A New Controller for the Acquisition and Guiding Unit for the Gemini South Telescope: New architecture and control schemes to use Power PMAC

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IN

DEGREE PROJECT

VEHICLE ENGINEERING, SECOND CYCLE,

30 CREDITS

,

STOCKHOLM SWEDEN 2016

A New Controller for the

Acquisition and Guiding Unit for

the Gemini South Telescope

New architecture and control schemes to use

Power PMAC

MATTHIEU BARREAU

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

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A New Controller for the Acquisition

and Guiding Unit for the Gemini South

Telescope

New architecture and control schemes to use Power PMAC

M A T T H I E U B A R R E A U

Master’s Thesis in Optimization and Systems Theory (30 ECTS credits) Degree Programme in Vehicle Engineering (300 credits) Royal Institute of Technology year 2016 Supervisor at Gemini Observatory: Luc Boucher Supervisor at KTH: Prof. Dr. Xiaoming Hu Examiner: Prof. Dr. Xiaoming Hu

TRITA-MAT-E 2016:10 ISRN-KTH/MAT/E--16/10--SE

Royal Institute of Technology School of Engineering Sciences

KTH SCI

SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

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Abstract

When the Gemini Telescopes were built 20 years ago, the control architecture for the high precision requirements was the same for all systems. It consisted of VME cards, Programmable Multi Axis Con-trollers (PMAC) with specific amplifiers and a central computer. It was the state of the art. Nevertheless such an infrastructure takes up a lot of space. It is also difficult to maintain mainly because of obsoles-cence and the lack of support of the engineers who do not consider this system adapted to the control.The code, documentation and wiring were complex and not fully understood.

Thus, it led the Gemini Observatory to consider the acquisition of a new controller for one of the most critical unit: the Acquisition and Guidance (A&G). To address these issues, I propose an alternative control system which will not only solve the current issues but also improve the performance.

The new generation of controller from Delta Tau is more efficient, more reliable and with a high level of integration. The main concern of compatibility with the current motors and encoders of the Acquisition and Guidance has been solved by testing 27 motors out of the 29 present in the unit. Results were obtained using a test bench and mechanical systems built during the internship. A fully functional test bench has been delivered.

Furthermore, a new control scheme for the backlash compensation is proposed. It consumes half the energy of the current one, is nearly two times faster and without oscillations. This will reduce the frequency of maintenance and the reliability of the unit. A cross gantry control for the skew compensation of the Science fold leads to a smarter control of the differences between the motors to prevent the mirror from breaking. Finally, an identification technique for the tilt mechanism provides more robustness and takes into account the ageing of the equipment.

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Acknowledgments

For this master study, I would like to thanks first Luc Boucher, my supervisor, for the work we have done together to try to catch a better understanding of the A&G unit, but also for the mounting of the test bench and the several advices he gave to me. Richard Naddaf, the Application Engineer from Delta Tau who came one week for a training on the Power PMAC and gave support by email during all this internship.

The mechanical design of the test bench has been done thanks to Gabriel Perez, mechanical engineer, but also the mechanical workshop of the Tololo Observatory. Hector Swett and Rolando Rogers gave an important help to me in the understanding part, providing access to some spares, to schemes and some support for the technical work.

Xiaoming Hu, my university supervisor, provided me some help for the redaction of this report, the proofreading, the structure of my work and help for the administrative work with KTH.

The forums on the Internet about Octave and the Octave development team for their contribution to all the simulations I have done during this work.

And last but not least, I want to give special thanks to my good friends Alison Centurion and Connor Grooms for their supports and the proofreading they have done.

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List of Figures

2.1 Simplified drawing of a telescope. . . 4

2.2 Two systems of focus widely used in telescopes’ optics. . . 4

2.3 Deformation of a wave emitted by a star. . . 5

2.4 Schematic of the adaptive optic system. . . 6

2.5 Drawings of the A&G system for Gemini. . . 7

3.1 Module 1 . . . 10

3.2 Electrical drawing for module 1. . . 11

3.3 Acquisition camera and PWFS . . . 12

3.4 Module 2 . . . 13

3.5 Electrical drawing for module 2 . . . 14

3.6 Module 3 . . . 15

3.8 Module 4 . . . 17

3.10 New architecture of the controller for the A&G. . . 20

4.1 Lumped-mass model of a telescope. . . 23

4.2 Block diagram of the lumped-mass model for the telescope. . . 25

4.3 Response time as a function of the angle . . . 26

4.4 Block diagram of a controller. . . 26

4.5 Adaptive controller . . . 27

4.6 Block diagram with Kalman filter . . . 28

4.7 Block diagram of the DDP system . . . 31

4.8 Block diagram of the regulation system . . . 32

4.9 Disturbance effect on the performance of the system. . . 33

4.10 Noise effect on the performance of the system. . . 33

4.11 Response time at 95% for the system. . . 34

4.12 Singular values of the system with and without DDP. . . 35

4.13 Response of the system to different wind steps. . . 35

4.14 Maximum acceleration and speed. . . 36

5.1 Motion of the Science Fold . . . 39

5.2 Schematic of the experiment . . . 39

5.3 Identification of the coefficients. . . 42

5.4 Linearisation of the system. . . 43

5.5 RST regulation . . . 44

5.6 Discretisation model. . . 45

5.7 Convergence of estimated parameters. . . 46

5.8 Model trajectory with estimated parameters. . . 46

5.9 The dead band algorithm. . . 47

5.10 Real implementation for 5 estimated θ. . . 47

5.11 Variation of ˆθc 4. . . 48

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List of Figures vi

5.12 Result of the RST regulation. . . 49

6.1 Description of backlash . . . 51

6.2 Schematic of the backlash experiment. . . 52

6.3 Backlash effect on the trajectory of the system. . . 53

6.4 Block diagram of regulation 1. . . 54

6.5 Explanation of the torque/counter-torque mechanism. . . 54

6.8 Algorithm for the adaptive counter-torque. . . 56

6.10 Relative energy used for the different regulations. . . 58

6.11 Modification of the linear stage. . . 60

6.12 Block diagram of cross-gantry control. . . 60

6.13 Block diagram of linear motion regulation. . . 61

6.14 Science Fold transfer function for linear drive. . . 61

C.1 Projection of the new thermal enclosure. . . 71

D.1 First Electrical drawing of the test bench. . . 73

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List of Tables

0.1 List of notations. . . ix

0.2 List of acronyms. . . x

0.1 Requirements for the A&G system of Gemini North and South. . . 7

0.1 Values for the lumped-mass model. . . 24

0.1 Values used for the simulation. . . 32

0.1 Estimation of ki. . . 48

0.1 θˆd _{for different values of d. . . .} ₄₈

0.1 Comparison of the different regulations . . . 57

0.1 Characteristics of a PVT movement for a bell velocity curve. . . 63

0.1 List of motors and encoders. . . 68

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Nomenclature

In this master thesis, the signification of each letter or symbol is summarized before or after the equation. To have a summary of all that will be discussed and an idea of the notations used, the following table gives the general meaning of each symbol listed by order of apparition. The same for the acronym is done after.

