Disturbance Rejection Control for the Green Bank Telescope

Home , Green Bank Telescope

DISTURBANCE REJECTION CONTROL FOR THE

GREEN BANK TELESCOPE

TRUPTI RANKA

Submitted in partial fulﬁllment of the requirements

for the degree of Doctor of Philosophy

Dissertation Advisor: Dr. Mario Garcia-Sanz

Department of Electrical Engineering and Computer Science

CASE WESTERN RESERVE UNIVERSITY

May, 2016 CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of Trupti Ranka

candidate for the degree of Doctor of Philosophy*.

Committee Chair Dr. Mario Garcia-Sanz

Committee Member Dr. Vira Chankong

Committee Member Dr. Sree N. Sreenath

Committee Member Dr. Christos Papachristou

Date of Defense

March 28, 2016

*We also certify that written approval has been obtained

for any proprietary material contained therein. Contents

List of Tables iv

List of Figures v

Acknowledgments xi

Abstract xiii

1 Introduction 1 1.1 The Green Bank Telescope ...... 1 1.2 Motivation ...... 2 1.3 Research contributions ...... 5 1.4 Assumptions ...... 6 1.5 Dissertation organization ...... 7

2 GBT analytical and experimental Modeling 8 2.1 Analytical Modeling ...... 9 2.1.1 GBT structure and the primary reﬂector servo system . . . . .9 2.1.2 Modeling with FEA data ...... 11 2.1.3 Relative Gain Analysis for axes coupling ...... 16 2.1.4 Model Reduction ...... 17 2.1.5 Uncertainty Analysis ...... 20

i CONTENTS

2.1.6 Simplified model of the subreflector servo system...... 25 2.2 Experimental Modeling ...... 29 2.2.1 Experimental setup ...... 29 2.2.2 Data Processing ...... 30 2.2.3 Model identification ...... 31 2.3 Modeling Results ...... 34 2.3.1 Comparison of the analytical model vs. the experimental model 34

3 Primary reﬂector servo control 39 3.1 Extended State Observer theory ...... 40 3.1.1 ESO for active disturbance rejection control ...... 40 3.1.2 Uncertainty reduction through ESO loop ...... 42 3.1.3 Quantitative feedback technique reformulation of the ESO . . 44 3.2 ESO for the GBT primary reﬂector position control ...... 49 3.2.1 QFT based ESO design ...... 50 3.2.2 Outer loop controller design ...... 55 3.2.3 Comparison with the legacy controller response ...... 61

4 Subreflector servo control 67 4.1 Extremum Seeking Control theory ...... 69 4.1.1 Effect of the initial condition ...... 70 4.1.2 ESC for a single parameter system with dynamics ...... 71 4.1.3 ESC for a multi-paramter system with dynamics ...... 72 4.1.4 ESC with disturbance feedforward signal ...... 73 4.2 ESC for the GBT subreflector ...... 77 4.2.1 Telescope gain and pointing offsets ...... 77 4.2.2 ESC loop for the subreflector ...... 78 4.2.3 The ESC design ...... 81

ii CONTENTS

4.2.4 Results ...... 83 4.2.5 Design limitations ...... 90 4.2.6 Implementation discussion ...... 90

5 Conclusion 92 5.1 Summary of the main results ...... 92

Appendix A: Subreﬂector actuator measured FRFs 94

iii List of Tables

2.1 Az and El structural modes below 2 Hz and Az motor-gearbox modes between 8 to 10 Hz (gray cells). Boldfaced modes are selected during model reduction ...... 18 2.2 Modeling error percentage for the azimuth axis ...... 35 2.3 Modeling error percentage for the elevation axis ...... 35

iv List of Figures

1.1 The Green Bank Telescope ...... 2 1.2 Encoder error due to az track joints at 460th and 690th seconds. The telescope is moving in az axis at a speed of 0.04 degrees/sec...... 4 1.3 Total tracking error measured through astronomical observations for az and el axes as a function of the mean wind speed...... 4

2.1 Schematic of the GBT structure showing the primary reflector (100 m diameter), feed arm, subreflector, azimuth track, elevation bull gear, az and el motors and encoders, position of the accelerometers 1 and 4 and the quadrant detector...... 10 2.2 Cascaded current, rate and position servo loops per axis for the primary reflector servo system...... 10 2.3 Bending of the telescope structure in the first bending mode at 0.60 Hz. Light gray: undeformed structure, Colored: mode deformation. . 11 2.4 Bending of the telescope structure in the second the bending mode at 0.88 Hz. Light gray: undeformed structure, Colored: mode deformation. 12 2.5 Bending of the telescope structure in the third bending mode at 1.02 Hz. Light gray: undeformed structure, Colored: mode deformation. . 12 2.6 Free-body diagram of the force generated by a single motor at the azimuth axis track and wheel contact...... 15

v LIST OF FIGURES

2.7 Free-body diagram of the force generated by a single motor at the elevation axis gearbox pinion and bullgear contact...... 15 2.8 FRF of (a) Az and (b) El models with the first 40 modes (blue) versus the reduced model (green) with dominant modes below 2 Hz...... 19 2.9 a: Measured closed velocity loop response of az axis at elevations 5◦, 30◦, 60◦, 95◦. b: FEA model open loop velocity bode plot of az axis at elevations of 0◦, 30◦, 60◦, 90◦ ...... 21 2.10 FRF of the azimuth axis open velocity loop at the elevations of 5, 30 , 60 and 90 degrees with upper and lower bounds of the FRFs as quantified by varying zeros of the transfer function of the system at 60 degrees...... 24 2.11 Side view of the GBT subreflector actuator mounting [15]...... 26 2.12 Conceptual representation of the 6 DOFs motion of the subreflector [15]. 27 2.13 Bode plot of the approximate model of closed position loop of a subreflector Stewart platform leg. -3 dB @ 1.33 Hz with phase = -134 degrees...... 27 2.14 Configuration of servo loop for each axis during system identification experiment. Each axis had its velocity loop closed while the position loop was kept open. The input signal for the experiment was a rate command while the output signals recorded were tachometer, encoder, and accelerometer signals...... 29 2.15 Flow chart for Eigen system realization algorithm [18] ...... 33 2.16 a: Az uncompensated open position loop FRF. b: Az open position loop response to velocity step command. Blue: Measured response. Red: Response simulated using the identified model. Green: Response simulated using the FEA based model...... 35

vi LIST OF FIGURES

2.17 a: Az closed velocity loop FRF. b: Az velocity loop step response. Blue: Measured response. Red: Response simulated using the identified model. Green: Response simulated using the FEA based model. . 36 2.18 a: Accelerometer 1 X-direction FRF to the Az axis excitation. b: Ac- celerometer 1 X-direction response to the Az velocity step command. Blue: Measured response. Red: Response simulated using the identified model. Green: Response simulated using the FEA based model. . 36 2.19 a: El uncompensated open position loop FRF. b: El open position loop response to the velocity step command. Blue: Measured response. Red: Response simulated using the identified model. Green: Response simulated using the FEA based model...... 37 2.20 a: El closed velocity loop FRF. b: El velocity loop step response. Blue: Measured response. Red: Response simulated using the identified model. Green: Response simulated using the FEA based model. . 37 2.21 a: Accelerometer 1 Y-direction FRF to the El axis excitation. b: Ac- celerometer 1 Y-direction response to the El velocity step command. Blue: Measured response. Red: Response simulated using the identified model. Green: Response simulated using the FEA based model. . 38

3.1 Magnitude response of the uncertain plant P(s) as given in eqn. 3.8 . 43 3.2 Magnitude response of the uncertain plant P(s) as given in eqn. 3.8

with ESO loop. a) ESO bandwidth ω0 = 1 rad/sec. b) ESO bandwidth

ω0 = 100 rad/sec ...... 43 3.3 ESO transfer function block diagram ...... 45 3.4 ESO transfer function block diagram with preﬁlter F (s)...... 47 3.5 QFT loop shaping of ESO for az axis on Nichols chart. Sensitivity bound violation between 3-6 rads/sec ...... 51

vii LIST OF FIGURES

3.6 Az axis a) QFT loop analysis for uncertainty reduction. Sensitivity bound violation between 3-6 rads/sec b)QFT loop analysis for model

matching after applying feedforward ﬁlter Fo(s)...... 52 3.7 QFT loop shaping of ESO for el axis on Nichols chart. Sensitivity bound violation between 4-6 rads/sec...... 53 3.8 El axis a) QFT loop analysis for uncertainty reduction. Sensitivity bound violation between 4-6 rads/sec. b)QFT loop analysis for model

matching after applying feedforward ﬁlter Fo(s)...... 54

3.9 a) Az axis Co(s) controller loop shaping. b)Az axis step response with overshoot of 10% and settling time of 12 secs...... 56

3.10 Az axis Co(s) controller a) Frequency domain worst-case stability bound analysis b) Preﬁlter F (s) tracking bound analysis ...... 57

3.11 a) El axis Co(s) controller loop shaping. b)El axis step response with overshoot of 15% and settling time of 17 secs...... 59

3.12 El axis Co(s) controller a) Frequency domain worst-case stability bound analysis b) Frequency tracking bound analysis...... 60 3.13 Az axis position loop. a) Step response. b) track joint response. c)x-el acceleration at feed arm tip due to step. d) x-el acceleration at feed arm tip due to track joint. e) Controller output for step. f) Controller output for track joint disturbance...... 63 3.14 Az axis position loop. a) Wind response at encoder end. b)x-el acceleration at feed arm tip due to wind disturbance as input torque disturbance. c) controller output for wind disturbance rejection. . . . 64

viii LIST OF FIGURES

3.15 El axis position loop. a) Step response. b) Wind disturbance response at encoder end. c) el acceleration at feed arm tip due to step. d) el acceleration at feed arm tip due to wind disturbance as input torque disturbance. e) Controller output for step. f) Controller output for wind disturbance...... 65

4.1 ESC loop for a nonlinear function f(θ(t)) which has a extremum at θ∗. Edited ﬁgure from [39] ...... 69 4.2 ESC loop for a nonlinear function f(θ(t)) which has a extremum at θ∗.

The loop is set at an initial value of θ(0) = θb ...... 70 4.3 ESC loop for a nonlinear function f(θ(t)). The extremum point is dynamic θ∗(t) and the loop has SISO linear plant dynamics. Edited ﬁgure from [39] ...... 71 4.4 ESC loop for a multi parameter, nonlinear function f(θ(t)). (θ(t)) is a vector with a dynamic extremum at θ∗(t), and MISO linear plant

dynamics. Edited ﬁgure from [39]. p = 1, ··· , l. For odd p, ωp+1 =

ωp, βp = 0 and βp+1 = π/2...... 72 4.5 ESC loop for a nonlinear function f(θ(t)). The extremum point is dynamic θ∗(t) and the loop has SISO linear plant dynamics. The loop

also has disturbance feedforward signal θf (t)...... 74 4.6 a) Ideal radio beam of a single dish telescope. I(ν, θ, φ) is the source intensity as a function of observing frequency ν and the pointing angles in 2-D θ, φ. A(ν, θ, φ) is the antenna gain b) Normalized Gain of the telescope as a function of pointing angle in one dimension. [40] . . . . 78 4.7 ESC loop for the subreﬂector...... 80 4.8 Power spectrum of a sample wind gust signal at the Green Bank site. The mean wind velocity is subtracted to give the power spectrum of only the wind velocity variation about the mean value...... 83

ix LIST OF FIGURES

4.9 Subreflector servo system with only QD disturbance feedforward correction. The QD calibration model has 20% gain error and 2.5 arcseconds offset error for both (θ(t), φ(t)) a) Θ∗(t) − Θ(t) on f(θ(t), φ(t)) contour map. b) Normalized RF signal received f(θ(t), φ(t)). . . . . 85 4.10 Subreflector servo system with only ESC feedback. a) Θ∗(t) − Θ(t) on f(θ(t), φ(t)) contour map. b) Normalized RF signal received f(θ(t), φ(t)). 86 4.11 Subreflector servo system with ESC feedback and QD disturbance feedforward correction. a) Θ∗(t) − Θ(t) on f(θ(t), φ(t)) contour map. b) Normalized RF signal received f(θ(t), φ(t))...... 87 4.12 θ(t), φ(t) command signal to the subreflector servo with both QD feedforward and ESC in effect...... 88 4.13 θ(t), φ(t) command signal to the subreflector servo zoom in. The perturbation amplitude at 10rads/sec is 1 arcsecond ...... 88

x Acknowledgments

I am grateful to Prof. Mario Garcia-Sanz for being my thesis adviser and allowing me to work on this project. He has always encouraged me to look into the theoretical as well as the application aspects of control system design. This has kept my research focus well balanced. He introduced me to the method of Quantitative Feedback Technique (QFT). QFT has strengthened my understanding of the classical frequency domain control design which often gets neglected at the graduate level programs in control engineering. He has also supported me ﬁnancially throughout my graduate school without which this work would not have been possible. I would like to thank all my committee members for their suggestions which led to the improvement of my ﬁnal dissertation document. The ground work for this thesis began when I was working at Giant Meterwave Radio Telescope (GMRT) in India between 2008-2010. I would like to thank Prof. Bhal Chandra Joshi at National Center for Radio Astronomy, India for giving me an opportunity to work at GMRT and for introducing me to the area of advanced control system design for large telescopes. I am forever indebted to John Ford at the Green Bank Telescope (GBT), for giving me an opportunity to work at GBT and making this thesis possible. I have been extremely fortunate to have worked with John. My experience at the GBT has certainly made me a better engineer. I would like to thank Arthur Symmes at National Radio Astronomy Observatory who provided me with the Finite Element Analysis (FEA) model of the telescope. In spite

xi ACKNOWLEDGMENTS of his very busy schedule, he patiently simulated the free-free FEA model for me and answered my numerous questions. I would like to acknowledge Tim Weadon’s help for my experiments at GBT and Joe Brandt’s help with the information regarding GBT position loop and subreflector servo system. I am also thankful to all the engineering staff and technicians at GBT who have helped me with the necessary engineering data and with my experiments. I would like to acknowledge Richard Prestage’s brief discussions regarding the GBT subreflector and GBT radio signals; as well as Prof. Kristic’s planeray talk on extremum seeking control at DSCC 2015 conference. Together these events resulted in the work presented in chapter 4 about the subreflector control. I am thankful to Wang Fa and Tony Joy for making my time in the lab at CRWU pleasant. I would also like to thank all my friends at International Students’ Fel- lowship, who made my stay in Cleveland fun and memorable. I am thankful to Sue Shears who made it possible for me to have a great time at GBT outside of work. I am deeply grateful to my family; my mother and my sisters, who made it financially possible for me to come and study here in the USA. They also egged me on to con- tinue my work even during the most taxing times of the graduate school, and helped me reach the finish line. The funding for my time at CWRU was sponsored through The Milton and Tamar Maltz Family Foundation and Cleveland Foundation. The Green Bank Telescope Observatory hosted and funded my stay at the GBT. The Green Bank Telescope is an observatory of the National Radio Astronomy Observatory (NRAO). NRAO is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc.

