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Distributed Control of a Segmented Telescope Mirror

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Distributed Control of a Segmented Telescope Mirror Distributed Control of a Segmented Telescope Mirror by Dan Kerley B.Eng., University of Victoria, 2004 A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF APPLIED SCIENCE in the Department of Mechanical Engineering Dan Kerley, 2010 University of Victoria All rights reserved. This thesis may not be produced in whole or in part, by photocopy or other means, without the permission of the author ii Distributed Control of a Segmented Telescope Mirror by Dan Kerley B.Eng., University of Victoria, 2004 Supervisory Committee Dr. Edward Park (Department of Mechanical Engineering) Supervisor Dr. Afzal Suleman (Department of Mechanical Engineering) Department Member Dr. Panajotis Agathoklis (Department of Electrical & Computer Engineering) Outside Member Ms. Jennifer Dunn (Herzberg Institute of Astrophysics) Additional Member iii SUPERVISORY COMMITTEE Dr. Edward Park (Department of Mechanical Engineering) Supervisor Dr. Afzal Suleman (Department of Mechanical Engineering) Department Member Dr. Panajotis Agathoklis (Department of Electrical & Computer Engineering) Non-Department Member Ms. Jennifer Dunn (Herzberg Institute of Astrophysics) Additional Member ABSTRACT As astronomers continue to examine fainter objects and farther back in time, they require increasingly large telescopes due to the fundamental diffraction of optical elements. Therefore several of the next generation optical telescopes will employ extremely large primary mirrors. However to realistically construct mirrors of these magnitudes they will need to be assembled as a collection of many smaller mirrors. This mirror segmentation leads to the additional challenge of aligning the smaller mirror elements with respect to one another, and maintain that alignment in the presence of disturbances on the optical surface and its supporting structure. To achieve this alignment and disturbance rejection, a complex active control system will be required. iv There are several possible solutions to the control problem ranging from fully decentralized control to a global control scheme. However since many of these segmented mirrors will be comprised of hundreds of mirror elements a global control scheme quickly becomes an intractable solution. On the other extreme, a highly scalable decentralized scheme is realizable, however, would lack any global sense of the system. Therefore an appealing solution is a scalable distributed network of controllers, where individual controllers ‘act locally’ yet ‘think globally’. This is achieved by coupling adjacent controller to one another, forming a lattice across the spatial extents of the system. v TABLE OF CONTENTS SUPERVISORY COMMITTEE .................................................................... iii ABSTRACT ................................................................................................... iii TABLE OF CONTENTS ................................................................................ v LIST OF FIGURES ...................................................................................... viii LIST OF TABLES ......................................................................................... xii NOMENCLATURE ..................................................................................... xiii ACKNOWLEDGEMENT ............................................................................ xiv DEDICATION ................................................................................................ ix CHAPTER 1 .................................................................................................... 1 INTRODUCTION ........................................................................................... 1 1.1 Motivation .................................................................................................................. 1 1.2 Project Background .................................................................................................... 4 1.3 Research Objectives ................................................................................................... 7 1.4 Scope of Thesis .......................................................................................................... 9 CHAPTER 2 .................................................................................................. 11 SEGMENTED MIRROR GEOMETRIC LAYOUT .................................... 11 2.1 Chapter Overview .................................................................................................... 11 2.2 Segment Layout ....................................................................................................... 12 2.3 Support Structure Layout ......................................................................................... 14 2.4 Actuator Layout ....................................................................................................... 17 2.4 Global Coordinate System ....................................................................................... 18 2.5 Sensor Layout .......................................................................................................... 19 CHAPTER 3 .................................................................................................. 21 SEGMENTED MIRROR MODELING ........................................................ 21 3.1 Chapter Overview .................................................................................................... 21 3.2 Finite Element Modeling ......................................................................................... 22 vi 3.2.1 Support Structure Modeling .............................................................................. 28 3.2.2 Segment Modeling ............................................................................................ 29 3.2.3 Actuator Modeling ............................................................................................ 31 3.3 System Plant Modeling ............................................................................................ 34 3.3.1 State Space Model Reduction ........................................................................... 41 3.4 Decentralized System Modeling .............................................................................. 44 3.5 Distributed System Modeling .................................................................................. 47 3.5.1 Distributed Unit Modeling of a Mirror Segment .............................................. 49 3.5.2 Distributed Model Reduction ............................................................................ 65 3.6 Edge Sensor System Modeling ................................................................................ 69 CHAPTER 4 .................................................................................................. 75 SEGMENTED MIRROR CONTROLLER DESIGN AND SYNTHESIS .. 75 4.1 Chapter Overview .................................................................................................... 75 4.2 H ∞ Controller Synthesis Overview .......................................................................... 76 4.3 Global H ∞ Control .................................................................................................... 80 4.4 Decentralized H ∞ Control ........................................................................................ 81 4.5 Distributed Control .................................................................................................. 82 4.6 Position Error Estimation ......................................................................................... 88 CHAPTER 5 .................................................................................................. 93 SEGMENTED MIRROR CONTROL SIMULATION ................................ 93 5.1 Chapter Overview .................................................................................................... 93 5.2 MATLAB/Simulink Simulation .............................................................................. 94 5.2.1 Disturbance Force Module ................................................................................ 95 5.2.2 Plant Module ..................................................................................................... 96 5.2.3 Sensor Module .................................................................................................. 96 5.2.4 Control System Module .................................................................................... 99 5.2.5 Output Data Module ......................................................................................... 99 5.2.6 Integral Module ............................................................................................... 100 5.3 Controller Synthesis ............................................................................................... 100 5.3.1 Global Controller ............................................................................................ 101 vii 5.3.2 Decentralized Controller ................................................................................. 103 5.3.1 Distributed Controller ..................................................................................... 105 5.4 System Analysis ..................................................................................................... 111 5.6 Closed Loop Simulation Results ............................................................................ 122 CHAPTER 6 ...............................................................................................
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