Development of Adaptive Optics Control Algorithms for Free Space Optical Communications

Development of Adaptive Optics Control

Algorithms for Free Space Optical

Communications

Joana Sofia do Sul da Mota Torres

Prof. Doutor Fernando Gomes de Almeida Mechanical Engineering Department

Supervisor at DLR: Dr. Andrew Paul Reeves

Institute for Communications and Navigation

A thesis submitted in fulfillment of the requirements for the degree of Master in Engineering

in the

Master in Mechanical Engineering

You are an aperture through which the universe is looking at and exploring itself.

Resumo

A tecnologia de comunicações óticas de espaço livre (Free Space Optical Communications ou FSOC) é necessária para assegurar ligações de elevado débito à próxima geração de satélites. A performance dos sistemas FSOC é afetada pela turbulência na atmosfera terrestre, que distorce o feixe de luz, causando cintilação na receção, o que limita severamente a taxa de transferência de informação. Em Astronomia, a atenuação dos efeitos da turbulência é conseguida através de sistemas de ótica adaptativa, que fazem uso de espelhos deformáveis para corrigir a distorção de uma frente de onda recebida. A mesma técnica pode ser utilizada em comunicação para minimizar a influência da turbulência e corrigir a frente de onda previamente ao acoplamento da luz na fibra ótica. No entanto, o controlo do loop de ótica adaptativa deve assegurar robustez de funcionamento em condições de turbulência extrema. Nesta dissertação, várias técnicas de controlo utilizadas em Astronomia para o loop de ótica adaptativa foram revistas em termos de adequação aos requisitos funcionais de comunicações óticas em espaço livre. Compararam-se qualitativamente as técnicas revistas para promover a escolha de uma alternativa de controlo. Optou-se por enveredar por Machine Learning de modo a prever a influência da turbulência na fase da frente de onda com base em dados reais de funcionamento recolhidos em campanha de medição com o demonstrador THRUST. A performance de redes neuronais foi comparada à de um modelo de regressão linear múltipla. Foi provado que ambos são úteis para previsão da distorção de fase na frente de onda.

Abstract

Free Space Optical Communications (FSOC) are required to provide high throughput data links to the next generation of satellites. The performance of FSOC systems is hindered by the turbulence in the atmosphere, which distorts the light beam causing scintillation at the receiver and hence severely limiting available data throughput. In Astronomy, turbulence mitigation is achieved via Adaptive Optics (AO) systems that use a deformable mirror to correct distortion in the incoming wavefront. The same technique can be used in communications to address the influence of turbulence and correct the wavefront before coupling the beam into the fiber. However, control of the adaptive optics closed loop must ensure robustness of communication in fast-changing turbulent conditions. In this dissertation, several control techniques used in AO for Astronomy are reviewed for feasibility in the context of FSOC requirements. A qualitative comparison is drawn between selected controllers in order to choose an appropriate alternative. A Machine Learning Control approach has been chosen to predict turbulence based on real-life data collected with the THRUST Demonstrator. The performance of neural networks was compared to a multiple linear regression model. It was proven both models are useful for prediction of wavefront distortion.

Acknowledgements

First and foremost, the most heartfelt thank you goes to my parents, for always being the wind behind my sails, especially through recent storms. I could never have done this without their support (and the occasional photos of my cats progressively destroying the house). An extended thank you goes to all my close family for being the best I could possibly ask for.

A very special thank you to Dr. Fernando Gomes de Almeida, whom I admire deeply, for the close guidance, even from a distance.

I am grateful beyond words to both Dr. Andrew Reeves and Dr. Rámon Mata-Calvo for the opportunity to work at the German Aerospace Center (DLR) in such a compelling topic. I am incredibly honored to have been a part of the amazing work developed at the Institute of Communications and Navigation. An extended thank you to all the AOT and COS family, not only for contributing towards a warm and healthy work environment, but also - of course! - for all the breakfasts we shared. I will forever appreciate how much I learned and grew as an individual while working here the last few months. I am looking forward to future endeavors at KN.

I cannot go without mentioning my office mates, and the cooperative, fun and produc-tive environment I was lucky enough to be a part of. Every single one has made an impact - quite literally , if we consider the occasional rubberband fights. A special thank you to Mario, for the experienced advice on Keras and guidance on Machine Learning Courses. To Rishi and Himani, for being the best housemates one could ask for, and to Ilija, as the sunset runs around the lake are not same without any of you. To Marvin, for tea, and keeping me sane through uncertainty.

Lastly, but most definitely not the least, to all the people I am lucky enough to call friends, for the outpouring support I was always met with. A special mention to Rebeca, Pedro and Tiago, for making the distance irrelevant and helping me laugh through tough times - and for reading my thesis at every little change I made. And to my sisters from other mothers, Rose and Sara, for the continued unwavering support and inspiration.

