recall that the control law and the adaptive law for the three parameter GAC is given by Control Law up = θ˙⊤φ + θ⊤ω 51 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER Adaptive Law θ˙ = −e1φ where θ⊤ = [L1 L2 P], ω⊤ = [yL y˙L r], e1 = yL − ym and φ = 1 s + p0 ω. 3.3.1 Experiments with opaque web Figures 3.8 to 3.11 show a representative sample of experimental results with the three parameter adaptive controller with the opaque web. In Figure 3.8 the top plot shows the sinusoidal disturbance observed by the Kamberoller when the guide is not actuated. The middle plot shows the performance of the PI controller. And the bottom plot shows the performance of the adaptive control scheme. A significant amount of the sinusoidal disturbance generated at the offset pivot guide has propagated to the Kamberoller. The three parameter guide adaptive controller is able to significantly attenuate the disturbance when compared to the PI controller. The guidepoint for the two experiments were not the same. In subsequent experiments the guidepoints for the adaptive scheme and the PI controller were matched to maintain consistency. In Figure 3.9 the performance of the adaptive controller with sine disturbance is shown in the top plot, the control effort is shown in the middle plot, and the parameter estimates are shown in the bottom plot. The control effort generated is well within the actuator limits which is ± 10 V. Even though the parameters may not converge to their true values, it is important to observe the evolution of the estimated parameters. Notice that the last two parameters (L2 and P) vary sinusoidally while the first parameter (L1) slowly increases. It was experimentally observed that L1 reaches a steadystate value while the other two parameters vary about zero sinusoidally. These observations are useful in improving the robustness properties of the adaptive controller. One such algorithmwhich can increase the robustness of the adaptive controller is parameter projection. For more information on parameter projection refer to page 565 in [18] and page 328 in [17]. The top plot in Figure 3.10 shows the performance of the PI controller with a pulse disturbance while the bottom plot shows the performance of the three parameter GAC. The adaptive controller is able to provide a faster response with better guiding performance. Similar performance is also seen at different transport speeds. The performance of the adaptive controller for stepreferencechange is shown in the Figure 52 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER 3.11. The adaptive controller is capable of following the reference model accurately. The transient response characteristics of the guiding system is similar to the reference model transient response characteristics. 0 5 10 15 20 25 30 2 3 4 5 Sine disturbance observed at the kamberoller guide Lateral Position (V) 0 5 10 15 20 25 30 2 3 4 5 Performance of A9 controller Lateral Position (V) 0 5 10 15 20 25 30 2 4 Performance of three parameter GAC Time (Seconds) Lateral Position (V) Figure 3.8: Performance Comparison: 3 Parameter, Sine Disturbance, 300 fpm, Opaque Web 3.3.2 Experiments with transparent web Figures 3.12 to 3.14 show the performance of the three parameter guide adaptive controller with the transparent web. Figure 3.12 compares the performance of the PI controller and the adaptive controller. The top plot shows the disturbance observed at the Kamberoller when the guide is not actuated. The middle plot shows the performance of the PI controller. Comparing the top and the middle plots, no disturbance attenuation is observed. The PI controller is not able to cope with the sensor gain change. The adaptive controller on the other hand is able to attenuate the sinusoidal disturbance. Notice that a small streak is observed in all the three plots. This streak is observed when the joint passes the sensor window. The web opacity variation at the joint is significant to cause a sudden jump in the sensor output which causes the streak. In Figure 3.13 the top plot shows the adaptive controller performance, the middle plot shows the control effort generated and the bottom plot shows the time history of the estimated parameters. Similar to the experiments with the opaque web, the last two parameters vary 53 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER 0 5 10 15 20 25 30 2 4 6 Three Parameter Adaptive Controller Performance Time (Seconds) Lateral Position (V) Web Position Reference Position 0 5 10 15 20 25 30 −10 0 10 Time (Seconds) Control (V) 0 5 10 15 20 25 30 −0.4 −0.2 0 Time (Seconds) Parameters Figure 3.9: Adaptive Controller: 3Parameter, Sine Disturbance, 300 fpm, Opaque Web 0 5 10 15 20 25 30 3 3.5 4 4.5 Performance of A9 controller Lateral Position (V) 0 5 10 15 20 25 30 3 3.5 4 4.5 Performance of three parameter GAC Time (Seconds) Lateral Position (V) Web Position Reference Position Figure 3.10: Performance Comparison: 3Parameter, Pulse Disturbance, 300 fpm, Opaque Web 54 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER 0 5 10 15 20 25 30 3.5 4 4.5 5 5.5 6 Performance of three parameter GAC (300 fpm) Lateral Position (V) 0 5 10 15 20 25 30 3.5 4 4.5 5 5.5 6 Performance of three parameter GAC (500 fpm) Time (Seconds) Lateral Position (V) Reference Model Output Reference Position Edge Position Figure 3.11: Adaptive Controller: 3Parameter, 300 and 500 fpm, Step Reference Changes, Opaque Web sinusoidally. The sinusoidal variation is centered around zero while the parameter L1 slowly increases and reaches a steady state value. In Figure 3.14 the performance of the PI controller and the adaptive controller for a pulse disturbance is shown. The top plot shows the performance of the PI controllerwhile the bottom plot shows the performance of the adaptive controller. Even though the results look similar, a careful examination would reveal that the PI controller is nonresponsive to the pulse disturbance. Due to the change in the sensor gain the PI controller is not capable of providing good guiding performance. In fact the response observed using the PI controller is exactly the same as the response when the Kamberoller is unactuated. To clearly visualize the performance of the adaptive controller it is necessary to examine the typical response for a pulse disturbance. The pulse disturbance is formed by a series of two steps. The first step has a positive magnitude and the second has the same magnitude but a negative sign. As soon as the first step passes the sensor, an error in lateral position is created. The controller tries to regulate the position by bringing the web back to its reference position. Therefore as soon as the pulse disturbance enters the sensor a sudden dip is observed and the guide is actuated to bring the web back to its reference position. The second step creates a 55 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Sine disturbance observed at the kamberoller guide Lateral Position (V) 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of A9 controller Lateral Position (V) 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of three parameter GAC Time (Seconds) Lateral Position (V) Figure 3.12: Performance Comparison: 3Parameter, 500 fpm, Sine Disturbance, Transparent Web 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Three Parameter Adaptive Controller Performance Time (Seconds) Lateral Position (V) Web Position Reference Position 0 5 10 15 20 25 30 −10 0 10 Time (Seconds) Control (V) 0 5 10 15 20 25 30 −0.4 −0.2 0 Time (Seconds) Parameters Figure 3.13: Adaptive Controller: 3Parameters, 500fpm, Sine Disturbance, Transparent Web 56 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of A9 controller Lateral Position (V) 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of three parameter GAC Time (Seconds) Lateral Position (V) Web Position Reference Position Figure 3.14: Performance Comparison: 3Parameter, 500 fpm, Pulse Disturbance, Transparent Web similar error but in the opposite direction, i.e., it creates a positive error. Therefore, when the pulse disturbance leaves the sensor, a spike is observed and the controller tries to regulate the web edge back to its reference. If the guide is unactuated the lateral position of the web is unchanged as the pulse passes the sensor. The complete pulse is observed in the sensor output. Observing the plots in Figure 3.14, we can notice that the PI controller is unable to regulate the lateral position in the presence of a pulse disturbance. While the adaptive controller is capable of providing good guiding performance in the presence of the pulse disturbance. Additional experimental results for the three parameter adaptive controller are presented in Appendix D section D.1. From the experimental results it is evident that the three parameter adaptive controller is suitable for web guiding. The controller is capable of compensating for the common disturbances and process variations. The adaptive algorithmis able to adapt to the process variations and is capable of providing the appropriate control effort to regulate the lateral edge position. The adaptive controller is also able to provide good guiding performance in the presence of measurement noise. 57 3.4. FOUR PARAMETER GUIDE ADAPTIVE CONTROLLER 3.4 Four Parameter Guide Adaptive Controller The same set of experiments with the two types of webmaterials were carried out with the four parameter guide adaptive controller. In this controller implementation only one measurement is needed, the lateral web position. The performance of the four parameter adaptive controller was similar to the three parameter adaptive controller. The adaptive controller provides good guiding perfromance in the presence of the common disturbances and process variations. A complete set of the experimental results is presented in Appendix D.2. Figure 3.15 shows the guiding performance of the adaptive controller at different guidepoints for a sinusoidal disturbance. Interestingly, at different guidepoints the performance of the same controller is different. When the guidepoint is around 3 V clear indication of the sinusoidal disturbance is observed. Additionally, the Kamberoller guide was positioned at different guidepoints and the magnitude of the sinusoidal disturbance was observed. The magnitude was higher at 3 V compare to a guidepoint of 5 V. This may be attributed to sensor nonlinearity. The slope of the sensor may not be linear along its entire window length. 0 5 10 15 20 25 30 2 2.5 3 3.5 4 4.5 5 5.5 6 Four Parameter Adaptive Controller Performance Lateral Position (V) Web Position Reference Position Figure 3.15: Adaptive Controller: 4Paramater, 300 fpm, Sine Disturbance, Opaque Web 58 3.5. GUIDE ADAPTIVE CONTROLLER 3.5 Guide Adaptive Controller The guide adaptive controller (with eight parameters) has a similar implementation as the four parameter GAC: it is a SISO system with eight parameters. Figure 3.16 shows a representative sample of the performance of the eight parameter adaptive controller and the PI controller. A complete set of the experimental results is presented in Appendix D.3. The experimental results indicate that the controller provides good guiding perfromance in the presence of the common disturbances and process variations. While implementing the adaptive controller on the realtime hardware no assumption on the values of the parameters were made. The initial values for all the parameters were set to zero. The plots in Figure 3.17 shows the performance of the adaptive controller as soon as it is started. The plots in Figure 3.18 show the performance of the adaptive controller after the adaptation is continued for a long time. Observe the bottom plots in Figures 3.17 and 3.18. The plots show the evolution of the estimated parameters. As the time progresses the estimated parameters reach steady state values. The guiding performance of the controller is unaffected during the evolution of the parameters. This clearly indicates that the variables associated with the system are bounded. Similar results were found with all the adaptive control schemes, and with all the disturbances that were discussed previously. Recall that the control law and the adaptive law for an eight parameter adaptive controller is given by: Control Law up = θ⊤ω + θ˙⊤φ (3.1a) φ = 1 s + p0 ω (3.1b) ω = [ω⊤ 1 ω⊤ 2 yL r] (3.1c) ω˙ 1 = Fω1 + gup, ω1(0) = 0 (3.1d) ω˙ 2 = Fω2 + gyp, ω2(0) = 0 (3.1e) Adaptive Law θ˙ = −e1φ, e1 = yL − ym (3.2) Notice that the control law not only depends on the actual value of the parameter vector θ, but also depends on the rate of change (θ˙) of the parameter vector. This is one of the reasons why the adaptive controller provides good guiding performance even when the parameters 59 3.6. SYSTEMATIC PROCEDURE FOR ADAPTIVE CONTROLLER IMPLEMENTATION does not reach a steadystate value. Once the parameters reach a steadystate value then the control will only be dependent on the parameter vector θ and the regressor vector ω. 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Sine disturbance observed at the kamberoller guide Lateral Position (V) 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of A9 controller Lateral Position (V) 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of eight parameter GAC Time (Seconds) Lateral Position (V) Figure 3.16: Performance Comparison: 8Parameter, 300 fpm, Sine Disturbance, Transparent Web 3.6 Systematic Procedure for Adaptive Controller Implementation The theory developed in the previous chapter does not impose any constraint on the value of design parameters. Most of the design parameters have to be positive in the case of scalars and positive definite in the case of matrices. When implementing the adaptive controller, it is important to consider the actuator rate constraints, the bandwidth of the actuator, etc., and the design parameters cannot be chosen arbitrarily. This section presents a systematic procedure for choosing various design parameters in the adaptive control schemes presented in the previous chapters. 1. Choose a suitable reference model: In a model reference adaptive controller design, the first step is to choose a reference model. The choice of the reference model is based on common performance characteristics like the settling time and the percentage overshoot. 60 3.6. SYSTEMATIC PROCEDURE FOR ADAPTIVE CONTROLLER IMPLEMENTATION 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Eight Parameter Guide Adaptive Controller Performance Lateral Position (V) Web Position Reference Position 0 5 10 15 20 25 30 −10 0 10 Control (V) 0 5 10 15 20 25 30 −0.5 0 0.5 Time (Seconds) Parameters Time history of estimated parameters q 1 q 2 q 3 q 4 q 5 q 6 q 7 c 0 Figure 3.17: Adaptive Controller: 8Parameter, 300 fpm, Sine Disturbance, Transparent Web 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Eight Parameter Guide Adaptive Controller Performance Lateral Position (V) Web Position Reference Position 0 5 10 15 20 25 30 −10 0 10 Control (V) 0 5 10 15 20 25 30 −5 0 5 Time (Seconds) Parameters Time history of estimated parameters q 1 q 2 q 3 q 4 q 5 q 6 q 7 c 0 Figure 3.18: Adaptive Controller: 8Parameter, 300 fpm, Sine Disturbance, SteadyState, Transparent Web 61 3.6. SYSTEMATIC PROCEDURE FOR ADAPTIVE CONTROLLER IMPLEMENTATION Choose a model which is well damped. The settling time can be chosen based on the performance requirements and the actuator rate constraints. 2. Perform computer simulations: Computer simulations are important because they provide a good starting point for practical implementation. Some basic information about the evolution of the estimated parameters, the gain matrices and the control effort generated can be obtained from the computer simulations. 3. Choose the polynomial (s): This polynomial is chosen only in the four parameter and the eight parameter case. A simple choice would be (s) = (s + a0)n (3.3) where n = 1 for the four parameter case and n = 3 for the eight parameter case. Notice that the polynomial (s) filters the control input signal up and the lateral position signal yL (refer equations (2.53) and (2.78)). This polynomial can be chosen based on the bandwidth of the motor. The polynomial is chosen such that the filtered versions of up and yL have the same bandwidth as the actuator. 4. Choose the parameter p0: Based on the necessary and sufficient conditions derived in section C.2, p0 is chosen according to the constraint 0 < p0 < 2ζωn. This parameter sets the bandwidth of the filter L(s)−1 which is defined as L(s)−1 = 1 s + p0 Recall that this filter is added to satisfy the SPR condition. The filter L(s)−1 filters all the regressor signals. Large values for the parameter p0 will significantly attenuate the magnitude of the regressor vector. This may lead to poor guiding performance due to very slow adaptation. Figure 3.19 shows the performance of the adaptive controller with different values of the parameter p0. The value of the parameter p0 is decreased after 15 seconds and the guiding performance improves with the decrease. Small values for the parameter p0 may help in quick adaptation but may also cause the system to be sensitive (equations (2.76b) and (3.1a)). 5. Choose gains based on simulation: The initial choice of the gains can be based on the simulation results. A positive definite diagonal matrix can be chosen as the gain matrix. As a starting point all the diagonal elements may have the same value. 62 3.6. SYSTEMATIC PROCEDURE FOR ADAPTIVE CONTROLLER IMPLEMENTATION 6. Observe the evolution of parameters: Observing the way in which the parameters adapt is important in the design parameter selection process. The experiments with pulse disturbance help in selecting the adaptation gains. The control effort supplied to the actuator is directly related to the magnitude and the rate at which the estimated parameters vary ((2.76b) and (3.1a)). When the parameters vary quickly, the magnitude of the control effort tends to increase. This may not be desirable in most cases. 7. Set a bound on the parameter estimates: Simple bounding of parameters improves the robustness of the controller. In order to set a bound, knowledge of the estimated parameters is required. Conservative bounds can be set based on the experiments with pulse disturbance. 8. Observe the control effort: The control effort supplied to the actuator can help us in choosing some parameters such as p0. If the parameter p0 is very small, then the guiding system may be sensitive. This can be clearly observed even without a pulse disturbance experiment. When the parameter p0 is small, noisy measurement may cause a ringing effect (the guide may vibrate with a very high frequency about the guidepoint). The parameter p0 may be increased to reduce this effect. 0 5 10 15 20 25 30 4 4.5 5 5.5 6 Performance of eight parameter GAC with different p0 Time (Seconds) Lateral Position (V) Figure 3.19: Performance Comparison: 8Parameter, 500 fpm, Pulse Disturbance, Effect of p0, Transparent Web Based on the observations from the experiments some guidelines for practical implementation are as follows: 63 3.6. SYSTEMATIC PROCEDURE FOR ADAPTIVE CONTROLLER IMPLEMENTATION • The estimated parameters reach a steadystate value after some time. How fast the parameters reach the steadystate value is dependent on the adaptation gains. Once the steadystate value is reached there is no significant change in the estimated parameters. Therefore, adaptation can be stopped or the estimated parameters can be frozen. Once the parameters are frozen, the controller behaves like a fixed gain controller. • When changes in the process parameters aremade, adaptation can be continued. This can be implemented by continuously monitoring the error variable. Once the error variable exceeds a predefined limit, the adaptation of the parameters can be continued. • Instead of stopping all the parameters, the estimated parameters can be selectively frozen. All the parameters except the last two parameters can be frozen. • The decision on when to stop the adaptation can be made based on the adaptive law. Notice that the adaptive law is given by: θ˙ = −e1φ, e1 = yL − ym (3.4) When the parameters reach a steadystate value, the vector θ˙ would be zero. Whenever all the elements of the vector θ˙ are close to zero, then the adaptation can be stopped. 64 Chapter 4 Friction Compensation inWeb Guides Friction is a type of phenomena that is found in almost all servomechanisms. Friction is inevitable in mechanisms with relative motion between parts which are in contact with each other. Although friction is essential in some mechanisms, such as braking, it is usually not desired in high precision motion control systems. Typically friction may cause steadystate errors in position regulation and tracking and may lead to limit cycle behavior. Hence it is important to consider the effect of friction when designing and implementing a closedloop control system. In this chapter we consider static friction models and analyze compensation techniques which are specific to web guides. 4.1 Static Models Early modeling of friction involved static models of friction which are simply a function of relative velocity between contacting surfaces. Static friction takes into account only a few properties of friction with the main idea being that the friction force opposes motion and the magnitude of friction is independent of the surface area of contact. A brief discussion of some well known static friction models are given below. Coulomb investigatedDa Vinci’s frictionmodel and proposed themodel in 1785 [26]. Coulomb friction in dynamic systems is modeled as a piecewise continuous function which is positive for positive velocities and negative for negative velocities. Coulomb friction is given by F = Fcsgn(v) (4.1) where F is the friction force, Fc is the Coulomb friction coefficient, sgn(.) denotes the sign function, and v is the relative velocity between contacting surfaces. 65 4.1. STATIC MODELS 0 0 Coulomb Friction Coefficient Fc Friction F Velocity v Figure 4.1: Static Friction Model with Coulomb Friction Effect Morin (1833) [26] introduced the idea of static friction in which the friction forces opposes the direction of motion when the relative velocity is zero. The equation describing the model is given by F = Fssgn(Ft) (4.2)
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Title  Model Reference Adaptive Control of Web Guides 
Date  20070501 
Author  Seshadri, Aravind 
Keywords  Adaptive Control; Web Guiding; Web Handling 
Department  Mechanical Engineering 
Document Type  
Full Text Type  Open Access 
Abstract  The term web is used to describe materials which are typically manufactured in a rolltoroll manner. A wide variety of materials such as textile, paper, plastics, composites and metals are manufactured in rolled form because rolltoroll manufacture of materials is convenient for transport and storage, saves time and reduces costs. As the web is transported in the processing machinery, inherent machine misalignment and process induced disturbance cause lateral fluctuations. These lateral fluctuations, if not controlled, can result in inferior quality of the finished product. The primary focus of this research is on the development and implementation of adaptive control strategies for controlling the lateral fluctuations (lateral control or lateral guiding) in a web processing line. New performance measures which can better assist machine operators in diagnosing lateral behavior were also investigated. A novel model reference adaptive controller, which is referred to as the ``Guide Adaptive Controller' (GAC), was developed for web guiding applications. The GAC was implemented on an experimental web line and the performance of the controller was compared with an existing industrial controller. The adaptive control strategy resulted in better performance compared to an industrial controller especially when used with difference web materials with opacity and gage variations. Compared to the existing industrial controller, the adaptive controller does not require retuning when the operating conditions change because the GAC adapts to process variations and attenuates disturbances. Additionally, the same adaptive controller can be used with different guide mechanisms. Two simplified approximations of GAC were also developed and implemented on an experimental web line in order to observe the adaptive behavior of the controllers. Based on these observations a systematic procedure for industrial implementation of the adaptive control strategy was developed. Extensive experimental results on commercial guides with different operating conditions and disturbances indicate that GAC can provide improved guiding performance when compared to an existing industrial controller. Therefore, GAC has a high potential to successfully replace existing guide controllers. A novel performance metric which provides a better characterization of the guiding performance when compared to existing metrics was developed. The developed metric is based on histograms. Commonly observed histograms were studied and their occurrence in web guiding applications were analyzed. The performance metric can be used as a diagnostic tool in the web processing industry to monitor the occurrence of different machine and process induced disturbances, and also to compare controllers. 