Symbol Unit Description

Chapter 4

t95 [s] Time taken by a signal s to reach a final value sf such that

∀t > t95, 0.95|sf| < |s(t)| < 1.05|sf|

Mi - Motor number i

Ji [kg.m2] Inertia of object i

θi [rad, deg, ”, ’] Angle of object i

τ [N.m] Torque

KT [N.m.rad−1] Elastic coefficient of the tube

BT / BD [N.m.rad−1.s−1] Dumping coefficient of the tube / drive

τW [N.m] Torque induced y the wind

fi [Hz] Frequency of the object i

(A, B, C, D) - Matrices of a state space representation

X Matrix 1 column State vector

Y Matrix 1 column Output vector

Z Matrix 1 column Observation vector

u - Input of a system

e - Error of the system (e = y − u)

Gi - Transfer function of a system

s - Continuous differential operator in Laplace Transform

kp, ki, kd - Proportional, integral or derivative coefficients of a PID

R - matrix of power for noise or disturbance

Im(f ) - Image of a function f

Ker(f ) - Kernel of a function f

hA + BF |Si - MCIS containing the subspace S with F a friend of B.

Ω∗

(A,B)(S) - Maximum controlled invariant subspace in S

V∗ _- _Ω∗

(A,B)(Ker(C))

ˆ

θ - Estimation of θ

Pw - Power of the disturbance (without unit)

Chapter 5

t [s] Continuous time

θc _- _{Continuous parameters of the equation}

θd _- _{Discrete parameters of the equation}

ψi - Functions of the state matrix

x Matrix 1 column State matrix

u - Command

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List of Tables ix

Mi - Motor number i

Ei - Encoder number i

Γ [N.m] Torque induced by the motors

Ni [1] Number of teeth for gear i

Js [kg.m2] Global inertia of the system

Jl [kg.m2] Inertia of the load

d [m] Distance between the center of mass of the load and the

sym-metrical axis

m [kg] Mass of the load

g [m.s−2_] _{Gravity constant, g = 9.81m.s}−2

B [N.s.rad−1_] _{Bearing coefficient}

α [rad, deg] Angle of the load

ts [s] Sampling period

k - Number of step (discrete equivalent of t)

P - Convergence matrix

ˆ

θ(k) - Estimation of parameter θ at step k

αe [rad, deg] Equilibrium position

||P || = tr(P PT₎ _- _{Norm of matrix P}

z - Z-Transform variable (z−1 _{is a delay of t}

s) G = B A - Transfer function ξ - Damping factor hαi - Average of α Chapter 6

δ [rad, deg] Angle of backlash

Γm [N.m] Torque induced by the motors

α [rad, deg] Angle of the load

τi [N.m] Torque induced by motor i

Table 0.1: List of symbols or operator used in this book.

Angles can be given in radian (rad), arc minute (’) or arc second (”). The list of acronyms is in alphabetical order.

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List of Tables x

Acronym Description

A & G Acquisition and Guidance Unit

AC Alternative Current

Alt Altitude axis

AO Adaptive Optics

Az Azimuth axis

D Drive of the telescope

DDP Disturbance Decoupling Problem

HRWFS High Resolution Wave Front Sensor

I/O Inputs/Outputs

L Load

LQG Linear Quadratic Gaussian controller

LV Low Voltage

MACRO Fiber Optic System

MCIS Minimal Controlled Invariant Subspace

PMAC Programmable Multi-Axis Controller

PA Probe Arm

PI Proportional Integral controller

PID Proportional Integral Derivative controller

PLC Programmable Logic Controller

PVT Position Velocity Time movement

PWFS Primary Wave Front Sensor

RT Rotary Table

SF Science Fold

T Tube of the telescope

ZOH Zero Order Hold

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Chapter 1

Introduction

This chapter will give a background to better understand the thesis but also clearly states the goals so that the conclusion can answer whether or not the intership has been successful. An outline is also provided.

1.1 Background

The mechanisms of the Acquisition and Guiding unit (A&G) at the Gemini Observatory are controlled, as many other critical mechanisms, with a Generation 1 VME format PMAC Controller. This generation of PMAC controller and the next one (Turbo PMAC familly) are progressively placed out of the production

lines and technical support by Delta-Tau1 _{(the VME format one being the first affected). Even if}

Delta-Tau supported in a case by case basis some sparing for special clients in the past, the recent industrial context (acquisition of Delta-Tau by OMRON in July 2015) will surely consolidate the rationalization of Delta-Tau’s production lines.

The last trade-off study for the A&G Upgrade project in terms of ”Motion Control System Upgrade” led to the last generation of Delta Tau’s PMAC controller with a versatile/expandable architecture. We indeed propose a MACRO RING (fiber optic) architecture of highly integrated Power Brick contollers (controller, amplifier inside a 1.4Ghz real time kernel linux machine) supervised by a POWER PMAC Etherlite. This will not only provide 20 more ”obsolescence free” years to the A&G control architecture system but also optimize its reliability, maintenance and efficiency.

1.2 Goals

The main goals for this thesis are first to identify the problems in the A&G unit of the Gemini South Observatory and clearly list the possible solutions for them. Then, the study of one solution has to be conducted. The purpose is thus to demonstrate the ability to control all the motors of the A&G. To reach this conclusion, simple control of some spare motors has to be done and more advanced functionalities like adaptive control or backlash/skew compensation need to be tested.

Another goal is to provide a functional and easy to use test bench, not only for the test of other spares but also for other teams to be involved in the upgrade process. Then, it is possible to show the compatibility with others systems of the telescope.

This report should also be seen as one of the main references for the development and operation teams of the telescope in the scope of the A&G unit and this controller.

1.3 Outline

This report briefly presents the telescope and focus more on the A&G unit design and current limitations. This will also provide the first drawings of the actual mechanisms with updated and more readable

1

The company selling the controllers used in the observatory.

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CHAPTER 1. INTRODUCTION 2 electrical schemes. This has been done to rationalize future improvements and implementation.

The new generation of Power PMAC enables us to use adaptive control. Then, it will be shown how important this new feature can be for the telescope pointing and guiding system and how to design it. The results for this part are only simulations as it is not possible to use the motors and the mechanical infrastructure of the telescope for experimentation purposes. This provide also a summary of the require-ments and give an overview and a first sight of how the telescope works and how to deal with this highly constrained environment.

The two last chapters will be focused especially on the problems of the A&G. The vibration of some elements is a main concern today because the unit is not easily accessible and then a manual intervention can take a very long time. Adaptive control seen as a parameters estimation will be introduced to solve this issue. A very general equation will be studied. This equation can be found in the tilt mechanism of the telescope for example. Results are shown using both simulations and a mechanical setup aimed at reproducing the movement.

Finally, some known problems as backlash and skew will be solved. These problems are partly due to the lack of power of the actual control system. It will be shown that they can be efficiently solved and improved so that the maintenance work could be easier. Results are direct implementations of the control schemes on some hand made mechanisms.

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Chapter 2

The Fundamentals of a Telescope

A telescope is an instrument that helps astronomers to look at the observable sky. There are two main kinds of telescopes depending on their location: ground or space. In this report, I will focus only the ground based ones and try to draw a simple but comprehensive outline of how a telescope works. The book [3] is the main reference for this chapter.

First, I will briefly describe a ground based telescope and give some examples. Then, I will make a review of the instruments inside Gemini’s telescopes.

