Modeling and Control of Antennas and Telescopes

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Mechanical Engineering Series Frederick F. Ling Editor-in-Chief

Mechanical Engineering Series Wodek Gawronski, Modeling and Control of Antennas and Telescopes Makoto Ohsaki and Kiyohiro Ikeda, Stability and Optimization of Structures: Generalized Sensitivity 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: Nonlinear Systems 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. Garc´ıa de Jal´on 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 Control of Antennas and Telescopes

123

Wodek Gawronski California Institute of Technology Jet Propulsion Laboratory Pasedena, CA, USA [email protected]

ISSN: 0941-5122 ISBN: 978-0-387-78792-3 e-ISBN: 978-0-387-78793-0 DOI: 10.1007/978-0-387-78793-0 Library of Congress Control Number: 2008922734 c 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission 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 in connection with any form of information storage and retrieval, electronic adaptation, computer software, 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 if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

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Mechanical Engineering Series Frederick F. Ling Editor-in-Chief

The Mechanical Engineering Series features graduate texts and research monographs to address the need for information in contemporary mechanical engineering, including areas of concentration of applied mechanics, biomechanics, computational mechanics, dynamical systems and control, energetics, mechanics of materials, processing, production systems, thermal science, and tribology.

Advisory Board/Series Editors Applied Mechanics

F.A. Leckie University of California, Santa Barbara D. Gross Technical University of Darmstadt

Biomechanics

V.C. Mow Columbia University

Computational Mechanics

H.T. Yang University of California, Santa Barbara

Dynamic Systems and Control/ Mechatronics

D. Bryant University of Texas at Austin

Energetics

J.R. Welty University of Oregon, Eugene

Mechanics of Materials

I. Finnie University of California, Berkeley

Processing

K.K. Wang Cornell University

Production Systems

G.-A. Klutke Texas A&M University

Thermal Science

A.E. Bergles Rensselaer Polytechnic Institute

Tribology

W.O. Winer Georgia Institute of Technology

Series Preface

Mechanical engineering, and engineering discipline born of the needs of the industrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of productivity and competitiveness that require engineering solutions, among others. The Mechanical Engineering Series is a series featuring graduate texts and research monographs intended to address the need for information in contemporary areas of mechanical engineering. The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and research. We are fortunate to have a distinguished roster of series editors, each an expert in one of the areas of concentration. The names of the series editors are listed on page vi of this volume. The areas of concentration are applied mechanics, biomechanics, computational mechanics, dynamic systems and control, energetics, mechanics of materials, processing, thermal science, and tribology.

Preface

This book is based on my experience with the control systems of antennas and radiotelescopes. Overwhelmingly, it is based on experience with the NASA Deep Space Network (DSN) antennas. It includes modeling the antennas, developing control algorithms, field testing, system identification, performance evaluation, and troubleshooting. My previous book1 emphasized the theoretical aspects of antenna control engineering, while this one describes the application part of the antenna control engineering. Recently, the increased requirements for antenna/telescope pointing accuracy have 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 other hand, the Large Millimeter Telescope will operate up to 200 GHz, which requires extremely accurate pointing. These requirements bring new challenges to the antenna control engineers. Classical PI controllers cannot assure the required accuracy, while model-based controllers (LQG, and H∞ ) increase the antenna accuracy by a factor of 10. These controllers are new for antenna/telescope engineering. This book describes their development and application. The book is addressed primarily to the antenna, telescope, and radiotelescope engineers, engineers involved in motion control, as well as students and researches in motion control and mechatronics (antenna is a combination of mechanical, power, and electronics subsystems). The book consists of two parts: Modeling (Chapters 2–5) and Control (Chapters 6–13). In the modeling part, Chapter 2 describes the development of the analytical 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 antenna. Chapter 3 describes the determination of the antenna model using field tests and the system identification approach. These models are quite accurate, and are used in the development of the model-based controllers. Because the order of models (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.

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for controller implementation, it is reduced to a reasonable size while preserving the critical properties of the full model. Chapter 4 briefly describes the reduction techniques as applied to antennas and telescopes. Finally, the wind disturbance models are developed in Chapter 5. Wind is the major disturbance source for antennas and telescopes. The steady (or static) wind model is presented, based on wind tunnel data and confirmed by the field data. Also, three wind gusts models are presented. The control part presents the performance criteria and shows how to transform the antenna model to be the most suitable for controller tuning. In Chapter 7, the book presents the development, properties, and limitations of the PI controller. It shows the impact of the proportional and the integral gains on the antenna closedloop performance. It also analyzes the limits of the PI controller performance. Chapter 8 describes the tuning process of the LQG controller. It analyzes the performance of the LQG controller, including its limits. It presents the graphical user interface (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 shown in 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 the case of the 34-m DSN antennas the servo error with LQG controller decreased by a factor of 6.5 when compared to the PI controller. In Chapter 9, H∞ controller tuning is presented: gain determination, closed-loop equations, limits of performance. In the following chapter, the non-linear control issues are addressed. The velocity and acceleration limits often interfere with antenna dynamics. Two solutions are proposed: a command preprocessor, and the anti-windup technique. Friction is the source 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 minimize it is presented. A typical antenna control system consists of two loops: velocity and position loops. A single-loop solution is studied in Chapter 11. The antenna control system uses the encoder measurements to estimate RF beam position, but it is only a rough estimate. The encoder measures the antenna angular position at its location, which is different from the RF beam location; thus, antenna structural compliance is the source of the error. Chapter 12 describes two techniques to control the RF beam position: monopulse and conscan. Finally, in Chapter 13 we describe an open-loop RF beam control: the look-up table that corrects for the uneven azimuth track that impacts the RF beam position. Analysis is a skill. However, even complex theories and the supporting analysis are, in a sense, simple because they assume certain properties that simplify the analysis, 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 rigorously by theoretical analysis, but not all. There are cases where engineering reality does not satisfy analytical assumptions. Hence, engineering solutions are often ad hoc solutions. This engineering dilemma is summarized by Scholnik: “Who cares how it works, just as long as it gives the right answer.”

Preface

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

r r r r r

LQG controllers cannot be applied to antennas, because the antenna model have poles 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 reduced model (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 control systems, including antennas and telescopes. Readers who would like to contact me with comments and questions are invited to do so. My e-mail address is [email protected] or w.gawronski@sbc global.net. W. Gawronski Pasedena, California

Acknowledgements Part of the work described in this book was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics 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 Ground Systems Section, Jet Propulsion Laboratory, for their support of the Deep Space Network antenna study. I would like to acknowledge the contributions of my colleagues who have had an influence on this work:

r r r r r

from the Jet Propulsion Laboratory: Harlow Ahlstrom, Mimi Aung, Farrokh Baher, 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): Angel Martin, 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 Erin Maneri whose work is also included in this book.

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

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Examples of Antennas and Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 NASA Deep Space Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Large Millimeter Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 ESA Deep Space Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Atacama Large Millimeter Array . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Thirty Meter Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6 Green Bank Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.7 Effelsberg Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Short Description of the Antenna Control System . . . . . . . . . . . . . . . . 1.2.1 Velocity Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Position-Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Antenna and Telescope Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 2 2 3 4 4 5 6 7 7 8

Part I Modeling 2 Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Rigid Antenna Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Structural Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Finite-Element Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Modal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 State-Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Models with Rigid Body Modes . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Discrete-Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Drive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Motor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Reducer Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Drive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Velocity Loop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Drive Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Drive Stiffness Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 11 12 12 14 17 19 21 24 24 25 25 26 26 27 xi

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2.5.2 Drive Inertia Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Models from Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 White Noise Testing of the Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Purpose and Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Test Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Test Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Data Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Basic Relationships for the Discrete-Time Data . . . . . . . . . . . 3.2 Identification of the Velocity Loop Model . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Description of the Velocity Loop Model . . . . . . . . . . . . . . . . . 3.2.2 Identification of the Velocity Loop Model . . . . . . . . . . . . . . . . 3.2.3 A Comparison of the Analytical and Identified Models . . . . . 3.2.4 Azimuth Model Depends on the Antenna Elevation Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Fundamental Frequency Depends on Antenna Diameter . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 31 32 32 34 37 38 39 39 41

4 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Why Reduction? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Balanced Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Modal Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Norms of a Single Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Norms of a Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Antenna Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 45 47 47 48 49 50

5 Wind Disturbance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Steady-State Wind Disturbance Model . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Dimensionless Wind Torques . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Obtaining Wind Torques from Field Data . . . . . . . . . . . . . . . . 5.1.3 Comparing Wind Tunnel Results and the Field Data . . . . . . . 5.2 Wind Gusts Disturbance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Model of Wind Forces Acting on the Dish . . . . . . . . . . . . . . . 5.2.2 Model of Wind Torque Acting at the Drives . . . . . . . . . . . . . . 5.2.3 Algorithm to Generate a Time Profile of Wind Gusts Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Model of Wind at the Velocity Input . . . . . . . . . . . . . . . . . . . . 5.2.5 Algorithm to Generate Time Profile of Wind at the Velocity Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.6 The Equivalence of Wind Torque and Wind Velocity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.7 Closed Loop Pointing Accuracy with Wind Gusts Disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 54 56 59 60 64

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Part II Control 6 Preliminaries to Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Performance Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Transformations of the Velocity Loop Model . . . . . . . . . . . . . . . . . . . . 6.2.1 Transformation into Modal Coordinates . . . . . . . . . . . . . . . . . 6.2.2 Antenna Position as the First State . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Augmentation with the Integral of the Position . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

73 73 77 78 78 78 79

7 PI and Feedforward Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Properties of the PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Closed Loop Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 The Proportional Gain Analysis . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 The Integral Gain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 PI Controller Tuning Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Closed Loop Equations of a Flexible Antenna with a PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Performance of the PI Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Limits of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Feedforward Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 83 83 85 86 87 87 87 90 91 93

8 LQG Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.1 Properties of the LQG Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.1.1 LQG Controller Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.1.2 Tracking LQG Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.1.3 Closed Loop Equations of the Tracking LQG Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.1.4 LQG Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 8.1.5 Resemblance of the LQG and PI Controllers . . . . . . . . . . . . . . 104 8.1.6 Properties of the LQG Weights . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.1.7 Limits of the LQG Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 8.2 LQG Controller Tuning Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 8.3 Performance of the LQG Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8.3.1 Summary of the Antenna Servo Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8.3.2 Performance of the DSN Antennas with LQG Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 8.3.3 Disturbance Rejection Properties and the Position-Loop Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 8.3.4 Performance Comparison of the PI and LQG Controllers . . . 115 8.3.5 Limits of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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Contents

8.4 Tuning a LQG Controller Using GUI . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.4.1 Selecting LQG Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.4.2 GUI for the LQG Controller Tuning . . . . . . . . . . . . . . . . . . . . . 119 8.4.3 Fine Tuning of the LQG Controller . . . . . . . . . . . . . . . . . . . . . 121 8.5 LQG Controller in the Velocity Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.5.1 Position Loop Bandwidth Depends on the Velocity Loop . . . 124 8.5.2 Four Control Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.5.3 PP Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.5.4 PL Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.5.5 LP Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.5.6 LL Control System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 9 H∞ Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.1 Definition and Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 9.2 Tracking H∞ Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.3 Closed-Loop Equations of the Tracking H∞ Controller . . . . . . . . . . . . 138 9.4 34-M Antenna Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 9.5 Limits of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 10 Single Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 10.1 Rigid Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 10.1.1 Rigid Antenna with Velocity and Position Loops . . . . . . . . . . 145 10.1.2 Rigid Antenna with Position Loop Only . . . . . . . . . . . . . . . . . 147 10.1.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 10.2 34-M Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 11 Non-Linear Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 11.1 Velocity and Acceleration Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 11.1.1 Command Preprocessor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 11.1.2 Anti-Windup Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 11.2 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 11.2.1 Dry Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 11.2.2 Low-Velocity Tracking Using Dither . . . . . . . . . . . . . . . . . . . . 170 11.2.3 Non-linear Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 173 11.3 Backlash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 11.3.1 Backlash and Its Prevention . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 11.3.2 The Velocity Loop Model with Friction and Backlash . . . . . . 177 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 12 RF Beam Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 12.1 Selecting the RF Beam Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 12.2 Monopulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

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12.2.1 Command following . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 12.2.2 Disturbance Rejection Properties . . . . . . . . . . . . . . . . . . . . . . . 188 12.2.3 Stability Due to the Gain Variation . . . . . . . . . . . . . . . . . . . . . . 190 12.2.4 Performance Simulations: Linear Model . . . . . . . . . . . . . . . . . 190 12.2.5 Performance Simulations: Nonlinear Model . . . . . . . . . . . . . . 191 12.3 Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 12.3.1 Conical Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 12.3.2 Sliding Window Conscan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 12.3.3 Lissajous Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 12.3.4 Rosette Scan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 12.3.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 13 Track-Level Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 13.1 Description of the Track-Level Problem . . . . . . . . . . . . . . . . . . . . . . . . 211 13.2 Collection and Processing of the Inclinometer Data . . . . . . . . . . . . . . . 213 13.3 Estimating Azimuth Axis Tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 13.4 Creating the TLC Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 13.5 Determining Pointing Errors from the TLC Table . . . . . . . . . . . . . . . . . 219 13.6 Antenna Pointing Improvement Using the TLC Table . . . . . . . . . . . . . 221 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Chapter 1

Introduction

This chapter presents examples of antennas and telescopes, shortly describes the antenna control system, and presents references on the antenna mechanical and control 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 by sending commands (uplink) and by receiving information from spacecraft (downlink). To assure continuous tracking during Earth’s rotation, the antennas are located at three sites: Goldstone (California), Madrid (Spain), and Canberra (Australia). The signal frequencies are 8.5 GHz (X-band) and 32 GHz (Ka-band). The dish size of the antennas is either 34 m or 70 m. An example of the 70-m antenna is shown in Fig. 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 respect to the vertical (or azimuth) axis. For the Ka-band frequency the required tracking accuracy is on the order of 1 mdeg. This requirement is a driver for the control system upgrade of the antennas. In [2], [3], and [4] you can find the description of 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 Deep Space 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 of ´ Massachusetts at Amherst and the Instituto Nacional de Astrof´ısica, Optica, y Electr´onica (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 telescope rotates with respect to elevation and azimuth axes. It is built atop of Sierra Negra W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 1,

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1 Introduction

Fig. 1.1 NASA/JPL 70-m antenna at Goldstone, CA (courtesy of NASA/JPL/Caltech)

(4640 m), a volcanic peak in the state of Puebla, Mexico. The LMT is a significant step forward in antenna design: in order to reach its pointing accuracy specifications, it must outperform every other telescope in its frequency range. The antenna designers expect that the telescope will point to its specified accuracy of 0.3 mdeg under conditions of low winds and stable temperatures, with radiofrequency up to 110 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; see http://www.esa.int/SPECIALS/ESOC/SEMZEEW4QWD 0.html. The antennas are designed for frequencies up to 35 GHz and a pointing accuracy of 6 mdeg. The first antenna (that moves in azimuth and elevation axes) has been installed in Australia and has proven its compliance to the specifications. The second antenna is under construction in Spain. The 35 m antenna incorporates a full motion pedestal with a beam 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 accuracy of 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 Millimeter Telescope (courtesy of the Large Millimeter Telescope Project)

accurate movement at sidereal tracking velocities as well as several extremely fast switching 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/infrared ground-based telescopes (of 30-m diameter of primary mirror) addressing one of the most compelling areas in astrophysics: the nature of dark matter, the assembly of galaxies, the growth of structure in universe, and the physical processes involved in 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, the 10-m Keck telescope. It will use adaptive optics system to allow diffraction-limited performance, resulting in spatial resolution 12.5 times sharper than is achieved by the Hubble Space Telescope. For more about TMT see http://www.tmt.org/ or http://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 aperture telescope (Fig. 1.3). It is located in Green Bank, West Virginia. The GBT has an unusual 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 diffraction that ordinarily complicate a telescope’s pattern of response. More information can 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 telescopes (Fig. 1.4). It is operated by the Max Planck Institute for Radio Astronomy in Bonn, Germany, at wavelengths from about 3.5 mm to 90 cm. The telescope is used to observe pulsars, cold gas and dust clusters, the sites of star formation, jets of matter emitted by black holes, and the nuclei of distant far-off galaxies. Its steel construction 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 Effelsberg telescope (courtesy of Max Planck Institute for Radio Astronomy, 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 antenna can rotate around azimuth (vertical) and elevation (horizontal) axes. The rotation is controlled by azimuth and elevation controllers. The combination of the antenna structure 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 two outputs (azimuth and elevation position), and the position loop is closed between the encoder outputs and the velocity inputs. The drives consist of gearboxes, electric motors, amplifiers, and tachometers. The antenna controller consists of two independent subsystems: azimuth and elevation controllers. Because both subsystems are independent, a single system is considered further, and azimuth or elevation system is specified, if necessary. Also,

6

1 Introduction

Fig. 1.5 The Deep Space Network antenna at Goldstone, California (courtesy of NASA/JPL/Caltech). It can rotate with respect to the azimuth (vertical axis) and with respect to elevation (horizontal axis)

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

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

uc

+ –

encoder, y

velocity controller

V

ANTENNA tach velocity

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

1.3 Antenna and Telescope Literature

7

VELOCITY LOOP command, r

position controller

uc

velocity controller

V

encoder, y

tach velocity encoder

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 controller is run by computer software that drives the antenna depending on the actual antenna position and the commanded position. The controller has two inputs: the encoder position y, and the commanded position (or shorter: command) r (deg). The controller 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 Deep Space Network, although the experience of others is included in the references. Obviously, there are too many references to be included here, but readers interested in further study of antenna and telescope control and pointing issues are referred 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 of structural dynamics and control, but includes many examples of antenna dynamics and antenna control systems. Karcher [6] gives an interesting overview of the telescope structure, mechanism, and control system design. The handbook [10] presents the control design of a simple (rigid) antenna. General antenna theory is presented in a comprehensible form in [9] and [11].

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1 Introduction

References 1. Biernson G. (1990). Optimal Radar Tracking Systems. Wiley, New York. 2. Gawronski W. (2001). Antenna control systems: from PI to H∞ . IEEE Antennas and Propagation 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, San Diego. 6. Karcher HJ. (2006). Telescopes as mechatronic systems. IEEE Antennas and Propagation Magazine 48 (2):17–37. 7. Kitsuregawa T. (1990). Advanced Technology in Satellite Communication Antennas: Electrical 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 I

Modeling

Chapter 2

Analytical Models

This chapter presents the development of the analytical model of the antenna velocity loop. The analytical model includes the antenna structure, its drives, and the velocity loop itself. First, a rigid antenna model is discussed. Next, the modal model of a flexible antenna structure is analyzed based on the finite-element data. The modal model is transferred into the state-space model, in both continuous- and discrete-time. Next, the drive model is derived, and combined into the velocity loop model. Finally, the impact of the drive parameters (gearbox stiffness and motor inertia) on the velocity loop properties is analyzed. It is worth to remind that the analytical model is mainly used in the design stage of the antenna. Due to limited accuracy of the analytical modeling, these models cannot be used in implementation, such as the model-based controllers.

2.1 Rigid Antenna Model In this section the velocity loop of a rigid-body antenna is analyzed. Such models represents an antenna without flexible deformations in the disturbance frequency band, and are well beyond antenna bandwidth. This is a model of an idealized antenna, with rigid gearboxes and a rigid structure. It might be applicable to small antennas, but in our case it serves as a tool for explaining and deriving basic velocity loop properties in a closed form (while for larger, flexible antennas the analysis is based on Matlab and Simulink simulations). This simple analysis is extended later to illustrate properties of a flexible antenna control system, and for the better insight into more complicated models and their properties. A rigid antenna in the open loop configuration has torque input and velocity output. The relationship between the torque τ and angular velocity ω follows from the Newton inertia law J ω˙ = τ

(2.1)

where J is the antenna inertia. The Laplace transform of the above equation is J sω(s) = τ (s); hence, the antenna transfer function is represented as an integrator W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 2,

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d u

ko

τ

rigid antenna 1 Js

+

ϕ

1 s

ϕ

Fig. 2.1 Velocity loop model of a rigid antenna

G(s) =

1 ω(s) = τ (s) Js

(2.2)

The velocity loop model of a rigid antenna with a proportional controller is shown in Fig. 2.1. The transfer function of the closed velocity loop system, from the velocity command u to the antenna velocity ϕ, ˙ is as follows: G rl (s) =

ko G(s) 1 ϕ(s) ˙ = = u(s) 1 + ko G(s) 1 + Ts

(2.3)

where T = J ko . The bandwidth of the velocity loop system is equal to B = 1 T = ko J rad/s. The gain, ko , is tuned to obtain the required bandwidth of the system. For example, if the required bandwidth is B = 20 rad/s, and the inertia is J = 1 Nms2 /rad, then ko = 20 Nms.

2.2 Structural Model The structural model is typically derived from the finite-element model. It is not the finite-element model per se, but its by-product, the modal model. Note that the modal model includes the rigid-body mode (antenna-free rotation). For the controller tuning purposes the modal model is represented in state-space form, which it is finally given in the discrete-time representation.

2.2.1 Finite-Element Model The analytical model of an antenna is obtained from its finite-element model. The finite element model of a 34-m antenna is shown in Fig. 2.2. It consists of hundreds of nodes and elements, and thousands of degrees of freedom. The model is characterized by the mass, stiffness, and damping matrices, and by the sensors and actuator locations.

2.2 Structural Model

13

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

1 .

Fig. 2.2 The finite-element model of the 34-m antenna

Let N be a number of degrees of freedom of the finite-element model, with antenna displacement and velocity at the encoder location as the output and force at the motor pinion attachment as the input. Antenna structure is represented by the following second-order matrix differential equation: M q¨ + D q˙ + K q = Bo u, ˙ y = Coq q + Cov q.

(2.4)

In this equation q is the N × 1 nodal displacement vector; q˙ is the N × 1 nodal velocity vector; q¨ is the N × 1 nodal acceleration vector; u is the input; y is the output; M is the mass matrix, N × N ; D is the damping matrix, N × N ; and K is the stiffness matrix, N × N . The input matrix Bo is N × 1, the output displacement matrix Coq is 1 × N , and the output velocity matrix Cov is 1 × N . The mass matrix is positive definite (all its eigenvalues are positive), and the

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2 Analytical Models

stiffness and damping matrices are positive semi-definite (all their eigenvalues are non-negative).

2.2.2 Modal Model Modal models of antennas are expressed in modal coordinates. Because these coordinates are independent, they simplify the analysis. The modal model is obtained by transforming equation (2.4) using a modal matrix; see [2]. Consider free vibrations of an antenna structure without damping, and without external excitation, that is, u = 0 and D = 0. The equation of motion (2.4) in this case turns into the following equation: M q¨ + K q = 0.

(2.5)

The solution of the above is q = φe jωt ; hence, the second derivative of the solution is q¨ = −ω2 φe jωt . Introducing q and q¨ into (2.5) gives (K − ω2 M)φe jωt = 0.

(2.6)

This set of homogeneous equations has a nontrivial solution if the determinant of K − ω2 M is zero, that is, det(K − ω2 M) = 0.

(2.7)

The above determinant equation is satisfied for a set of n values of frequency ω. These frequencies are denoted ω1 , ω2 , ..., ωn , and their number n does not exceed the number of degrees of freedom, i.e., n N . The frequency ωi is called the ith natural frequency. Substituting ωi into (2.6) yields the corresponding set of vectors {φ1 , φ2 , ..., φn } that satisfy this equation. The ith vector φi corresponding to the ith natural frequency is called the ith mode shape. The natural modes are not unique, since they can be arbitrarily scaled. Indeed, if φi satisfies (2.6), so does αφi , where α is an arbitrary scalar. For a notational convenience define the matrix of natural frequencies ⎡

ω1 0 ⎢ 0 ω2 ⍀=⎢ ⎣ ··· ··· 0 0

⎤ ··· 0 ··· 0 ⎥ ⎥ ··· ··· ⎦ · · · ωn

(2.8)

2.2 Structural Model

15

and the matrix of mode shapes, or modal matrix ⌽, of dimensions N × n, which consists of n natural modes of a structure ⎡

⌽=

φ1 φ2 ... φn

⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

φ11 φ21 φ12 φ22 ... ... φ1k φ2k ... ... φ1n d φ2n d

... ... ... ... ... ...

φn1 φn2 ... φnk ... φnn d

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦

(2.9)

where φi j is the jth displacement of the ith mode, that is, ⎫ ⎧ ⎪ ⎪ ⎪ φi1 ⎪ ⎪ ⎬ ⎨ φi2 ⎪ . φi = .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ φin

(2.10)

The modal matrix ⌽ has an interesting property: it diagonalizes mass and stiffness matrices M and K, Mm = ⌽T M⌽,

K m = ⌽T K ⌽.

(2.11)

The obtained diagonal matrices are called modal mass matrix (Mm ) and modal stiffness matrix (K m ), respectively. The same transformation, applied to the damping matrix Dm = ⌽T D⌽,

(2.12)

gives the modal damping matrix Dm , which is a diagonal if it is, for example, proportional to the stiffness matrix. In order to obtain the modal model a new variable, qm , is introduced. It is called modal displacement, and satisfies the following equation: q = ⌽qm .

(2.13)

Equation (2.13) is introduced to (2.4), and the latter additionally left-multiplied by ⌽T , obtaining Mm q¨ m + Dm q˙ m + K m qm = ⌽T Bo u, y = Coq ⌽qm + Cov ⌽q˙ m .

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2 Analytical Models

using (2.11), and (2.12) notations. Next, the left multiplication of the latter equation by Mm−1 , which gives q¨ m + 2Z⍀q˙ m + ⍀2 qm = Bm u, y = Cmq qm + Cmv q˙ m .

(2.14)

In (2.14) ⍀ is a diagonal matrix of natural frequencies, defined before, and Z is the modal damping matrix. It is a diagonal matrix of modal damping, ⎡

ζ1 0 ⎢ 0 ζ2 Z=⎢ ⎣ ··· ··· 0 0

⎤ ··· 0 ··· 0 ⎥ ⎥. ··· ··· ⎦ · · · ζn

(2.15)

where ζi is the damping of the ith mode. This matrix is obtained using the following relationship Mm−1 Dm = 2ZΩ, thus, −1

−1

Z = 0.5Mm−1 Dm Ω −1 = 0.5Mm 2 K m 2 Dm .

(2.16)

The modal input matrix Bm in (2.14) is as follows Bm = Mm−1 ⌽T Bo .

(2.17)

Finally, in (2.14) the following notations is used for the modal displacement and velocity matrices: Cmq = Coq ⌽,

(2.18)

Cmv = Cov ⌽.

(2.19)

Note that (2.14) (a modal representation of a structure) is a set of uncoupled equations. Indeed, due to the diagonality of ⍀ and Z, this set of equations can be written, equivalently, as q¨ mi + 2ζi ωi q˙ mi + ωi2 qmi = bmi u i = 1, . . . , n, yi = cmqi qmi + cmvi q˙ mi , n y= yi ,

(2.20)

i=1

where bmi is the ith row of Bm and cmqi , cmvi are the ith columns of Cmq and Cmv , respectively. In the above equations yi is the system output due to the ith mode dynamics. Note that the structural response y is a sum of modal responses yi , which is a key property used to derive structural properties in modal coordinates. Theoretically the determination of the input and output matrices as in (2.17), (2.18), and (2.19) requires large matrix ⌽ (of size: number of degrees of freedom

2.2 Structural Model

17

by number of modes). In fact, one needs only one row of it for each input/output matrix. Indeed, the input matrix Bo is all zero, except the location of the actuator (pinion) to the structure. The same, the output matrix Cmq is all zero, except the location of the sensor (encoder) to the structure. Let the location of the sensor is at the kth degree of freedom of the finite-element model. The modal matrix is as in (2.9). Because Cmq is all zero but the kth entry, therefore ⎡

Cmq

⎢ ⎢

⎢ = Coq ⌽ = 0 0 ... 1 ... 0 ⎢ ⎢ ⎢ ⎣

φ11 φ21 φ12 φ22 ... ... φ1k φ2k ... ... φ1n d φ2n d

... ... ... ... ... ...

φn1 φn2 ... φnk ... φnn d

⎤ ⎥ ⎥ ⎥

⎥ = φ1k φ2k ... φnk ⎥ ⎥ ⎦ (2.21)

that is, Cmq consists of modal displacement at the encoder location. Similarly, if the actuator (pinion) is located at the kth degree of freedom, the corresponding input matrix is ⎤ φ1k /m m1 ⎢ φ2k /m m2 ⎥ ⎥ =⎢ ⎣ ··· ⎦ φnk /m mn ⎡

Bmq

(2.22)

where m mi is the ith modal mass. Thus, Bmq consists of modal displacement at the actuator (pinion) location scaled by the modal masses. In summary, from comparatively few parameters of the finite-element model, such as natural frequencies, modal masses, modal damping, and modal displacements at the actuator and sensor locations one creates the antenna structural model in equation (2.14). Note that this structural dynamics model of an antenna is much simpler than its static finite-element model. As an example, determine the first four natural modes and frequencies of the antenna presented in Fig. 2.2. The modes are shown in Fig. 2.3. For the first mode the natural frequency is ω1 = 13.2rad/s (2.10 Hz), for the second mode the natural frequency is ω2 = 18.1 rad/s (2.88 Hz), for the third mode the natural frequency is ω3 = 18.8 rad/s (2.99 Hz), and for the fourth mode the natural frequency is ω4 = 24.3 rad/s (3.87 Hz).

2.2.3 State-Space Model State-space representation is the standard representation of the control system models. Thus, the antenna structure needs to be represented in this form, to allow the use of control system software such as Matlab or Simulink. The modal state-space

18

2 Analytical Models

(a)

(b)

(c)

(d)

Fig. 2.3 Antenna modes: (a) First mode (of natural frequency 2.10 Hz); (b) second mode (of natural frequency 2.87 Hz); (c) third mode (of natural frequency 2.99 Hz); and (d) fourth mode (of natural frequency 3.87 Hz). For each mode the nodal displacements are sinusoidal, have the same frequency, and the displacements are shown at their extreme values. Gray color denotes undeformed state

representation of the antenna structure is a triple ( Am , Bm , Cm ) characterized by the block-diagonal state matrix, Am ; see [2] ⎡

× × 0 0 · · · · · · 0 0 ⎢ × × 0 0 · · · · · · 0 0 ⎢----------------------------⎢ ⎢ 0 0 × × · · · · · · 0 0 ⎢ ⎢ 0 0 × × · · · · · · 0 0 ⎢ Am = diag(Ami ) = ⎢ - - - - - - - - - - - - - - - - - - - - - - - - - - - - ⎢··· ··· ··· ··· ··· ··· ··· ··· ⎢ ⎢··· ··· ··· ··· ··· ··· ··· ··· ⎢ ⎢----------------------------⎣ 0 0 0 0 ··· ··· × × 0 0 0 0 · · · · · · × ×

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

i = 1, 2, . . . , n,

(2.23)

where Ami are 2 × 2 blocks (their nonzero elements are marked with ×), and the modal input and output matrices are divided, correspondingly,

2.2 Structural Model

19

⎡

⎤ Bm1 ⎢ Bm2 ⎥ ⎢ ⎥ Bm = ⎢ . ⎥ , ⎣ .. ⎦

Cm = Cm1 Cm2 · · · Cmn ,

(2.24)

Bmn The blocks Ami , Bmi , and Cmi are as follows: 0 ωi 0 , Bmi = , Ami = −ωi −2ζi ωi bmi

Cmi =

cmqi cmvi ; ωi

(2.25)

The state x of the modal representation consists of n independent components, xi , that represent a state of each mode ⎧ ⎫ x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 ⎪ ⎬ (2.26) x= .. , ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎩ ⎭ xn The ith state component is as follows: ωi qmi xi = , q˙ mi

(2.27)

The ith component, or mode, has the state-space representation ( Ami , Bmi , Cmi ) independently obtained from the state equations x˙ i = Ami xi + Bmi u, yi = Cmi xi , n y= yi .

(2.28)

i=1

This decomposition is justified by the block-diagonal form of the matrix Am , The poles of a structure are the zeros of the characteristic equations . The equation s 2 + 2ζi ωi s + ωi2 = 0 is the characteristic equation of the ith mode. For small damping the poles are complex conjugate, and in the following form: (2.29) s1,2 = −ζi ωi ± jωi 1 − ζi2 . The plot of the poles is shown in Fig. 2.4, which shows how the location of a pole relates to the natural frequency and modal damping.

2.2.4 Models with Rigid Body Modes Antenna structure is unrestrained—it can free rotate with respect to azimuth and elevation axes. Modal analysis for such structures shows that they have zero natural frequency, and that the corresponding natural mode shows structural displacements

20

2 Analytical Models Im s1

2

ωi sqrt(1– ζi ) ωi –arcsin(ζi)

–ζiωi

0

Re

ωi s2

– ωi sqrt(1– ζi ) 2

Fig. 2.4 Pole location of the ith mode of a lightly damped structure: It is a complex pair with the real part proportional to the ith modal damping; the imaginary part approximately equal to the ith natural frequency; and the radius is the exact natural frequency

but no flexible deformations. A mode without flexible deformations is called a rigidbody mode. Corresponding zero frequency implies that the zero frequency harmonic excitation (which is a constant force or torque) causes rigid-body movement of the structure. Structural analysts sometimes ignore this mode, as there is no deformation involved. However, it is of crucial importance for a control engineer, because this mode is the one that allows a controller to move a structure and track a command. The rigid-body modes are obtained by solving the same eigenvalue problem as presented for the standard models. Because the natural frequency is zero, the modal equation becomes det(K ) = 0, that is, the stiffness matrix becomes singular. The corresponding rigid-body mode φr b is the one that satisfies the equation K φr b = 0.

(2.30)

The modal equations for the rigid-body modes follow from (2.20) by assuming ωi = 0, that is, q¨ mi = bmi u, yi = cmqi qmi + cmvi q˙ mi , y=

n i=1

yi .

(2.31)

2.2 Structural Model

21

The state-space modal model for a rigid-body mode is as follows,

Ami

01 = , 00

0 , = bmi

Bmi

Cmi = cmqi cmvi ,

(2.32)

For the experienced engineer the rigid-body frequency and mode are not difficult to determine: rigid-body frequency is always zero, and rigid-body mode can be predicted as a structural movement without deformation. The importance of distinguishing it from “regular” modes is the fact that they make a system unstable, and thus a system that requires special attention. The Deep Space Network antenna has two rigid-body modes: rigid-body rotation with respect to the azimuth (vertical) axis, and rigid-body rotation with respect to the elevation (horizontal) axis. Figure 2.5 shows the azimuth rigid-body mode. Figure 2.5(a) presents the initial position from the top view; Fig. 2.5(b) presents the modal displacement (rigid-body rotation with respect to the azimuth axis) from the top view,

2.2.5 Discrete-Time Model For the antenna performance simulations and for the controller tuning purposes the discrete-time models are required. The discrete time model is obtained from a continuous-time state-space representation (A,B,C). Because the discrete-time sequences of this model are sampled continuous-time signals, that is,

xk = x(k⌬t),

(a)

u k = u(k⌬t), and yk = y(k⌬t),

(2.33)

(b)

Fig. 2.5 Antenna in neutral position (a), and the azimuth rigid-body mode (b), where no flexible deformations are observed

22

2 Analytical Models

for k = 1, 2, 3 . . ., thus the corresponding discrete-time representation for the sampling time ⌬t is (Ad , Bd , Cd , Dd ), where Ad = e

A⌬t

,

Bd =

⌬t

e Aτ Bdτ ,

Cd = C,

(2.34)

0

and the corresponding state-space equations are xk+1 = Ad xk + Bd u k , yk = C xk .

(2.35)

The discretization can be carried out numerically using the c2d command of Matlab.

Next, the discrete-time models is presented in modal coordinates. Assume small damping and the sampling rate sufficiently fast, such that the Nyquist sampling theorem is satisfied (i.e., ωi ⌬t π for all i), see, for example, [1, p. 111], then the state matrix in modal coordinates Adm is block-diagonal, Adm = diag(Admi ),

i = 1, . . . , n.

(2.36)

The 2 × 2 blocks Admi are in the form (see [4]), Admi = e−ζi ωi ⌬t

cos(ωi ⌬t) − sin(ωi ⌬t) , sin(ωi ⌬t) cos(ωi ⌬t)

(2.37)

where ωi and ζi are the ith natural frequency and the ith modal damping, respectively. The modal input matrix Bdm consists of 2 × 1 blocks Bdmi , ⎡

Bdm

⎤ Bdm1 ⎢ Bdm2 ⎥ ⎢ ⎥ = ⎢ . ⎥, ⎣ .. ⎦

(2.38)

Bdmn where Bdmi = Si Bmi ,

1 Si = ωi

sin(ωi ⌬t) −1 + cos(ωi ⌬t) , sin(ωi ⌬t) 1 − cos(ωi ⌬t)

(2.39)

2.2 Structural Model

23

and Bmi is the continuous-time modal representation. The discrete-time modal matrix Cdm is the same as the continuous-time modal matrix Cm . The poles of the matrix Adm are composed of the poles of matrices Admi , I = 1,. . ., n. For the ith mode the poles of Admi are s1,2 = e−ζi ωi ⌬t (cos(ωi ⌬t) ± sin(ωi ⌬t)).

(2.40)

The location of the poles is shown in Fig. 2.6, which is quite different from the continuous-time system, cf., Fig. 2.5. For a stable system they should be inside the unit circle. The question arises how to choose the sampling time ⌬t. Note that from the Nyquist criterion the ith natural frequency is recovered if the sampling rate is at least twice the natural frequency in Hz ( f i = ωi 2π ), that is, if 1 2 fi ⌬t or, if ωi ⌬t π

⌬t

or

π . ωi

(2.41)

Considering all modes, the sampling time will be smaller than the smallest π ωi , ⌬t

π . max(ωi )

(2.42)

i

Im

–ri = exp(–ζiωiΔ t) s1

ri sin(ωiΔ t) ωiΔt

ri cos(ωiΔ t) r=1

–ri sin(ωiΔ t)

Re

s2

Fig. 2.6 Pole location of the ith mode of a lightly damped structure in discrete time: It is a complex pair with angle proportional to natural frequency and magnitude close to 1

24

2 Analytical Models

2.3 Drive Model The drive model consists of motor, reducer, amplifiers, and tachometers.

2.3.1 Motor Model The motor model is shown in Fig. 2.7. The motor position (θm ) is controlled by the armature voltage (v a ) va = L a

di o + Ra i o + kb ωm dt

(2.43)

where Ra is motor resistance, L a is motor inductance, and kb is the armature constant. The motor torque (To ) is proportional to the motor current (i o ) To = km i o

(2.44)

where km is the motor torque constant. The motor torque, To is in equilibrium with the remaining torque acting on the rotor; therefore, To = Jm θ¨m + T

(2.45)

where T is the drive output torque, and Jm is a total inertia of the motor and the brake. The above equations give the following (after Laplace transform, where s is the Laplace variable) v a − kb ωm L a s + Ra 1 (To − T ) ωm = Jm s

io =

motor Ra io va

La

reducer Tm θm

N Jm

Tg θg

Fig. 2.7 Motor and reducer model

T kg

2.3 Drive Model

25

2.3.2 Reducer Model The reducer model is shown in Fig. 2.7 as well. The torque T at the output of the gearbox can be expressed with the torsional deformation of the gearbox T = k g (θg − θ p )

(2.46)

θ p is the angle of rotation of the pinion, k g is the stiffness of the reducer and drive shaft, N is the reducer gear ratio, θg is the angle of rotation of the reducer output shaft θg =

θm N

(2.47)

2.3.3 Drive Model Based on the above equations the drive model (e.g., Simulink) is created. Besides motor and reducer, amplifiers are added. A block-diagram of the drive is created; see Fig. 2.8. The parameters of this model are given in Table 2.1. AMPLIFIERS

ARMATURE

v2

vo

kr(1 + τ2 s) + s(1 + τ3 s)

k1k5 v1 1 + τ1 s + _

_

kc

kf ki(1 + τ4 ) s(1 + τ5 s) + _

vs

MOTOR AND REDUCER

TN

io

_

1

Ra + La s

km +

kb

ωm

kt

1/N

1

ωm

Jm s +

_

ωm

N

Fig. 2.8 Block diagram of the drive model

Table 2.1 Drive parameters k1 = 716.2 km = 1.787 kb = 1.79 ks = 0.8 kt = 0.0384 kr = 80 kc = 0.1266

[V s/rad] [Nm/A] [V s/rad] [-] [V s/rad] [V/s/V] [V/A]

kf = 54 [-] kg = 1.7× 106 Jm = 0.14 Ra = 0.456 La = 0.011 N = 354

[-] [Nm/rad] [Nm/s2 ] [⍀] [H] [-]

τ1 τ2 τ3 τ4 τ5 τ6

= 0.00637 = 0.094 = 0.002 = 0.00484 = 0.0021 = 0.7304

kg Ns

[s] [s] [s] [s] [s] [s]

T

ω

26

2 Analytical Models Velocity Command

1

Torque

Velocity Command

Pinion velocity

y(n) = Cx(n)+Du(n) x(n+1) = Ax(n)+Bu(n)

1 Encoder

Antenna Structure

Drive

pinion velocity

Fig. 2.9 Velocity loop model

magntiude

100

10–1

10–2

10–3

from analysis from identification

10–1

100 frequency, Hz

101

Fig. 2.10 The magnitude of the transfer function of the velocity loop model

2.4 Velocity Loop Model The Simulink velocity loop model is a combination of the structural and drive models; see Fig. 2.9. The drive torque moves the antenna. The pinion velocity is fed back to the drive to calculate the reducer flexible deformation. The velocity command controls the antenna velocity, and the encoder measures its position. The magnitude of the velocity loop transfer function is shown in Fig. 2.10, solid line.