xii Disturbance Rejection Control for The Green Bank Telescope

Abstract by TRUPTI RANKA

The GBT is a single dish, receiving radio telescope. It is capable of receiving radio waves in the frequency range of 300 MHz to 115 GHz. The off-axis primary reflector of the telescope is 100 meters in diameter. A truss boom (feedarm) extends about 60 meters perpendicular to the primary reflector and is supported at the edge of the reflector. A subreflector is placed at the tip of the feedarm, which directs the focused radio waves from the primary reflector to the radio receivers placed on the feedarm. At high radio frequencies of observation, the uncorrected pointing and tracking errors become limiting factors for making useful scientific observations. The primary reflector and subreflector servo systems need to reduce the pointing and tracking errors due to torque disturbances acting on the system. The overall aim of this research is to redesign the servo control systems such that they are able to give a superior disturbance rejection performance. The 4 contributions of this research are: 1) Verifying the dynamical model of the structure using system identification experiments. 2) The unique reformulation of the extended state observer (ESO) design as a quantitative feedback design problem in frequency domain and splitting the design of the ESO as a feedback observer and a feedforward filter. This formulation gives a more systematic way of designing an ESO as compared to the current technique used for the ESO design. This method is then used to design the ESO based controller for the primary reflector position loop. The ESO based controller provides more than 50% improvement in disturbance rejection in the primary reflector servo loop, as compared to the legacy PID controller. 3) The innovative use of extremum seeking controller (ESC) with a disturbance feedforward signal. We investigate the use of disturbance feedforward with ESC and show that

xiii Abstract disturbance feedforward improves the speed of the ESC loop by improving the initial condition of the ESC loop and by reducing the magnitude of the error dynamics. 4) The formulation of the subreﬂector control as an extremum seeking problem, and using the ESC with disturbance feedforward for the subreﬂector control. This method gives more than 40% improvement in the tracking of a point source against feedarm swaying in wind gusts.

xiv Chapter 1

Introduction

1.1 The Green Bank Telescope

The Green Bank Telescope (GBT) Figure 1.1, situated in Green Bank, West Virginia, is a fully steerable, single dish, radio telescope. The optical design of the telescope is off-axis Gregorian style, with an altitude-azimuth mount to steer the telescope. Its primary reflector is an off axis paraboloid and is 100 meters in diameter. The dish surface is made of 2004 actuated panels. The panels can be actively oriented such that together they form a paraboloid surface. The radio receivers are placed on a truss boom, called the feed arm (FA). The FA is supported perpendicular to the dish near its edge. The radio waves collected by the primary reflector are directed towards the receivers with the help of an 8 meter elliptical reflector called the subreflector which is placed at the tip of the FA. The vertical axis of the primary reflector, controlled by the servo system, is called the azimuth (az) axis. The horizontal axis of the primary reflector, controlled by the servo system, is called the elevation (el) axis. The axis perpendicular to both the azimuth and the elevation axes is called the cross-elevation (x-el) axis. The primary reflector can not rotate about the x-el axis and it is not servo controlled.

1 CHAPTER 1. INTRODUCTION

Figure 1.1: The Green Bank Telescope

The subreﬂector is mounted on Stewart platform (Hexapod) and can be actuated in 6 degrees of freedom (DOFs).

1.2 Motivation

The telescope is capable of receiving radio signals in the frequency range of 300 MHz to 115 GHz. The diffraction limited resolution of the telescope is inversely proportional to its observation frequency. For making useful astronomical observations it is important that the astronomical source lies at least within the half width of this resolution angle. At high frequencies above 50 GHz, the diffraction limited radio beam of the telescope becomes less than 15 arcseconds. Hence pointing and tracking accuracy of the telescope become the limiting factors for making useful scientific observations [1]. Systematic errors in tracking are caused mainly due to gravitational and thermal deformations of the structure. The bending of the FA and the dish under gravity and

2 CHAPTER 1. INTRODUCTION thermal gradients is carefully calibrated using astronomical data. The data is then used to derive the pointing models. The subreflector is moved to the dish focus using these pointing models. Offsets in the primary reflector servo loop command are also set using these models. The parabolic shape of the dish is corrected by moving the dish panels using a technique known as out of focus holography [1]. Through these careful corrections of the systematic errors, a pointing error of 5 arcseconds is achievable for the wind speeds of less than 3m/s. This is half the width of the useable radio beam width at 90 GHz. Hence, at 90 GHz and above it becomes absolutely necessary to compensated for wind induced pointing and tracking errors, through the primary reflector and subreflector servo systems. Currently, high frequency use of the telescope is limited to only very calm wind conditions. By correcting for wind induced errors, the observation time available for the telescope can be doubled. It will also improve the dynamic range (signal to noise ratio) of the telescope at low frequencies of observations. Other potential sources of error are the torque disturbances caused due to the joints in azimuth tracks and motor cogging. Figure 1.2. shows the effect of the az track joints on the encoder error when the azimuth axis is moving at the rate of 150 arcseconds/sec. Figure 1.3. shows the total error in the az and el tracking performances, as measured using astronomical sources, as a function of the mean wind speed 1.

1The data for this graph was collected by Paul Ries between 2008-11

3 CHAPTER 1. INTRODUCTION

Figure 1.2: Encoder error due to az track joints at 460th and 690th seconds. The telescope is moving in az axis at a speed of 0.04 degrees/sec.

Figure 1.3: Total tracking error measured through astronomical observations for az and el axes as a function of the mean wind speed.

4 CHAPTER 1. INTRODUCTION

1.3 Research contributions

The telescope primary reflector, the mount structure (also called the alidade), and the feed arm together form a large flexible structure, which requires a high precision servo control. Improvement in the telescope performance against the torque disturbances can be achieved by carefully redesigning the servo control system of the primary reflector and the subreflector. Hence, all the work in the research is driven by the goal of proving disturbance rejection solutions for the GBT servo systems. The first unique contribution of the research involves reformulating the Extended State Observer(ESO) as a Quantitative Feedback Technique (QFT) design problem in frequency domain, and splitting the design of the ESO as a feedback observer and a feedforward filter. This makes the design of ESO more systematic as compared to the current design method, which involves just specifying the bandwidth of the observer. The QFT formulation allows the designer to systematically specify and evaluate the effect of ESO at each frequency and its interaction with plant dynamics. We then design the ESO based controller for the primary reflector servo system and show that it provides about 50% improvement in disturbance rejection in the primary reflector position servo loop compared to the PID controller in use. This improvement is seen as reduction in position error as seen by the axis encoder, and reduction of vibration at the feedarm tip. The second unique contribution of this research is in the area of extremum seeking control (ESC). It is known that ESC can be very slow in its response against external disturbances, hence making it less effective in compensating for fast changing extremum. We investigate the use of extremum seeking control in combination with disturbance feedforward. We show that disturbance feedforward improves the speed of the ESC loop by improving the initial condition of the ESC loop and by reducing the magnitude of the error dynamics. This is done by analyzing the periodic, linear, time varying system, associated with the ESC loop error dynamics. As the third

5 CHAPTER 1. INTRODUCTION unique contribution of this research, we formulate the subreflector control problem as an extremum seeking control problem when tracking a point source in the sky. We apply ESC with disturbance feedforward and show that this method gives more than 40 % improvement in the tracking of a point source against feed arm swaying in wind gusts, as compared to current method in use. The current method in use allows only disturbance feedforward. The fourth important contribution of the research is the verification of the dynamical model of the structure and primary subreflector servo system with system identification experiments performed on the telescope. The uncertainty in the model, due to changing orientation of the telescope is modeled using interval analysis method. The work on the analytical and experimental modeling of the GBT was published in the papers [2], [3], [4]. The work on the design of the ESO based controller for the GBT primary reflector was published in [5].

1.4 Assumptions

Throughout this dissertation we use small signal approximation of the models. That is, the servo command signals are always in the range of few 100s of arc seconds and the models used are good approximations for this command range. This is a justifiable assumption, since for the actual operation of the telescope servo command amplitudes of only a few 100s of arcsecond are used during astronomical source tracking. Even the huge wind torques acting on the structure result in position and velocity error of only a few 100s of arcseconds. Complete effect of wind on a structure like GBT can be simulated only through computational fluid dynamics (CFD) model. Since CFD simulation is beyond the scope of this work, we simulate only the effect of wind torque disturbance as seen by the servo system. This is a justifiable assumption, as the control system can correct

6 CHAPTER 1. INTRODUCTION only for disturbances observed by the sensor system in the servo loop.

1.5 Dissertation organization

We begin by presenting the analytical model and the experimental model of the system in chapter 2. Chapter 3 discusses the QFT formulation of the ESO and the use of the ESO for primary reﬂector position loop control. Chapter 4 presents the analysis of ESC with disturbance feedforward signal. It then discusses the design of extremum seeking control for the subreﬂector servo system. Chapter 5 summaries the main results of the thesis.

7 Chapter 2

GBT analytical and experimental Modeling

In this chapter, we first develop the model of the system to be controlled about the azimuth and elevation axes of the primary subreflector. We provide the detailed analytical model using the Finite Element Analysis (FEA) data. As the FEA models can have a very large model order, we perform model reduction to reduce the model order. We verify these models using experimental data. We also discuss the uncertainty in the model, as it will have a direct impact on the robustness of the control system stability and performance. We also discuss the modeling of the subreflector servo system. The subreflector servo is a 6 × 6 MIMO system. By using some simplifying assumptions, we convert the model into 6 SISO loops. Of these 6 SISO loops we will use only 2 SISO loops for subreflector control against the feedarm swaying in wind gusts.

8 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

2.1 Analytical Modeling

2.1.1 GBT structure and the primary reﬂector servo system

Figure 2.1 shows the schematic of the GBT structure at an elevation angle of 66◦. The telescope is driven in the azimuth axis by 16 motors, each of 30 hp. Four motors are coupled together in one truck. Four such trucks move along the track of the azimuth axis. In the elevation axis, there are eight motors, each of 40 hp. The motors in both azimuth and elevation axes are preloaded to remove the gear train backlash. The telescope has an off-axis geometric focus. The receivers are placed at the focus with the support of a feed arm as shown in Figure 2.1. There are five accelerometers mounted on the structure. Accelerometer 1 is placed on the top of the receiver cabin near the feed arm tip, and accelerometer 4 is near the feed arm base. Accelerometers 2 and 3 are on the elevation axle, and accelerometer 5 is on the alidade structure. Each accelerometer has three outputs; in their local X, Y, and Z directions. The telescope is equipped with a position sensitive detector called the quadrant detector (QD) which senses the relative motion between the primary reflector and the feed arm. The illuminator of the QD is at the tip of the feed arm, and its detector is on the elevation axle. For each axis, the servo system consists of a cascaded control system with three loops: current, velocity, and position loops, as shown in Figure 2.2. The current loop closes at each individual motor in the motor servo drives. Each motor also has its own tachometer. The velocity loop controls the axis velocity using the average tachometer feedback. The position loop controls the axis position using 24 bit absolute encoders as the feedback element.

9 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.1: Schematic of the GBT structure showing the primary reﬂector (100 m diameter), feed arm, subreﬂector, azimuth track, elevation bull gear, az and el motors and encoders, position of the accelerometers 1 and 4 and the quadrant detector.

Figure 2.2: Cascaded current, rate and position servo loops per axis for the primary reﬂector servo system.

10 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.3: Bending of the telescope structure in the ﬁrst bending mode at 0.60 Hz. Light gray: undeformed structure, Colored: mode deformation.