2.1 Fundamental of a telescope

Insight into the working

The main purpose of a telescope is to collect radiations from space. They are emitted from cosmic bodies in the form of electromagnetic waves of varying lengths. Longer emitted waves can be studied by telescopes like the Atacama Large Milimeter Array and shorter, or optical wave lengths can be studied by places like Gemini. The latter usually presents the same structure as the Keck Telescope depicted in the figure 2.1.

In the previous figure, it is easy to see the focus mechanism with two mirrors. The focus system can be called Gregorian or Cassegrain depending on the shape of the second mirror. Both of them are shown in figure 2.2. No matter the case, the primary mirror always receives waves from space and focus them on the secondary mirror which transmits the beam to the instruments inside the telescope. The diameter of the mirrors can change depending on what kind of observing the telescope is designed for. A large mirror can look at very distant objects because of its ability to catch more radiation waves. These larger mirrors also have a better resolution than their smaller counterparts. However the bigger the telescope is, the more difficult the construction is. By way of consequence, they are more expensive. This is why there are very few large telescopes. Also, in order to build these large telescopes, political agreements are required from various countries. The large telescopes also lead to new engineering problems.

There are two kinds of mounts for a telescope: the altitude-azimuth (alt-az) or altitude-altitude (alt-alt). I will only describe the first one, more information can be found in [3] for the two systems. See figure 2.1 for an example of an alt-az telescope. The telescope is mounted on a pier and can turn around one axis called azimuth axis. The angle around this axis is denoted β. The altitude axis is related to the elevation ring. This ring can be seen on the drawing between the primary mirror and the tube. It can rotate around one axis and the angle between one parallel of the ground and the normal of the elevation ring is the altitude angle denoted α. The line of sight can then be oriented on the direction of the target. Observing from the Earth has several other problems. Most of the time, there are background sources behind the object of interest that can prevent us from analysing the radiation correctly. Ground based telescopes also have atmospheric deformation problems. There is atmospheric extinction depending on the wavelength but also emission. They usually cannot be ignored for the study as they can have a high impact on the signal over noise ratio. With these spectrum modifications come a refraction effect and addition of turbulences. These turbulences can be driven by many factors including the wind. In figure

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CHAPTER 2. THE FUNDAMENTALS OF A TELESCOPE 4

Figure 2.1: A drawing of the Keck telescope, taken from [25]. Units on this figure will be described later.

(a) Cassegrain (b) Gregorian

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Figure 2.3: Schematic of the difference between the signal emitted by the target body and the one received by the telescope.

2.3, it is possible to have a general idea of the turbulences and distortion of a signal compared to the original one.

Unfortunately, the waves received by the telescope will become more disturbed after reflection on the mirrors. These mirrors can be deformed due to thermal effect and vibrations of the telescope pier. For mirrors bigger than 4 meters, a gradient of the gravity vector ~g will also affect their shape. Then the image will become even more deformed.

Most of these distortion problems can be solved using active and adaptive optics. The first technique will correct the gravitational and thermal effects so that the alignment of mirrors remains the same. Adaptive optics will correct the effect of the atmosphere on the wave. A simple way to understand how it works is to draw a schematic like the one in figure 2.4.

In short, waves focused by the secondary mirror are transmitted to another mirror which can rotate around its center in two directions. The first direction is called tip while the second is tilt. This rotation can correct the gravitational and thermal distortions. It can also deal with noise and internal vibrations of the telescope using a LQG control [19]. The wave is then directed to a deformable mirror. This is a flexible mirror with actuators which can change its shape. Finally, the light is divided in two. One part goes to scientific instruments and the other one goes to a wave front sensor to digitalize it. Then a computer will try to identify the distortion caused by the atmosphere and compute commands to the deformable and tip/tilt mirror to correct it. The tip/tilt modifications are real-time and the shape of deformable mirrors may not be the same between the beginning and the end of the observation sequence.

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Figure 2.4: Schematic of the adaptive optic system. Arrows mean a wave or a signal and boxes are physical components. The wave received by the telescope is directed to another mirror which can do a tip/tilt deformation and to the deformable mirror. Both of them are controlled by the wave front sensor with a computer.

About the Acquisition and Guidance Unit

The intensity of the target can be very low. Thus, to catch some details, the needed exposition time must be high. Then, problems like wind turbulences or rotation of the earth lead to a poor quality image, indeed unusable for astronomers. The mechanism to counter-act these effects is called AG for acquisition and guidance unit.

This unit has several purposes:

1. Do some movements (pointing, guiding and tracking). The pointing seems quite simple. It consists in an alignment between a target and the line of sight. The guiding operation is the use of a celestial body to track motion. That means the guiding object must remain at the same position on the image. The target may never have been studied and its characteristics like diameter, distance from the earth, intensity, etc. are usually unknown. It is then impossible to determine whether or not the target is deformed. Adaptive optics cannot use properties of the target and will be less effective. However, there are very bright stars that have all their characteristics known. They can be used for this guiding and are known as guide stars. A guider is a dedicated object of the telescope aiming at tracking another star in another focal plan. There are some problems with this technique, the star must be bright enough and sometimes, there is no star inside the ”guide star catalogue” in an accessible field of view, preventing the telescope from using it. A laser can also be used to simulate a ”guide star” but this is usually not enough for the correction and fainter, natural guide stars even less bright need to be used.

2. Redirect the light beam. The A&G has mirrors so that the light beam can be directed in other instruments connected to the side of the A&G.

3. Do some active corrections. The focus for cameras is done in the unit and some treatments on the image are also calculated like the use of filters or calibration.

2.2 Insight of Gemini

There are two Gemini’s telescopes. One in the North hemisphere in Hawaii and the other one in Chile close to La Serena. Each of them are built on the same design but they have different instruments. The A&G system: this system is in the first version of each telescope since 1994 [17]. The requirements for this system are presented in table 0.1 [6,26]. We can see that the maximum acceleration is low because

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CHAPTER 2. THE FUNDAMENTALS OF A TELESCOPE 7 Axe Range (o₎ Pointing accuracy Tracking accuracy Max speed Maximum acceleration t95 (s) 1′ ₁₀o _max Az ±270 _{6” − 8”} _0.1” ₃o_/s _0.05o_/s2 < 5 < 30 < 300

Alt 15 − 89.5 faster than az

Table 0.1: Requirements for the A&G system of Gemini North and South.

(a) Modules of the A&G

PWFS No.1

(Module 4)

PWFS No.2

& AO Fold

(Module 3)

Science Fold

Mirror

(Module 2)

HRWFS/AC

(Module 1)

(b) A&G position in the telescope

Figure 2.5: Drawings of the A&G system for Gemini.

of several reasons. The mass of the telescope is very important and the greater the acceleration is, the more energy it requires to stop the rotation. A sudden stopping can damage the gears or motors which will handle a high torque and the demand of energy may not be supplied preventing the deceleration and thus, damage on the telescope can occur. Moreover, the pointing can be slow while it is not done very often and the tracking does not require great variation in speed so the acceleration is low. To understand

the table, t95is the time needed for the angle to go from 0 to the desired position with an error less than

5%.

The A&G system is split into 4 modules which lay one on the other as shown in figure 2.5.

The module 4 is a first wavefront sensor. The third is another wavefront sensor with an adaptive optic mirror. On the module 2 and 3, it is possible to plug instruments on the side. The module 2 is the science fold. It is a tip/tilt mirror with a rotation and translation movement. The light beam can then go to the side-looking instruments or to up-looking ones. The module 1 is an acquisition camera/high resolution

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CHAPTER 2. THE FUNDAMENTALS OF A TELESCOPE 8 wavefront sensor. Chapter 2 goes further into details about the A&G system.