Note  Thesis 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  MODEL REFERENCE ADAPTIVE CONTROL OF WEB GUIDES By ARAVIND SESHADRI Bachelor of Engineering University of Madras Tamil Nadu, India 2003 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May 2007 MODEL REFERENCE ADAPTIVE CONTROL OF WEB GUIDES Thesis Approved: Dr. Prabhakar R. Pagilla Thesis Advisor Dr. Karl N. Reid Dean of the College of Engineering, Architecture & Technology Dr. Gary E. Young Committee Member Dr. A. Gordon Emslie Dean of the Graduate College ii ACKNOWLEDGMENTS I wish to express my deepest gratitude to my advisor, Dr. Prabhakar R. Pagilla, for his intelligent supervision, friendship and support throughout my graduate program. I am indebted to him for his motivation and encouragement, which kept me focused on my goals. I would like to thank him for supporting me as a research assistant, during which time I was fortunate to learn a lot about the latest advancements in web handling and control systems. I would like to extend my warmest thanks to my committee members: Dr. Karl N. Reid and Dr. Gary E. Young for their support, suggestions and time. I would like to thank my colleagues, at Oklahoma State University, Anil Abbaraju, Mauro Cimino, Ramamurthy Dwivedula, Reza Jafari, Seshadri Kuppuswamy, Pranav Kumar Peddi Ravi, Ryan Ratliff, Nilesh Siraskar, Diao Yu, and Yunfei Zou for their timely support and suggestions. Finally, I greatly appreciate the support, patience and motivation from my parents and my brother. Without them I would have never reached this level in my life. iii TABLE OF CONTENTS Chapter Page 1 Introduction 1 1.1 Lateral Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.1 Remotely Pivoted Guide (Steering Guide) . . . . . . . . . . . . . . . . . . . 6 1.1.2 Offset Pivot Guide (Displacement Guide) . . . . . . . . . . . . . . . . . . . 8 1.2 Lateral Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Need for a Different Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Adaptive Control Design forWeb Guiding 13 2.1 Introduction to Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Online Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.1 Parameter Estimation: An Example . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.1 Adaptive Regulation using a Reference Model: An Example . . . . . . . . 22 2.4 Simplified Guide Adaptive Controller Design . . . . . . . . . . . . . . . . . . . . . 24 2.4.1 Three parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . 26 2.4.2 Simplified GAC with an Estimator . . . . . . . . . . . . . . . . . . . . . . . 29 2.4.3 Four Parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . . 31 2.5 Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.5.1 Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5.2 Adaptive Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.5.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 iv 3 Experimental Results 45 3.1 Experimental Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.1 Process Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.2.2 Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Three Parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . 51 3.3.1 Experiments with opaque web . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.2 Experiments with transparent web . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 Four Parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . 58 3.5 Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.6 Systematic Procedure for Adaptive Controller Implementation . . . . . . . . . . . 60 4 Friction Compensation inWeb Guides 65 4.1 Static Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Friction Compensation Based on Static Models . . . . . . . . . . . . . . . . . . . . 69 4.2.1 Friction Compensation forWeb Guides Based on a Static Model . . . . . . 70 4.2.2 Friction Parameters Identification . . . . . . . . . . . . . . . . . . . . . . . 71 4.3 Adaptive Static Friction Compensation forWeb Guides . . . . . . . . . . . . . . . 76 4.4 Adaptive Friction Compensation with Web Dynamics . . . . . . . . . . . . . . . . 85 4.5 Adaptive Friction Compensation using RLS Algorithm . . . . . . . . . . . . . . . 89 4.5.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.5.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5 Histogram : A New Performance Metric for Web Guiding 94 5.1 Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.1.1 Normally Distributed Histograms . . . . . . . . . . . . . . . . . . . . . . . 96 5.1.2 Symmetric, NonNormal, ShortTailed Histograms . . . . . . . . . . . . . 101 5.1.3 Symmetric, NonNormal, LongTailed Histograms . . . . . . . . . . . . . 102 5.1.4 Symmetric Bimodal histogram . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.1.5 Skewed NonNormal histogram . . . . . . . . . . . . . . . . . . . . . . . . 105 5.1.6 Symmetric Histogram with Outliers . . . . . . . . . . . . . . . . . . . . . . 107 5.1.7 Ideal Error Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 v 6 Summary and Future Work 110 BIBLIOGRAPHY 113 A Offset Adaptation 117 B Mathematical Preliminaries 120 B.1 Continuous Functions and their Limits . . . . . . . . . . . . . . . . . . . . . . . . . 120 B.2 InputOutput Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 B.3 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 B.4 Positive Real and Strictly Positive Real Transfer Functions . . . . . . . . . . . . . 126 C Model Reference Adaptive Control: Supplement 128 C.1 MRAC for Relative Degree 1 System . . . . . . . . . . . . . . . . . . . . . . . . . . 128 C.2 Choice of parameter p0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 D Additional Experimental Results 132 D.1 Three Parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . 132 D.1.1 Experiments with opaque web . . . . . . . . . . . . . . . . . . . . . . . . . 132 D.1.2 Experiments with transparent web . . . . . . . . . . . . . . . . . . . . . . . 134 D.2 Four Parameter Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . 136 D.2.1 Experiments with opaque web . . . . . . . . . . . . . . . . . . . . . . . . . 136 D.2.2 Experiments with transparent web . . . . . . . . . . . . . . . . . . . . . . . 141 D.3 Guide Adaptive Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 D.3.1 Experiments with opaque web . . . . . . . . . . . . . . . . . . . . . . . . . 145 D.3.2 Experiments with transparent web . . . . . . . . . . . . . . . . . . . . . . . 150 vi LIST OF TABLES Table Page 4.1 Kamberoller Guide Motor Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Coulomb and Viscous Friction Estimates . . . . . . . . . . . . . . . . . . . . . . . . 73 vii LIST OF FIGURES Figure Page 1.1 A Web Material used for Packaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Experimental Web Handling System . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 An Example of aWeb Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 End Pivoted Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Center Pivoted Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Offset Pivot Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7 Remotely Pivoted Guide or Steering Guide . . . . . . . . . . . . . . . . . . . . . . 5 1.8 A Schematic of a Remotely Pivoted Guide . . . . . . . . . . . . . . . . . . . . . . . 7 1.9 A Schematic of an Offset Pivot Guide . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 Adaptive Gain Scheduling Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Adaptive Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Model Reference Adaptive Control System . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Online Parameter Adjustment Mechanism . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Adaptive Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.6 Simulation Results for 3Parameter GAC . . . . . . . . . . . . . . . . . . . . . . . 30 2.7 Simulink Block Diagram for 3Parameter GAC . . . . . . . . . . . . . . . . . . . . 30 2.8 Simulation Results for 3Parameter GAC with an Estimator . . . . . . . . . . . . . 32 2.9 Simulink Block Diagram for 3Parameter GAC with an Estimator . . . . . . . . . 32 2.10 Simulation Results for 4Parameter GAC . . . . . . . . . . . . . . . . . . . . . . . 38 2.11 Simulation Results for 4Parameter GAC with a Pulse Disturbance . . . . . . . . 39 2.12 Simulink Block Diagram for 4Parameter GAC . . . . . . . . . . . . . . . . . . . . 40 2.13 Simulation Results for Guide Adaptie Controller . . . . . . . . . . . . . . . . . . . 43 2.14 Simulation Results for Guide Adaptie Controller with a Pulse Disturbance . . . . 43 2.15 Simulink Block Diagram for Guide Adaptie Controller . . . . . . . . . . . . . . . 44 viii 3.1 Experimental Web Handling Platform . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Line Schematic of the Experimental Web Handling Platform . . . . . . . . . . . . 46 3.3 offset pivot Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Remotely Pivoted Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.5 Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.6 Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.7 The Effect of Opacity on Sensor Gain for an Infrared Sensor . . . . . . . . . . . . 50 3.8 Performance Comparison: 3 Parameter, Sine Disturbance, 300 fpm, Opaque Web 53 3.9 Adaptive Controller: 3Parameter, Sine Disturbance, 300 fpm, Opaque Web . . . 54 3.10 Performance Comparison: 3Parameter, Pulse Disturbance, 300 fpm, Opaque Web 54 3.11 Adaptive Controller: 3Parameter, 300 and 500 fpm, Step Reference Changes, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.12 Performance Comparison: 3Parameter, 500 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.13 Adaptive Controller: 3Parameters, 500fpm, Sine Disturbance, Transparent Web . 56 3.14 Performance Comparison: 3Parameter, 500 fpm, Pulse Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.15 Adaptive Controller: 4Paramater, 300 fpm, Sine Disturbance, Opaque Web . . . 58 3.16 Performance Comparison: 8Parameter, 300 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.17 Adaptive Controller: 8Parameter, 300 fpm, Sine Disturbance, Transparent Web . 61 3.18 Adaptive Controller: 8Parameter, 300 fpm, SineDisturbance, SteadyState, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.19 Performance Comparison: 8Parameter, 500 fpm, Pulse Disturbance, Effect of p0, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.1 Static Friction Model with Coulomb Friction Effect . . . . . . . . . . . . . . . . . . 66 4.2 Static Friction Model with Coulomb and Viscous Effects . . . . . . . . . . . . . . . 67 4.3 Static Friction Model with Coulomb, Viscous and Stiction Effects . . . . . . . . . 68 4.4 Static Friction Model with Stribeck Effect . . . . . . . . . . . . . . . . . . . . . . . 68 4.5 Friction Compensation using Estimated Friction . . . . . . . . . . . . . . . . . . . 69 ix 4.6 Velocity Output for a Sinusoidal Input Voltage . . . . . . . . . . . . . . . . . . . . 73 4.7 Zero Velocity Crossing Friction Effect . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.8 Estimation of Stiction Using a Ramp Input . . . . . . . . . . . . . . . . . . . . . . 74 4.9 Stribeck Velocity Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.10 Simulink Block Diagram: Adaptive Friction Compensation . . . . . . . . . . . . . 80 4.11 Initial Condition Response (Angular Position) . . . . . . . . . . . . . . . . . . . . 81 4.12 Parameter Estimated with Initial Condition Response (Angular Position) . . . . . 81 4.13 Initial Condition Response Angular Velocity . . . . . . . . . . . . . . . . . . . . . 82 4.14 Parameter Estimated with Initial Condition Response (Angular Velocity) . . . . . 82 4.15 Simulink Block Diagram: Position Regulation . . . . . . . . . . . . . . . . . . . . . 83 4.16 Adaptive Static Friction Compensationwith Position regulation. θdes = 2, θ(0) = 0 and θ(˙0) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.17 Parameter Estimates for Adaptive Static Friction Compensation with Position Regulation. θdes = 2, θ(0) = 0 and θ(˙0) = 0 . . . . . . . . . . . . . . . . . . . . . . 84 4.18 Model Reference Adaptive Control with Adaptive Friction Compensation . . . . 85 4.19 Performance Comparision: 3Parameter, with and without Friction Compensation 92 4.20 Performance Comparision: 4Parameter, with and without Friction Compensation 93 4.21 Performance Comparision: 8Parameter, with and without Friction Compensation 93 5.1 Performance Comparison Based on Regulation Error: Clear Distinction . . . . . . 95 5.2 Performance Comparison Based on Regulation Error: Difficult to Compare . . . 95 5.3 Performance Comparison using Histograms: . . . . . . . . . . . . . . . . . . . . . 97 5.4 Performance Comparison using Histograms: . . . . . . . . . . . . . . . . . . . . . 97 5.5 Normally distributed Histogram, with Zero Mean . . . . . . . . . . . . . . . . . . 98 5.6 Mean of a Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.7 Variance of a Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.8 Experimental Data: Bottom Plot Shows Normal Distribution . . . . . . . . . . . . 100 5.9 ShortTailed Histogram Characteristic with “Fat” Body . . . . . . . . . . . . . . . 101 5.10 Experimental Data: Top Plot Indicates ShortTailed Distribution . . . . . . . . . . 102 5.11 LongTailed Histogram Characterized by “Lean” Body and Long Tails . . . . . . 103 5.12 Experimental Data: Bottom Plot Shows LongTailed Distribution . . . . . . . . . 103 x 5.13 Symmetric Bimodal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.14 Experimental Data: Top Plot Indicates Bimodal Distribution . . . . . . . . . . . . 105 5.15 A Right Skewed Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.16 Experimental Data: Skewed Histogram . . . . . . . . . . . . . . . . . . . . . . . . 106 5.17 A Histogram with Outliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.18 Experimental Data: Presence of Outliers . . . . . . . . . . . . . . . . . . . . . . . . 108 5.19 Experimental Data: The Ideal Distribution for Guiding Applications . . . . . . . 109 A.1 Output voltage range and typical curves for opaque and transparent webs . . . . 118 A.2 Offset adaptation on the opaque web . . . . . . . . . . . . . . . . . . . . . . . . . . 119 A.3 Offset adaptation on the transparent web . . . . . . . . . . . . . . . . . . . . . . . 119 D.1 Performance Comparison: 3Parameter, 500 fpm, Sine Disturbance, Opaque Web 132 D.2 Adaptive Controller: 3Parameter, 500 fpm, Sine Disturbance, Opaque Web . . . 133 D.3 Performance Comparison: 3Parameter, 500 fpm, Pulse Disturbance, Opaque Web 133 D.4 Performance Comparison: 3Parameter, 300 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 D.5 Adaptive Controller: 3Parameter, 300 fpm, Sine Disturbance, Transparent . . . . 135 D.6 Performance Comparison: 3Parameter, 300 fpm, Pulse Disturbance, Transparent 135 D.7 Performance Comparison: 4Parameters, 300 fpm, Sine Disturbance, Opaque Web 136 D.8 Adaptive Controller: 4Parameters, 300 fpm, Sine Disturbance, Opaque Web . . . 137 D.9 Performance Comparison: 4Parameters, 500 fpm, Sine Disturbance, Opaque Web 137 D.10 Adaptive Controller: 4Parameters, 500 fpm, Sine Disturbance, Opaque Web . . . 138 D.11 Performance Comparison: 4Parameters, 300 fpm, Pulse Disturbance, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 D.12 Adaptive Controller : 4Parameters, 300 fpm, Pulse Disturbance, SteadyState, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 D.13 Performance Comparison: 4Parameters, 500 fpm, Pulse Disturbance, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 D.14 Performance Comparison: 4Parameters, 500 fpm, Pulse Disturbance, Steady State, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 xi D.15 Adaptive Controller: 4Parameter, 300 and 500 fpm, Step Reference Changes, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 D.16 Performance Comparison: 4Parameters, 300 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 D.17 Adaptive Controller: 4Parameters, 300 fpm, Sine Disturbance, Transparent Web 142 D.18 Performance Comparison: 4Parameters, 500 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 D.19 Adaptive Controller: 4Parameters, 500 fpm, Sine Disturbance, Transparent Web 143 D.20 Performance Comparison: 4Parameters, 300 fpm, Pulse Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 D.21 Performance Comparison: 4Parameters, 500 fpm, Pulse Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 D.22 Adaptive Controller: 4Parameters, 300 and 500 fpm, Step Reference Changes, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 D.23 Performance Comparison: 8Parameters, 300 fpm, Sine Disturbance, Opaque Web 145 D.24 Adaptive Controller: 8Parameters, 300 fpm, Sine Disturbance, Opaque Web . . . 146 D.25 Adaptive Controller: 8Parameters, 300 fpm, Sine Disturbance, SetadyState, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 D.26 Performance Comparison: 8Parameters, 500 fpm, Sine Disturbance, Opaque Web 147 D.27 Adaptive Control: 8Parameters, 500 fpm, Sine Disturbance, Opaque Web . . . . 