Mechanical Engineering SeriesFrederick F. LingEditor-in-Chief

Mechanical Engineering Series

Wodek Gawronski, Modeling and Control of Antennas and Telescopes

Makoto Ohsaki and Kiyohiro Ikeda, Stability and Optimization of Structures: GeneralizedSensitivity Analysis

A.C. Fischer-Cripps, Introduction to Contact Mechanics, 2nd ed.

W. Cheng and I. Finnie, Residual Stress Measurement and the Slitting Method

J. Angeles, Fundamentals of Robotic Mechanical Systems: Theory Methods and Algorithms,3rd ed.

J. Angeles, Fundamentals of Robotic Mechanical Systems:Theory, Methods, and Algorithms, 2nd ed.

P. Basu, C. Kefa, and L. Jestin, Boilers and Burners: Design and Theory

J.M. Berthelot, Composite Materials:Mechanical Behavior and Structural Analysis

I.J. Busch-Vishniac, Electromechanical Sensors and Actuators

J. Chakrabarty, Applied Plasticity

K.K. Choi and N.H. Kim, Structural Sensitivity Analysis and Optimization 1: Linear Systems

K.K. Choi and N.H. Kim, Structural Sensitivity Analysis and Optimization 2: NonlinearSystems and Applications

G. Chryssolouris, Laser Machining: Theory and Practice

V.N. Constantinescu, Laminar Viscous Flow

G.A. Costello, Theory of Wire Rope, 2nd ed.

K. Czolczynski, Rotordynamics of Gas-Lubricated Journal Bearing Systems

M.S. Darlow, Balancing of High-Speed Machinery

W. R. DeVries, Analysis of Material Removal Processes

J.F. Doyle, Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability

J.F. Doyle, Wave Propagation in Structures:Spectral Analysis Using Fast Discrete Fourier Transforms, 2nd ed.

P.A. Engel, Structural Analysis of Printed Circuit Board Systems

A.C. Fischer-Cripps, Introduction to Contact Mechanics

A.C. Fischer-Cripps, Nanoindentation, 2nd ed.

J. Garcıa de Jalon and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems:The Real-Time Challenge

W.K. Gawronski, Advanced Structural Dynamics and Active Control of Structures

W.K. Gawronski, Dynamics and Control of Structures: A Modal Approach

G. Genta, Dynamics of Rotating Systems

D. Gross and T. Seelig, Fracture Mechanics with Introduction to Micro-mechanics

K.C. Gupta, Mechanics and Control of Robots

R. A. Howland, Intermediate Dynamics: A Linear Algebraic Approach

D. G. Hull, Optimal Control Theory for Applications

J. Ida and J.P.A. Bastos, Electromagnetics and Calculations of Fields

M. Kaviany, Principles of Convective Heat Transfer, 2nd ed.

M. Kaviany, Principles of Heat Transfer in Porous Media, 2nd ed.

(continued after index)

Wodek Gawronski

Modeling and Controlof Antennas and Telescopes

123

Wodek GawronskiCalifornia Institute of TechnologyJet Propulsion LaboratoryPasedena, CA, USAwodek.k.gawronski@jpl.nasa.gov

ISSN: 0941-5122ISBN: 978-0-387-78792-3 e-ISBN: 978-0-387-78793-0DOI: 10.1007/978-0-387-78793-0

Library of Congress Control Number: 2008922734

c© 2008 Springer Science+Business Media, LLCAll rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even ifthey are not identified as such, is not to be taken as an expression of opinion as to whether or notthey are subject to proprietary rights.

Printed on acid-free paper

springer.com

Mechanical Engineering Series

Frederick F. LingEditor-in-Chief

The Mechanical Engineering Series features graduate texts and research monographsto address the need for information in contemporary mechanical engineering, in-cluding areas of concentration of applied mechanics, biomechanics, computationalmechanics, dynamical systems and control, energetics, mechanics of materials, pro-cessing, production systems, thermal science, and tribology.