GeMS/ALTAIR: these two instruments are respectively the south/north adaptive optic systems. GMOS: this acronym means Gemini Multi-Object Spectrographs. It is a multi-object spectroscopy and imaging system for the North and South.

Other scientific instruments:

1. Flamingos-2 is a near-infrared wide field imager and multi-object spectrometer

2. GPI is a planet imager for extrasolar planet. It is aimed not only to detect exoplanets but to determine their mass and composition.

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Chapter 3

Acquisition and Guiding Unit

The design of the telescopes began in the early 1990, nearly 25 years ago. The first generation of instruments had been selected in 1994. The control systems were documented by a german subcontrator Carl Zeiss Jena GmbH. So in general, the telescope is an old system and the documentation is sometimes not existing nor updated.

In this chapter, a summary of the A&G unit with drawings of the 4 modules previously discussed will be drawn. Finally, a description of the main objectives for the update will be provided and some problems highlighted.

3.1 Drawings of the system

As explained before, the A&G is split into 4 modules and is situated just under the first mirror as shown previously. In this section, each module will be briefly described and a drawing is done to see how it is organized and how it works. This part is essentially from the documentation [4, 12]. All the motors are described in appendix A.

Module 1: HRWFS and Acquisition Camera

The module is drawn in figure 3.1 and the electrical connections can be seen in figure 3.2. It can be split into 2 main parts: the acquisition camera and the probe arm. This is the last one in the column and there is no rotary table.

The acquisition camera

The acquisition camera with high resolution wave front sensor is depicted in figure 3.3 and is made up of:

• Focus: On linear bearings, there is the focus mechanism. This is driven by a DC motor (M8) with

two limit switches. The internal encoder of the motor helps for the position loop. The zero position is set using an inductive sensor.

• Calibration source: this is an halogen lamp driven accurately. The field stop is on the same slide

with the motor M6. The position is determined using a tape encoder (BB3). The aperture stop

limits the rays that enter the telescope while the field stop limits the extent of the image [3].

• Neutral density and Colour filter wheels: driven by M5and M7respectively.

• HRWFS1 _{collimator /Acquisition camera: driven by M}

3. The encoder head if BB1.

• The pupil image: this one is driven by the motor M4and is on the same slide as M3. This is similar

to the field stop but just before the camera.

1

stands for High Resolution Wave Front Sensor.

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CHAPTER 3. ACQUISITION AND GUIDING UNIT 10

Figure 3.1: Module 1: This is the latest module. It is made up of 2 main elements: the acquisition camera and a probe arm.

Nearly all the motors here have limits and switches to prevent them from hurting another component. To deal with contact between elements of the A&G, four levels of security exist:

1. The control system should deal with collisions and they should never happen.

2. Software limit: In case of problem on the control input or on the system, switches inform the software of a limit and then a command is applied to stop the motor.

3. Hardware limit: another switch directly stops the motor without any intervention of the control system.

4. The physical stop is a mechanical stop that may damage one motor but prevents contact. The probe arm

The probe arm HRWFS is driven by a gearless motor M1. The clamping mechanism driven by M2 is

here to fix the position. There is no servo neither encoder for each clamping system. The purpose of probe arms is to redirect the light beam. So they hold a mirror and can rotate around one axis.

The home and park positions are detected by an inductive switch. The park position is used because it is safer for the mirror in any orientation of the A&G. Moreover, there is an easier access for cleaning or maintenance.

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(a) Acquisition camera (b) PWFS

Figure 3.3: (a) Insight of the HRWFS of module 1. Double arrows show the movement induced by the closest motor. (b) is a PWFS of module 3.

Module 2: Science Fold Mirror

The module 2 is made up of a mirror. For it to move in every direction, there are one rotary table, a linear drive and a tilt mechanism. The mechanical drawing is in figure 3.4 while the electrical one is in figure 3.5. A more advanced description of this module is provided in chapters 5 and 6.2.

The rotary table

The rotary table is driven by two motors, one is controlled in speed while the other has a constant torque for each action (tracking or pointing). Their own movements are opposite, this is the torque/counter torque mechanism. This mechanism has some advantages and drawbacks [9]:

• it prevents motors from working around the 0 torque area which is less accurate.

• for pointing, movements are larger and then the two motors can move together to be more efficient. • for tracking, the counter torque is high enough so that no loss of contact between gears should

happen, this is the anti-backlash system. • the mechanical loads and stresses may be high.

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Figure 3.4: Module 2

• there is a need to differentiate which mode is used (pointing, tracking, guiding)

The expected range is ±220o_{. The same mechanism is used for the rotation of probe arms.}

In module 2, the motors are M12 and M13. This movement can be called tip of the science fold.

The linear drive

The linear drive moves the science fold mirror from the park position to the optical beam. There is only

one motor M9driving two belts which move the two motors M10 and M11. The position is obtained on

one slide using a tape BB4 and with the help of the velocity encoder included in M9.

The use of one motor is to be sure that the two slides move symmetrically. But the use of only one encoder and one motor can lead to issues. If the non controlled slide is stuck by something, there is a risk of breaking the science fold mirror due to high stresses the mirror structure. Another problem visible on the electrical drawing of figure 3.5 in the tilt part is the use of only one amplifier for two motors. Tilt movement

This movement is useful to send the light beam into the instruments. It is a rotation of the mirror.

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Figure 3.5: Electrical drawing for module 2

improvement with a new controller will be to decouple the two motors and use a follow mode between them.

Module 3: PWFS2 and AO-Fold

Module 3 as shown in figure 3.6 is made up of one probe arm as explained in module 1. The motor for the

rotation is M17, there is no counter torque there. The other element in this module is a PWFS as shown

in the picture 3.3. It is really similar to the acquisition camera with a torque/counter torque mechanism

for the rotation. The motor M17 for the rotation of the AO-fold is a preloaded AC gear motor. Two

arms must share the same space but one can move only if the other one is in home position. The electrical drawing is presented in figure 3.7.

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Figure 3.6: Module 3

Module 4: PWFS1

This module contains only a rotary table with motors M28 and M29, another PWFS and a detector

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3.2 Reasons and Purposes for the Upgrade

Main Reasons

The A&G upgrade started around 2010. Obsolescence, weakening and overload are examples of problems the company is facing today. Because the A&G is required for most observations, these critical units must be replaced.

Obsolescence Lots of motors initially used are not produced any more. New motors have been chosen but compatibility problems can occur with old parts of the telescope. This is also the case for the controllers. Bugs are known on some libraries but without a new architecture, programmers must deal with them. New engineers are not aware of these old technologies and may not be as efficient as they would have been with more modern controllers. Moreover, the code used to drive motors is written in a very low level language called PLC. Every program runs in sequences with a datum algorithm which is not the most efficient. It requires a times to understand what is happening and which variable is linked to which motor.

Weakening Because of wear and tear, these heavily used units require some days off each year for corrective and preventive maintenance. The backlash is one of the reason of these maintenance work. The anti-backlash system is described in chapter 6.1.

Overload The actual controller has a limited amount of external connections. There are more

mecha-nisms than servo channels and some motors are then not directly plugged to the controller and share the same amplifier. This structure leads to an overload of some electronic devices even if everything looks fine from the user point of view.