2.5 Drive Parameter Study In this section the effect of the drive inertia and stiffness on the antenna velocity loop properties is investigated. The inertia of the drive includes motor, brake, and gearbox, and the drive stiffness includes gearbox stiffness and shaft stiffness. Because the motor dominates the drive inertia, the motor inertia is investigated, and because gearbox stiffness dominates the drive stiffness, the drive stiffness is investigated.

2.5 Drive Parameter Study

27

102

elevation rate

101

100

10–1 kg 10–2

10*kg 100*kg

10–3 –2 10

10–1

100 frequency, Hz

101

102

Fig. 2.11 The magnitudes of the transfer function of the velocity loop model for the nominal and increased gearbox stiffness

It is intuitively obvious that a rigid gearbox and a powerful but weightless motor are the best choice for the antenna drive (if they exist). So, what is the smallest allowable gearbox stiffness, or the largest allowable motor inertia? The 34-m antenna velocity loop bandwidth is simulated as a function of motor inertia, and as a function of the gearbox stiffness [3], using the model as in Fig. 2.8 and Fig. 2.9. The nominal motor inertia is 0.14 [Nms2 ], and the nominal gearbox stiffness is 1.7 × 106 [N m/rad]; see Table 2.1. The transfer function, from the velocity command to the output velocity for the nominal values, is shown in Fig. 2.11, solid line.

2.5.1 Drive Stiffness Factor The impact of the gearbox stiffness on the velocity loop properties is simulated, for the nominal gearbox stiffness and for increased stiffness by factors of 10 and 100, respectively. The corresponding transfer functions are shown in Fig. 2.11, dotted (for factor 10) and dot-dashed lines (for factor 100), respectively. One can see that the transfer function bandwidth changed when k g incresed to 10k g . The resonant peak, related to the drive vibration mode shifted to higher frequencies. At the nominal value of the stiffness, the resonant frequency of the drive mode coincided with the structural fundamental frequency. The plots show that the nominal stiffness is

28

2 Analytical Models 102

elevation rate

101

100

10–1

10–2

kg 0.1*kg 0.01*kg

10–3 10–2

10–1

100 frequency, Hz

101

102

Fig. 2.12 The magnitudes of the transfer function of the velocity loop model for the nominal and decreased gearbox stiffness

slightly too low, because the inreased gearbox stiffness increased the velocity loop bandwidth. Next, the gearbox stiffness is decreased by factors of 0.1 and 0.01, and the corresponding transfer functions are shown in Fig. 2.12 as dotted and dot-dashed lines, respectively. One can see that the transfer function bandwidth changed significantly. The resonant peak, related to the drive vibration mode is shifted to lower frequencies, impacting the bandwidth. The plots show that the nominal stiffness should not be decreased, because the lower gearbox stiffness impacts the velocity loop bandwidth.

2.5.2 Drive Inertia Factor Here, the impact of the motor inertia on the velocity loop properties is investigated. The transfer function from the velocity command to the output velocity for the nominal inertia is shown in Fig. 2.13, solid line. First, the motor inertia is decreased by factors 0.1 and 0.01, and the corresponding transfer functions are shown in Fig. 2.13, dotted and dot-dashed lines, respectively. One can see that the transfer function bandwidth changed insignificantly. The resonant peak, related to the drive vibration mode shifted to higher frequencies. The plots show that the nominal inertia

2.5 Drive Parameter Study

29

102

elevation rate

101

100

10–1

Jm 10–2

0.1*Jm 0.01*Jm

10–3 10–2

10–1

100 frequency, Hz

101

102

Fig. 2.13 The magnitudes of the transfer function of the velocity loop model for the nominal and decreased motor inertia

102

elevation rate

101

100

10–1

10–2

Jm 10*Jm 100*Jm

10–3

10–2

10–1

100 frequency, Hz

101

102

Fig. 2.14 The magnitudes of the transfer function of the velocity loop model for the nominal and increased motor inertia

30

2 Analytical Models

is sufficient, because the lower motor inertias does not impact the velocity loop bandwidth. Next, the motor inertia is increased by factors 10 and 100, and the corresponding transfer functions are shown in Fig. 2.14, dotted and dot-dashed lines, respectively. In this case the transfer function bandwidth changed significantly. The resonant peak related to the drive vibration mode is shifted to the lower frequencies, impacting the bandwidth. Note that the antenna fundamental frequency has not been changed. The plots show that the nominal inertia cannot be increased, because it impacts the velocity loop bandwidth. This simple investigation shows that the gearbox stiffness has its lower limit that should not be violated, because it impacts the velocity loop bandwidth and the closed loop performance. Similarly, the motor inertia has its upper limit. The limit is defined by the antenna fundamental frequency: drive natural frequency should be at least the fundamental frequency of the structure.

References 1. Franklin GF, Powell JD, Workman ML. (1992). Digital Control of Dynamic Systems. AddisonWesley, Reading, MA. 2. Gawronski W. (2004). Advanced Structural Dynamics and Active Control of Structures. Springer, New York. 3. Gawronski W, Mellstrom JA. (1992). A Parameter and Configuration Study of the DSS-13 Antenna Drives. TDA Progress Report, No.42-110, available at http://ipnpr.jpl.nasa.gov/ progress report/42-110/110S.PDF. 4. Lim KB, Gawronski W. (1996). Hankel Singular Values of Flexible Structures in Discrete Time,” AIAA J. Guidance, Control, and Dynamics, 19(6):1370–1377.

Chapter 3

Models from Identification

This chapter describes the process of deriving the velocity loop model from field test data. It includes the description of the white noise testing of an antenna, test conditions, input and output signals, open- and closed-loop testing, the instrumentation set-up, and the system identification test. Analytical background for the data description and data processing are also included. Next, the dependency of the antenna model on its elevation position, and on the dish diameter is discussed. Finally, analytical and identified models are compared.

3.1 White Noise Testing of the Antenna The white noise signal is applied at the input of the velocity loop to excite the antenna dynamic, and the antenna reaction is measured at the encoder. The white noise is a random signal of even spectrum within the antenna bandwidth; hence each frequency component of the antenna dynamics is excited evenly. It should guarantee to obtain a meaningful transfer function and trustworthy model of the velocity loop.

3.1.1 Purpose and Conditions The purpose of the white noise testing of the antenna is the determination of the antenna velocity loop model. The accurate model is necessary to obtain a stable and precise model-based controller. Other signals, beside the white noise, can be used for the testing and model identification, for example, sweep-sine. It was found, however, that the white noise testing is the most effective and least damaging when applied to antennas. The sweep-sine excites antenna resonances, causing wear, and excites antenna backlash since resonance forces exceed anti-backlash torques. During the tests the antenna is positioned at 45 deg elevation, azimuth position is arbitrary, and wind shall be mild (below 12 km/h).

W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 3,

31

32

3 Models from Identification

3.1.2 Test Input and Output The following signals are generated and/or measured during the test: • The test input signal, denoted u o It is a discrete white noise, generated by the data collection computer. The noise is converted into analog signal (voltage). • The antenna input signal, denoted u. It is the total voltage at the velocity input. • The antenna output signal, denoted y. It is the encoder reading (digital signal). The common characteristics of the input and output signals are their sampling rate, the test duration, signal amplitude, and offset. • Sampling rate is obtained from the Nyquist criterion, described later. Typical sampling rate for 34-m antennas is 30 Hz. It can be chosen, however, between 25 and 50 Hz. • Duration. The test duration determines the system identification accuracy, and the lowest frequency of the obtained transfer function. Because the input is a random signal, the accuracy of the transfer function depends on the number of averaging of the collected data. The longer is the collected data file, the smoother is the average. Also, the longer is the data segment, the lower is the frequency of the obtained transfer function. Thus, a large data file is needed. But the limiting factor for data size is the computer memory (less and less important nowadays), and patience of a test engineer (more and more important nowadays). Our experience shows that 40,000 samples are sufficient, which for 30-Hz sampling rate translates into 20–30 min of test time. • Amplitude. Amplitude of the test noise shall be large enough to excite antenna dynamics, and be larger than the disturbances (mostly wind gusts). However, it shall be low enough in order to prevent antenna wear and damage; thus the white noise amplitude shall be at least 5% of maximal allowable amplitude. • Velocity offset. A non-zero input signal offset is applied in order to overcome antenna static friction. The nonzero velocity offset forces an antenna to move with a constant velocity. The offset is about 1% of the maximum velocity. A typical white noise input is shown in Fig. 3.1a. Note that the amplitude is 0.5 V, and the offset is –0.1 V. The output signal (encoder reading) is a digital signal of the same sampling rate as the input signal. Due to the input offset, the output signal shows a slope proportional to the input offset; see Fig. 3.1b.

3.1.3 Test Configuration The test can be conducted with the antenna position loop open or closed; see Fig. 3.2 and Fig. 3.3. Open loop test. The open loop test configuration is shown Fig. 3.2. By setting the antenna in the manual mode the position loop is broken, the position controller

velocity input, V

3.1 White Noise Testing of the Antenna

33

offset

0.5

0

–0.5 0

20

40

60

80 100 120 sample number

140

160

180

200

20

40

60

80 100 120 sample number

140

160

180

200

EL encoder, deg

35.55 35.54 35.53 35.52 0

Fig. 3.1 White noise input signal of 0.14 V offset, the first 200 samples, and antenna response to the white noise signal the first 200 samples

is disconnected, and the white noise input (u) is injected into the velocity loop. The white noise is recorded by the data computer. The encoder digital signal (y) and tachometer voltage (v t ) are recorded synchronously by the computer.

encoder (y) test signal (uo) command +_

u = uo Antenna velocity loop

Position controller velocity input

Fig. 3.2 Open loop test configuration (u = u o )

encoder

34

3 Models from Identification

encoder (y ) test signal (uo) command +_

Position controller

u = uo + uf Antenna velocity loop

uf

encoder

velocity input

Fig. 3.3 Closed loop test configuration (u = u o )

Closed loop test. The closed loop tests are useful when the antenna is unstable or drifting when the position loop is open. An interesting review of the closed loop identification is given in [5]. The closed loop test configuration is shown Fig. 3.3. The white noise input (u o ) is injected into the velocity loop, with the position loop closed. The total signal (u) after the summing junction is recorded by the data computer. Note that the signal after the summing junction is not identical with the test input signal u o , because it is a sum of u o and the feedback signal u f , that is, u = u o + u f . The encoder digital signal (y) is recorded synchronously by the computer.

3.1.4 Data Processing The collected input (u) and output (y) data are used for the determination of the velocity loop transfer function and for the system identification (i.e., the velocity loop state-space model). The open loop transfer function is determined using the following steps; see [1]. Let the u and y records of length of 20,000 samples be divided into n =10 segments of length T = 2000 Δt s. The segments are denoted u i and yi , where i =1,. . .,n. The FFT algorithm applied to the segments gives u i and yi spectra, denoted Ui and Yi , respectively. The mean value of their product represents the estimate of the cross-spectral density function Puy (ω): Puy (ω) =

n 2 ∗ U (ω)Yi (ω) nT i=1 i

(3.1)

where Ui∗ denotes the complex conjugate of Ui . The estimate is smoothed through averaging (this was the reason for cutting the records into segments).

3.1 White Noise Testing of the Antenna

35

Similarly, one obtains a smooth estimate of the input power spectral density function Pu (ω) Pu (ω) =

n 2 |Ui (ω)|2 nT i=1

(3.2)

and the output power spectral density function Py (ω) Py (ω) =

n 2 |Yi (ω)|2 nT i=1

(3.3)

ˆ From these functions one obtains an estimate G(ω) of the open loop transfer function G(ω) as follows Puy (ω) ˆ G(ω) = Pu (ω)

(3.4)

The above transfer function is an approximate one. The coherence function is an indicator of the quality of the approximation. It is defined as Puy (ω) γ (ω) = (3.5) Pu (ω)Py (ω) Its value varies between 0 and 1. The unity value indicates that a perfect linear relationship exists between the input and the output, and that no disturbances were recorded. A non-linear behavior of the antenna, and/or a presence of excessive disturbances (wind) cause low value of coherence. The transfer function and coherence are calculated using the Matlab command p=spectrum(u,y,n) where n is the length of the segment, and P consists of eight columns, such that Pu ( f ) is the first column of p, Py ( f ) is the second column of p, ˆ f ) is the fourth column of p, G( γ ( f ) is the fifth column of p, For the sampling frequency of the data is f s , the frequency range f is f =

fs n 1: (HZ ) n 2

The magnitude and phase of the velocity loop transfer function obtained from the data presented in Fig. 3.1 is shown in Fig. 3.4. The coherence is shown in Fig. 3.5. The mean value of the coherence is larger than 0.8 for frequencies between 0.06 Hz

36

3 Models from Identification

magnitude

0

0.01

0.0001 10–2

10–1

100

101

100

101

frequency, Hz 0

phas e, deg

–100 –200 –300 –400 10–2

10–1 frequency, Hz

Fig. 3.4 The magnitude and phase of the 34-m antenna transfer function obtained from the test data 1

coherence

0.8

0.6

0.4

0.2

0 10–2

10–1

100 frequency, Hz

Fig. 3.5 The coherence function for the data in Fig. 3.1

101

3.1 White Noise Testing of the Antenna

37

and 6 Hz. That is, the obtained transfer function is considered accurate within this frequency interval. Note that for selected frequencies (e.g., 2.8 Hz and 4.6 Hz) the value of coherence is low. These two frequencies correspond to the anti-resonances of the transfer function (cf. Fig. 3.4), where antenna does not respond to the excitation.

3.1.5 Basic Relationships for the Discrete-Time Data Let u i = u(iΔt) be sampled data of N samples, i = 1, . . . , N ; see Fig. 3.6a. The sampling time of a discrete-time data is denoted Δt. The corresponding sampling rate f s is the inverse of the sampling time: fs =

1 Δt

(3.6)

The record length is T. The number of samples in the record is N, thus T = N Δt =

N fs

(3.7)

(a)

data, ui

Δt

T time, t[s]

data spectrum, Ui

(b) fn Δf

frequency, f [Hz]

Fig. 3.6 (a) Discrete time signal and (b) its spectrum: Δt, sampling time; T, record length; Δf , frequency resolution; and f n , Nyquist frequency

38

3 Models from Identification

Denote Ui the spectrum of the data, which is shown in Fig. 3.6b. The resolution of lines of the frequency spectrum is Δf . It is obtained either from the record length or from the sampling time or from the sampling rate Δf =

fs 1 1 = = T N N ⌬t

(3.8)

It is also the lowest frequency of the spectrum. For N = 2048 samples, and for sampling frequency f s = 30 Hz one obtains the resolution Δf = 30/2048 = 0.015 Hz. The range of the spectrum f n is called the Nyquist frequency, obtained as fn =

1 1 N = = fs 2T 2⌬t 2

(3.9)

It is the maximum frequency of a digital signal. It is equal to the half of the sampling frequency of this signal. It is also called the cut-off frequency, or the bandwidth of the signal. Because the desired range is usually known, the sampling frequency is obtained from the above equation as fs = 2 fn

(3.10)

The maximal frequency must be larger then the velocity loop bandwidth, f b , fn > fb

(3.11)

fs > 2 fb

(3.12)

thus

The sampling rate, fs , must be at least twice the bandwidth of the velocity loop. For the 34-m antenna the velocity loop bandwidth is below 10 Hz; hence 30 Hz satisfies the Nyquist criterion (3.9). High sampling rates (above 50 Hz) add no new information in the test; however, they significantly increase the size of data files.

3.2 Identification of the Velocity Loop Model In this section the process of obtaining an antenna velocity loop model from the collected input and output data is described.

3.2 Identification of the Velocity Loop Model

39

3.2.1 Description of the Velocity Loop Model The antenna control system is shown in Fig. 3.7. It consists of the antenna open loop system (velocity loop), position controller, and velocity and acceleration limiters. The controller output u represents the commanded velocity. The antenna angular position (measured at the encoder) is the output of the velocity loop system. The antenna velocity loop model is obtained from field tests and the system identification in the state-space form x(i + 1) = Ax(i) + Bu(i) + w(i), y(i) = C x(i),

(3.13)

where x is the state, w is the disturbance, y is the antenna position, and (A,B,C) is the state-space triple. The size of the matrices and vectors in Eq. (3.13) is as follows: x(i) is n × 1, w(i) is n × 1, A is n × n, B is n × 1, C is 1 × n, u(i) is a scalar, y(i) is a scalar, n is the order of the system, and i is the sample number. The order n is the input parameter to the system identification program, selected by a user. Typically, it is selected n = 40, which is oversized, but necessary in order to assure the inclusion of all significant system dynamics in the model. Later, however, the order is reduced to n ≤ 14, suitable for the DSN antenna controller tuning purposes.

3.2.2 Identification of the Velocity Loop Model The determination of the antenna velocity loop model is part of the LQG controller tuning process (as the LQG estimator). Thus, the accuracy of the model and its coordinate selection are the factors that influence the controller performance. The accuracy is solved by determining the model through field testing of the velocity loop antenna, applying system identification, and comparing the identified model transfer function with the transfer function obtained from the data. Although the w r controller (K) y

uo

u rate limit

antenna (G)

y

acceleration limit

Fig. 3.7 Antenna control system: G, velocity loop system; K, position controller; r, position command,; y, antenna angular position; u, limited velocity command; uo , unlimited velocity command; w, disturbance

40

3 Models from Identification

antenna transfer function has been already determined, there is no equation that describes the velocity loop dynamics. These equations, or a mathematical model of the velocity loop, are obtained using the system identification software, such as SOCIT program of the NASA Langley Research Center, see [4], or Matlab program n4sid. The complete description of the SOCIT algorithm can be found in [3], or in [2], Ch. 9. In our case the identified model is determined in the form of the state-space representation (A,B,C).

One also can use the Matlab system identification program (n4sid). In this case, having the input (u) and output (y) signals one obtains the state-space representation (A,B,C,D) as follows DAT = iddata (y,u,dt); MODEL = n4sid(DAT,40, CovarianceMatrix , None ); A=MODEL.a; B=MODEL.b; C=MODEL.c;

The transfer function plot, using the identified models (A,B,C) is shown in Fig. 3.8, solid line, and compared with the transfer function obtained from the Fourier transform of the data (dashed line). The estimated and identified transfer functions show good coincidence.

100

magnitude

10–1

10–2

10–3 from id model from data 10–1

100 frequency, Hz

101

Fig. 3.8 Magnitude of the transfer function, from the test data (dashed line) and from the identified model (solid line)

3.2 Identification of the Velocity Loop Model

41

3.2.3 A Comparison of the Analytical and Identified Models Figure 2.10 (dashed line) shows the magnitude of the azimuth transfer function of the 34-m antenna obtained from the identification. The transfer function plotted in Fig. 2.10, solid line, was obtained from the analytical model of the same antenna. One can see that the character of the transfer function is the same; however, there are also differences. The first resonant frequency of the identified model is lower than the analytical model. Amplitudes of the resonances at higher frequencies in the identified model are significantly lower than in the analytical model.

3.2.4 Azimuth Model Depends on the Antenna Elevation Position In the system identification process the antenna azimuth and elevation models are determined separately. The models are independent of each other because there is a very weak coupling between the axes. However, the azimuth model depends on the elevation position of the antenna. The antenna structural properties, as measured at the elevation axis, do not depend on the elevation position. Thus, the elevation structural modes and frequencies are also independent on the antenna elevation position. This is shown in Fig. 3.9, with the measurements of the elevation transfer function at the antenna elevation position 13 deg, 45 deg, and 84 deg. The magnitudes of the transfer function are identical at all three positions. A different situation is observed for azimuth models. Azimuth model properties depend on the antenna elevation position because the antenna inertia and stiffness with respect to azimuth axis depend on the elevation position. It is visible with the measurements of the azimuth transfer function at the antenna elevation position 10, 30, 45, 60, and 90 deg; see Fig. 3.10.

elevation magnitude

100

10–1

10–2

10–3 10–1

EL = 13 deg EL = 45 deg EL = 84 deg 100 frequency, Hz

Fig. 3.9 Elevation transfer function does not depend on antenna elevation position

101

42

3 Models from Identification (a) natural frequencies: first

second third fourth

azimuth magnitude

100

10–1

10–2

10–3

EL = 10 deg EL = 30 deg EL = 45 deg EL = 60 deg EL = 90 deg 100

101

(b) natural frequencies: first second

third fourth

azimuth magnitude

100

10–1

10–2

10–3

EL = 10 deg EL = 30 deg EL = 45 deg EL = 60 deg EL = 90 deg 100

101

frequency, Hz

Fig. 3.10 Azimuth transfer function depends on antenna elevation position: (a) 34-m HEF antenna, and (b) 34-meter BWG antenna

3.2 Identification of the Velocity Loop Model

43

In Fig. 3.10a the damping of the first mode decreases with an increase of the elevation position, while the natural frequency of the third mode increases with an increase of the elevation position. In Fig. 3.10b the damping of the first mode does not change with the elevation position, while the natural frequency of the third mode decrease with the increase of the elevation position. Both antennas are 34-m antennas, but different types. Thus, there is no obvious pattern in the dynamical properties of the antennas. For the controller tuning purposes the mean model is chosen, the one at elevation position of 45 deg.

3.2.5 Fundamental Frequency Depends on Antenna Diameter Antenna natural frequencies depend also on the antenna size. It is a general tendency that the natural frequencies decrease with the increase of the antenna size (the structure becomes “softer”). The lowest natural frequency (called the fundamental frequency) is considered a measure of the compliance of a structure. The data of fundamental frequencies of many antenna structures have been collected by D.D. Pidhayny, A.R. Lewis, and S.R. Bandel of Aerospace Corporation. In these data the decreasing tendency of frequencies with the increase of antenna dish size is observed. Based on these data, the best-fit line is determined (in logarithmic scale). It relates the antenna dish diameter (in meters) with the antenna fundamental frequency (in Hertz). It is given by the following equation

antenna fundamental frequency, Hz

f = 20 d −0.7

(3.14)

10

1 1

10 antenna diameter, m

100

Fig. 3.11 The best-fit line that fits the Aerospace Corp. chart of the antenna fundamental frequencies

44

3 Models from Identification

The line is presented in Fig. 3.11. This equation represents the average natural frequency for a given antenna diameter. Equation (3.14) allows evaluating the structural soundness of a particular antenna: if the fundamental frequency of the considered antenna structure is higher than the frequency obtained from Eq. (3.14), the structure is stiffer than the average (thus, better pointing performance is expected); if it is lower, the structure is softer than average (thus causing inferior pointing performance).

References 1. Bendat, JS, Piersol AG. (1986). Random Data. John Wiley, New York. 2. Gawronski W. (2004). Advanced Structural Dynamics and Active Control of Structures. Springer, New York. 3. Juang JN. (1994). Applied System Identification, Prentice-Hall, Englewood Cliffs, NJ. 4. Juang JN, Horta LG, Phan M. (1992). System/observer/controller identification toolbox. NASA Tech Memo 107566. 5. Landau ID. (2001). Identification in closed loop: a powerful design tool (better design models, simpler controllers). Control Engineering Practice 9: 51–65.

Chapter 4

Model Reduction

The velocity loop model usually consists of too many state variables (i.e., a smaller number would reflect the system dynamics with similar precision). There are different methods of reducing the number of variables in the velocity loop models; see, for example, [3]. The balanced and modal model reduction methods have been selected, which are well tested in the antenna environment. Also, the reduction of a system with rigid-body modes (such as antennas) is discussed; it requires special attention because the standard approaches fail.

4.1 Why Reduction? Analytical models have an excess of state variables that describe the antenna dynamics, due to the large number of degrees of freedom of the structural model and variables of the drives. For example, some drive variables may be important in the drive dynamics, but less important in the overall velocity loop dynamics. Thus, the model order reduction is recommended. The identified model also has excessive order, because the unimportant structural modes (weakly observed at the output and/or weakly excited at the input) are part of the model. Additionally, the measurement noise is included in the model. Besides the reduction of the model complexity, there is another important factor that requires model reduction: the model-based controller property. The order (and complexity) of the controller is equal to the order of the antenna model. In the implementation one requires an accurate model, which is as simple as possible (but not simpler!) to ease the implementation. This is the task of model reduction.

4.2 Balanced Model Reduction The order of the antenna model obtained from the system identification is too high for the implementation, and it includes dynamics that do not represent the antenna, but possible disturbances and measurement noise. The mode order has to be reduced. The reduced model shall include all relevant antenna dynamics in order to assure a stable LQG controller. W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 4,

45

46

4 Model Reduction

In order to reduce the model the model (e.g., obtained from the system identification) is transformed into the balanced representation. The balanced representation is described in [3].

It can be executed in Matlab as follows: [sysb, g] = balreal(sys); or, equivalently [Ab, Bb, Cb, g] = balreal(A, B, C); where sys is the original system (or with the state space representation (A, B, C)), while sysb is the balanced system (or with the state space representation (Ab, Bb, Cb)), and g is the vector of the Hankel singular values (HSV).

As part of the transformation the Hankel singular values (HSV, also called Hankel norms) of the system are obtained. The model is presented in descending order of the HSV, as follows: ⎧ ⎫ x1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x2 ⎪ ⎬ x= .. ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ xn

(4.1)

where xi is the ith state and x1 is the state with the highest HSV, while xn is the state with the lowest HSV. The reduced-order model is obtained by truncating the least important states. Because the states with the smallest HSVs are the last ones in the state vector, a reduced-order model is obtained here by truncating the last states in the state vector. How many such states? Enough that the remaining states represent the significant portion of the antenna dynamics (e.g., the part represented in the magnitude of the transfer function above a selected significance or noise level). “The significant portion of the antenna dynamics” is an ambiguous statement, but it is a part of the art of engineering, which includes experience and intuition. Let (Ab , Bb , Cb ) be the balanced representation corresponding to the state vector x as in (4.1). Let x be partitioned as follows: x=

xr , xt

(4.2)

where xr is the vector of the retained states and xt is a vector of truncated states. If there are k < n retained states, xr is a vector of k states, and xt is a vector of n − k

4.3 Modal Model Reduction

47

states. Let the state triple (Ab , Bb , Cb ) be partitioned accordingly,

Ar Ar t Br Ab = , Bb = , Cb = Cr Ct . Atr At Bt

(4.3)

The reduced model is obtained by deleting the last n − k rows of Ab , Bb , and the last n − k columns of Ab , Cb , obtaining the reduced triple ( Ar , Br , Cr ), marked boldface in equation (4.3).

4.3 Modal Model Reduction Similar results can be obtained using the modal model described in Chapter 2. For this, a value of each mode should be obtained to decide which mode should be retained or removed in the reduction process. Each mode can be “valued” in three different ways: using H2 , H∞ , or Hankel norms.

4.3.1 Norms of a Single Mode For structures in the modal representation, each mode is independent; thus the norms of a single mode are independent as well (they depend on the mode properties, but not on other modes). The H2 norm of the ith mode is Bmi 2 Cmi 2 G i 2 ∼ √ = 2 ζi ωi

(4.4)

The H∞ norm is estimated as Bmi 2 Cmi 2 G i ∞ ∼ = 2ζi ωi

(4.5)

The Hankel norm is determined from Bmi 2 Cmi 2 G i h ∼ = 4ζi ωi

(4.6)

where Bmi and Cmi are defined in Chapter 2; equations (2.32). For single-input single-output systems—which is the case for antennas or telescopes—the above norms are easily determined from the magnitude of the transfer function. Namely, the H∞ norm is of the ith mode and is the height of its resonant peak; see Fig. 4.1. The Hankel norm is half of this peak. The determination of the H2 norm of a mode is a little more complicated. Let Δωi be a half-power frequency at the ith resonance, Δωi = 2ζi ωi ; see [1], [2]. It is the width of the

48

4 Model Reduction 5

magnitude

4

G1

3

∞

G1

= 2 G1 h mode 1 ∞

2

2

mode 2

1

Δω1 = 2ζ 1ω1 0

0

1

2

3

4 5 frequency, Hz

6

7

8

Fig. 4.1 The determination of the half-power frequency, H2 , Hankel and H∞ norm for the first

shaded area in √ this figure, obtained as a cross section of the resonance peak at the height of h i / 2, where h i is the height of the resonance peak. In practice, the norm can be evaluated visually through the inspection of the magnitude of the transfer function. When Hankel or H∞ norm is used, the reduction is carried out by eliminating modes with lowest resonances; when H2 norm is used, the height and the width of the resonances is used. Comparing (4.4)–(4.6) the approximate relationships between H2 , H∞ , and Hankel norms is obtained G i ∞ ∼ = 2 G i h ∼ =

ζi ωi G i 2 .

(4.7)

The above relationship is illustrated in Fig. 4.2

4.3.2 Norms of a Structure Next, the norms of the whole structure are expressed in terms of the norms of its modes. The structure H2 norm is approximately the rms sum of the modal norms G2 ∼ =

n

G i 22 ,

i=1

where n is the number of modes, and G i 2 is given by (5.21).

(4.8)

4.4 Antenna Model Reduction

49

H2 Hinf

102

modal norm

Hankel

100

10–2

10–3

10–2

10–1

100

101

102

ζiωi, rad/s

Fig. 4.2 Modal norms versus ζi ωi

The structure H∞ norm is the largest of the mode norms, i.e., G∞ ∼ = max G i ∞ , i = 1, . . . , n. i

(4.9)

where n is the number of modes, and G i ∞ is given by (5.22). The Hankel norm of the structure is the largest norm of its modes, and it is half of the H∞ norm, that is, Gh ∼ = max G i h = 0.5 G∞ , i

(4.10)

where n is the number of modes, and G i h is given by (5.23).

4.4 Antenna Model Reduction Antennas and telescopes have rigid-body modes, that is, they have poles at zero. But, the H2 , H∞ , and Hankel norms for systems with poles at zero do not exist as their values tend to infinity; see equations (4.4)–(4.6) for ωi = 0. However, the infinite values of the norms of modes are not an obstacle in the reduction process. These values indicate that the corresponding states should be retained in the reduced model, regardless of the norms of other modes. Here a simple approach is used for the reduction of systems with poles at zero. In this approach, the system with a pole at zero is represented in modal coordinates by the following system triple:

Br 0 0 , B= , C = Cr C o , (4.11) A= Bo 0 Ao

50

4 Model Reduction 100

magnitude

10–1

10–2

10–3

full model reduced model

10–1

100 frequency, Hz

101

Fig. 4.3 Magnitudes of the AZ transfer functions of the 40-state full model (solid line) and 10-state balanced reduced model (dashed line)

where the triple ( Ao , Bo , Co ) has no poles at zero, but is itself in modal coordinates. The vector of the corresponding modal norms is denoted h o . This vector is arranged in descending order, and the remaining infinite norms are added h = {∞, h o }

(4.12)

to obtain the vector of norms of the (A, B, C) representation. The system is reduced by truncation, as described at the beginning of this chapter. Figure 4.3 shows the transfer functions of the 40-state antenna AZ model, and the transfer functions of the reduced (14-state) model using balanced representation. One can see that the antenna rigid-body motion (represented by the straight line at frequencies up to 1 Hz) and the antenna major resonances have been included in the reduced-order model, while the high-frequency dynamics, above 9 Hz, have been eliminated.

References 1. Clough RW, Penzien J. (1975). Dynamics of Structures. McGraw-Hill, New York. 2. Ewins DJ. (2000). Modal Testing. Research Studies Press, Baldock, England. 3. Gawronski W. (2004). Advanced Structural Dynamics and Active Control of Structures. Springer, New York.

Chapter 5

Wind Disturbance Models

Wind is the main source of antenna/telescope disturbance. It loads the antenna structure and causes pointing problems. The wind load is divided into mean (or static load) and variable load (gusts). The load depends on the wind speed, on direction of the wind with respect to the antenna structure (yaw), and on the antenna elevation position (pitch). This chapter discusses models of steady-state wind, and a model of wind gusts. The steady-state model is obtained from wind tunnel data for different yaw angles and different antenna elevation positions. The model is verified with the 34-m antenna wind data obtained in the field. The wind gust models include forces acting on the dish, wind torques at the drives, and wind disturbance at the velocity input of the velocity loop.

5.1 Steady-State Wind Disturbance Model The antenna wind loads are typically separated into two components: steady-state (or mean, or static) and dynamic (gusts) load. Both the static and dynamic wind loads are used to estimate antenna pointing accuracy. The dynamic loads are discussed in [5] and [6]. The mean wind loads are used to size antenna drives and motors. They are typically obtained by scaling the wind tunnel data. The primary source of such data is the result of the wind tunnel tests of antenna wind loading. One of the tests was conducted over 40 years ago at California Institute of Technology, as reported in [1–3]. The wind tunnel data presented in the CP-6 Memorandum [1] are regarded as the most reliable because they were collected using model of the entire antenna rather than the dish-only as in [2] and [3]. In addition, the ground effects and a 25% porosity of the outer portion (25% of the radius) of the antenna were modeled during the testing described in [1]. As a result, the modeled antenna represented geometrical properties of a typical radiotelescope (including the DSN antennas). The geometrical scale employed in wind tunnel testing described in the CP-6 Memorandum was 1:75, which resulted in the 0.46 m (diameter) model of the antenna 34-m reflector. W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 5,

51

52

5 Wind Disturbance Models

5.1.1 Dimensionless Wind Torques Wind-induced torque in an antenna drive, Tw , depends on the antenna geometry and orientation relative to the wind, and also on the wind characteristics. The principal geometrical parameters affecting this loading are shown in Fig. 5.1, where the elevation and plan views of the antenna are schematically depicted. These parameters include the dish diameter, D; the distance, d, of the dish vertex from the elevation axis; the elevation angle, βe ; the yaw angle βa , and the antenna porosity. The wind-induced torques in the antenna drives are represented in a dimensionless form as torque coefficients τw ; see [10], and [11], τw =

Tw p AD

(5.1)

where Tw is the on-axis (drive) torque (Nm), p is the wind dynamic pressure (N/m2 ), 2 and A = π D 4 is the frontal area of the dish (m2 ). Defined in this way the dimensionless torques are independent of the antenna size and wind speed, and their dependence on yaw angle (wind direction with respect to the antenna azimuth position) and on the antenna elevation (or pitch) angle. They have been obtained from the wind tunnel tests, see [1], and are shown in Figs. 5.2, 5.3, and 5.4, solid line. When the torque coefficient, τw , is known (e.g., determined from wind-tunnel testing), the actual torque for the prototype antenna can be obtained from equation (5.1) Tw = p ADτw

(5.2)

which takes into account the size of the antenna (D, A), and the wind speed in form of the dynamic wind pressure (p).

(a)

(b)

wind

d

D

βe Fl

βa wind

Fig. 5.1 Antenna configuration with respect to wind: (a) side view and (b) top view

5.1 Steady-State Wind Disturbance Model

53

0.30

dimensionless yaw torque

0.25 0.20 0.15 0.10 0.05 0.00 –0.05 –0.10

0

20

40

60 80 100 120 140 160 180 yaw angle, deg

Fig. 5.2 Field and wind tunnel yaw torque coefficients, elevation angle 10 deg

dimensionless yaw torque

0.10 0.08 0.06 0.04 0.02 0.00 –0.02

0

20

40

60 80 100 120 140 160 180 yaw angle, deg

dimensionless EL torque

Fig. 5.3 Field and wind tunnel yaw torque coefficients, elevation angle 60 deg

0.20 0.15 0.10 0.05 0.00 –0.05

0

10

20

30

40 50 EL position, deg

60

Fig. 5.4 Field and wind tunnel elevation (pitch) torque coefficients

70

80

90

54

5 Wind Disturbance Models

5.1.2 Obtaining Wind Torques from Field Data It has been recognized that wind tunnel test results need verification, because the tests were run on small models of antennas (Reynolds number effects), and the wind tunnel approach flow characteristics (mean wind profile, turbulence) were not necessarily the same as those in the field wind conditions. The wind tests were conducted on the 34-m antenna located at Goldstone, CA; see [8] and [9]. During the test the antenna orientation was changed in azimuth at a constant velocity of 0.05 deg/s, and the wind speed and direction were recorded at the rate of 40 samples per second, along with the currents in the antenna motor drives in the azimuth and elevation axes. The currents were used to calculate the antenna torques. The dynamic wind pressure can be expressed in terms of the reference wind speed, U, typically the mean wind speed, at the reference location (the elevation of the geometrical center of the antenna) p = 0.5ρU 2

(5.3)

where ρ = 1.29 kg/m3 is the air mass density. For the 34-m antenna, AD = 30,874 m3 , which (when combined with equation (5.3)) gives equation (5.1) as follows τw = αt

Tw U2

(5.4)

where αt = 0.506 × 10−4 (for Tw in Nm and U in m/s). In the field, the torque was measured while moving the antenna in azimuth direction with a constant velocity of 0.05 deg/s. Due to antenna movement the measured torque, Tm , comprised of the torque generated by wind, Tw , and of the torque due to the internal friction, Tf , thus, Tw = Tm − T f

(5.5)

The friction torque was measured when slewing the antenna with the same azimuth velocity and in quiet weather, with the wind speed close to zero. Hence, the dimensionless torque in equation (5.4) is modified to read τw = αt

Tw − T f U2

(5.6)

The dimensionless torques depend on the porosity of the antenna dish; the distance, d, of the dish vertex from the elevation axis; the elevation angle of the antenna, β e ; and the yaw angle, β a . The distance, d, has the same effect on the torques of the 34-m antenna and of the CP-6 Memorandum model. To prove it, note

5.1 Steady-State Wind Disturbance Model

55

that the total azimuth axial torque is a sum of the dish torque, Td , and the torque, Tl , generated by the lateral force, Fl (see Fig. 5.1). From this figure, one obtains Tl = Fl d cos βe

(5.7)

fl d cos βe D

(5.8)

or dividing by pDA one obtains τl =

where βe is the elevation angle, τl is the dimensionless lateral torque, and fl is the dimensionless lateral force, τl =

Tl , p AD

fl =

Fl pA

(5.9)

The ratios: d/D = 0.14 for the CP-6 Memorandum case, and 0.11 for the 34-m antenna; are close enough so that from equation (5.8) it follows that the dimensionless lateral force is about the same in the CP-6 experiment and the 34-m antenna experiment. The porosity of the 34-m antenna dish was about 25% for the outer 25% of the dish radius, which is similar to the CP-6 model. The torque was tested with respect to the two remaining parameters: elevation angle, βe , and yaw angle, βa . Thus, the dimensionless torque was a function of βe and βa , that is, τw = τw (βe , βa ). The dimensionless yaw torques from the CP-6 Memorandum are shown in Figs. 5.2 and 5.3 (for the elevation angle 10 and 60 deg, respectively). Figure. 5.4 shows the pitch torques. It will be useful to express the wind tunnel data in an analytical form, to make them easily available (CP-6 memo is not needed). Thus, the wind-tunnel torques are approximated with Fourier series, and the analytical model is obtained. For azimuth torques the polynomial order was 5, and for elevation torque it was 1. These approximations can be written for both the azimuth and elevation cases as follows tw (α) = ao +

5 i=1

ai sin(2πi

α α ) + bi cos(2πi ) αo αo

(5.10)

where the Fourier coefficients are given in Table 5.1. The variable α is the wind yaw angle if equation (5.10) is used as the yaw torque equation, and α is the antenna elevation position if equation (5.10) is used as the elevation torque equation. The wind tunnel data and the approximating functions are shown in Fig. 5.5. In summary, the torque comparison of field measurements with the wind tunnel data showed a good agreement for the steady-state winds. The differences between the wind-induced steady-state dimensionless torques on the 34-m antenna and the wind tunnel data were less than 10% of the field-measured torques.