Advisory Board/Series Editors

Applied Mechanics F.A. LeckieUniversity of California,Santa Barbara

D. GrossTechnical University of Darmstadt

Biomechanics V.C. MowColumbia University

Computational Mechanics H.T. YangUniversity of California,Santa Barbara

Dynamic Systems and Control/ D. BryantMechatronics University of Texas at Austin

Energetics J.R. WeltyUniversity of Oregon, Eugene

Mechanics of Materials I. FinnieUniversity of California, Berkeley

Processing K.K. WangCornell University

Production Systems G.-A. KlutkeTexas A&M University

Thermal Science A.E. BerglesRensselaer Polytechnic Institute

Tribology W.O. WinerGeorgia Institute of Technology

Series Preface

Mechanical engineering, and engineering discipline born of the needs of theindustrial revolution, is once again asked to do its substantial share in the callfor industrial renewal. The general call is urgent as we face profound issues ofproductivity and competitiveness that require engineering solutions, among others.The Mechanical Engineering Series is a series featuring graduate texts and researchmonographs intended to address the need for information in contemporary areas ofmechanical engineering.

The series is conceived as a comprehensive one that covers a broad range of con-centrations important to mechanical engineering graduate education and research.We are fortunate to have a distinguished roster of series editors, each an expert inone of the areas of concentration. The names of the series editors are listed on pagevi of this volume. The areas of concentration are applied mechanics, biomechanics,computational mechanics, dynamic systems and control, energetics, mechanics ofmaterials, processing, thermal science, and tribology.

Preface

This book is based on my experience with the control systems of antennas andradiotelescopes. Overwhelmingly, it is based on experience with the NASA DeepSpace Network (DSN) antennas. It includes modeling the antennas, developingcontrol algorithms, field testing, system identification, performance evaluation, andtroubleshooting. My previous book1 emphasized the theoretical aspects of antennacontrol engineering, while this one describes the application part of the antennacontrol engineering.

Recently, the increased requirements for antenna/telescope pointing accuracyhave been imposed. In the case of DSN antennas, it was the shift from the S-band(4 GHz) and X-band (8 GHz) communication to Ka band (32 GHz). On the otherhand, the Large Millimeter Telescope will operate up to 200 GHz, which requiresextremely accurate pointing. These requirements bring new challenges to the an-tenna control engineers. Classical PI controllers cannot assure the required accuracy,while model-based controllers (LQG, and H∞) increase the antenna accuracy by afactor of 10. These controllers are new for antenna/telescope engineering. This bookdescribes their development and application.

The book is addressed primarily to the antenna, telescope, and radiotelescopeengineers, engineers involved in motion control, as well as students and researchesin motion control and mechatronics (antenna is a combination of mechanical, power,and electronics subsystems). The book consists of two parts: Modeling (Chapters2–5) and Control (Chapters 6–13).

In the modeling part, Chapter 2 describes the development of the analyti-cal model of an antenna, including its structure (finite-element model) and drives(motors, reducers, amplifiers). This modeling is useful in the design stage of an an-tenna. Chapter 3 describes the determination of the antenna model using field testsand the system identification approach. These models are quite accurate, and areused in the development of the model-based controllers. Because the order of mod-els (analytical as well as from the identification) is often too high for the analysis and

1 Gawronski W. (2004). Advanced Structural Dynamics, and Active Control of Structures.Springer, New York.

vii

viii Preface

for controller implementation, it is reduced to a reasonable size while preserving thecritical properties of the full model. Chapter 4 briefly describes the reduction tech-niques as applied to antennas and telescopes. Finally, the wind disturbance modelsare developed in Chapter 5. Wind is the major disturbance source for antennas andtelescopes. The steady (or static) wind model is presented, based on wind tunneldata and confirmed by the field data. Also, three wind gusts models are presented.