Space and complexity The actual system is made up of electronic cards. They are stored in cases called thermal enclosures. As we can see on the electrical drawings of this chapter, the controllers talk to amplifiers which drive the motors. Some analogue to digital converter and interpolation boxes are needed. This organization is complex and took a lot of space. This is also difficult to have a comprehensive look at drawings.

New Architecture

These problems are a main concern and they must be solved soon. To do so, the company wishes to use a new generation of controller. But changing the whole system is an important upgrade and has to be done in several steps. A study of which controller should be used has been conducted a few years ago [10] and resulted in a choice of a central controller (Power PMAC Etherlite) with some bricks (Power Brick AC or LV) which can drive several axis at the same time. They are made by the same company as the actual ones: Delta Tau. The scheme of the new architecture is shown in figure 3.10.

In the figure 3.10, one can see that the Power PMAC Etherlite is the master of the ring of controllers. The ring is closed using the MACRO technology which is a fiber optic protocol. Users can send commands to it and it talks to the bricks. Each brick is configured in link with the motors (axes) it will drive. These bricks can be considered as amplifiers also so that each motor has its own characteristic taken in charge

by the associated brick. 3 Power Brick AC may be needed to drive up to 202_{AC motors while 2 Power}

Brick LV driving up to 16 ”small” motors are needed. An extra Power Controller to receive the data of tape encoders is also required.

In the appendix A, a list of motors is proposed. There are 15 AC/power motors controlled with an AC brick and 14 small mechanisms. But there are at least 32 encoders, the use of a Power Controller to connect the extra encoders is a solution. Then, the Power Etherlite will link each encoder to the correct motor, no matter on which brick it is connected. Care must be taken when motors will be assigned to

2

The third Power Brick AC can be a 4, 6 or 8 axis. Then 20 motors can be driven for a 4 axis Power Brick AC, 22 if it is a 6 axis and 24 for an 8 axis.

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Figure 3.10: New architecture of the controller for the A&G.

a brick so that in case of a MACRO Ring failure, each brick can behave independently and park all the instruments.

With such an architecture, the expected benefits are [10]: 1. High level of integration.

2. Simplify the wiring (please see appendix C. 3. Increase reliability.

4. Ease maintenance of the thermal enclosure. 5. Tackle the obsolescence issue.

6. Have a full access to telemetry.

7. More analogue and digital I/O to communicate.

Other Improvements

Other improvements can be done to correct some recurrent problems. They are all discussed in the following chapters:

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2. Use the increased number of axis to control M10and M11 independently (chap. 5).

3. Improve the backlash compensation (chap. 6.1).

Validation

This new architecture has been studied in the Instrument Lab of SBF3_{. The test bench that has been}

realised uses 1 Power Brick AC and 1 Power PMAC Etherlite. These controllers have been chosen so that the fiber optic ring can be tested and small or big mechanisms can be tested also. Indeed, the Power Brick AC is designed to be used with high power motors or AC driven but the small mechanisms (like focus or filters) can also be studied even if they will not be controlled in a real implementation with this kind of controller.

The test bench is described more in the appendix D. It is divided into two parts. The controller part has been designed to be easily moved and fully operational. DIN rails for an easier integration and connection are the main connectors. Some outputs have been wired to test as many functions as possible. The second part is the motor ones. In this test bench, the following motors/encoders have been tested:

1. Harmonic Drive AC with its own digital encoder: used inside the probe arms and rotary table. This is the main high power motor.

2. Faulhaber with a magnetic encoder: for the linear drive of the science fold. 3. Faulhaber with an optical encoder: for small mechanisms.

4. ROD 280: a high resolution sinusoidal encoder for probe arms. It is used with an interpolation box in the A&G and with the internal interpolation of the Power Brick in the test bench.

5. Devantech motors: these motors are not in the A&G but they are useful for experimentation purposes because high power and cheap.

All this has been successfully tuned, driven and controlled [2]. That means 27 motors over the 29 of the A&G can be used with this controller. The two last are Maccon used in the tilt mechanism of the science

fold (M10and M11). There should be no problem with them. The tape encoder for position has not been

tested yet because it is highly expensive to get a spare. This is a high precision position encoder (LIDA 10C) but it should be readable. The only concern is about the Heidenhain 11µApp. This is a current encoder and further studies should be done to read it but Richard Naddaf told us that a resistor and an adapted choice of voltage should make this encoder readable.

As a conclusion for this part. A test bench had been realised to test most of the motors and encoders of the A&G. Some works still need to be done for compatibility of very specific products but the versatility of the Power Brick AC will give positive results.

3

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Chapter 4

Variable Structure Law Controller for

Pointing and Guiding

This chapter will focus on the control design for the pointing and guiding mechanism of the A&G. This design is aimed at showing that adaptive control seen as a variable structure law controller can be interesting. Indeed, it is possible with the new controller to use such a feature.

Firstly, a modelling of the telescope is explained. A variable structure law controller is defined and the context of its utilization will be justified Thus, a methodology usable in most cases is explained and applied to the telescope. Finally, some results and a discussion are proposed to complete the chapter.

4.1 Modelling of a telescope

Background

A LTI1_{system is an object which reacts linearly to an input. A control problem is a modification of the}

input so that the final system will behave as expected. Here for example, a telescope can be considered as a first approximation as a LTI with two inputs that correspond to the torque on the drive and the torque implied by the wind. The output will be the position of the tube. For big telescopes, the wind turbulence cannot be avoided as it has an important impact on the position of the tube.

A possible modelling of the figure 2.1 uses a lumped mass model. That means the telescope is compared to masses linked by springs. There are five notable advantages to a lump mass system, these advantages consist of [3]:

1. the model is simple allowing for a fast data processing.

2. by increasing the number of masses, it is possible to have more dominant frequencies. As a controller does not deal with more than 3 or 4, the model will remain simple and close to the expected result. 3. it can model the global evolution of the system if the local ones do not have too much impact on

the final behaviour. For a telescope, the structure is stiff enough to not have a prevalent effect. 4. once the lumped mass model is created, it can be easily used for other telescopes because just by

changing values of masses, dumping and elastic coefficients.

5. it is also possible to identify the sense of each spring and mass so that the addition of non-linearities such as backlash, saturation or time-delay can be made.

Nevertheless, the controllers that will be designed on this model cannot be applied directly on the telescope. Once the result is ”good enough”, a medium-size lumped-mass model can be derived. The stiffness is taken into account with a better modelling of internal objects. The last step is then a full simulation. Each time, the controller needs to be adjusted to provide the correct telescope performances.

1

for Linear Time Invariant

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CHAPTER 4. POINTING AND GUIDING CONTROL 23

Figure 4.1: Lumped-mass model of a telescope. The first mass D is the drive and θDis its position. BD

is the dumping coefficient between the ground and the drive. τ is the torque provided by the motors to

the drive. JDis the inertia of the drive. T is the tube and θT is its position. KT and BT are the elastic

and dumping coefficient of the mechanical linkage between the tube and the the drive respectively. τW is

the torque applied on the tube by the wind. The dumping coefficient represents Coulomb friction while K is for the flexibility of the telescope.

Using [25], a first order lumped-mass model of a telescope will have the equivalent shown on figure 4.1. Two modellings will be made, one for the azimuth and another for the altitude.