56

5 Wind Disturbance Models Table 5.1 Fourier coefficients for the dimensionless azimuth and elevation torque

Fourier coefficient

AZ torque for EL = 10 deg

AZ torque for EL = 60 deg

EL torque

␣o ao a1 a2 a3 a4 a5 b1 b2 b3 b4 b5

180 0.06197 –0.05198 0.005744 –0.01203 –0.001404 0.002014 –0.07083 –0.008286 0.01181 –0.005739 0.003556

180 0.03538 –0.02233 –0.004266 0.0003286 –0.001649 –0.000888 –0.009118 –0.005024 0.0007024 0.0005609 –0.0007596

120 0.06066 –0.03338 0 0 0 0 0.005447 0 0 0 0

5.1.3 Comparing Wind Tunnel Results and the Field Data In the following, the field and wind tunnel data are compared. The dimensionless torques were obtained for the range of yaw angles, from 0 to 180 degrees, with the elevation angle fixed; thus, τw = τw (βa , i), where i is the sample number. They were also measured for a range of elevation angles, from 10 to 89 degrees, with the yaw angle fixed, thus τw = τw (βe , i). Denote the dimensionless torque from the CP-6 Memorandum τw6 (βa ) if it depends on yaw angle or tw6 (βe ) if it depends on elevation angle. The new dimensionless (rescaled) torques, τwn , are obtained through the following linear transformation τwn (β) = a1 τw6 (β) + a2

(5.11)

where β = βa or β = βe . Define the error, ε, between the CP-6 Memo data and the field data as ε(a1 , a2 ) =

i

(τw (β, i) − τw6 (β))2

(5.12)

The coefficients a1 and a2 are determined such that the error is minimal. The parameter a1 is the scaling coefficient that adjusts the CP-6 Memorandum curve τw6 to best fit the field data. The parameter a2 , the shifting coefficient, shifts the field data to compensate for undetermined friction forces. The wind data were collected when the antenna rotated by 360 deg in azimuth with a fixed elevation angle (at 10 or 60 deg) and the azimuth torques were measured. The torques were measured for the average wind speed of 16.7 m/s and an elevation angle of 10 deg. The wind direction was 251 deg with respect to the azimuth zero position. The yaw angle is the difference between the antenna azimuth position and the wind direction. The wind speed ranged from 13 to 19 m/s, as illustrated in Fig. 5.6. The plot of the azimuth torque versus yaw angle is shown in Fig. 5.7. It was obtained by averaging the torques every 5 s, and by subtracting the friction torque,

5.1 Steady-State Wind Disturbance Model

57

0.15 (a)

EL = 10 deg

yaw torque coefficient

0.1

EL = 60 deg 0.05

0

–0.05

0

20

40

60

80 100 yaw angle, deg

120

140

160

180

EL torque coefficient

0.12 (b)

0.1 0.08 0.06 0.04 0.02 0 10

20

30

40 50 60 elevation angle, deg

70

80

mean wind speed, m/s

Fig. 5.5 Polynomial approximations of yaw and elevation torque coefficients

20 18 16 14 12

–150

–100

–50

Fig. 5.6 Mean wind speed versus yaw angle

0 50 yaw angle, deg

100

150

90

58

5 Wind Disturbance Models

azimuth torque, Nm

which was measured for a non-windy day at 17.9 Nm. This plot is expected to be anti-symmetric, and a small departure from the anti-symmetry is caused by the varying wind speed during this experiment. The mean wind speed (a linear fit to the wind speed data) is shown in Fig. 5.6. In order to obtain the results for the constant wind speed, say Uo , the data were scaled by the factor (Uo /U )2 . By further rescaling the torques according to equation (5.6), the dimensionless torque was obtained (Fig. 5.2, dots). The dimensionless torque from the CP-6 Memorandum was best fitted to this data (Fig. 5.2, solid line). It required a scaling factor of a1 = 1.016 and a shifting factor of a2 = 0.0066, which means that the wind tunnel data are 1.6% apart from the field data. The friction torque estimation is about 10% error, because the shift a2 = 0.0066 corresponds to 1.8 Nm friction torque. Two other experiments for an elevation angle of 10 deg required scaling of 1.07 and 1.11; thus, the tunnel data for this case can be considered to be accurate to within 11% or better. A similar experiment was conducted for an elevation angle of 60 deg. The averaged field-measured torques (dots) and the rescaled CP-6 Memorandum data (solid line) are shown in Fig. 5.3. For this case, the scaling was a1 = 1.08 and the shifting was a2 = 0.0008; thus, the difference between the field and the CP-6 Memorandum data was within 8%, and the friction torque correction was about 1%. The elevation (pitch) dimensionless torques were verified for winds blowing from behind the antenna (at yaw angle of 180 deg). The torques were measured for elevation angles ranging from 10 to 89 deg. The friction torques were 5.2 Nm, and the unbalanced torques were 2.0 Nm, with opposite direction to the friction torques (the antenna dish was driven down). These torques were scaled to obtain the dimensionless torques, and the dimensionless torques from the CP-6 Memorandum were fitted to the field data; see Fig. 5.4. The fitting coefficients were: scaling a1 = 0.87, and shifting a2 = –0.013, showing that the CP-6 and field measurements are within a 13% error margin. A slightly larger a2 was due to poorer friction torque estimation (friction torques were obtained for different velocities, and the unbalance torque was not known exactly).

50

0

–50 –150

–100

–50

0 50 yaw angle, deg

Fig. 5.7 Azimuth torque versus yaw angle, elevation angle 10 deg

100

150

5.2 Wind Gusts Disturbance Models

59

5.2 Wind Gusts Disturbance Models There are three ways to model wind gusts acting on an antenna structure: • The first model, where the wind gust disturbances are modeled as a force acting on the antenna dish; see Fig. 5.8, and [5–7]. The white noise of unit standard deviation is filtered by the Davenport filter (on the spectrum of the Davenport wind gusts, see [11]), and appropriately scaled, with scale factor kf . This model is used when the antenna finite-element model is available, and includes forces at the dish panels. It is mainly used during an antenna design stage. • The second model is presented as a time-varying torque, Tw , acting at the antenna drives; see also Fig. 5.8. The wind torque adds-up to the antenna drive torque Tc , producing the total torque T acting at the structure. The Davenport filter is again used to shape white noise into wind gusts, but a different scaling factor (kt ) is now used. This model is often used in antenna analyses, giving adequate

wind 3

wind 2

velocity input

command CONTROLLER

+

+

wind 1 force

torque

+

+ DRIVE

_

encoder

+

velocity feedback position feedback

wind 1

wind 2

wind 3

white noise

white noise

white noise

Davenport filter

Davenport filter

Davenport filter

Δvo

Δvo

Δvo kt

kf

kt Tw

force

Tw

Filter, F velocity input

Fig. 5.8 Wind gust models: (wind 1) wind forces acting on the dish, (wind 2) wind torques acting at the drive motors, and (wind 3) wind velocity acting at the velocity input

60

5 Wind Disturbance Models

approximation of servo errors. It is used, for example, when antenna structure and antenna drives are modeled as separate units. • The third model is used when the antenna and drives are inseparable. It is the case of the antenna velocity loop model obtained from system identification, where an integrated model of the antenna structure and the drives is obtained. In this model, as shown in Fig. 5.8, the velocity command is the input to the model, the encoder reading is its output, and the drive torque is an internal variable. In this situation, the only possible entry point for the wind gusts disturbances is the velocity input. This model is obtained by adding a filter F; see Fig. 5.8. The wind torques are now filtered to obtain an equivalent velocity input signal. The filter transfer function F is such that the action of white noise (see the model in Fig. 5.8) causes the same servo error as the action of white noise in the model shown in the same figure. This chapter presents the development of the Davenport filter, the determination of force (kf ) and torque (kt ) gains, and the determination of the filter transfer function (F) for the 34-m antenna. It also compares servo errors in wind gusts produced by the three wind models and measured at the antenna sites. The Davenport wind gusts model is used; however, other models (e.g., Kaimal, Antanoui) may be more appropriate, depending on the terrain conditions.

5.2.1 Model of Wind Forces Acting on the Dish In this model, as shown in Fig. 5.8, the wind gust is represented as a uniformly distributed force acting on the antenna dish, either from its front, its back, or its side. The gust force is obtained from the gust velocity Δv o of the unit standard deviation, and the velocity Δv o , on the other hand, is obtained from the Davenport spectrum. Consider first the determination of wind velocity Δv o . In this step, wind velocity from the Davenport spectrum is derived. The wind velocity v is a combination of a steady-state, or mean velocity v m , and a turbulence (gust) Δv, that is, v = v m + Δv

(5.13)

The gust component is a random process with zero mean and with a spectrum, called the Davenport spectrum. The Davenport spectrum Sv (ω) depends on average wind speed and terrain roughness, and is given by the following equation: Sv (ω) = 4800v m κ

βω 4

(1 + β 2 ω2 ) /3

(5.14)

600 where β = πv , and κ is the surface drag coefficient, obtained from the roughness m of the terrain; see [11]:

κ=

1 (2.5 ln(z/z o ))2

(5.15)

5.2 Wind Gusts Disturbance Models

61

In these equations, z is the distance from the ground to the antenna dish center, and zo is the height of the terrain roughness (e.g., z o = 0.1 to 0.3 m at Goldstone, CA, where the DSN antennas are located). The wind gust velocity Δv o of the unit standard deviation is obtained by applying a white noise input of unit standard deviation to a filter that approximates the Davenport spectrum. The unit standard deviation is obtained by appropriate scaling of filter gain. This filter is further called a Davenport filter. The filter transfer function is of fourth order and was obtained in [6] and [7] by adjusting the filter parameters such that the magnitude of the filter transfer function best fits the Davenport spectrum within the antenna bandwidth of [0.001, 20] Hz. The resulting filter transfer function is

H=

3.9021s 3 + 230.1426s 2 − 686.3151s + 3.4197 0.331s 4 + 38.2997s 3 + 224.7118s 2 + 22.7788s + 0.3538

(5.16)

The corresponding digital filter for sampling time of 0.02 s is as follows

Hd =

0.1584z3 − 0.3765z2 + 0.2716z − 0.0534 z4 − 2.9951z3 + 3.0893z2 − 1.1930z + 0.0988

(5.17)

This filter is scaled such that applying white noise of unit standard deviation one obtains an output Δv o of unit standard deviation as well. The plot of the square root of the Davenport spectrum and of the magnitude of the filter transfer function is shown in Fig. 5.9, and a sample of the wind speed generated by the filter is shown in Fig. 5.10. In the next step wind force from wind velocity is obtained. In order to do this, consider first a steady wind for which the quadratic law relates its velocity (v n ) and force (Fn ) Fn = k F v n2

(5.18)

The constant k F depends on the scaling of the structural model. In our case modes were scaled such that 44.7 m/s wind corresponds to a force of 4.4482 N. Thus, from the above equation, for this case one obtains k F = 0.0022 N/(m/s)2

(5.19)

with velocity in m/s, and force in N. Equation (5.18) represents a steady (or static) wind force.

62

5 Wind Disturbance Models 102 square root of Davenport spectrum and filter t.f.

Davenport spectrum filter

101

100

10–1 10–4

10–3

10–2

10–1

100

101

frequency, Hz

Fig. 5.9 The Davenport spectrum of the wind velocity and magnitude of the filter transfer function

The next step is to obtain time-varying forces generated by wind gusts using the velocity time history Δv o (t). Consider a long enough time interval (e.g., of 200 s or more). The wind speed over this interval can be decomposed into its constant 4 3

Davenport filter output

2 1 0 –1 –2 –3 –4

0

10

20

30

40

50 time, s

Fig. 5.10 Generic wind speed of unit standard deviation

60

70

80

90

100

5.2 Wind Gusts Disturbance Models

63

(or mean) component v m and variable component Δv of zero mean value, see equation (5.13). The corresponding wind forces are similarly decomposed: F = Fm + Fw

(5.20)

Variable Fm represent the steady-state (static force) component. The wind gust variations are 10%–20% of the static force. The wind gust force variations (Fw ) are related to wind velocity variations Taylor ! " (Δv) by expanding! in " series equation (5.18), obtaining Fw = ⭸F ⭸v v=vm Δv. Because ⭸F ⭸v v=vm = 2v m ,, therefore Fw = 2k F v m Δv

(5.21)

The wind gust Δv obtained in the previous section is scaled to obtain its unit standard deviation, that is, the speed Δv o (t) is such that Δv o (t) =

Δv(t) σv

(5.22)

where σv is the standard deviation of Δv. However, the standard deviation of the wind gust is proportional to the mean wind speed; see [8]: σv = αv n

(5.23)

Combining (5.22) and (5.23) one obtains Δv = αv m Δv o

(5.24)

In the above equations α=

√

6κ

(5.25)

and κ is the surface drag coefficient, defined in equation (5.15). Introducing (5.24) to (5.21) one obtains the final relationship between wing gust velocity and force Fw = k f Δv o

(5.26)

k f = 2k F αv n2

(5.27)

where

For the 34-m k F = 0.00223 Ns2 /m2 , and α = 0.20, thus k f = 0.000892v m2

64

5 Wind Disturbance Models

with velocity in m/s, and force in N. For 8.94 m/s wind the force gain is k f = 0.0713 Ns/m.

5.2.2 Model of Wind Torque Acting at the Drives In this model the wind torque disturbance is added to the drive torque. It is a time function, Tw (t), determined from the velocity gusts Δv o . This model is shown in Fig. 5.8, where the white noise is applied to the Davenport filter. The filter output is the velocity gust Δv o of unit standard deviation, which is consequently scaled to obtain wind torque that is added to the antenna drive torque. In this model the Davenport filter is identical with the filter presented in previous section. The scaling factor, kt , from the velocity to torque is obtained from the wind quadratic law for torques (Tn ) and for steady wind speed v m : Tn = k T v m2

(5.28)

The constant k T in this equation is particular for an antenna, antenna elevation position, antenna site, terrain profile, and wind direction. It is determined as follows. First, the wind torque depends on wind pressure, and the pressure–torque relationship was determined experimentally in wind tunnels [1–3] and in field tests [8] Tn = ct ADpn

(5.29)

where D is the antenna dish diameter (m), and A is the antenna dish frontal area, A = π D2 /4 (m2 ), and ct is a dimensionless torque coefficient. This coefficient depends on the wind direction and the antenna elevation position, and varies from –0.05 to 0.25. Next note that the dynamic pressure of wind, p, depends of wind velocity, similarly to the torque, see [3] and [8]: pn = α p v m2

(5.30)

where α p is the static air density, α p = 0.6126 Ns2 /m4 (pressure is in N/m2 , and velocity is in m/s). Introducing (5.30) to (5.29) one obtains Tn = ct α p ADv m2

(5.31)

Comparing (5.30) and (5.28) the quadratic law coefficient k T is obtained k T = ct α p AD = ct α p

π D3 4

(5.32)

For the 34-m antenna D = 34 m, and for ct = 0.25 (front and side wind) one obtains k T = 4661 Ns2 /m.

5.2 Wind Gusts Disturbance Models

65

The torque equation (5.28) is valid for steady wind only. However, the wind gust torque can be derived from it, using a linear expansion (again, the gust part of the wind is between 10% and 20% of the steady wind that justifies the linearization). Indeed, it is determined from the velocity gusts Δv o by linearizing equation (5.28), which gives ΔT = 2k T v m Δv

(5.33)

The above represents the antenna axis torque. The wind torque at the pinion axis, Tw , is the axis torque ΔT divided by the axis-to-pinion ratio Np ; thus the pinion torque is also proportional to the velocity Tw =

ΔT 2k T v n = Δv Np Np

(5.34)

In simulations the wind speed model with unit standard deviation Δv o (t) is used. It was previously derived that Δv o (t) is related to an arbitrarily scaled wind speed as in equation (5.24). Introducing (5.24) to (5.34) one obtains Tw = kt Δv o

(5.35)

2k T α 2 v Np m

(5.36)

where kt =

For the 34-m antenna k T = 4661 Ns2 /m, α = 0.20 and Np = 42, thus kt = 44.4v m2

(5.37)

where velocity is in m/s, and torque in Nm. For the elevation axis one obtains kt = 294.7v m2

(5.38)

5.2.3 Algorithm to Generate a Time Profile of Wind Gusts Torque 1. Apply white noise to the filter H, with the following transfer function Hd =

0.1584z3 − 0.3765z2 + 0.2716z − 0.0534 z4 − 2.9951z3 + 3.0893z2 − 1.1930z + 0.0988

and obtain time series of wind Δv(t). The above filter represents the Davenport wind gusts model. 2. Calculate ␴v the standard deviation of Δv(t).

66

5 Wind Disturbance Models

3. Divide Δv(t) by ␴v , obtaining wind speed with unit standard deviation Δv o (t) = Δv(t)/σv The standard deviation of Δv o (t) is 1. 4. Scale Δv o (t) by kt to obtain Tw (t) Tw (t) = kt Δv o (t) where kt =

2kα 2 v Np m

and vn is the mean wind speed, Np is the gear ratio between antenna √ main 6κ, and axis and the drive pinion, k is the “wind quadratic law,” α = κ = (2.5 ln(z/z o ))−2 . In the latter equation, z is the distance from the ground to the antenna dish center, and zo is the height of the terrain roughness.

5.2.4 Model of Wind at the Velocity Input In this model the wind gusts is applied at the velocity input. It is shown in Fig. 5.8, where the white noise is applied to the Davenport filter; its output is the velocity gust Δv o of unit standard deviation, appropriately scaled with gain kt to obtain wind torque. Next, the torque is filtered with filter F that produces a velocity, which is added to the velocity input of the antenna. In this model the Davenport filter and the scaling factor, kt , are identical with the filter and factor presented in the previous section. The task of this section is to find the filter transfer function F such that the encoder response to the wind disturbances is almost the same as the response for the wind torque acting at the drives. In order to have the antenna response the same as for the wind torque acting at the drives the transfer function of the filter should be the inverse of the drive transfer function. The drive transfer function Fd (s), from the velocity input to the torque output is shown in Fig. 5.11, solid line. Within the antenna bandwidth it can be approximated with the integrator, that is, with Fd (s) = ksd , where kd is the drive gain. The magnitude of the approximate transfer function is shown in Fig. 5.11, dashed line. Thus, the filter transfer function, as an inverse of the approximate drive transfer function, is a derivative with the inverse gain F(s) =

s kd

(5.39)

5.2 Wind Gusts Disturbance Models

67

El drive approx. EL drive

109

torque, Nm

gain = 1.1*108

108

f = 1/2π

107

106 –2 10

10–1

100

101

frequency, Hz

Fig. 5.11 Transfer function of the EL drive (solid line), and its approximate (dashed line)

The corresponding discrete-time filter of sampling time Δt is as follows F(z) =

1 z−1 kd z Δt

(5.40)

For the 34-m antenna the drive gain is kd = 1.1 × 108 [Nms/deg].

5.2.5 Algorithm to Generate Time Profile of Wind at the Velocity Input 1. Follow the steps of the algorithm in Section 5.2.3 to obtain Tw (t). 2. Filter Tw (t) to obtain uw (t). Use filter F with the transfer function as follows F(z) =

1 z−1 kd z Δt

where kd is the drive gain from input u to torque T, and Δt is the sampling time.

5.2.6 The Equivalence of Wind Torque and Wind Velocity Models The nature of the first model, which consists of wind forces acting on the dish, is different than the second and third models, which consists of torque and velocity signals acting on the antenna drives. Thus, the first and the second model and the first and the third model can be only compared in statistical terms, such as standard deviation of the resulting antenna servo error. However the second and the third

68

5 Wind Disturbance Models

(a) 3 EL encoder, mdeg

2 1 0 –1 –2 –3

0

10

20 time, s

30

40

(b) EL encoder, mdeg

3 2 1 0 –1 –2 –3

0

1

2

3

4

5 time, s

6

7

8

9

10

Fig. 5.12 Open loop EL encoder response to the wind: torque input (solid line) and velocity input (dashed line): (a) 40 s response, and (b) 10 s segment

models are closely related and they can be compared directly. Thus, the antenna positions excited by the wind gusts generated by the second model (wind torque), and by the third model (wind input velocity) are compared. In order to do this, one applies the white noise as marked in Fig. 5.8, Wind 2 and Wind 3, and with the position loop open. The resulting elevation encoder reading is shown in Fig. 5.12a,b. The plots show that the encoder outputs for the two wind cases are very close, indicating close equivalence of the second and the third models.

5.2.7 Closed Loop Pointing Accuracy with Wind Gusts Disturbances The wind gusts using all three models were simulated. Then the antenna servo error in azimuth and elevation for the front and side wind directions were determined, and compared with the field data. The Simulink model was developed for the 34-m antennas, and allows for simulation three wind models; shown in Fig. 5.13a. The elevation drive model is shown in Fig. 5.13b. A sample of simulation results is shown in Fig. 5.14. It represents the azimuth and elevation position errors obtained from model of wind velocity acting at the velocity input for 32 km/h wind

5.2 Wind Gusts Disturbance Models

69

(a) EL wind 3

EL wind 2

side wind

5

3

6 EL encoder 1 EL pinion rate

EL

4

2 EL point

servo error vel. comm.

1

y(n) = Cx(n) + Du(n) x(n+1) = Ax(n) + Bu(n)

EL drive

EL comm

m

EL controller 3 Structure

8

XEL point

servo error vel. comm.

AZ comm

8

AZ controller

AZ drive

AZ pinion rate 4 AZ encoder

4

2

7

AZ wind3

AZ wind2

front wind

AZ

(b) kcur

1

k1*ks

KTsz

KTsz

rate to V

KTsz

kf*ki

kr

rate

1/N

z-1

z-1

1/Ra

1/Jm

km

KTsz

z-1

kg/N z-1

MOTOR AMPLIFIER

1 Torque

GEARBOX kb

ktach

N

2 Pinion Rate

Fig. 5.13 (a) The Simulink model of the 34-m antenna with three wind model inputs: front wind1 and side wind1 for model 1, AZ wind2 and EL wind2 for model2, and AZ wind3 and EL wind3 for model3; (b) EL drive Simulink model composed of amplifier, motor and gearbox models 2

elevation servo error, mdeg

1.5 1 0.5 0 –0.5 –1 –1.5 –2 –2

–1

0 1 azimuth servo error, mdeg

2

Fig. 5.14 34-m antenna azimuth and elevation position in 32-km/h wind gusts blowing from its side

70

5 Wind Disturbance Models Table 5.2 Standard deviations of 34-m antenna servo errors due to 32 km/h wind gusts Front wind Field data1 Model 1 Model 2 Model 3 1

Side wind

AZ (mdeg)

EL (mdeg)

AZ (mdeg)

EL (mdeg)

0.23–0.48 0.33 0.32 0.44

0.6–1.2 1.73 1.20 1.21

0.23–0.48 0.30 0.32 0.44

0.9–1.5 1.02 1.20 1.07

From [4]

gusts blowing from the side of 34-m antenna. The simulation results are shown in Table 5.2, where they are compared to the measured servo errors from [4]. The servo errors were measured repeatedly, and the table shows the minimal and maximal values of the standard deviation of the errors. The table shows that the simulation results are within the range of measurements; except for the elevation error in front wind of model 1, which exceeds the measurements.

References 1. Blaylock RB. (1964). Aerodynamic Coefficients for Model of a Paraboloidal Reflector Directional Antenna Proposed for a JPL Advanced Antenna System. JPL Memorandum CP-6 (internal document), Jet Propulsion Laboratory, Pasadena, CA. 2. Fox NL. (1962). Load Distributions on the Surface of Paraboloidal Reflector Antennas. JPL Memorandum CP-4 (internal document), Jet Propulsion Laboratory, Pasadena, CA. 3. Fox NL, Layman, Jr. B. (1962). Preliminary Report on Paraboloidal Reflector Antenna Wind Tunnel Tests. JPL Memorandum CP-3 (internal document), Jet Propulsion Laboratory, Pasadena, CA. 4. Gawronski W. (1995). Wind Gust Models Derived from Field Data. TDA Progress Report 42-123, pp. 30–36. Available at http://ipnpr.jpl.nasa.gov/progress˙report/42-123/123 G.pdf. 5. Gawronski W. (2004). Modeling Wind Gusts Disturbances for the Analysis of Antenna Pointing Accuracy. IEEE Antennas and Propagation Magazine, 46(1): 50–58. 6. Gawronski W, Bienkiewicz B, Hill RE. (1994). Wind-Induced Dynamics of a Deep Space Network Antenna. Journal of Sound and Vibration, 178(1). 7. Gawronski W, Mellstrom JA. (1994). Control and Dynamics of the Deep Space Network Antennas. In: Control and Dynamics Systems, ed. C.T. Leondes, vol. 63, Academic Press, San Diego, pp. 289–412. 8. Gawronski W, Mellstrom JA. (1994). Field Verification of the Wind Tunnel Coefficients. TDA Progress Report 42-119, pp. 210–220. Available at http://ipnpr.jpl.nasa.gov/ progress˙report/42-119/119 G.pdf. 9. Gawronski W, Mellstrom JA, Bienkiewicz B. (2005). Antenna Mean Wind Torques: A Comparison of Field and Wind Tunnel Data. IEEE Antennas and Propagation Magazine, 47(5). 10. Rae, Jr., WH, Pope A. (1984). Low-Speed Wind Tunnel Testing. John Wiley, New York. 11. Simiu E, Scanlan RH. (1996). Wind Effects on Structures. John Wiley, New York.

Part II

Control

Chapter 6

Preliminaries to Control

Before starting the controller tuning process, one has to know the criteria that the closed loop system should satisfy. Also, one has to prepare the plant model (in our case the velocity loop model) such that it is the convenient configuration for the controller tuning. This chapter defines the performance criteria used in the controller tuning process. Also, before starting the tuning process, the plant model (antenna velocity loop model) is transformed and augmented to make the tuning more straightforward, and analysis physically interpretable.

6.1 Performance Criteria The controller performance is evaluated using the following criteria: • Settling time of a small step response, shown in Fig. 6.1. Settling time is defined as the time at which the antenna encoder output remains within ±3% threshold of the nominal value of the step command. For example, a 5.1 s settling time due to a 0.02 deg step command is illustrated in Fig. 6.1. The settling time indicates how fast antenna reacts to the command. One does not use the settling time obtained at large position offsets (larger than 0.1 deg), because at these offsets antenna reaches acceleration and velocity limits. • Overshoot of a small step response is illustrated in Fig. 6.1. Overshoot (in percent) is the relative difference between the maximal encoder output and the commanded step with respect to the value of the commanded step. In Fig. 6.1 the overshoot is 20%. It is measured at small position offsets (e.g., 0.02 deg). • Overshoot of a large step response is measured in absolute values (deg) rather than in percentage. • Steady-state error in velocity offsets; see Fig. 6.2. This represents a lagging of the antenna when commanded at constant velocity. • Bandwidth of the closed loop transfer function, illustrated in Fig. 6.3. Bandwidth is the frequency at which the magnitude drops 3 dB, or to 70.7% below zero dB level. This is illustrated in Fig. 6.3, where the bandwidth is 0.37 Hz. The wider is the bandwidth, the faster and more precise is the antenna. W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 6,

73

74

6 Preliminaries to Control 0.025 X: 5.1 Y: 0.02055

overshoot

antenna pos ition, deg

0.02 settling time +/– 3% threshold 0.015

0.01

0.005

0

0

1

2

3

4

5 time, s

6

7

8

9

10

antenna c ommand and pos ition, deg

Fig. 6.1 Settling time (5.1 s) and overshoot (20%) in a step response

12 10

servo error

8 6 4 command antenna position

2 0 0

2

4

6

8

10

12

14

16

18

time, s

Fig. 6.2 Steady-state (mean) servo error in velocity offsets

• Amplitude and settling time of a disturbance step; see Fig. 6.4. Wind disturbance in a form of rapid (stepwise) action is suppressed by the controller counteraction. The time required to suppress it, and amplitude of the antenna movement, are measures of the controller performance (the smaller is the amplitude and reaction time, the better is the controller). • Magnitude of the disturbance transfer function; see Fig. 6.5. The lower is the magnitude, the better are the wind disturbance rejection properties of the controller.

6.1 Performance Criteria

75

100

magnitude

X: 0.37 Y: 0.7

10–1 bandwidth

10–2 –3 10

10–2

10–1 frequency, Hz

100

101

Fig. 6.3 Determining bandwidth (0.3 Hz) of the closed loop system from the magnitude of its transfer function 0.8

antenna pos ition, deg

0.7 0.6 0.5 dmax

0.4 0.3 0.2 0.1 0

0

1

2

3

4

5 time, s

6

7

8

9

10

Fig. 6.4 Max servo error due to step disturbance

• Root-mean-square servo error in 10 m/s wind gusts; see Fig. 6.6. In this figure the wind gust simulations are used to evaluate the servo performance in wind. • Phase and gain stability margins; see Fig. 6.7. Note that gain crossover is the frequency at which the open loop magnitude first reaches the value of unity, and phase crossover is the frequency at which the open loop phase angle first reaches the value of –180 deg. Thus, gain margin is the factor by which the open loop magnitude must be multiplied to destabilize the system, and phase margin is the

76

6 Preliminaries to Control

magnitude

100

10–1

10–2 10–3

10–2

10–1 frequency, Hz

100

101

Fig. 6.5 Disturbance transfer function

3

elev at ion s erv o error, mdeg

2

1

0

–1

–2

–3 –3

–2

–1 0 1 azimuth servo error, mdeg

2

3

Fig. 6.6 AZ and EL servo errors in wind gusts

number of degrees of delay to destabilize the system. The gain and phase margins are measures of stability robustness. They show how much the system gain can be changed to destabilize the closed loop system, and how much signal delay the closed loop system can tolerate.

6.2 Transformations of the Velocity Loop Model

77

101 gain = 1

magnitude

100

gain margin

10–1

gain crossover

10–2

10–3 –2 10

–1

0

10

1

10

10

frequency, Hz 0

phas e, deg

–100

phase = –180 deg

phase crossover

phase margin

–200

–300

–400

10–2

10–1

100

101

frequency, Hz

Fig. 6.7 Gain and phase margins

6.2 Transformations of the Velocity Loop Model The selection of the suitable coordinates for the antenna velocity loop model is crucial in the tracking controller tuning process. The state-space model (as obtained from the system identification) is transformed into coordinates that are convenient for the tuning of the antenna controller. The proposed velocity loop model transformation consists of three stages: 1. transformation into modal coordinates (makes the structural modes independent); 2. transformation to coordinates with the antenna position as the first state (allows for addressing the tracking requirements); 3. augmentation with the integral of the position (eliminates lagging in constant velocity tracking).

78

6 Preliminaries to Control

6.2.1 Transformation into Modal Coordinates In the first step, the velocity loop model (A, B, C) is transformed into the model in modal coordinates (Am , Bm , Cm ), with the modal state vector xm ; for the details see Chapter 2. The modal state-space representation is characterized by the blockdiagonal state matrix Am , and by the weakly coupled modal states, xmT = T T T [x # 1m , x2m , ·$· ·], where the ith component xim is the state of the ith mode, xim = xi1m xi2m . The weak coupling allows adjusting gains for each modal state independently, which vastly simplifies the controller tuning process.

6.2.2 Antenna Position as the First State Antenna position is a variable that defines antenna performance; therefore it shall be directly controlled. It could be achieved if the position is a part of the antenna state-space representation. Thus, the modal model is transformed further, to obtain the new state x p , where the first state is the antenna position, while the remaining states are unchanged. The new state, after transformation, is as follows y (6.1) xp = xf where y is the antenna angular position, and x f are the remaining (unchanged) states. The new state-space representation (A p , B p , C p ) is obtained using the following transformation x p = P xm

(6.2)

where, for n states, one has

Cm P= , Po

(6.3)

where Po = 0n−1,1 In−1 , and 0n−1,1 is zero vector (n − 1) × 1, and In−1 is the identity matrix of order n − 1. The transformation was derived by noting that y = Cm xm , and partitioning Cm as follows: Cm = [Cm1 Cm2 ], where Cm1 is the first component of Cm , and Cm2 are the remaining components.

6.2.3 Augmentation with the Integral of the Position The model is also augmented with the integral of the position, as recommended in [1, 2, 3], to eliminate the steady-state errors in constant-velocity tracking and to improve the antenna disturbance rejection properties. Thus the integral of the

References

79

position (denoted yi ) is added to the state-space model as a new state, obtaining the state xo ⎧ ⎫ ⎨ yi ⎬ yi y = xo = (6.4) xp ⎩ ⎭ xf The new state yi satisfies the following equation y˙ i = y, that is, y˙ i = C p x p

(6.5)

which combined with the antenna state space representation ( A p , B p , C p ) x˙ p = A p x p + B p u y = Cpxp gives the new representation ( Ao , Bo , Co ) 0 Cp 0 ; Bo = ; Ao = 0 Ap Bp

(6.6)

Co = 0 C p

(6.7)

The state xo as in (6.4) is the desired state vector of the antenna velocity loop system, and (6.7) is the preferred velocity loop model of the antenna. The velocity loop state consists of the integral of the position, the position, and the flexible deformations in modal coordinates.

References 1. Athans M. (1971). On the Design of PID Controllers Using Optimal Linear Regulator Theory. Automatica, vol. 7: 643–647. 2. Johnson CD. (1968). Optimal Control of the Linear Regulator with Constant Disturbances. IEEE Trans. Automatic Control, vol. 13: 416–421. 3. Porter B. (1971). Optimal Control of Multivariable Linear Systems Incorporating Linear Feedback. Electronic Lett., vol. 7: 170–174.

Chapter 7

PI and Feedforward Controllers

Tuning process of the antenna controller is discussed in two steps: 1. analysis of a simple (i.e., rigid) antenna and simple (PI) controller (to derive basic properties of the closed loop system), 2. extension of the properties of a simple system to the real (flexible) antenna and a complex (LQG or H∞ ) controller. The first step is discussed in this chapter, the second step in the next two chapters. The combination of PI controller and rigid antenna allows for the closed-form analysis. The analysis includes the relationship between the proportional and integral gains and the closed loop properties of the antenna. This leads to the tuning method for the PI controller and to the closed loop equations of a flexible antenna with the PI controller. Next, the performance of the PI controller and its limits are discussed. Finally, the feedforward controller is analyzed.

7.1 Properties of the PI Controller The block diagram of the antenna control system is shown in Fig. 7.1a. For a rigid antenna the position proportional-and integral (PI) controller is selected, as shown in Fig. 7.1b. The velocity loop of a rigid antenna is a pure integrator, while the flexible antenna velocity loop is designed so that it is approximately an integrator, as illustrated in Fig. 7.2. The magnitude of the transfer function of a perfect integrator is shown in Fig. 7.2 as a straight dashed line sloping at –20 dB/dec. The magnitude of the transfer function of the 34-m flexible antenna obtained from the field test is shown in Fig. 7.2, solid line, as a straight line sloping at –20 dB/dec for low frequencies (up to 1 Hz), and showing flexible deformations (resonances) at higher frequencies. (Structural and drive flexibility that cause deformations are not a part of the integration.) The closed loop system is shown in Fig. 7.1.a, where K denotes the controller transfer function, and G is the antenna transfer function. A rigid antenna as a pure W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 7,

81

82

7 PI and Feedforward Controllers (a)

w r

controller (K)

y

antenna (G)

uo

y

(b) PI controller

kp

r

+

e

+ _

y

∫

ei

ki

uo

+

Fig. 7.1 Antenna position loop (a) and its PI controller (b): k p , proportional gain; ki , integral gain; r, command; y, antenna position; e. servo error; ei , integral of the servo error; uo . velocity command

102

101

magnitude

flexible antenna 100

10–1 rigid antenna

10–2

10–3 10–2

10–1

100

101

frequency, Hz

Fig. 7.2 Magnitudes of the transfer function of the velocity loop model of the 34-m antenna (solid line), and a rigid antenna (dashed line)

7.1 Properties of the PI Controller

83

integrator and the PI position controller have the following transfer functions G=

1 ki and K = k p + s s

(7.1)

where k p is the proportional gain and ki is the integral gain of the PI controller. Also in Fig. 7.1, r is a position command, uo is the controller output (called velocity command), w is wind disturbance, y is the antenna angular position, and e = r – y is a servo error.

7.1.1 Closed Loop Transfer Functions Besides the variables listed in Fig. 7.1, also the commanded acceleration a (a derivative of the controller input uo ) is considered, in order to explain the impact of the command r and disturbances w on the antenna position y and the commanded acceleration a. Thus, the following transfer functions are analyzed: • • • •

Try – from the command to the angular position encoder; Twy – from the disturbance to the encoder; Tra – from the command to the acceleration; and Twa – from the disturbance to the acceleration. From the block diagram in Fig. 7.1 one obtains Tr y =

GK G and Twy = 1 + GK 1 + GK

(7.2)

Tra =

sK −s K G and Twa = 1 + KG 1 + KG

(7.3)

and introducing (7.1) – (7.3) gives Tr y =

k p s + ki s and Twy = 2 s 2 + k p s + ki s + k p s + ki

(7.4)

Tra =

(k p s + ki )s 2 −(k p s + ki )s and Twa = 2 2 s + k p s + ki s + k p s + ki

(7.5)

7.1.2 The Proportional Gain Analysis The controller tuning starts with the selection of the proportional gain, thus assuming ki = 0 in the above transfer functions, one obtains 1 and Twy = Tr y = 1 s +1 kp

1 kp 1 s kp

+1

(7.6)

84

7 PI and Feedforward Controllers

Tra =

s2 and Twa = 1 s+1 kp

−s +1

(7.7)

1 s kp

The magnitudes of the above transfer functions, for different values of the proportional gain, are shown in Fig. 7.3. One can see from this figure that the increase of the proportional gain: 1. increases the bandwidth of the transfer function Tr y (from the command to the antenna position); see Fig. 7.3a, 2. improves the disturbance rejection properties of the antenna by lowering the magnitude of the disturbance rejection transfer function Twy ; see Fig. 7.3b, 3. increases the impact of the command on the antenna acceleration (increases the magnitude and the bandwidth of the acceleration transfer function Tra ); see Fig. 7.3c, 4. increases the impact of disturbances on the antenna acceleration (increases the magnitude and the bandwidth of the acceleration transfer function Twa ); see Fig. 7.3d. 10

1

10

1

10

10

10

(b) magni tude, Twy

magnitude, Try

(a) 0

–1

kp –2

10

10

–2

0

10 frequency, Hz

10

10 10

–3 –2

(d)

–2 0

10 frequency, Hz

0

10 frequency, Hz

10

2

10

10

10

10

2

2

kp

0

–2

kp

–2

10

2

10

–1

10

magni tude, Twa

magni tude, Tra

10

2

10

4

(c) 10

10

10

0

kp

1

0

–1

10

–2

0

10 frequency, Hz

10

2

Fig. 7.3 Magnitude of transfer functions of the proportional controller for kp = 1 (solid line), for kp = 4 (dashed line), and for kp = 16 (dotted line): (a) Try , from the command to the encoder, (b) Twy , from the disturbance to the encoder, (c) Tra , from the command to the acceleration, and (d) Twa , from the disturbance to the acceleration

7.1 Properties of the PI Controller

85

The first two transfer functions show the improvement of the antenna performance with the increase of the proportional gain. However, the last two functions show a potential problem: antenna acceleration increases at high frequencies, both due to command and due to disturbances. The increased acceleration indicates that the antenna can hit the acceleration limit and enter a nonlinear regime; consequently its performance will deteriorate, leading even to instability. Thus, the proportional gain increase is limited by the acceleration limits imposed at the antenna drives.