147 D.28 Adaptive Control: 8Parameters, 500 fpm, SineDisturbance, SteadyState, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 D.29 Performance Comparison: 8Parameters, 300 fpm, Pulse Disturbance, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 D.30 Performance Comparison: 8Parameters, 500 fpm, Pulse Disturbance, Opaque Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 D.31 Performance Comparison: 8Parameters, 500 fpm, Sine Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 D.32 Adaptive Controller: 8Parameters, 500 fpm, Sine Disturbance, Transparent Web 151 D.33 Adaptive Controller: 8Parameters, 500 fpm, Sine Disturbance, SteadyState, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 xii D.34 Performance Comparison: 8Parameters, 300 fpm, Pulse Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 D.35 Performance Comparison: 8Parameters, 500 fpm, Pulse Disturbance, Transparent Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 xiii NOMENCLATURE SYMBOLS Cm : transmission ratio e : error e1 : tracking error E : modulus of elastcity of web ǫ1 : estimation error F : friction force Fc : Coulomb friction coefficient Fs : static friction coefficient Fv : viscous friction coefficient γ : gain : gain matrix i : current I : moment of inertia J : rotor inertia km : motor parameter or high frequency gain for a reference model kp : high frequency gain for a plant model K = r T EI : web span parameter Ke : back e.m.f constant Kt : torque constant/sensitivity xiv SYMBOLS L : inductance or length of span L1 : distance from the guide roller to instant center L : Laplace operator L−1 : inverse Laplace operator μ : mean n∗ : relative degree ω : regresor vector ωn : natural frequency Wm(s) : reference model transfer function φ : filtered regressor vector r : reference command R : resistance Rm(s) : denominator polynomial of reference model Rp(s) : denominator polynomial of plant model R : set of all real numbers sgn(.) : signum function σ : standard deviation σ2 : variance T : torque or Tension τ : time constant θ : angular position or parameter vector θ∗ : true parameter vector θ0 : roller misalignment u,Up : input to a plant v : velocity vs : Stribeck velocity constant xv SYMBOLS x : state variable x1 : distance from the guide roller to the instant center y : output of a plant ˆy : estimator output Y0 : initial lateral position misalignment yL, YL : lateral edge position ym : output of a reference model ζ : damping ratio Z : guide position Zm(s) : numerator polynomial of reference model Zp(s) : numerator polynomial of plant model xvi ABBREVIATIONS BIBO : Boundedinput boundedoutput I/O : InputOutput LMS : Least Mean Square LTI : Linear Time Invariant MRAC : Model Reference Adaptive Control OPG : Offset Pivot Guide P : Proportional PD : ProportionalDerivative PE : Persistent Excitation PI : ProportionalIntegral PR : Positive Real RLS : Recursive Least Square RPG : Remotely Pivoted Guide SISO : Single Input Single Output SPR : Strictly Positive Real xvii Chapter 1 Introduction The term web is used to describe materials which have their length considerably larger than their width and width considerably larger than thickness. Webs are materials which are manufactured and processed in a continuous, flexible strip form. Webs consist of a broad spectrum of materials that are used extensively in every day life such as plastics, paper, textile, metals and composites. Typically web materials are manufactured into rolls since it is easy to transport and process the materials in the rolled form. An example of a web material is shown in Figure 1.1. Figure 1.1: A Web Material used for Packaging Web handling is a term that is used to refer to the study of the behavior of the web while it is transported and controlled through the processing machinery from an unwind roll to a 1 rewind roll. A typical operation involves transporting a web in rolled, unfinished form from an unwind roll to a rewind roll through processing machinery where the required processing operations are performed. An example of such a process is commonly seen in the metals industries. The web (metal strip) that is to be processed is transported on rollers to various sections where different operations like coating, painting, drying, slitting, etc., are performed. The process line generally has unwind and rewind rolls, many idle rollers and one or more intermediate driven rollers. An experimental web handling system with unwind and rewind sections is shown in Figure 1.2. REWIND SECTION UNWIND SECTION DRIVEN ROLLER IDLE ROLLERS WEB Figure 1.2: Experimental Web Handling System The longitudinal dynamics of theweb is the behavior of theweb in the direction of transport of the web. Web transport velocity and web tension are two key variables of interest that affect the longitudinal behavior of the web. The lateral dynamics of the web is the behavior of the web perpendicular to the direction of transport of the web and in the plane of web. Several parameters which affect the lateral web dynamics include web material, tension, transport velocity, and web geometry, etc. The quality of the finished web depends on how well the web is handled on the rollers during transport. The longitudinal and lateral control of the web on rollers play a critical role in the quality of the finished product. 2 The focus of thiswork is on control of lateral dynamics of aweb. Adaptive control strategies that are capable of providing the required performance in the presence of the variations in the process and web parameters are investigated. The suitability of these control strategies and their ability to provide the required performance are studied in detail, both from theoretical and experimental perspectives. Web guiding (also called as lateral control) involves controlling web fluctuations in the plane of the web and perpendicular to web travel. Web guiding is important because rollers in any web handling machinery tend to have inherent misalignment problems and this may cause the web to move laterally on the rollers. The lateral movement of the web on the rollers may produce wrinkles or slackness in the web, or the web may completely fall off the rollers. A number of web processes like printing, coating, winding may get affected severely due to the web lateral motion and it becomes important to maintain the lateral position of the web. Web guides are used to maintain the lateral position of the web on rollers during transport. EDGE SENSOR GUIDE ROLLER LATERAL LONGITUDINAL WEB GUIDE MECHANISM ACTUATOR Figure 1.3: An Example of aWeb Guide Aweb guidemechanismtypically consist of a roller sitting on a pivoted base, whosemotion is controlled to change the axis of rotation of the roller. The working principle of all web guides is based on awell understood fundamental principle inweb handling – a web approaching a roller 3 will always tend to orient itself perpendicular to the axis of rotation of the roller. The lateral motion of the web is controlled by changing the axis of rotation of the guide roller. The lateral position of the web is measured using an edge sensor. Based on this measurement as feedback the axis of rotation of the guide roller is controlled tomaintain the lateral position at the required location. Figure 1.3 shows a web guide mechanism with an electromechanical actuator and an infrared sensor. Web guides are positioned at different locations in an industrial process line where guiding is required. Guides located at either ends in a process line are usually called terminal guides [1]. An unwind guidemaintains the lateral position of thewebwhich is fed into the processing line, whereas a rewind guide maintains the lateral position of the processed web which is wound onto a roll in the rewind section. Apart fromterminal guiding, web guides are extensively used in the intermediate process sections and they are referred to as intermediate guides. The intermediate web guides are classified based on the way in which the axis of rotation of the guide roller is changed. Figure 1.4 shows an end pivoted guide where the change in the axis of rotation of the roller is about a pivot point which is at one end of the roller. Similarly the center pivoted guide shown in Figure 1.5 has its pivot point in the center of the guide roller. An offsetpivot guide (shown in Figure 1.6) utilizes a pair of rollers to change the axis of rotation while in a remotely pivoted guide (shown in Figure 1.7) the guide roller moves along a curved path to change its axis. These are some commonly used intermediate web guides in the web handling industry. WEB PIVOT POINT EDGE SENSOR GUIDE ROLLER AXES OF ROTATION Figure 1.4: End Pivoted Guide WEB PIVOT POINT EDGE SENSOR GUIDE ROLLER AXES OF ROTATION Figure 1.5: Center Pivoted Guide 4 PIVOT CARRIER MOUNTING BASE ENTERING SPAN EXITING SPAN FIXED ENTERING ROLLER FIXED EXITING ROLLER WEB PIVOT POINT EDGE SENSOR AXES OF ROTATION WEB Figure 1.6: Offset Pivot Guide WEB GUIDE ROLLER FIXED ENTERING IDLER ROLLER CENTERLINE OF MACHINE AND GUIDE ASSEMBLY CENTER OF ROTATION OF GUIDE ROLLER (INSTANT CENTER) AXIS OF ROTATION GUIDE ROLLER WEB EDGE SENSOR ENTERING SPAN PREENTERING SPAN EXITING SPAN Figure 1.7: Remotely Pivoted Guide or Steering Guide 5 1.1. LATERAL DYNAMICS 1.1 Lateral Dynamics In order to control aweb guide it is important to understand the physics behind the transport of web on rollers. Lateral and longitudinal dynamics of a moving web are dependent on various process parameters like transport velocity, web tension, web material, and the geometry of the web material, etc. Understanding the lateral dynamics of the web is important for designing an effective lateral controller for the web guide mechanism. Lateral dynamics of the web was first modeled in [2], based on the assumption that the web behaves like a string. A major improvement to the mathematical model for the lateral dynamics of the web was described by J. J. Shelton [3]. A first order model of an ideal web was presented in [4] followed by a secondorder model [5] by considering the web as an Euler beam. A model for the web lateral dynamics for a multiroll system was developed in [6] based on the Timoshenko beam theory. A stochastic modeling formulation for the lateral dynamics was developed in [7]. An overview of the lateral and longitudinal dynamics along with a historic perspective of modeling and control of moving webs was presented in [8]. A detailed modeling of the lateral dynamics of the web for different types of intermediate guides was presented in [9, 10]. The two types of intermediate guides which are considered in this work are a remotely pivoted guide (also called as a steering guide) and an offsetpivot guide (also called as a displacement guide). The web span lateral dynamics for the two guides are similar and hence the same controller design can be implemented on both the guides. Even though the focus of this research is on these two intermediate guides, the theory developed can be adapted to other guides as well. 1.1.1 Remotely Pivoted Guide (Steering Guide) The action of the remotely pivoted guide on the lateral position of the web is given by YL(s) = G1s(s)Z(s) + G2s(s)θ0(s) + G3s(s)Y0(s) (1.1) where YL(s) is the Laplace transform of the web lateral position, Z(s) is the input to the guide in the lateral direction, θ0(s) denotes the entering span roller misalignment and Y0(s) is the initial lateral position misalignment (see Figure 1.8). The transfer functions in equation (1.1) are given by G1s(s) = s2 + β2s + β1 s2 + β2s + β0 , G2s(s) = β′3 s2 + β2s + β0 , and G3s(s) = −β3s + β0 s2 + β2s + β0 (1.2) 6 1.1. LATERAL DYNAMICS Y0 X Y L v Z YL 0 L x1 INSTANT CENTER ORIGINAL CENTER q q Figure 1.8: A Schematic of a Remotely Pivoted Guide where β0 = 1 τ 2 (KL)2(coshKL − 1) KLsinhKL − 2(coshKL − 1) , 1 τ 2 f1(KL) β1 = L τ 2x1 KL(KLcoshKL − sinhKL) KLsinhKL − 2(coshKL − 1) , L τ 2x1 f2(KL) β2 = 1 τ KL(KLcoshKL − sinhKL) KLsinhKL − 2(coshKL − 1) , 1 τ f2(KL) β3 = 1 τ KL(sinhKL − KL) KLsinhKL − 2(coshKL − 1) , 1 τ f3(KL) β′3 = L τ 2 KL(sinhKL − KL) KLsinhKL − 2(coshKL − 1) , L τ 2 f3(KL) where the web span parameter K is defined as K2 = T EI , E is the modulus of elasticity of web, I is the moment of inertia of the web, T is the web tension, L is the length of the entering span, τ = L/v is the time constant, v is the web transport velocity, and x1 is the distance from the guide roller to its instant center of rotation. The variables θ0(s) and Y0(s) are considered as the disturbances, and the objective of the web guide is to reject these disturbances to maintain the lateral position downstream of the web guide. Thus the effect of the input guide displacement, Z(s), to the lateral position of the web, YL(s), is given by YL(s) = s2 + β2s + β1 s2 + β2s + β0 Z(s) (1.4) 7 1.1. LATERAL DYNAMICS 1.1.2 Offset Pivot Guide (Displacement Guide) Y 3 3 YL Y L B X3 3 A v L A B C D L1 0 Y 1 YL X Y L v L 3 Z L B PIVOT AXIS (ROLLER POSITION) L C PIVOT AXIS Figure 1.9: A Schematic of an Offset Pivot Guide The lateral dynamics of the web with a displacement guide is influenced by the input to the guide mechanism, Z(s), and the initial lateral position misalignment, Y0(s), and is given by YL(s) = G1dZ(s) + G2dY0(s) (1.5) where G1d(s) = − f3(KL) τ s + f1(KL) τ 2 s2 + f2(KL3) τ3 s L1 − L L1 s2 + f2(KL3) τ3 s + f1(KL3) τ 2 3 s2 + f2(KL) τ s + f1(KL) τ 2 + s2 + f2(KL) τ s + f2(KL)L τ 2L1 s2 + f2(KL) τ s + f1(KL) τ 2 G2d(s) = − f3(KL3) τ3 s + f1(KL3) τ 2 3 − f3(KL) τ s + f1(KL) τ 2 s2 + f2(KL3) τ3 s + f1(KL3) τ 2 3 s2 + f2(KL) τ s + f1(KL) τ 2 with L3 as the span length as shown in Figure 1.9 and τ3 = L3/v. Displacement guides are typically installed such that the length of the guide span (denoted by L) is very close to the distance from the pivot axis to the second guide roller (denoted by L1) 8 1.2. LATERAL CONTROL (see Figure 1.9). When L ≈ L1, the transfer function G1d(s) is simplified [9], and the structure is same as the transfer function G1s(s) in the dynamics of the remotely pivoted guide given by equation (1.2). With this simplification, the dynamics from the guide input to the web lateral position is given by YL(s) = s2 + β2s + β′1 s2 + β2s + β0 Z(s) (1.6) where β′1 = f2(KL)L τ 2L1 . 1.2 Lateral Control Lateral control involves the design of a closedloop control system for regulating the lateral position of the web in a process line using a web guide mechanism. The guide mechanism includes an actuator which provides the input to the system and a feedback sensor which is used to measure the lateral position of the web. Current industrial controllers for web guiding are simple and do not consider the web lateral dynamics in the controller design. A typical industrial controller consists of three loops. A very fast current loop, a velocity loop and an outer position loop. The two inner loops are designed based on the actuator dynamics, and they regulate the current and velocity of the actuator. The actuator dynamics is assumed to be completely known. The outer position loop is designed to regulate the lateral position of the web, and the compensation is usually based on a Proportional (P) controller or a ProportionalIntegral (PI) controller. The error in the lateral position of the web drives the two inner loops. Hence all the three loops work in unison to regulate the lateral position of the web. Since the inner two loops are driven by the outer position loop, the lateral positionmeasurement becomes critical. Depending on the web handling application, the type of sensor used to measure the lateral position of the web varies. Applications which handle opaque webs commonly use infrared sensors while transparent film applications employ ultrasonic sensors. The position loop is dependent on this sensor measurement and the sensor gain affects the position loop. Both analog and digital industrial controllers commonly have a manual variable gain, which can be used for tuning the controller gains based on the type of the web. Based on the sensor gain, the controller gain is adjusted appropriately. A number of strategies have been presented in the literature to control the lateral position of the web. The strategies include Proportional (P) control [11] and [7], ProportionalDerivative 9 1.3. NEED FOR A DIFFERENT CONTROL STRATEGY (PD) control [11], a state estimationproportional gain strategy [12], a state variable feedback control strategy [13], estimated velocity feedback control [14] and [15], controller based on frequency domain design [16], etc. In most of the control strategies cited above, it is assumed that the parameters that affect the lateral dynamics of the web are known. Some of the key parameters that affect the lateral dynamics are the web material properties, web geometry, transport velocity, web tension, etc. These parameters may vary due to process condition variations or due to the processing of different web materials in the same process line. Unless these parameters are measured regularly it is difficult to know the exact dynamic model of the web for each situation. 