It is possible to see on this modelling that the drive and the tube are two different mechanical components. The oscillation of the tube and its influence on the drive are the essential characteristic that will be studied. A mechanical study can be made using for example Lagrangian mechanics and the equation for this system is [25]:

τ = JDθ¨D+ BD˙θD+ BT( ˙θD− ˙θT) + KT(θD− θT)

τW = JTθ¨T+ BL( ˙θT − ˙θD) + KL(θT − θD)

(4.1)

A state space representation can then be obtained with X = θD ˙θD θT ˙θT :

                                 ˙ X =     0 1 0 0 −KT JD − BT+BD JD KT JD BT JD 0 0 0 1 KT JT BT JT − KT JT − BT JT     | {z } A X +     0 1 JD 0 0     | {z } B τ +     0 0 0 1 JT     | {z } P τW y =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1     | {z } C X (4.2)

This model will be called model 1 in all this chapter.

Measurable States

Such a lumped model is not documented for the Gemini’s telescope. Values from [25] of the Keck telescope

will be used. Table 0.1 presents the necessary values used here. It can be easily adapted if the inertias JT,

JB, BD and BT are known2. Indeed, there are two resonances fT and fD which can be experimentally

2

This is not really an issue as inertias can be obtained using a finite elements modelling and the dumping coefficients are linked to Coulomb friction and depends on the material and lubrication used in gears and motors [3].

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Parameters [units] Azimuth Elevation

JD [kg.m2] 829 17.2

JT [kg.m2] 192 1358

KT [N.m.rad−1] 5.27e5 1.413e6

BD [N.m.rad−1.s−1] 7.3 17.4

BT [N.m.rad−1.s−1] 1.95 8.8

Table 0.1: Values for the lumped-mass model obtained from [25].

measured. They are obtained by the vibrations of the tube and drive respectively when motors are locked.

fT and fDare obtained by the following set of equations [3]:

   fT =_2π1 q KT JT fD= _2π1 q KT JT + KD JD (4.3)

Then, elastic coefficients can be estimated and all the parameters in the model are known.

The vector X is not fully measurable. That means here that only part of the X vector can be measured

by captors. A tachometer and an encoder can measure θD and ˙θD. An accelerometer on the tube can

measure ¨θT. To take this into account, the modelling of the previous part must be adapted. A transfer

function G which is a matrix 4x2 can be obtained using the formula G = (sI4− A)−1 B P with s

the differential operator as defined in [11] and B P is the concatenation of B and P . This transfer

function is between u = τ τW

T

and X. Then, it is possible to derive a new transfer function G2

such that the acceleration of the tube ¨θT = s ˙θT can be obtained:

G2= G sG(4, :) (4.4)

G(4, :) means the columns of the 4th _{line of G.}

The vector u is made of τ which is the control command and τW which is the possibly known wind

turbulence. To take into account the robustness a third input can be added to the model. An unknown

disturbance w which affects each line of G2can model uncertainties [11]. w is a white noise of intensity

Rw. The accelerometer used in Gemini as a color spectrum [22, 23] which can be modelled with the

following transfer function Gw:

Gw=10

−6_{(s − 120π)}

s (4.5)

Then, a new transfer function G3 is derived with1n,m a n − m matrix full of one:

G3= G 14,1 sG(4, :) Gw (4.6) Some noise n can be added to the three outputs. This noise is supposed to be a white one with intensity

matrix R2.

To work properly with this system, it is possible to compute a minimal state-space representation sys3 =

(A3, B3, C3, D3). For more information, [11] can be used. C3∗ is computed so that the only outputs are

θD, ˙θDand ¨θT. Then C3∗= C3((1, 2, 5), :) where C3((1, 2, 5), :) is a new matrix with 3 lines corresponding

to the lines 1, 2 and 5 of C3.

It is possible to check that D3 and D2 are null on the output y for this case. The full output is then:

y = C3(1 : 4, :)X3= CX (4.7)

with X from equation 4.2 and X3the state vector from model 3. As C = I4, that means:

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Telescope dynamic

Figure 4.2: Block diagram of the lumped-mass model for the telescope. The cross circle is an addition of

two signals. n = n1 n2 n3

T

is a white noise of intensity R2 while ω is a white noise of intensity

Rw. Axe Range (o₎ Pointing accuracy Tracking accuracy Max speed Maximum acceleration t95(s) 1′ ₁₀o _max Az ±270 _{6” − 8”} _0.1” ₃o_/s _0.05o_/s2 < 5 < 30 < 300

Alt 15 − 89.5 faster than az

Table 0.1: The table of requirement for the A&G (repeated from page 7).

That there is a change of basis from 3 to 1 that can be modelled by the non square matrix C3(1 : 4, :) =

P3→1.

To conclude, one extra feature can be added: saturation of the motors. Indeed, in order to do not damage the motors, there is a saturation of the current that leads to a torque limits. This one has been set to 200 N.m per motor [3]. With a gear ratio of 3 and 8 motors, that implies a torque τ in the range

[−4800; 4800] = [−τmax; τmax]. Saturation is not a linear phenomenon so it must be controlled separately.

Figure 4.2 is a summary of this part.

4.2 A Variable Structure Law Controller

The purpose of this chapter is to find the control τ = C(s)X + u with C a matrix if necessary so that the requirements summarized in table 0.1 could be fulfilled. The two next sections will define the process

used to find the matrix C such that θT will equal u with the requirements specified in table 0.1.

Definitions

PID

Lots of control problems can be solved by using a proportional integrator derivative (PID) controller. Such a controller can be used in a feedback loop. A scheme of this concept is presented in figure 4.4. The output y can be seen as the product of G and τ . Then, the final equation that links u and y is

y = (1 + GC)−1_{GCu. The purpose of C is to help G to reduce the difference between y and u. For a}

PID, C has a specific shape C(s) = kp+ ki1_s+ kds.

The kp is directly linked to the speed of the system. s−1 has the role of an integrator here so s−1e is

the integration of e. An integration can be useful to make the system faster and without static error. Indeed, such a controller will increase τ if y < u. It will stop increasing if y is oscillating around u.

Then, an integrator can increase the oscillation but lead to a faster system. kd is the derivative gain.

Due to oscillations of the telescope and uncertainties on the signal due to noise, a derivation can have a destabilizing effect. It will not be used in this chapter.

There are methods which can help to determine kp, kiand kdso that the requirements are fulfilled. This

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CHAPTER 4. POINTING AND GUIDING CONTROL 26 0 50 100 150 200 250 300 350 101 ₁₀2 ₁₀3 ₁₀4 ₁₀5 ₁₀6 ₁₀7 T im e (s ) Angle (”)

Time requirement for the pointing system

Requirements Square root fitting curve

Figure 4.3: Plots of the response time as a function of the command. It can be easily seen that this not linear and fits more to a square root curve.

Figure 4.4: Block diagram of a controller C(s) with a system G. u is the reference signal and y is the

output. Here, u will be given by the user of the telescope to point the desired star while y = θT will be

the position of the tube. e = y − u is the error, that means the difference between the requested angle u and the real angle y. τ is the signal computed after C which is the controller. tau is also the input in the system, here the torque delivered by the motors.