7.1.3 The Integral Gain Analysis Before the impact of the integral gain on antenna dynamics is analyzed, two critical values are introduced: the critical integral gain and the critical frequency. Critical integral gain. It is well known that large integral gain causes lowfrequency oscillations of the closed loop system. Consider the poles s1 and s2 of the closed loop system (7.4) and (7.5), that is, the roots of the polynomial s 2 + k p s + ki : (7.8) s1,2 = 0.5 −k p ± k 2p − 4ki √ The system is non-oscillatory if poles are real, which happens when k p > 2 ki , or when ki ≤ 0.25k 2p

(7.9)

kic = 0.25k 2p

(7.10)

Thus

is the upper limit of the integral gain, called the critical integral gain. Critical frequency. For the critical integral gain the denominator of the transfer functions shown in (7.4) and (7.5) is as follows: s 2 + k p s + 0.25k 2p = (s + 0.5k p )2 . At frequency (7.11) ωo = 0.5k p = ki the slope of the transfer function Tr y drops by –40 dB/dec. This is the critical frequency of the closed loop system that determines the antenna bandwidth. In the following, the frequencies significantly smaller than ωo are called low frequencies; frequencies significantly larger than ωo are called high frequencies; and frequencies in the neighborhood of ωo are called medium frequencies. By considering low, medium, and high frequencies in (7.4) and (7.5) one can show how the transfer functions depend on the integral gain. Note first that for medium frequencies the variations of all four transfer functions are minimal (see Fig. 7.4) because the integral gain is smaller than the critical integral gain. For low and high frequencies the transfer functions behave as follows:

86

7 PI and Feedforward Controllers 1

0

10

10

(b) magnitude, Twy

magnitude, Try

(a) 0

10

kp

–1

10

–2

10

–2

0

10

kp

10–2

10–3 –2 10

2

10 frequency, Hz

10–1

10

0

2

10 frequency, Hz

10

2

10

(c)

magnitude, Tra

10

(d) magnitude, Twa

4

kp

2

10

0

10

–2

10

1

10

0

kp

10

–1

–2

10

0

10 frequency, Hz

2

10

10

–2

10

0

10 frequency, Hz

2

10

Fig. 7.4 Magnitude of transfer functions of the PI controller for kp = 16, and ki = 1 (solid line), for kp = 16, and ki = 4 (dashed line), for kp = 16, and ki = 16 (dotted line), and for kp = 16,: (a) Try , (b) Twy , (c) Tra , and (d) Twa

1. The transfer function Tr y does not depend on ki , because for low frequencies Tr y ∼ = 1 and for high frequencies Tr y ∼ = k p /s, as shown in Fig. 7.4a. 2. The transfer function Twy is inversely proportional to ki for low frequencies, because Twy ∼ = s/ki ; for high frequencies it does not depend on ki because Twy ∼ 1/s, as shown in Fig. 7.4b. = 3. The transfer function Tra does not depend on ki , because for low frequencies Tra ∼ = s 2 , for high frequencies Tra ∼ = k p s, as shown in Fig. 7.4c. 4. The transfer function Twa does not depend on ki , because for low frequencies Twa ∼ = −s; for high frequencies Twa ∼ = −k p as shown in Fig. 7.4d. The above analysis showed that the integral gain impacts the disturbance rejection transfer function Twy only, and that the impact is at low frequencies.

7.2 PI Controller Tuning Steps The PI controller tuning involves: 1. Tuning the proportional gain. Increase the gain until the antenna reaches the acceleration limits at typical commands and at expected disturbances, and set it at 75% of this value.

7.4 Performance of the PI Controller

87

2. Tuning the integral gain. Increase the gain until oscillations or undershoot appear, and set it at 75% of this value. It should be smaller than the critical integral gain. In summary, the proportional gain shapes the bandwidth of the transfer function Tr y . The larger is the gain, the wider is the bandwidth. For a rigid antenna the limit for the proportional gain is set by the antenna acceleration limits, because the increase of proportional gain increases antenna acceleration caused by commands and disturbances; see Fig. 7.3c,d. The integral gain shapes the disturbance transfer function at low frequencies. It is increased to obtain better disturbance rejection properties. But there is a limit: the integral gain should not exceed the critical integral gain (i.e., should satisfy condition (7.9), to prevent antenna oscillations.

7.3 Closed Loop Equations of a Flexible Antenna with a PI Controller Consider a flexible antenna with the state-space representation of the velocity loop (A, B, C). For the system as in Fig. 7.1, the closed loop equations are as follows x˙ cl = Acl xcl + Bcl r + Bw w y = Ccl xcl

(7.12)

where xcl =

ei , x

1 , Bcl = Bk p

Acl =

0 −C , Bki A − Bk p C

0 Bw = , I

Ccl = 0

(7.13)

C

(7.14)

7.4 Performance of the PI Controller The performance of the closed loop system depends on the PI controller gains. However, the performance characteristics of the controller, such as settling time, bandwidth or the rms servo error in wind gusts, are inter-related.

7.4.1 Performance Characteristics Our analysis includes a rigid antenna. The rigid antenna is considered to be a perfect antenna, with no flexible deformations, for which the controller needs to overcome inertia forces only. The closed loop transfer function, from the command to the

88

7 PI and Feedforward Controllers 14 ki = 0.2 kp ki = 0.4 kp ki = 0.6 kp

12

setting time, s

ki = 0.8 kp

10

(a) 8 6 4 2 0.5

1

1.5

2

2.5

3

2

2.5

3

kp

bandwidth, Hz

ki = 0.2 kp

0.6

ki = 0.4 kp ki = 0.6 kp

0.5

ki = 0.8 kp

(b) 0.4 0.3 0.2 0.1 0.5

1

1.5 kp

rms servo error in wind gusts, mdeg

5 ki = 0.2 kp

4.5

ki = 0.4 kp ki = 0.6 kp

4

ki = 0.8 kp

3.5 3

(c) 2.5 2 1.5 1 0.5 0.5

1

1.5

2

2.5

3

kp

Fig. 7.5 The performance parameters of the rigid antenna with PI controller as a function of the controller gains: (a) settling time, (b) bandwidth, and (c) rms servo error in 20-mph wind gusts

7.4 Performance of the PI Controller

89

antenna position, and from the wind disturbance to the antenna position is given by (7.12). Using these equations, the step responses, frequency responses and wind errors simulated (see [1]) for the proportional gain from the range of k p = [0.5–3.0], and the integral gain, from the range ki = [0.2kp – 0.9 kp ]. As a result, the settling time, bandwidth and servo error in wind gusts (wind gusts are modeled as a random process with zero mean and a Davenport spectrum) to obtained. The results are plotted in Fig. 7.5a–c. The plots show that

0.6 ki = 0.2kp ki = 0.4kp ki = 0.6kp ki = 0.8kp

bandwith, Hz

0.5

0.4

(a) 0.3

0.2 3

4

5

6

7 8 setting time, s

9

10

11

12

rms servo error in wind gusts, mdeg

4 ki = 0.2kp ki = 0.4kp ki = 0.6kp ki = 0.8kp

3.5 3

(b) 2.5 2 1.5

1

3

4

5

6

7 8 setting time, s

9

10

11

12

Fig. 7.6 The performance parameters of the rigid antenna with PI controller as a function of the settling time: (a) bandwidth, and (b) rms servo error in 20-mph wind gusts

90

7 PI and Feedforward Controllers

• the settling time significantly decreases with the increase of the proportional gain, and also decreases with the increase of the integral gain. • The bandwidth increases linearly with the proportional and integral gains. • The servo error in wind decreases with the increase of the proportional gain and with the increase of the integral gain. Figures 7.5a–c can be recombined to show how the bandwidth and the servo error in wind gusts depend on the settling time; see Fig. 7.6a,b. The bandwidth decreases and the servo error increases with the increase of the settling time.

7.4.2 Limits of Performance It was shown in the above sections that the performance of the PI controller improves with the increase of its proportional and integral gains. Unfortunately, when applied to a flexible antenna, the increase of the gains has its limits: high proportional gain causes instability. Indeed, assume a zero integral gain, and the proportional gain k p = 0.6, then the response of the closed loop system to a 20-mdeg step command is shown in Fig. 7.7, dash line. It has no overshoot and a settling time is 6 s. However, increasing the gain to 1.6 produces 2 s settling time, but the system becomes unstable, with undying vibrations (see Fig. 7.7). Thus, antenna compliance is the source of the controller instability. A controller that suppresses the antenna vibrations when the gains are increased would significantly improve antenna precision. Model-based controllers, considered in subsequent chapters, act in the abovedescribed manner. 25

antenna position, mdeg

20

kp = 1.6

15 kp = 0.6 10

5

0

0

1

2

3

4

5 time, s

6

7

8

9

10

Fig. 7.7 Step responses of the PI controller with flexible antenna: low gain controller (dashed line), and high gain controller (solid line)

7.5 Feedforward Controller

91

7.5 Feedforward Controller The feedforward loop is added to the PI controller to improve its tracking accuracy at high velocities [2]. In the feedforward loop the command is differentiated and forwarded to the velocity loop input, as a velocity command, see Fig. 7.8. Consider the open loop system with feedforward and velocity loop in series; see Fig. 7.8b. The transfer function of this system is approximately equal to 1. Indeed, the derivative is an approximate inversion the velocity loop transfer function, so the series of the two gives transfer function equal to 1. It is illustrated with the magnitude of the velocity loop transfer function G r is shown in Fig. 7.9, approximated (up to 1 Hz) with an integrator (G ra = 1/s), shown in the same figure, dashed line. The feedforward transfer function is a derivative (G f f = s) shown in Fig. 7.9, dashdotted line. Thus the overall transfer function is a series of the feedforward and the velocity loop G o = G r G f f , which is approximately equal to 1 up to the frequency 1 Hz (dashed line). The position feedback is added to compensate disturbances and system imperfections. Note, however, that the disturbance transfer function is not affected by the feedforward loop. Thus, feedforward improves command following, but not disturbance rejection properties. The impact of the feedforward loop on the 34-m antenna performance is illustrated as follows. The step response of the 34-m antenna with a PI controller is shown in Fig. 7.10, solid line. The settling time is 15 s. Adding the feedfordward controller makes the antenna response to the command significantly faster (4 s settling time), but response is more oscillatory. The latter is eliminated by using command pre-processor, discussed in Chapter 10. Note also that the feedforward (a)

d/dt

+ command

–

velocity command

position controller

+

+

velocity loop

position

(b) command

d/dt

velocity command

velocity loop

position

Fig. 7.8 Antenna control system with the feedforward controller: (a) closed loop, and (b) open loop

92

7 PI and Feedforward Controllers 2

10

1

10

0

10

–1

10

velocity-loop velocity-loop approx. feedforward velocity-loop+feedforward

–2

10

–3

10

–2

–1

10

0

10

1

10

10

frequency, hz

Fig. 7.9 The feedforward action is illustrated with the magnitudes of the transfer functions: the velocity loop transfer function G r , the velocity loop transfer function approximation, G ra , the feedforward loop transfer function, G f f , and G o = G f f G r is the transfer function of the series connection of feedforward and velocity loop

loop increases the bandwidth of the antenna transfer function from the command to the encoder from 0.1 Hz to 1.4 Hz; see Fig. 7.11. However, the transfer function from the disturbance to the encoder remains unchanged.

0.03

azimuth position, deg

0.025 0.02 0.015 0.01 0.005 0

with feedforward without feedforward 0

5

10

15

20

Fig. 7.10 Step responses of the antenna with the PI controller, and the antenna with PI and feedforward controller

References

93 1

10

0

magnitude

10

–1

10

–2

10

–3

10

with feedforward without feedforward –4

10

–2

10

–1

10

0

10

1

10

Fig. 7.11 Magnitudes of the transfer functions from the command to the encoder of the antenna with PI controller, and the antenna with PI and feedforward controller

References 1. Gawronski W. (2007). Servo Performance Parameters of the NASA Deep Space Network Antennas. IEEE Antennas and Propagation Magazine, 49(6). 2. Gawronski W, Mellstrom JA. (1994). Antenna Servo Design for Tracking Low-Earth-Orbit Satellites. AIAA J. Guidance, Control Dynamics, 17(6):1179–1184.

Chapter 8

LQG Controller

This chapter presents the tuning process of the LQG controller. Its applications in the antenna industry are discussed in many papers, but scarcely implemented. LQG controllers are implemented at the 34- and 70-m Deep Space Network antennas. The discussion begins with the short description of the LQG controller, and its tracking version, gives the closed loop equations of a flexible antenna with the LQG controller, and shows the similarity of the PI and LQG controllers. It discusses how to select the LQG weights to obtain the required performance, present the controller tuning steps, and show the limits of the performance. Next, it presents two graphical user interfaces (GUI) to tune the LQG controllers. The GUIs do not require detailed knowledge of the tuning process to obtain excellent closed loop performance. The chapter also discusses the LQG controller in the velocity loop.

8.1 Properties of the LQG Controller The LQG controllers are well established; see, for example, [4–6], [8], [10], [12–17]. However, their tuning (called also design) for antenna tracking purposes is a tricky process. The controller must address the antenna tracking requirements (such as minimization of the antenna servo error in wind gusts, and fast response to commands) while complying with antenna limitations (such as acceleration and velocity limits). Thus, blindly applying the standard LQG tuning steps can lead to an optimal controller that minimizes the LQG performance index but might not satisfy the performance requirements, because “optimal” does not mean “satisfactory.” One has to remember that every LQG controller is an optimal controller because it minimizes the LQG performance index, defined later. But, the LQG index seldom includes engineering requirements, thus the LQG controller with poorly defined index would have non-satisfactory performance requirements.

8.1.1 LQG Controller Description A block diagram of the antenna LQG control system is shown in Fig. 3.7. Assuming the command zero, and no limits imposed on the velocity and acceleration, the W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 8,

95

96

8 LQG Controller

Fig. 8.1 The LQG closed loop system: G, antenna velocity loop; K, controller; u, velocity input; and y, encoder

u

G

y

K

control system is reduced to the one in Fig. 8.1. It consists of the antenna velocityloop (G) and controller (K). The antenna output y is the encoder measurement and is supplied to the controller. Using the output y the controller determines the control signal u that drives the antenna. The inside structure of the antenna and the controller is shown in Fig. 8.2. The plant is described by the following state-space equations: x˙ = Ax + Bu + v, y = C x + w,

(8.1)

as shown in Fig. 8.2. In the above description the plant (antenna velocity loop) state vector is denoted x. The plant is perturbed by random disturbances, denoted v, and its output is corrupted by noise w. The noise v, called process noise, has covariance V = E(vv T ); the noise w is called measurement noise, and its covariance is W = E(ww T ). Both noises are assumed uncorrelated, that is, E(vw T ) = 0, Antenna (G)

A

v

w u

B

+

+

–

x

x

∫

+

C

+

y

y u Controller (K )

C

yˆ −

A u

Kc

xˆ

+ Ke

∫

xˆ +

+ –

u

B

Fig. 8.2 The inner structure of the LQG closed loop system

y

8.1 Properties of the LQG Controller

97

where E(.) is the expectation operator. Without loss of generality, it is assumed that the covariance of the measurement noise is unity, that is, W = I The controller is driven by the plant output y. The controller produces the control signal u that drives the plant. This signal is proportional to the plant estimated state denoted xˆ , and the gain between the state and the controlled signal u is the controller gain (K c ). The estimated state xˆ rather than the actual state x is used, because the latter is not available from measurements (except for the antenna position). The estimated state is obtained from the estimator, which is part of the controller, as shown in Fig. 8.2. The estimator equations follow from the block-diagram in Fig. 8.2: x˙ˆ = A xˆ + Bu + K e (y − C xˆ ).

(8.2)

Assuming that the plant model (A,B,C) is known exactly (e.g., from system identification), one can see that the estimated state is an exact copy of the actual state, except for the initial (transient) dynamics. From the above equation it follows that in order to determine the estimator one has to determine the estimator gain, K e . The controller gains are obtained by minimizing the performance index J, ⎛

⎞ ∞ J 2 = E ⎝ (x T Qx + u T Ru) dt ⎠

(8.3)

0

In the above definition Q is a positive semi-definite weight matrix and R is a positive scalar. Assume R = 1 because the case R = 1 is equivalent to R = 1 with the scaled weight matrix Q/R. The minimum of J is obtained for the feedback u = −K c xˆ

(8.4)

K c = BoT Sc ,

(8.5)

with the gain K c obtained as

and Sc is the solution of the controller algebraic Riccati equation (called CARE) A T Sc + Sc A − Sc B B T Sc + Q = 0.

(8.6)

One can see that the controller gain K c depends solely on the weight matrix Q (A and B are fixed). The optimal estimator gain is given by K e = Se C T ,

(8.7)

98

8 LQG Controller

where Se is the solution of the estimator (or filter) algebraic Riccati equation (called FARE) ASe + Se A T − Se C T C Se + V = 0.

(8.8)

Using (8.2) and the block-diagram in Fig. 8.2 the controller state-space equations from input y to output u are derived: x˙ˆ = (A − B K c − K e C)xˆ + K e y, u = −K c xˆ .

(8.9)

From these equations the controller state-space representation (Alqg ,Blqg ,Clqg ), are obtained Alqg = A − B K c − K e C, Blqg = K e , Clqg = −K c .

(8.10)

The controller gain K c is obtained in Matlab as follows: Kc = lqr(A,B,Q,R) and in the discrete-time case Kc = dlqr(A,B,Q,R) Similarly, the estimator gain K e is obtained in Matlab as follows: Ke = lqe(A,I,C,Q,R) or, in the discrete time case, Ke = dlqe(A,I,C,Q,R) The matrices A, B, C, Q, and R are described earlier, and I is the identity matrix.

8.1.2 Tracking LQG Controller Above, the LQG controller was described for the zero (or constant) command. However, the antenna must follow a variable command, and the controller should

8.1 Properties of the LQG Controller

99

Structure (G) A

v +

u

+ +

B

x

w x

∫

+

C

+

y

u

y Controller (K ) +

–

+

ki

kf

C

Cf

A

kp

e

xˆ

e

∫

yˆ −

∫

+y

Ke xˆ +

+ +

r +

−

Cp u

B

Fig. 8.3 The tracking LQG controller with an integral upgrade

assure zero steady-state tracking error. It is achieved by adding an integral of the plant position to the plant state-space representation; see Section 6.2.3. The closed loop system configuration of the tracking LQG controller is shown in Fig. 8.3. In this figure (A,B,C) is the plant state-space triple, x is the state, xˆ is the estimated state, xˆ f is the estimated state of a flexible part, r is the command, u is the control input, y is the output, yˆ is the estimated output, e = r − yˆ is the servo error, ei is the integral of servo error, v is the process noise of intensity V, and the measurement noise w is of intensity W. Both v and w are uncorrelated: E(vw T ) = 0, V = E(vv T ), W = E(ww T ) = I , E(v) = 0, and E(w) = 0. Similarly to the actual state x, see (8.4), the estimated state consists of three components: estimated integral of the position yˆ i , estimated position yˆ , and estimated flexible mode state xˆ f : ⎧ ⎫ ⎨ yˆ i ⎬ xˆ = yˆ (8.11) ⎩ ⎭ xˆ f In order to make a stabilizing controller into a tracking controller, the antenna estimated position yˆ is replaced with the antenna servo error e = r − )y, and the integral of position is replaced with the integral of the servo error ei = e dt; see [2, pp. 603–604], thus ⎧ ⎫ ⎨ eˆ i ⎬ xˆ = eˆ (8.12) ⎩ ⎭ xˆ f

100

8 LQG Controller (a)

w r controller ( K)

y

antenna (G)

uo

y

(b)

LQG controller PI controller

kp

r y

+

e

+ _

∫

ei

+ ki

u

+

Flexible Mode controller

Kf

xˆ f

u Estimator

yˆ _ +

eˆ

Ke

Fig. 8.4 Antenna position-loop (a) and its LQG controller (b): k p , proportional gain; ki , integral gain; K f , flexible mode gain; K e , estimator gain; r, command; y, antenna position; e, servo error, ei , integral of the servo error; u, velocity command; y, antenna angular position; yˆ , antenna estimated position; eˆ , estimation error; and xˆ f , estimated flexible mode states

The antenna position loop with the LQG controller is shown in Fig. 8.4a (the same structure as with the PI controller), and the LQG controller structure is shown in Fig. 8.4b. The LQG controller structure corresponds to the above representation. In this representation the controller gain is divided into the proportional gain k p , integral gain ki , and flexible mode gain K f , that is, K c = [K i K p K f ], which gives (8.9) in the form u = −K i ei − K p e − K f xˆ f . The missing part of the controller is the estimated state xˆ in (8.4). It is obtained from the estimator, which is a computer model of the antenna. Its equation is as follows: x˙ˆ = A xˆ + Bu + K e (y − C xˆ ). with the estimator gain given by (8.7). The controller gain is divided according to the state (8.12), that is,

(8.13)

8.1 Properties of the LQG Controller

101

K c = ki

kp

Kf

(8.14)

Introducing (8.14) and (8.12) to (8.4) one obtains u = −ki ei − k p e − K f xˆ f

(8.15)

8.1.3 Closed Loop Equations of the Tracking LQG Control System For the system as in Fig. 8.3 the closed loop equations are as follows x˙ cl = Acl xcl + Bcl r + Bv v y = Ccl xcl + w

(8.16)

where ⎧ ⎫ ⎨ei ⎬ xcl = x , ⎩ ⎭ xˆ

⎡

0 Acl = ⎣ Bki Bki ⎡

⎤ 1 Bcl = ⎣ Bk p ⎦ , Bk p

⎤ −C ⎦ , (8.17) −Bk p C − Bk f C f A − ke C − Bk p C − Bk f C f

0 A ke C

⎡ ⎤ 0 Bv = ⎣ I ⎦ , 0

Ccl = 0

C

0

(8.18)

8.1.4 LQG Weights For the open loop state-space representation (A,B,C ) of a flexible structure the state vector x is divided into the tracking, xt , and flexible, x f , parts, that is, x=

xt . xf

(8.19)

The tracking part includes the position error (e), and its integral (ei ), xt =

ei e

(8.20)

102

8 LQG Controller

while the flexible mode part includes modes of deformation. For this division the system triple can be presented as follows:

At A= 0

At f , Af

Bt B= , Bf

C = Ct

0 .

(8.21)

The gain, K c , the weight, Q, and solution of CARE, Sc , are divided similarly to x, K c = K ct Q=

Kc f ,

Qt 0 , 0 Qf

Sct Sct f . Sc = T Sct f Sc f

(8.22)

The tuning process is carried out such that the tracking system is of low authority, that is, such * that * the flexible weights are much smaller than the tracking ones: Q t * Q f *. It was shown in [1] that for Q f = 0 one obtains Sc f = 0 and Sct f = 0. This means that the gain of the tracking part, K ct , does not depend on the flexible part. And, for the low-authority tracking system (with small Q f ), one obtains weak dependence of the tracking gains on the flexible weights, due to the continuity of the solution. Similar conclusions apply to the FARE equation (8.8). This property can be validated by observation of the closed loop transfer functions for different weights. Consider the transfer function of the Deep Space Network antenna, as in Fig. 8.5. Denote by In and 0n the identity and zero matrices of order n, then the magnitude of the closed loop transfer function (azimuth angle to azimuth command) for Q t = I2 and Q f = 010 is shown as a dashed line, for Q t = 8× I2 and Q f = 010 as a dot–dashed line, and for Q t = 8I2 and Q f = 5 × I10 as a solid line in Fig. 8.5. It follows from the plots that variations in Q f changed the properties of the flexible subsystem only, while variations in Q t changed the properties of both subsystems. Note, however, that large Q f increases dependency of the gains on the flexible system; only quasi-independence in the final stage of controller tuning is observed, while separation in the initial stages of controller tuning is still strong. The tuning consists therefore of the initial choice of weights for the tracking subsystem, and determination of the controller gains of the flexible subsystem. It is followed by the adjustment of weights of the tracking subsystem, and a final tuning of the flexible weights, if necessary. In conclusion, in modal coordinates the LQG weight matrix is selected as a diagonal matrix (due to independence of states in modal coordinates), that is, Q = diag(qi , q p , q f )

(8.23)

8.1 Properties of the LQG Controller

103

magnitude

100

10–1

10–2

Qt = I2, Qf = 010 Qt = 8*I2, Qf = 010 Qt = 8*I2, Qf = 5*I10

10–3 10–3

10–2

10–1 frequency, Hz

100

101

Fig. 8.5 Magnitudes of the transfer function of a closed loop system for different LQG weights: Q f impacts the flexible modes (higher frequencies), while Q t impacts the low and high frequencies

where qi is the integral weight, q p is the proportional weight, and q f is a vector of flexible mode weights. It is convenient to present the LQG weights in the vector form as the LQG weight vector q ⎧ ⎫ ⎨ qi ⎬ q = qp ⎩ ⎭ qf

(8.24)

The weight vector q corresponds to the state vector xo , and the flexible mode weights are divided into weights of each mode ⎫ ⎧ ⎪ ⎬ ⎨q f 1m ⎪ q f = q f 2m ⎪ ⎭ ⎩ .. ⎪ .

(8.25)

$ # with each mode weight q Tfim = q f i q˙ f i corresponding to the two-state mode. For the antenna model in modal coordinates the modal states are weakly coupled. They are also almost independent from the antenna position and integral of the position. Thus, the corresponding weights act independently on each flexible mode, and almost independently on the position and on the integral of the position states. This adds to the flexibility to the controller tuning.

104

8 LQG Controller

8.1.5 Resemblance of the LQG and PI Controllers It was shown in Chapter 7 that for a rigid antenna the increase of the proportional gain improves antenna bandwidth and the disturbance rejection properties. However, an increase of proportional gain, when applied to a flexible antenna, is drastically limited: even a moderate gain can excite structural vibrations and cause instability. However, because the LQG controller consists of the PI part and the flexible mode part, see Fig. 8.4, the flexible mode controller is able to suppress antenna vibrations. In this way, the increased proportional gain does not cause instability: a flexible antenna behaves approximately as a rigid one. Therefore the controller tuning approach used for a rigid antenna with a PI controller can be extended (with certain limitations) for tuning the LQG controller of a flexible antenna. The limitations are formulated as follows: the action of the flexible mode gains should not be excessive; the gains should be large enough to assure vibration suppression. Such a controller is a low-authority LQG controller. The following example shows the similarity between the rigid antenna with a PI controller and a flexible antenna with a LQG controller. Consider the 34-m antenna velocity loop model with a transfer function from the velocity input to the position output shown in Fig. 7.2, solid line. At lower frequencies the transfer function is identical to the transfer function of a rigid antenna, and at higher frequencies it shows flexible mode resonances. The LQG controller to this antenna is applied as follows. First, the weights of three LQG controllers are selected, with their integral gain zero, and proportional gains 1, 4, and 16, respectively. For these cases the plots of magnitudes of the transfer functions Try , Twy , Tra , and Twa are shown in Fig. 8.6. Comparing Fig. 8.6 and Fig. 7.3 one can see similarities between the rigid antenna with a PI controller and a flexible antenna with a LQG controller. Namely, the plots of Try show the expanding bandwidth with the increase of the proportional gain. The plots of Twy show the decreasing antenna response to disturbances with the increase of the proportional gain. The plots of Tra and Twa show increased acceleration response at high frequencies. Next, the weights of the LQG controller are selected to obtain a fixed proportional gain, k p = 16 and the integral gains 1, 4, and 16, respectively. Note from (8.16) that the critical integral gain is 64 in this case. The plots of magnitudes of the transfer functions Try , Twy , Tra , and Twa for the above three cases are shown in Fig. 8.7. Comparing Fig. 8.7 and Fig. 7.4 one can observe similarities between the rigid antenna with a PI controller and a flexible antenna with a LQG controller. The integral gain impacts significantly the disturbance rejection properties (Twy ) only, and there is no significant impact on the closed loop bandwidth (see Try plot) or on the system acceleration; see the plots of Tra and Twa . Finally, the impact of flexible mode weights on antenna dynamics is analyzed. Figure 8.8 presents the magnitudes of the transfer functions Try , Twy , Tra , and Twa for fixed proportional and integral gains (k p = 9.5 and ki = 6.3) and for small flexible mode weights (solid lines) and for large flexible mode weights (dashed lines). The plots show that the excessive flexible mode weights reduce the

8.1 Properties of the LQG Controller

105

101

magnitude, Twy

magnitude, Try

kp 100

10–1

10–2 10–2

kp

100 frequency, Hz

10

10–2

102

10–2

104

100 frequency, Hz

102

102

102

magnitude, Twa

magnitude, Tra

0

kp

100

101

100 kp

10

–1

–2

10–2

100 frequency, Hz

102

10

10–2

100 frequency, Hz

102

Fig. 8.6 Magnitude of transfer functions of the LQG controller for ki = 0: kp = 1 (dot-dash line), kp = 4 (dotted line), and kp = 16 (solid line): (a) Try , (b) Twy , (c) Tra , and (d) Twa .

closed-loop bandwidth (Fig. 8.8a), and deteriorate the disturbance rejection properties (Fig. 8.8b).

8.1.6 Properties of the LQG Weights The above comparison indicates that the LQG weights have similar impact on a flexible antenna performance as PI gains on a rigid antenna performance, assuming the proper choice of coordinate system. The following list summarizes the properties of the LQG weights: • The increase of the flexible mode weights causes antenna vibration suppression. A single mode weight impacts only states corresponding to this particular mode (the flexible mode coordinates are weakly coupled). • The increase of the proportional weight increases the closed loop bandwidth and improves the disturbance rejection properties.

106

8 LQG Controller 100

magnitude, Twy

magnitude, Try

101

100

10–1

10–2 10–2

100 frequency, Hz

102

10–1

ki 10–2

10–3 10–2

100

10–2

102

100 frequency, Hz

102

102

magnitude, Twa

magnitude, Tra

105

100 frequency, Hz

100 frequency, Hz

102

101

100

10–1 10–2

Fig. 8.7 Magnitude of transfer functions of the LQG controller for kp = 16: ki = 1 (dash-dot line), ki = 4 (dot line), and ki = 16 (solid line): (a) Try , (b) Twy , (c) Tra , and (d) Twa

• The increase of the integral weight improves the disturbance rejection properties, but does not impact the bandwidth.

The position and integral of the position weights are coupled, but they are easily manageable because the coupling involves two variables only. Finally, Figs. 8.9 and 8.10 show the dependency of the proportional and integral gains on the proportional and integral weights of the 34-m antenna. First, one can see that the increase of the proportional weight increases both proportional and integral gains. Similarly, the increase of the integral weight increases simultaneously proportional and integral gains. Next, one can see that the weight-gain dependency relies on the ratio of the integral-to-proportional gain (except for the relationship between the proportional weight and integral gain). The plots may serve as a guide in selection of the proportional and integral weights.

8.1 Properties of the LQG Controller

107

(a)

(b)

magnitude, Twy

magnitude, Try

101

100

10–1

10–2 10–2

100 frequency, Hz

102

(c)

10–2

10–2

100 frequency, Hz

102

100 frequency, Hz

102

(d)

102 magnitude, Twa

104 magnitude, Tra

100

102

100

10–2 –2 10

100 frequency, Hz

102

101

100

10–1 10–2

Fig. 8.8 Magnitude of transfer functions of the LQG controller for kp = 9.5 and ki = 6.3: flexible weights small (solid line), and flexible weights large; overdamped modes (dotted line): (a)Try , (b) Twy , (c) Tra , and (d) Twa

8.1.7 Limits of the LQG Gains The increases of the proportional, integral, and flexible mode gains have their limits, namely: • Increase of flexible mode weights should not be excessive. Large weights lead to the overdamped dynamics and deteriorate overall antenna performance (reduced bandwidth, depreciated disturbance rejection properties). • The position weight is restricted by the antenna acceleration limit: Large position weight causes excessive acceleration of the antenna that hits the acceleration limit, and leads to non-linear dynamics and deterioration of the performance. • The integral weight should not exceed the critical integral gain in order to prevent low-frequency oscillations (the frequency of these oscillations is lower than the antenna fundamental frequency).

108

8 LQG Controller 3

2.5

kp

2

1.5 ki = 0.2kp ki = 0.4kp ki = 0.6kp ki = 0.8kp

1

0.5

0

1

2

3 proportional weight

4

5

6

3

2.5

kp

2

1.5

ki = 0.2kp ki = 0.4kp ki = 0.6kp ki = 0.8kp

1

0.5

0

1

2

3

4 5 integral weight

6

7

8

Fig. 8.9 The dependency of the proportional gain on the proportional and integral weights

8.2 LQG Controller Tuning Steps Based on the above analysis, the LQG controller tuning involves: 1. Tuning the flexible mode weights. Apply small weights of the integral of the position and position (which result in small PI gains), and also apply small flexible

8.2 LQG Controller Tuning Steps

109

2.5

2

ki

1.5

1 ki = 0.2 kp ki = 0.4 kp ki = 0.6 kp ki = 0.8 kp

0.5

0

0

1

2

3 4 proportional weight

5

6

7

2.5

ki = 0.2 kp ki = 0.4 kp ki = 0.6 kp ki = 0.8 kp

2

ki

1.5

1

0.5

0

0

1

2

3 4 integral weight

5

6

7

Fig. 8.10 The dependency of the integral gain on the proportional and integral weights

mode weights. Check the closed loop transfer function for the appearance of flexible mode resonances. If they are excessive, increase the corresponding flexible mode weights. Do not apply unnecessarily large weights, because overdamped modes impact the antenna tracking performance. 2. Tuning the proportional weight. Increase the position weight: the proportional gain increases accordingly. The increase of the position gain causes the expansion

110

8 LQG Controller

of the closed loop bandwidth. Increase the weight until bandwidth reaches the antenna fundamental frequency. 3. Tuning the integral weight. Increase the weight the position integral, causing the increase of the integral gain. The weight should increase until oscillations appear. The integral gain should satisfy the condition (8.15). 4. Correction of the flexible mode weights. Check the flexile mode dynamics. If resonances resurface after tuning the proportional and integral parts, increase the corresponding flexible mode weights.

8.3 Performance of the LQG Controller The primary measure of antenna control system performance is the antenna servo error while tracking in wind gusts. However, the error is not easily measured, and antenna performance is frequently poorly estimated. The difficulty of evaluating the servo error in wind gusts led to the search for alternative (or indirect) measures of the antenna pointing performance in the wind. The control systems of the 34-m and 70-m Deep Space Network antennas [7] is analyzed. It was observed that the servo performance parameters of the antennas, such as settling time and bandwidth, are related to the servo error in wind gusts. Qualitatively, the shorter the settling time, or the wider the bandwidth, the smaller is the servo error in wind gusts. In this section, the qualitative knowledge is transformed into a quantitative one. The relationships between controller gains and performance parameters have been established, as well as between the parameters themselves. They allow one to estimate the antenna servo error in wind (which is not easy to measure) using the settling time (which is simple to measure). Because the results of the analysis are similar for the 34-me and 70-m antennas, it is presumed that they are valid for many other antennas and radio-telescopes.

8.3.1 Summary of the Antenna Servo Performance Characteristics In this section, the antenna servo performance is characterized by the following parameters: • Settling time (s) • Bandwidth (Hz) • Root-means-square (rms) servo error in wind gusts (mdeg) The ultimate measure of antenna servo performance is the rms servo error (called further servo error, for simplicity) in wind gusts during spacecraft tracking. It consists of the constant (mean) component and the variable component. The constant component is required to be zero. The variable component is characterized by its standard deviation (or rms error).

8.3 Performance of the LQG Controller

111

The servo error measurements are straightforward, but require long logging time to obtain reliable statistical estimates. They depend on the wind velocity, wind direction with respect to the antenna position (yaw angle), and antenna elevation position. The important factor, not to be ignored, is the availability of wind gusts of significant power during measurements. For this reason, measurements of the servo error in wind gusts require time, patience, and significant data processing, which explain why they are rather scarce. Occasional servo performance measurements and analysis results indicated a connection between the servo error and other antenna parameters, such as settling time or bandwidth, and also a connection between the servo error and antenna servo proportional and integral gains. The importance of these facts lays in the fact that measurement of the settling time or bandwidth is much simpler and less time consuming than measurement of the error in wind gusts, while the servo gains do not need to be measured—they are given.

8.3.2 Performance of the DSN Antennas with LQG Controllers In this section only the elevation axis controller of the 34-m DSN antennas is analyzed. For the azimuth axis the relationships are similar; see [7]. The performance parameters have been determined as follows: For a selected proportional and integral gain, the LQG controller was tuned and the performance parameters for this controller ware simulated (using Matlab). The process was repeated for another set of proportional and integral gain values. Figure 8.11a–d show the dependency of the settling time, overshoot, bandwidth, and servo error in wind gusts on the proportional and integral gains of the LQG controller. Figure 8.12a,b show the dependency of the bandwidth and the servo error in wind gusts on the antenna settling time. In Figure 8.13a–c, the performance parameters of a rigid antenna, a 34-m antenna, and a 70-m antenna is compared. The comparison is presented for the integral gain being half of the proportional gain, ki = 0.5 k p , which is typical for the DSN antennas. The figures show that for the same gains • The settling time is smallest for the rigid antennas. However, for other antennas, it is similar; the difference does not exceed 0.5 s. • The 34-m and 70-m antenna bandwidths slightly exceed the rigid antenna bandwidth. • The servo error in wind gusts is definitely smallest for the rigid antenna. • All the DSN antennas have very similar servo error in wind gusts. The unexpected result of the lowest bandwidth of the rigid antenna (compared to 34-m and 70-m antennas) is due to the integral action of the controllers. For the rigid antenna, the magnitude of the transfer function is less “inflated” by the integral gain, giving slightly lower bandwidth.

112

8 LQG Controller 14 ki = 0.2 kp ki = 0.4 kp ki = 0.6 kp ki = 0.8 kp

12

settling time, s

10 8

(a)

6 4 2 0.5

1

1.5

2

2.5

3

2

2.5

3

kp

ki = 0.2 kp ki = 0.4 kp ki = 0.6 kp ki = 0.8 kp

bandwidth, Hz

0.6 0.5

(b) 0.4 0.3 0.2

rms servo error in wind gusts, mdeg

0.1 0.5

1

1.5

kp

5 ki = 0.2 kp ki = 0.4 kp ki = 0.6 kp ki = 0.8 kp

4.5 4 3.5

(c)

3 2.5 2 1.5 1 0.5

1

1.5

2

2.5

3

kp

Fig. 8.11 The performance parameters of the 34-m antenna (elevation axis) ith LQG controller as a function of the controller gains: (a) settling time, (b) bandwidth, and (c) rms servo error in 20 mph wind gusts

8.3 Performance of the LQG Controller

113

ki = 0.2 kp ki = 0.4 kp ki = 0.6 kp ki = 0.8 kp

0.55

bandwidth, Hz

0.5 0.45 0.4 (a)

0.35 0.3 0.25 0.2 0.15

3

4

5

6

7 8 settling time, s

9

10

11

12

4.5

rms servo error in wind gusts, mdeg

ki = 0.2 kp 4

ki = 0.4 kp ki = 0.6 kp

3.5

ki = 0.8 kp

3 (b)

2.5 2 1.5 1

3

4

5

6

7

8

9

10

11

12

settling time, s

Fig. 8.12 The performance parameters of the 34-m antenna (elevation axis) with LQG controller as a function of the settling time: (a) bandwidth, and (b) rms servo error in 20-mph wind gusts

8.3.3 Disturbance Rejection Properties and the Position-Loop Bandwidth It has been observed that the expansion of the bandwidth of a telescope improves its disturbance rejection properties (resulting in smaller servo errors in wind disturbances). It can be explained using Fig. 8.4a. The position loop transfer function

114

8 LQG Controller (a)

8 70 m antenna 34 m antenna rigid antenna

7.5

settilng time, s

7 6.5 6 5.5 5 4.5 4

0.5

1

1.5 2 proportional gain,kp

2.5

3

2.5

3

(b) 0.6 70 m antenna 34 m antenna rigid antenna

0.55

bandwidth, Hz

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.5

1

1.5 2 proportional gain,kp

rms servo error in wind gusts, mdeg

(c) 2.5 70-meter antenna 34-meter antenna rigid antenna

2

1.5

1

1

1.5

2 proportional gain,kp

2.5

3

Fig. 8.13 The comparison of elevation axis performance of the rigid antenna, 34-m antenna, and 70-m antenna: (a) settling time, (b) bandwidth, and (c) rms servo error in 20-mph wind gusts

8.3 Performance of the LQG Controller

115

Tr y (from r to y) and the disturbance transfer function Twy (from w to y) are as follows: Tr y =

GK , 1 + GK

Twy =

G , 1 + GK

where K is the controller transfer function and G is the velocity loop transfer function. It follows from the above equations that Tr y +

Twy = 1, G

which, consequently, gives Twy = G(1 − Tr y ). Note that within the bandwidth, by definition, Tr y ∼ = 1, which implies – from the above equation – that Twy ∼ = 0, i.e., the disturbance error is small within the bandwidth. Thus, if the bandwidth covers the wind disturbance spectrum, the telescope servo error due to wind is small. Note, however, that G is a part the above equation, and that for telescopes G is large for low frequencies and small for high frequencies. Thus, Tr y ∼ = 1 should be tight at low frequencies, and not necessarily tight at high frequencies. The bandwidth of Tr y is limited by the fundamental frequency of the telescope. Thus, if it reaches the fundamental frequency, there is little room left for the improvement of the disturbance rejection properties. In conclusion, the following rules of thumb can be introduced: 1. The bulk improvement of the disturbance rejection properties is due to the bandwidth expansion. 2. The fundamental frequency of the telescope limits its disturbance rejection properties.