1.3 Need for a Different Control Strategy Some potential shortcomings of the existing lateral control strategies are given in the following. • Existing industrial strategies do not consider the web lateral dynamics in the analysis and design of the lateral controllers. Inclusion of the lateral web dynamics in the control design process has the potential to significantly improve the guiding performance in the presence of process variations as well as many machine induced lateral disturbances. • Sensor gain changes as a result of web material variations cause poor guiding with existing fixed gain controllers. • Parameters of the dynamic model are not known. In a model based controller design for lateral guiding, the knowledge of the parameters like tension, web transport velocity, web material geometry, etc., are important to achieve better tracking performance. But in most industrial applications these process parameters are not known to the guide controller. Additionally, most industrial process lines are designed to process different web materials under different operating conditions. The goal is to find a controller that is capable of providing specified guiding performance which has the ability to overcome many of the limitations of the existing fixed gain controllers, including the ones given above. To achieve this goal, adaptive control strategies are investigated in this thesis. 10 1.4. CONTRIBUTIONS 1.4 Contributions Two main contributions of this work are summarized below: 1. Model reference adaptive control designs that are applicable to web guiding are developed. A systematic approach for industrial implementation of these new adaptive strategies are developed. Additionally, practical industrial implementation guidelines are proposed. 2. A new performance metric that clearly highlights the web guiding performance is developed. The newmetric is based on histograms. Profiles of commonly observed histograms are studied and their occurrence in guiding situations are analyzed. This novel performance metric can be used as a metric for tuning controllers as well as a diagnostic tool for lateral web guiding applications. In addition to the primary contributions, several other secondary contributions are summarized below: 1. Simplified adaptive controllers are developed based on the approximation of the lateral dynamics by reduced order models. 2. Extensive experimentation of the adaptive strategies on an experimental platform containing various intermediate web guides is carried out. Further, an often used industrial control strategy is simultaneously implemented for all situations, and the results are compared. 3. Commonly used friction models are investigated and an adaptive friction compensation scheme based on static friction model is proposed for web guiding applications. A simplified adaptive controller with friction compensation is proposed and an indirect friction compensation scheme based on recursive least squares is implemented. 4. The output voltage of an optical sensor depends on the opacity of theweb. A newmethod to determine the range of the optical sensor is developed (see Appendix A). The rest of the document is organized as follows. In Chapter 2, a detailed description of the design of model reference adaptive control strategies suitable for web guiding is given. The results of the experiments carried out on an experimental web handling platform with the 11 1.4. CONTRIBUTIONS proposed adaptive control strategies are presented in Chapter 3. A detailed discussion of the results along with practical industrial implementation guidelines based on the experimental observations are also presented in Chapter 3. Chapter 4 gives friction compensation techniques that are applicable to web guiding. In Chapter 5, a new performance metric for web guiding based on histograms is discussed. Chapter 6 summarizes the thesis and provides suggestions for future work. 12 Chapter 2 Adaptive Control Design for Web Guiding 2.1 Introduction to Adaptive Control To adapt means to change to meet requirements or adjust to new circumstances. An adaptive controller is a scheme that adapts to changes in process dynamics and disturbances. Adaptive controllers are commonly used when there is a considerable change in the process dynamics and disturbances, which may not be compensated by using fixed gain controllers. One practical application of an adaptive control scheme is the autopilot guidance system in aircrafts. The process dynamics of an aircraft depends on various parameters such as speed, altitude, head wind, tail wind, etc., and the aircraft also experiences various external disturbances. It is difficult to design a linear fixed gain feedback controller which can perform well under different operating conditions and in the presence of various disturbances. In the early autopilot design, the adaptive scheme that was commonly used was gain scheduling. Various parameters that affect the process dynamics of an aircraft have a direct relationship with process outputs (or measured variables), and hence these outputs can be used to change the controller parameters directly. Hence it is possible to determine a suitable constant gain linear feedback control strategy for each operating condition. The model is thus linearized around those operating conditions. This is called gain scheduling because the control scheme determines the operating condition based on the process outputs and consequently an appropriate controller is chosen to compensate for the process dynamics (or process gain). The system has basically two loops as shown in Figure 2.1. The inner loop is a standard feedback loop while the outer loop adjusts the controller in the inner loop based on the operating conditions. Gain scheduling can be regarded as a mapping from process parameters to controller parameters [17]. 13 2.1. INTRODUCTION TO ADAPTIVE CONTROL CONTROLLER PLANT CONTROL SIGNAL GAIN SCHEDULE REFERENCE COMMAND OUTPUT CONTROLLER PARAMETERS OPERATING CONDITIONS Figure 2.1: Adaptive Gain Scheduling Scheme Gain scheduling is possible only when the variations (or the process dynamics) are directly correlated to the measured variables. In other words, gain scheduling is possible only when the operating conditions could be characterized based on these measured variables. Variation in disturbance characteristics cannot be compensated using gain scheduling and hence adaptive controllers are used. The modern autopilot for aircrafts employs adaptive controllers to compensate for various disturbances that act on the plane. The modern autopilot system is not limited to level flight. Current systems include different autopilot schemes for taxi, takeoff, ascent, level flight, descent, landing and taxi back to the terminal. Since the disturbance changes frequently, it is reasonable to adjust the controller parameters to cope with the disturbance characteristics. A common adaptive control scheme is illustrated in Figure 2.2. Similar to gain scheduling, there are two loops. The inner loop is a standard feedback loop with a controller. The outer loop has a parameter adjustment mechanism which modifies the controller based on variations. Direct and Indirect Adaptive Control An adaptive controller consists of two main subsystems. A parameter estimator and a controller based on the parameter estimator. The parameter adjustment block in Figure 2.2 is the parameter estimator which estimates the unknown parameters, based on the output of the plant, reference command, control signal, etc. The estimated parameters are then used by the controller to compute the control signal to the plant. The way in which the parameters are estimated, also referred to as the adaptive law, along with the way in which the control signal is calculated, gives rise to two main kinds of adaptive control schemes, indirect and direct. Indirect 14 2.1. INTRODUCTION TO ADAPTIVE CONTROL CONTROLLER PLANT CONTROL SIGNAL PARAMETER ADJUSTMENT REFERENCE COMMAND OUTPUT CONTROLLER PARAMETERS Figure 2.2: Adaptive Control System adaptive control involves the estimation of actual plant parameters which are used to compute the controller parameters. The controller parameters are computed indirectly and hence the name indirect adaptive control. In direct adaptive control, the plant model is parametrized in terms of the controller parameters which are estimated directly without intermediate calculations involving plant parameter estimates [18]. Indirect adaptive control is also referred to as explicit adaptive control while direct adaptive control is also referred as implicit adaptive control because the control design is based on explicit or implicit plantmodel estimation. Both types of control schemes can be used for minimumphase plants, but difficulty arises when designing an indirect adaptive control scheme for nonminimumphase plants [18]. In direct adaptive control scheme, since the plant parameters are parametrized in terms of the controller parameters, the convergence of parameters to their true values is not of utmost importance. The parameter estimation is usually driven by the output error. Although the parameters do not converge to their true values, the controller is capable of meeting the performance requirement on the output error convergence. On the contrary, convergence of parameters to their true values is of utmost importance while using indirect adaptive control. The convergence of parameters and its importance will be discussed later in this chapter. Model Reference Adaptive Control In a Model Reference Adaptive Control (MRAC) scheme the control law and adaptive law are designed such that the closedloop Input/Output (I/O) properties of the plant exactly match a reference model. The reference model is any dynamic model which meets a desired closedloop 15 2.2. ONLINE PARAMETER ESTIMATION performance requirements for the plant. Hence the control law for the plant is designed such that the closedloop dynamics matches the dynamics of the reference model. There are two kinds ofMRAC strategies, direct and indirectMRAC. A schematic of directMRAC is shown in Figure 2.3. CONTROLLER PLANT CONTROL SIGNAL PARAMETER ADJUSTMENT REFERENCE COMMAND OUTPUT CONTROLLER PARAMETERS REFERENCE MODEL ERROR _ + Figure 2.3: Model Reference Adaptive Control System In this thesis a direct model reference adaptive control scheme is investigated for web guides. In the following sections, the design and analysis of common adaptive control schemes will be presented along with their application to web guides. Online parameter estimation, a key component in adaptive control design, is discussed in section 2.2. A simple adaptive control example based on online parameter estimation is discussed in section 2.3. Model reference adaptive control schemes suitable for web guiding are developed in sections 2.4 and 2.5. Model reference adaptive control schemes, based on a simplified models for the web dynamics, are proposed in section 2.4. Amodel reference adaptive scheme based on the complete web dynamic model is presented in 2.5. 2.2 Online Parameter Estimation Online parameter estimation is required for any adaptive controller design. Both in direct and indirect adaptive control schemes some form of parameter estimation is carried out. In 16 2.2. ONLINE PARAMETER ESTIMATION indirect adaptive control the plant parameters are estimated. But in direct adaptive control the controller parameters are estimated. In both the cases parameters are estimated in realtime and the controller is adjusted appropriately. PARAMETER ESTIMATOR _ + PLANT OUTPUT ESTIMATED OUTPUT ESTIMATION ERROR CONTROL r(t) ^ y(t) REFERENCE y(t) u(t) Figure 2.4: Online Parameter Adjustment Mechanism In an online parameter estimation problem a parameter adjustment mechanism is utilized to estimate the unknown plant parameters, as shown in Figure 2.4. The parameter estimator is designed such that its structure is similar to the plant structure. The estimator is driven by the input to the plant u(t), reference command r(t) and the output of the plant y(t). The output of the estimator ˆy(t) is constantly compared with the output of the plant. The difference between the estimator output and the actual plant output is called the estimation error. The parameters in the estimator are constantly adjusted so that the estimation error is minimized. Standard optimization techniques like the gradientdescent, leastsquares, Newton’s method, etc., can be used to minimize the estimation error in realtime (see Appendix B in [18]). CONTROLLER PLANT CONTROL SIGNAL PARAMETER ADJUSTMENT REFERENCE COMMAND OUTPUT CONTROLLER PARAMETERS Figure 2.5: Adaptive Control System 17 2.2. ONLINE PARAMETER ESTIMATION An online estimation problem involve three key steps. First the parameter estimator has to be parametrized appropriately in terms of the unknown parameters. The second step involves the design of the adaptive law which defines the way in which the parameters are updated. The adaptive law is designed such that the optimization techniques used to minimize the estimation error results in a stable system. The final step involves the design of the plant inputs such that the estimates converge to their true values. The final step is important only for identification problems. For direct adaptive control problems the minimization of the output error is important rather than the convergence of parameters. In the following section a simple scalar plant of unknown parameters will be considered and the procedure for designing an online parameter estimation scheme will be discussed. The example is based on the procedure described in [18]. Additionally, various considerations for designing the input and the stability properties of the adaptive system will be discussed. 2.2.1 Parameter Estimation: An Example Consider a firstorder system described by x˙ = ax + bu, x(0) = x0 (2.1) where x is the output, u is the input and a, b are unknown constants. We also assume that the system is stable, i.e., a < 0 and the input u is bounded. The objective is to develop an online estimation algorithmfor estimating the unknown parameters using themeasured signals x and u. For estimation and adaptive control problems, an effective parametrization of estimator in terms of unknown plant parameters is extremely important. A simple parametrization would be x(s) = [ a b ] x(s) s u(s) s (2.2) where x(s) and u(s) are the Laplace transforms of x(t) and u(t). This parametrization is not desirable due to the presence of the integrator. A similar parametrization which is implementable can be obtained by using a low pass filter for the measured signals x and u. Let the plant be parametrized by adding and subtracting amx, where am > 0, am ∈ R1. The 18 2.2. ONLINE PARAMETER ESTIMATION parameter am is the low pass filter parameter whose value will be chosen later. x˙ = −amx + (a + am)x + bu (2.3a) x = 1 s + am [(a + am)x + bu] (2.3b) = θ∗⊤φ (2.3c) where θ∗⊤ = [b (a + am)] is the unknown parameter vector and φ = [ 1 s+am u 1 s+am x]⊤ is filtered measurement. Define the estimate of x as ˆx = θ⊤φ (2.4) where θ is the estimate of the unknown parameter vector in equation (2.3c), i.e., θ = [ˆb (ˆa + am)]. Define the estimation error as ǫ1 = x − ˆx (2.5) The estimation error dynamics is given by ˙ ǫ1 = −amǫ1 − ˜ax −˜ bu (2.6) where ˜a , ˆa − a and ˜b , ˆb − b. Equation (2.6) describes the behavior of the estimator which is influenced by the parameters and their estimates. If the parameter estimates converge to the true parameter values, i.e., ˆa = a,ˆb = b, then ǫ1 → 0 exponentially1, since am > 0. The adaptive law for updating the parameter estimates, ˆa and ˆb , can be derived using the estimation error ǫ1 as ˙ˆ a = f1(ǫ1, x, ˆx, u), ˙ˆb = f2(ǫ1, x, ˆx, u) (2.7) Notice that f1 and f2 are functions of the measured signals which are chosen such that the equilibrium state ˆae = a, ˆb e = b, ǫ1e = 0 (2.8) of equations (2.6) and (2.7) is uniformly stable or asymptotically stable or exponentially stable. Let us consider a Lyapunov function candidate as a function of the three states ǫ1, ˜a,˜b V (ǫ1, ˜a,˜b ) = 1 2 (ǫ21 + ˜a2 +˜b 2) (2.9) 1The choice of am determines the rate of convergence of ǫ1, since the decay is as per e−amt. 19 2.2. ONLINE PARAMETER ESTIMATION The function V is positive definite, decrescent and radially unbounded in R3. The time derivative of V along the trajectories of equations (2.6) and (2.7) is ˙V (ǫ1, ˜a,˜b ) = ǫ1(−amǫ1 − ˜ax −˜ bu) + ˜af1 +˜ bf2 (2.10a) = −amǫ21 − ˜axǫ1 −˜ buǫ1 + ˜af1 +˜ bf2 (2.10b) If f1 = ǫ1x, f2 = ǫ1u, we have ˙V = −amǫ21 ≤ 0. (2.11) The function ˙V is negative semidefinite since for all ˜a,˜b ∈ R and ǫ1 = 0, ˙V = 0. The parameter update law is given by ˙ˆ a = ǫ1x, ˙ˆb = ǫ1u (2.12) where ˆx is generated using equation (2.4). 