Contents

Abstract v

Acknowledgements vii

Contents ix

List of Figures xi

List of Tables xiii

1 Introduction 1

1.1 Why Study Adaptive Optics Control? . . . 1

1.2 Research Objectives and Results . . . 2

1.3 Summary of this Thesis . . . 2

2 Background on Adaptive Optics 3 2.1 Free Space Optical Communications . . . 3

2.2 Adaptive Optics . . . 4

2.2.1 Historical Context . . . 5

2.2.2 Definition and Principles . . . 5

2.2.3 Shack-Hartmann Wavefront Sensor . . . 6

2.2.4 Deformable Mirror . . . 7

2.2.5 Wavefront Reconstruction . . . 9

2.2.6 Performance Metrics and Error Sources . . . 10

2.3 Drawing Parallels from Astronomy . . . 12

2.4 Requirements for the Controller . . . 13

2.5 Fibre Coupling . . . 13

2.6 Scintillation and Fading . . . 14

3 Study of Control Techniques 15 3.1 Classical Control . . . 15 3.1.1 PID Control . . . 15 3.1.2 Smith Predictor . . . 17 3.1.3 Summary . . . 18 3.2 Robust Control . . . 18 3.2.1 H∞ Control . . . 18

3.2.2 Variable Structure Control (Sliding Mode Control) . . . 19

3.2.3 Summary . . . 20

3.3 Optimal Control . . . 20

x

3.3.2 LQG Control . . . 20

3.3.3 Data-Driven H2 Optimal Control . . . 21

3.3.4 Summary . . . 22

3.4 Adaptive Control . . . 22

3.5 Machine Learning Control . . . 23

3.5.1 Artificial Neural Networks . . . 23

3.5.2 Convolutional Neural Networks . . . 24

3.5.3 Wavefront Sensorless Optimization Algorithms . . . 25

3.6 Qualitative Assessment and Comparison . . . 26

4 Machine Learning Control 27 4.1 Overview of Machine Learning Control . . . 27

4.2 Neural Networks . . . 27

4.2.1 Introduction . . . 27

4.2.2 Architecture and Mathematical Background . . . 28

4.2.3 Activation Functions . . . 29

4.2.4 Backpropagation and Stochastic Gradient Descent . . . 29

4.2.5 Data Processing . . . 31

4.3 Establishing Benchmarks . . . 31

4.4 Mapping Control Objectives . . . 32

5 Measurement Campaign 33 5.1 The THRUST Project . . . 33

5.2 Current Setup . . . 34

5.3 Procedure . . . 34

5.4 Results . . . 35

6 Synthesizing a Controller 37 6.1 Neural Network A: Single Value Prediction . . . 37

6.1.1 Architecture and Results . . . 38

6.1.2 Discussion . . . 40

6.2 Neural Network B: Extending Prediction to All the Grid . . . 41

6.2.1 Architecture and Results . . . 42

6.2.2 Discussion . . . 43

6.3 Neural Network C: Assessing Datasets Influence on Performance . . . 44

6.3.1 Architecture and Results . . . 45

6.3.2 Discussion . . . 45

6.4 Neural Network D: Generalized NN for Whole Grid Prediction . . . 47

6.4.1 Architecture and Results . . . 47

6.4.2 Discussion . . . 49

6.5 Neural Network AA: Generalized Single Value Prediction . . . 51

6.5.1 Architecture and Results . . . 51

6.5.2 Discussion . . . 53

6.6 Summary of Results . . . 54

List of Figures

1 CANARY, ESA’s Optical Ground Station at Tenerife, Spain. (ESA/D. Lopez) 3 2 Graphic representation of correction of phase aberrated wavefront by a

deformable mirror [11] . . . 4

3 Graphic representation of adaptive optics closed loop system [11]. . . 5

4 Conceptual AO closed-loop operation. . . 6

5 Graphic representation of a Shack-Hartmann WFS [11]. . . 7

6 Schematic illustrating deformable mirror membrane in three different cor-rection configurations [18]. . . 8

7 ALPAO Deformable Mirror schematic illustrating voice coil actuators [18]. . 8

8 Rendering of the first 21 Zernike Modes [19]. . . 9

9 Interaction Matrix. . . 9

10 Point Spread Function [20]. . . 10

11 Illustration of uplink and downlink in satellite communication [21]. . . 12

12 AO control loop detailing MIMO decoupling into SISO servo-systems. . . . 17

13 Common WFSless AO loop [50]. . . 25

14 Basic architecture of a generic neural network. . . 28

15 Weights for each input learned from training on the fourth iteration at quoted times. . . 29

16 SGD represented for one pair of w and b. [58] . . . 30

17 Model loss comparing training and validation set. . . 31

18 . . . 33

19 Evidence of optical paths. . . 34

20 Measurement campaign setup topography. . . 34

21 Temporal Power Spectra coupled into the fiber, comparing perfomance with only tip/tilt correction and with a closed AO loop. . . 35