8.3.4 Performance Comparison of the PI and LQG Controllers The servo errors were measured at the 34-m antenna at the Madrid Deep Space Communication Complex, during 15–16 km/h wind; see [9]. The measurements were taken with the PI coefficients and with the LQG coefficients, one after another, so that similar wind conditions were observed. The wind data were obtained one sample per minute, while the servo data were obtained 10 samples per second. The mean wind velocity during the PI test was 15.0 km/h, and during the LQG test was 16.2 km/h. The measured servo errors were scaled to wind velocity 16 km/h. ! "2 = 1.15, and for the LQG The scaling factor for the PI measurements was 16 15 ! 16.0 "2 measurements it was 16.2 = 0.99. The scaled measured servo errors are plotted in Fig. 8.14. The plots show that the elevation servo errors are significantly larger than the azimuth servo errors, which is

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15

EL servo error, mdeg

10 5 0 –5 –10 –15 –20 –20

–15

–10

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10

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15

20

20 LQG controller

15

EL servo error, mdeg

10 5 0 –5

–10 –15 –20 –20

–15

–10

–5 0 5 AZ servo error, mdeg

Fig. 8.14 The servo errors in 16.1 km/h (10 mph) wind, for the DSS55 antenna with the PI coefficients (upper plot), and with the LQG coefficients (lower plot)

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Table 8.1 Servo errors in 16.1 km/h wind for 34-m antenna with PI and LQG controllers Controller

AZ rms error mdeg

EL rms error mdeg

Mean radial error mdeg

PI LQG Error ratio (PI/LQG)

0.32 0.066 4.9

2.69 0.42 6.5

2.71 0.42 6.5

typical for the 34-m antennas. They also show that the servo errors for the antenna with LQG coefficients are significantly smaller than the errors with the PI coefficients (4.9 times smaller in azimuth and 6.5 times smaller in elevation); see Table 8.1. The mean radial error (rms of azimuth and elevation errors) dropped by factor 6.5.

8.3.5 Limits of Performance It is not difficult to obtain a high-gain LQG controller with outstanding wind disturbance rejection properties. However, simulations and measurements results indicate that such a controller can be implemented only if the drives are powerful enough. The power of a drive is expressed in the antenna acceleration limits. Because the bulk of the drive power is used to accelerate the antenna, the acceleration limit thus expresses the drive capability to react to the antenna dynamics. The limit (on the input signal u that drives the antenna) is applied to prevent overheating of the motors. For the 34-m antenna, the acceleration limits are ±0.4 deg/s2 . But the acceleration limit effectively cancels the benefits of a high-gain controller. Even small motion may cause the limit violation. Consider the high-gain LQG controller in response to a 20-mdeg step offset. It accelerates up to 10 deg/s2 . Because the acceleration is limited to 0.4 deg/s2 (Fig. 8.15b), the antenna control system is in a nonlinear regime that causes the oscillations, see Fig. 8.15a.

8.4 Tuning a LQG Controller Using GUI A user-friendly graphical user interface (GUI) was developed to simplify the tuning process and create an environment in which one with limited control engineering background may “play” to obtain a tracking algorithm that ensures the desired performance [15]. In this section two Matlab based–software packages for the controller tuning—basic LQG tuning and fine-tuning of the algorithm—are described.

8.4.1 Selecting LQG Weights The process of determining the LQG gains is explained in detail in Section 8.1. Here, the approach is explained in a heuristic way. The controller gains depend

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azimuth position, mdeg

40 30 20 10 0

0

1

2

3

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5

4

5

azimuth acceleration, deg/s2

time, s 0.5 accel. limit 0.4 deg/s2

0

accel. limit 0.4 deg/s2 –0.5

0

1

2

3 time, s

Fig. 8.15 Step response and acceleration of a high-gain LQG controller

on the weight matrix Q and covariance matrix V. The weight matrix shapes the optimization index, a positive variable to be minimized. The covariance matrix specifies the noise in the system, and it impacts the estimator gains. The difficulty arises when one tries to express the performance of the antenna (such as the rms servo error in wind, closed loop bandwidth, etc.) through the values of Q and V. A closed-form relationship does not exist between those matrices and the required behavior of an antenna. Thus, an immediate solution is a “trial-and-error” approach. However, there are too many parameters in Q and V to make this approach effective. Moreover, Q and V depend on the choice of the state-space coordinates, with some coordinates more useful than others. Physical coordinates, such as structural displacements, motor torques or currents, although easy to interpret, are highly coupled; they therefore create undesirable difficulties in the tuning process. The modal coordinates simplify the tuning because they are weakly coupled (i.e., modifications of one of them weakly impact the remaining ones). In consequence, Q and V matrices are diagonal (therefore there is a smaller number of parameters to control the closed loop dynamics). Additionally, Q = V is assumed. This simplification eases the search for the “best” controller and is rather opportunistic, with the goal to simplify the GUI approach. Experience shows, however, that the results obtained

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119

using this assumption are satisfactory or even exceed the expectations. The reader shall note that in individual cases, the tuning of Q and V separately may improve the performance. Taking into account the above considerations, the matrices Q and V are equal and diagonal, that is, Q = V = diag(q), where q is a weighting vector. Note that components of q correspond to the components of the state vector x. Thus, for the state vector as in (8.24), the weighting vector has the form similar to x # q = qei

qe

qf1

qf1

···

qfn

qfn

$

(8.26)

where qei is the weight of the integral of the servo error, qe is the weight of the servo error, q f 1 is the weight of the first natural mode, and q f n is the weight of the nth natural mode. Now, one has to specify 2n + 2 weights, which, for a typical number n = 4 it makes 10 weights. The critical fact is that the weights have weak dependence amongst each other: the ith modal weight, q f i , impacts mostly the ith mode, that is, state variables, x f i1 , and x f i2 . Also, the “tracking” weights, qe , and qei , impact mostly the “tracking” states, e, and ei . The tuning process is illustrated in the next section.

8.4.2 GUI for the LQG Controller Tuning The GUI display is shown in Fig. 8.16. The GUI allows for simple manipulations of the tuning parameters and observations of the antenna performance to make tuning decisions. 1. On the left-hand side of the interface, there is a short description of the tool. 2. The frame below the description contains a Matlab’s data file that holds the A, B, and C matrices. 3. The eight sliders and their three displays are the user tools to modify the controller’s performance. 4. Each slider (except the wind-speed slider) ranges from 0 to 100 (these are dimensionless numbers). 5. In the bottom right corner of the GUI screen are the simulation results and the values of the proportional and integral gains. 6. The upper-left plot is the step response to a one-degree offset. The upper-right plot is the magnitude of the transfer function of the closed loop system. 7. The “simulate” button is used to execute the simulation with the parameters on the screen. On the GUI, the antenna performance is displayed, such as settling time, bandwidth, overshoot, rms error in wind gusts for the selected wind speed, steady-state error when the antenna is moving with a constant velocity. Also, the proportional and integral gains are displayed.

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Step response

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0

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80 70 60 50 40

Wind Speed 1

30

0.5 0

0

10

20

30

40

50

20 10 0

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0.5

1

Weights

Results

Fig. 8.16 Interface of the LQG tuning

The performance of the antenna depends on all controller weights. Generally, changing a single weight impacts more than one performance feature. However, in the modal coordinates of the antenna model each weight influences predominantly one or two performance features, and is weakly coupled with the remaining ones. This helps to follow the improvement of the antenna performance, because each slider predominantly addresses a single performance feature. Table 8.2 summarizes these relationships. In the 34-m antenna example the following steps correspond to the interfaces pictured in Fig. 8.17a–b. These steps reflect antennas’ common features, and can be followed as a guideline to the controller tuning process: 1. The tuning starts with the input of the state space representation of the open loop antenna model (A,B,C,D), a file abc.mat at the directory c:/lqg/gui/az. In the lower left window enter: load c:/lqg/gui/az/abc.mat. 2. Next, select arbitrary, but small, values of proportional and integral weights (say qei = 0.1 and qe = 0.3) and zero for frequency weights see Fig. 8.17a. After simulation the step response shows excessive settling time and overshoot (8.52 s and 18%, respectively), visible flexible oscillations and an unacceptable maximal disturbance step value (0.82 deg). The magnitude of the transfer

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121

Table 8.2 Relationship between LQG weights and performance Slider movement (LQG weight)

Result (performance)

Proportional weight Integral weight Frequency weight 1 Frequency weight 2 Frequency weight 3 Frequency weight 4

Settling time, bandwidth, disturbance rejection Overshoot, velocity offset error, disturbance rejection Vibration amplitude of the lowest (fundamental) frequency Vibration amplitude of the second frequency Vibration amplitude of the third frequency Vibration amplitude of the fourth frequency

function shows low bandwidth (0.21 Hz), sharp resonance peaks, and a largerthan-desired magnitude of the disturbance transfer function. 3. The most excessive oscillations come from the first (or fundamental) mode. Therefore the weight of the first flexible mode is increased to 5, and other weights are unchanged, as shown in Fig. 8.17a. The results are visible in the step responses where the oscillations are now invisible and in the transfer function where the first resonant peak disappeared. 4. The proportional weight is increased to 10, and the integral gain was increased to 3, c.f., Fig. 8.17b. This move reduced the settling time to 4 s, overshoot to 13%, and expanded bandwidth to 0.8 Hz.

8.4.3 Fine Tuning of the LQG Controller Certain features of the controller obtained through the approach described above can be improved by further modifying the controller weights. For example, the oscillations of the closed loop response can be further reduced or the closed loop response can be modified such that the wind disturbance does not exceed its specification. However, at this stage, the closed loop states are not as easily decoupled as the open loop states; therefore the relationship between the weights and the closed loop dynamics is less clear, and often difficult to track. For this reason a constrained optimization approach is used to tune the already tuned controller. In this approach the weights are tuning parameters, and the following variables are either optimized or constrained: 1. steady state servo error in the 0.1 deg/s velocity offset, 2. max value of the servo error due to the unit step disturbance (deg), edmaz = max(|ed |), 3. rms servo error in 24 km/h wind gusts, ew (mdeg), for 0 ≤ t ≤ 100 s, 4. overshoot of the unit step response, eo (%), 5. settling time of the unit step response, ts (s), 6. bandwidth, f o (Hz), 7. magnitude of the closed loop transfer function (from the command to the encoder) for the high-frequency range (above 3.5 Hz), that is, m h = max m( f ), for f ≥ 3.5Hz

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8 LQG Controller

Fig. 8.17 Interfaces for the 34-m antenna example

8. magnitude of the closed loop transfer function (from the disturbance to the encoder), i.e., m d max = max m d , for f ≥ 0Hz. The optimization index (a positive function to be minimized) is defined as follows:

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123

f = w1 er +w2 ed max +w3 ew +w4 eo +w5 ts +w6 (3− f o )+w7 m h +w8 m d max (8.27) Each variable in the function is weighted. The weight, wi , indicates the relative importance of each variable. If wi = 0, the ith variable is not optimized. In the above-defined function, the bandwidth is maximized by using 3 − fo variable rather than the bandwidth fo itself. For the 34-m antennas the bandwidth will not exceed 3 Hz; therefore the minimization of 3 − fo leads to expansion of fo . The variables above are functions of the LQG weights; therefore, f is function of weights as well. The initial values of the LQG weights are taken from the tuning part of GUI. This tool is quite similar to the previously described tool, see Fig. 8.18. The two plots of the fine-tuning GUI are identical to those of the tuning GUI. The user is given a line on which she/he enters the file containing the discrete time state space representation of the velocity loop model. In this GUI, there are eight variables explained above that can be either minimized or constrained. Different variables and their changes are displayed and can be tracked throughout. The Matlab optimization function, constr, is used here to find a local minimum of a constrained, nonlinear, multivariable function. Before attempting to minimize a function, it will ensure that the constraints have been met. Two main decisions must be made when using the constr function. One is whether to optimize or constrain a

Transfer Function

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10–2

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–1

10

Frequency (Hz)

Weights

Fig. 8.18 The fine-tuning interface

Constrain or optimize

0

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1

10

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8 LQG Controller

certain variable. The other is how many iterations are necessary to run in a block. Too many iterations may require long simulation time and give the user the feeling of being out of control; too few requires frequent re-starting. The following steps are recommended to tune the controller using the GUI: 1. For each variable verify the starting values by clicking on the “optimization weights” (rather than “constrain”) radiobutton, and set the weights to zero. 2. Construct the weight vector: ⎤ ⎡ I ntegral ⎢ Pr opor tional ⎥ ⎥ ⎢ ⎢ Special ⎥ ⎥ ⎢ ⎢ Fr equency1 ⎥ ⎥ ⎢ ⎢ Fr equency2 ⎥ ⎥ ⎢ ⎣ Fr equency3 ⎦ Fr equency4 3. Using the last values from the tuning GUI, run the optimizer by pressing the “Optimize” button. 4. Keep all of the “optimize” radiobuttons on, but scale the different variables according to your priorities, that is, change the weights to nonzero prioritized values. Run this optimization in increments of about 300 iterations, or until the system is optimized. 5. After the system is optimized (“Converged Successfully” is printed in the Matlab command window), look through the values of the variables and decide which performance values are acceptable and which are not. Of the ones that are not, choose their “constrain” radiobutton and select constraint values tighter than the current values. Run the optimization until it converges successfully. The display in Fig, 8.18 show the optimization results for the 34-m antenna.

8.5 LQG Controller in the Velocity Loop The LQG controllers presented above were located in the position loop, while the velocity loop was controlled by a simple proportional controller. The question arises, What happens to the antenna performance when the LQG controller is placed in the velocity loop? The advantage of placing the LQG controller in the velocity loop is first explained with a rigid antenna and proportional controller, and later analyzed with the LMT radiotelescope.

8.5.1 Position Loop Bandwidth Depends on the Velocity Loop In this section a simple (rigid) antenna and a simple (proportional) controller is analyzed, to obtain the impact of the velocity loop bandwidth on the position loop bandwidth. Consider a rigid antenna, the transfer function of which is an integrator,

8.5 LQG Controller in the Velocity Loop

125

1 . s

When the velocity loop of is closed with a proportional controller, its transfer function is as follows, see Chapter 2, equation (2.3): G(s) =

1 Ts + 1

(8.28)

where T is inverse proportional to the gain of the velocity loop. The time constant T obviously defines the velocity loop bandwidth. Figure 8.19 shows the magnitudes of the transfer functions for T = 0.5/␲ and for T = 5/␲, with bandwidth 0.1 Hz and 1 Hz, respectively. Now, if the position loop is closed with the proportional controller of gain k, its transfer function H is H (s) =

k/T s 2 + s/T + k/T

(8.29)

The plot of this transfer functions and step responses for k = 5, T = 0.5/␲, for k = 0.5, T = 5/␲ and for k = 5, T = 5/␲ is shown in Fig. 8.20a. The figures show that the antenna with low velocity loop bandwidth (0.1 Hz, for T = 5/␲) has low position loop bandwidth, when compared to the antenna with high velocity bandwidth (1 Hz, for T = 0.5/␲). Increasing the low bandwidth by increasing the position controller gain (from k = 0.5 to k = 5) causes antenna oscillations, and a resonance peak appears; see Fig. 8.20b. This concludes that the bandwidth of the position loop depends on the bandwidth of the velocity loop. Thus, wide bandwidth of the velocity loop is essential to the position loop performance. 101 T = 5/π T = 0.5/π

magnitude

100

10–1

10–2 –2 10

10–1

100

101

frequency, Hz

Fig. 8.19 Magnitudes of the velocity loop transfer functions, for T = 5/␲ (bandwidth 0.1 Hz) and for T = 0.5/␲ (bandwidth 1 Hz)

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magnitude

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10–1

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100

101

frequency, Hz

(b) 1.6 1.4

position, deg

1.2 1 0.8 0.6 0.4

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0.2

T = 0.5/π, k = 5

0

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2

4

6

8

10 time, s

12

14

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Fig. 8.20 Position loop for T = 5/␲, k = 0.5 (bandwidth 0.1 Hz), for T = 5/␲, k = 5 (bandwidth 0.4 Hz but oscillatory response) and for T = 0.5/␲, k = 5 (bandwidth 1 Hz): (a) magnitudes of the transfer functions and (b) step responses

8.5 LQG Controller in the Velocity Loop

127

8.5.2 Four Control Configurations The LQG controller placed at different loops is analyzed. The problem is investigated at the Large Millimeter Telescope (LMT); see Chapter 1. The control system of the LMT consists of velocity- and position loops, as shown in Fig. 8.21. Four control systems are analyzed [11]. They have the following structure: • PP control system, with PI controller in the position loop, and PI controller in the velocity loop, • PL control system, where the PI controller is in the velocity loop, and LQG controller is in the position loop. • LP control system, where the LQG controller is in the velocity loop, and PID controller is in the position loop • LL control system, where the LQG controller is in the velocity loop, and LQG controller is in the position loop. The PP control system is a typical telescope control system configuration. The PL case is the configuration of the Deep Space Network antenna control system, which has been considered by MAN Technologie [3], [12]. The LP and LL control systems have not yet been implemented. In this section the performance analysis (in terms of bandwidth, step responses, and wind disturbance rejection properties) of the four control systems as applied to the LMT is analyzed. This analysis shall help to evaluate and select the control system, not only for the LMT but also for other antennas and radiotelescopes of a similar design.

8.5.3 PP Control System The PP control system consists of the PI controller in the position loop, and the PI controller in the velocity loop. Its Simulink model is shown in Fig. 8.22a, and the velocity loop subsystem in Fig. 8.22b. The controller is shown in Fig. 8.22c,

position command

velocity command

position controller

velocity controller

torque

position

structure and drives

velocity position

Fig. 8.21 Four control systems of the LMT: (1) PP control system, where velocity controller is PI, and position controller is PI, (2) PL control system, where velocity controller is PI, and position controller is LQG, (3) LP control system, where velocity controller is LQG, and position controller is PID, and (4) LL control system, where velocity controller is LQG, and position controller is LQG.

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rp

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r

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(b) AZ drive velocity AZ vel. controller AZ vel.

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Antenna structure FEM model

4 EL wind

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(c) Integrator K Ts z–1

ki K*u

kp 1 u

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rp

+

kf

–

K*u

1 +–

2 y

u

y(n) = Cx(n)+Du(n) x(n+1) = Ax(n)+Bu(n) Estimator

Fig. 8.22 The Simulink model of (a) the position loop, and (b) the velocity loop system, and (c) the controller (for kf = 0 it is a PI controller, for k f = 0 it is an LQG controller)

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129

assuming that the flexible mode gain is zero, kf = 0. Here, the position loop PI controller is complemented with the feedforward (FF) loop to improve the tracking properties, especially at high velocities, and with a command preprocessor (CPP) to avoid large overshoots during target acquisition and limit cycling during slewing. The velocity loop model is shown in Fig. 8.22b. It consists of the finite-element model of the telescope structure, which includes the drives and azimuth and elevation velocity loop controllers. It is a discrete-time (digital) control system, with 0.001-s sampling time. The proportional and integral gains of the azimuth controller are 300, and for the elevation controller proportional gain is also 300, and the integral gain is 400. The bandwidth of the velocity loop transfer function is 1.0 Hz, both in azimuth and in elevation. The position loop model is shown in Fig. 8.22a. It consists of the velocity loop model, PI, and feedforward controllers in azimuth and elevation, command preprocessors in azimuth and elevation, and velocity and acceleration limiters in azimuth and elevation. The telescope velocity limit is 1.0 deg/s, and the acceleration limit is 0.5 deg/s2 , both in azimuth and elevation. The PI controller gains were selected to minimize settling time and servo error in wind gusts. They also guarantee zero steady-state error for constant velocity tracking. The proportional gain is 3.0, and the integral gain is 1.0. The telescope performance is illustrated with the elevation axis performance (azimuth axis performance is similar). The position loop transfer function is shown in Fig. 8.23, dotted line. It follows from this figure that the bandwidth is 1.8 Hz. In order to evaluate settling time a small (0.01 deg) step response was simulated, shown in Fig. 8.24. From the plot one can see that there is no overshoot and that the settling time is 3.0 s. The wind gusts time history was obtained from the wind spectrum; see Chapter 5. The plots of the servo error in azimuth and elevation are shown in Fig. 8.25a. The PP control system analysis showed also that LMT is a sturdy structure. Its fundamental frequency is 1.7 Hz (it is expected 1.3 Hz for a 50-m radiotelescope, as shown in Fig. 3.12). The more than usual stiffness of the LMT structure (expressed as a higher-than-expected fundamental frequency) allows for higher gains in the PP controller, thus enhancing pointing performance.

8.5.4 PL Control System The PL control system consists of the LQG controller in the position loop, and PI controller in the velocity loop. In this case the velocity loop model of telescope is as in Fig. 8.22b. It consists of the structure and drive model, and azimuth and elevation velocity loop controllers (PI type). The position loop is presented in Fig. 8.22, where the PI controllers are replaced with the LQG controllers, as in Fig. 8.22c. It consists of the same velocity loop model as PP control system, the LQG controller with feedforward loop, command

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–150 –200 –250 –2 10

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2

10

10

Fig. 8.23 Telescope position loop transfer functions: (a) magnitude and (b) phase

0.014

EL telescope position, deg

0.012 0.01 0.008 0.006 LL LP PL PP

0.004 0.002 0

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0.5

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Fig. 8.24 Telescope responses to 0.01 deg step

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8.5 LQG Controller in the Velocity Loop

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0

–0.5 –0.5

0 AZ servo error, mdeg

0.5

Fig. 8.25 Telescope servo error in 12 m/s wind gusts: with PP, PL, LP, and LL control systems

preprocessor, and velocity and acceleration limits. The CPP parameters are as follows: kv = 6, ko = 0.6, and β = 20, for azimuth and elevation. The performance of the PL control system was evaluated, using settling time, bandwidth, and servo error in wind gusts. The step response for small step (0.01 deg) is shown in Fig. 8.24, dash-dot line. The figure shows 1.2 s settling time and 35% overshoot. The position loop transfer function is shown in Fig. 8.23, dash-dot line, showing wide bandwidth of 1.5 Hz. The wind gusts simulations show 0.15 mdeg rms servo error in azimuth and 0.74 mdeg rms servo error in elevation, as shown the servo error plot in Fig. 8.25b. These numbers compared with the PP control system (0.35 mdeg in azimuth and 1.4 mdeg in elevation) show that the LQG controller improves the servo error in wind over the PI controller by factor 2.3 in azimuth, and by factor 1.9 in elevation. The bandwidth of the PL system is smaller than the PP system, due to the PP system transfer function hump seen in Fig. 8.23, which extends the bandwidth but deteriorates the system performance.

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8 LQG Controller

8.5.5 LP Control System The LP control system consists of the PID (proportional, integral, and derivative) controller in the position loop, and the LQG controller in the velocity loop. Its Simulink model is shown in Fig. 8.22a, and the velocity loop subsystem in Fig. 8.22b. The Simulink model of the open loop telescope is shown in Fig. 8.22b, with the velocity feedback removed. The open loop model is scaled to obtain maximal velocity of 1 deg/s for 10 V command (a standard input to motor drives). For this open loop model an LQG controller was tuned and then evaluated the performance of the velocity loop LQG controller by using step responses and transfer functions of the elevation axis. For this velocity loop the settling time is 0.2 s and the bandwidth is 1.8 Hz. The position loop is as in Fig. 8.22, where the velocity controllers are now of the LQG type. Besides the velocity loop, it consists of the PID controller with feedforward loop, the command preprocessor, and velocity and acceleration limiters. The feedforward loop forwards the command velocity to the velocity loop input. The following PID gains were selected: proportional gain 10, integral gain 6, and derivative gain 5. The CPP parameters are as follows: kv = 6, ko = 0.93, and β = 30. The position loop step responses for small step (0.01 deg) is shown in Fig. 8.24, dashed line, showing 0.6 s settling time. The position loop transfer function is shown in Fig. 8.23, dashed line, showing wide bandwidth of 20 Hz. The wind gust simulations to 12 m/s wind are plotted in Fig. 8.25c. The figure shows 0.012 mdeg rms servo error in azimuth and 0.150 mdeg rms servo error in elevation. These small numbers show that the LQG controller in the velocity loop improves the servo error in wind over the PP control system by factor 30 in azimuth, and by factor 10 in elevation.

8.5.6 LL Control System Finally, the telescope control system with the LQG controller in the velocity- and position loops was tuned. The velocity loop of the LL system is identical to that of the LP control system. The position loop controller was tuned to minimize the servo error in the wind gusts. The position loop characteristics are plotted in Figs. 8.23 and 8.24. From Fig. 8.24 it follows that the system settling time is 0.5 s, and there is no overshoot. From Fig. 8.23 one can find that the bandwidth is 40 Hz. Finally, the wind gusts simulations to 12 m/s wind are plotted in Fig. 8.25d. The figure shows 0.0012 mdeg rms servo error in azimuth and 0.0057 mdeg rms servo error in elevation, which gives the total rms error of 0.0058 mdeg. It is 250 times smaller than the error of the PP control system. Thus, the LL control system performance is the best of all presented system, although the system is the most complex and will require careful tuning of both velocity and position loop LQG controllers in order to obtain the predicted performance.

References

133 Table 8.3 Performance of the PP, LP, PL and LL control systems Control system

Settling time s

Overshoot %

Bandwidth Hz

Servo error in wind mdeg

PP LP PL LL

3.0 0.6 1.4 0.5

20 0 20 0

1.2 20 1.4 20

1.48 0.15 0.76 0.004

The LMT control systems performance is summarized in Table 8.3. Based on the performed analysis one concludes that: • The PP control system shows improved pointing accuracy when compared to similar control systems applied to typical antennas or telescopes. It was achieved because the analysis showed the exceptionally rigid LMT structure. • The PL control system has twice better pointing precision in wind than the PP system. • The analysis shows that pointing accuracy in the wind of the LP control system is ten times better than the PP system. This significant reduction was achieved because of the expanded bandwidth of the velocity loop. • The analysis shows that its pointing accuracy in the wind of the LL control system is 250 times better than PP system. Finally, some comments on the obtained performance estimates of the telescope are necessary. The estimates include unknown factors. First, the presented telescope performance is based on the analytical models the structure and the drives, which do not represent an accurate dynamics of the telescope. To improve the accuracy a model shall be derived from the system identification and data collected at the real telescope. Next, the wind disturbance torques are applied to the drives, while in reality the wind acts on the entire structure, including the dish surface. Finally, the RF beam movement, the ultimate goal of the control, was not simulated. Instead azimuth and elevation encoders were used, which only partially reflected the beam position. The encoders—although relatively precise—cannot exactly measure the actual beam position due to their distant location from the beam focal point, which is the RF beam location.

References 1. Collins, Jr., EG, Haddad WM, Ying SS. (1994). Construction of Low-Authority, Nearly Non-Minimal LQG Compensators for Reduced-Order Control Design. 1994 IEEE American Control Conference, Baltimore, MD. 2. Dutton K, Thompson S, Barraclough B. (1997). The Art of Control Engineering. AddisonWesley, Harlow. 3. Eisentraeger P, Suess M. (2000). Verification of the Active Deformation Compensation System of the LMT/GMT by End-to-End Simulations. In: Radio Telescopes, Proceedings of SPIE, Vol. 4015. 4. Gawronski W. (1994). Design of a Linear Quadratic Controller for the Deep Space Network Antennas. AIAA J. Guidance, Control, and Dynamics, vol. 17.

134

8 LQG Controller

5. Gawronski W. (2001). Antenna Control Systems: From PI to H∞ . IEEE Antennas and Propagation Magazine, 43(1). 6. Gawronski W. (2004). Advanced Structural Dynamics and Active Control of Structures, Springer, New York. 7. Gawronski W. (2007). Servo Performance Parameters of the NASA Deep Space Network Antennas. IEEE Antennas and Propagation Magazine, 49(6). 8. Gawronski W, Mellstrom JA. (1994). Control and Dynamics of the Deep Space Network Antennas. In: Control and Dynamics Systems, ed. Leondes, CT vol. 63, Academic Press, San Diego, CA, pp. 289–412. 9. Gawronski W, Perez-Zapardiel P. (2007). Performance Comparison of the LQG and PI Controllers in Wind Gusts.IPN Progress Report, 42–171. Available at http://ipnpr.jpl.nasa.gov/progress report/42-171/171D.pdf. 10. Gawronski W, Racho C, Mellstrom JA. (1995). Application of the LQG and Feedforward Controllers for the DSN Antennas. IEEE Trans. Control Systems Technology, vol. 3. 11. Gawronski W, Souccar K. (2005). Control Systems of the Large Millimeter Telescope. IEEE Antennas and Propagation Magazine, 47(4). 12. Kaercher HJ, Baars JWM. (2000). The Design of the Large Millimeter Telescope/Gran Telescopio Milimetrico (LMT/GTM). In: Radio Telescopes, Proceedings of SPIE, vol. 4015. 13. Li K, Kosmatopoulos EB, Ioannou PA et al. (1998). Control Techniques for a Large Segmented Reflector. Proc. 37 th IEEE Conf. Decision and Control, Tampa, FL. 14. Li K, Kosmatopoulos EB, Ioannou PA et al. (2000). Large Segmented Telescopes: Centralized, Decentralized and Overlapping Control Designs. IEEE Control Systems Mag., 20(5): 59–72. 15. Maneri E, Gawronski W. (2000). LQG Controller Design Using GUI: Application to Antennas and Radio-Telescopes. ISA Transactions, 39(2): 243–264. 16. Olberg M, Lindeborg C, Seyf A et al. (1995). A Simple Robust Digital Controller for the Onsala 20-m Radio Telescope. Proc. SPIE, 2479: 257–265. 17. Whorton M, Angeli G. (2003). Modern Control for the Secondary Mirror of a Giant Segmented Mirror Telescope. Proc. SPIE, Future Giant Telescopes, vol. 4840.

Chapter 9

H∞ Controller

This chapter presents the tuning process of the H∞ controller. It describes the H∞ controller modified for tracking purposes, gives the closed loop equations, and presents the 34-m antenna example, with the limits of the performance.

9.1 Definition and Gains Application of the H∞ controllers to antennas and telescopes is discussed in [1, 2, 3, 4, 5, 6, 7, 8]. The structure of an H∞ controller is similar to that of the LQG controller, although its parameters are obtained from a different algorithm. The algorithm minimizes the system H∞ norm, which is, in the case of a SISO system, the maximal magnitude of its transfer function. Also, in the algorithm the control (u) and the disturbance (w) inputs of a system are separated (see Fig. 9.1). The control (y) and performance (z) outputs are also parted. In our case the plant-controlled input (u) is the velocity input to the drives, the disturbance input (w) is the wind pressure at the antenna structure, the controlled output (y) is the encoder reading, and the performance output is the RF beam position. A significant portion of the antenna tracking error is generated by the antenna vibrations excited by wind gusts. The LQG controller improves its tracking, but the disturbance rejection properties have not been addressed directly in the tuning process. The H∞ controller allows for addressing simultaneously its tracking and disturbance rejection properties, as it will be shown in the following. The closed loop system architecture is shown in Fig. 9.1. In this figure G is the transfer function of an antenna, while K is the transfer function of a controller. For a closed loop system as in Fig. 9.1 the plant transfer function G(s) and the controller transfer function K(s) are such that + , + , z(s) w(s) = G(s) , y(s) u(s) (9.1) u(s) = K (s)y(s), where u, w are control and exogenous inputs and y and z are measured and controlled outputs, respectively. The related state-space equations of a structure are as follows: W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 9,

135

136

9 H∞ Controller w

Fig. 9.1 The H∞ closed loop system configuration: G, plant; K, controller; u, velocity input; w, disturbances; y, encoder; and z, RF error

z

G u

y

K

x˙ = Ax + B1 w + B2 u, z = C1 x + D12 u,

(9.2)

y = C2 x + D21 w. Hence, the state-space representation in the H∞ controller description consists of the quintuple (A, B1 , B2 , C1 , C2 ). For this representation, (A, B2 ) is stabilizable and (A, C2 ) is able to be detectable, and the conditions

T C1 D12 = 0 I , D12 (9.3)

T = 0I D21 B1T D21 are satisfied. When the latter conditions are satisfied the H∞ controller is called the central H∞ controller. These are quite common assumptions, and in the H2 control T C1 = 0), they are interpreted as the absence of cross terms in the cost function (D12 T and the process noise and measurement noise are uncorrelated (B1 D21 = 0). The H∞ control problem consists of determining controller K such that the H∞ norm of the closed loop transfer function G wz from w to z is minimized over all realizable controllers K, that is, one needs to find a realizable K such that G wz (K )∞

(9.4)

is minimal. Note that the LQG control system depends on y and u rather than on w and z, as above. The solution says that there exists an admissible controller such that G wz ∞ < ρ, where ρ is the smallest number such that the following four conditions hold: 1. S∞c ≥ 0 solves the following central H∞ controller algebraic Riccati equation (HCARE), S∞c A + A T S∞c + C1T C1 − S∞c (B2 B2T − ρ −2 B1 B1T )S∞c = 0.

(9.5)

2. S∞e ≥ 0 solves the following central H∞ estimator (or filter) algebraic Riccati equation (HFARE), S∞e A T + AS∞e + B1 B1T − S∞e (C2T C2 − ρ −2 C1T C1 S∞e = 0.

(9.6)

9.1 Definition and Gains

137

3. λmax (S∞c S∞e ) < ρ 2 ,

(9.7)

where λmax (X ) is the largest eigenvalue of X. 4. The Hamiltonian matrices A ρ −2 B1 B1T − B2 B2T , −C T C1 −A T 1T A ρ −2 C1T C1 − C2T C2 , T −B1 B1 −A

(9.8)

do not have eigenvalues on the jω-axis. With the above conditions satisfied the optimal closed loop system is presented in Fig. 9.2, and the controller state-space equations, from the input y to the output u, are obtained from the block-diagram in Fig. 9.2, x˙ˆ = (A + ρ −2 B1 B1T S∞c − B2 K c − K e C2 )xˆ + K e y, u = −K c xˆ .

w

Structure (G)

z D21

B1

d

A + u

B2

+ –

x

x

∫

+

C2

+

y

C1 +

+

D12

y

u

Controller (K) C2

yˆ –

A u

xˆ

Kc

u

Fig. 9.2 An H∞ closed loop system

+

Ke

∫

xˆ +

+

+

dˆ

–

ρ–2B1B1TSα c

B2

y

(9.9)

138

9 H∞ Controller

According to the above equations the H∞ controller state-space representation (A∞ , B∞ , C∞ ) is as follows: A∞ = A + ρ −2 B1 B1T S∞c − B2 K c − K e C2 , B∞ = K e ,

(9.10)

C∞ = −K c , where K c = B2T S∞c

(9.11)

and K e = So S∞e C2T , So = (I − ρ −2 S∞e S∞c )−1 .

(9.12)

The gain K c is called the controller gain, while K e is the estimator gain. The order of the controller state-space representation is equal to the order of the plant. Note that the form of the H∞ solution is similar to the LQG solution. However, the LQG gains are determined independently, while the H∞ gains are coupled through the inequality, and through the component So in as in the above equation.

9.2 Tracking H∞ Controller For tracking purposes the plant is upgraded with the integrator to maintain zero steady-state error for constant-rate command, as was already discussed in the LQG controller tuning. An H∞ tracking controller with an integral upgrade is presented in Fig. 9.3.

9.3 Closed-Loop Equations of the Tracking H∞ Controller The closed loop equations for the controller as presented in Fig. 9.3 are as follows, x˙ cl = Acl xcl + Bcl r + Bw w y = Ccl xcl + D21 w

(9.13)

z = C1 x + D12 u where ⎧ ⎫ ⎨ei ⎬ xcl = x ⎩ ⎭ xˆ

(9.14)

9.4 34-M Antenna Example

w

139

Structure (G)

z D21 B1

d

A u

+

B2

+

x

+

∫

x

C2

+

+

y

C1 +

+

D12

z

y

u

Controller (K) u

–

Kf

C2

C1

yˆ –

+

y

+ +

A

kp

ki ∫

ei

r

xˆ

e

yˆ +

C2

– u

∫ ρ–2B1B1TSαc

Ke xˆ + +

dˆ

+ +

B2

Fig. 9.3 An H∞ tracking controller with an integral upgrade

⎡

⎤ −C2 ⎦, −B2 k p C2 − B2 k f C1 A − ke C2 − B2 k p C2 − B2 k f C1 + ρ −2 B1 B1T S∞c (9.15) ⎡ ⎡ ⎤ ⎤ 1 0

Bcl = ⎣ B2 k p ⎦ , (9.16) Bw = ⎣ B1 ⎦ , Ccl 0 C p 0 B2 k p 0

0 Acl = ⎣ B2 ki B2 ki

0 A ke C 2

9.4 34-M Antenna Example In Chapter 5, the wind spectra were determined from the wind field data. Based on these spectra, and using the antenna model in the modal representation, the wind filter is added by an appropriate scaling of the input matrix B1 of the antenna. The scaling factors are the filter gains at the natural frequencies of the antenna. The H∞ controller for the elevation axis is obtained and simulated. Its tracking performances are and compared with the LQG and PI controller performances. First, the responses to small (10 mdeg) step commands are compared; see Fig. 9.4. The settling time of the PI controller is 15 s, LQG controller is 2 s, and the H∞ controller is 1 s. The responses to the step disturbance are shown in Fig. 9.5. The integral over the

140

9 H∞ Controller 14

s tep res pons e, mdeg

12 10 8 6 4

PI Hinf

2

LQG 0

0

2

4

6

8

10 time, s

12

14

16

18

20

Fig. 9.4 Comparison of the step responses of the PI, LQG and H∞ controllers

dis turbanc e s tep res pons e, mdeg

20 PI Hinf

15

LQG

10

5

0

–5

0

2

4

6

8

10 time, s

12

14

16

18

20

Fig. 9.5 Comparison of the disturbance step responses of the PI, LQG and H∞ controllers

absolute value of the responses is 98, 3.2, and 0.9, for the PI, LQG, and H∞ controller, respectively. The transfer functions from the command input to the encoder output are shown in Fig. 9.6. The plot shows improved tracking performance of the H∞ controller (the bandwidth is 6 Hz for the H∞ controller, 1.2 Hz for the LQG controller, and 0.09 Hz for the PI controller). The wind disturbance rejection properties are represented by the transfer functions from the wind disturbance input to the encoder output (Fig. 9.7), where the H∞ controller disturbance transfer function is about a decade lower than (factor 0.1) the LQG controller, and three decades lower (factor 0.001)

9.5 Limits of Performance

141

1

10

0

magnitude

10

–1

10

–2

PI Hinf

10

LQG –3

10

–2

10

–1

0

10

1

10

10

frequency, Hz

Fig. 9.6 Comparison of the transfer functions of the PI, LQG and H∞ controllers 1

10

PI Hinf magnitude, dis turbanc e

0

10

LQG

–1

10

–2

10

–3

10

–2

10

–1

0

10

10

1

10

frequency, Hz

Fig. 9.7 Comparison of the disturbance transfer functions of the PI, LQG and H∞ controllers

than the PI controller, showing improved disturbance rejection properties of the H∞ controller.

9.5 Limits of Performance The simulations and measurements show that the high-gain H∞ controller is hard to implement due to acceleration limits imposed on the velocity input signal u that drives the antenna. The limit is imposed to prevent overloading of the motors. For the 34-m antennas the acceleration limits are ±0.4 deg/s2 . The acceleration

142

9 H∞ Controller (a)

v eloc ity , deg/s

1.5 1 0.5 0 –0.5

0

0.2

0.4

ac c eleration, deg/s 2

0.6

0.8

1

0.6

0.8

1

time, s

(b)

50

0

–50

0

0.2

0.4 time, s

Fig. 9.8 H∞ controller response to 10-mdeg step offset: (a) velocity at the antenna input, (b) acceleration at the antenna input

of the input u during 10 mdeg step offset are shown in Fig. 9.8. The acceleration reaches the large value of 40 deg/s2 . The acceleration limiter reduces these values to ±0.4 deg/s2 , but this non-linear operation destabilizes the system. In order to avoid the excessive accelerations a command pre-processor is implemented. It commands the antenna with max velocity and/or acceleration if the velocity and/or accelerations are exceeded, and commands with the original command if the limits are not exceeded. This helps in no-wind weather. However, in windy weather the antenna is moved not only by the command, but also by the wind gusts. In this scenario the input u is a combination of the command and the feedback of the wind disturbance. Although the command, due to the preprocessor, is below the acceleration limits, the encoder signal from the feedback—due to the strong reaction of the H∞ controller— exceeds the acceleration limits and destabilizes the system. This behavior shows that the improvement of the H∞ controller performance is tied to the acceleration limits imposed on the antenna drives. The higher are the limits the smaller is the servo error.

References 1. Erm T, Bauvir B, Hurak Z. (2004). Time to Go H-infinity? Proc. SPIE, Advanced Software, Control, and Communication Systems for Astronomy, Glasgow, UK, vol. 5496. 2. Gawronski W. (1996). H∞ Controller for the DSS13 Antenna with Wind Disturbance Rejection Properties. TDA Progress Report, vol. 42–127, 1996. Available at http://ipnpr.jpl.nasa.gov/ progress report/42-127/127 G.pdf 3. Gawronski W. (2001). Antenna Control Systems: From PI to H∞ . IEEE Antennas and Propagation Magazine, 43(1). 4. Gawronski W. (2004). Advanced Structural Dynamics and Active Control of Structures, Springer, New York.