2.2.1.1 Analysis Applying Theorem B.2, we conclude that V is a Lyapunov function and the equilibrium given by equation (2.8) is uniformly stable. Also V > 0, ˙V ≤ 0 implies that V ∈ L∞ which further implies that ǫ1, ˜a, ˜b ∈ L∞. Since ǫ1 = x − ˆx ∈ L∞ and x is bounded, ˆx ∈ L∞. Hence all the signals in the system described by equation (2.6) are bounded. Additionally, ǫ122 = Z ∞ 0 ǫ21 (τ )dτ = − 1 am Z ∞ 0 ˙Vdτ = − 1 am (V∞ − V0), V ∈ L∞ ⇒ ∃V∞ lim t→∞ V = V∞ ⇒ ǫ122 exists ⇒ ǫ1 ∈ L2 From equation (2.6), since ǫ1, ˜a, x,˜b, u ∈ L∞, ˙ ǫ1 ∈ L∞. Since ǫ1 ∈ L2 ∩ L∞ and ˙ ǫ1 ∈ L∞, using Lemma B.2, ǫ1 → 0 as t → 0. Fromequation (2.12), ˙ˆ a, ˙ˆb → 0 as t → ∞. Nowwe have established that ǫ1, ˙ˆ a, ˙ˆb → 0, as t → ∞ but this does not imply that ˜a,˜b → 0 as t → ∞. From equation (2.9) it is clear that limt→∞ V = V∞ = 1 2 (˜a2 +˜b 2) but this does not mean that ˜a,˜b have a limit. Notice that there is no restriction on the input u apart from being bounded. To guarantee parameter convergence additional conditions have to be imposed on the input u. 2.2.1.2 Sufficiently Rich Signals In this section the properties of the input signal for parameter convergence are discussed in detail. The notion of sufficiently rich signal and persistent excitation are introduced. 20 2.2. ONLINE PARAMETER ESTIMATION The general solution for the differential equation (2.1) is x(t) = eatx0 + Z t 0 ea(t− )bu(τ )dτ (2.14) The first part of equation (2.14) decays to zero as t → ∞ since a < 0 . Taking Laplace transform on both sides for equation (2.1) with zero initial condition we get X(s) = b s − a U(s) = G(s)U(s) (2.15) If u = 0, then the solution carries information about the parameter a alone, and if x0 = 0, information about both the parameters a, b cannot be extracted from the input/output signals. Similarly if u = c, c 6= 0, a constant, then u(t) = c ⇒ U(s) = c s ⇒ X(s) = b c s(s − a) = −b c/a s + b c/a s − a ⇒ x(t) = −b c a + b c a eat x(t) → −bc a as t → ∞. Hence, the input/output signals only have the information about the ratio b a , and not about the individual values of a and b. If u(t) = sin ω0t, then X(s) = ω0 b (s2 + ω2 0)(s − a) ⇒ x(t) = k1eat + L−1{ k2s + k3 s2 + ω2 0 } where k1, k2, k3 are constants which depend on a and b. Notice that as a < 0 and t → ∞, x(t) = L−1{ k2s + k3 s2 + ω2 0 } = G(jω0) sin(ω0t + ̺), ̺ = ∠G(jω0) G(jω0) = p b ω2 0 + a2 , ∠G(jω0) = tan−1 ImgG(jω0) ReG(jω0) By observing the magnitude and the phase of the output signal at steadystate, the unknown constants a and b can be determined. Hence, with a sinusoidal input the unknown parameters can be estimated. Such a signal is a sufficiently rich signal for this system. The property of the signal u to be sufficiently rich is called persistent excitation (PE). Persistent excitation is very important in identification problems where parameter convergence is of utmost importance. 21 2.3. ADAPTIVE CONTROL For adaptive control problems where output error convergence to zero is desired, it is not critical. For more information on persistent excitation refer to [17] (pages 63 – 73), [18] (pages 177 – 180) and [22] (Chapter 6). 2.3 Adaptive Control In this section the design of model reference adaptive control is introduced by considering a scalar tracking problem. The stability analysis and convergence properties use the mathematical preliminaries given in Appendix B. 2.3.1 Adaptive Regulation using a ReferenceModel: An Example Consider a scalar plant with dynamics described by x˙ = ax + u, x(0) = x0 (2.19) where a is the unknown plant parameter. It is desired to apply a bounded control input, u, such that the closed system is stable and x → xm as t → ∞, where xm is the output of the reference model given by x˙m = −amxm, xm(0) = xm0, am > 0 (2.20) 2.3.1.1 Control Law Let u = −k∗x be the control law such that a − k∗ = −am. In other words if the parameter a is known, then one can choose k∗ = a+am so that the closedloop dynamics of equation (2.19) is same as the reference model (2.20). If x0 = xm0, then x(t) = xm(t) for all t ≥ 0. And if x0 6= xm0 then the transient response of the closedloop system will differ from the reference model, at steady state x(t) = xm(t). Because the parameter a is unknown, the feedback gain k∗ cannot be calculated and hence the control law cannot be implemented. A control law based on the estimate of the unknown parameter is chosen as u = −k(t)x (2.21) where k(t) is the estimate of k∗. 22 2.3. ADAPTIVE CONTROL 2.3.1.2 Adaptive Law An adaptive law is now developed to estimate the unknown parameter by posing the problem as an online estimation problem. Let us define the tracking error as e1 = x − xm. Therefore, e˙1 = x˙ − x˙m = (a − k)x + amxm (2.22a) = ax + amxm − kx = ax + amxm − kx + k∗x − k∗x = ax −{zk∗x}+amxm + k∗x{−z kx} = −amx {+z amxm}−˜kx, ˜k = k − k∗ e˙1 = −ame1 − ˜kx (2.22b) ⇒ e1 = 1 s + am (−˜kx) = Gc(s)(−˜kx) The transfer function Gc(s) is SPR from Theorem B.6. Hence, SPRLyapunov2 design scheme is considered. All the signals in equation (2.22b) can bemeasured except ˜k which is a function of time. The error dynamics in equation (2.22b) is in a suitable form to choose the adaptive law for k(t) such that e1 → 0 as t → ∞. Let ˙˜ k(t) = f1(e1, x, u) where f1 is a function to be chosen. Let V (e1, ˜k) = e21 2 + ˜k2 2γ (2.23) be a Lyapunov function candidate with γ > 0 for the system described by ˙ e1 and ˙k . Choosing f1 = γe1x, we get ˙V = −ame21 ≤ 0 and ˙k = γe1x, k(0) = k0 (2.24) 2.3.1.3 Analysis Because V > 0 and ˙V ≤ 0, V is bounded i.e., V ∈ L∞ ⇒ e1 ∈ L∞, ˜k ∈ L∞ (from (2.23)). e1 = x − xm ⇒ x ∈ L∞ (since xm ∈ L∞). Additionally, Z ∞ 0 e21 (τ )dτ = Z ∞ 0 ˙V (τ )dτ = V (0) − V (∞) ⇒ ˙V ∈ L1 ⇒ e1 ∈ L2 (2.25) 2In this design the estimation error is related to the parameter by a SPR transfer function. Once in this form, the KYP ( Lemma B.5) or MKY (Lemma B.6) Lemmas can be invoked to choose an appropriate Lyapunov function V such that ˙V 0. Refer to chapter 4 (OnLine Parameter Estimation) in [18]. 23 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN ˜k ∈ L∞ ⇒ k ∈ L∞ since, k∗ is constant. From equation (2.22b) we get e˙1 ∈ L∞. Therefore, all the signal in the closedloop system are bounded. Because e˙1 ∈ L∞ and e1 ∈ L2 ∩ L∞, from lemma B.2, e1(t) → 0 as t → ∞. u = −kx ⇒ u ∈ L∞. Hence, all the signals in the closedloop system are bounded and the tracking error, e1, converges to zero. It has to be noted that, even though k ∈ L∞, ˙k → 0 as t → ∞, ˜k may not converge to zero. Convergence of ˜k to zero can be gauranteed only when ˙k ∈ L∞, k ∈ Lp ∩ L∞ where p ∈ [1,∞] (using lemma B.2). 2.4 Simplified Guide Adaptive Controller Design In the rest of this chapter model reference adaptive controllers called Guide Adaptive Controller (GAC) for web guiding applications are developed. In this section a simplified approximation of the GAC is designed to regulate the position of the web by considering a simple lateral web dynamic model. Recall that the transfer functions for both the steering guide and the displacement guide are of relative degree n∗ = 0 and of the order n = 2. By approximating the system with a reduced order model, the number of estimated parameters in the adaptive control design is reduced and the controller implementation is simpler. The transfer function for a steering guide (between the guide position and the lateral web position) is given by yL(s) = s2 + β2s + β1 s2 + β2s + β0 Z(s) (2.26) The transfer function between the voltage input and the lateral web position is given by yL(s) = kmCm(s2 + β2s + β1) s(s + a)(s2 + β2s + β0) up(s) (2.27) where km and a are motor parameters and Cm is the transmission ratio between the actuator position and the guide position. Ignoring the dynamics of the web and considering only the static gain of the lateral dynamics, we get yL(s) = kmCmβ1 s(s + a)β0 = K s(s + a) up(s) , kp Zp Rp up(s). (2.28) The transfer function kp Zp(s) Rp(s) represents a general transfer function where Zp(s), Rp(s) are monic polynomials and kp is a constant. The complexity of the adaptive controller depends on the degree of the polynomials Zp(s) and Rp(s). The knowledge of the sign of the constant kp simplifies the adaptive controller design. 24 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN The dynamics in equation (2.28) in time domain is given by y¨L + ay˙L = Kup. (2.29) Define state variables y1 = yL and y2 = y˙L. The dynamics in the state space form is y˙1 y˙2 = 0 1 0 −a y1 y2 + 0 K up (2.30a) = Ay + Bup (2.30b) yL = [1 0]y = C⊤y (2.30c) Notice that since K > 0, a > 0, the system is controllable. Since the system is controllable, a suitable reference model can be chosen such that the closedloop plant dynamics matches the referencemodel dynamics. A referencemodel with the same relative degree as the plantmodel is chosen to be ym r = ω2n s2 + 2ζωns + ω2n = km Zm Rm = Wm(s) (2.31) The reference model in state space form is x˙ 1 x˙ 2 = 0 1 −ω2n −2ζωn x1 x2 + 0 ω2n r (2.32a) x˙ = Acx + Bcr (2.32b) ym = [1 0]x = C⊤ c x (2.32c) Two different types of adaptive controllers can be designed for the simplified system given in equation (2.28), each of which has a different number of estimated parameters. By reducing the number of estimated parameters, the controller is simple. But with very few parameters, it may not be possible to sufficiently capture the dynamic behavior of the system. Theoretical and simulation results are presented for the two adaptive controllers designed based on the reduced order lateral dynamicmodels. The results indicate that for lateral guiding applications the simplified adaptive controllers are able to provide good tracking performance in the presence of process variations and disturbances. Experimental results that illustrate the effect of the number of the estimated parameters on guiding performance will be discussed in the next chapter. 25 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN 2.4.1 Three parameter Guide Adaptive Controller 2.4.1.1 Control Law Our objective is to design an adaptive control law such that the closedloop plant matches the reference model. Consider a control law with a feedback term and a feed forward term. up = L∗y + P∗r = [L∗1 L∗2 P∗] y1 y2 r = θ∗⊤ω (2.33a) where L∗1 ,L∗2 and P∗ are the parameter values which make the closedloop system match the reference model. Substituting up in equation (2.30) y˙ = 0 1 0 −a y + 0 K [L∗1 L∗2 P∗] y1 y2 r = 0 1 0 −a y + 0 0 KL∗1 KL∗2 y + 0 KP∗ r = 0 1 KL∗1 KL∗2 − a y + 0 KP∗ r (2.34) Comparing the closedloop system in equation (2.34) with the desired reference model given by equation (2.32a) , the desired control parameters that match the closedloop system with the reference model are L∗1 = − ω2n K L∗2 = a − 2ζωn K P∗ = ω2n K 2.4.1.2 Adaptive Law Since the parameters K and a are unknown, it is not possible to find the exact values of the control parameters. The adaptive control problem is posed as an estimation problem with an objective of minimizing the estimation error. Let us estimate L∗1 ,L∗2 , P∗ and use the estimates (θ⊤ = [L1 L2 P]) in the control input. 26 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN Consider the openloop system y˙ = 0 1 0 −a y + 0 K up (2.35) Adding and subtracting the desired control input we get y˙ = 0 1 0 −a y + 0 K h up + θ∗⊤ω − θ∗⊤ω i (2.36) where θ∗⊤ = [L∗1 L∗2 P∗] , ω⊤ = [y1 y2 r] y˙ = 0 1 0 −a y + 0 K [up + L∗1 y1 + L∗2 y2 + P∗r − L∗1 y1 − L∗2 y2 − P∗r] (2.37) = 0 1 0 −a y + 0 K [L∗1 y1 + L∗2 y2 + P∗r]  {z } + 0 K [up − L∗1 y1 − L∗2 y2 − P∗r]  {z } (2.38) = 0 1 −ω2n −2ζωn y + 0 ω2n r  {z } + 0 K h up − θ∗⊤ω i  {z } (2.39) = A cy {+zBcr}+ 0 K h up − θ∗⊤ω i  {z } (2.40) Denote the state estimation error by ǫ , y−x and the tracking error by e1 , yL −ym. Then, ˙ ǫ = Acǫ + B h up − θ∗⊤ω i = Acǫ + B˜θ⊤ω, ˜θ = θ − θ∗ (2.41) e1 = [1 0]ǫ = C⊤ c ǫ (2.42) Therefore, e1 = C⊤ c [sI − Ac]−1 0 K h up − θ∗⊤ω i (2.43) Since Wm(s) = C⊤ c [sI − Ac]−1 0 KP∗ , we have (2.44) e1 = Wm(s)ρ∗ ˜θ⊤ω (2.45) 27 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN where ρ∗ = 1 P∗ . One can think of an adaptive law based on the SPRLyapunov approach and hence could take advantage of the KYP (B.5) and MKY lemmas (B.6). Since the relative degree of the reference model is 2, Wm(s) is not SPR and hence a modification to the system has to be made so that the error e1 relates to the parameters through a SPR transfer function. With this modification the adaptive controller design procedure is similar to a relative degree one system (controller design for a relative degree one system is presented in Appendix C.1). If the controller can have differentiators, one can choose a transfer function L(s) = (s + p0) such that Wm(s)L(s) is SPR and follow the same procedure as in Appendix C.1. Narendra and Valavani [23] introduced a new control structure which was able to make Wm(s)L(s) SPR without using differentiators in the controller. Since Wm(s) has a relative degree 2, let us use the operator PL(θ) , L(s)θL(s)−1 in equation (2.45) and follow the same modification to the control structure and the error equation as in [23] to find the adaptive control law. The error equation changes to ˙ ǫ = Acǫ + Bρ∗ ˜θ⊤φ, B = (s + p0)BP∗ (2.46a) e1 = Wm(s)(s + p0)  {z } ρ∗ ˜θ⊤φ, φ = 1 s + p0 ω (2.46b) = Wm1(s)ρ∗ ˜θ⊤φ, where Wm1(s) is SPR (2.46c) = Wm1(s)ρ∗(θ⊤φ − θ∗⊤φ) (2.46d) = Wm1(s)ρ∗(L(s)−1θ⊤ω − θ∗⊤φ) (2.46e) Let the estimate of tracking error be ˆe1 = Wm1(s)ρ(L(s)−1up − θ⊤φ) (2.47) Notice that equation (2.47) is similar to equation (C.3). In order to use the procedure as in Appendix C.1, the estimation error ǫ1 = e1 − ˆe1 should be e1, hence the controller has to be modified such that the term (L(s)−1up − θ⊤φ) is zero. up = L(s)θ⊤φ = (s + p0)θ⊤φ = ˙ θ⊤φ + θ⊤ ˙φ + p0θ⊤φ = ˙ θ⊤φ + θ⊤ (˙φ + p0φ)  {z } = θ˙⊤φ + θ⊤ω (2.48a) 28 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN With the appropriate modifications to the estimation error dynamics and the controller, the dynamics of the system resemblems a relative degree one system. The procedure in Appendix C.1 is followed to find the adaptive law that results in a stable system with limt→∞ e1(t) = 0. The adaptive law is θ˙ = −e1φsgn(ρ∗) = −e1φ ∵ ρ∗ > 0 (2.49) Notice that the controller has a differentiator, but θ˙ can be obtained from the adaptive law given in equation (2.49). This modified structure was first introduced in [23]. For a detailed analysis of this problem (relative degree = 2) refer to chapter 5, pages 199  226, [22] and the article by Narendra and Valavani [23]. 2.4.1.3 Simulation In order to understand the validity of the simplified model, the GAC based on the simplified model was simulated in Simulink using the complete plant model. The lateral web dynamics block in Figure 2.7 represents the second order model for the web dynamics. The adaptive law and the control law are based on the simplifiedmodel. Figure 2.6 shows the performance of the controller with a sinusoidal reference trajectory. The top plot shows the tracking performance, the middle shows the controller output while the bottom plot shows the time history of the estimated parameters. Notice that the estimated parameters reach a steady value but these parameter values are not necessarily their true values. The parameter update stops as soon as the error reaches zero. The Simulink block diagram for this simulation is shown in Figure 2.7. The simulation results indicate that the adaptive control scheme based on the approximate model is capable of regulating the position of the web. A sinusoidal reference with a bias was used as the output of the reference model in order to observe the transient, steadystate and tracking performance of the controller. 2.4.2 Simplified GAC with an Estimator In consideration of the implementation of the controller in practice note that it is not possible to obtain the measurement of the lateral velocity y˙L (Notice that the control input depends on ω which depends on y˙L). One possible way to obtain the lateral velocity y˙L would be to use the finite difference approximation from the lateral position measurement yL. One can also setup an estimator for the measurement y˙L based on the previous assumption that the dynamics of 29 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 −10 −5 0 5 10 15 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −1 −0.5 0 0.5 1 1.5 Time history of estimated parameters Time in seconds Parameter values y m y p L1 L2 P Figure 2.6: Simulation Results for 3Parameter GAC Web and Actuator Dynamics Sensor Gain Adaptive Controller Adaptive Controller Regressor Adaptive Law ym r error lateral position yp control up 1 yp ym u e e omega up omega r yp 1 Reference Model lateral position yp reference r yp omega 1 yp1 du/dt 2 1 Regressor Adaptive Law −Gamma error omega theta dot theta control up phi 1 1 th s 1 s K*u 2 K*u 1 Figure 2.7: Simulink Block Diagram for 3Parameter GAC 30 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN the web is a constant and if the motor velocity measurement is available. The relationship between the motor velocity and the lateral velocity can be approximated as y˙L = Kw θ˙ = Kwvm (2.50) where y˙L is the lateral web velocity, Kw is a constant which is the approximated lateral web dynamics and θ˙ = vm is the motor velocity. Filtering both left and right hand side of equation (2.50) by a low pass filter F(s) = 1 s+1 and taking the Laplace transform we get sF(s)yL = KwF(s)vm (2.51) Let an intermediate system be defined as Z = ϕ∗⊤φ (2.52) where Z , F(s)vm, φ = sF(s)yL and ϕ∗ = 1 Kw . Hence φ is obtained by filtering the lateral position measurement using the filter s s+1 and Z is obtained by filtering the motor velocity vm by using the filter 1 s+1 . Let e be the estimation error y − ϕ⊤φ with ϕ being the estimate of ϕ∗. Notice that this is a linear estimation problem and hence the minimum can be reached in a single step if the update is in the direction of negative gradient of the cost function. It is common to choose the cost function as J(ϕ) = e2 2 and hence the gradient is▽J(ϕ) = ee˙ = −eφ. Hence the update law for the ϕ is ϕ˙ = γeφ, where γ > 0 and ϕ(0) = ϕ0. From the estimate of ϕ = 1 Kw , we can use the equation (2.50) to find the estimated lateral velocity of the web. 2.4.2.1 Simulation Figure 2.8 shows the simulation result for the simplified guide adaptive controller with the estimated lateral velocity. The top plot shows the tracking performance of the controller while the middle plot shows the control effort and the bottom plot shows the estimated parameters. The simulation results indicate a similar performance as observed with the previous controller. The Simulink model is presented in shown in Figure 2.9. 