22 MSE obtained over prediction on each dataset (A1). . . 38

23 MSE obtained for each dataset (A6). . . 39

24 Estimator comparison against real measured value. . . 39

25 Model loss for the fourth iteration, similar in most datasets. . . 40

26 Weights for each input learned from training on the fourth iteration at quoted times. . . 41

27 Model loss for B1 at quoted times. . . 43

28 Model loss for B2 at quoted times. . . 43

29 Learned weights for the input layer (B1). . . 44

30 Weights for each input learned from training on C2 at quoted times. . . 46

xii

32 Prediction error comparison for C1. . . 46

33 Prediction error comparison for D10. . . 49

34 Learning metrics for model D10. . . 49

35 Learning metrics for model D11. . . 50

36 Model loss plots for iteration AA6 from different train/test iterations. . . . 53

37 Model loss plots for iteration AA6 from different train/test iterations. . . . 53

38 Prediction error comparison for AA6. . . 54

List of Tables

1 Overall error metrics for all architectures of NN A. . . 40

2 Overall error metrics for architectures that considered 9 previous measure-ments to predict the next for NN B. . . 42

3 Overall error metrics for architectures that considered 5 previous measure-ments to predict the next for NN B. . . 42

4 Overall error metrics for architectures that considered 5 previous measure-ments to predict the next two for NN B. . . 43

5 Overall error metrics for all architectures for NN C. . . 45

6 Overall error metrics for iterations 1 to 6 of NN C. . . 48

7 Overall error metrics for iterations 7 to 12 of NN C. . . 48

8 Error metrics to compare influence of number of inputted previous time-frames on measurement error for the the NN AA. . . 51

9 Error Metrics for NN AA, iterations 4 to 7. . . 52

10 Error Metrics for NN AA, iterations 8 and 9. . . 53

Nomenclature

Ψ Wave Complex Amplitude

A Amplitude of wave

ϕ Phase of the field fluctuation ϕtur Turbulent Phase

ϕcor Corrected Phase ϕres Residual Phase

~

d Deformable Mirror Control Command Vector ~

m Wavefront Sensor Measurement Vector ~ w Measurement Noise ¯ M Calibration Matrix ¯ W Reconstruction Matrix S Strehl Ratio

Imeas Intensity of Measured Aberrated Beam

Iperfect Intensity of a Perfect Beam

σp Mean Squared Phase Error

r0 Fried Parameter

C_N2(h) Refractive Index Coefficient

γ Zenith Angle λ Wavelength τ0 Coherence Time

˜ν Weighted Average of the Turbulence Layer

σ_fit2 Fitting Error p Actuator Pitch σ_T2(τ) Temporal Error AO Adaptive Optics

AOT Advanced Optical Technologies Group at DLR CCD Charged Coupled Device

DoF Degrees of Freedom

DLR Deutsches Zentrum für Luft- und Raumfahrt DM Deformable Mirror

FSO Free Space Optics

FSOC Free Space Optical Communications GEO GEostationary Orbit

Laser Light Amplification by Stimulated Emission of Radiation LOS Line Of Sight

LTI Linear Time Invariant ML Machine Learning

xvi

MIMO Multi-Input Multi-Output NN Neural Network

OLS Ordinary Least Squares

PID Proportional Integral Derivative Control RF Radio Frequency

SISO Single-Input Single-Output SNR Signal-to-Noise Ratio

THRUST Terabit-Throughput Satellite System Technology WFS WaveFront Sensor

Chapter 1

Introduction

In this chapter, the concept of Adaptive Optics (AO) is introduced and this revolutionary imaging technology is contextualized in the range of Free Space Optical Communications (FSOC), thus setting the motivation for a study of control techniques. Main goals and

results are presented, and the remainder of the thesis is outlined.

1.1

Why Study Adaptive Optics Control?

We live in an increasingly automated world, with the ever growing Internet of Things standing as a looming staple in everyday applications that could astoundingly increase citizens’ quality of life, as well as being the center point of the next industrial revolution: Industry 4.0. As a result, society is increasingly dependent on high throughput broadband connection, of which current worldwide availability is a lot scarcer than required. According to the International Telecommunication Union, at the end of 2018 almost 50% of the world was still without broadband internet connection [1].