References

143

5. Li K, Kosmatopoulos EB, Ioannou PA et al. (2000). Large Segmented Telescopes: Centralized, Decentralized and Overlapping Control Designs. IEEE Control Systems Magazine, 20(5): 59–72. 6. Li K, Kosmatopoulos EB, Ioannou PA et al. (1998). Control Techniques for a Large Segmented Reflector. Proc. 37th IEEE Conf. on Decision and Control, Tampa, FL. 7. Schoenhoff U, Klein A, Nordmann R. (2000). Attitude Control of the Airborne Telescope SOFIA: ␮-Synthesis for a Large-Scaled Flexible Structure. Proc. 39th IEEE Conf. Decision and Control, Sydney, Australia. 8. Whorton M, Angeli G. (2003). Modern Control for the Secondary Mirror of a Giant Segmented Mirror Telescope. Proc. SPIE, Future Giant Telescopes, vol. 4840.

Chapter 10

Single Loop Control

Control systems of most antennas and radiotelescopes consist of two feedback loops: the velocity and position loops. The velocity feedback is an inner loop that controls motor velocity. The position feedback is an outer loop that controls the antenna position. The questions arise: Is the velocity loop necessary, and for what reason? Is the antenna pointing performance degraded when the velocity feedback is removed? This chapter tries to answer these questions. Simplicity is the reason for the proposed removal of the velocity feedback, because removing the velocity feedback simplifies the hardware, lowers the cost of the antennas, and lowers the cost of their maintenance. A trend toward eliminating the velocity feedback loop is observed in spacecraft and robot control. Recent papers (e.g., [1] and [2]) show that spacecraft can be stabilized and controlled without velocity feedback, and that robots can be controlled without velocity measurements; see [3], [4], [6], and [7]. The analysis starts with an ideal (or rigid) antenna to obtain initial assessment of the performance in a closed-form and then proceeds to a 34-m antenna that displays drive and structural flexibility [5]. The performance is compared for antennas with and without a velocity feedback loop, including their responses to commands and disturbances.

10.1 Rigid Antenna The rigid-body antenna model is a model of an idealized antenna, which allows for analysis of basic properties in a closed form.

10.1.1 Rigid Antenna with Velocity and Position Loops A typical control system for a rigid-body antenna consists of the velocity and position loops, as shown in Fig. 10.1a. A proportional controller is used to control the velocity loop. The transfer function of the closed velocity loop system, from velocity command u to antenna velocity ϕ˙ is as follows W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 10,

145

146

10 Single Loop Control (a) d r

e

K(s)

u

τ

ko

(b)

rigid antenna

1 Js

+

ϕ

1 s

ϕ

d r

e

K(s)

τ

1

+

ϕ

Js2 rigid antenna

Fig. 10.1 Rigid antenna control system: (a) with the velocity and position feedback, and (b) with the position feedback only: r, position command; e, servo error; τ , torque; d, disturbance; φ, antenna position; ϕ, ˙ antenna velocity; ko , gain of the velocity controller; K(s), position controller transfer function

G rl (s) =

ko G(s) 1 ϕ(s) ˙ = = u(s) 1 + ko G(s) 1 + Ts

(10.1)

where T = J / ko . The bandwidth of the velocity loop system is equal to B = 1 / T = ko / J rad/s. The gain, ko , is tuned to obtain the required bandwidth of the system. The required bandwidth is B = 10 rad/s, the inertia is J = 1 Nms2 /rad, thus ko = 10 Nms. The position loop is closed using a proportional-and-integral (PID) controller. The PID controller transfer function is K (s) = kd s + k p +

ki s

(10.2)

where kd , k p , and ki are derivative, proportional, and integral gains, respectively. The transfer function of the closed position loop (from the command r to the encoder position φ) is as follows G cl (s) =

ko (kd s 2 + k p s + ki ) K (s)G rl (s)/s ϕ(s) = = r (s) 1 + K (s)G rl (s)/s J s 3 + ko (1 + kd )s 2 + ko k p s + ko ki (10.3)

This system is stable for k p > T ki . The above equation shows that for low frequencies (s → 0) the magnitude of the transfer function is unity; thus, for these frequencies ϕ = r , i.e., the antenna follows exactly the command. For high

10.1 Rigid Antenna

147

frequencies (s → ∞) the magnitude of the transfer function tends to zero, that is, the antenna does not respond to the command or to the high-frequency noise. The transfer function from the disturbance d to encoder position φ for the PI controller is as follows (see Fig. 10.1a): G d (s) =

s ϕ(s) = d(s) J s 3 + ko s 2 (1 + kd ) + ko k p s + ko ki

(10.4)

It shows that for low frequencies (s → 0) the magnitude of the transfer function tends to zero and for high frequencies (s → ∞) the magnitude of the disturbance transfer function tends to zero as well, that is, for these frequencies the disturbances are completely rejected.

10.1.2 Rigid Antenna with Position Loop Only When the velocity loop feedback is removed, the antenna transfer function from the torque input to the position output is G(s) =

1 ϕ(s) = τ (s) J s2

(10.5)

The closed loop control without the velocity feedback is shown in Fig. 10.1b. By closing the position feedback with a PI controller the system becomes unstable. Indeed, the closed loop transfer function is G cl (s) =

k p s + ki K (s)G(s) ϕ(s) = = 3 r (s) 1 + K (s)G(s) J s + k p s + ki

(10.6)

From Routh–Hurwitz criterion the above system is unstable for arbitrary positive gains, that is, for k p > 0 and ki > 0. However, adding a derivative gain to the PI controller one obtains the PID controller. For this controller the position-only loop is stable. The transfer function of the PID controller is K (s) = ksi +k p +kd s; thus the closed loop transfer function is G cl (s) =

kd s 2 + k p s + ki K (s)G(s) ϕ(s) = = r (s) 1 + K (s)G(s) J s 3 + kd s 2 + k p s + ki

(10.7)

From the Routh–Hurwitz criterion, this system is stable for kd > 0. The above transfer function shows that for low frequencies (s → 0) the magnitude of the transfer function is unity; thus, for these frequencies one has ϕ = r , that is, the antenna follows the command exactly. For high frequencies (s → ∞) the magnitude of the transfer function tends to zero, that is, the antenna does not respond to highfrequency noise.

148

10 Single Loop Control

The disturbance transfer function is derived from Fig. 10.1b, obtaining G d (s) =

s ϕ(s) = 3 2 d(s) J s + kd s + k p s + ki

(10.8)

It shows that for low frequencies (s → 0) as well as for high frequencies (s → ∞) the magnitude of the disturbance transfer function tends to zero |G d | ≈ 0, that is, for these frequencies the disturbances are rejected.

10.1.3 Simulation Results The rigid-body antenna with and without velocity loop feedback is simulated. The gains of the PID controller for the rigid antenna with and without velocity loop are given in Table 10.1. For these controllers the closed loop system responses to the 1 deg step command and to the 1 deg/s disturbance step were simulated. They are shown in Fig. 10.2a,b, respectively. The figures show that the settling time of the antenna with the velocity loop and the antenna without a velocity loop are approximately the same (3.5 s). However, the disturbance rejection properties of the antenna with the velocity loop are much worse than of the antenna without the velocity loop (the amplitude and the duration of the response of the antenna with a velocity loop are much larger than the antenna without the velocity loop). Magnitudes of the transfer functions of the closed loop systems, from the command to the encoder, are shown in Fig. 10.3a and from the disturbance to the encoder in Fig. 10.3b. The first figure shows that the bandwidth of the antenna with and without the velocity loop are the same (0.6 Hz). However, the magnitude of the disturbance transfer function of the antenna without the velocity loop in Fig. 10.3b is smaller that the antenna with the velocity loop (for all frequencies). It shows again that the antenna without a velocity loop has better disturbance rejection properties than the antenna with the velocity loop. Finally, note that the controller gains of the antenna without a velocity loop are higher than the ones of the antenna with the velocity loop, indicating that the control effort (torques) may be higher in the case of antennas without a velocity loop. In summary the tracking performance of the rigid antenna with and without velocity feedback are similar, but the antenna without the velocity loop has better disturbance rejection properties, and possibly requires more control effort.

Table 10.1 Controller gains of the rigid antenna Control system

ko

kd

kp

ki

With velocity feedback Without velocity feedback

10 0

0 130

2.2 300

1.4 240

10.2 34-M Antenna

149

(a)

c ommand s tep res pons e

1.2 1 0.8 0.6 0.4 with velocity loop without velocity loop

0.2 0

0

2

4

6

8

10

time, s (b)

dis turbanc e s tep res pons e

0.04 with velocity loop without velocity loop 0.03

0.02

0.01

0

0

2

4

6

8

10

time, s

Fig. 10.2 Step responses of the rigid antenna with a velocity loop (solid line) and without a velocity-loop (dashed line): (a) responses to the unit command step, and (b) responses to the unit disturbance step

10.2 34-M Antenna The 34-m antenna model, as described in [5] and Chapter 2, is used for this analysis. Our task is to tune and compare the LQG controllers for the antenna with velocity feedback and without velocity feedback. The description of the LQG controller for the antenna with velocity feedback was given in Chapter 8. The LQG control loop is shown in Fig. 10.4, which includes the antenna drive. Consider two kinds of drives: one with the velocity feedback, and another without the velocity feedback. An antenna drive with a velocity loop feedback is shown in Fig. 10.5. In this figure the ktach gain represents the velocity (tachometer) feedback

150

10 Single Loop Control (a) 0

magnitude

10

–1

10

–2

10

with velocity loop without velocity loop –3

10

–3

–2

10

10

–1

0

10 10 frequency, Hz

1

2

10

10

dis turbanc e magnitude

(b)

–2

10

–4

10

with velocity loop without velocity loop –6

10

–3

–2

10

10

–1

0

10 10 frequency, Hz

1

2

10

10

Fig. 10.3 Magnitudes of the transfer functions of the rigid antenna with a velocity loop (solid line) and without a velocity loop (dashed line): (a) from the command input, and (b) from the disturbance input

gain. The same drive without velocity feedback is shown in Fig. 10.6. It was obtained from the original drive by breaking the velocity feedback (by setting the feedback gain to zero, ktach = 0) and by replacing the Amplifier1 with the gain kr . wind disturbance CONTROLLER

DRIVE

STRUCTURE

command

position servo error

velocity

velocity Pinion velocity

Fig. 10.4 Control system of the 34-m antenna

encoder torque

torque

Pinion velocity

2

Amplifier 1

tau3.s2+s

tau2*kr.s+kr

ktach

Amplifier 2

tau5.s2+s

ki*tau4.s+ki

Fig. 10.5 The 34-m antenna drive with velocity loop feedback

Voltage

kf

kc

Motor Armature

La.s+Ra

1 km

1

kb

Motor

Jm.s

+ –

N

1

1/N

Pinion Velocity

Gearbox

kg s

1/N

1 Torque

10.2 34-M Antenna 151

kr

Amplifier 2

tau5.s2+s

ki*tau4.s+ki kf

Fig. 10.6 The 34-m antenna drive without velocity loop feedback

2 velocity input

kc

Motor Armature

La.s+Ra

1

kb

km

10

Motor

Jm.s

N

Gearbox

s

kg*100(s)

1/N

1/N

1

1

pinion velocity

torque

152 10 Single Loop Control

10.2 34-M Antenna

153

The value of the latter gain is determined such that the open loop gain (from velocity input to the antenna velocity output) is equal to 1. The LQG controller gains for the antenna with the velocity feedback are determined in [5]. A similar controller was tuned for the antenna without a velocity loop, and the performance of the two systems was compared. The command step responses, disturbance step responses, command transfer functions, and the disturbance transfer functions have been compared. The command step responses are shown in Fig. 10.7a. The figure shows very similar responses of the antenna with and without a velocity loop (i.e., similar overshoot and settling time). The disturbance step responses (representing a rapid increase of wind torque) are shown in Fig. 10.7b. The plot shows an improved

(a) 1.2

antenna pos ition, deg

1 0.8 0.6 0.4 with velocity loop without velocity loop

0.2 0

0

2

4

(b) 2 antenna pos ition, deg

6

8

10

time, s x 10

–4

with velocity loop without velocity loop

1

0

–1

–2

0

2

4

6

8

10

time, s

Fig. 10.7 Step responses of the 34-m antenna with a velocity loop (solid line) and without a velocity loop (dashed line): (a) responses to the unit command step, and (b) responses to the unit disturbance step

154

10 Single Loop Control (a) 0

magnitude

10

–1

10

–2

10

with velocity loop without velocity loop –3

10

–2

10

–1

10

0

10 frequency, Hz

1

10

(b) –2

magnitude, dis turbanc e

10

with velocity loop without velocity loop

–3

10

–4

10

–5

10

–6

10

–2

10

–1

10

0

10 frequency, Hz

1

10

Fig. 10.8 Magnitudes of the transfer functions of the 34-m antenna with a velocity loop (solid line) and without a velocity loop (dashed line): (a) from the command input, and (b) from the disturbance input

response of the antenna without velocity feedback (although not as dramatic as at the rigid antenna). Figure 10.8a shows the antenna transfer function from the position command to the antenna encoder position. The transfer functions (of the antenna with and without velocity feedback) are similar. Figure 10.8b shows the antenna disturbance transfer function, where the antenna without velocity feedback has slightly better performance (the magnitude of the transfer function is smaller than the one for the antenna with velocity feedback). In summary, the tracking performance of the 34-m antenna with and without velocity feedback are similar, but the antenna without velocity feedback has better wind disturbance rejection properties.

References

155

Safety is one of the reasons that the velocity feedback is implemented at the antennas. In case of failure of the position loop the velocity feedback controls antenna velocity, maintaining it at zero value. Note, however, that for the safety reason the velocity loop is not necessary: in the case of failure of the control system the antenna remains motionless because the gearboxes oppose antenna motion and brakes are applied automatically. Another reason for implementing velocity feedback is its ability to increase damping in the system. This is justified in a case when a simple (PI type) controller is in the position loop. However, if LQG controller implemented in the position loop significantly increases the damping of the closed loop system, and the velocity loop is not necessary.

References 1. Akella MR. (2000). Rigid-Body Attitude Tracking without Angular Velocity Feedback. In: Spaceflight Mechanics 2000, American Astronautical Society, San Diego. 2. Caccavale F, Villani L. (1999). Output Feedback Control for Attitude Tracking. System Control Lett., vol. 38. 3. Chang YC, Chen BS, and Lee TC. (1996). Tracking Control of Flexible Joint Manipulators Using Only Position. Int. J. Control, 64(4). 4. Chang YC, Lee CH. (1999). Robust Tracking Control for Constrained Robots Actuated by DC Motors without Velocity Measurements. IEE Proc. Control Theory and Applications,146(2). 5. Gawronski W. (2002). Single loop Antenna Control. IPN Progress Report 42-151. Available at: http://ipnpr.jpl.nasa.gov/progress report/42-151/151D.pdf 6. Lim SL, Dawson DM, Hu J et al. (1997). An Adaptive Link Position Tracking Controller for Rigid-Link Flexible-Joint Robots without Velocity Measurements. IEEE Trans. System, Man, Cybernetics, Part B: Cybernetics, 27(3). 7. Loria A, Panteley E. (1999). Force/Motion Control of Constrained Manipulators without Velocity Measurements. IEEE Trans. Aut. Control, 44(7).

Chapter 11

Non-Linear Control

This chapter discusses three main sources of the antenna non-linear behavior: velocity and acceleration limits, dry friction, and backlash. The velocity and acceleration limits are neutralized by using the command preprocessor, or anti-windup technique. The friction model is presented and the dither is analyzed as a cure for the antenna sticking at low velocities. The backlash model is given and the performance of an antenna with applied counter-torque is analyzed.

11.1 Velocity and Acceleration Limits The antenna control system is shown in Fig. 3.7. Besides the controller, it includes the acceleration and velocity limiters. The acceleration limiters are implemented because the motor currents must be limited to avoid overheating (the currents are proportional to the antenna acceleration). Antenna velocity is limited for safety reasons. Those limits are not affected during normal tracking, but are reached during slew operations. When the limits are reached, the antenna is in a non-linear regime and starts limit cycling. The limit cycling can be prevented by the following means: • • • •

applying two different controllers, one for tracking and another for slewing; implementing an anti-windup technique, see [18, 21]; applying a controller with variable gains [24, 25]; using a trajectory calculated in advance, such that it never exceeds the velocity and acceleration limits [32]; • using a command preprocessor, see [14, 15].

11.1.1 Command Preprocessor The limit cycling during antenna slewing operations and target acquisitions is caused by the violation of the antenna velocity and acceleration limits that cause, in turn, the increase of the integral error. This problem can be avoided by introduction of a command that does not exceed the limits; see [14] and [15]. A command W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 11,

157

158

11 Non-Linear Control w r

CPP

rp controller

uo

u

y

antenna

velocity acceleration limit limit

y

Fig. 11.1 Antenna control system with CPP

preprocessor (CPP) is computer software that generates a modified command, identical to the original one, if the velocity and accelerations of the command are within the limits; and a command of maximal (or minimal) velocity and acceleration when the limits are met or violated. The location of the preprocessor in the antenna control system is shown in Fig. 11.1. CPP Description. The block diagram of the CPP is shown in Fig. 11.2. It consists of a derivative, an integrator, and velocity and acceleration limiters. The proportional feedback loop has a variable gain ki that depends on the preprocessor error ei . The sampling time is denoted Δt, where Δt = 0.02s, the command at the ith time instant t = iΔt is denoted ri , the command velocity is νi , the preprocessed command is denoted r pi , the input to the integrator is u i , and the preprocessor error is ei = ri − r pi . Consider a case where the command does not exceed the velocity and acceleration limits. In this case the system is linear, and the velocity and acceleration limiters in Fig. 11.2 are replaced with a unit gain. For the linear case the equations are: • for the integrator: r pi = r pi−1 + Δtu i

(11.1)

• for the derivative νi =

ri − ri−1 Δt

(11.2)

d/dt

ri

ei

+ –

kpi

+

variable gain controller

+

ui

accel. limit

velocity limit

∫

rpi

rigid antenna

Fig. 11.2 Command preprocessor: derivative, integrator, velocity limiter, and acceleration limiter

11.1 Velocity and Acceleration Limits

159

• for the error ei = ri − r pi

(11.3)

u i = ki ei−1 + v i

(11.4)

• and for the integrator input

Combining the above equations one obtains r pi − r pi−1 + Δtki r pi−1 = ri − ri−1 + Δtki ri−1

(11.5)

The above equation shows that ri = r pi for zero error initial conditions. As a consequence, if the preprocessed command reaches the original command it follows the latter one exactly. In the following the transient motion of CPP is investigated. In order to do this, equation (11.5) can be rewritten as ei = αi ei−1

(11.6)

where αi = 1 − Δtki . This is an equation of the transient dynamics of the CPP error. From (11.6) it follows that the system is stable if |α1 | < 1. For 0 < αi < 1 there is no overshoot, and for −1 < αi < 0 the transient is oscillatory, i = 1, 2, . . . ,. Consider further positive αi only, the case with no overshoot of the preprocessed command over the original command. Note that in this case the smaller is the gain αi , the quicker the error dies down. The gain ki controls the value of αi ; therefore, for large-gain ki the transient between the original and the preprocessed command is strongly damped. Non-tracking activities consist of slewing and of target acquisition. Antenna slews and targets are acquired by inserting step commands. For this reason step inputs are used to determine the CPP gain. Two kinds of steps are distinguished: small steps (that do not violate the velocity and acceleration limits) and large steps (that evidently violate the limits). For the antenna sampling time Δt = 0.02s and the velocity limit vmax =0.8 deg/s, the maximal step that does not violate the velocity limit is v max Δt = 0.016deg. The field test results show that steps larger than 0.150 deg exhibit significant non-linear behavior. Thus, a step of 0.01 deg is considered small, and one of 10 deg is considered large. First, assume small gain, ki = 1. For this gain the response to a large step is almost optimal. The steady value is reached with the maximal velocity and acceleration, and without overshoot. For small steps, however, the steady value is reached after 4 s, which is too slow for the antenna. Assume now a CPP with a large gain, ki = 6. It performs outstandingly for small steps (settling time is 1 s), but for large steps the CPP exhibits limit cycling. This is illustrated in the simulations presented

160

11 Non-Linear Control

in the next section. These two mental experiments indicated that small gain is useful for large steps, and large gain is needed for fast response to small steps. In order to help the CPP to tell the difference between small and large steps, note that steps are tied to the CPP tracking error: large steps produce large error, and, not surprisingly, small steps produce small error. Thus, in order to produce different gains for different steps, the gain value shall depend on the error value. The gain shall be larger for small error, and smaller for large error. It is achieved by using the following relationship between the gain and the error: ki = ko + kν e−β|ei |

(11.7)

where ko is the constant part of the gain, kν is the variable part of the gain, and β is the error exponential. The plot of ki (ei ) for ko = 1, kv = 5, and for β = 10, 20, 40, and 100 is shown in Fig. 11.3. It was already determined that ki = 1 is recommended for small errors, and ki = 6 for large errors. In equation (11.7) the gain ko = 1 sets the lower value of ki , and kv = 5 sets the upper value of ki . The parameter α shapes the transition of ki from its minimal to its maximal value, within the error interval of [0.016, 0.150] deg. β = 20 was chosen because for this value the gain covers the error segment of [0.016, 0.150] deg (marked gray in Fig. 11.3) where the antenna transfers from linear to non-linear behavior. Additionally, this choice of β was verified with the simulations of the CPP performance, as illustrated in the following sections. Finally the small gain ko needs additional correction, because for the antenna operational purposes it is preferable that the CPP does not have overshoots for large steps. The overshoots may appear during the deceleration phase due to the 7 linear

nonlinear

6

gain ki

5 4 3 2 1 0 10–4

β = 10 β = 20 β = 40 β = 100 10–3

Fig. 11.3 CPP gain vs. CPP error

10–2 error ei, deg

10–1

100

11.1 Velocity and Acceleration Limits

161

0.16

ko = 0.95

0.14

ko = 0.90

overshoot, deg

0.12

ko = 0.85

0.1 0.08 0.06 0.04 0.02 0

–0.02 0.8

0.82

0.84

0.86

0.88 0.9 0.92 0.94 acceleration reduction, α

0.96

0.98

1

Fig. 11.4 Overshoots for large steps vs. acceleration limit reduction and CPP gain ko

finite sampling time. The magnitude of overshoot with respect to the value of the acceleration limit, and the CPP gain ko , as in equation (11.7), was investigated. The maximum value of the acceleration was reduced by factor α, where α varied from 0.8 to 1.0, while the gain ko varied from 0.85 to 0.95. The plot of overshoots for various large steps with respect to ko and α is shown in Fig. 11.4. The figure shows that for small acceleration and small gain the overshoot is small too, and eventually it disappears. Because a 10% margin of acceleration should be preserved to guarantee a stable closed-loop of the antenna, α = 0.9 was chosen. For this acceleration the gain ko = 0.88 produces no overshoot, as seen in Fig. 11.5. CPP Dynamics. The CPP dynamics are checked in the two scenarios, typical for antennas and radiotelescopes: (1) Step responses, small and large. Small steps do not violate the limits; large steps do. (2) Acquiring and tracking a typical trajectory. The velocity and acceleration limits of the 34-m antenna are 0.8 deg/s and 0.4 deg/s2 , respectively, and the acceleration limits of the CPP were 90% of the antenna limits, while the velocity limit of the CPP was equal to the antenna velocity limit. For the large step of 10 deg, the preprocessed command, its velocity and acceleration are shown in Fig. 11.6a–c the preprocessed command begins with the maximal acceleration until it reaches the maximum velocity, then continues with the maximal (and constant) velocity, and finally slows down with the minimal deceleration. After reaching the steady-state value of 10 deg the error between the original and the preprocessed command is zero.

162

11 Non-Linear Control

0.1

overshoot, deg

0.08 0.06 0.04 0.02 0 –0.02 0.8

0.82

0.84

0.86

0.88 0.9 0.92 0.94 acceleration reduction, α

0.96

0.98

1

Fig. 11.5 Overshoots for large steps vs. acceleration limit reduction, and CPP gain ko = 0.88

Finally, a typical azimuth trajectory acquisition and tracking by the CPP is shown in Fig. 11.7a. The antenna position at the initial time is 10 deg, while the target position is at 24 deg. The target is acquired in 20 s with the maximal speed (see Fig. 11.7b) and maximal acceleration (see Fig. 11.7c), and also with very small overshoot (c.f. Fig. 11.7a). The CPP error (the difference between the original and preprocessed trajectory) after the acquiring is virtually zero. Antenna Dynamics. The 34-me antenna step response is shown in Fig. 11.8a. The velocity and acceleration of the antenna are shown in Fig. 11.8b and c, respectively. Velocity is within the imposed limits (0.8 deg/s) and acceleration initially hits the limit (0.4 deg/s2 ), but does not destabilize the system. For large steps, however, the antenna hits velocity and acceleration limits (see Fig. 11.9a,b,c), which causes limit cycling. This happens during antenna slewing. The response of the CPP to the small step input of 0.01 deg is shown in Fig. 11.8, dashed line. It is a rapid response of less than 1 s settling time. The antenna follows the preprocessed command with very little overshoot. The response of the antenna to a large-step input of 10 deg is shown in Fig. 11.10. For comparison, the response of the same antenna to a non-processed step is shown in Fig.11.9. Clearly, an unstable limit cycling disappeared for the antenna with CPP.

11.1.2 Anti-Windup Technique The anti-windup technique is another option to avoid limit cycling of antennas. The description of this approach one can be found in [12, 18, 19, 21, 27]. Here a simple,

11.1 Velocity and Acceleration Limits

163

(a)

12 position, deg

10 8 6 4 command pre-processed command

2 0

0

2

4

6

8

10 time, s

12

14

16

18

20

0

2

4

6

8

10 time, s

12

14

16

18

20

0

2

4

6

8

10 time, s

12

14

16

18

20

(b)

1

rate, deg/s

0.8 0.6 0.4 0.2 0 (c)

accel., deg/s2

0.5

0

–0.5

Fig. 11.6 Preprocessed large-step commands: (a) command, (b) velocity, (c) acceleration

but effective application of the anti-windup technique to the antenna control system is described. The antenna control system is shown in Fig. 11.11, solid line. Besides the controller it includes the acceleration and velocity limiters. When the limits are reached the antenna is in a non-linear regime and starts limit cycling. Figure 11.12, dashed line, shows the antenna oscillatory response to 2 deg position offset. During slewing and re-targeting, when a large position offset is commanded, it causes saturation of the velocity and/or acceleration limiters. Thus, the antenna input u is smaller than the controller output uo. Consequently, the antenna response y is slower, and the servo error decreases slowly, as well. The slow process impacts

164

11 Non-Linear Control

(a) 25

position, deg

command antenna position 20

15

10

0

20

40

60

80 time, s

100

120

140

160

0

20

40

60

80 time, s

100

120

140

160

0

20

40

60

80 time, s

100

120

140

160

(b)

velocity, deg/s

1

0.5

0

–0.5

(c)

accel., deg/s2

0.5

0

–0.5

Fig. 11.7 Azimuth trajectory: original (dashed line), and preprocessed (solid line): (a) command, (b) velocity, (c) acceleration

mostly the integral term of the controller and the integral error becomes large. When eventually y approaches r the controller output uo is still saturated due to the large integral error, pushing the antenna position to pass the command. This is the beginning of the limit cycling. One can limit the integral term, but it causes unacceptably large overshoot. The increase of the integral term can be slowed down by applying the additional feedback loop that feeds the difference between controller output and antenna input,

11.1 Velocity and Acceleration Limits

165

position, deg

(a)

0.01

0.005 antenna CPP 0

0

0.5

1

velocity, deg/s

(b)

1.5 time, s

2

2.5

antenna CPP

0.04 0.02 0

0

0.5

1

(c)

1.5 time, s

2

2.5

0.4 accel., deg/s2

3

3

antenna CPP

0.2 0 –0.2 0

0.5

1

1.5 time, s

2

2.5

3

Fig. 11.8 Closed-loop step response to 0.01 deg step: (a) antenna position, (b) antenna velocity, and (c) antenna acceleration

uo – u, through a compensator. G aw Because the integral term has to be reduced, the compensator is an integral itself: G aw =

kaw s

(11.8)

This is the “classical” anti-windup approach. The only parameter to be determined is the compensator gain kaw , which should guarantee required and stable performance. Figure 11.12 presents the antenna responses to the 10 deg position offset, one without the anti-windup feedback (broken line) and one with the feedback (solid line), for the feedback gain kaw = 2. It is clear that for large position offsets, without the anti-windup feedback, the antenna is unstable, while the feedback produces a stable response with minimal overshoot.

166

11 Non-Linear Control

position, deg

(a) 10

5

0

0

5

10

15

20 time, s

25

30

35

40

(b) velocity, deg/s

1 0.5 0 –0.5 –1

0

5

10

15

20 time, s

25

30

35

40

(c)

accel., deg/s2

0.5

0

–0.5

0

5

10

15

20 time, s

25

30

35

40

Fig. 11.9 Closed-loop step response to 10 deg step: (a) antenna position, (b) antenna velocity, and (c) antenna acceleration

11.2 Friction Many telescopes cannot precisely point at a certain part of the sky that requires tracking with very low azimuth angular velocities (approximately 0.0004 deg/s or lower). For such low velocities, dry rolling friction is observed at the telescope drives that cause an unwanted increase of pointing error. (Pointing error is defined as a difference between the target location and the RF beam position.) In this section the National Radio Astronomy Observatory’s Green Bank Telescope located in West Virginia (see Fig. 1.1) is analyzed. The telescope’s size, weight, and especially configuration create difficulties in precision tracking. One of them is observed during tracking at low velocities. Namely, for velocities lower than 0.0003 deg/s a non-smooth telescope motion with breakaways may occur. The peak-to-peak pointing error due to friction is 1.4 mdeg.

11.2 Friction

167

12

antenna position, deg

10

8 6

4 2

0

0

5

10

15 time, s

20

25

30

Fig. 11.10 Antenna responses to preprocessed 10 deg step command

Gaw – r antenna controller y

+

+

uo

u

+

y

antenna

velocity acceleration limit limit

Fig. 11.11 Block diagram of the antenna control system (solid line) and anti-windup controller (broken line): uo , controller output; u, antenna input

A high-frequency external signal injected into a system (its frequency much higher than the system dynamics) is called a dither [2, 11]. It has long been known that injection of high-frequency signals into a non-linear system results in eliminating the system limit cycles [2, 11, 23]. This phenomenon was detected in electrical circuits by Appleton in 1922. Although dither is a well-known remedy for the non-linear behavior of control systems, little is known of its impact on the dynamics of large flexible structures. It is shown here that dither improves the telescope pointing at low tracking velocities and that high-frequency dither signal excites local vibrations only (at the wheels), causing the breaking of the frictionstiction phenomena. The vibrations are not transmitted through the telescope structure; thus they do not impact its pointing performance. Also, in this section a friction

168

11 Non-Linear Control 12

position, deg

10 8 6 4 2 0

with AW without AW 0

5

10

15

20 time, s

25

30

35

40

Fig. 11.12 The 34-m antenna responses to 10 deg position offset: with and without anti-windup (AW) feedback

model frequently used in the antenna industry is presented. It includes the velocity threshold and the determination of the applied torque within the threshold.

11.2.1 Dry Friction Model The telescope’s non-linear dynamics, at low velocities, is caused by the dry friction phenomenon. Friction is a torque, or a force, that depends on the relative velocity of the moving surfaces. In the Coulomb friction model it is constant after the motion begins, and this constant value is called the Coulomb friction torque. At zero speed, the friction torque is equal and opposite of the applied torque, unless the latter one is larger than the stiction torque. In this case, the friction torque is equal to the stiction torque. The stiction torque is a torque at the moment of breakaway and is larger than the Coulomb torque. A diagram of the friction torque versus relative velocity is shown in Fig. 11.13. Many friction models have been developed; see, for example, [1–3], [7], and [8]. They reflect different aspects of the friction phenomena; their usefulness depends on application purposes. The model presented below combines basic physical properties of the dry friction with the numerical features that improve digital simulations. Its accuracy for antenna tracking purposes has been tested on many existing telescopes and antennas. In this model, denote v the telescope wheel velocity, and νt > 0 a wheel velocity threshold that is a small positive number. Denote Tc the Coulomb friction torque, and Ts the stiction torque; then the friction torque model, T, is defined as follows:

11.2 Friction

169 Tf

velocity threshold

Ts Tc

–νt

νt

ν

–Tc –Ts

Fig. 11.13 Friction torque versus velocity

T =

−Tc sign(ν) for − min(|Td |, Ts )sign(Td ) for

|ν| > νt |ν| ≤ νt

(11.9)

where sign(ν) = 1, 0, −1 for ν > 0, ν = 0, and ν < 0, respectively, and Td denotes the total applied torque. In this model, if the surfaces in a contact develop a measurable relative velocity, such that |ν| > νt , the friction torque is constant, directed opposite to the relative speed. If the relative velocity is small, namely, within the threshold, (|ν| ≤ νt ) the torque does not exceed the stiction torque and the applied torque, and is directed opposite to the applied torque. The velocity threshold νt is implemented for numerical purposes, because numerically the zero state does not exist. It follows from equation (11.9) that in order to determine the friction torque T one has to know the following: • • • • •

the Coulomb friction torque Tc , the stiction (breakaway) torque Ts , the applied torque Td , the wheel velocity ν, and the wheel velocity threshold νt .

Each variable is determined as follows: The Coulomb friction torque is proportional to force F, which is normal to the surface Tc = μr F

(11.10)

where r is the wheel radius and μ is the friction coefficient. For hard steel μ = 0.0012–0.002. The stiction (breakaway) torque is most often assumed to be 20%–30% higher than the Coulomb friction that is

170

11 Non-Linear Control

Ts = αTc

(11.11)

where α = 1.2 ÷ 1.3. The total applied torque Td is determined from the plant dynamics as follows. Let the discrete state-space equation of the plant (which includes the telescope structure and its drives) be x(i + 1) = Ad x(i) + Bdr r (i) + Bd f T (i), ν(i + 1) = Cd x(i + 1)

(11.12)

In this model Δt, denotes sampling time, i denotes the ith sample, v(i) is the wheel velocity at time instant iΔt, x(i) is the plant state at the instant iΔt, r(i) is the telescope angular input velocity, and T(i) is the friction azimuth torque. Additionally, Ad is the telescope discrete-time state matrix, Bdr nd Bd f are telescope velocity and friction torque input matrices, and Cd is the wheel velocity output matrix. Left-multiplying the first equation (11.12) by Cd gives ν(i + 1) = Cd x(i + 1) = Cd Ad x(i) + Cd Bdr r (i) + Cd Bd f T (i)

(11.13)

According to the friction model, for the wheel velocity being within the threshold, that is, such that |ν(i + 1)| ≤ νt one obtains ν(i + 1) = 0. Thus, from equation (11.13) it follows that T (i) = −

Cd (Ad x(i) + Bdr r (i)) Cd Bd f

(11.14)

and the applied torque Td has opposite sign to the above friction torque T. The wheel velocity threshold νt was assumed to be 0.67 mdeg/s.

11.2.2 Low-Velocity Tracking Using Dither At low velocities friction significantly impacts the antenna dynamics, and consequently the pointing precision. There are many ways to reduce the system dynamics due to friction. Most of these methods are based on closed-loop compensation [2, 6, 10, 11, 28]. Here, an open-loop technique is applied by dithering the driving torque [17]. The block diagram of the closed-loop system with the dry friction and the velocity dither is shown in Fig. 11.14. In this diagram, according to equation (11.9), the dry friction torque T is a non-linear function of the azimuth wheel velocity v and the drive torque Td . Consider a torque at the azimuth wheel. For low velocities, the driving torque Td is smaller than the dry friction torque Tc , causing the telescope to stop. While resting, the error between the commanded position and the actual telescope position

11.2 Friction

171 dither +

command +

Controller

+

Drive

encoder

Td +

–

+ T Dry friction

Antenna structure

pinion velocity

Fig. 11.14 Azimuth control system with dry friction and dither

increases. In the closed-loop configuration the increased error causes an increase of the driving torque Td , and eventually the movement of the telescope is observed. The cycle repeats itself and is called limit cycling. The plots of the telescope pointing error in this limit cycling are shown in Fig. 11.15. A harmonic dither of amplitude do and period to , d(t) = do sin(2π t/to ),

(11.15)

is introduced at the telescope velocity input. When dither is implemented the torque level at the wheel is raised. This increase is not a constant; it varies harmonically. If the amplitude of the driving torque Td exceeds the friction torque, the telescope is moving continuously and the limit cycling is overcome. Due to the high frequency of the dither signal (high, when compared to the telescope dynamics), the harmonic movement is a local phenomenon at the wheels, which is shown later in this section. It is not propagated through the telescope structure, and has very low impact on the structural dynamics, and consequently on the telescope pointing. The above heuristic explanation of the dither action can be derived more formally. Consider the continuous-time telescope model with the non-linear friction torque T (ν), driven by the command velocity, r, and dither,d

Fig. 11.15 Pointing error for the simulated tracking, no dither

172

11 Non-Linear Control

x˙ = Ax + Br (r + d) + B f T (ν)

(11.16)

The parameters ((A, Br , B f , C)) are the continuous-time counterparts of the discrete-time parameters (( Ad , Bdr , Bd f , Cd )) as in equation (11.12), and x(t) is the state variable of the plant. This equation is averaged over the dither period to . The average value xa of x is defined as 1 xa = to

t+to x(τ )dτ

(11.17)

t

Note that in equation (11.16), the average value of the velocity command, is almost the same as the instantaneous value, because the command changes insignif icantly over the period to , that is, r (t) ∼ = r (t +to ). The average value T (ν) of the nonlinear torque T (ν) is obtained from the dry friction torque as in equation (11.9). The velocity threshold νt in this equation is assumed to be zero (the non-zero threshold was previously introduced to avoid numerical difficulties in simulations). Thus, the wheel friction torque is given as T = −Tc sign(ν + d). The average torque Tˆ is called the smooth image of the dry friction torque. The smooth image Tˆ is defined t+t )o T dt; therefore one obtains as Tˆ = t1o t

1 Tˆ (ν) = to

t+to t

1 T (ν)dt = − to

t+to Tc sign(ν + d)dτ = − t

ν 2Tc arcsin( ) (11.18) π do

The plot of Tˆ with respect to ν do for Tc = 1 is shown in Fig. 11.16. One can see from this figure that although the dry friction T is a discontinuous function of the velocity, its smooth image Tˆ is, by definition, a smooth function of the velocity. It also follows from Fig. 11.16 that the smooth image exists only for the dither amplitudes that extend the wheel velocity, that is, for do > ν. This is quite understandable because for the dither amplitude smaller than the wheel velocity there is no change in the friction torque. Because the function Tˆ is smooth it can be linearized for small velocity variations. Consequently, for small wheel velocity ν that is proportional to the velocity of the command r, ν = kr r , one obtains Tˆ (do ) ∼ = ko r,

ko = −

2Tc kr π do

(11.19)

The plot of linearized Tˆ in Fig. 11.16, shown as a dashed line, reveals a good coincidence with Tˆ for ν < 0.5do .

11.2 Friction

173 1 0.8 0.6 0.4

T

0.2 0 –0.2 –0.4 –0.6 –0.8 –1 –1

–0.5

0 v/do

0.5

1

Fig. 11.16 Smooth image of dry friction (solid line) and the lineraized image (broken line)

Introducing equation (11.19) to the averaged equation (11.16) one obtains the following linear system x˙ a = Axa + Br o r + Br d,

ya = C xa

(11.20)

with the input matrix Br o in the form Br o = Br + ko Bn . The latter equation proves that the system dynamics with dither is linear one. Notice that the dither input has no significant impact at the telescope pointing. Let us write the pointing ya as a superposition of the pointing yar due to the input r and the pointing yad due to the dither input d, that is, ya = yar + yad . Notice that the dither is of high frequency; therefore, the response yad is negligible when compared to yar ; thus ya ∼ = yar . The latter shows that the dither action makes the telescope dynamics linear, but it does not show itself at the output; thus the telescope pointing performance is not affected.

11.2.3 Non-linear Simulation Results Typically, the dither signal should be injected just ahead of the non-linearity. In the case of the Green Bank Telescope it is a velocity command at the telescope azimuth drives. In this case the dither is simply added to the feed-forward command generated by the controller computer. The dither amplitude and frequency are determined as follows. The frequency must be much higher than the telescope dynamics. Because no significant telescope

174

11 Non-Linear Control

Fig. 11.17 Pointing error for the simulated tracking with dither

dynamics is observed above 10 Hz, the dither frequency 30 Hz is chosen. The dither amplitude depends on the level of friction torque. Dry friction torque for tracking at a velocity of 0.0003 deg/s is smaller than 5600 Nm. For this friction level the dither amplitude was selected to be 0.18 deg/s using the telescope pointing simulations with various dither amplitudes and showing that the pointing error was the smallest for the dither of amplitude 0.18 deg/s. The plot of the pointing error for the above dither amplitude is shown in Fig. 11.17. The plot shows that the maximum pointing error dropped 18-fold, from 1.4 to 0.08 mdeg.