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Figure 4.5: Block diagram of an adaptive controller C(s) with a system G. u is the reference signal and

y is the output. Here, u will be given by the user of the telescope to point the desired star while y = θT

will be the position of the tube. τ is the signal computed after C which is the controller. tau is also the input in the system, here the torque delivered by the motors.

Adaptive Control

Adaptive control is a kind of control which require not only e but y and u. The simplified block diagram of an adaptive control is shown in figure 4.5.

In the next chapter adaptive control will be intensively used but here, the focus will be on a special type of adaptive controller, the structure variable law ones. They are PID but gains can vary depending on the input. It can then become a non-linear controller. A general structure of C(s) seen as a structure

variable law controller is C(s) = kp(u) + ki(u)1_s+ kd(u)s.

Pros and Cons of a Variable Structure Law Controller

Such a structure can be useful in different situations. To fulfil a complex set of requirements

In the table 0.1 (page 25), it is easy to see that a linear controller will not be optimal. Indeed, if the

system is linear, then the multiplicative properties imply that all the t95 will remain the same whatever

the input is. Then, if one wants to fulfil all the requirements with a linear controller, then t95 will be

5s. For an amplitude of 270o_{, by applying the mean value theorem, the acceleration will be at a time}

tc∈ [0; 5] equals to 270₂₅ = 10.8o/s2which is a lot more than the maximum allowed.

Then, a linear controller cannot be designed and a structure variable controller seems a good idea as it is a simple non-linear controller.

To make a system more stable

It appears that non saturation of the input (here τ ) is a classical problem [21]. To prevent it from growing to much, the integral part of the PID can be removed but then, the 0 steady state error may never be achieved. One solution may be to use an anti-windup system. The integral effect is saturated if it exceeds a specific value. Such a controller perform usually quite well in the literature [21].

A variable structure controller can deal with this problem by playing on the coefficient Ki. For important

values of the error, the integration coefficient will be small and get bigger as the error decreases. Then, the integral effect is naturally saturated.

For a telescope

Another solution to this problem should be to design different controllers, each optimized in its field of action. The problem with that is the manual switch that occurs when the error between the reference angle and tube angle is ”small enough”. In case of wind, the switch can occur often if the angle is too far from its reference position. To counteract this phenomenon, a simple solution is the structure variable controller which can adapt without human intervention [21].

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Telescope dynamic

Kalman Filter

Figure 4.6: Block diagram with Kalman filter for the estimation of X and outputs.

4.3 Methodology and implementation

Now, the telescope system can be studied. This part will only deal with the azimuth control but a similar methodology can be applied to all the systems. First, a Kalman filter will be designed so that one can use the simple model 1. To decrease the influence of the wind, a disturbance decoupling problem will be studied. Finally, a position and speed proportional integral controller will be designed.

State reconstruction

To get rid of the noise on the output, correct the disturbance on the accelerometer and evaluate the

position of the tube θT, a Kalman filter can be used. This is a linear system that will reconstruct the

X variable and estimate the output according to the measurable outputs (¨θT, ˙θD, θD) and the inputs,

here τ and τW. The choice of τW as an input will be discussed later. The Kalman filter requires also

a knowledge about intensities of white noises. The priori intensities must be higher so that the Kalman filter will be less sensitive to noise [11]. But, the higher they are, the slower the approximation will be.

As our system is quite slow and the instruments are designed to be precise, the intensities of noise R2

are not high. The disturbance intensity Rw represents the fidelity of the model. To test the robustness,

several values of Rwhave to be tested. The cross coupling coefficients between noises and the disturbance

are assumed to be 0. Then, R2is a diagonal matrix of rank 3 while Rwis a strictly positive real number.

For the Kalman filter to be optimal in the sense of minimization of the variance, some assumptions need to be checked [11]:

1. R2 is symmetric positive definite which is the case.

2. ˜R = Rw is positive semi-definite. This affirmation is also true.

3. (A3, C3∗) is observable.

The condition 3 does not always need to be fulfilled. Indeed, if (A3, C3∗) is not observable, the Ricati

equation that leads to the Kalman filter will have more than one positive definite solution but there is only

one leading to the optimal observer. In this case, the observer always exists and X3can be reconstructed.

To get X, it is possible to use P3→1 defined in the previous part. By denoting with ˆ˙ the estimation of

one variable, the figure 4.6 will summarize this subsection.

The Disturbance Decoupling Problem

The disturbance decoupling problem (DDP) [15] is to find F as defined below such that the output y is

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CHAPTER 4. POINTING AND GUIDING CONTROL 29 Geometric Control

All this subsection is based on the book [15]. To keep it simple, the system on which I will work is the first one described by the set of equation 4.2.

For a linear system (A, B, C, D), an invariant subspace is the set ΩA of all X so that Ax remains in

ΩA. An invariant-controlled subspace Ω(A,B)(F ) is a generalized invariant subspace extended to the set

(A, B) with the feedback matrix F . For all X in Ω(A,B)(F ), (A + BF )X is also in Ω(A,B)(F ).

If it is possible to compute Ω(A,B)(F ) for a given F , this set will represent all the possible X the system

can experience. This powerful definition can help to choose F such that the X can remain in a specific set V for example. In other words, more mathematically, that means the minimal controlled invariant subspace (MCIS) containing a generating set, usually Im(B) is included in V. To do so, we will denote hA + BF |Im(B)i the MCIS containing Im(B).

To solve the disturbance decoupling problem, one wants the MCIS generated by the disturbance so here Im(P ) to be included into Ker(C). That means find a F so that all the X generated by w are inside Ker(C). Then, they have no influence on the output y. This is a solution of the DDP: hA + BF |Im(P )i ⊆ Ker(C). This condition is equivalent [15] to another one:

Im(E) ⊆ V∗ (4.9)

where V∗_{= Ω}∗

(A,B)(Ker(C)) represents the maximum controlled invariant subspace in Ker(C).

The problem is that all F cannot be chosen as some may lead to an unstable system. It is possible to compute the reachability subspace which is the subspace of all X such that one can choose the poles of the system. In other word, there is enough degree of freedom on F so that the eigenvalues of A + BF can be chosen arbitrarily. We will assume here that a subspace S is reachable if and only if hA + BF |Im(B) ∩ Si = S. That means the maximum controlled invariant subspace generated by Im(B) ∩ S is the subspace itself. For further information on the reachability subspace, please see [15]. Saying that hA + BF |Im(P )i must be a reachable subspace is the same as:

Im(E) ⊆ R∗⊆ V∗ (4.10)

where R∗_{is the maximal controllability subspace included in V}∗_{. This condition is a solution of the DDP}

and one can choose the poles of the system. Then, it is possible to make the system stable. The condition expressed by equation 4.10 is sufficient for the DDP with stability.

Nevertheless, this is a very strong condition and it is often possible to solve the DDP without the reachability condition and still be stable. But the subset that will allow these condition may not be linear. This is the boundary of this theory.

In short, if condition 4.10 is true, then the DDP has a solution. If not, then one must check if condition 4.9 is correct. If it is, then a search for a stable solution can be done manually and the DDP is solvable. If 4.9 is not fulfilled, the problem is not solvable.