The control part presents the performance criteria and shows how to transformthe antenna model to be the most suitable for controller tuning. In Chapter 7, thebook presents the development, properties, and limitations of the PI controller. Itshows the impact of the proportional and the integral gains on the antenna closed-loop performance. It also analyzes the limits of the PI controller performance.Chapter 8 describes the tuning process of the LQG controller. It analyzes the per-formance of the LQG controller, including its limits. It presents the graphical userinterface (GUI) that allows us to tune the antenna LQG controller without analysis,but by playing with the GUI sliders and buttons (“LQG for dummies”). It is shownin this chapter that the performance of the LQG controller depends on its location:either at the position loop or at the velocity loop, or at both. It shows also that in thecase of the 34-m DSN antennas the servo error with LQG controller decreased by afactor of 6.5 when compared to the PI controller. In Chapter 9, H∞ controller tun-ing is presented: gain determination, closed-loop equations, limits of performance.In the following chapter, the non-linear control issues are addressed. The velocityand acceleration limits often interfere with antenna dynamics. Two solutions areproposed: a command preprocessor, and the anti-windup technique. Friction is thesource of a deteriorated pointing. Dither can be a solution, as presented in this book.Finally, gearbox backlash is described and the counter-torque solution to minimizeit is presented.

A typical antenna control system consists of two loops: velocity and positionloops. A single-loop solution is studied in Chapter 11. The antenna control systemuses the encoder measurements to estimate RF beam position, but it is only a roughestimate. The encoder measures the antenna angular position at its location, whichis different from the RF beam location; thus, antenna structural compliance is thesource of the error. Chapter 12 describes two techniques to control the RF beamposition: monopulse and conscan. Finally, in Chapter 13 we describe an open-loopRF beam control: the look-up table that corrects for the uneven azimuth track thatimpacts the RF beam position.

Analysis is a skill. However, even complex theories and the supporting analysisare, in a sense, simple because they assume certain properties that simplify the anal-ysis, or to make it possible. Still, although the theoretical path delivers an answer,engineers have to ask if the answer is applicable to the real environment.

Engineering is an art. Some aspects of engineering can be described rigorouslyby theoretical analysis, but not all. There are cases where engineering reality doesnot satisfy analytical assumptions. Hence, engineering solutions are often ad hocsolutions. This engineering dilemma is summarized by Scholnik: “Who cares howit works, just as long as it gives the right answer.”

Preface ix

The art of engineering often includes features that theoretically cannot work. Forexample:

� LQG controllers cannot be applied to antennas, because the antenna model havepoles at zero (rigid-body mode). But the LQG controllers do control the antennas.

� The noise in the tuning of the LQG controller should be the Gaussian noise,which is not the case of antennas or telescopes.

� Rigidly applied model reduction algorithms do not produce the best reducedmodel (best in terms of the antenna performance).

� Every LQG controller is an optimal controller, but not every one is acceptable.� In the H∞ controller tuning the plant uncertainity should be either additive or

multiplicative. The antenna uncertainity, which depends on its elevation position,is neither additive nor multiplicative.

Despite these difficulties, engineers succeed in developing acceptable controlsystems, including antennas and telescopes.

Readers who would like to contact me with comments and questions are invitedto do so. My e-mail address is wodek.k.gawronski@jpl.nasa.gov or w.gawronski@sbcglobal.net.

W. GawronskiPasedena, California

Acknowledgements

Part of the work described in this book was carried out at the Jet Propulsion Labora-tory, California Institute of Technology, under contract with the National Aeronau-tics and Space Administration. I thank Alaudin Bhanji, Mark Gatti, Pete Hames,Wendy Hodgin, Andre Jongeling, Scott Morgan, Jean Patterson, Daniel Rascoe,Christopher Yung, and Susan Zia, the managers at the Communications GroundSystems Section, Jet Propulsion Laboratory, for their support of the Deep SpaceNetwork antenna study.