11.3 Backlash Gearboxes and gears are two critical components of the antenna drives. Measurements at the drives of the DSN antennas indicated that the small gap between gear teeth was causing backlash at the gearboxes and at elevation via the bull gear. A backlash phenomenon is observed when one gear rotates through a small angle without causing a corresponding movement in the gear it is driving. Left uncorrected, backlash deteriorates the antenna’s pointing precision. Backlash is observed not only in antennas and telescopes [29, 30], but also in precision instruments [20– 23], and robots [24, 25]. Often, to reduce backlash impact on the closed-loop precision, anti-backlash controllers are designed [9, 13, 26]. For antennas and telescopes the backlash is eliminated by implementing two identical drives that impose two non-identical torques [16]. The torque difference is called a torque bias, or counter-torque. The difference between these two torques depends on the antenna load and is shaped by the drive electronic circuits. With a two-motor configuration (see Fig. 11.18), backlash clearance will occur at one drive while the other is still coupled. The antenna dynamics will be controlled by the latter drive. The effectiveness of the two-motor approach depends on the amount of torque bias applied at the drives, which depends on the antenna load. It should be large enough to lead the antenna through the gap for the maximal allowable

11.3 Backlash

175 (a)

Drive 2

Drive 1

(b)

Tachometer

Motor Gearbox

Fig. 11.18 Azimuth drive of the 34-m antenna: (a) two-drive configuration, (b) close look at a single drive

torque load, but small enough that it will not cause excessive local stress, friction and wear. High and steady loads do not need a torque bias because the backlash is observed for low and reversing loads only. Time-varying loads, such as wind gusts, can produce high torques that become very low within a short period of time, which causes a backlash gap when the torque bias dynamics are too slow. Reversing loads from wind gusting were observed and recorded at the DSN antennas.

11.3.1 Backlash and Its Prevention Consider a simple gearbox with two gears rotating in opposite directions. Let the angle of rotation of the first gear be β1 , the second gear be β2 , the gearbox ratio be N, and the gearbox stiffness (measured at the second gear) be k. In this case the relationship between the gearbox rotation and the torque T at the second gear is as follows

176

11 Non-Linear Control

⎧ ⎪ ⎨0 T = k(Δβ − b) ⎪ ⎩ k(Δβ + b)

for |Δβ| ≤ b for Δβ > b for Δβ < −b

(11.21)

where Δβ = β2 − β1 N , and b is the size of the backlash gap measured at the second gear. A plot of torque versus angle difference Δβ is shown in Fig. 11.19 for b =1, and k = 20. It is clear from equation (11.21) that if the angle difference of two gears is smaller than the backlash gap b, the gear motion is discontinuous and the gear teeth impact each other. Implementing a drive with two gears of torques T1 and T2 (that differ by the amount of ) instead of a single drive will minimize the impact at the discontinuity. This difference, called torque bias or counter-torque, will continually drive the antenna even if one gear is in backlash. This is the principle of torque sharing. The question remains as to how large the torque bias must be to prevent backlash. If the stiffness of the gearbox is k and the backlash gap is b, the torque bias ΔT should be greater than 2kb. But ΔT is a function of the load applied to the gears as well: a bias is not required if torque load is high (T1 ∼ = T2 ΔT ) because the angle difference is large and backlash is not observed (even when ΔT = 0). Plots of the existing profile of motor torque vs. axial load (as percentage of the maximal load) are shown in Fig. 11.20, for 10, 20, and 30 percent of the bias. The bias is shaped such that it is the largest for the low loads, and phases out for higher loads. In the event of dynamic loading, such as wind gusts, the appropriate magnitude of the torque bias is not obvious. During dynamic loading, the torque difference determined for the steady-state case may not be large enough to prevent the backlash, and assuming a higher counter-torque may lead to premature wear. Additionally, 200 150 100

T

50 0 –50 –100 –150 –200 –10

–5

0

5

10

Δβ

Fig. 11.19 The backlash function: relationship between gear angle difference and the torque

11.3 Backlash

177

1 0.8

motor torque/max. motor torque

0.6

30% bias 20% bias

0.4

10% bias 0.2 0 –0.2 –0.4 –0.6 –0.8 –1 –1

–0.8

–0.6

–0.4 –0.2 0 0.2 0.4 axis torque/max. axis torque

0.6

0.8

1

Fig. 11.20 Motor torque vs. axial torque, for 10% (dashed line), 20% (solid line), and 30% (dotdashed line) torque bias. Upper lines represent drive 1, lower lines represent drive 2

quickly varying loads with small steady components may cause backlash in both gears simultaneously, despite the non-zero torque bias. Thus, the torque bias time response is also an important design factor.

11.3.2 The Velocity Loop Model with Friction and Backlash The Simulink model of the velocity loop system is shown in Fig. 11.21. The model contains the antenna structure with the velocity input. The outputs are the encoder, antenna velocity, and pinion velocity. The antenna structure model is obtained from the finite-element model, as described in Chapter 2. The drive model which consists of the velocity input, pinion velocity inputs, and torque output, is shown in Fig. 11.21b. The drive consists of two motors (with gearboxes) and the torque share circuit. The block diagram of the drive subsystem is shown in Fig. 11.21c. It consists of two amplifiers, a motor armature, and a gearbox. The gearbox model includes non-linear friction and backlash models; see Fig. 11.21d. The friction torque in this model depends on the motor torque and motor speed, as described in equation (11.9). In the backlash model the torque depends on the difference between the motor and the pinion angle, as in equation (11.21). The torque share circuit is shown in Fig. 11.21b.

178

11 Non-Linear Control

(a)

Pinion velocity y(n) = Cx(n) + Du(n)

Torque

1

x(n + 1) = Ax(n) + Bu(n)

m 1

Velocity Comm.

velocity command

antenna position

Antenna Structure

Drive

(b)

Motor 1 Pinion Rate

Torque

Command Voltage

Current

Current 1

1 Pinion velocity

num(z)

2 Velocity Comm.

Bias Voltage Fade Out

den(z) Filter1

Current 2

–8

1 Torque

Fade Out

Torque Bias Circuit

Command Voltage

Current

Pinion Rate

Torque

Motor 2

(c) 2 Current 2 Command Voltage

Command Voltage Amp1 Voltage Motor Speed

Current Amp 2 Voltage Amp 1 Voltage

Amplifier 2

Amplifier 1

Amp 2 Voltage

Current

Motor Torque Motor Speed

Motor Armature

1 Pinion Rate

Motor Torque Motor velocity Pinion Rate

Torque

Motor and Gearbox

(d)

1 Torque

-KGain 2 K Ts

2

- K-

Motor Torque1

- Kz–1

1/J _ friction _

1 Pinion Rate

K Ts z

z–1

Integrator 2 u

Backlash

Gain 4

2 Torque

1 Motor velocity

- KGain 3

Fig. 11.21 Simulink model of the antenna backlash: (a) velocity loop, (b) drive, (c) motor, gearbox, and amplifiers (d) motor and gearbox with backlash

11.3 Backlash

179

The accuracy of the model was verified experimentally, showing significant accuracy of the simulations; see [16]. Open-loop tests were conducted at the 34-m antenna to compare the measured antenna dynamics with the simulated dynamics of the model that includes backlash and friction. The test data were used to determine the amount of friction and the backlash angle. The velocity loop experiments were conducted with a square-wave input of period 6.3 s and amplitude 0.013 deg/s. Two tests were conducted: one with zero torque bias, and another with a torque bias of 15% of the maximal motor torque (the maximal torque is 308 kGm; thus torque bias was set at 46 kGm). With zero torque bias the simulated motor current (excited by the square wave) and encoder reading are shown in Figs. 11.22 and 11.23, respectively. These plots coincide with the field data in [16]. In addition, the motor currents, which are proportional to the motor torques, were used to determine the frictional torques. The constant current in Fig. 11.22 measures ±1 A, and corresponds to the constant antenna velocity because inertial forces are not present and the motor’s effort is totally dedicated to overcoming the frictional forces. The 1 A current corresponds to a 61 kGm motor torque, or 9.1 × 105 kGm axis torque, which is the measure of frictional torque. For 15% torque bias the plots of measured and simulated motor currents and encoder readings are given in Figs. 11.24 and 11.25, respectively. This situation differs from that of the zero torque bias case in that the encoder shows less chaotic

Fig. 11.22 Motor currents for zero torque bias

180

11 Non-Linear Control

Fig. 11.23 Encoder readings for zero torque bias

movement of the antenna (c.f. Figs. 11.23 and 11.25) and the motor torque plots indicate the presence of the torque bias (their mean values are non-zero and have opposite sign; c.f. Figs. 11.22 and 11.24).

Fig. 11.24 Motor currents for 15% torque bias

References

181

Fig. 11.25 Encoder readings for 15% torque bias

References 1. Armstrong-Helouvry B, Dupont P, Canudas de Wit C. (1994). Friction in Servo Machines: Analysis and Control Methods. Appl. Mechanics Rev., 47(7): 275–305. 2. Armstrong-Helouvry B, Dupont, P, Canudas de Wit, C. (1994). A Survey of Models, Analysis Tools and Compensation Methods for the Control of Machines with Friction. Automatica, 30(7): 1083–1138. 3. Bliman PAJ. (1992). Mathematical Study of the Dahl’s Friction Model. European J. Mechanics, A/Solids, 11(6): 835–848. 4. Boddeke FR, VanVliet LJ, Young IT. (1997). Calibration of the Automated z-Axis of a Microscope Using Focus Function. Journal of Microscopy, 186 (3). 5. Bridges MM, Dawson DM, Hu J. (1996). Adaptive Control for a Class of Direct Drive Robot Manipulators. Int. J. Adaptive Control and Signal Processing, 10(4). 6. Cai L, Song G. (1994). Joint Stick-Slip Friction Compensation of Robot Manipulators by Using Smooth Robust Controllers. J. Robotic Systems, 11(6): 451-469. 7. Canudas de Wit C, Olsson H, Astrom KJ. (1995). A New Model for Control of Systems with Friction. IEEE Trans. on Aut. Control, 40(3): 419–425. 8. Dahl PR. (1976). Solid Friction Damping of Mechanical Vibrations. AIAA J., 14(12): 1675–1682. 9. Dhaouadi R, Kubo K, Tobise MI. (1994). Analysis and Compensation of Speed Drive Systems with Torsional Loads. IEEE Trans. Industry Applications, 30(3): 760–766. 10. Dupont PE. (1994). Avoiding Stick-Slip Through PD Control. IEEE Trans. Aut. Control, 39(5): 1094–1097. 11. Dupont PE, Dunlap EP. (1995). Friction Modeling and PD Compensation at Very Low Velocities. J. Dynamic System, Measurement, and Control, 117(1): 8–14. 12. Edwards C, Postlethwaite I. (1998). Anti-Windup and Bumpless-Transfer Schemes. Automatica, 34(2): 199–210. 13. Friedland B, Davis L. (1997). Feedback Control of Systems with Parasitic Effects. Proc. American Control Conference, Albuquerque, NM.

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14. Gawronski W. (1999). Command Preprocessor for the Beam-Waveguide Antennas. TMO Progress Report, vol. 42-136. Available at http: //ipnpr.jpl.nasa.gov/progress˙report/42-136/ 136A.pdf . 15. Gawronski W, Almassy W. (2002). Command Pre-Processor for Radiotelescopes and Microwave Antennas. IEEE Antennas and Propagation Magazine, 44(2). 16. Gawronski W, Brandt JJ, Ahlstrom, Jr., HG et al. (2000). Torque Bias Profile for Improved Tracking of the Deep Space Network Antennas. IEEE Antennas and Propagation Magazine, 42(6): 35–45. 17. Gawronski W, Parvin B. (1998). Radiotelescope Low Rate Tracking Using Dither. AIAA J. Guidance, Control, and Dynamics, 21: 349–352. 18. Glattfelder AH, Schaufelberger W. (2003). Control Systems with Input and Output Constraints, Springer, London. 19. Grimm G, Hatfield J, Postlethwaite I et al. (2001). Experimental Results in Optimal Linear Anti-Windup Compensation. Proc. 40th IEEE Conf. on Decision and Control, Orlando, FL. 20. Hale LC, Slocum AH. (1994). Design of Anti-Backlash Transmission for Precision Position Control Systems. Precision Engineering, 16(4). 21. Hippe P. (2006). Windup in Control, Its Effects and Their Prevention, Springer, London. 22. Ku SS, Larsen G, Cetinkunt S. (1998). Fast Tool Servo Control for Ultra-Precision Machining at Extremely Low Feed Rates. Mechatronics, 8(4). 23. Lee S, Meerkov SM. (1983). Generalized Dither. International Journal of Control, 53(3): 741–747. 24. Mancini D, Brescia M, Cascote E et al. (1997). A Neural Variable Structure Controller for Telescopes Pointing and Tracking Improvement. Proc. SPIE, vol. 3112. 25. Mancini D, Brescia M, Cascote E et al. (1997). A Variable Structure Control Law for Telescopes Pointing and Tracking. Proc. SPIE, vol. 3086. 26. Mata-Jimenez MT, Brogliato B, Goswami A. (1997). On the Control of Mechanical Systems with Dynamics Backlash, Proc. 36th Conf. Decision and Control, San Diego, CA. 27. Peng Y, Vrancic D, Hanus R. (1996). Anti-Windup, Bumpless, and Conditioned Transfer Techniques for PID Controllers. IEEE Control Systems Magazine, August, 48–57. 28. Southward SC, Radcliffe CJ, MacCluer CR. (1991). Robust Non-linear Stick-Slip Friction Compensation. J. Dynamic Systems, Measurement, and Control, 113: 639–645. 29. Stark AA, Chamberlin RA, Ingalls JG et al. (1997). Optical and Mechanical Design of the Antarctic Submillimeter Telescope and Remote Observatory. Rev. Sci. Instrum., 68(5). 30. Tickoo AK, Koul R, Kaul SK et al (1999) Drive-Control System for the TACTIC gamma-ray telescope. Experimental Astronomy, vol.9, no.2. 31. Trautt TA, Bayo E (1999) Inverse Dynamics of Flexible Manipulators with Coulomb Friction or Backlash and Non-Zero Initial Conditions. Dynamics and Control, vol.9, no.2. 32. Tyler SR. (1994). A Trajectory Preprocessor for Antenna Pointing. TDA Progress Report, 42-118, pp. 139–159. Available at: http: //ipnpr.jpl.nasa.gov/progress˙report/42-118/118E. pdf. 33. Yeh TJ, Pan YC. (2000). Modeling and Identification of Opto-mechanical Coupling and Backlash Non-linearity in Optical Disk Drives. IEEE Trans. Consumer Electronics, 46(1).

Chapter 12

RF Beam Control

This chapter discusses the detection and control of the position of the radio frequency (RF) beam. The position of the beam is slightly different from the antenna position as measured by the encoders. This happens due to structural deformations caused either by loads (e.g., gravity, wind) or by the temperature gradient, atmospheric distortion, or azimuth track unevenness. We also discuss how the beam position is detected and controlled to minimize pointing error. First, the RF controller to be used in the RF control is selected. Next, the monopulse detection and control technique are analyzed, showing its performance in its linear and nonlinear models. Finally, the scanning techniques, such as conical scan, Lissajous, rosette, and sliding window conical scan are discussed.

12.1 Selecting the RF Beam Controller In this section, based on [9], the RF beam position controller (RF controller, for short) is selected. The RF feedback loop is closed over the antenna position loop (see the block diagram in Fig. 12.1). The RF control system consists of the antenna position loop, the RF controller, and the RF receiver. The receiver is either a conical scan RF detector or a monopulse receiver, discussed later. Two RF control systems are tuned and evaluated: the first one with the PI controller in the position loop, and the second one with the LQG controller in the position loop. In the tuning process the following conditions were assumed: 1. Azimuth and elevation control loops are independent. 2. Noises in elevation and cross-elevation channels are independent. The properties of the RF loop depend on the properties of the position loop. The magnitudes and phases of the position-loop transfer functions are shown in Figs. 12.2 and 12.3 (solid lines). The position loop is tuned such that its transfer function for low frequencies (up to the frequency denoted f o ) is approximately equal to 1. The frequency f o is a bandwidth, and for the PI controller f o = 0.1Hz, and for the LQG controller f o = 1.0 Hz. For frequencies higher than f o the position loop transfer function rolls-off, although it contains resonance peaks that reflect W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 12,

183

184

12 RF Beam Control command RF beam

eb

RF beam controller

cc + +

–

POSITION-LOOP

Position loop controller

Velocity loop

encoder RF beam

eb = beam error cc = command correction

Fig. 12.1 RF beam controller (a)

magnitude

100

10–2

actual simplified

10–2

10–1

100

101

100

101

frequency, Hz

(b)

phase, deg

0 –100 –200 –300 10–2

actual simplified 10–1 frequency, Hz

Fig. 12.2 The transfer function of the PI position control loop: (a) magnitude and (b) phase

the antenna flexible deformations. For the RF controller tuning purposes the peaks are ignored, and the high-frequency part of the position-loop transfer function is approximated with the slope of –20 dB/dec. Therefore, for the RF controller tuning purposes the position loop is approximated with the first-order transfer function: G(s) =

1 1 + Ts

(12.1)

It has a unit gain and the time constant reciprocal to the bandwidth, T = 2π1fo . The time constant is different for the PI and LQG controllers, namely, T =1.592 s

12.1 Selecting the RF Beam Controller

185

(a)

magnitude

100

10–2

actual simplified

10–2

10–1

100

101

100

101

frequency, Hz

(b)

phase, deg

0 –100 –200

actual simplified

–300 10–2

10–1 frequency, Hz

Fig. 12.3 The transfer function of the LQG position control loop: (a) magnitude and (b) phase

for the PI system, and T = 0.159 s for the LQG system. The magnitudes and phases of the transfer functions of the simplified models are shown in Figs. 12.2 and 12.3 (dashed lines). Based on the above model the RF controller is tuned. It includes the determination of its transfer function F(s) and the tracking and disturbance rejection properties. The block diagram of the simplified RF control system is presented in Fig. 12.4. In this diagram, α denotes the target location, y is the beam position, e is beam error, and r is the command (or predict). Denote the transfer function from α to y by H, that from r to y by Hr , and the transfer function of the position loop by G. From Fig. 12.4 one obtains

command e

+

F(s)

+

+

G(s)

–

Fig. 12.4 A simplified block diagram of the RF control system

y

186

12 RF Beam Control

H (s) =

GF G y(s) y(s) = and Hr (s) = = α(s) 1 + GF r (s) 1 + GF

(12.2)

Using the plant transfer function G(s) as in equation (12.1), one arrives at H (s) =

F(s) 1 and Hr (s) = 1 + T s + F(s) 1 + T s + F(s)

(12.3)

The controller transfer function F(s) is determined by shaping the tracking properties of the RF system, H (s). For good tracking properties it is required that the magnitude of the transfer function within the bandwidth 0 ≤ ω ≤ 1/T be equal to 1, that is, |H (ω)| = 1, for 0 ≤ ω ≤

1 T

(12.4)

From equation (12.3), it follows that the above condition is satisfied for F 1 for 0 ≤ ω ≤

1 T

(12.5)

On the other hand, outside the bandwidth, for ω > 1/T it is required that |H (ω)| → 0 for ω > 1/T.

(12.6)

From equation (12.3) it follows that the above condition is satisfied for |F(ω)| → 0, for ω > 1/T

(12.7)

Finally, for the reasonable stability margin, the roll-off rate of F at the crossover frequency should be −20 dB/dec; see [5, p. 172]. It is easy to see that the transfer function of an integrator F(s) =

k s

(12.8)

satisfies all the above conditions. Thus, an integrator is chosen as a RF controller. Its gain k was chosen to obtain acceptable tracking properties. Namely, k = 0.75 was determined for the PI position loop system, and k = 1.0 for the LQG position loop system. The selection was backed with extensive simulations of the tracking properties of the closed-loop system.

12.2 Monopulse

187

12.2 Monopulse The principle of the monopulse technique is simple. The feedhorns of the monopulse tracker are slightly displaced so that each receives the signal from a slightly different position, that is, at slightly different power. The received power of the opposite horns is added, to form a sum beam, and subtracted, to form a difference beam. The difference beam characterizes the pointing error. If the difference beam is zero, the antenna is at the target. If the difference beam is nonzero, the pointing error is generated. The difference beam is linear near the origin; thus the monopulse sensor is linear for small antenna deviations. The PI and LQG controllers in the position loop are analyzed and compared. The selection of the monopulse controller is presented, along with the analysis of the monopulse tracking errors in a noisy environment. The simulations tried to reflect the real antenna environment. For example, the antenna model was derived from the field test data using system identification test; encoder error disturbances, servo noise, and wind gusts were obtained from the field data; and received noise was obtained by propagating white noise from the antenna input to the output of the digital receiver.

12.2.1 Command following Introducing equation (12.8) to equation (12.2), one obtains the closed-loop transfer function H(s) of the second order: H (s) =

ωo2 s 2 + 2ς ωo s + ωo2

(12.9)

√ where ωo = k/T and ζ = 2√1kT . The monopulse closed-loop parameters are compared in Table 12.1. The plots of the simplified (dashed line) and full-order (solid line) transfer functions of the monopulse closed-loop system are shown in Fig. 12.5 (with the PI position controller) and in Fig. 12.6 (with the LQG position controller), showing good coincidence. Note that the monopulse system with the PI controller is narrowbanded ( fo = 0.11 Hz) and underdamped, (the damping ratio is smaller than the critical one, ζ = 0.46 < ζcritical , where ζcritical = 0.71), while the bandwidth of the monopulse system with LQG controller is wider (fo = 0.40 Hz) and is overdamped, (the damping ratio is larger than the critical one, ζ = 1.25 > ζcritical ). Thus, the monopulse system with the PI position loop exhibits overshoot and longer settling Table 12.1 Monopulse closed-loop parameters Controller

T [s]

k[1/s]

ωo [rad/s]

fo [Hz]

␨[–]

PI LQG

1.592 0.159

0.75 1.0

0.69 2.51

0.11 0.40

0.46 1.25

188

12 RF Beam Control

(a)

magnitude

100

10–2

actual simplified

10–3

10–2

10–1 frequency, Hz

100

101

10–1 frequency, Hz

100

101

(b)

phase, deg

0

–100

–200

actual simplified

–300 10–3

10–2

Fig. 12.5 The transfer function of the monopulse control system with the PI position control loop: (a) magnitude and (b) phase.

time in the step response, while the monopulse system with the LQG position loop has no overshoot, and shorter settling time. Within the bandwidth the simplified and the full-order systems show good coincidence in terms of the properties used in the controller tuning and stability analysis. However, only the full-order system can give reliable error estimates of the pointing errors with the required precision.

12.2.2 Disturbance Rejection Properties The transfer function Hr (s) from the command to the encoder describes the system disturbance rejection properties and can be obtained by introducing equation (12.8) to the second transfer function in equation (12.3): Hr (s) =

2ζ ωo s s 2 + 2ς ωo s + ωo2

(12.10)

The plots of the transfer function from the wind gusts input to the encoder output are given in Fig. 12.7: the solid line for the PI position loop and the dashed line for

12.2 Monopulse

189

(a)

magnitude

100

10–2

actual simplified

10–2

10–3

10–1 frequency, Hz

100

101

10–1 frequency, Hz

100

101

(b)

phase, deg

0

–100

–200

actual simplified

–300 10–3

10–2

Fig. 12.6 The transfer function of the monopulse control system with the LQG position control loop: (a) magnitude and (b) phase

Hw magnitude

101 100 10–1 10–2 10–3 –3 10

LQG PI

10–2

10–1 frequency, Hz

100

101

Fig. 12.7 A comparison of the wind disturbance transfer functions of the monopulse control systems with the PI position control loop and LQG position control loop

the LQG position loop. It is easy to see that the monopulse system with the LQG position loop has wind-disturbance rejection properties of an order of magnitude better than has the monopulse with the PI position loop. This will be illustrated later with the results of the wind gusts simulations.

190

12 RF Beam Control

12.2.3 Stability Due to the Gain Variation In practice, the monopulse gain, k, is slowly but significantly varying. Therefore, it is important to check the maximal gain for which the system is stable. The critical gains for which the system is still stable were obtained from simulations and are given in Table 12.2. For the PI position loop, the critical gains are 8.3 in azimuth and 16.7 in elevation. Because the nominal value of the gain for the PI controller is k = 0.75, the stability margin is large enough. For the LQG position loop the critical gains are 5.9 in azimuth and 7.1 in elevation. Because the nominal value of the gain for the LQG controller is k = 1.0, this is also an acceptable stability margin. However, one should notice that the LQG controller is the model-based one and that the velocity loop model varies slightly with the antenna elevation angle. These variations impact mainly the azimuth loop, and the stability margins for this loop can be slightly lower than those obtained from the simulations. The performance under the varied gain is discussed in the next section.

12.2.4 Performance Simulations: Linear Model The full-order model of the antenna obtained from the system identification (rather than the simplified one), is used to evaluate the monopulse closed-loop performance. The following disturbances were applied in the simulations: 1. The 25 km/h wind gusts. The wind disturbance model was taken from the field measured data, see [7] and Chapter 5. 2. Servo noise, generated by the white noise input. This noise generates encoder jitters of standard deviation about 0.1 mdeg 3. Azimuth encoder error consists of the radial run-out error, jitters, and rapid changes due to the gaps between the encoder segments. 4. Monopulse receiver noise. It is a white noise of 1 mdeg standard deviation. The results of pointing simulations due to all of the above-mentioned disturbances are given in Fig. 12.8 (solid line for the LQG and dashed line for the PI). They are obtained for the nominal monopulse gains (k = knom , where knom = 0.75 for the PI controller and knom = 1.0 for the LQG controller). The simulations also were performed for the varying gain, k = αknom , i.e. for the reduced gain (α = 0.5), and for increased gains (α = 2.0 and α = 4.0). The figures show that for the monopulse system with the PI controller the total error is 1.23 mdeg for the nominal gain. The bulk of this error is due to wind gusts. However, the LQG controller

Table 12.2 Critical gains Axis

PI

LQG

Azimuth Elevation

8.3 16.7

5.9 7.1

12.2 Monopulse

191 (b)

total rms error, mdeg

total rms error, mdeg

(a)

1

0.5

0

0

1 2 3 gain, multiple of nominal

4

1

0.5

0

0

1 2 3 gain, multiple of nominal

4

Fig. 12.8 Monopulse system error under total disturbances: (a) azimuth, and (b) elevation

suppresses the wind disturbances significantly, with the rms error below 0.2 mdeg for the nominal gain (see Fig. 12.8).

12.2.5 Performance Simulations: Nonlinear Model The monopulse pointing-error model combines an antenna, feed, and low-noise amplifier. This model is a non-linear one, and the elevation and cross-elevation errors are not independent. The non-linear monopulse pointing-error detector is incorporated into the control system model. The combined control system model that includes both axes and the non-linear monopulse model is shown in Fig. 12.9(a). It consists of the antenna velocity loop model, the position controller, the monopulse controller with variable gain k, the monopulse sum g(θ, φ) and difference h(θ, φ) functions, the monopulse processing block (BVR), and the conversion block of elevation and cross-elevation errors into elevation and azimuth errors (θ and φ are magnitude and phase of the error, respectively). This model includes the following signals: encoder errors, commands, servo noises, wind disturbances, and main and error signal noises. The finite null depth of the detector function was modeled by adding a coherent signal to the difference pattern h(θ, φ). The imbalance between the main and the error channels was modeled by adding phase error ⌬φcalib . BVR noises (noises in error detection) ηel , ηxel , and the receiver demodulation noise ηφ were added to the BVR monopulse processing block. The block diagram of the monopulse pointing error detector is shown in Fig. 12.9(b). This nonlinear device transforms the azimuth and elevation pointing errors into the elevation and cross-elevation pointing errors. It consists of the converter of azimuth and elevation angles into (θ and φ parameters, √ nonlinear functions g(θ, φ) and h(θ, φ), and other nonlinear functions (such as (.), sin(.), cos(.)). The detector functions g(θ, φ) and h(θ, φ) are shown in Fig. 12.10a,b for the 15 dB signal-to-noise-ratio (SNR). They depend on the size of the null-depth. Additionally, the detector model consists of disturbances and noises that represent its

192

12 RF Beam Control

(a)

Az Encoder Error enc_err2 y

y(n) = Cx(n) + Du(n) x(n + 1) = Ax(n) + Bu(n)

AZ correction AZ err

r

wind_az

AZ Velocity Loop (Antenna)

AZ Position, Controller

AZ Wind

AZ monopulse controller AZ Noise servo_na

azerr AZ (deg)

.001

g_off

10

deg_el

conversion fromXEL to AZ 10

R1

1

th_calib

25

Monopulse Pointing Error Detector

bl_bvr

1

EL(deg)

deg_el factor

xel_off

50

pc_n0

1/25

T_update

50

gamma

el_off

elerr EL Noise servo_ne

EL Velocity Loop (Antenna)

wind_el

y(n) = Cx(n) + Du(n) x(n + 1) = Ax(n) + Bu(n)

EL monopulse controller

EL Position, Controller

EL Wind

r

EL correction

EL err

y

enc_err2 EL Encoder Error

(b) nxel

5

-1 1 xel_off 1 azerr

az theta

6 elerr

el

theta g

-K-

sqrt

R/delta phi

(AZ,EL) to (theta,phi)

phi

h

2 el_off

cos

(g,h) sin

dphi_calib

3

4

2

nel

nphic

Fig. 12.9 Block diagrams of (a) the nonlinear monopulse control model and (b) the monopulse pointing model detector

imperfections or environmental interactions: the bore sight shift, ⌬φcalib ; the BVR noises, ηel and ηxel ; and the receiver demodulation noise, ηφ . The detector couples azimuth and elevation errors; therefore the errors cannot be determined separately. As a consequence, the nonlinear control system model

12.2 Monopulse

193

(a)

(b)

102

200 100

g

h,deg

100 10–2

0 –100

10–4 –0.02 –0.01

0 θ, deg

0.01

0.02

–200 –0.02 –0.01

0

0.01

0.02

θ, deg

Fig. 12.10 The detector functions for an SNR of 15 dB: (a) g and (b) h

combines azimuth and elevation loops together, as in Fig. 12.9(a). This model includes, besides the detector, converters from the cross-elevation, and elevation coordinates to azimuth, elevation coordinates, time delays (of 0.1 s), variable loop gains kaz and kel , the monopulse controllers, position controller (LQG), and azimuth and elevation velocity loop models. This closed-loop system is subject to wind disturbances, servo noises, encoder errors, and is driven by the azimuth and elevation commands. Extensive simulations of the non-linear closed-loop system that depend on the above-listed parameters have been performed. The results are summarized in Fig. 12.11, showing the elevation and cross-elevation errors for the 5 mdeg step offset in elevation and cross-elevation. The target of 5 mdeg in elevation and 5 mdeg in cross-elevation is acquired. The error depends on the SNR. For a high SNR of 40 dB the radius of the circling of the destination point of (5, 5) mdeg is small, about 0.5 mdeg. For an SNR of 12 dB it increases about 5 mdeg.

10 XEL error, mdeg

XEL error, mdeg

10

5

0

0

5 EL error, mdeg

10

5

0

0

5 EL error, mdeg

10

Fig. 12.11 Elevation and cross-elevation errors for the 5-mdeg step offset in elevation and crosselevation, and for an SNR of: (a) 40 dB, (b) 12 dB

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12 RF Beam Control

12.3 Scanning The spacecraft trajectory (its position versus time) is typically known with high accuracy, and this trajectory is programmed into the antenna, forming the antenna command. However, due to environmental disturbances such as temperature gradient, wind and gravity forces, and manufacturing imperfections, the antenna does not point precisely towards the spacecraft. Scanning techniques are used to determine the spacecraft position. While tracking, antenna besides following the target performs scanning movements. Scanning movements of an antenna are small harmonic axial movements added to the antenna trajectory. They are used to estimate the true spacecraft position, because the scanning motion produces power variations of the received signal, which are used to estimate the spacecraft position. Three different scanning patterns (conical scan, Lissajous scan, and rosette scan) are presented and analyzed in this chapter. The analysis includes the evaluation of the estimation errors due to random or harmonic variation of the antenna position and due to random and harmonic variations of the power level. Typically, the estimation of the spacecraft position is carried out after completing a full scanning cycle. However, it could be done more frequently using the sliding window scanning, where the spacecraft position estimation is carried out in an almost-continuous manner. The sliding window scanning is analyzed, showing that it reduces estimation time by half.

12.3.1 Conical Scan A technique commonly used for the determination of the true spacecraft position is the conical scanning method (conscan). During conscan, circular movements are added to the antenna command as shown in Fig. 12.12. These circular movements cause sinusoidal variations in the power of the signal received from the spacecraft by the antenna, and these variations are used to estimate the true spacecraft position. The radius of the conscan movement is typically chosen such that the loss of the signal power is of 0.1 dBi (dB power relative to isotropic source). Thus, it depends on the frequency of the receiving signal. For Ka-band signals (32 GHz), the conscan radius is 1.55 mdeg. Depending on the radius, sampling rate, antenna tracking capabilities, and desired accuracy, the period of the conscan typically varies from 30 to 120 s (in our case it is 60 s). Finally, the sampling frequency was chosen as 50 Hz to satisfy the Nyquist criterion, which says that the sampling rate shall be at least twice the antenna bandwidth (of 10 Hz). The conscan technique is used for antenna and radar tracking; see [2, 4]. For spacecraft applications it is described in [1, 6, 11]. For missile tracking is described in [10]. Papers [1] and [5] present the least square and Kalman filter techniques, respectively. In [3] the nonlinear estimation techniques were used to estimate spacecraft position using conscan, and in [8] there are analyses of different pattern of scanning.

195

elevation

12.3 Scanning

trajectory trajectory with conscan azimuth

Fig. 12.12 Schematic diagram of spacecraft trajectory and antenna conscan

Power variation during conscan. Define a coordinate system with its origin located at the antenna command position (i.e., translating with the antenna command). The coordinate system consists of two components: the elevation rotation of the dish and the cross-elevation rotation of the dish. The first component is defined as a rotation with respect to a horizontal axis orthogonal to the boresight, and to the second as a rotation with respect to a vertical axis orthogonal to the boresight and the elevation axis (see Fig. 12.13). Because the spacecraft position in this coordinate system is measured with respect to the antenna boresight, and the spacecraft trajectory is accurately known, its position deviations are predominantly

XEL

boresight

EL

Fig. 12.13 Elevation and cross-elevation coordinate system

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12 RF Beam Control

caused by either unpredictable disturbances acting on antenna (e.g., wind pressure) or by antenna deformations (such as thermal deformations, and un-modeled atmospheric phenomena such as refraction). During conical scanning the antenna moves in a circle of radius r, with its center located at the antenna command position. The conscan data are sampled with frequency of 50 Hz; thus the sampling time is ⌬t = 0.02s. The antenna position ai at time ti = i⌬t consists of the elevation component, aei , and the cross-elevation component, axi , as shown in Fig. 12.14. The position is described by the following equation r cos ωti aeli = (12.11) ai = axeli r sin ωti Plots of aei and axi for conscan radius r = 1.55 mdeg, and conscan period T = 60 s, ω = 2π /T is conscan frequency (ω = 0.1047 rad/s) are shown in Fig. 12.14. Also, T = n⌬t

(12.12)

where n is the number of samples per 1 circle, n = 3000. The target position is denoted si , with the elevation and cross-elevation components sei and sxi s (12.13) si = ei sxi where sei is the elevation component of the target position at i⌬t, and sxi is the cross-elevation component of target position at i⌬t (Fig. 12.15), The antenna position error is defined as the difference between the target position and the antenna position, that is, ei = si − ai

(12.14)

1.5 XEL EL

EL and XEL, mdeg

1 0.5 0 –0.5 –1 –1.5

0

10

20

30 time, s

Fig. 12.14 Elevation and cross-elevation components of conscan

40

50

60

12.3 Scanning

197 XEL

axi

sxi sˆxi 0

ai

si

trajectory

sˆi aei sei sˆei EL

Fig. 12.15 Antenna position, target position, and estimated target position during conscan

Like the antenna position, it has two components, elevation and cross-elevation position errors: sei − aei eei = (12.15) ei = exi sxi − axi The total error at ti = i⌬t is described as the position rms error, that is, εi = eiT ei = ei 2 (12.16) Combining (12.11), (12.14), and (12.16), one obtains εi2 = siT si − 2aiT si + aiT ai = siT si − 2aiT si + r 2

(12.17)

Next, we describe how the error impacts the beam power. The carrier power pi is a function of the error εi , and its Gaussian approximation is expressed as pi = poi exp(−

μ 2 ε ) + vi h2 i

(12.18)

where poi is the maximum carrier power, v i is the signal noise, h is the half-power beamwidth (17 mdeg for Ka-band), and μ = 4 ln(2) = 2.7726. Note that although a spacecraft can move relatively quickly with respect to fixed coordinates, it typically moves slowly in the selected coordinate frame (this relative movement is caused by slowly varying disturbances). For example, thermal deformations have period of several hours, while conscan period is only one minute. Therefore, it is safe to assume that the target position is constant during the conscan period, that is, si ∼ = s. It is also safe to assume that the power is constant during the conscan period, that is, poi = po .

198

12 RF Beam Control

Using the approximation exp(x) ∼ = 1 + x one obtains equation (12.18) as follows pi = po (1 −

μ 2 ε ) h2 i

(12.19)

Substituting (12.17) into the above equation, and assuming si = s, poi = po , one obtains pi = po −

po μ 2 2 po μ T (r + s T s − 2aiT s) + v i = pm + a s + vi h2 h2 i

where pm is mean power. Using equation (12.11) one arrives at pi = pm +

2 po μr (se cos ωti + sx sin ωti ) + v i h2

(12.20)

In the above equations pm is the mean power, defined as μ pm = po 1 − 2 (r 2 + s T s) h

(12.21)

(see [6]). The plot of pi is shown in Figs. 12.16 and 12.17. Additionally, in these figures the plot of the power as a function of the antenna position is marked. These

antenna off target

antenna on target

1 0.8 0.6 0.4 0.2 0 20 10 0 cross-elevation, mdeg

20 10 –10 –20

–10

0 elevation, mdeg

–20

Fig. 12.16 Carrier power and the conscan power for perfectly pointed antenna and for elevation error

12.3 Scanning

199

(a)

power/powermax

1 0.98 0.96 0.94 rosette

conscan

0.92 0.9

0

10

20

Lissajous

30 time, s

(b)

40

50

60

power/powermax

1 0.98 0.96 0.94 rosette

0.92 0.9

0

10

Lissajous

20

30 time, s

conscan

40

50

60

Fig. 12.17 A power variation (with respect to maximum power) for the conscan, Lissajous, and rosette scans: (a) antenna on target, and (b) 0.7 mdeg elevation error

plots are presented for the case of the antenna perfectly pointed at the target, as well as for the case of an error in antenna elevation position. It can be seen that for the perfectly pointed antenna the received power is constant and smaller than the maximum power. For a mis-pointed antenna the received power varies in sinusoidal fashion, as it is derived in the following paragraph. The algorithm presented above is a corrected Alvarez algorithm [1]. Here the Taylor expansion was taken with respect to the error εi , which produces the maximum power rather than mean power in the second component in equation (12.20). Denoting the variation from mean power as dpi = pi − pm one obtains from equation (12.20) dpi = gse cos ωti + gsx sin ωti + v i

(12.22)

where g=

2 po μr h2

(12.23)

In this equation, g, and ω are known parameters, the power variation dpi is measured, and the spacecraft coordinates se , sx are to be determined. If no noise

200

12 RF Beam Control

were present, the spacecraft position could be obtained from the amplitude and phase of the power variation. Because the received power signal is noisy, the leastsquare technique is applied. Estimating spacecraft position from the power measurements. Denoting

ki = g cos ωti sin ωti

(12.24)

equation (12.22) can be written as dpi = ki s + v i where s =

se . For an entire conscan circle/period, sx ⎧ ⎫ ⎧ ⎫ ⎧ ⎫ k1 ⎪ v1 ⎪ dp1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨k2 ⎪ ⎨v 2 ⎪ ⎨dp2 ⎪ ⎬ ⎬ ⎬ , K = . , V = . , dP = .. .⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪.⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎩ ⎭ ⎭ dpn kn vn

(12.25)

(12.26)

and equation (12.25) is obtained in the following form dP = Ks + V

(12.27)

The estimated spacecraft position sˆ is the least-square solution of the above equation: sˆ = K + d P

(12.28)

where K + = (K T K )−1 K T .