2.4.3 Four Parameter Guide Adaptive Controller Notice that the adaptive controllers derived in the previous section require two measurements, the lateral position yL and the velocity y˙L. It is possible to derive a model reference adaptive 31 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 0 5 10 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −1 0 1 2 Time history of estimated parameters Time in seconds Parameter values y m y p L1 L2 P Figure 2.8: Simulation Results for 3Parameter GAC with an Estimator Web and Actuator Dynamics Sensor Gain Adaptive Controller Regressor ym r error Lateral Position yp control up Tachometer Feedback yp dot estimator 1 yp ym u e e omega up omega r yp yp dot 2 1 Adaptive Controller Adaptive Law −Gamma error omega theta dot theta control up phi 1 th 1 s 1 s K*u 2 K*u 1 Regressor Adaptive Law lateral position yp reference r yp r omega estimated lateral velocity 3 1 2 1 Figure 2.9: Simulink Block Diagram for 3Parameter GAC with an Estimator 32 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN control law for the system described by equation (2.30) using one measurement, the lateral position measurement yL. The following section describes the design process. 2.4.3.1 Control Law Let us define the following control law for the system described by equation (2.30) so that the dynamics of the closedloop plant matches the dynamics of the reference model given in equation (2.32a) . up = θ∗ 1 1 (s) up + θ∗ 2 1 (s) yL + θ∗ 3yL + c∗0 r (2.53) where (s) = s + a0 be a Hurwitz polynomial and θ∗ i ’s, c∗0 are the true parameters which make the closedloop system dynamics match the reference model dynamics. The usefulness of parametrizing the control law in this manner will be evident later. Rewriting the control law in terms of the feedback and feedforward terms we get up = θ∗ 3s + (θ∗ 2 + a0θ∗ 3) s + (a0 − θ∗ 1) yL + (s + a0)c∗0 s + (a0 − θ∗ 1) r (2.54) Substituting the control law given by equation (2.54) in the system described by equation (2.28) we get s2yL + asyL = K θ∗ 3s s + (a0 − θ∗ 1) yL + θ∗ 2 + a0θ∗ 3 s + (a0 − θ∗ 1) yL + (s + a0)c∗0 s + (a0 − θ∗ 1) r (2.55) (s2 + as)(s + (a0 − θ∗ 1)) − Kθ∗ 3s − K(θ∗ 2 + a0θ∗ 3) yL = K(s + a0)c∗0 r (2.56) yL r = K(s + a0)c∗0 s3 + (a0 − θ∗ 1 + a)s2 + [a(a0 − θ∗ 1) − Kθ∗ 3] s − K(θ∗ 2 + a0θ∗ 3) = w2n s2 + 2ζωns + ω2n (2.57) Notice that in order to have the closedloop system dynamics match the reference model dynamics, there has to be a polezero cancellation. The cancellation will occur at −a0 and hence the filter (s) = s+a0 has to be Hurwitz. If (s) is not Hurwitz, then the polezero cancellation will occur in the righthalf plane which is not desirable. If Kc∗0 = ω2n then, (s + a0)(s2 + 2ζωns + ω2n ) = s3 + (a0 − θ∗ 1 + a)s2 + [a(a0 − θ∗ 1) − Kθ∗ 3] s − K(θ∗ 2 + a0θ∗ 3) (2.58a) 33 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN Equating both we obtain the true parameters to be θ∗ 1 = a − 2ζωn (2.59a) θ∗ 3 = 1 K (aa0 − a2 + 2ζωna − ω2n − 2ζωna0) (2.59b) θ∗ 2 = − a0ω2n K − a0θ∗ 3 (2.59c) c∗0 = ω2n K (2.59d) It is not always possible to find the desired control law which can make the closedloop system match the reference model. A few conditions on the type of the plant model and the reference model have to be made [18]. 1. The polynomial Zp(s) should be Hurwitz and monic. 2. The order, n, of the plant should be known or at least an upper bound should be known. 3. The relative degree of the plant (n∗) should be known. 4. The sign of kp should be known. 5. Zm(s),Rm(s) should be monic and Hurwitz and the order of the reference model should not be greater than the order of the plant. 6. The relative degree of the reference model should be same as that of the plant model. From equation (2.57) it can be seen that if the zeros of the reference model are not the same as the plant zeros, then the plant zeros will be cancelled by the control in order to match the closedloop system dynamics to the reference model dynamics. If Zp(s) is not Hurwitz, then polezero cancellation would occur in the right half plane which is not desirable. If Zp(s) is not monic, then the constant termof the leading coefficient can be pulled out and augmented to the gain kp. The requirement for the sign of kp to be known will be evident from the adaptive law derivation. Without the knowledge of the relative degree of the plant, it would not be possible to design an adaptive control law. The order of the plant is important because it is not always possible to find a parametrized control law such that the closedloop systemdynamicsmatches the reference model. But we can use the Bezout Identity to overcome the problem (refer to [22] sections 5.4.1 and 5.4.2). In order to obtain proper parametrization of the controller parameters θ∗ i ’s, both the numerator and denominator polynomials of the referencemodel transfer function should be Hurwitz. 34 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN In order to have the minimum number of controller parameters, it is necessary that both polynomials of the transfer function of the reference model be monic. Since the parameters a and K are unknown it is not possible to use the control law in equation (2.54) and hence the adaptive control problem is posed as an estimation problem with the objective of minimizing the parameter estimation error. We can follow the same procedure of adding and subtracting the desired control effort, but first we need to represent the control law in the statespace form in order to simplify the analysis. We shall also define an augmented system which is useful for our analysis both in this section as well as in the next section. up = θ∗ 1 1 s + a0 up + θ∗ 2 1 s + a0 yL + θ∗ 3yL + c∗0 r up = θ∗⊤ω where θ∗⊤ = [θ∗ 1, θ∗ 2, θ∗ 3, c∗0 ] and ω⊤ = 1 s + a0 up, 1 s + a0 yL, yL, r . ω˙ 1 = Fω1 + gup, ω1(0) = 0, F = −a0 (2.60a) ω˙ 2 = Fω2 + gyp, ω2(0) = 0, g = 1 (2.60b) The statespace representation of the closedloop system with the desired control effort can be obtained by augmenting the states y of the system described by equation (2.30) with the states of the controller in equation (2.60). ˙Y = AsY + Bsc∗0 r (2.61a) yL = C⊤ s Y (2.61b) where Y = y⊤, ω1, ω2 ⊤ and y˙ = Ay + Bup = Ay + B [θ∗ 1ω1 + θ∗ 2ω2 + θ∗ 3yL + c∗0 r] (2.62a) ω˙ 1 = Fω1 + gup = Fω1 + g [θ∗ 1ω1 + θ∗ 2ω2 + θ∗ 3yL + c∗0 r] (2.62b) ω˙ 2 = Fω2 + gyp = Fω2 + gC⊤y (2.62c) ∴ ˙Y = A + Bθ∗ 3C⊤ Bθ∗ 1 Bθ∗ 2 gθ∗ 3C⊤ F + gθ∗ 1 gθ∗ 2 gC⊤ 0 F Y + c∗0 B g 0 r (2.63a) yL = [C⊤ 0 0]Y (2.63b) 35 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN From equation (2.57) yL(s) r(s) = K(s + a0)c∗0 s3 + (a0 − θ∗ 1 + a)s2 + [a(a0 − θ∗ 1) − Kθ∗ 3] s − K(θ∗ 2 + a0θ∗ 3) = Wm(s) (2.64) and from equation (2.61) yL(s) r(s) = C⊤ s (sI − As)−1Bsc∗0 = Wm(s) = ym r (2.65) The reference model can be described by the following state space representation. ˙Y m = AsYm + Bsc∗0 r (2.66a) ym = C⊤ s Ym (2.66b) Similarly, the augmented closedloop system with the control up can be obtained as ˙Y c = A0Yc + Bsup (2.67a) yL = C⊤ s Yc (2.67b) where Yc = y⊤, ω1, ω2 ⊤ . y˙ = Ay + Bup (2.68a) ω˙ 1 = Fω1 + gup (2.68b) ω˙ 2 = Fω2 + gyp = Fω2 + gC⊤y (2.68c) Therefore, ˙Y c = A 0 0 0 F 0 gC⊤ 0 F Yc + B g 0 up (2.69a) yL = [C⊤ 0 0]Yc (2.69b) 2.4.3.2 Adaptive Law Our objective now is to obtain an estimate of θ∗ so that the closedloop system described in equation (2.69) matches the reference model in equation (2.66). Adding and subtracting the 36 2.4. SIMPLIFIED GUIDE ADAPTIVE CONTROLLER DESIGN desired control effort θ∗⊤ω we get ˙Y c = A0Yc + Bsup + Bsθ∗⊤ω − Bsθ∗⊤ω (2.70a) = A0Yc + Bsθ∗⊤  {z ω}+Bs(up − θ∗⊤ω) (2.70b) = AsYc + Bsc∗0 r + Bs(up − θ∗⊤ω) (2.70c) yL = C⊤ s Yc (2.70d) Let ǫ = Yc − Ym be the state estimation error and e1 = yL − ym be the tracking error. Hence the error equations can be obtained from equations (2.70) and (2.66) as ˙ ǫ = Asǫ + Bs(up − θ∗⊤ω), ǫ(0) = ǫ0 (2.71a) e1 = C⊤ s ǫ (2.71b) Notice that e1 up − θ∗⊤ω = C⊤ s (sI − As)−1Bs (2.72) and Wm(s) = C⊤ s (sI − As)−1Bsc∗0 (2.73) ⇒ e1 = Wm(s)ρ∗(up − θ∗⊤ω), ρ∗ = 1 c∗0 (2.74) The error equation is similar to the form seen in equation (C.5) and the same procedure can be continued if Wm(s) is SPR. Since the relative degree of the reference model is 2, Wm(s) is not SPR and hence we shall use the operator PL(θ) as before and the modified error equation is ˙ ǫ = Asǫ + Bcρ∗ ˜θ⊤φ (2.75a) e1 = C⊤ s ǫ (2.75b) which is similar to the error equation (2.46). The same procedure is followed and the adaptive law and control law which result in a stable system with e1(t) → 0 as t → ∞ are given by θ˙ = −e1φsgn(ρ∗) = −e1φ (2.76a) up = θ⊤ω + θ˙⊤φ (2.76b) 2.4.3.3 Simulation Figure 2.10 shows the performance of the adaptive controller. The plant used in this simulation is the complete plantmodel instead of the simplified plant. The top plot shows the tracking performance, the middle plot shows the control input, while the bottom plot shows the estimated 37 2.5. GUIDE ADAPTIVE CONTROLLER parameters. The Simulink block diagram for the simulation is shown in Figure 2.12. Similar to the previous controller the adaptive controller with four parameters is capable of regulating the lateral position of the web. By observing the control effort for the two controllers one can notice that the controller with four parameters exhibits a better transient response behavior. This can be attributed to the fact that the controller with three parameters has no knowledge of the control effort generated. The estimated parameters settle to a steady state value after the plant starts tracking the sinusoidal reference. A common disturbance which helps in characterizing the transient response is a pulse disturbance. The performance of the controller with a pulse disturbance is shown in Figure 2.11 and the plots indicate that the controller is capable of rejecting the pulse disturbances. 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 0 5 10 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −1 0 1 2 3 Time history of estimated parameters Time in seconds Parameter Values y m y p q 1 q 2 q 3 c 0 Figure 2.10: Simulation Results for 4Parameter GAC 2.5 Guide Adaptive Controller In the previous section a simplified guide adaptive controller was designed based on the assumption that the dynamics of the web is a constant. Here in this section we shall consider the complete dynamics of the web and develop a guide adaptive controller. The transfer function for a steering guide is given by equation (2.27) and its state space form is y˙ = Ay + Bup, yL = C⊤y (2.77) 38 2.5. GUIDE ADAPTIVE CONTROLLER 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 −10 −5 0 5 10 15 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −2 −1 0 1 2 3 Time history of estimated parameters Time in seconds Parameter values y m y p q 1 q 2 q 3 c 0 Figure 2.11: Simulation Results for 4Parameter GAC with a Pulse Disturbance Consider the reference model in equation (2.32a) and the objective is to design a control law such that the closedloop system matches the reference model. So far our first step in designing an adaptive controller has been to find the desired control input which can ensure that the closedloop system matches the reference model. Then we would pose the problem as an estimation problem and use the estimated parameters in the control law. In a general model reference adaptive control problems with relative degree n∗ = 1, it is simple to choose a parametrized control law which can algebraically render the closedloop system to have the same roots as the reference model. But in a general adaptive control problem with relative degree n∗ ≥ 2, finding the desired control is not simple. In the previous section, we were able to choose a desired control (θ∗⊤ω) which resulted in a closedloop system that matched the reference model. As discussed in [22] (section 5.4.1 and 5.4.2), Bezout Identity can be used to show that it is always possible to find a parameter vector θ∗ so that the following control law [18] will result in a closedloop system whose dynamics is same as the reference model dynamics: up = θ∗ 1 ⊤ α(s) (s) up + θ∗ 2 ⊤ α(s) (s) yL + θ∗ 3yL + c∗0 r (2.78) where α(s) = [s2 s 1]⊤, (s) is an arbitrary monic Hurwitz polynomial of degree 3 which has to be chosen. Similar to the procedure in the previous section (section (2.60) on page 35) we can 39 2.5. GUIDE ADAPTIVE CONTROLLER Web and Actuator Dynamics Sensor Gain Adaptive Controller Adaptive Controller Regressor Adaptive Law ym r error lateral position yp control up 1 yp ym u e e omega up omega r yp 1 Reference Model Regressor Adaptive Law −Gamma error omega theta dot theta control up phi 1 1 th s 1 s K*u 2 K*u 1 Control up lateral position yp reference r omega 1 omega 2 yp r omega 1 1 s 1 s g g K*u K*u 3 2 1 Figure 2.12: Simulink Block Diagram for 4Parameter GAC 40 2.5. GUIDE ADAPTIVE CONTROLLER represent the control law in the statespace form as ω˙ 1 = Fω1 + gup, ω1(0) = 0 (2.79a) ω˙ 2 = Fω2 + gyp, ω2(0) = 0 (2.79b) up = θ∗⊤ω (2.79c) where ω1, ω2, θ1, θ2 ∈ R3, θ∗ = [θ∗ 1 ⊤ θ∗ 2 ⊤ θ∗ 3 c∗0 ] and ω = [ω⊤ 1 ω⊤ 2 yL r]. The matrix F and g are realized as F = −λ2 −λ1 −λ0 1 0 0 0 1 0 , g = 1 0 0 (2.80) where (s) = s3 + λ2s2 + λ1s + λ0. The adaptive control design is similar to the design in the previous section but with more estimation parameters which are a result of considering the complete web lateral dynamic model. Notice that in equation (2.59) we have shown the existence of the true parameters with the desired control given by equation (2.53). Similarly, it is possible to show the existence of the true parameters for the adaptive controller with the desired control given in equation (2.78). Same procedure as in the previous section can be followed to obtain the adaptive law and control law (refer to chapter 6 in [18] for a comprehensive formulation). 2.5.1 Control Law up = θ⊤ω + θ˙⊤φ (2.81a) φ = 1 s + p0 ω (2.81b) ω = [ω⊤ 1 ω⊤ 2 yL r] (2.81c) ω˙ 1 = Fω1 + gup, ω1(0) = 0 (2.81d) ω˙ 2 = Fω2 + gyp, ω2(0) = 0 (2.81e) 2.5.2 Adaptive Law θ˙ = −e1φ, e1 = yL − ym (2.82) 41 2.5. GUIDE ADAPTIVE CONTROLLER 2.5.3 Analysis It has been shown in [18] that the above adaptive law along with the control law results in a stable closedloop system with e1(t) → 0 as t → ∞. The proof is similar to the one described in section C.1.0.1. The chosen Lyapunovlike function is V (˜θ, ǫ) = ǫ⊤Pcǫ 2 + ˜θ⊤−1 ˜θρ∗ 2 and with the above control law and adaptive law the time derivative of V is given by ˙V = − ǫ⊤qq⊤ǫ 2 − ǫ⊤νcL⊤c ǫ 2 where q is a vector and νc > 0 is a scalar and Lc is a positive definite matrix. Because V > 0 and ˙V ≤ 0, V is bounded or V ∈ L∞ ⇒ ǫ, θ, ˜θ ∈ L∞. With ǫ ∈ L∞ we get Yc, Ym ∈ L∞. In addition to proving that u = θ⊤ω ∈ L∞, we need to prove that u = θ⊤ω + θ˙⊤φ ∈ L∞. Since φ is a filtered version of ω and F being a stable filter, it is clear that u ∈ L∞. 2.5.4 Simulation Figures 2.13 and 2.14 show the performance of guide adaptive controller. The top plot in each figure shows the tracking performance, the middle plot shows the controller output while the bottom plot shows the estimated parameters. In Figure 2.14 in addition to the sinusoidal reference, a pulse disturbance of 0.25 V amplitude is introduced at two time instants, 4 and 12 seconds. The controller is capable of rejecting the pulse disturbance. The Simulink block diagram for the GAC is shown in Figure 2.15. 