2.1.2 Modeling with FEA data

The analytical model of the telescope was derived using FEA data. Figures 2.3, 2.4, 2.5 illustrate the first three flexible modes of vibration of the structure at an elevation of 90 deg. The first and second flexible modes of vibration have calculated frequencies of 0.60 and 0.88 Hz, respectively, and represent the maximum deformation at the tip of feedarm with bending in the cross elevation direction. Deformation of the feed arm in this cross elevation direction is predominately affected by telescope motion about the azimuth axis. The third vibration mode at 1.02 Hz is characterized by bending of the feedarm about the elevation axis. The first substantial deformation of the primary reflector structure occurs at the fifth vibration mode with a frequency of 1.45 Hz. To derive the dynamic response of the structure to the servo system, we augment the FEA model with motor, gearbox, servo drives, and servo loop models. Since the motor and servo drive dynamics (containing motor current loop) are much faster than the structural dynamics, we approximate them as a constant. The equations for the velocity loop controller are derived from the GBT internal servo documents. We have

11 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.4: Bending of the telescope structure in the second the bending mode at 0.88 Hz. Light gray: undeformed structure, Colored: mode deformation.

Figure 2.5: Bending of the telescope structure in the third bending mode at 1.02 Hz. Light gray: undeformed structure, Colored: mode deformation.

12 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING assumed that the torque bias in the velocity loop cancels the eﬀect of backlash. The FEA model with free-free analysis for Az and El axes, is simulated in NAS- TRAN software, and is imported into MATLAB using the third-party toolbox SD- Tools [6]. The FEA model is simulated for elevation angles of 0◦, 30◦, 60◦, and 90◦. Here, we provide the detailed results of the model with elevation angle of 60 deg. We use the models at the four elevation angles for uncertainty analysis. Both axes are always free while deriving the model. The FEA model has 489396 nodes. Each node can have up to 6 DOFs. The state-space model in nodal coordinates can be written as,

x˙ n = Axn + BFn (2.1)

y = Cxn + DFn

and where xn is nodal degrees of freedom, y is the vector of outputs which for each axis and includes the axis encoder at the load end, average axis speed at motor end, accelerometer 1 output in X, Y and Z directions and Fn is the force applied to the structure by motor-gearbox. The motor torque is applied to the structure model where the Az wheel node (nc) touches the azimuth track as shown in Fig. 2.6. Similarly, the motor torque in the elevation axis is applied where the gearbox pinion touches the bullgear nc. For the El axis, the reaction of the force to the alidade structure is transmitted through node no[7, 8] as shown in Fig. 2.7. For the Az axis, we assume that the four motors on each truck are tightly coupled hence it is replaced by an equivalent motor with four times the force applied at node no. Similarly, for the El axis, we consider the two opposing motors to be tightly coupled and replace it with force equivalent to two motors. We model the output of the encoder and the accelerometer by dynamic motion of the nodes closest to these sensors, which is represented by matrix C. Matrix D represents the direct feedforward of force for the accelerometer measurement. The model is ﬁrst derived in nodal coordinates and

13 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING then is transformed into modal coordinates using the mode shape data matrix Φ. The FEA model modal analysis is done up to 5 Hz, which accounts for the ﬁrst 40 modes of the structure. Through the transform xn = Φxq we get the model in modal coordinates as follows,

n×q Φ ∈ R , Rank(Φ) = q

T T Φ xn = Φ Φxq (2.2) ΦT Φ is q × q, hence invertible

T −1 T † xq = (Φ Φ) Φ xn = Φ xn

Using equation 2.1, † † x˙ q = Φ AΦxq + Φ BFn (2.3)

y = CΦxq + DFn

Where xq is the vector of the modal degrees of freedom. The model in Eq. (2.3) is then augmented with the motor and gearbox model which is used for model reduction. For all experiments on the telescope the velocity loop is always closed. Hence, to compare the model with measured data, we further augment the model with the approximate velocity controller for each axis. The FEA analysis does not provide damping for each mode. For each mode a damping ratio of 0.005 is used to match the FRF of the FEA model with the measured FRF. From the control design perceptive, a lower damping assumption might lead to conservative controller design but it should also lead to controllers with better stability if the actual system has more damping. Since the ﬁrst few modes are very close to each other, there is a possibility of complex damping which is modeled with a non-diagonal damping matrix. But for simplicity of analysis we consider only the diagonal damping matrix.

14 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.6: Free-body diagram of the force generated by a single motor at the azimuth axis track and wheel contact.

Figure 2.7: Free-body diagram of the force generated by a single motor at the elevation axis gearbox pinion and bullgear contact.

15 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

2.1.3 Relative Gain Analysis for axes coupling

Through FEA data, one can derive a single state-space 2×2 MIMO model for both Az and El axes. If the axis are only weakly coupled, then we can develop a SISO control loop for each axis without compromising on the control performance. Decoupled axes also allows us to have a separate model for each axis and allows us to choose fewer modes (smaller model order) per axis during model reduction. We can investigate the coupling between the two axes using a method called as Relative Gain Analysis (RGA). In transfer function form, the model of the telescope from the torque inputs to the structure to the axes angular position can be given as,

Y = P(s)2×2U (2.4)

T Where Y = [θaz θel] is the angular position of each axis at the axis encoder end

T and U = [Taz Tel] is the combined motor torque input to each axis. The 2 × 2 transfer function matrix P(s) is obtained by converting the state space model into transfer function model. Let,

λ = P(s)|s=0 (2.5)

Then the RGA which measures the interaction between the inputs and the outputs is given by [9],[10], −1 R = λT ⊗ λ (2.6)

From the input torque to output position, the gain of the system at s = 0 is inﬁnity as the system has 2 poles at zero. Hence instead of gain at s = 0, we calculate the gain at s = 1. This eﬀectively gives us the gain between input torque and the torque

16 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

at the output. Hence, for the two axes of GBT, the RGA using equation 2.6 is,

  1.0001e + 000 −115.8516e − 006   R =   (2.7) −115.8516e − 006 1.0001e + 000

We see that only the diagonal terms of R are dominant. Hence there is little interaction between the az and el axes, and we can disregard the cross coupling between the axes. Now we are justiﬁed to use separate state space model for the two axes and will consider SISO loop designs for each axis.

2.1.4 Model Reduction

When the structural model augmented with motors and gearbox is represented in modal coordinates, the state-space matrices are of the form:

    Am1 0 0 Bm1      .   .  A =  0 .. 0  ,B =  .  ,C = C ··· C (2.8)     m1 mn     0 0 Amn Bmn

th Where Ami,Bmi,Cmi are matrices corresponding to the i mode. Using modal

th damping ζi and natural frequency ωi of the i mode. The Hankel Singular Value (HSV) of each mode can be approximately calculated as [11],

||Bmi||2||Cmi||2 γi = (2.9) 4ζiωi

For each axis we select the structural modes up to 2 Hz whose HSV is above a threshold of 0.1, for the model augmented with motor and gearbox. The rigid body mode has ωi = 0 giving γ0 = ∞. Table 2.1 gives the HSV for Az and El axis structural modes up to 2 Hz. We select the modes whose values are boldfaced in the table. For

17 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Table 2.1: Az and El structural modes below 2 Hz and Az motor-gearbox modes between 8 to 10 Hz (gray cells). Boldfaced modes are selected during model reduction Az (Hz) Az HSV El (Hz) El HSV 10.2724 24.1868 0.9260 6.9306 9.3585 0.0038 0.8148 0.0917 8.9891 0.5069 0.5991 0.0189 8.3649 0.0903 1.1427 0.6305 0.8109 0.4378 0.0019 Inf 0.5983 0.1543 1.6487 0.0260 1.1669 0.0031 2.0698 0.5168 1.6433 0.3072 1.9003 0.7776 0.0010 Inf 2.0266 1.9508 1.0386 0.0008 1.9356 0.0731 1.7921 2.3414 0.0000 Inf 2.0639 0.1825 1.9709 0.0078 1.9765 0.0057 1.9709 0.0003 0.0000 Inf 0.0000 Inf 0.0000 Inf - - the Az axis the motor gearbox resonance modes are between 8 to 10 Hz. Table 2.1 also lists the four motor-gearbox modes corresponding to the Az motor-gearbox. We select the most dominant motor-gearbox mode at 10.27 Hz. Fig. 2.8 compares the open velocity loop FRF of the structure connected to motors, with all 40 modes up 5 Hz and the reduced model with dominant structural modes below 2 Hz. After model reduction, we employ only six flexible modes in the Az axis and five flexible modes in the El axis.

18 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.8: FRF of (a) Az and (b) El models with the ﬁrst 40 modes (blue) versus the reduced model (green) with dominant modes below 2 Hz.

19 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

2.1.5 Uncertainty Analysis

Figure 2.9a. shows the measured closed velocity loop bode plot of az axis at 5◦, 30◦, 60◦, 95◦ of elevation. Figure 2.9b shows the FEA model open loop velocity bode plot of az axis at 0◦, 30◦, 60◦, 90◦. It is evident that we see the same elevation angle dependent change in analytical and measured system FRF. We can model this variation as a parametric uncertainty using interval models. Below we present the algorithm for ﬁnding the interval model set from a given set of transfer functions and apply it to the az axis open velocity loop. This will help us to quantify the uncertainty in the system dynamics. The method to ﬁnd an interval model from a given set of FRFs is described in [12], [13], [14]. Consider the transfer function of an uncertain plant,

n sm + n sm−1 + ··· + n s + n m m−1 1 0 (2.10) G(s) = m m−1 s + dm−1s + ··· + d1s + d0

Let the nominal plant be given by,

0 m 0 m−1 0 0 n0(s) nms + nm−1s + ··· + n1s + n0 g0(s) = = m 0 m−1 0 0 d0(s) s + dm−1s + ··· + d1s + d0 0 m (2.11) n0 = n0 ··· n0 0 m d0 = d0 ··· d0

We choose the structure of the family of uncertain plants as follows,

( Pm+1 ) n0(s) + i=1 αiri(s) − + − + G(s) = g(s) = Pm : αi ∈ [αi , αi ]; βj ∈ [βj , βj ] (2.12) d0(s) + j=1 βjqj(s)

The aim is to ﬁnd the interval for αi, βj and polynomials ri(s), qj(s). Let the

20 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.9: a: Measured closed velocity loop response of az axis at elevations 5◦, 30◦, 60◦, 95◦. b: FEA model open loop velocity bode plot of az axis at elevations of 0◦, 30◦, 60◦, 90◦

21 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING transfer function of ith plant be given as,

i m i m−1 i i nms + nm−1s + ··· + n1s + n0 gi(s) = m i m−1 i i s + dm−1s + ··· + d1s + d0 i i (2.13) ni = nm ··· n0 i i di = dm−1 ··· d0

Then the parameter perturbation for the ith transfer function is given by,

∆ni = ni − n0 (2.14)

∆di = di − d0

We consider the method of interval analysis for the denominator coefficients. Same method is repeated for the numerator coefficients. Let there be p number of plant transfer functions in the set. Define,

∆D = ∆d1 ··· ∆dp (2.15)

The SVD factors ∆D as,

∆D = USV T (2.16)

T Let ∆B = ∆β1 ··· ∆βp = U ∆D. Then,

+ βj = max1≤i≤p(∆βi(j)), j = 1, ··· , m (2.17) − βj = min1≤i≤p(∆βi(j)), j = 1, ··· , m

th Where βi(j) is the j element of the vector ∆βi. The ﬁxed polynomials qj(s) are composed of the basis vector of U,

m X m−j qi(s) = ui(j)s (2.18) j=1

22 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

th th Where ui(j) is the j element of the i vector. In our case p < m, hence we choose only first p columns of U. That is i = 1, ··· , p. Change in the elevation angle of the telescope changes the mode shapes and mode frequencies of the structure. It also changes the location of the actuators and sensors w.r.t the structure. Both these effects result in uncertainty in the system model. We find that for GBT the effect of the change in location of the actuators and sensors w.r.t the structure causes a much bigger change in the model than the change in mode shapes and frequencies. Change in actuator/sensor location is quantified by uncertainty in B and C matrix of the state space model or the zeros of the transfer function model. Hence, we use the above algorithm to only quantify the uncertainty in numerator parameters (zeros of the transfer function) and keep the denominator

+ − ﬁxed to d0(s) by setting βj = βj = 1 and qj(s) = d0(s). This helps in reducing the number of uncertain parameters by half. Figure 2.10 shows the resulting magnitude and phase bounds on the az open loop velocity bode plot. We see that by changing the numerator parameters alone, most of the uncertainty in both the magnitude and phase response can be quantiﬁed.

23 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.10: FRF of the azimuth axis open velocity loop at the elevations of 5, 30 , 60 and 90 degrees with upper and lower bounds of the FRFs as quantiﬁed by varying zeros of the transfer function of the system at 60 degrees.

24 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

The model of the structure with the servo drives is then augmented with the velocity loop controller derived from the GBT servo manuals. These models are then verified using system identification experiments on the telescope. The verified, open, uncompensated, position loop models for az and el axes are then used for controller design. The same method for interval analysis described in the last section for open, uncompensated, velocity loop can be applied for the open position loop, to find the set of models. This model set is then used for robust position loop controller design, as detailed in chapter 3.