DDP on the telescope

As we can see from the previous section, the matrix C plays a great role. With the Kalman filter we have made last time, it is possible to shape this matrix the way we want to influence the result of the DDP. Of course, the choice of C as the identity matrix means that the disturbance does not affect the telescope at all. It seems unrealistic and further calculation will show that it is not possible. The choice of C = 0 0 1 0 0 0 0 1 will lead to y = θ_˙θT T

. Apply the DDP on this would say there is no influence of the wind on the tube but only on the drive. This would be the ideal solution. That means simply that effect of the wind on the tube will be immediately compensated by a new position on the drive. But this is proven to be infeasible by means of calculations. One thing that can still be interesting is the effects of the wind on the drive. We can assume that the structure is good enough so that the position of the drive should stay close to the position of the tube. This implies:

C = 1 0 0 0 0 1 0 0 and y = θ_˙θD D (4.11)

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A simple way to find V∗ _{is to do step by step. V}

0 = Ker(C) = Span(e3, e4) where (ei)i∈[1,4] is the

canonical basis of R4_{. Then, it is possible to find V}

1: V1= {X ∈ Ker(C)|AX ∈ V0+ Im(B)} A e3 e4 =     0 0 KT JD BT JD 0 1 −KT JT − BT JT    ∈ (V0+ Im(B)) 2 (4.12)

Then, V1= V0, that means V∗= Ker(C) and Im(E) ∈ V∗ so the DDP is solvable.

It is also possible to compute R∗ = hA + BF |Im(B) ∩ V∗i. Im(B) ∩ V∗ = 0 leads to R∗ = 0. It is not

possible to choose the pole but we can check the stability and try to find one possible solution.

To do so, eigenvalues of the matrix A − BF must be negative. Then, F as to be found so that V∗ _{is a}

controlled invariant subspace. That means (A + BF )V∗ _{⊂ V}∗_{with F =} _f

1 f2 f3 f4 and F ∈ R4_, in other words: (A + BF ) e3 e4 =     0 1 0 0 −KT JD + f1 JD − BT+BD JD + f2 JD KT JD + f3 JD BT JD + f4 JD 0 0 0 1 KT JT BT JT − KT JT − BT JT     e3 e4 =     0 0 KT JD + f3 JD BT JD + f4 JD 0 1 −KT JT − BT JT    ∈ Span e3, e4 ⇒    f1, f2∈ R f3= −KT f4= −BT (4.13)

Then, it is possible to compute characteristic polynomial χA+BF(X) of A + BF :

χA+BF(X) = |XI4− A − BF | = X −1 KT−f1 JD X + BT+BD−f2 JD · X −1 KT JT X + BT JT =X2₊BT+BD−f2 JD + KT−f1 JT X2₊BT JT X + KT JT (4.14)

Then, as BJ is small, the second polynomial as two complex roots with a negative real part −B_J_TT. There

are two others values and the Routh-Hurwitz criterion [1] will help to find conditions on f1 and f2 to

make the system stable if possible. That leads to the following set of equations:

BT + BD− f2>0

KT− f1>0 (4.15)

The choice of f1= KT− ε and f2= −BT− BL− ε with ε a positive number will satisfy the inequalities

and then make a stable DDP system.

Figure 4.7 is a summary of this section. A static gain K has to be applied on the input to scale the final

value [11]. To do so, a bode plot of the system can be drawn and the static gain G0 of the system with

feedback will be the inverse of K so that K = G−10 . If A† denotes the pseudo-inverse of A, then another

solution is the computation of the matrix K = C(BF + A)−1_B† _{that should give the same result.}

Speed and position PI

To reach the corrected final angle, the speed and the position must be controlled. A proportional integral controller (PI) is used for each of them.

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Telescope dynamic

Kalman Filter

Figure 4.7: Block diagram of the DDP. The reference signal is then ˙θc and is here the speed reference.

Then, ˙θc−ˆ˙θD −−−→

t→∞ 0. For others drawing to be simple, a new system called Geq is created. It will

represent the solution of the DDP problem with Kalman filter and represent here the dashed system. Speed

The PI controller for speed is found manually. It must converge without steady-state error and the time

at 95% must faster than the t95 of the position.

A structure variable controller will not be used in this case. There is no need for it as the reaction in speed can be as fast as possible. The only noticeable problem will be the acceleration limit that may reach its limits. This will be discussed later.

Coefficients Kspeed

p and Kispeed can be adapted manually to find the curve that fits ou the requirements

the best. Then, P Ispeed = Kpspeed+ K

speed i

1 s.

Position

The position needs to be an adaptive controller for all the reasons expressed in the previous section. The

strategy used here is quite simple. The design of two PI will be made, one for 270o_{and the other one for}

60 arc-seconds. Then as advised in [21], a structure for the variable controller will be as follows where k can be the proportional or integral gain:

( k(e) = k0 1+c|e|n n = loghK0 Kf − 1 c−1i_{log 220} π 180 (4.16)

where K0 is the gain for small angles, Kf refers to the larger angles and c is a coefficient to be chosen

according to how fast the transition should be. An index p and i will denote the proportional and integral

gain. Then, using the linear notation, P Ipos= kp(e) + ki(e)1_s. The design with the two PI is drawn on

figure 4.8.

Acceleration requirement

Such a system has one important issue. The maximum acceleration may be too high. This is mainly due to a step in the reference signal. Some simulations will show that the maximum acceleration of the tube is reached in the 10 first seconds. One way of dealing with that can be a low-pass filter such as a feedforward controller. Another solution is a modification of the speed coefficient so that big differences in speed will be damped while small differences still remain undamped. A possible solution is a modification of the controller to be time dependent.

The new controller denoted kn _{is then:}

kn(e, t) = 1.15 · k(e) 1 − exp − t

τa

p |e|

!!

A New Controller for the Acquisition and Guiding Unit for the Gemini South Telescope: New architecture and control schemes to use Power PMAC

IN

DEGREE PROJECT

VEHICLE ENGINEERING, SECOND CYCLE,

30 CREDITS

,

STOCKHOLM SWEDEN 2016

A New Controller for the

Acquisition and Guiding Unit for

the Gemini South Telescope

New architecture and control schemes to use

Power PMAC

MATTHIEU BARREAU

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

A New Controller for the Acquisition

and Guiding Unit for the Gemini South

Telescope

New architecture and control schemes to use Power PMAC

M A T T H I E U B A R R E A U

Abstract

Acknowledgments

Table of contents

List of Figures

List of Tables

Nomenclature

Chapter 1

Introduction

1.1

Background

1.2

Goals

1.3

Outline

Chapter 2

The Fundamentals of a Telescope

2.1

Fundamental of a telescope

Insight into the working

About the Acquisition and Guidance Unit

2.2

Insight of Gemini

PWFS No.1

PWFS No.2

& AO Fold

Science Fold

Mirror

HRWFS/AC

Chapter 3

Acquisition and Guiding Unit

3.1

Drawings of the system

Module 1: HRWFS and Acquisition Camera

Module 2: Science Fold Mirror

Module 3: PWFS2 and AO-Fold

Module 4: PWFS1

3.2

Reasons and Purposes for the Upgrade

Main Reasons

New Architecture

Other Improvements

Validation

Chapter 4

Variable Structure Law Controller for

Pointing and Guiding

4.1

Modelling of a telescope

Background

Measurable States

4.2

A Variable Structure Law Controller

Definitions

Pros and Cons of a Variable Structure Law Controller

4.3

Methodology and implementation

State reconstruction

The Disturbance Decoupling Problem

Speed and position PI

Acceleration requirement