42 2.5. GUIDE ADAPTIVE CONTROLLER 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 −1 0 1 2 3 4 5 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −0.5 0 0.5 1 1.5 Time history of estimated parameters Time in seconds Parameter values y m y p Figure 2.13: Simulation Results for Guide Adaptie Controller 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 Actual vs Reference Model output Position (V) 0 2 4 6 8 10 12 14 16 18 20 −10 −5 0 5 10 Control effort Control input (V) 0 2 4 6 8 10 12 14 16 18 20 −1 −0.5 0 0.5 1 1.5 Time history of estimated parameters Time in seconds Parameter values y m y p Figure 2.14: Simulation Results for Guide Adaptie Controller with a Pulse Disturbance 43 2.5. GUIDE ADAPTIVE CONTROLLER Web and Actuator Dynamics Sensor Gain Adaptive Controller Adaptive Controller Regressor Adaptive Law ym r error lateral position yp control up 1 yp ym u e e omega up omega r yp 1 Reference Model Regressor Adaptive Law −Gamma error omega theta dot theta control up phi 1 1 th s 1 s K*u 2 K*u 1 Control up lateral position yp reference r omega 1 omega 2 yp r omega 1 1 s 1 s g g F F 3 2 1 Figure 2.15: Simulink Block Diagram for Guide Adaptie Controller 44 Chapter 3 Experimental Results In the previous chapter the design and analysis of model reference adaptive control strategies suitable for web guiding were presented. Three different guide adaptive controllers were developed based on the number of estimated parameters in the control law. Two adaptive controllers were designed based on a simplified model for the lateral web dynamics and one was based on the complete model. In this chapter the controller designs are implemented on an experimental web platform containing two intermediate web guides and their performance under different operating conditions are evaluated. Guidelines for implementation of the adaptive algorithms are given and discussed. 3.1 Experimental Platform The experimental web handling platform used for implementing the designed controllers is shown in Figure 3.1. A line schematic of the platform is shown in Figure 3.2. The platform is an endless web line with several idle rollers and one driven roller (Master Speed Roller). The platform does not have an unwind or a rewind section; the web runs in a loop around the rollers. The transport velocity of the web is set by the speed of the driven roller. In order to provide traction between the driven roller and the web, a nip roller is used on the driven roller. The nip roller applies pressure at the contact surface to maintain web traction on the driven roller during the startup of the line. The platform is equipped with a passive dancer and an active dancer which can be used to control the tension of the web in the platform [24, 25]. In an active dancer, the dancer roller position is controller by an actuator. The tension in the web line can be increased or decreased by varying the position of the dancer roller. In a passive dancer mechanism the dancer roller is 45 3.1. EXPERIMENTAL PLATFORM Remotely Pivoted Guide OffsetPivot Guide Figure 3.1: Experimental Web Handling Platform MOTOR LOAD CELL LOAD CELL REMOTELY PIVOTED GUIDE NIP ROLLER ACTIVE DANCER SYSTEM PASSIVE DANCER AIR PISTON AIR PISTON MASTER SPEED ROLLER OFFSETPIVOT GUIDE EDGE SENSOR INFRARED SENSOR Figure 3.2: Line Schematic of the Experimental Web Handling Platform 46 3.1. EXPERIMENTAL PLATFORM Ultrasonic Sensor Web Figure 3.3: offset pivot Guide Infrared Sensor Web Figure 3.4: Remotely Pivoted Guide free to move about a pivot or on a linear slide [25]. The dancer roller in a passive dancer is not actuated but floats due to the pressure from an air cylinder. There are two web guides in the platform; an offset pivot guide and a remotely pivoted guide. The offset pivot guide is a Fife Narrow web guide and the remotely pivoted guide is a Fife Kamberoller guide. The offset pivot guide is equipped with an ultrasonic sensor and the remotely pivoted guide is equipped with an infrared sensor. The actuators (servo motors) and the sensors of each guide mechanism are connected to a dSPACE DS1103 realtime board through Fife controller hardware. The Kamberoller guide and its infrared sensor is connected to the realtime hardware through a Fife A9 analog controller. Electrical provisions are made such that the guide can be controlled either with the A9 controller or with the dSPACE hardware. The offset pivot guide is connected to the dSPACE hardware through a Fife DP01 controller. The offset pivot guide can only be controlled using the dSPACE hardware. The feedback signals from the sensors and the control signals to the actuators are processed by the dSPACE hardware. The DS1103 hardware has eight A/D converters for a total of 20 A/D channels, of which 16 channels are multiplexed by four converters. The digital output from the realtime hardware is converted into analog output by 8 D/A converters. The input and output range of the dSPACE board is ± 10 V. The dSPACE board is driven by a PowerPC CPU running at 400 MHz. The realtime control software is written in C using dSPACE RTLib realtime libraries. The control software is driven by interrupts which occur every sampling period. At each sampling time the following operations are performed by the realtime software: 1. the sensor feedback signals are read; 47 3.2. EXPERIMENTAL PROCEDURE 2. the control algorithm is executed and the required control effort is calculated; and 3. the control effort is then supplied to the actuators. The dSPACE board is installed inside a host computer running Microsoft Windows 2000 operating system. The dSPACE ControlDesk software is utilized to communicate between the realtime hardware and the host computer. The realtime hardware is capable of buffering the data using its internal memory. Trace variables can be setup so that the buffered data in the realtime control hardware can be stored/displayed in the host computer. These trace variables can also be used to change the variable values in the realtime control software while executing in realtime. 3.2 Experimental Procedure Experiments were carried out to observe the guiding performance of the designed adaptive controllers for process variations and disturbances. The experimental results presented in this thesis cover only the Kamberoller guide. Experiments were also conducted on the offset pivot guide and similar results were observed. The Kamberoller is located four spans downstream of the offset pivot guide. Any arbitrary disturbance can be created using the offset pivot guide and the Kamberoller can be used to reject the disturbance. On the contrary, since the web has to travel 13 spans fromthe Kamberoller to the offset pivot guide, the lateral disturbances generated at the Kamberoller do not propagate to the offset pivot guide. The Kamberoller guide is connected to the dSPACE hardware through the Fife A9 controller. Electrical modifications to the A9 controller were made so that either the A9 controller or the controller implemented on the dSPACE hardware can be used to control the Kamberoller guide. This setup provides a convenient way to compare the two controllers. Such an arrangement is not available with the offset pivot guide. The adaptive control schemes were compared with an A9 analog controller (PI controller). The A9 controller is the only industrial controller in the experimental platform which can be directly compared with the adaptive control schemes. Modifications to the existing setup are required in order to incorporate other industrial controllers in the experimental platform. The following sections present the process variations and the disturbances thatwere created for controller evaluation. 48 3.2. EXPERIMENTAL PROCEDURE 3.2.1 Process Variations The guiding performance of the developed adaptive control schemes were evaluated with different process parameters. Two main process parameters that affect the coefficients of the web lateral dynamic model are the web span parameter K and the longitudinal velocity of the web v. The parameter K is affected by the properties of the web material. Two different web materials with distinct physical characteristics were used; an optically opaque magnetic film like material (Figure 3.5) and an optically transparent polyethylene polymer web (Figure 3.6). The webs used were of different dimensions. The experiments were conducted with different web transport velocities and with different tensions in the web line. Figure 3.5: Opaque Web Figure 3.6: Transparent Web Whenever the web material is changed from opaque to transparent web, the gain of the infrared sensor changes. The full scale voltage range changes from 0  6 V (opaque) to approximately 5  6 V (transparent) as illustrated in Figure 3.7. One of the drawbacks of the existing controllers is their inability to cope with sensor gain variations. Due to sensor gain variations the guidepoint also changes. The guidepoint is the physical reference position of the web inside the sensing window. Notice from Figure 3.7, the minimum voltage output from the sensor 49 3.2. EXPERIMENTAL PROCEDURE is 5 V for a transparent web. This is due to the fact that the maximum signal attenuation with the transparent web is only 1 V. ACTUAL WEB POSITION SENSOR OUTPUT 0 V 0 in 0.7 in 6 V 5 V Transparent Web Opaque Web Figure 3.7: The Effect of Opacity on Sensor Gain for an Infrared Sensor Experimentswere conductedwith the twowebmaterials to evaluate the performance of the controllers for sensor gain variations. All the parameters of the controllers including the gain matrix and the filter parameters remained constant with both webs. Similarly the PI controller gains remained constant with both webs. Since the guidepoint changes with the opacity of the web material, different guidepoints were used with the two webs. For example, the guidepoint for the opaque was 3 V while the guidepoint with the transparent web was 5.4 V. Both guidepoints correspond to the same physical position of the web. A new technique for automatic guidepoint detection called offsetadaptation is presented in Appendix A. 3.2.2 Disturbances Some common lateral disturbances in industrial lines are the result of misaligned rollers, telescoped unwind rolls, splicing, improper web edges, wrinkled web, etc. To mimic these disturbances in the experimental web platform, sinusoidal, step and pulse disturbances were created by the offset pivot guide and with changes to the web edge. A common disturbance observed in an industrial web line is due to misaligned rollers. 50 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER It is seldom possible to have exact alignment for all the rollers in a web line. The angular misalignment of a series of rollers may produce lateral disturbances which can be periodic. Telescoping of rolls may produce periodic disturbances when unwound. In order to simulate a periodic disturbance, the offset pivot guide in the web line is used. The offset pivot guide, which is located before the Kamberoller in the web line (refer Figure 3.2), was made to follow a sinusoidal reference. As the sinusoid propagates, a periodic disturbance is created at the Kamberoller. In order to characterize the transient response, the performance of the controller with a pulse disturbance was evaluated in the experiments. The pulse disturbance was generated by adding a small strip of web material about one feet long and 0.2 inches wide to the edge of the web. Since the strip is of constant length, the duration of the pulse is dependent on the transport velocity of the web. As the web transport velocity increases, the pulse width reduces and tends to an impulse. In order to observe the performance characteristics such as the percentage overshoot and settling time, stepreferencechange experiments were conducted. The guidepoint or the lateral position reference was changed and the performance of the controllers for this change was evaluated. The sensor signals from both the infrared sensor and the ultrasonic sensor were noisy. In industrial controllers, analog or digital signal conditioning are carried out on the raw signal to filter the noise. The error signal for the adaptive controllers were unfiltered in order to observe the performance of the controllers with noisy measurements. 3.3 Three Parameter Guide Adaptive Controller The performance of the three parameter guide adaptive controller under different operating conditions and disturbances are presented in this section. The three parameter GAC requires an additional measurement, i.e., the lateral web velocity. While implementing the controller using the realtime hardware, the lateral velocity measurement was approximated by the finite difference of the lateral position. Recall that the control law and the adaptive law for the three parameter GAC is given by Control Law up = θ˙⊤φ + θ⊤ω 51 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER Adaptive Law θ˙ = −e1φ where θ⊤ = [L1 L2 P], ω⊤ = [yL y˙L r], e1 = yL − ym and φ = 1 s + p0 ω. 3.3.1 Experiments with opaque web Figures 3.8 to 3.11 show a representative sample of experimental results with the three parameter adaptive controller with the opaque web. In Figure 3.8 the top plot shows the sinusoidal disturbance observed by the Kamberoller when the guide is not actuated. The middle plot shows the performance of the PI controller. And the bottom plot shows the performance of the adaptive control scheme. A significant amount of the sinusoidal disturbance generated at the offset pivot guide has propagated to the Kamberoller. The three parameter guide adaptive controller is able to significantly attenuate the disturbance when compared to the PI controller. The guidepoint for the two experiments were not the same. In subsequent experiments the guidepoints for the adaptive scheme and the PI controller were matched to maintain consistency. In Figure 3.9 the performance of the adaptive controller with sine disturbance is shown in the top plot, the control effort is shown in the middle plot, and the parameter estimates are shown in the bottom plot. The control effort generated is well within the actuator limits which is ± 10 V. Even though the parameters may not converge to their true values, it is important to observe the evolution of the estimated parameters. Notice that the last two parameters (L2 and P) vary sinusoidally while the first parameter (L1) slowly increases. It was experimentally observed that L1 reaches a steadystate value while the other two parameters vary about zero sinusoidally. These observations are useful in improving the robustness properties of the adaptive controller. One such algorithmwhich can increase the robustness of the adaptive controller is parameter projection. For more information on parameter projection refer to page 565 in [18] and page 328 in [17]. The top plot in Figure 3.10 shows the performance of the PI controller with a pulse disturbance while the bottom plot shows the performance of the three parameter GAC. The adaptive controller is able to provide a faster response with better guiding performance. Similar performance is also seen at different transport speeds. The performance of the adaptive controller for stepreferencechange is shown in the Figure 52 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER 3.11. The adaptive controller is capable of following the reference model accurately. The transient response characteristics of the guiding system is similar to the reference model transient response characteristics. 