2.1.6 Simpliﬁed model of the subreﬂector servo system.

We intend to close an outer extremum seeking loop about the existing GBT subreflector servo as explained in chapter 4, so that the subreflector can direct the focused radio waves on the radio receivers even in windy conditions. The GBT subreflector is mounted on a Stewart platform which is capable of moving in 6 DOFs. Report [15] gives brief explanation of the subreflector actuator motion. Figure 2.11 shows the side view of the subreflector actuators and Fig. 2.12 shows the conceptual representation of the 6 DOFs. The x axis has 2 actuator, the y axis has 3 actuators, and the z axis has 1 actuator. For feed arm swaying compensation, we intend to move the subreflector only in the direction of primary reflector el and x-el axes. The RF beam can be moved about the x-el axis by translation of the subreflector along z axis or by rotation about x-axis. The RF beam can be moved about the el axis by translating the subreflector along x axis or by rotating the subreflector about the z axis. We consider a simplified subreflector model derived from commissioning data [16]. Appendix A shows the closed position loop bode plot for each of the 6 legs of the subreflector Stewart platform. It is seen that each Stewart platform leg has almost the same closed position loop transfer function. This is true except for X1 where there seems to be an additional -20 degrees phase shift. From the bode plots (neglecting

25 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.11: Side view of the GBT subreﬂector actuator mounting [15].

the -20 degrees phase shift), a common transfer function can be approximated as,

33600(s + 2) G (s) = (2.19) act (s + 4)(s + 7)(s + 10)(s + 15)(s + 16)

Figure 2.13 shows the bode plot for closed position loop model of a single subre- ﬂector actuator, which can be compared with the measured FRFs of the subreﬂector in appendix A. Since we intend to move the Stewart platform dynamically by only a few 100s of acrseconds in el and x-el directions, we assume that the inverse kinematics of the platform can be given by some linear mapping,

Xl = RkXs (2.20)

Where Xs vector represents the motion in 6 DOFs of the Stewart platform and

Xl is the vector of leg motions. Rk is a 6 × 6 matrix transforming subreﬂector DOFs

Xs to actuator length DOFs Xl, in a small area around the origin. The solution for

26 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.12: Conceptual representation of the 6 DOFs motion of the subreﬂector [15].

Figure 2.13: Bode plot of the approximate model of closed position loop of a subre- ﬂector Stewart platform leg. -3 dB @ 1.33 Hz with phase = -134 degrees.

27 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

the forward kinematics for a Stewart platform is not analytically solvable. For GBT,

an iterative Newton-Raphson method is used to transform from Xl to Xs. But for small motion we can approximate the forward kinematics by a linear map,

Xs = FkXl (2.21)

where Fk is a 6×6 forward kinematic matrix transforming actuator length Xl into subreﬂector DOFs Xs in a small area around the origin. The complete dynamical model of the 6 DOF Stewart platform servo system with Xsc as the command vector and Xso as the output vector in terms of Stewart platform DOFs, for a small motion can be given as,

Xso = FkG(s)RkXsc (2.22)

Where G(s) is a 6×6 transfer function matrix. As a result of the above discussions, these 2 assumptions are justiﬁable.

1. Each actuator leg has same dynamic response. That is G(s)(i, j) = Gact(s).

2. As the motion is very small FkRk = I.

Due to above 2 assumptions, we can see that the 6 input 6 output servo loop is mostly decoupled in individual SISO loops such that,

Xso(i) = Gact(s)Xsc(i) (2.23)

th Where Xsc(i) is the command to i subreﬂector DOF and Xso(i) is the output motion in the ith DOF.

28 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.14: Conﬁguration of servo loop for each axis during system identiﬁcation experiment. Each axis had its velocity loop closed while the position loop was kept open. The input signal for the experiment was a rate command while the output signals recorded were tachometer, encoder, and accelerometer signals.

2.2 Experimental Modeling

We now discuss the experimental modeling and model validation of the structure augmented with the primary reﬂector servo system.

2.2.1 Experimental setup

For each axis, the telescope is in a closed velocity (rate) loop while the position loop is kept open. Figure 2.14 shows the configuration of servo system used in the identification experiment. Two types of rate commands signals are given. The main identification signal is the swept sine signal from 31.125 mHz to 6.25 Hz for azimuth, and 31.125 mHz to 12.5 Hz for elevation. The peak to peak amplitude of the signal is 0.3 volts, with a DC offset of 0.5 volts. The DC offset ensures that the effect of stiction is minimized [17]. The input signal is generated using a HP Dynamic signal analyzer. A square wave signal of 0.4 volts peak to peak, with a DC offset of 0.4 volts, is also given as the rate command to measure the step response of the velocity loop. The step response data is used for verification of the identified model. The data is recorded using the GBT in-house data acquisition hardware and software. The data is collected at 200 Hz. Since the input excitation signal is only up to 6.25 Hz and 12.5 Hz we cannot measure the FRF above these frequencies. Hence we

29 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

first anti-alias the signal using low pass filter with cutoff frequency at 6.25 Hz and 12.5 Hz for azimuth and elevation axis receptively. The anti-aliased signal was then sampled at 12.5 Hz and 25 Hz respectively for further analysis. The small DC offset in the input signal ensures that stiction is not present. We assume that the preloaded motors cancel backlash hence it does not affect the identification experiment. Together these two things reduce the effect of non-linearities on the system response, hence we can try and fit a linear dynamical model to the system. We also assume that the two axes are independent with no cross coupling, hence separate models can be derived for each axis. For the azimuth axis the experiment is performed at elevation angles of 5, 30, 60 and 95 degrees.

2.2.2 Data Processing

The input and the output data are detrended to remove linear trends in the data. For each experiment 10 swept sine signals are used. The analysis is performed on each length of swept sine. The result is then averaged over the 10 swept sines. As the swept sine signal generated by the dynamic signal analyzer is periodic, the FFT of the signal does not suﬀer from spectral leakage. Hence we do not window the data to reduce the spectral leakage. The frequency response functions (FRFs) are derived by dividing the input-output cross power spectral density with input power spectral density, and averaging over all the data sets in the experiment. The coherence ratio γ, is deﬁned using the average power spectral density and the average cross power density as given in by Eq. 2.24. The coherence ratio of the FRF tells us if the FRF is reliable in the given frequency range. A coherence ratio close to 1 indicates that the

30 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING output is highly correlated to the input and the FRF at that frequency is reliable.

S FRF = yu Suu ∗ (2.24) SyuS γ2 = yu SuuSyy

2.2.3 Model identiﬁcation

The goal is to identify a discrete state-space model Eq. (2.25) from the experimental data.

x(k + 1) = Ax(k) + Bu(k) (2.25) y(k) = Cx(k) + Du(k)

We use the Observer/Kalman identiﬁcation (OKID) algorithm [18] to identify the model in Eq. (2.25). In OKID, we ﬁrst identify Markov parameters (impulse response) of an associated system observer with an observer gain G as given in Eq. (2.26). The observer Markov parameters are then used to identify the Markov parameters (impulse response) of the system model in Eq. (2.25).

xˆ(k + 1) = (A + GC)ˆx(k) + (B + GD)u(k) − Gy(k) (2.26) yˆ(k) = Cxˆ(k) + Du(k)

The observer Markov parameters are

Y¯ = DCBC¯ A¯B¯ ··· CA¯p−1B¯ (2.27) where A¯ = A + GC and B¯ = [B + GD, −G] for the observer in Eq. (2.26). These parameters are calculated directly from the input and output data. The observer Markov parameters are used to ﬁnd the Markov parameters of the system in Eq.

(2.25). The system Markov parameters given by Y = [ D CB CAB CA2B ··· ]

31 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

as detailed in reference [18]. The Eigensystem Realization Algorithm (ERA) uses system Markov parameters to form the Hankel matrix as given by Eq. (2.28)

  Yk+1 Yk+2 ··· Yk+ξ      Yk+2 Yk+3 ··· Yk+ξ+1  H(k) =   (2.28)  . . . .   . . .. .      Yk+ν Yk+ν+1 ··· Yk+ν+ξ−1

th Where Yk is the k Markov parameter of the system, ν, ξ depend on the number of inputs and outputs and the order of the system to be identiﬁed. To identify an nth order system the singular value decomposition of H(0) is performed as H(0) = RΣS . The largest n singular values, and the corresponding left and right singular vectors,

th are used to approximate H(0) ≈ RnΣnSn, and will lead to a n order model. For r inputs and m outputs, with an identity matrix Ik of order k, and Ok zero matrix T of order k, let Er = Ir Or ··· Or ,Em = Im Om ··· Om , then the

identiﬁed state-space matrices (A0,B0,C0,D0) are given by,

−1/2 T −1/2 1/2 T A0 = Σn Rn H(1)SnΣn ,B0 = Σn Sn Er, (2.29) T 1/2 C0 = EmRnΣn ,D0 = Y0

Figure 2.15 shows the ﬂowchart for the algorithm steps.

32 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Input and output Observer impluse time histories response

Feedforward matrix System impluse Observer Observer (D) response parameters gain

System matrices Hankel matrix (A,B,C,D) H(0)

Left Singular value Right Eigenvectors decomposition Eigen vectors

Time shifted Observability Hankel matrix H(1) Controllability matrix matrix

Transition matrix Output matrix (A) Input matrix (C) (B)

Eigen Mode solution Mode shapes Amplitude Narural frequencies and damping

Reduced system (A,B,C,D)

Figure 2.15: Flow chart for Eigen system realization algorithm [18]

33 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

2.3 Modeling Results

2.3.1 Comparison of the analytical model vs. the experimen-

tal model

For each axis, the model was identified from the detrended swept sine response data using the OKID algorithm. Along with the encoder and the tachometer data we used the accelerometer data as the model outputs. Only the accelerometer whose data had the best coherence ratio was considered in identification for each axis. We choose accelerometer-1 x-axis (orientation along cross elevation axis) for the azimuth axis and for elevation we use accelerometer-1 y-axis (orientation along elevation axis) data. The accelerometer signals are used to improve the identification of resonant modes of the structure. Though generally the system impulse response (Markov parameters) is infinite, we select only the first 60 Markov parameters. Model order of n = 40 was found sufficient to describe the dynamics for both the axes. We compare the measured FRF, the FRF of the model derived from the FEA data, and the FRF of the model derived from the OKID algorithm. The experimental data was detrended before deriving the identified model, hence we lose information of the model gain at 0 Hz. Additionally, instead of a pole at zero for the rigid dynamics of the position output a small non–zero pole is identified using experimental data. While validating the position loop output this pole is forced to zero and the DC gain is adjusted to match the experimental data. The velocity loop step response of the identified model, and the FEA model, are compared with the measured step response for the purpose of model validation. For model validation the percentage error between the measured response ym and the

identiﬁed model, or the ﬁrst principles model response ys is given by Eq. (2.30).

||y − y || e% = m s 2 × 100 (2.30) ||ym||2

34 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.16: a: Az uncompensated open position loop FRF. b: Az open position loop response to velocity step command. Blue: Measured response. Red: Response simulated using the identiﬁed model. Green: Response simulated using the FEA based model.

Table 2.2: Modeling error percentage for the azimuth axis Encoder Tach X1 Ident. model 34 6.4119 36.371 FEA. model 18 6.5784 36.299

Figures 2.16, 2.17, 2.18, show the modeling results for the az axis and figures 2.19, 2.20, 2.21, show the results for the el axis. Tables 2.2 and 2.3 give the modeling errors for the Az and El axis. We see that both, the analytical and the identified models, give a good fit to the experimental data with similar error values. Hence either the analytical derived using FEA data, or the identified model derived using measured data, can be used for controller design and simulations as discussed in the next chapter.

Table 2.3: Modeling error percentage for the elevation axis Encoder Tach Y1 Ident. model 5.89 8.89 28.9 FEA. model 6.13 9.23 55.52

35 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.17: a: Az closed velocity loop FRF. b: Az velocity loop step response. Blue: Measured response. Red: Response simulated using the identiﬁed model. Green: Response simulated using the FEA based model.

Figure 2.18: a: Accelerometer 1 X-direction FRF to the Az axis excitation. b: Ac- celerometer 1 X-direction response to the Az velocity step command. Blue: Measured response. Red: Response simulated using the identiﬁed model. Green: Response simulated using the FEA based model.

36 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.19: a: El uncompensated open position loop FRF. b: El open position loop response to the velocity step command. Blue: Measured response. Red: Response simulated using the identiﬁed model. Green: Response simulated using the FEA based model.

Figure 2.20: a: El closed velocity loop FRF. b: El velocity loop step response. Blue: Measured response. Red: Response simulated using the identiﬁed model. Green: Response simulated using the FEA based model.

37 CHAPTER 2. GBT ANALYTICAL AND EXPERIMENTAL MODELING

Figure 2.21: a: Accelerometer 1 Y-direction FRF to the El axis excitation. b: Ac- celerometer 1 Y-direction response to the El velocity step command. Blue: Measured response. Red: Response simulated using the identiﬁed model. Green: Response simulated using the FEA based model.

38 Chapter 3

Primary reﬂector servo control

In this chapter, we develop the control design for the primary reflector servo system. For GBT it will be easiest to modify the position loop as it is the only loop implemented in software. Additionally, the final aim is to improve the position tracking accuracy of the telescope hence the disturbance rejection performance of the position loop is the most important. For past three decades, various control methods to improve the tracking performance of large telescopes have been explored. The most discussed amongst these have been the use of Linear Quadratic Gaussian control and H-infinity control [19],[20],[21],[22], [23]. Recently, the Extended State Observer (ESO) control technique was used for a 20 meter aperture telescope [24]. It led to a significant improvement in the disturbance rejection performance. ESO was first proposed by Han in 1990s [25]. Since then, it has been successfully used for many control applications [24], [26], [27]. The original theory, which was proposed by Han, used nonlinear time-varying observer and prefilter in its control design. The linear version of the theory was discussed and analyzed in papers [27],[28], [29]. Linear ESO based controller is easier to develop and implement as compared to full state feedback based controller like LQG or H-infinity controllers. In this chapter, we also propose Quantitative Feedback Technique (QFT) based

39 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL

design method for the ESO controller. This new formulation makes the design procedure more transparent and allows to prescribe specifications to be satisfied by the ESO at the beginning of the design. Additionally, we recommend the use of a feedforward pre-filter to the ESO loop to reduce the effect of increased ESO loop sensitivity

around the bandwidth frequency ω0. We then use this new formulation to design ESO based controller for the az and el axes position loop.