I would like to acknowledge the contributions of my colleagues who have hadan influence on this work:

� from the Jet Propulsion Laboratory: Harlow Ahlstrom, Mimi Aung, FarrokhBaher, Abner Bernardo, John Cucchissi, Jeffery Mellstrom, Martha Strain;

� from the NASA Goldstone Deep Space Communication Complex (California):Michael Winders and Jeffrey Frazier;

� from the NASA Madrid Deep Space Communication Complex (Spain): AngelMartin, David Munoz Mochon, and Pablo Perez-Zapardiel;

� from the NASA Canberra Deep Space Communication Complex (Australia):Paul Richter and John Howell;

� visiting students: Jason Brand, Emily Craparo, Brandon Kuczenski, and ErinManeri whose work is also included in this book.

x Preface

I also acknowledge the help of Kamal Souccar in the study of the Large Milli-meter Telescope; Bogusz Bienkiewicz of Colorado State University in the wind dis-turbance study; and Toomas Erm of the European Southern Observatory, Munich,Germany, for many interesting discussions on control systems of antennas andtelescopes.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Examples of Antennas and Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 NASA Deep Space Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Large Millimeter Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 ESA Deep Space Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.4 Atacama Large Millimeter Array . . . . . . . . . . . . . . . . . . . . . . . 21.1.5 Thirty Meter Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.6 Green Bank Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.7 Effelsberg Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Short Description of the Antenna Control System . . . . . . . . . . . . . . . . 51.2.1 Velocity Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2.2 Position-Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Antenna and Telescope Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Part I Modeling

2 Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Rigid Antenna Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Finite-Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Modal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.3 State-Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.4 Models with Rigid Body Modes . . . . . . . . . . . . . . . . . . . . . . . . 192.2.5 Discrete-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Drive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.1 Motor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2 Reducer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.3 Drive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Velocity Loop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Drive Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.1 Drive Stiffness Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

xi

xii Contents

2.5.2 Drive Inertia Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Models from Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1 White Noise Testing of the Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Purpose and Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.2 Test Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.3 Test Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.4 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.5 Basic Relationships for the Discrete-Time Data . . . . . . . . . . . 37

3.2 Identification of the Velocity Loop Model . . . . . . . . . . . . . . . . . . . . . . . 383.2.1 Description of the Velocity Loop Model . . . . . . . . . . . . . . . . . 393.2.2 Identification of the Velocity Loop Model . . . . . . . . . . . . . . . . 393.2.3 A Comparison of the Analytical and Identified Models . . . . . 413.2.4 Azimuth Model Depends on the Antenna

Elevation Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.5 Fundamental Frequency Depends on Antenna Diameter . . . . 43

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1 Why Reduction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Balanced Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 Modal Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.1 Norms of a Single Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.2 Norms of a Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Antenna Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Wind Disturbance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.1 Steady-State Wind Disturbance Model . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.1.1 Dimensionless Wind Torques . . . . . . . . . . . . . . . . . . . . . . . . . . 525.1.2 Obtaining Wind Torques from Field Data . . . . . . . . . . . . . . . . 545.1.3 Comparing Wind Tunnel Results and the Field Data . . . . . . . 56

5.2 Wind Gusts Disturbance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2.1 Model of Wind Forces Acting on the Dish . . . . . . . . . . . . . . . 605.2.2 Model of Wind Torque Acting at the Drives . . . . . . . . . . . . . . 645.2.3 Algorithm to Generate a Time Profile of Wind

Gusts Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2.4 Model of Wind at the Velocity Input . . . . . . . . . . . . . . . . . . . . 665.2.5 Algorithm to Generate Time Profile of Wind

at the Velocity Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2.6 The Equivalence of Wind Torque and Wind

Velocity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2.7 Closed Loop Pointing Accuracy with Wind Gusts

Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Contents xiii

Part II Control

6 Preliminaries to Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.1 Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2 Transformations of the Velocity Loop Model . . . . . . . . . . . . . . . . . . . . 77

6.2.1 Transformation into Modal Coordinates . . . . . . . . . . . . . . . . . 786.2.2 Antenna Position as the First State . . . . . . . . . . . . . . . . . . . . . . 786.2.3 Augmentation with the Integral of the Position . . . . . . . . . . . . 78

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7 PI and Feedforward Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.1 Properties of the PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.1.1 Closed Loop Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . 837.1.2 The Proportional Gain Analysis . . . . . . . . . . . . . . . . . . . . . . . . 837.1.3 The Integral Gain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.2 PI Controller Tuning Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867.3 Closed Loop Equations of a Flexible Antenna

with a PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.4 Performance of the PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7.4.1 Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.4.2 Limits of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.5 Feedforward Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

8 LQG Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.1 Properties of the LQG Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

8.1.1 LQG Controller Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 958.1.2 Tracking LQG Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988.1.3 Closed Loop Equations of the Tracking LQG

Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018.1.4 LQG Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018.1.5 Resemblance of the LQG and PI Controllers . . . . . . . . . . . . . . 1048.1.6 Properties of the LQG Weights . . . . . . . . . . . . . . . . . . . . . . . . . 1058.1.7 Limits of the LQG Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

8.2 LQG Controller Tuning Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088.3 Performance of the LQG Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.3.1 Summary of the Antenna Servo PerformanceCharacteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

8.3.2 Performance of the DSN Antennaswith LQG Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

8.3.3 Disturbance Rejection Properties and the Position-LoopBandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

8.3.4 Performance Comparison of the PI and LQG Controllers . . . 1158.3.5 Limits of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

xiv Contents

8.4 Tuning a LQG Controller Using GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.4.1 Selecting LQG Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178.4.2 GUI for the LQG Controller Tuning . . . . . . . . . . . . . . . . . . . . . 1198.4.3 Fine Tuning of the LQG Controller . . . . . . . . . . . . . . . . . . . . . 121

8.5 LQG Controller in the Velocity Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.5.1 Position Loop Bandwidth Depends on the Velocity Loop . . . 1248.5.2 Four Control Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278.5.3 PP Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278.5.4 PL Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298.5.5 LP Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1328.5.6 LL Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9 H∞ Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359.1 Definition and Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359.2 Tracking H∞ Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389.3 Closed-Loop Equations of the Tracking H∞ Controller . . . . . . . . . . . . 1389.4 34-M Antenna Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399.5 Limits of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

10 Single Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14510.1 Rigid Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

10.1.1 Rigid Antenna with Velocity and Position Loops . . . . . . . . . . 14510.1.2 Rigid Antenna with Position Loop Only . . . . . . . . . . . . . . . . . 14710.1.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

10.2 34-M Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

11 Non-Linear Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15711.1 Velocity and Acceleration Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

11.1.1 Command Preprocessor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15711.1.2 Anti-Windup Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

11.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16611.2.1 Dry Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16811.2.2 Low-Velocity Tracking Using Dither . . . . . . . . . . . . . . . . . . . . 17011.2.3 Non-linear Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 173

11.3 Backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17411.3.1 Backlash and Its Prevention . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17511.3.2 The Velocity Loop Model with Friction and Backlash . . . . . . 177

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

12 RF Beam Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18312.1 Selecting the RF Beam Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18312.2 Monopulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Contents xv

12.2.1 Command following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18712.2.2 Disturbance Rejection Properties . . . . . . . . . . . . . . . . . . . . . . . 18812.2.3 Stability Due to the Gain Variation . . . . . . . . . . . . . . . . . . . . . . 19012.2.4 Performance Simulations: Linear Model . . . . . . . . . . . . . . . . . 19012.2.5 Performance Simulations: Nonlinear Model . . . . . . . . . . . . . . 191