12.3.2 Sliding Window Conscan The spacecraft position estimation technique described up to this point has used data collected during the whole scanning period. Thus, the spacecraft position estimate is updated every period T, which typically ranges from 30 to 120 s. This is a rather slow update, causing a significant lag in antenna tracking if the assumption of slowly varying target position is incorrect. This lag can be improved using a technique known as sliding window scanning. In sliding window scanning, spacecraft position is estimated every time period ⌬T, where ⌬T < T . The update moments are shown in Fig. 12.18 for ⌬T = 13 T . In this case the data used in equation (12.27) do not start at the beginning of every circle; rather, they begin at times T, T + ⌬T , T + 2⌬T ,

12.3 Scanning

201 (a) 0

T

2T

3T

…

time

(b) 0

T

2T

…

time

Fig. 12.18 Scans: (a) regular and (b) sliding window, for ⌬T = 13 T

T + 3⌬T , etc. Note that the first estimation is at T rather than ⌬T because an entire circle is required to begin the estimation process. To see the usefulness of the sliding window technique in the estimation process, assume that the target position changes rapidly by 0.15 mdeg at t = 150 s. This type of shift may be caused by a sudden disturbance in antenna position, such as a large gust of wind. This is an extreme situation, because in the closed-loop configuration the control system would “soften” the impact of the gusts. The shift is illustrated in Fig. 12.19, along with simulated responses of antennas using the traditional conscan method and the sliding window method. The simulations show that for the conscan period T = 60 s, the time of 120 s is required to reach the target, whereas the sliding window conscan with ⌬T = 5 s reaches the target in half the time, that is, in 60 s. This is especially important when antenna dynamics is involved, because faster sensors improve the pointing accuracy.

12.3.3 Lissajous Scan In the Lissajous scanning pattern, the antenna position ai at time ti = i⌬t consists of the elevation component, aei , and the cross-elevation component, axi , described by the following equation ai =

aei axi

=

r sin nωti r sin mωti

(12.29)

where n and m are natural numbers. Again, the components are harmonic functions, which are most desirable for the antenna motion because they do not result in jerks or rapid motions. The Lissajous curve for r = 1.55 mdeg, n = 3, and m = 4 is shown in Fig. 12.20, and the individual components’ plots (aei and axi ) are shown in Fig. 12.21. The radius was chosen such that the mean power loss was equal to the mean power loss resulting from a conscan sweep with r = 1.55 mdeg. For this scanning pattern, the power variation is obtained as follows. With the antenna position error defined as in equation (12.14), the total error at ti = i⌬t is described as the position rms error:

202

12 RF Beam Control

elevation mdeg

0.15 target position

0.1

sliding conscan conscan

0.05

0

0

20

40

60

80 time, s

100

120

140

160

cross-elevation mdeg

0.15 target position

0.1

sliding conscan conscan

0.05

0

0

20

40

60

80 time, s

100

120

140

160

Fig. 12.19 Estimated spacecraft position for regular conscan and sliding window conscan

εi2 = siT si − 2aiT si + aiT ai

(12.30)

Using the carrier power pi as in equation (12.19), and assuming that the spacecraft position and the carrier power are constant during conscan period, that is, that si = s and poi = po , one obtains po μ T (a ai + s T s − 2aiT s) + v i h2 i 2μ = pm exp ( 2 aiT s ) + v i h 2μr = pm exp ( 2 (se sin nωti + sx sin mωti ) ) + v i h

pi = po −

(12.31)

Using equation (12.29) one obtains pi = pm +

2 po μr (se sin nωti + sx sin mωti ) + v i h2

(12.32)

12.3 Scanning

203 2 1.5

cross-elevation, mdeg

1 0.5 0 –0.5 –1 –1.5 Lissajous courve

–2 –2

–1.5

conscan circle –1

–0.5 0 0.5 elevation, mdeg

1

1.5

2

Fig. 12.20 Lissajous curve for R = 1.55 mdeg, n = 3, and m = 4 2 XEL EL

EL and XEL, mdeg

1.5 1 0.5 0 –0.5 –1 –1.5 –2

0

10

20

30 time, s

40

50

60

Fig. 12.21 Elevation and cross elevation components of Lissajous scanning pattern for n=3, m=4

where pm is the mean power defined as pm = po ( 1 −

μ T (a ai + s T s) ) h2 i

(12.33)

Denoting the variation from mean power as dpi = pi − pm one obtains the power variation as a function of the spacecraft position, se and sx

204

12 RF Beam Control

dpi = gse sin nωti + gsx sin mωti + v i

(12.34)

In this equation g and ω are known parameters, dpi is measured, and se and sx are spacecraft coordinates to be determined. The plot of the power variation dpi is shown in Fig. 12.17. It is seen from this figure that power variation occurs even when the antenna is perfectly pointed, unlike the conscan situation. The Lissajous radius, however, was chosen such that in average the loss of the received power is the same as during conscan. Also, unlike the conscan situation, the maximum power is reached during the cycle because the Lissajous curve crosses through the origin. As in the conscan case, the spacecraft position estimate is obtained from equation (12.28). In the Lissajous case, the matrix K is changed such that its ith row is

ki = g sin nωti sin mωti

(12.35)

12.3.4 Rosette Scan The rosette scanning is used in missile tracking (see [10]) and in telescope infrared tracking (see [12]). In the rosette scanning, the antenna movement is described by the following equation ai =

aei axi

=

r cos nωti + r cos mωti r sin nωti − r sin mωti

(12.36)

The plots of the elevation component aei , and the cross-elevation component, axi are shown in Fig. 12.22 for the rosette curve of radius r = 1.10 mdeg, n = 1, and m = 3. The rosette curve itself is shown in Fig. 12.23 (the radius was chosen such that the mean power loss is the same as for the conscan with r = 1.55 mdeg). Note that the rosette curve, unlike the conscan circle, crosses the origin and thus receives peak power if the target and boresight position coincide. Following the derivation as above, the antenna power is obtained as follows 2 po μr (se (cos nωti + cos mωti ) + sx (sin nωti − sin mωti )) + v i h2 (12.37) therefore the power variation is pi = pm +

dpi = gse (cos nωti + cos mωti ) + gsx (sin nωti − sin mωti ) + v i

(12.38)

where g is given by equation (12.23). The plots of variations of dpi are shown in Fig. 12.17. The plots show that received power varies for the perfectly pointed

12.3 Scanning

205

EL and XEL, mdeg

2 XEL EL

1

0

–1

–2

0

10

20

30 time, s

40

50

60

Fig. 12.22 Rosette elevation and cross-elevation components

Fig. 12.23 Rosette curve for r = 1.1 mdeg, n = 1, and m = 3

2 1.5

cross-elevation, mdeg

1 0.5 0 –0.5 –1 –1.5 –2 –2

conscan circle

rosette courve –1.5

–1

–0.5 0 0.5 elevation, mdeg

1

1.5

2

antenna. Also, like the Lissajous case, the maximum power is reached during the cycle, because the rosette curve crosses through the origin. The spacecraft position estimate is determined from the above equation, using equation (12.28), where the ith row of the matrix K in this equation is ki = g cos nωti + cos mωti

sin nωti − sin mωti

(12.39)

206

12 RF Beam Control

12.3.5 Performance Evaluation The scanning performance is evaluated in the presence of disturbances. First, the antenna position is disturbed by random factors, such as wind gusts, and by deterministic disturbances, which can be decomposed into harmonic components. Secondly, the carrier power is also modeled as disturbed by random noise (e.g., receiver noise), and deterministic variations, such as spacecraft spinning. The latter disturbances are decomposed into harmonic components. Position disturbances are caused by antenna motion. These disturbances may be purely random, and as such are modeled as a white Gaussian noise of given standard deviation, or more or less deterministic disturbances, which are modeled as harmonic components within a bandwidth up to 10 Hz (i.e., antenna dynamics bandwidth). The random disturbances were simulated separately in the elevation and crosselevation directions. The results are shown in Tables 12.3 and 12.4. They show that the scanning algorithms have quite effective disturbance rejection properties. In conscan, for example, a disturbance in XEL direction of 0.3-mdeg standard deviation will cause approximately 0.005 mdeg of error in the EL estimation and 0.01 mdeg of error in the XEL estimation. Next, the antenna position xa was disturbed by a harmonic motion in the EL direction of frequency f and amplitude 0.1 mdeg. The disturbance’s impact on estimation accuracy was analyzed for frequencies ranging from 0.0001 Hz (very slow motion) to 10 Hz (antenna bandwidth). Note that the scanning frequency is fo = 1/60 = 0.0167 Hz. Consider the plots of the estimation error in EL and XEL as shown in Fig. 12.24a,b. They show that for frequencies higher that the scanning frequency the disturbances are quickly suppressed. The slope of the error magnitude drops as follows: for conscan, −60 dB/dec in EL and −40 dB/dec in XEL; for Lissajous, scan −20 dB/dec in EL and −60 dB/dec in XEL; and for rosette, −20 dB/dec in EL and −40 dB/dec in XEL. For low frequencies, the disturbance level is constant in EL for all scans, and drops down in XEL: −20 dB/dec for conscan and rosette scan, and –40 dB/dec for Lissajous scan. Random power variations are also simulated. Variations have standard deviations ranging from 0.1 to 10% of the maximum power. For all three scans, the EL Table 12.3 Estimation error (mdeg) due to 1-mdeg noise in EL Direction Elevation Cross-elevation

Conscan

Lissajous

Rosette

0.025 0.013

0.025 0.017

0.037 0.010

Table 12.4 Estimation error (mdeg) due to 1-mdeg noise in XEL Direction Elevation Cross-elevation

Conscan

Lissajous

Rosette

0.015 0.027

0.019 0.029

0.010 0.037

12.3 Scanning

207

(a) 10–1

elevation error, mdeg

10–2

10–3

10–4

10–5

10–6 –4 10

conscan Lissajous rosette 10–3

10–2 10–1 frequency, Hz

100

101

(b)

cross-elevation error, mdeg

10–1 conscan Lissajous rosette

10–2

10–3

10–4

10–5

10–6 –4 10

10–3

10–2 10–1 frequency, Hz

100

101

Fig. 12.24 Estimation errors in response to elevation harmonic disturbance of amplitude 0.1 mdeg: (a) elevation error, and (b) cross-elevation error

and XEL estimation error was proportional to the variation of power with gain of 0.080–0.089 mdeg per 10% of power variation, as shown in Table 12.5. This is very effective suppression of power noise, because the10% standard deviation of power variation causes less than 0.1 mdeg estimation error.

208

12 RF Beam Control Table 12.5 Estimation error (mdeg) due to power noise Direction

Conscan

Lissajous

Rosette

Elevation 0.089 0.083 0.080 Cross-elevation 0.089 0.080 0.088 The standard deviation of the power noise is 10% of peak power.

Next, the impact of pulsating power on the estimation of the EL and XEL position was analyzed. The results are shown in Fig. 12.25a,b. The harmonic power variations were of frequencies ranging from 0.0001 Hz to 10 Hz, and of amplitude (a) 101 conscan Lissajous rosette

elevation error, mdeg

100

10–1

10–2

10–3

10–4 –4 10

10–3

10–2 10–1 frequency, Hz

100

101

cross-elevation error, mdeg

(b) 101 conscan Lissajous rosette

100

10–1

10–2

10–3

10–4 –4 10

10–3

10–2

10–1

100

101

frequency, Hz

Fig. 12.25 Estimation errors in response to harmonic power variation: (a) elevation error, and (b) cross-elevation error

References

209

0.1 (10% of maximal power). The plots show that the maximal estimation error of 3 mdeg amplitude was observed for frequencies near the scan frequency, and that for frequencies lower and higher frequencies the amplitude of the estimation errors quickly drops. Thus, all three scanning algorithms act as effective filters for this kind of disturbance.

References 1. Alvarez L. (1993). Analysis of Open-Loop Conical Scan Pointing Error and Variance Estimators. TDA Progress Report 42-115, Jet Propulsion Laboratory, Pasadena, CA. Available at http://ipnpr.jpl.nasa.gov/progress report/42-115/115 g.pdf 2. Biernson G. (1990). Optimal Radar Tracking Systems, Wiley, New York. 3. Chen L, Fathpour N, Mehra R. (2007). Comparing Antenna Conical Scan Algorithms for Spacecraft Position Estimation. J Guidance, Control and Dynamics, 30(4): 1186–1188. 4. Damonte JB, Stoddard DJ. (1956). An Analysis of Conical Scan Antennas for Tracking. IRE National Convention Record, 4(1). 5. Dutton K, Thompson S, Barraclough B. (1997). The Art of Control Engineering. AddisonWesley, Harlow. 6. Eldred DB. (1994). An Improved Conscan Algorithm Based on Kalman Filter. TDA Progress Report 42-116, Jet Propulsion Laboratory, Pasadena, CA. Available at http://ipnpr.jpl.nasa.gov/ progress report/42-116/116r.pdf. 7. Gawronski W. (1995). Wind Gust Models Derived from Field Data. TDA Progress Report 42-123, Jet Propulsion Laboratory, Pasadena, California, pp. 30–36. Available at http://ipnpr. jpl.nasa.gov/progress report/42-123/123G.pdf. 8. Gawronski W, Craparo E. (2002). Antenna Scanning Techniques for Estimation of Spacecraft Position. IEEE Antennas and Propagation Magazine, 44(6). 9. Gawronski W, Gudim MA. (1999). Design and Performance of the Monopulse Control System. IEEE Antennas and Propagation Magazine, 41(6). 10. Lee HP, Hwang HY. (1997). Design of Two-Degree-of Freedom Robust Controllers for a Seeker Scan Loop System. Int. J. Control, 66(4). 11. Ohlson JE, Reid MS. (1976). Conical-Scan Tracking with 64-m Diameter Antenna at Goldstone. JPL Technical Report, 32-1605, Jet Propulsion Laboratory, Pasadena, CA. 12. Wan H, Liang Z, Zhang Q et al. (1996). A Double-Band Infrared Image Processing System Using Rosette Scanning. Detectors, Focal Plane Arrays, and Applications, SPIE Proceedings, vol. 2894.

Chapter 13

Track-Level Compensation

The track-level compensation technique is an open-loop control of the RF beam position. The beam position is calculated with respect to the antenna azimuth and elevation position from the look-up table, and the antenna position is corrected to reduce the pointing error. The whole antenna structure rotates in azimuth on a circular azimuth track. It is manufactured with the level of precision of ±0.5 mm for the 34-m DSN antennas. For more discussion of the azimuth track design, see [1]. The pointing accuracy of the antennas and radiotelescopes is impacted by the unevenness of the antenna azimuth track. The track unevenness causes repeated antenna tilts, hence repeatable pointing errors. In this chapter, track-level errors are described along with their compensation using the look-up table. Also, the creation of the table is described, which includes the collection and processing of the inclinometer data, and determining azimuth axis tilt. Next, the antenna pointing errors are derived from the table. Finally, pointing improvement is discussed using the table.

13.1 Description of the Track-Level Problem The track is shown in Fig. 13.1. Manufacturing and installation tolerances, as well as the foundation compliance and the gaps between the segments of the track, are the sources of the pointing errors that reach over 20 mdeg peak-to-peak magnitude. This chapter discusses the improvement of the pointing accuracy of the antennas by implementing the track-level-compensation look-up table; see [2, 3]. The table consists of three axis rotations of the antenna lower structure (alidade) as a function of the azimuth position. The development of the table is presented, based on the measurements of the inclinometer tilts, and describe the processing the measurement data. Also the determination of the elevation and cross-elevation errors of the antenna is presented as a function of the alidade rotations. The pointing accuracy of the antenna with and without a table was measured using various RF beam pointing techniques. It was shown that the pointing error decreased when the table was used, from 7.5 mdeg to 1.2 mdeg in elevation, and from 20.4 mdeg to 2.2 mdeg in crosselevation. W. Gawronski, Modeling and Control of Antennas and Telescopes, C Springer Science+Business Media, LLC 2008 DOI: 10.1007/978-0-387-78793-0 13,

211

212

13 Track-Level Compensation (a)

alidade track

(b)

Fig. 13.1 The 34-m antenna: (a) alidade and azimuth track, and (b) azimuth track

Track-level compensation (TLC) look-up tables were created for the DSN antennas in Goldstone, California, in Canberra, Australia, and in Madrid, Spain. However, the James Clerk Maxwell Telescope1,2 used inclinometers to perform track profile measurements to overcome possible systematic errors, but the results have not been published. The track-level unevenness compensation is planned for the

1 http://www.jach.hawaii.edu/ets/mech/mech

recent.html

2 http://www.jach.hawaii.edu/JCMT/telescope/pointing/20011006.html

13.2 Collection and Processing of the Inclinometer Data

213

Sardinia Radio Telescope [5]. The Green Bank Telescope memo3 reports on the pointing errors due to the azimuth track-level unevenness. GBT says4 that “in the antenna engineering and operations area work on the Green Bank Telescope azimuth track was seen as the most important.” Inclinometers were used also for the thermal deformation of the IRAM telescope [4].

13.2 Collection and Processing of the Inclinometer Data The TLC system hardware consists of four inclinometers, the interface assembly, and a PC computer. Four digital inclinometers (model D711 of Applied Geomechanics) are mounted on the antenna. The inclinometers are located on the alidade, as shown in Fig. 13.2. Each inclinometer measures tilt in two axis, denoted x and y. The manufacturer describes the inclinometer rotation as tilts. Note that the x-axis tilt is equivalent to the y-axis rotation, and vice versa, as shown in Fig. 13.3. First, the track profile was measured. A shim of 2.5 mm was placed on the track, and the antenna wheel positioned on the shim. The inclinometer tilts were measured to determine the relationship between the inclinometer tilt and the azimuth track unevenness. The shim caused 14.8 mdeg x-tilt of the inclinometer No. 3. Based on this scaling and the continuous records of the lower inclinometer x-tilt measurements during the antenna constant velocity slewing the azimuth track profile was determined, and is shown in Fig. 13.4. It is seen from this plot that the maximum, Z rotation

y2

2

L

x2

1

x1

Y rotation

x4

4

y1

X rotation

y4 3

x3

y3 H

Fig. 13.2 The location of the inclinometers at the alidade and X, Y, and Z rotations of the alidade

3 http://wwwlocal.gb.nrao.edu/ptcs/ptcspn/ptcspn40/AzTrackSpec.pdf 4 http://www.nrao.edu/news/newsletters/nraonews94.pdf

214

13 Track-Level Compensation

Fig. 13.3 x-axis tilt is a rotation with respect to y-axis

x-tilt x y-rotation

y

Fig. 13.4 Azimuth track profile of the 34-m antenna

peak-to-peak, track profile variation is 1.2 mm, slightly higher than the specification (1 mm). Next, the inclinometer data were collected to determine the alidade rotations. The antenna moves at constant azimuth axis velocity of 0.05 deg/s. Due to the environmental disturbances the inclinometer data are extremely noisy. Take for example the x-axis measurement of the inclinometer 1 shown in Fig. 13.5. The unfiltered data are represented by the gray line. Using a zero-phase filter to prevent filtering delay, the data is smoothed, as represented by the black line. 20

inclinometer 1, x-tilt, mdeg

15 10 5 0 –5 –10 –15 –20

0

50

100

150 200 azimuth encoder, deg

250

Fig. 13.5 Raw inclinometer data (gray line) and the filtered data (black line)

300

350

13.3 Estimating Azimuth Axis Tilt

215

13.3 Estimating Azimuth Axis Tilt The additional processing includes the removal of the azimuth axis tilt from the data. The tilt is present in the inclinometer data as harmonic functions in x- and y-axis tilt, of period 360 deg; see Fig. 13.6a,b. Its amplitude (a) and phase (ϕ) need to be determined. Let α1x (i) and α1y (i) be the ith sample of the x- and y-tilts of the inclinometer 1, and e(i) be the ith sample of the azimuth encoder. The inclinometer harmonics caused by the azimuth axis tilt are described as α1x (i) = a cos(e(i) + ϕ), and α1y (i) = a sin(e(i) + ϕ)

(13.1)

or, in short, as α1x (i) = ac ci − as si and α1y (i) = ac si − as ci

(13.2)

inc linometer x -tilt, mdeg

10 5 0 –5 –10

0

50

100

150 200 azimuth position, deg

250

300

350

0

50

100

150 200 azimuth position, deg

250

300

350

inc linometer y -tilt, mdeg

10 5 0 –5 –10

Fig. 13.6 Removing the azimuth axis tilt from the inclinometer data (solid line, inclinometer data; dash-dot line, inclinometer tilt caused by the azimuth axis tilt; and dashed line, inclinometer data after azimuth axis tilt removal)

216

13 Track-Level Compensation

where ac = a cos(ϕ), as = a sin(ϕ), ci = cos(e(i)), and si = sin(e(i)). For n samples define the following vectors and matrices

α1 =

α1x , where α1x α1y

P=

c s

⎧ ⎧ ⎫ ⎫ α1x (1)⎪ α1y (1)⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨α1x (2)⎬ ⎨α1y (2)⎪ ⎬ and α1y = = .. .. ⎪ ⎪ . ⎪ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭ ⎭ α1x (n) α1y (n)

⎧ ⎫ ⎧ ⎫ ls1 ⎪ lc1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎨ s2 ⎪ ⎨ ⎬ c 2 −s and s = , where c = . . .. ⎪ c ⎪ ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ ⎪ ⎭ ⎭ cn sn

(13.3)

(13.4)

and A=

ac as

(13.5)

For the above notations equations (13.2) can be rewritten in a compact form P A = α1

(13.6)

The least-square solution A of equation (13.5) is as follows A = (P T P)−1 P T α1

(13.7)

But, from equation (13.5) it follows that a cos(ϕ) = ac , and a sin(ϕ) = as

(13.8)

therefore a=

+ , as ac2 + as2 and ϕ = tan−1 ac

(13.9)

Based on several sets of data one obtains from (13.9) the following amplitude and phase of the azimuth axis tilt a = 4.2 mdeg

and

ϕ = 274.5 deg

13.4 Creating the TLC Table

217

that is, the tilt magnitude 4.2 mdeg and phase 274.5 deg. The x- and y-axis movements of the inclinometer 1 after the tilt removal is shown in Fig. 13.6a,b (dashed line).

13.4 Creating the TLC Table The TLC look-up table consists of X, Y, and Z rotations of the alidade, as shown in Fig. 13.2. They are obtained from the inclinometer tilts. Namely, a rotation with respect to the antenna x-axis, denoted X, is a rotation with respect to the antenna elevation axis. It is measured as the y-tilt of the second inclinometer (α2y ): X = α2y

(13.10)

The Y rotation is a tilt of the elevation axis. It is an average of the x-tilts of the inclinometers 1 and 2, that is, Y = 0.5(α1x + α2x )

(13.11)

The Z rotation of the alidade is a twist of the alidade, and is not directly measured by the inclinometers. It is determined from x-tilts of inclinometers 3 and 4, as follows. From Fig. 13.7, which represents the view from the top of the alidade, one has Z=

d3 − d4 L

(13.12)

where d3 and d4 are horizontal displacements of the locations of inclinometers 3 and 4, and L = 12.396 m is the distance between the two inclinometers. The displacements d3 and d4 are determined from the tilts of inclinometers 3 and 4, respectively, by assuming that the horizontal displacement of the alidade side due to azimuth track unevenness is caused predominantly by the rigid-body motion of each side of the alidade. This assumption has been checked with the finite-element model of the alidade, giving a 93% accuracy in the estimation of displacements d3 and d4 . It was

Z d4 Inclinometer 4

Fig. 13.7 Top view on the inclinometers 3 and 4

d3

L Inclinometer 3

218

13 Track-Level Compensation

also confirmed by the comparison of the rotations of the inclinometers located at the bottom, the middle, and the top of the alidade. The rigid-body angle is measured as the x-tilt of the inclinometers 3 and 4 (denoted as α3x and α4x , respectively), therefore d3 = H α3x

and

d4 = H α4x

(13.13)

where H is the height at which the inclinometers are located, H = 9.292 m. Introducing (13.13) to (13.12) one obtains the Z rotation of the alidade as Z=

H (α3x − α4x ) L

(13.14)

where H is the alidade height and L is the distance between the inclinometers 1 and 2. Because for the 34-m antennas L = 12.39 m, the ratio is H / L = 0.75, therefore Z = 0.75(α3x − α4x )

(13.15)

The X, Y, and Z alidade rotations obtained from the inclinometer data, for azimuth angles varied from 0 to 360 deg, are shown in Fig. 13.8. The plots show that the X rotation (the elevation correction) is comparatively small, and that the largest is the

X , mdeg

5

0

–5

0

50

100

150

200

250

300

350

0

50

100

150

200

250

300

350

0

50

100

250

300

350

Y, mdeg

10

0

–10

Z, mdeg

10 0 –10 150 200 azimuth position, deg

Fig. 13.8 The TLC look-up table of the 34-m antenna

13.5 Determining Pointing Errors from the TLC Table

219

Z rotation. But, it will be shown later that the Z rotation is compensated for by the azimuth encoder and hence it is not a part of pointing error.

13.5 Determining Pointing Errors from the TLC Table The antenna elevation and cross-elevation pointing errors are determined. The elevation error Δ E L is simply determined as the alidade X rotation ΔE L = X

(13.16)

The cross-elevation error, Δ X E L , depends on the antenna elevation position, EL, and on the alidade Y and Z rotations as illustrated in Fig. 13.9 Δ X E L = Z cos(E L) − Y sin(E L)

(13.17)

However, the Z-rotation contributions are assumed zero in the TLC table because this error is measured by the azimuth encoder and therefore eliminated by the azimuth servo. The following experiment at the 34-m antenna was conducted at the Madrid Deep Space Communication Complex to verify this hypothesis. With the antenna dish positioned at EL = 30 deg a shim of 1 mm thick was placed on the azimuth track, as shown in Fig. 13.10. The antenna was then moved slowly with constant speed in azimuth over the shim. The same antenna movement was repeated when the shim was removed. The difference between azimuth encoder reading with and without the shim is plotted in Fig. 13.11. It shows the azimuth position rising sharply (section A) when the antenna is climbing the shim. But the azimuth servo compensates for the shim disturbance (section B), and the azimuth position returns z xel

Z

Z cos(EL) EL

Δ XEL

Y y Y sin(EL)

Fig. 13.9 The relationship between the cross-elevation error and the X and Y rotations of the alidade

220

13 Track-Level Compensation

Fig. 13.10 Azimuth wheel crosses 1 mm shim

to the initial position (section C). As result the antenna does not need correction in z-axis, and the Z component of the TLC table shall be zero. Based on the above experiment the following equation ΔXEL = −Y sin(E L)

(13.18)

is the resulting formula for the cross-elevation error. 10 8

AZ enc oder, mdeg

6 4 2 0 –2 –4 –6

shim profile

–8 –10 0

A 0.5

B

C 1

1.5 AZ track angle, deg

2

2.5

3

Fig. 13.11 The azimuth encoder reading when crossing the shim. Section A shows a sharp rise in encoder reading at the beginning of the shim; Section B shows an azimuth servo correction to the shim disturbance; and Section C shows the stabilized azimuth position

13.6 Antenna Pointing Improvement Using the TLC Table

221

13.6 Antenna Pointing Improvement Using the TLC Table

elev at ion pos it ion, deg

The improvement of pointing accuracy with the look-up table was evaluated using the RF pointing data. The following RF beam measurement techniques were used: boresight, monopulse, and conscan. The data were measured with the installed TLC table (“TLC table on”) and without the TLC table (“TLC table off” ). Both methods are useful in the validation of the effectiveness of the TLC table. Namely, when the table is on, the pointing errors should be significantly smaller than the errors predicted from the TLC table (or the errors obtained for the same track with the TLC table off ). When the table is off, the RF pointing errors should match the errors predicted from the TLC table. The measurements with the TLC table off were taken for the trajectory shown in Fig. 13.12. Figure 13.13a shows that the measured elevation pointing errors and the errors predicted by the look-up table coincide. The elevation error predicted from the TLC table varies by 7 mdeg, from –3 to 4 mdeg (dashed line). Figure 13.13b shows that the cross-elevation errors (predicted and measured) coincide when the antenna elevation position is below 72 deg, and that a deterministic residual is uncompensated when the antenna elevation position is above 72 deg The cross-elevation error predicted from the TLC table vary by 12 mdeg, from –5 to 7 mdeg (dashed line); they show a “deterministic” error (for AZ < 240 deg, where antenna elevation position above 72 deg). The measurements of the radio beam position with a TLC table have a standard deviation of 0.41 mdeg (or 1.2 mdeg peak-to-peak), while the radio beam data with the TLC table shows its standard deviation of 0.72 mdeg (or 2.2 mdeg peak-to-peak), for the antenna at elevation position below 72 deg. Table 13.1 summarizes the antenna tracking accuracy with the TLC table on and off. The elevation pointing error decreased 6-fold, and the cross-elevation pointing error decreased 10-fold.

80 70 60 50 40 140

160

180 200 220 azimuth position, deg

Fig. 13.12 The 34-m antenna tracking trajectory

240

260

280

222

13 Track-Level Compensation (a)

EL error, mdeg

5

0

–5 100

120

140

160

180

200

220

240

260

280

100

120

140

160 180 200 220 azimuth position, deg

240

260

280

X E L error, mdeg

(b)

5

0

–5

Fig. 13.13 The 34-m antenna pointing errors, measured (black solid line) and predicted from the TLC table (dashed line): (a) the elevation pointing error, and (b) the cross-elevation pointing error

Table 13.1 Peak-to-peak pointing errors of the 34-m antenna Without TLC table With TLC table

Elevation error

Cross-elevation error

7.5 mdeg 1.2 mdeg

20.4 mdeg 2.2 mdeg

References 1. Antebi J, Kan FW. (2003) Precision Continuous High-Strength Azimuth Track gor Large Telescopes. Proc. of SPIE, Future Giant Telescopes, 4840: 612–623. 2. Gawronski W, Baher F, Quintero O. (2000). Azimuth track-level Compensation to Reduce Blind Pointing Errors of the Deep Space Network Antennas. IEEE Antennas and Propagation Magazine, vol. 42. 3. Gawronski W, Baher F, Gama E. (2006). Track-Level Compensation Look-Up Table Improves Antenna Pointing Precision. IPN Progress Report 42–164. Available at http://ipnpr.jpl. nasa.gov/progress report/42-164/164E.pdf 4. Greve M, Bremer J, Penalver P, et al. (2005). Improvement of the IRAM 30-m Telescope from Temperature Measurements and Finite-Element Calculations. IEEE Trans. Antennas and Propagation, 53(2). 5. Pisanu T, Morisani M, Pernechele C, et al. (2004). How to Improve the High-Frequency Capabilities of SRT. Proc. 7th European VLBI Network Symposium, Toledo, Spain, 2004.

Index

A Acceleration limit, 39, 85–87, 107, 117, 129–132, 141–142, 157–165 Antenna model reduction, 45–50, 133, 161–162 rigid, 11–12 Anti-windup technique, 157, 162–163 Atacama Large Millimeter Array, 2–3 Augmentation of the model, 73, 77, 78–79 Azimuth axis tilt, 211, 215–217 Azimuth model, 41–43 B Backlash equation, 177–181 Balanced model reduction, 45–47 Balanced representation, 46, 50 Bandwidth, 11, 12, 27–28, 30, 31, 38, 61, 66, 73, 75, 84–85, 87–90, 92, 104–107, 110–115, 118–119, 121, 123, 124–127, 129, 131–133, 140, 146, 148, 183–188, 194, 206 Bias, torque, 174–181 C Carrier power, 197, 198, 202, 206 Closed-loop test, 31 Coherence, 35–37 Command preprocessor with antenna, 157–162 description, 158 dynamics, 155, 161–162 gains, 160 gain selection, 158–165 Conical scan, see Conscan Conscan estimated spacecraft position, 200, 202 power variation, 195–200, 201, 203–204, 206–208

sliding window, 200–201 trajectory, 195, 197 Coulomb friction torque, 168–170 Counter-torque, 168–169 CPP, see Command preprocessor Critical frequency, 85 Critical integral gain, 85, 87, 104, 107 Cross-elevation coordinate, 195 Cross-spectral density function, 34–35 D Damping matrix, 13, 15–16 Data processing, 31, 34–37, 111 Davenport filter, 59–62, 64, 66 Davenport spectrum, 60–62, 89 Deep Space Network, 1, 5–7, 102, 127 34 meter antenna, 95, 110 70 meter antenna, 95, 110 Discrete-time data, 37–38 Discrete-time model, 21–23 Disturbance step, 74, 120, 140, 148–149, 153 Disturbance transfer function, 74, 76, 87, 91, 115, 121, 140, 141, 147–148, 153–154, 189 Dither, 157, 167, 170–174 Drive inertia, 26, 28–30 Drive model, 11, 24–26, 68, 129, 177 Drive stiffness, 26, 27–28 Dry friction model, 168–170 Dynamic pressure of wind, 64 E Effelsberg telescope, 4–5 ESA deep space antennas, 2 F Feedforward controller, 81–92, 129 Finite element model, 12–13, 17, 59, 129, 177 Friction dry friction model, 168–170

223

224 Friction torque smooth image, 172–173 Fundamental frequency, 27, 30, 43–44, 107, 108, 110, 115, 121, 129 G Gain stability margin, 75–76 Green Bank Telescope, 4, 166, 173, 213 H H2 norm, 47–49 H∞ controller closed-loop equations, 138–139 definition, 135–138 design, 136–138 with DSN antenna, 139–141 gains, 135–138 limits of performance, 141–142 tracking, 138 H∞ norm, 47–49, 135 Hankel norm, 46, 47, 48, 49 I Identification, velocity loop model, 26, 39–44 Inclinometer data, 211, 213–218 Integral gain, 81, 83, 85–90, 100, 104, 106, 107, 110–111, 119, 121, 129, 132, 146 K Ka band, 1, 194, 197 L Large Millimeter Telescope control systems, 1–3, 127 Limit acceleration, 39, 85–87, 107, 117, 129, 131, 132, 141–142, 157–165 velocity, 73, 95, 129, 157–165 Limit cycling, 129, 157, 159, 162, 163, 164, 171 Lissajous scan, 194, 201–204, 206 Low velocity tracking, 170–173 LQG controller in the position loop closed-loop equations, 101 description, 95–98 fine tuning, 121–124 gains from Matlab, 98 limits of the gains, 107–108 limits of performance, 117 performance, 110 performance with DSN antennas, 111–113 performance vs PI performance, 115–117 properties of weights, 105–107 resemblance of PI controller, 104–105

Index tracking, 98–101 tuning steps, 108–110 tuning using GUI, 117 weight properties, 105–107 weights, 101–103 LQG controller in the velocity loop, 124–132 M Mass matrix, 13–15 Matlab, 11, 17, 22, 35, 40, 46, 98, 111, 117, 119, 123, 124 Modal displacement, 15–17, 21 Modal model, 11, 12, 14–15, 21, 45, 78 reduction, 47–48 Model reduction, 45–50 Modes, 14, 15, 17, 18, 20, 21, 23, 41, 45, 47, 48, 49, 61, 77, 102, 103, 107 Monopulse command following, 187–188 disturbance rejection, 188–189 performance, linear model, 190–191 performance, nonlinear model, 191–193 principle, 187 stability, 190 Motor model, 24 N Natural frequencies, 14, 16, 17, 42, 43, 139 Natural modes, 14, 15, 17 Norm, single mode, 47–48 Norm, structure, 48–49 Nyquist frequency, 37–38 O Open-loop test, 179 Overshoot, 73–74, 90, 111, 119–121, 129, 131–133, 153, 159–162, 164, 165, 187–188 P Performance criteria, 73–77 Phase stability margin, 75–77 PI controller closed-loop equations, 87 closed-loop transfer function, 83 limit of performance, 90 performance, 87–90 tuning steps, 86–87 Pointing accuracy, closed loop, 68–70 Poles, 19, 23, 49–50, 85

Index Position loop, 5–7, 32–34, 68, 82, 100, 113–115, 124–132, 145–147, 155, 183–185, 187, 188–190 Power spectral density function, 35 Proportional gain, 82–90, 100, 104, 106, 108, 114, 115, 129, 132 R Reducer model, 24–25 Reduction, 45–50, 133, 161–162 RF beam controller Selection, 183–186 Riccati equations, 97, 98, 136 Rigid antenna model, 11–12 Rigid body model, 11–12, 19–21, 45, 49–50, 145, 148, 217, 218 Rms servo error, 87–89, 110, 112–115, 118, 131–132 Rosette scan, 194, 199, 204–205, 206 S Sampling rate, 22, 23, 32, 37–38, 194 Sampling time, 22, 23, 37–38, 61, 67, 129, 159, 161, 170, 196 Scanning performance evaluation, 206–208 Settling time, 73–74, 88–91, 110–113, 115, 120, 121, 129, 131, 132, 138, 139, 140, 148, 153, 159, 162, 188 Single-loop control 34-meter antenna, 149–154 rigid antenna, 145–149 Stability margin, 75, 186, 190 State-space model, 11, 17–19, 34, 77, 79 Steady-state error, 73, 78, 119, 129, 138 Stiction torque, 168–169 Stiffness matrix, 13, 15, 20 Structural model, 12–23, 45, 61 Surface drag coefficient, 60, 63 T Test configuration, 32–34 duration, 32 input, 32–34 input amplitude, 32 output, 32 velocity offset, 32 Thirty Meter Telescope, 3 Tilt, 211, 213–218 Time profile of wind gusts, 65–66 TLC, see track level compensation Torque bias, 174–181

225 Tracking H∞ controller, 138 Tracking LQG controller, 101 Tracking PI controller, 91–92 Track level compensation collecting data, 213–214 creating look-up table, 217–219 description, 211–213 determining pointing errors, 219–220 estimating azimuth axis tilt, 215–217 inclinometer locations, 213–214 pointing errors, 219–220 pointing improvement, 221–222 processing data, 213–214 table, 217–219 Track profile, 212–214 Transfer function, 11, 12, 26–30, 31, 32, 34–37, 39–42, 47, 48, 50, 60–62, 66–67, 73–76, 81–88, 91–93, 102–107, 109, 111, 113, 115, 119–132, 135, 140–141, 143, 146–148, 150, 153–154, 183–189 Transfer function from data, 34–40 Transformations of the velocity loop model, 34–37 Tuning LQG controller, 108–110 PI controller, 86–87 V Velocity limit, 73, 95, 129, 158–159, 161, 163, 167 Velocity loop, 5–7, 11–12, 26–30, 31, 33–35, 38–43, 45, 51, 60, 73, 77–79, 81–83, 87, 91–92, 95–96, 104, 115, 123–132, 145–155, 177–179, 184, 190–193 Velocity loop model rigid antenna, 11–12 Velocity offset, 32, 72–73, 121 W White noise, 59–61, 64–66, 68, 187, 190 testingm, 31–34 Wind filter, 139 Wind forces, 59, 60–64, 67–68 Wind model gusts, 59–70 static, 51 Wind torque, 51, 52, 54, 59–60, 64–68, 153 Wind tunnel data, 51, 55–56, 58 Wind velocity, 59–64, 67, 111, 115, 128 X X-band, 1

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E.N. Kuznetsov, Underconstrained Structural Systems P. Ladev`eze, Nonlinear Computational Structural Mechanics: New Approaches and NonIncremental Methods of Calculation P. Ladev`eze and J.-P. Pelle, Mastering Calculations in Linear and Nonlinear Mechanics A. Lawrence, Modern Inertial Technology: Navigation, Guidance, and Control, 2nd ed. R.A. Layton, Principles of Analytical System Dynamics F.F. Ling, W.M. Lai, D.A. Lucca, Fundamentals of Surface Mechanics: With Applications, 2nd ed. C.V. Madhusudana, Thermal Contact Conductance D.P. Miannay, Fracture Mechanics D.P. Miannay, Time-Dependent Fracture Mechanics D.K. Miu, Mechatronics: Electromechanics and Contromechanics D. Post, B. Han, and P. Ifju, High Sensitivity and Moir´e: Experimental Analysis for Mechanics and Materials R. Rajamani, Vehicle Dynamics and Control F.P. Rimrott, Introductory Attitude Dynamics S.S. Sadhal, P.S. Ayyaswamy, and J.N. Chung, Transport Phenomena with Drops and Bubbles A.A. Shabana, Theory of Vibration: An Introduction, 2nd ed. A.A. Shabana, Theory of Vibration: Discrete and Continuous Systems, 2nd ed. Y. Tseytlin, Structural Synthesis in Precision Elasticity