3.1 Extended State Observer theory

3.1.1 ESO for active disturbance rejection control

Consider a generic mechanical system with position x1 and velocityx ˙ 1 = x2. The state space equation for the generic mechanical system can be written as,

x˙ 1 = x2

x˙ 2 = f(x) + bu (3.1)

y = x1

where u is the torque input, y is the measured output and f(x) is both the unknown system dynamics and torque disturbance. f(x) is known as the total disturbance acting on the system. If we can estimate f(x) as fˆ(x), then the input (u − fˆ(x)) u = r will give us the double integrator, b

x˙ 1 = x2

x˙ 2 = ur (3.2)

y = x1

This can be controlled by either Proportional-Derivative controller(PD) or some other control method. Since fˆ(x) also contains the estimation of the disturbance torque

40 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL

(u − fˆ(x)) aﬀecting the output, u = r will also aid in the disturbance rejection. b

We can build an observer for f(x) as follows. Let f(x) = x3 the extended state, and f˙(x) = h(x), assuming f(x) is continuous. Then,

x˙ 1 = x2

x˙ 2 = x3 + bu (3.3)

x˙ 3 = h(x)

y = x1

This can be written as,

          x˙ 1 0 1 0 x1 0 0                     x˙  = 0 0 1 x  + b u + 0 h(x)  2    2               (3.4) x˙ 3 0 0 0 x3 0 1 f(x) = 0 0 1 x

Written as,

x˙ = Ax + B1u + B2h(x) (3.5) f(x) = Cx

From Eqn. 3.4 and 3.5 it is seen that the pair (A, C) is observable, hence f(x) can T be estimated. Consider the linear observer gain l = l1 l2 l3 . Then the extended state observer is given as,

          ˙ xˆ1 −l1 1 0 xˆ1 0 l1                     xˆ˙  = −l 0 1 xˆ  + b u + l  y  2  2   2    2 (3.6)  ˙          xˆ3 −l3 0 0 xˆ3 0 l3 fˆ(x) = 0 0 1 xˆ

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That is, ˙ xˆ = Aoxˆ + Bo1u + Bo2y (3.7) ˆ f(x) = Coxˆ

We can now use this fˆ(x) to close the observer feedback loop. This in essence describes the design of a linear ESO. Now we consider some additional properties of the ESO loop.

3.1.2 Uncertainty reduction through ESO loop

By studying the frequency domain characteristics of the ESO we can see that the ESO also helps in reducing uncertainty of the plant to be controlled. Consider a plant with multiplicative uncertainty,

P (s) = P0(1 + W (s)∆(s)), |∆(s)| ≤ 1 200 P0(s) = (3.8) s(s + 3) 1.59s + 1 W (s) = 0.32s + 1

Figure 3.1 shows the magnitude plot of the uncertain plant. We design an ESO loop T T 2 3 for this plant with l1 l2 l3 = 3ω0 3ω0 ω0 . Figure 3.2a shows the magnitude response of the plant with ESO with ω0 = 1 rad/sec and 3.2b with ω0 = 10 rad/sec. It is evident that the ESO causes reduction in uncertainty in the original, at frequencies below the ESO bandwidth. It also cause peak in response around the bandwidth frequency ω0. Also, higher the bandwidth of the ESO (higher value of ω0) , bigger is the frequency range of the uncertainty reduction. We can use this information for to reformulate the design problem in frequency domain.

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Figure 3.1: Magnitude response of the uncertain plant P(s) as given in eqn. 3.8

Figure 3.2: Magnitude response of the uncertain plant P(s) as given in eqn. 3.8 with ESO loop. a) ESO bandwidth ω0 = 1 rad/sec. b) ESO bandwidth ω0 = 100 rad/sec

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3.1.3 Quantitative feedback technique reformulation of the

ESO

Figure 3.3 shows the transfer function block diagram of the extended state observer.

Here H1(s) is transfer function for state-space system (Ao,Bo1,Co), and H2(s) is transfer function for state space system(Ao,Bo2,Co), deﬁned in eqn. 3.6 and 3.7. The equivalent plant Pe(s), controlled by controller C(s) can be given by,

P Pe = 1 + H1 + PH2 0 PH1 = 0 (3.9) 1 + PH1H2 0 1 H1 = 1 + H1

Current technique of designing a linear ESO involves just specifying ω0 to give disturbances rejection for a frequency range upti ω0. This design gives little insight as to how the ESO inﬂuences the design of controller C(s). The ESO bandwidth can be made as large as possible so that it gives maximum reduction in uncertainty and maximum disturbance rejection. In this case C(s) has to worry about only meeting the stability and tracking speciﬁcations of the control system. On the other hand, one can keep a moderate ESO bandwidth and C(s) can share the burden of uncertainty reduction and disturbance rejection. The method also does not tell much about the interaction of the ESO with the plant dynamics, which decides the magnitude of the peak frequency response around ω0. To quantify these characteristics, we develop frequency domain bounds that the ESO loop need to satisfy in order to get a prescribed equivalent plant Pe(s) for the controller C(s) loop. The ESO loop and the controller loop can be considered as a cascade system with ESO loop as the inner loop and the controller loop as the outer loop. We then use

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Figure 3.3: ESO transfer function block diagram

45 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL the following technique as described in [30], Let the plant to be controlled be,

P = P0(1 + ∆(s)), |∆(s)| ≤ r(ω) P − P (3.10) ∆(s) = 0 P0

After designing ESO, let the equivalent plant be,

Pe = Pe0(1 + ∆e(s)), |∆e(s)| ≤ |∆(s)| (3.11)

Where Pe0 is the nominal equivalent plant and ∆e(s) is the uncertainty in the equivalent plant. We can write ∆e(s) as,

Pe − Pe0 ∆e(s) = Pe0 0 0 PH1 P0H1 0 − 0 1 + PH1H2 1 + P0H1H2 = 0 P0H1 0 (3.12) 1 + P0H1H2 1 P − P0 = 0 1 + PH1H2 P0 1 = 0 ∆(s) 1 + PH1H2

If we design H1(s),H2(s) such that,

1 0 ≤ Wt(ω) (3.13) 1 + PH1H2

Then we can have, |∆e(s)| ≤ Wt(ω)r(ω). Hence we can prescribe the reduction in uncertainty for the equivalent plant for the outer loop as a frequency domain bound on ESO design parameters. One more goal of the ESO is to convert a type 0 plant into an ideal single integrator, or a type 1 plant as a double integrator, and so on, in a given frequency range which depends on the ESO bandwidth. This also directly relates the total disturbance

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Figure 3.4: ESO transfer function block diagram with preﬁlter F (s) rejection achieved by the ESO. We can quantify this characteristic as the following frequency domain bound on the ESO,

b0 Pe(s) − ≤ B(ω), 0 ≤ ω ≤ ωh sn+1 0 (3.14) PH1 b0 0 − n+1 ≤ B(ω), 0 ≤ ω ≤ ωh 1 + PH1H2 s Where n is the type of the original system. Using frequency domain bounds defined in eqns. 3.13 and 3.14 one can do quantitative feedback technique (QFT) based loop shaping of ESO on Nichols chart to give a predefined Pe to be controlled by C(s). QFT is a robust controller design technique proposed by Issac Horowitz in 1980s [31]. For an uncertain plant set, a robust controller is designed in QFT, by shaping the compensated open loop transfer function of a nominal plant on Nichol’s chart [32], [33], [34]. The shaping is done to satisfy stability and performance bounds on the uncertain plant set in frequency domain. These bounds are expressed as quadratic inequalities to be satisfied by the plant set in frequency domain [35]. We tackle the problem of the increased uncertainty at the ESO bandwidth frequency by adding a ESO feedforward filter Fo(s). Suppose the outer loop controller

C(s) is split into 2 parts C(s) = Co(s)Fo(s) as shown in ﬁgure 3.4. This feedforward

ﬁlter Fo(s) can be designed to suppress the eﬀect of increased sensitivity of the ESO

47 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL

loop around its bandwidth frequency ωo. If Fo(s) is now considered as a feedforawrd ﬁlter for the inner ESO loop, then the model matching bound in equation 3.14 can be written as,

0 PH1Fo b0 0 − n+1 ≤ B(ω), 0 ≤ ω ≤ ωh (3.15) 1 + PH1H2 s

This bound can be implmented as

0 P1H1Fo 0 1 + P2H1H2 0 ≤ δ(ω),P1,P2 ∈ P P2H1Fo 0 1 + P2H1H2 (3.16) P1 0 1 + P2H1H2 i.e ≤ δ(ω) P2 0 1 + P2H1H2

0 For loop shaping of P through H1H2, this bound is same as tracking bound for traditional QFT control. It can be solved in the same way as described in [33] or [36].

Hence the steps for designing ESO and controller Co(s) are as follows,

1. Deﬁne the speciﬁcations for ESO loop as given in equation 3.13 and 3.15.

0 2. Select the ESO Bandwidth ω0 and calculate H1(s)H2(s).

0 3. Use H1(s)H2(s) as the controller for ESO loop and check if the loop shaping satisfy the bounds using the QFT technique. If the bounds are not satisﬁed,

0 then select a new controller bandwidth ω0. On can also design a new H1(s)H2(s) by including the known higher order dynamics of the plant in the ESO.

4. Design the feedfoward ﬁlter Fo(s) for ESO loop, if needed.

5. Design the outer loop controller Co(s) to satisfy the overall control speciﬁcations, using any method of choice.

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Note: For systems of type 1 or higher, the closed ESO loop will always have more N(s) than one pole at 0. For example, for type 1 system, P (s) = . Let the open e s2D(s) 0 loop transfer function used for loopshaping be L = PH1H2. Then from deﬁnition of N(s) L Pe, 2 = . Hence, the L plotted on the Nichols chart, is of the type s D(s) H2(1 + L) H2N L = 2 . s D − H2N Though we may convert a stable plant P into an unstable equivalent plant Pe, the overall system with the outer loop controller Co + the ESO loop is always designed to be stable. Even though Pe is unstable, the unstable poles are at 0 and not RHP. Hence according to the Bode sensitivity integral [37] we will not add severe limitations to the system performance. Also both loops (ESO and Co loop) depend on same sensor feedback and result in only one actuator output. Hence any hardware failure will not get the system into only the unstable ESO closed loop operation.

3.2 ESO for the GBT primary reﬂector position

control

The legacy position controller for the GBT az and el axes position loop is a parallel PID controller with a ﬁlter for the derivative term. The controller is implemented in digital form with a sampling time of Ts = 0.02 seconds. The PID is implemented using backward Euler transform. The legacy controller equation is given as,

z N vcmd = KpPe + KiTs + Kd z (3.17) z − 1 1 + NTs z−1

Where Kp = 16.33,Ki = 4.387,Kd = −3.787,N = 4.137 for az axis. The values for el axis are Kp = 35.688,Ki = 12.3568,Kd = −8.2554,N = 4.137.

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3.2.1 QFT based ESO design

We now design ESO based disturbance rejection controller described in the previous section, and compare the controller performance against the legacy PID controllers through simulations. The QFT loop shaping is done using the Matlab toolbox QFTCT [38]. We used the reduced, open, uncompensated, position loop model developed in chapter 2 for the controller design. The uncertainty in the reduced model is calculated using interval analysis method as described in chapter 2. We start by first defining the specifications for ESO loop. We design the ESO loop such that it is effective upto a frequency of 3 rads/second (just below the first resonance frequency of the structure at 0.6 Hz). The ESO should reduce plant uncer-

tainty by 30 % below 3 rads/sec. The equivalent plant Pe(s) should match a double integrator upto 3 rads/sec. The uncertainty bound is set by a filter whose gain is 0.7 for frequencies less than 3rads/sec. The filter’s gain increases for frequency above 3rads/sec, so as to to relax the bound above 3rads/sec. We add zeros and poles to the model matching bounds above or at 3 rads/sec so that it relaxes the bound on model matching for frequencies above 3rads/sec.The ESO specifications for both az and el axis are given as,

3 2 1 1.6593s + 14.933s + 44.8s + 44.8 0 ≤ 3 2 1 + PH1H2 s + 12s + 48s + 64 0 2 (3.18) 0.1982 PH1Fo 0.00367s + 0.02202s + 0.03303 2 2 ≤ 0 ≤ 2 s (s + 8s + 16) 1 + PH1H2 s

Figures 3.5 and 3.6 show the results of the QFT loop shaping of the ESO loop for the az axis. Figures 3.7 and 3.8 show the results of the QFT loop shaping of the ESO loop for the el axis. We chose the ESO loop bandwidth parameter ω0 = 4rads/sec for az axis and 3.5 rads/sec for el axis, as compared to ω0 = 3rads/sec which would have been used according to the traditional method ESO design. The QFT analysis also

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Figure 3.5: QFT loop shaping of ESO for az axis on Nichols chart. Sensitivity bound violation between 3-6 rads/sec

quickly show increase in ESO loop sensitivity around ω0, which produces a peak in

frequency response around 4.5 rads/sec. This eﬀect is removed by design Fo(s) as a

notch ﬁlter with parameters for the az axis as ωn = 3.84rads/sec, a = 0.01, b = 1. For

the el axis, the parameters for Fo(s) notch ﬁlter are ωn = 4.5rads/sec, a = 0.05, b = 2. Thus QFT allows a more systematic and transparent design of the ESO loop

as compared to the traditional method by guiding the selection of ω0 according to predefined specifications. By adding the feedforward filter Fo(s) we also roubstly reduce the effect of the resonance around 4.5 rads/sec. .