12.3 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19412.3.1 Conical Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19412.3.2 Sliding Window Conscan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20012.3.3 Lissajous Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20112.3.4 Rosette Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20412.3.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

13 Track-Level Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21113.1 Description of the Track-Level Problem . . . . . . . . . . . . . . . . . . . . . . . . 21113.2 Collection and Processing of the Inclinometer Data . . . . . . . . . . . . . . . 21313.3 Estimating Azimuth Axis Tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21513.4 Creating the TLC Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21713.5 Determining Pointing Errors from the TLC Table . . . . . . . . . . . . . . . . . 21913.6 Antenna Pointing Improvement Using the TLC Table . . . . . . . . . . . . . 221References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Chapter 1Introduction

This chapter presents examples of antennas and telescopes, shortly describes theantenna control system, and presents references on the antenna mechanical andcontrol engineering.

1.1 Examples of Antennas and Telescopes

1.1.1 NASA Deep Space Network

The NASA Deep Space Network (DSN) antennas communicate with spacecraft bysending commands (uplink) and by receiving information from spacecraft (down-link). To assure continuous tracking during Earth’s rotation, the antennas are locatedat three sites: Goldstone (California), Madrid (Spain), and Canberra (Australia). Thesignal frequencies are 8.5 GHz (X-band) and 32 GHz (Ka-band). The dish size ofthe antennas is either 34 m or 70 m. An example of the 70-m antenna is shown inFig. 1.1. The antenna dish rotates with respect to the horizontal (or elevation) axis.The whole antenna structure rotates on a circular track (azimuth track) with respectto the vertical (or azimuth) axis. For the Ka-band frequency the required trackingaccuracy is on the order of 1 mdeg. This requirement is a driver for the controlsystem upgrade of the antennas. In [2], [3], and [4] you can find the descriptionof the DSN antenna control systems, and at the webpage http://ipnpr.jpl.nasa.gov/index.cfm the DSN antennas research reports, including control systems. The DeepSpace Network webpage is at the webpage deepspace.jpl.nasa.gov/dsn/.

1.1.2 Large Millimeter Telescope

The Large Millimeter Telescope (LMT) is the joint effort of the University ofMassachusetts at Amherst and the Instituto Nacional de Astrofısica, Optica, yElectronica (INAOE) in Mexico (Fig. 1.2). The LMT is a 50-m diameter telescope,designed for operation at wavelengths between 1 mm and 4 mm. The telescoperotates with respect to elevation and azimuth axes. It is built atop of Sierra Negra

W. Gawronski, Modeling and Control of Antennas and Telescopes,DOI: 10.1007/978-0-387-78793-0 1, C© Springer Science+Business Media, LLC 2008

1

2 1 Introduction

Fig. 1.1 NASA/JPL 70-mantenna at Goldstone, CA(courtesy ofNASA/JPL/Caltech)

(4640 m), a volcanic peak in the state of Puebla, Mexico. The LMT is a significantstep forward in antenna design: in order to reach its pointing accuracy specifica-tions, it must outperform every other telescope in its frequency range. The antennadesigners expect that the telescope will point to its specified accuracy of 0.3 mdegunder conditions of low winds and stable temperatures, with radiofrequency up to110 GHz. More about LMT, see www.lmtgtm.org/.

1.1.3 ESA Deep Space Antennas

For use in deep space, high elliptical orbit missions, and future missions to Mars,the European Space Agency (ESA) erected 35-m deep space ground stations; seehttp://www.esa.int/SPECIALS/ESOC/SEMZEEW4QWD 0.html. The antennas aredesigned for frequencies up to 35 GHz and a pointing accuracy of 6 mdeg. The firstantenna (that moves in azimuth and elevation axes) has been installed in Australiaand has proven its compliance to the specifications. The second antenna is underconstruction in Spain. The 35 m antenna incorporates a full motion pedestal with abeam waveguide system.