0 5 10 15 20 25 30 2 3 4 5 Sine disturbance observed at the kamberoller guide Lateral Position (V) 0 5 10 15 20 25 30 2 3 4 5 Performance of A9 controller Lateral Position (V) 0 5 10 15 20 25 30 2 4 Performance of three parameter GAC Time (Seconds) Lateral Position (V) Figure 3.8: Performance Comparison: 3 Parameter, Sine Disturbance, 300 fpm, Opaque Web 3.3.2 Experiments with transparent web Figures 3.12 to 3.14 show the performance of the three parameter guide adaptive controller with the transparent web. Figure 3.12 compares the performance of the PI controller and the adaptive controller. The top plot shows the disturbance observed at the Kamberoller when the guide is not actuated. The middle plot shows the performance of the PI controller. Comparing the top and the middle plots, no disturbance attenuation is observed. The PI controller is not able to cope with the sensor gain change. The adaptive controller on the other hand is able to attenuate the sinusoidal disturbance. Notice that a small streak is observed in all the three plots. This streak is observed when the joint passes the sensor window. The web opacity variation at the joint is significant to cause a sudden jump in the sensor output which causes the streak. In Figure 3.13 the top plot shows the adaptive controller performance, the middle plot shows the control effort generated and the bottom plot shows the time history of the estimated parameters. Similar to the experiments with the opaque web, the last two parameters vary 53 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER 0 5 10 15 20 25 30 2 4 6 Three Parameter Adaptive Controller Performance Time (Seconds) Lateral Position (V) Web Position Reference Position 0 5 10 15 20 25 30 −10 0 10 Time (Seconds) Control (V) 0 5 10 15 20 25 30 −0.4 −0.2 0 Time (Seconds) Parameters Figure 3.9: Adaptive Controller: 3Parameter, Sine Disturbance, 300 fpm, Opaque Web 0 5 10 15 20 25 30 3 3.5 4 4.5 Performance of A9 controller Lateral Position (V) 0 5 10 15 20 25 30 3 3.5 4 4.5 Performance of three parameter GAC Time (Seconds) Lateral Position (V) Web Position Reference Position Figure 3.10: Performance Comparison: 3Parameter, Pulse Disturbance, 300 fpm, Opaque Web 54 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER 0 5 10 15 20 25 30 3.5 4 4.5 5 5.5 6 Performance of three parameter GAC (300 fpm) Lateral Position (V) 0 5 10 15 20 25 30 3.5 4 4.5 5 5.5 6 Performance of three parameter GAC (500 fpm) Time (Seconds) Lateral Position (V) Reference Model Output Reference Position Edge Position Figure 3.11: Adaptive Controller: 3Parameter, 300 and 500 fpm, Step Reference Changes, Opaque Web sinusoidally. The sinusoidal variation is centered around zero while the parameter L1 slowly increases and reaches a steady state value. In Figure 3.14 the performance of the PI controller and the adaptive controller for a pulse disturbance is shown. The top plot shows the performance of the PI controllerwhile the bottom plot shows the performance of the adaptive controller. Even though the results look similar, a careful examination would reveal that the PI controller is nonresponsive to the pulse disturbance. Due to the change in the sensor gain the PI controller is not capable of providing good guiding performance. In fact the response observed using the PI controller is exactly the same as the response when the Kamberoller is unactuated. To clearly visualize the performance of the adaptive controller it is necessary to examine the typical response for a pulse disturbance. The pulse disturbance is formed by a series of two steps. The first step has a positive magnitude and the second has the same magnitude but a negative sign. As soon as the first step passes the sensor, an error in lateral position is created. The controller tries to regulate the position by bringing the web back to its reference position. Therefore as soon as the pulse disturbance enters the sensor a sudden dip is observed and the guide is actuated to bring the web back to its reference position. The second step creates a 55 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Sine disturbance observed at the kamberoller guide Lateral Position (V) 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of A9 controller Lateral Position (V) 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of three parameter GAC Time (Seconds) Lateral Position (V) Figure 3.12: Performance Comparison: 3Parameter, 500 fpm, Sine Disturbance, Transparent Web 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Three Parameter Adaptive Controller Performance Time (Seconds) Lateral Position (V) Web Position Reference Position 0 5 10 15 20 25 30 −10 0 10 Time (Seconds) Control (V) 0 5 10 15 20 25 30 −0.4 −0.2 0 Time (Seconds) Parameters Figure 3.13: Adaptive Controller: 3Parameters, 500fpm, Sine Disturbance, Transparent Web 56 3.3. THREE PARAMETER GUIDE ADAPTIVE CONTROLLER 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of A9 controller Lateral Position (V) 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of three parameter GAC Time (Seconds) Lateral Position (V) Web Position Reference Position Figure 3.14: Performance Comparison: 3Parameter, 500 fpm, Pulse Disturbance, Transparent Web similar error but in the opposite direction, i.e., it creates a positive error. Therefore, when the pulse disturbance leaves the sensor, a spike is observed and the controller tries to regulate the web edge back to its reference. If the guide is unactuated the lateral position of the web is unchanged as the pulse passes the sensor. The complete pulse is observed in the sensor output. Observing the plots in Figure 3.14, we can notice that the PI controller is unable to regulate the lateral position in the presence of a pulse disturbance. While the adaptive controller is capable of providing good guiding performance in the presence of the pulse disturbance. Additional experimental results for the three parameter adaptive controller are presented in Appendix D section D.1. From the experimental results it is evident that the three parameter adaptive controller is suitable for web guiding. The controller is capable of compensating for the common disturbances and process variations. The adaptive algorithmis able to adapt to the process variations and is capable of providing the appropriate control effort to regulate the lateral edge position. The adaptive controller is also able to provide good guiding performance in the presence of measurement noise. 57 3.4. FOUR PARAMETER GUIDE ADAPTIVE CONTROLLER 3.4 Four Parameter Guide Adaptive Controller The same set of experiments with the two types of webmaterials were carried out with the four parameter guide adaptive controller. In this controller implementation only one measurement is needed, the lateral web position. The performance of the four parameter adaptive controller was similar to the three parameter adaptive controller. The adaptive controller provides good guiding perfromance in the presence of the common disturbances and process variations. A complete set of the experimental results is presented in Appendix D.2. Figure 3.15 shows the guiding performance of the adaptive controller at different guidepoints for a sinusoidal disturbance. Interestingly, at different guidepoints the performance of the same controller is different. When the guidepoint is around 3 V clear indication of the sinusoidal disturbance is observed. Additionally, the Kamberoller guide was positioned at different guidepoints and the magnitude of the sinusoidal disturbance was observed. The magnitude was higher at 3 V compare to a guidepoint of 5 V. This may be attributed to sensor nonlinearity. The slope of the sensor may not be linear along its entire window length. 0 5 10 15 20 25 30 2 2.5 3 3.5 4 4.5 5 5.5 6 Four Parameter Adaptive Controller Performance Lateral Position (V) Web Position Reference Position Figure 3.15: Adaptive Controller: 4Paramater, 300 fpm, Sine Disturbance, Opaque Web 58 3.5. GUIDE ADAPTIVE CONTROLLER 3.5 Guide Adaptive Controller The guide adaptive controller (with eight parameters) has a similar implementation as the four parameter GAC: it is a SISO system with eight parameters. Figure 3.16 shows a representative sample of the performance of the eight parameter adaptive controller and the PI controller. A complete set of the experimental results is presented in Appendix D.3. The experimental results indicate that the controller provides good guiding perfromance in the presence of the common disturbances and process variations. While implementing the adaptive controller on the realtime hardware no assumption on the values of the parameters were made. The initial values for all the parameters were set to zero. The plots in Figure 3.17 shows the performance of the adaptive controller as soon as it is started. The plots in Figure 3.18 show the performance of the adaptive controller after the adaptation is continued for a long time. Observe the bottom plots in Figures 3.17 and 3.18. The plots show the evolution of the estimated parameters. As the time progresses the estimated parameters reach steady state values. The guiding performance of the controller is unaffected during the evolution of the parameters. This clearly indicates that the variables associated with the system are bounded. Similar results were found with all the adaptive control schemes, and with all the disturbances that were discussed previously. Recall that the control law and the adaptive law for an eight parameter adaptive controller is given by: Control Law up = θ⊤ω + θ˙⊤φ (3.1a) φ = 1 s + p0 ω (3.1b) ω = [ω⊤ 1 ω⊤ 2 yL r] (3.1c) ω˙ 1 = Fω1 + gup, ω1(0) = 0 (3.1d) ω˙ 2 = Fω2 + gyp, ω2(0) = 0 (3.1e) Adaptive Law θ˙ = −e1φ, e1 = yL − ym (3.2) Notice that the control law not only depends on the actual value of the parameter vector θ, but also depends on the rate of change (θ˙) of the parameter vector. This is one of the reasons why the adaptive controller provides good guiding performance even when the parameters 59 3.6. SYSTEMATIC PROCEDURE FOR ADAPTIVE CONTROLLER IMPLEMENTATION does not reach a steadystate value. Once the parameters reach a steadystate value then the control will only be dependent on the parameter vector θ and the regressor vector ω. 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Sine disturbance observed at the kamberoller guide Lateral Position (V) 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of A9 controller Lateral Position (V) 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Performance of eight parameter GAC Time (Seconds) Lateral Position (V) Figure 3.16: Performance Comparison: 8Parameter, 300 fpm, Sine Disturbance, Transparent Web 3.6 Systematic Procedure for Adaptive Controller Implementation The theory developed in the previous chapter does not impose any constraint on the value of design parameters. Most of the design parameters have to be positive in the case of scalars and positive definite in the case of matrices. When implementing the adaptive controller, it is important to consider the actuator rate constraints, the bandwidth of the actuator, etc., and the design parameters cannot be chosen arbitrarily. This section presents a systematic procedure for choosing various design parameters in the adaptive control schemes presented in the previous chapters. 1. Choose a suitable reference model: In a model reference adaptive controller design, the first step is to choose a reference model. The choice of the reference model is based on common performance characteristics like the settling time and the percentage overshoot. 60 3.6. SYSTEMATIC PROCEDURE FOR ADAPTIVE CONTROLLER IMPLEMENTATION 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Eight Parameter Guide Adaptive Controller Performance Lateral Position (V) Web Position Reference Position 0 5 10 15 20 25 30 −10 0 10 Control (V) 0 5 10 15 20 25 30 −0.5 0 0.5 Time (Seconds) Parameters Time history of estimated parameters q 1 q 2 q 3 q 4 q 5 q 6 q 7 c 0 Figure 3.17: Adaptive Controller: 8Parameter, 300 fpm, Sine Disturbance, Transparent Web 0 5 10 15 20 25 30 4.8 5 5.2 5.4 5.6 Eight Parameter Guide Adaptive Controller Performance Lateral Position (V) Web Position Reference Position 0 5 10 15 20 25 30 −10 0 10 Control (V) 0 5 10 15 20 25 30 −5 0 5 Time (Seconds) Parameters Time history of estimated parameters q 1 q 2 q 3 q 4 q 5 q 6 q 7 c 0 Figure 3.18: Adaptive Controller: 8Parameter, 300 fpm, Sine Disturbance, SteadyState, Transparent Web 61 3.6. SYSTEMATIC PROCEDURE FOR ADAPTIVE CONTROLLER IMPLEMENTATION Choose a model which is well damped. The settling time can be chosen based on the performance requirements and the actuator rate constraints. 2. Perform computer simulations: Computer simulations are important because they provide a good starting point for practical implementation. Some basic information about the evolution of the estimated parameters, the gain matrices and the control effort generated can be obtained from the computer simulations. 3. Choose the polynomial (s): This polynomial is chosen only in the four parameter and the eight parameter case. A simple choice would be (s) = (s + a0)n (3.3) where n = 1 for the four parameter case and n = 3 for the eight parameter case. Notice that the polynomial (s) filters the control input signal up and the lateral position signal yL (refer equations (2.53) and (2.78)). This polynomial can be chosen based on the bandwidth of the motor. The polynomial is chosen such that the filtered versions of up and yL have the same bandwidth as the actuator. 4. Choose the parameter p0: Based on the necessary and sufficient conditions derived in section C.2, p0 is chosen according to the constraint 0 < p0 < 2ζωn. This parameter sets the bandwidth of the filter L(s)−1 which is defined as L(s)−1 = 1 s + p0 Recall that this filter is added to satisfy the SPR condition. The filter L(s)−1 filters all the regressor signals. Large values for the parameter p0 will significantly attenuate the magnitude of the regressor vector. This may lead to poor guiding performance due to very slow adaptation. Figure 3.19 shows the performance of the adaptive controller with different values of the parameter p0. The value of the parameter p0 is decreased after 15 seconds and the guiding performance improves with the decrease. Small values for the parameter p0 may help in quick adaptation but may also cause the system to be sensitive (equations (2.76b) and (3.1a)). 5. Choose gains based on simulation: The initial choice of the gains can be based on the simulation results. A positive definite diagonal matrix can be chosen as the gain matrix. As a starting point all the diagonal elements may have the same value. 62 3.6. SYSTEMATIC PROCEDURE FOR ADAPTIVE CONTROLLER IMPLEMENTATION 6. Observe the evolution of parameters: Observing the way in which the parameters adapt is important in the design parameter selection process. The experiments with pulse disturbance help in selecting the adaptation gains. The control effort supplied to the actuator is directly related to the magnitude and the rate at which the estimated parameters vary ((2.76b) and (3.1a)). When the parameters vary quickly, the magnitude of the control effort tends to increase. This may not be desirable in most cases. 7. Set a bound on the parameter estimates: Simple bounding of parameters improves the robustness of the controller. In order to set a bound, knowledge of the estimated parameters is required. Conservative bounds can be set based on the experiments with pulse disturbance. 8. Observe the control effort: The control effort supplied to the actuator can help us in choosing some parameters such as p0. If the parameter p0 is very small, then the guiding system may be sensitive. This can be clearly observed even without a pulse disturbance experiment. When the parameter p0 is small, noisy measurement may cause a ringing effect (the guide may vibrate with a very high frequency about the guidepoint). The parameter p0 may be increased to reduce this effect. 0 5 10 15 20 25 30 4 4.5 5 5.5 6 Performance of eight parameter GAC with different p0 Time (Seconds) Lateral Position (V) Figure 3.19: Performance Comparison: 8Parameter, 500 fpm, Pulse Disturbance, Effect of p0, Transparent Web Based on the observations from the experiments some guidelines for practical implementation are as follows: 63 3.6. SYSTEMATIC PROCEDURE FOR ADAPTIVE CONTROLLER IMPLEMENTATION • The estimated parameters reach a steadystate value after some time. How fast the parameters reach the steadystate value is dependent on the adaptation gains. Once the steadystate value is reached there is no significant change in the estimated parameters. Therefore, adaptation can be stopped or the estimated parameters can be frozen. Once the parameters are frozen, the controller behaves like a fixed gain controller. • When changes in the process parameters aremade, adaptation can be continued. This can be implemented by continuously monitoring the error variable. Once the error variable exceeds a predefined limit, the adaptation of the parameters can be continued. • Instead of stopping all the parameters, the estimated parameters can be selectively frozen. All the parameters except the last two parameters can be frozen. • The decision on when to stop the adaptation can be made based on the adaptive law. Notice that the adaptive law is given by: θ˙ = −e1φ, e1 = yL − ym (3.4) When the parameters reach a steadystate value, the vector θ˙ would be zero. Whenever all the elements of the vector θ˙ are close to zero, then the adaptation can be stopped. 64 Chapter 4 Friction Compensation inWeb Guides Friction is a type of phenomena that is found in almost all servomechanisms. Friction is inevitable in mechanisms with relative motion between parts which are in contact with each other. Although friction is essential in some mechanisms, such as braking, it is usually not desired in high precision motion control systems. Typically friction may cause steadystate errors in position regulation and tracking and may lead to limit cycle behavior. Hence it is important to consider the effect of friction when designing and implementing a closedloop control system. In this chapter we consider static friction models and analyze compensation techniques which are specific to web guides. 4.1 Static Models Early modeling of friction involved static models of friction which are simply a function of relative velocity between contacting surfaces. Static friction takes into account only a few properties of friction with the main idea being that the friction force opposes motion and the magnitude of friction is independent of the surface area of contact. A brief discussion of some well known static friction models are given below. Coulomb investigatedDa Vinci’s frictionmodel and proposed themodel in 1785 [26]. Coulomb friction in dynamic systems is modeled as a piecewise continuous function which is positive for positive velocities and negative for negative velocities. Coulomb friction is given by F = Fcsgn(v) (4.1) where F is the friction force, Fc is the Coulomb friction coefficient, sgn(.) denotes the sign function, and v is the relative velocity between contacting surfaces. 65 4.1. STATIC MODELS 0 0 Coulomb Friction Coefficient Fc Friction F Velocity v Figure 4.1: Static Friction Model with Coulomb Friction Effect Morin (1833) [26] introduced the idea of static friction in which the friction forces opposes the direction of motion when the relative velocity is zero. The equation describing the model is given by F = Fssgn(Ft) (4.2) 



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