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Figure 3.6: Az axis a) QFT loop analysis for uncertainty reduction. Sensitivity bound violation between 3-6 rads/sec b)QFT loop analysis for model matching after applying feedforward ﬁlter Fo(s).

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Figure 3.7: QFT loop shaping of ESO for el axis on Nichols chart. Sensitivity bound violation between 4-6 rads/sec.

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Figure 3.8: El axis a) QFT loop analysis for uncertainty reduction. Sensitivity bound violation between 4-6 rads/sec. b)QFT loop analysis for model matching after applying feedforward ﬁlter Fo(s)

54 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL

3.2.2 Outer loop controller design

We now design the outer loop controller Co(s) using QFT design method, this time applied to the equivalent plant Fo(s)Pe(s). The QFT design allows design of 2-DOF controller with feedback controller Co(s) and prefilter F (s). For the outer loop design, same specifications are use for the az and el axes. The specifications are chosen so that closed loop system has stability gain margin of 4dB and phase margin of 45 degrees. For tracking performance we have overshoot limit of 20 % and settling time of 15 seconds. The tracking specifications are chosen to match the tracking performance of the legacy PID controller tracking performance. These specifications for the outer loop in frequency domain translate to,

Stability speciﬁcation

Pe(s)Co(s) ≤ 1.3 1 + Pe(s)Co(s) Tracking speciﬁcation

2.481 F (s)Pe(s)Co(s) 0.1162s + 0.02906 3 2 ≤ ≤ 2 s + 20.67s + 13.46s + 2.481 1 + Pe(s)Co(s) 0.25s + 0.1176s + 0.02906 (3.19)

The controller Co(s) and Preﬁlter F (s) for the az axis are found to be,

10(s/0.3 + 1) C (s) = o (s/4.3 + 1) (3.20) 1 F (s) = (s/0.52 + 1)

Figure 3.9a shows Nichols chart loop shaping for az axis. Figure 3.9b) shows the step response of the system for az axis. Figure 3.10 shows frequency domain analysis of the system for the stability bound and the tracking bound.

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Figure 3.9: a) Az axis Co(s) controller loop shaping. b)Az axis step response with overshoot of 10% and settling time of 12 secs.

56 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL

Figure 3.10: Az axis Co(s) controller a) Frequency domain worst-case stability bound analysis b) Preﬁlter F (s) tracking bound analysis

57 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL

The controller Co(s) and Preﬁlter F (s) for the el axis are found to be,

14(s/0.2 + 1) C (s) = o (s/4.3 + 1) (3.21) (s/8 + 1) F (s) = (s/0.54 + 1)

Figure 3.11a shows Nichols chart loop shaping for el axis. Figure 3.11b shows the step response of the system for el axis. Figure 3.12 shows frequency domain analysis of the system for the stability bound and the tracking bound.

58 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL

Figure 3.11: a) El axis Co(s) controller loop shaping. b)El axis step response with overshoot of 15% and settling time of 17 secs.

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Figure 3.12: El axis Co(s) controller a) Frequency domain worst-case stability bound analysis b) Frequency tracking bound analysis.

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3.2.3 Comparison with the legacy controller response

In this section we compare the legacy PID controller given in eq. 3.17 with the ESO based controller designed in last section. We test these controllers on the whole model which has 88 degrees of freedom. All the position loop controllers are discretized at a sampling time of 0.02 secs which is the sampling time of the position loop of GBT. We present the results when the telescope is at an elevation of 0 degrees. In figure 3.13a we first compare the step response of the two controllers. In figure 3.13b we compare the response of the azimuth servo system to track joint jerk disturbance at the encoder end. We also compare the effect of the controller on the tip of the feedarm for both step input and track joint disturbance input in figures 3.13d and e. Finally we compare the controller output for the position loop step response and track joint disturbance response in the figures 3.13 e and f. From the figures 3.13 a, c and e we see more 50 % reduction in step response overshoot, corresponding reduction in the acceleration at the tip of feed arm, and the controller output reduction. From figure 3.13 b and we see more 50 % reduction track joint reduction and about 30 % reduction in feed arm tip acceleration. For track joint disturbance rejection we see a 50% increase in controller output. This should be acceptable as long as the output is well below the rate and saturation limits of the system. Wind is a low frequency torque disturbance acting on the structure. The frequency content of this disturbance will be similar to the general wind velocity spectrum at the GBT site. The position loop will be able to reject only the part of the disturbance seen at the servo motor end. Hence, the wind disturbance is simulated as a frequency limited random torque acting on the torque input to the structure. This disturbance is generated by shaping white noise with a filter H(s) (eq 3.22) such that the spectrum of the output of the filter approximately matches the wind spectrum. The output is then scaled and applied as an input disturbance torque to the structure. Figure 3.14a

61 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL compares the response of the two controllers to wind input torque disturbance. We see more than 50% reduction in wind induced position error at the encoder end. The feed arm tip acceleration also reduces as shown in fig 3.14b. Also we don’t see any significant increase in the controller output as shown in fig 3.14c.

s3 + 58.93s2 − 178.5s + 8.778 H(s) = (3.22) 80s5 + 9284s4 + 5.755e04s3 + 2.469e04s2 + 2790s + 93.85

In figure 3.15a we first compare the step response of the two controllers for elevation axis position loop. We again see more than 50% reduction in the step response overshoot with a slight increase in the selling time. In figure 3.15b we compare the response of the elevation axis to the wind torque input disturbance at the encoder end. The wind torque input disturbance is simulated in the same way as the done for azimuth axis. We again see more than 50 % reduction in wind torque induced pointing error at the encoder end. In the figures 3.15c and d we compare the effect of the controllers on the tip of the feedarm for the step input and the wind torque input disturbance. We see that the ESO based controller reduced the feed arm tip acceleration for both step input and wind torque disturbance. Finally, we compare the controller output for the position loop step response and wind disturbance response in figure 3.15 e and f. The controller output for step input reduces significantly while for wind disturbance rejection, the controller output stays almost the same. In summary, we see that adding ESO in the position loop improves the disturbance rejection of the loop by more than 50 % as seen at the axis encoder end. The response near the tip of the feedarm also improves which is crucial for improving the performance of the telescope. By using a 2 DOF controller for the position loop, designed using the classical SISO QFT method, we greatly improve the tracking response of the telescope position loop as well as its effect at the feedarm tip. This

62 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL

Figure 3.13: Az axis position loop. a) Step response. b) track joint response. c)x-el acceleration at feed arm tip due to step. d) x-el acceleration at feed arm tip due to track joint. e) Controller output for step. f) Controller output for track joint disturbance.

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Figure 3.14: Az axis position loop. a) Wind response at encoder end. b)x-el acceleration at feed arm tip due to wind disturbance as input torque disturbance. c) controller output for wind disturbance rejection.

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Figure 3.15: El axis position loop. a) Step response. b) Wind disturbance response at encoder end. c) el acceleration at feed arm tip due to step. d) el acceleration at feed arm tip due to wind disturbance as input torque disturbance. e) Controller output for step. f) Controller output for wind disturbance.

65 CHAPTER 3. PRIMARY REFLECTOR SERVO CONTROL again should improve the overall astronomical performance of the telescope. We also showed the usefulness of using QFT based loop shaping for ESO and decomposing the ESO design as a observer plus ESO feedforward filter. Analyzing the ESO design on Nichols chart in presence of frequency domain constraints quickly tells if the chosen ESO bandwidth is enough for the given specifications. The ESO feedforward filter is used to suppress the increase in uncertainty of the equivalent plant near the bandwidth frequency of the ESO. A properly designed ESO with feedforward filter makes the outer loop design very simple and effective.

66 Chapter 4

Subreﬂector servo control

A large part of the tracking error, caused by the swaying of the feed arm in wind, cannot be detected, and hence corrected by the primary reflector control system. The GBT subreflector, which is located at the tip of the feed arm, sits on a Stewart platform, and can move in 6 DOFs. One of the aims of the subreflector is to reposition itself dynamically so that it can direct the focused radio waves on to the receivers, in spite of the feed arm swaying. The swaying of the feedarm can be detected using an optical position sensitive detector called the quadrant detector (QD). The optical source of QD is fixed near the tip of feed arm below the subreflector. The detector is located below the primary reflector on the elevation axle. As the feedarm moves, the position of the light beam on the detector changes, hence sensing the feed arm motion. A calibration model is used to convert the signal from the detector to the corresponding correction needed in the subreflector motion to compensate for the FA swaying. The FA deforms with change in temperature and change in the orientation of the telescope. Thus the calibration model will always give corrections which are slightly offset as compared to the true correction needed. This correction can be wrong by a factor of 2 in some cases. As a result a new calibration model will be required after every few hours.

67 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Also the motion of the subreflector does not affect the QD output, as QD optical source is located independent of the subreflector unit. Hence, the QD can give only a disturbance feed forward correction and cannot be used in a feedback loop with the subreflector servo system. Since correction aided by the QD may be insufficient, new sensors need to be installed, or a different method to detect the FA motion and control of the subreflector needs to be developed. We propose that one can cancel the effect of the feed arm swaying by measuring the change in received radio signal and use this information to move the subreflector. This essentially means adding an outer loop to the subreflector servo system, which is closed using the received radio signal. We use extremum seeking control to achieve the loop closure with the RF signal feedback. Extremum seeking control is used in systems which essentially have a nonlinear gain with a local extremum, in its feedback loop. The goal of ESC is to achieve optimal output by forcing the system to stay at this local extremum. In the case of GBT the telescope gain is a 2-D Gaussian curve w.r.t to the motion of FA in el and x-el direction. Hence the ESC will try to move the subreflector such that the telescope gain reaches at the maximum of the 2-D Gaussian. We note that the method will work only when the telescope is pointing somewhere in the vicinity of the astronomical source, that is the source lies on the 2-D Gaussian gain of the RF beam and the signal has not dropped to zero (or background noise). Hence we assume that the primary reflector control, systematic error corrections of the telescope using pointing models, and the QD disturbance feedforward corrections first brings the source on some point of the 2-D Gaussian gain. The ESC then moves the subreflector so that the telescope gain is at the maximum of the 2-D Gaussian curve.

68 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.1: ESC loop for a nonlinear function f(θ(t)) which has a extremum at θ∗. Edited ﬁgure from [39]

4.1 Extremum Seeking Control theory

Figure 4.1 shows the basic ESC loop for a nonlinear function f(θ(t)) with an extremum at θ∗. The aim of the ESC loop is to drive θˆ → θ∗ so as to optimize y = f(θ(t)). We consider f(θ(t)) to be convex or concave. The loop is perturbed by a signal a sin(ωt). The high pass filter s/(s + h) and the demodulation with sin(ωt) takes the approximate derivative of y . This gives the direction in which θ should move to reach the optimum value of f(θ(t)). The approximate derivative ξ is then integrated by −k/s to give the optimization correction step. The step size is determined by the gain k and the perturbation amplitude by a. For optimizing a dynamic function f(θ(t)) one can use a control signal u(t) derived R T by solving the Euler-Lagrange equation for the functional 0 f(θ(t)). Instead, in ESC, by calculating the gradient of y, we use a gradient based optimization scheme. Integrating the gradient collapses the gradient function to single value of θˆ(t) for the instantaneous time t . This is in contrast with the calculation of the function u(t) in Euler-Lagrange based optimal control, which is done for a finite time t ∈ [0,T ] or for infinite time as T → ∞. Let θ˜ = θ∗ − θˆ, where θˆ is defined in figure 4.1. It is shown in [39], that for a

69 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.2: ESC loop for a nonlinear function f(θ(t)) which has a extremum at θ∗. The loop is set at an initial value of θ(0) = θb suﬃciently large value of ω and h,

−kaf 00 θ˜˙ ≈ θ˜ (4.1) 2

That is, if kaf¨ ≥ 0, then θˆ → θ∗.

4.1.1 Eﬀect of the initial condition

Figure 4.1 shows the basic ESC loop for a nonlinear function f(θ(t)) with an extremum

∗ at θ . The loop starts with an oﬀset value set to θb. For θb = 0 the ESC loop is same as the ESC loop shown in 4.1. θb helps set the initial value for the optimization in the ESC loop. Equation 4.1 gives the approximate homogeneous dynamics of the error term θ˜ = θ∗ − θˆ [39]. The ESC problem can be restated as the problem of driving the homogeneous error dynamics to its equilibrium point at 0. The speed at ˜ ˜ which the θ → 0 depends on the loop gain and the value of θ|t=0. Smaller is the ˜ ˜ value of θ|t=0, smaller will be the time taken by the ESC loop to drive θ to 0. If ˜ ∗ ˜ ∗ θb = 0, then θ|t=0 = θ , otherwise it is θ|t=0 = θ − θb. For appropriate value of

∗ ∗ ∗ θb, θ − θb < θ . That is, θb sets the system closer to the extremum point θ and improves the response time of the ESC loop. Similar idea will be used later to show the eﬀectiveness of disturbance feedforward signal on the performance of the ESC

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Figure 4.3: ESC loop for a nonlinear function f(θ(t)). The extremum point is dynamic θ∗(t) and the loop has SISO linear plant dynamics. Edited ﬁgure from [39] loop.