1.1.4 Atacama Large Millimeter Array

The Atacama Large Millimeter Array (ALMA) is an international astronomy facility.A brief description can be found at www.alma.nrao.edu/info. The pointing accuracyof the 12 m ALMA telescope is 0.16 mdeg. The control system must handle very

1.1 Examples of Antennas and Telescopes 3

Fig. 1.2 Large MillimeterTelescope (courtesy of theLarge Millimeter TelescopeProject)

accurate movement at sidereal tracking velocities as well as several extremely fastswitching functions. To do this, the drives are designed to accelerate up to 24 deg/s2,which is very unusual for a telescope of this size.

1.1.5 Thirty Meter Telescope

The Thirty Meter Telescope (TMT) will be the first of the giant optical/infraredground-based telescopes (of 30-m diameter of primary mirror) addressing one ofthe most compelling areas in astrophysics: the nature of dark matter, the assemblyof galaxies, the growth of structure in universe, and the physical processes involvedin star and planet formation. TMT will operate over 0.3–30 �m wavelength range,providing nine times the collecting area of the current largest optical telescope, the10-m Keck telescope. It will use adaptive optics system to allow diffraction-limitedperformance, resulting in spatial resolution 12.5 times sharper than is achievedby the Hubble Space Telescope. For more about TMT see http://www.tmt.org/ orhttp://www.astro.caltech.edu/observatories/tmt/.

4 1 Introduction

1.1.6 Green Bank Telescope

The National Radio Astronomy Observatory operates the Green Bank Telescope(GBT), the world’s largest (along with the Effelsberg telescope) single aperturetelescope (Fig. 1.3). It is located in Green Bank, West Virginia. The GBT has anunusual design. Unlike conventional telescopes, the GBT’s aperture is unblocked(using an off-axis feed arm) so that incoming radiation meets the surface directly.This increases the useful area of the telescope and eliminates reflection and diffrac-tion that ordinarily complicate a telescope’s pattern of response. More informationcan be found at http://www.gb.nrao.edu/.

1.1.7 Effelsberg Telescope

The Effelsberg 100-m telescope is one of the world’s largest fully steerable tele-scopes (Fig. 1.4). It is operated by the Max Planck Institute for Radio Astronomyin Bonn, Germany, at wavelengths from about 3.5 mm to 90 cm. The telescope isused to observe pulsars, cold gas and dust clusters, the sites of star formation, jetsof matter emitted by black holes, and the nuclei of distant far-off galaxies. Its steelconstruction weighs 3,200 tons. Its azimuth velocity is 0.5 deg/s, and 0.25 deg/s in

Fig. 1.3 The Green Bank Telescope (courtesy of National Radio Astronomy Observatory)

1.2 Short Description of the Antenna Control System 5

Fig. 1.4 The Effelsbergtelescope (courtesy of MaxPlanck Institute for RadioAstronomy, Bonn, Germany)

elevation. More information can be found at http://www.mpifr-bonn.mpg.de/public/index e.html.

1.2 Short Description of the Antenna Control System

A Deep Space Network antenna with a 34-m dish is shown in Fig. 1.5. This antennacan rotate around azimuth (vertical) and elevation (horizontal) axes. The rotationis controlled by azimuth and elevation controllers. The combination of the antennastructure and its azimuth and elevation drives makes the velocity loop of the antenna.The velocity loop plant has two inputs (azimuth and elevation velocities) and twooutputs (azimuth and elevation position), and the position loop is closed betweenthe encoder outputs and the velocity inputs. The drives consist of gearboxes, electricmotors, amplifiers, and tachometers.