4.1.2 ESC for a single parameter system with dynamics

Figure 4.3 shows the block diagram of the ESC loop for single parameter θ(t) system

∗ with dynamic extremum θ (t) and linear plant dynamics. λθΓθ(s) and λf Γf (s) are the

∗ ∗ transfer functions for θ (t) and f (t). λθ, λf are transfer function gains whose value

is not explicitly needed in ESC design. Fi(s) is the input dynamics, and Fo(s) is

the output dynamics. We assume Fi(s),Fo(s), are asymptotically stable and proper.

Γf (s), Γθ(s) are strictly proper functions, and unstable poles of Γθ(s) are not zeros

of Fi(s). Ci(s) and Co(s) are designed according to the following guidelines given in [39],

1. Perturbation frequency ω is suﬃciently large.

2. Zeros of Γf (s) that are not asymptotically stable are also zeros of Co(s).

3. Poles of Γθ(s) that are not asymptotically stable are not zeros of Ci(s).

71 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.4: ESC loop for a multi parameter, nonlinear function f(θ(t)). (θ(t)) is a vector with a dynamic extremum at θ∗(t), and MISO linear plant dynamics. Edited ﬁgure from [39]. p = 1, ··· , l. For odd p, ωp+1 = ωp, βp = 0 and βp+1 = π/2.

Co(s) 4. Let Hi(s) = Ci(s)Γθ(s)Fi(s) and Ho(s) = k Fo(s), φ = ∠(Fi(jω)), L(s) = Γf (s) af 00 Re{ejφF (jω)}H (s). The ESC will have a stable region of attraction to- 4 i i 1 wards the extremum, if C (s) and are asymptotically stable. o 1 + L(s)

4.1.3 ESC for a multi-paramter system with dynamics

The design for a single parameter system with dynamic extremum θ∗(t) can be extended to a multi-parameter system as shown in ﬁgure 4.4. Consider θ(t) =

T ∗ T ∗ [θ1(t) ··· θl(t)] and the non-linearity f(θ(t)) = (θ(t) − θ (t)) P(θ(t) − θ (t)) where P is a l × l matrix. We consider a case where this non linearity is a part of a MISO system. Figure 4.4 shows an ESC loop for such a system.

∗ T ∗ L{θ (t)} = [λ1Γθ1(s) ··· λlΓθl(s)] , L{f (t)} = λf Γf (s), φp = ∠(Fip(iωp)). The

output dynamics Fo(s) is SISO dynamics. The input dynamics Fi(s) are MIMO dynamics but only diagonal terms are considered for diagonal compensation. We make the following assumptions about the system [39].

T 1. Fi(s) = [Fi1(s) ··· Fil(s)] (only diagonal part of Fi(s)) and Fo(s) are asymp-

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totically stable and proper.

2.Γ θ(s) and Γf (s) are strictly proper, and Fip(s) does not contains zeros equal to

the poles in Γθp(s) that are not asymptotically stable.

3. ωp + ωq 6= ωr for any p, q, r = 1 ··· l.

The ESC control also needs to satisfy the following criteria to have a stable region of attraction towards the extremum. Let us deﬁne,

δ = 1 if p is odd, δ = −1 if p is even

Hip(s) = Cip(s)Γθp(s)Fip(s) C H (s) = k op F (s) op p Γ (s) o f (4.2) 1 L (s) = H (s)Re{ejφp F (jω )} p 4 ip i,p p 1 j(φ +δ π ) M (s) = Re{e p 2 F (jω )} p 4 i,p+δ p

Xpq(s) = PpqapLp(s) + Pp+δ,qap+δMp(s)

Where Xpq(s) is the (p, q) element of the matrix X(s). It has been proved in [39] that the system has asymptotically stable region of attraction towards the ex- 1 tremum, if Cop(s) are asymptotically stable for all p = 1, ··· , l and is det(X(s) + Il) asymptotically stable.

4.1.4 ESC with disturbance feedforward signal

In this section, we investigate the eﬀect of the disturbance feedforward signal on the ESC loop and show that the ESC loop performance improves with an appropriate disturbance feedforward signal. Consider a SISO system with a nonlinearity as shown in ﬁgure 4.5. The nonlinearity has a dynamic extremum θ∗(t). We assume that θ∗(t) changes due to some disturbance acting on the system. The ESC loop needs to compensate against this disturbance by appropriately tracking θ∗. Generally, it is

73 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.5: ESC loop for a nonlinear function f(θ(t)). The extremum point is dynamic θ∗(t) and the loop has SISO linear plant dynamics. The loop also has disturbance feedforward signal θf (t). found that the response of ESC loop is slow and it cannot satisfactorily cancel out the eﬀect of a fast changing disturbance. If we have a way to approximately measure the disturbance, we can use it to generate a disturbance feedforward signal θf (t). We will now analyze the eﬀect of θf (t) on the ESC loop. We follow the analysis on the same lines as the analysis for proposition 1.5 in [39]. Note In the derivation we trace the signal as it passes through various linear

filters. If Ffilt(s) is the transfer function of the filter, u(t) is the input to the filter, and r(t) is the output, then we say,

r(t) = Ffilt(s)[u(t)] (4.3)

That is u(t) ﬁltered through Ffilt(s) gives r(t).

74 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Let, ∗ L{θ (t)} = λθΓθ(s)

∗ L{f (t)} = λf Γf (s)

L{θf (t)} = λθf Γθf (s) (4.4)

Hi(s) = Ci(s)Γθ(s)Fi(s)

Co(s) Ho(s) = k Fo(s) Γf (s) From ﬁgure 4.5 we can say,

f” y = F (s) f ∗(t) + (θ0 − θ∗(t))2 o 2

θ = Fi(s)[a sin(ωt) − Ci(s)Γθ(s)[ξ]] (4.5) 0 θ = Fi(s)[a sin(ωt) − Ci(s)Γθ(s)[ξ] + θf ] = θ + Fi(s)[θf ] C (s) ξ = k sin(ωt − φ) o [y + n] Γf (s)

For the purpose of analysis we further deﬁne,

˜ ∗ 0 θ(t) = θ (t) − θ (t) + θ0 + Fi(s)[θf ]

˜ ˜ ∗ θf (t) = θ − Fi(s)[θf ] = θ + Hi(s)[ξ] (4.6)

θ0 = Fi(s)[a sin(ω(t))]]

∗ y˜ = y − Fo(s)[f (t)]

The objective of the extremum seeking control is to drive the output errory ˜ to a small value, by driving θ0(t) close to θ∗(t).

For n = 0 and system with non-zero feedforward signal θf (t),

˜ ∗ θf = θ + Hi(s)[ξ] f” (4.7) θ˜ = θ∗ + H (s) sin(ωt − φ)H (s) f ∗(t) + (θ0 − θ∗(t))2 f i o 2

75 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

0 ∗ ˜ From equation 4.6, θ − θ = θ0 − θf . Hence,

f” θ˜ = θ∗ + H (s) sin(ωt − φ)H (s) f ∗(t) + (θ − θ˜ )2 f i o 2 0 f (4.8) f” θ˜ = θ∗ + H (s) sin(ωt − φ)H (s) f ∗(t) + (θ2 − 2θ θ˜ + θ˜2) f i o 2 0 0 f f

˜2 We neglect the θf term as it is assumed to be small. By design, we consider stable dynamics for Hi(s) and Ho(s). Hence,

∗ −1 −t −t sin(ωt − φ)Ho(s)[f (t)] = λf sin(ω(t) − φ)L {Ho(s)Γf (s)} = sin(ωt) =

2 2 2 −t sin(ωt − φ)Ho(s)[θ0] = C1a sin(ωt + µ1) + C2a sin(3ωt + µ2) + (4.9)

Where C1,C2, µ1, µ2 are constants depending on frequency of sine perturbation. We f” deﬁne, u = a2 [C sin(ωt + µ ) + C sin(3ωt + µ )]. Then the tracking error 13 2 1 1 2 2 dynamics can be approximated as,

˜ ∗ −t ˜ θf = θ + Hi(s)[u13(t)] + − f” sin(ωt − φ)Ho(s)[θ0θf ]] (4.10) ˜ −t −t ˜ den{Hi(s)}[θf ] = + num{Hi(s)}[[u13] + − f” sin(ωt − φ)Ho(s)[θ0θf ]]

In equation 4.10 the θ∗ term is dropped as it decays exponentially when operated upon by denΓθ(s). The equation 4.10 can be written represents a time varying linear system whose dynamics can be written as,

x˙ = A(t)x + Bu13(t),A(t + T ) = A(t),T = 2π/ω (4.11) ˜ x = θf

The dynamics of the periodic LTV system in equation 4.11 are stable if the eigenvalues of the transition matrix Φ(T, 0) lie within the unit circle. From equation 4.10 we see

that the value of system matrices A(t),B depends on Hi(s),Ho(s), a sin(ωt), φ and

76 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

f” which are independent of the feedforward signal and perturbation signal. At this point we make 2 observations,

1. For the same perturbation signal u13 and system matrices A(t),B, the speed at ˜ ˜ ˜ which θf → 0 depends on size of ||θf ||. Smaller is the magnitude of ||θf || faster ˜ is the rate of convergence to zero. From deﬁnition of θf in equation 4.6, we see ∗ ˜ ˜ that for disturbance feedforward in the direction of θ (t), ||θf || < ||θf ||θf (t)=0. Hence the speed of convergence to 0 increases with appropriate disturbance feedforward signal.

2. As shown for static nonlinearity, the initial condition for system in equation 4.11,

∗ with no feedforward signal, is x0 = θ |t=0. This initial condition for optimization

∗ can be improved by appropriately setting x0 = θ |t=0 − Fi(s)[θf (t)]|t=0 and starting the optimization closer to the extremum point.

As the response time of the ESC depends both on the initial condition for opti- ˜ mization and magnitude of error dynamics ||θf ||, through this analysis we show that appropriate disturbance feedforward signal improves the response time of the ESC. This result can be similarly extended to multi-parameter extremum seeking case.

4.2 ESC for the GBT subreﬂector

Now we apply ESC to the GBT subreﬂector control. First we establish the functional blocks of the ESC loop for the subreﬂector and then design the controller.

4.2.1 Telescope gain and pointing oﬀsets

Figure 4.6a shows the conceptual radio beam of a single dish radio telescope. The angle θhw is the point where the gain of the telescope is 0.5 times its maximum gain and this angle is called full width half maximum (FWHM). It is generally proportional

77 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.6: a) Ideal radio beam of a single dish telescope. I(ν, θ, φ) is the source intensity as a function of observing frequency ν and the pointing angles in 2-D θ, φ. A(ν, θ, φ) is the antenna gain b) Normalized Gain of the telescope as a function of pointing angle in one dimension. [40] to λ/D where λ is wavelength of observation and D is the telescope diameter. Hence, at high observing frequency and large primary reﬂector diameter, as in the case of

GBT, θhw can be a few tens of arcseconds. For observing a source, it should lie within

FWHM of the beam, and the source must be within 0.1θhw to detect 95% of the source brightness. A pointing error of 0.3θhw can cause a source loss of 22%. The gain of the radio beam is a 2 dimensional Gaussian curve depending on pointing angles in el and x-el . Figure 4.6b shows the Gaussian curve of the gain in one dimension. For controlling the subreﬂector of the GBT, this gain curve forms the non linear gain f(θ) for the ESC loop. The maximum point θ∗ of f(θ) changes with feedarm swaying and the ESC loop will try to follow it.

4.2.2 ESC loop for the subreﬂector

We now consider the relationship between the subreﬂector servo mechanical system described in section 2.1.6, the telescope gain, the Quadrant detector signal, and the received radio signal to formulate the ESC loop for the subreﬂector. Figure 4.7 shows

78 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

the qualitative relationship between various blocks to form an ESC loop. Feedarm swaying causes a shift of the optimal position θ∗(el) and φ∗ (x-el), of the non-linear telescope gain function A(t, ν, θ(t), φ(t)) where ν is the observation frequency. The maximum value of the gain function is also aﬀected by things like changing weather and receiver gains. t accounts for change in this gain function with time due to weather and receiver gain changes. We consider that these changes due to the weather and the receiver gains are much slower than the feed arm swaying in the wind and the ESC loop response. Hence the ESC can quickly settle on the optimum value of the dynamic A(t). The received signal of the telescope is a convolution of the antenna gain A and the source intensity I. We consider the source to be a point source, that is the angular area of the source in the sky is less than the FWHM of the beam in θ and φ direction. Hence, the ﬁnal non linearity with a maximum in the ESC loop is f = AI. The radio signals received are very weak and need to be integrated over a span of few minutes to several hours. Hence the output dynamics are given 1 by F (s) = . The feed arm swaying is detected by the QD, which generates an o s

approximate disturbance feed forward corrective motion θD(t), φD(t), needed by the subreflector. This feed forward signal along with the ESC correction is fed to the subreflector servo-mechanical system, whose simplified model was discussed in 2.1.6. For correction against the feedarm swaying we need to move the subreflector only in the el and x-el directions. As discussed in 2.1.6, this roughly corresponds to x and z axes linear motion of the subreflector or z-rotation and x-rotation motion of the subreflector. As explained in [15],[41] the rotation motions allow higher acceleration limits in corresponding el and x-el directions. Even though the subreflector has 6 DOFs with corresponding simplified SISO loops, we consider only the x and z rotation DOFs, and the corresponding 2 SISO loops for subreflector ESC control input dynamics.

T That is Fi(s) = [Fi1(s) Fi2(s)] , Fi1(s) = Fi2(s) = Gact(s), where Gact(s) is given by Eq. 2.19. Without aﬀecting the ﬁnal controller design result, we consider that A

79 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.7: ESC loop for the subreﬂector. and f don’t explicitly change with time, that is f(t, θ(t), φ(t)) = f(θ(t), φ(t)) Let the maximum of the Gaussian curve be G. Without loss of generality, we consider a the 2-D Gaussian curve with no cross terms between θ and φ. Let Θ(t) = [θ(t) φ(t)]T , and dynamic optimum of Θ(t) = Θ∗(t) = [θ∗(t) φ∗(t)]T . Then f(Θ(t)) can be given by,

∗ T ∗ f(Θ(t)) = Ke−(Θ(t)−Θ (t)) Pe(Θ(t)−Θ (t)) (4.12)

Where K is some constant. This constant will depend on source intensity I and the receiver gains. Since we are considering only the qualitative model of the RF gain, we assume K to be 1. In the actual system, the ESC gain kp and ap can be adjusted according to the approximate value of K determined from the system signal ﬂow. For joint, independent, 2 parameter, Gaussian contours, P is diagonal. But as noted in [42] for GBT the motion in x and z of the subreﬂector don’t map exactly to el and x-el but has some cross-correlation. Hence P will have some small non-diagonal terms.

80 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

4.2.3 The ESC design

We simulate the system for a radio beam at 90 GHz observation frequency (the highest observing frequency at the time of this writing) with a FWHM of 9 arcseconds or 4.06e-5 radians, in the el and x-el directions. We consider that the RF beam is ideally positioned at (θ∗, φ∗) = (1 arcmin, 1 arcmin) w.r.t the subreflector. Figure 4.8 shows the power spectrum of a typical wind velocity profile at GBT. The mean wind velocity is subtracted to give the power spectrum of only the wind velocity variation about the mean value. Equation 4.13 gives the approximate filter fit H(s), to this power spectrum. H(s) gives the dynamics of the disturbance acting on the FA. We generate the the dynamic behavior of (θ∗(t), φ∗(t)) be perturbing (θ∗, φ∗) with a scaled white noise signal filtered through H(s). We assume a QD disturbance feerdorward signal with 20% gain error and offset error of 2.5 arcseconds, in both (θ, φ) correction. The source intensity is considered as a constant 1. Since the z-rotation and x-rotation of the subreflector don’t exactly align with the el and x-el motion, we consider a 10% cross-correlation between the parameters (θ, φ) of the Gaussian function.

s3 + 58.93s2 − 178.5s + 8.778 H(s) = (4.13) 80s5 + 9284s4 + 5.755e04s3 + 2.469e04s2 + 2790s + 93.85

We simulate the qualitative system block diagram as shown in ﬁgure 4.6 using the

81 CHAPTER 4. SUBREFLECTOR SERVO CONTROL following parameters,

T T Fi(s) = [Fi1(s) Fi1(s)] = [Gact(s) Gact(s)] 1 F (s) = o s Θ∗ = [θ∗ φ∗]T = [2.9089e − 04 2.9089e − 04], RF beam position in el, x-el

Θ∗(t) = Θ∗ + H(s)w(t)[1 1]

w(t) = Uniformly distributed white noise with a peak amplitude of 6e-5 radians

I(ν, θ, φ) = 1, Source intensity   1/4.0667e − 052 0.1/4.0667e − 052   Pe = 4ln(2)   0.1/4.0667e − 052 1/4.0667e − 052

K = 1

∗ T ∗ A(t, ν, θ, φ) = Ke−([Θ(t)−Θ (t)] Pe[Θ(t)−Θ (t)])

f(t, θ, φ) = AI

P = IKPe

T ∗ ΘD(t) = [θD(t) φD(t)] = 0.8Θ (t) + 1.2044e − 04(QD feedforward input) (4.14) The ESC controller is simulated using the following design parameters,

ω = 10 rads/sec

kp = 0.04

a1 = a2 = 1e − 10 1 Γ (s) = f s (4.15) 1 C (s) = o s + 100 1 Γ (s) = Γ (s) = Θi1 Θi2 s 0.5 C (s) = C (s) = i1 i2 (s + 0.5)

82 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.8: Power spectrum of a sample wind gust signal at the Green Bank site. The mean wind velocity is subtracted to give the power spectrum of only the wind velocity variation about the mean value.

4.2.4 Results

We simulate subreﬂector servo system with the ESC loop using the qualitative representation of the system and the controller design parameters as described in the previous section. Figure 4.9a plots the trajectory of Θ∗(t) − Θ(t) on the contour of f(θ(t), φ(t)), with just QD disturbance feed forward correction. Figure 4.9b shows the corresponding f(θ(t), φ(t)), which is the normalized RF signal received by the radio telescope. With 20% error in QD calibration gain and 2.5 arcseconds calibration oﬀset error in both (θ(t), φ(t)), the mean signal drops from 1 to 0.56. Figure 4.10a plots the trajectory of Θ∗(t) − Θ(t) on the contour of f(θ(t), φ(t)), with just ESC feedback. Figure 4.10b shows the corresponding f(θ(t), φ(t)). We see that the ESC is not fast enough to respond to changing Θ∗(t) and the mean signal drops to 0.47. Figure 4.11a plots the trajectory of Θ∗(t) − Θ(t) on the contour of f(θ(t), φ(t)), with ESC feedback and QD disturbance feedforward. Figure 4.11b shows the corre-

83 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

sponding f(θ(t), φ(t)). With both ESC feedback and QD feed forward the mean signal drops to only 0.9675. Hence as expected, a combination of both QD feedforward and ESC feedback is the best strategy to compensate for the feedarm swaying. Figure 4.12 shows the command signal sent to subreflector servo mechanical system when both ESC and QD feedforward are in effect. Figure 4.13 shows the zoom in on the θ(t), φ(t) command to the subreflector. At 10 rads/sec frequency of perturbation, the amplitude of the perturbation in the command is about ±1 arcseconds. The maximum acceleration of each leg of the Hexapod is 0.4 inch/sec2. Using data in [15], [41] the limit for x rotation amplitude at 10 rads/sec is 19.6 arcseconds. Simi- larly, the limit for z rotation perturbation amplitude at 10 rads/sec is 15.8 arcseconds. Hence the perturbation is well within the acceleration limits of the subreflector. The perturbation frequency is 10 rads/sec i.e 1.59 Hz is above the resonant frequencies of the FA which are 0.6 and 0.8 Hz in x-el bending and 1.02 Hz in el bending. Hence we should not expect any resonance excitation of the FA. According to stability conditions stated in section 4.1.3, the poles of transfer 1 function are, det(I + X(s))

84 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.9: Subreﬂector servo system with only QD disturbance feedforward correction. The QD calibration model has 20% gain error and 2.5 arcseconds oﬀset error for both (θ(t), φ(t)) a) Θ∗(t) − Θ(t) on f(θ(t), φ(t)) contour map. b) Normalized RF signal received f(θ(t), φ(t)).

85 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.10: Subreﬂector servo system with only ESC feedback. a) Θ∗(t) − Θ(t) on f(θ(t), φ(t)) contour map. b) Normalized RF signal received f(θ(t), φ(t)).

86 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.11: Subreﬂector servo system with ESC feedback and QD disturbance feedforward correction. a) Θ∗(t) − Θ(t) on f(θ(t), φ(t)) contour map. b) Normalized RF signal received f(θ(t), φ(t)).

87 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

Figure 4.12: θ(t), φ(t) command signal to the subreﬂector servo with both QD feedforward and ESC in eﬀect.

Figure 4.13: θ(t), φ(t) command signal to the subreﬂector servo zoom in. The perturbation amplitude at 10rads/sec is 1 arcsecond

88 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

  −16.8128e + 000      −16.5097e + 000 + 762.0050e − 003i       −16.5097e + 000 − 762.0050e − 003i         −15.6656e + 000 + 1.2297e + 000i       −15.6656e + 000 − 1.2297e + 000i       −14.5422e + 000 + 1.1166e + 000i         −14.5422e + 000 − 1.1166e + 000i       −13.7633e + 000       −10.4024e + 000         −9.8926e + 000 + 328.4220e − 003i       −9.8926e + 000 − 328.4220e − 003i       −9.7528e + 000         −7.1847e + 000 + 195.1438e − 003i    (4.16)    −7.1847e + 000 − 195.1438e − 003i       −6.8552e + 000 + 151.2602e − 003i         −6.8552e + 000 − 151.2602e − 003i       −4.0181e + 000 + 27.9087e − 003i       −4.0181e + 000 − 27.9087e − 003i         −3.9669e + 000 + 26.9678e − 003i       −3.9669e + 000 − 26.9678e − 003i      −500.0000e − 003 + 804.5669e − 009i       −500.0000e − 003 − 804.5669e − 009i      −484.9954e − 003 + 19.0746e − 003i       −484.9954e − 003 − 19.0746e − 003i         −14.4526e − 003 + 18.4152e − 003i    −14.4526e − 003 − 18.4152e − 003i

89 CHAPTER 4. SUBREFLECTOR SERVO CONTROL

All the roots are negative and hence the ESC loop has a stable region of attraction towards the extremum.

4.2.5 Design limitations

We note that the controller gains and the ESC loop stability will depend on the source intensity, hence we will need diﬀerent tuning parameters for bright sources versus dim sources. The controller tuning will also change with change in observation frequency. We also note that the method will work for only point (unresolved) sources whose angular area in the sky is smaller than the FWHM of the RF beam. As only in this case the non linearity in the loop will have a clear maximum which we would like to track. For extended sources the non-linearity f(t, θ, φ) = A ∗ I will have several maximums and minimums and the ESC loop will drive the subreﬂector only to the local maximum thus failing to map the entire source.

4.2.6 Implementation discussion

For the subreflector, the assumption that all legs of the Stewart platform have the same dynamical behavior needs to be verified experimentally. If this is not true the measured dynamic response of different legs should be used to design the ESC loop. The behavior of the ESC loop in case of actuator limit needs to be studied. The range of gains, according to changing source intensity, for a given observing frequency needs to be calculated experimentally. We can setup a gain scheduling scheme which will allow different tuning parameters according to different source intensity ranges. In simulation the perturbation signals is 1 arcsecond peak-to-peak. This may be below the positioning accuracy of the Stewart platform. In this case the perturbation maybe increased above the accuracy limit of the subreflector (but still well below the acceleration limit of the subreflector) and the loop maybe tuned for the new

90 CHAPTER 4. SUBREFLECTOR SERVO CONTROL perturbation amplitude. We envisage that the primary reflector servo system will first move the telescope to a point close enough to the source, so that the source will be somewhere on the RF beam of the telescope. Only after the transients due to the telescope slewing command have decayed, the offsets of the pointing model for both the primary reflector and the subreflector have been commanded and the telescope is in steady state tracking state, should the subreflector ESC loop be activated. This will ensure that the ESC is not reacting to the transient response of the telescope primary reflector servo system. Finally, the ESC scheme will have to tested for point source tracking and source scanning trajectories like daisy pattern (Lissajous pattern) and on-the-fly mapping.

91 Chapter 5

Conclusion

5.1 Summary of the main results

We provided the detailed modeling of the GBT structure and the primary reflector servo system. The modeling is a huge improvement over the Lagrange formulation based model available with GBT. Using FEA data allowed us to define multiple outputs at various points on the structure, like the tip of the feedarm. This allowed us to monitor the effect of the servo system and external disturbances at the points on the structure most important for the astronomical performance of the telescope. The model was verified by system identification experiments. Uncertainty quantification is important to analyze the robustness of the control design. The uncertainty in the model was effectively captured using interval analysis method. The last 2 points give us confidence in the simulation models and performance improvement due to proposed control design. We proposed the QFT formulation of the ESO which systematically and transpar- ently helps in choosing the ESO bandwidth. The formulation was done by capturing the uncertainty reduction ability of the ESO as a sensitivity bound on the observer loop and by setting a model matching bound on the observer loop. We also made

92 CHAPTER 5. CONCLUSION the use of a feedforward filter along with ESO to remove the effects of increased ESO loop sensitivity at the observer bandwidth frequency. The method was then applied to primary reflector position loop control and showed a remarkable 50% improvement in disturbance rejection. We used a robust 2 DOF controller for final position loop controller in each axis. This helped in robust tracking performance improvement of the servo loop. We investigated the innovative use of ESC with disturbance feedforward signal and showed that it leads to an increase in the speed of the ESC loop. At least for tracking of point sources, we showed that a subreflector servo system with the quadrant detector disturbance feed forward signal and an extremum seeking loop closed with the measured radio signal can help to dynamically reposition the subreflector to compensated for feedarm swaying in windy conditions. Closing of the ESC loop about the radio signal could lead to more than 40% error correction in the telescope gain in windy conditions. The use of ESO for the primary axes servo system and ESC with disturbance feedfowrd for subreflector control should help in significant improvement of GBT tracking and pointing accuracy against torque disturbances, especially wind torque disturbances. This should allow astronomical observations at high frequencies in windy conditions, increase the dynamic range (signal to noise ratio) of the telescope at low frequencies and double the observation time available to the telescope.

93 Appendix A: Subreﬂector actuator measured FRFs

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