The antenna controller consists of two independent subsystems: azimuth andelevation controllers. Because both subsystems are independent, a single system isconsidered further, and azimuth or elevation system is specified, if necessary. Also,

6 1 Introduction

Fig. 1.5 The Deep SpaceNetwork antenna atGoldstone, California(courtesy ofNASA/JPL/Caltech). It canrotate with respect to theazimuth (vertical axis) andwith respect to elevation(horizontal axis)

the antenna control system consists of the open loop (or velocity loop) system andthe closed loop (or position loop) system.

1.2.1 Velocity Loop

The velocity loop includes the structure and the drives (motors, gearboxes andamplifiers). It is driven by the velocity input signal uc (deg/s), see Fig. 1.6, while theencoder reading y (deg) is the output. The elevation and azimuth loops are similar.The velocity-loop allows for the manual control of antenna.

encoder, y

tach velocity

uc

–

+ velocitycontroller

V

ANTENNA

Fig. 1.6 The velocity loop model of the Deep Space Network antenna

1.3 Antenna and Telescope Literature 7

command, r encoder, y

VELOCITY LOOP

uc

tach velocity

V

encoder

velocitycontroller

positioncontroller

Fig. 1.7 The velocity- and position-loops of the Deep Space Network antenna

1.2.2 Position-Loop

The block diagram of the closed loop (or position loop) system is shown in Fig. 1.7.It consists of the antenna velocity loop and the position controller. The controlleris run by computer software that drives the antenna depending on the actualantenna position and the commanded position. The controller has two inputs: theencoder position y, and the commanded position (or shorter: command) r (deg). Thecontroller output is the velocity uc (deg/s) that drives the antenna.

1.3 Antenna and Telescope Literature

This book is based predominantly on the author’s experience with the NASA DeepSpace Network, although the experience of others is included in the references.Obviously, there are too many references to be included here, but readers inter-ested in further study of antenna and telescope control and pointing issues arereferred to:

• the IEEE Antennas and Propagation Magazine,• the Proceedings of SPIE (the International Society for Optical Engineering),

Optical Engineering (SPIE Journal),• the Interplanetary Network Progress Report (JPL), http://ipnpr.jpl.nasa.gov/

index.cfm

There are three books that address antenna pointing and control issues, see Refs.[1], [7], and [8], as well as this author’s book [3], which addresses general issues ofstructural dynamics and control, but includes many examples of antenna dynamicsand antenna control systems. Karcher [6] gives an interesting overview of the tele-scope structure, mechanism, and control system design. The handbook [10] presentsthe control design of a simple (rigid) antenna. General antenna theory is presentedin a comprehensible form in [9] and [11].

8 1 Introduction

References1. Biernson G. (1990). Optimal Radar Tracking Systems. Wiley, New York.2. Gawronski W. (2001). Antenna control systems: from PI to H∞. IEEE Antennas and Propa-

gation Magazine, 43 (1).3. Gawronski W. (2004). Advanced Structural Dynamics and Active Control of Structures.

Springer, New York.4. Gawronski W. (2007). Control and pointing challenges of large antennas and telescopes. IEEE

Trans. Control Systems Technology, 15, (2).5. Gawronski W, Mellstrom JA. (1994). Control and dynamics of the Deep Space Network

antennas. In: Control and Dynamic Systems, vol. 63, C.T. Leondes, ed. Academic Press, SanDiego.

6. Karcher HJ. (2006). Telescopes as mechatronic systems. IEEE Antennas and PropagationMagazine 48 (2):17–37.

7. Kitsuregawa T. (1990). Advanced Technology in Satellite Communication Antennas: Elec-trical and Mechanical Design. Artech House, Boston.

8. Levy R (1996) Structural Engineering of Microwave Antennas. IEEE Press, New York.9. Macnamara T. (1995). Handbook of Antennas for EMC, Artech House, Boston.

10. Nise NS (1995) Control System Engineering. The Benjamin/Cummings Publishing Company,Redwood City, CA.

11. Toomay JC. (1989). Radar Principles for the Non-Specialist. Van Nostrand Reinhold,New York.

Part IModeling