Optical satellite communications have the potential to enable global internet access at high data rates with the use of a few GEO (geostationary orbit) satellites. This would require an optical feeder-link in order to achieve higher bandwidths than those provided currently in the Radio Frequency (RF) domain [2]. The use of optical communications links provides higher carrier frequencies, leading to a high modulation bandwidth, enhanced security, and freedom from interference [3]. The feeder-links are provided via optical ground stations, which means the laser beam has to travel through the Earth’s atmosphere. The performance of current FSOC systems is hindered by turbulence in the troposphere, which distorts the light beam causing scintillation at the receiver and hence severely limiting available data throughput.

AO technology is one possible way of addressing turbulence mitigation in laser com-munications, by use of a deformable mirror for real-time correction of the incoming light wavefront. AO systems are common in Astronomy, and are to be featured in upcoming Extremely Large Telescopes [4]. The control of the deformable mirror should allow for correction of the wavefront phase, therefore addressing the distortion caused by the turbu-lence in the atmosphere. Thus, scintillation becomes minimized and more light is coupled into the fibre, which will ultimately enhance data throughput. However, classical control becomes unstable when facing the high rate of change of the turbulence. This fast changing turbulence doesn’t affect communication per se, as the rate of change is significantly lower than the communication speed. Hence, the main drive behind this thesis lies in the need

2 Chapter 1. Introduction for turbulence mitigation to achieve high data throughput within the realm of optical communications. To tackle this issue, an adaptive controller for the AO system is to be developed, and must ensure real-time control, which would strongly benefit from prediction of wavefront distortion.

The work hereby detailed was supervised by Dr. Andrew Reeves, and was conducted within the Advanced Optical Technologies (AOT) group of the Institute for Communica-tions and Navigation at the German Aerospace Center (DLR – Deutsches Zentrum für Luft-und Raumfahrt), led by Dr. Rámon Mata-Calvo. It stands as part of the development of experimental laser communication systems for optical feeder-links, technology which will hopefully improve global internet access.

1.2

Research Objectives and Results

Modern control theory has prompted the existence of several advanced control schemes, but AO control still relies mostly on classical theory, which is not entirely adequate for such complex systems. The main goal of this dissertation is the selection of an ade-quate and reliable control technique that would both accommodate for characteristics of the AO system and allow for real-time correction of the wavefront. Thus, a possi-ble solution is prediction of the phase distortion caused by atmospheric turbulence at the next timeframe, based on previous and current timeframe data. In order to do so, an in-depth review of concepts is necessary for further understanding of the system’s characteristics and requirements. Subsequently, an extensive overview of used control techniques in AO is compulsory to allow for an informed choice and further implementation. Based on this study, Machine Learning Control was selected and a neural network was developed for prediction of the distortion. The solution was compared to defined benchmarks. Results have shown that prediction of a single value in one subaperture is done best by an artificial neural network, but when considering the whole grid, a multiple linear regression achieves better estimation.

1.3

Summary of this Thesis

This thesis intends to discuss and explore alternatives for control of the AO loop in detail with the aim of implementing a chosen technique in order to mitigate turbulence effects. First, the required theoretical background is introduced, and an in-depth overview of control techniques that have been used in the context of AO is discussed. Each approach is then considered in the scope of FSOC. Upon commentary and comparison, Machine Learning Control is then focused as an appropriate choice, and the topic is further explored, with an emphasis on Neural Networks. An artificial neural network is implemented and its performance is optimized with real data recording from a measurement campaign. Results of the network are presented and its performance discussed.

Chapter 2

Background on Adaptive Optics

In this chapter, a brief review of concepts necessary to the understanding of this text is given. The concept of Free Space Optical Communications (FSOC) is explained, as well as Adaptive Optics (AO) systems and their potential relevance in FSOC. The differences between Astronomy and Communications requirements are established and a further understanding of turbulence, scintillation and fading is detailed. Lastly, the coupling of light into the fibre is explained.

2.1

Free Space Optical Communications

FSOC refers to the transport of data via laser or light through free space. It stands as a line-of-sight (LOS) technology based on transmission of modulated light pulses through free space to ensure broadband communications. The first uses of FSOC report back to the military and space aviation during the cold war [5].

Figure 1: CANARY, ESA’s Optical Ground Station at Tenerife, Spain.

(ESA/D. Lopez)

A FSOC link involves two optical transceivers, i.e. a laser transmitter and a receiver at each terminal in order to provide full duplex capability. A FSOC transceiver operates similarly to a fibre optics one, but without the physical support of the fibre, which entails a significantly higher path loss [6]. An optical communications link must ensure transmission

4 Chapter 2. Background on Adaptive Optics of a modulated beam with minimal distortion of the modulated signal. Use through the atmosphere will result in distortion as the phase of the beam is randomly varied over the propagation path [3]. However, phase distortions are much slower than communications signal and therefore don’t affect the data directly.

Currently, it stands as an attractive alternative to RF communications, mainly due to a much higher bandwidth and hence a significantly higher data throughput, as well as unregulated frequency spectrum. Optical communications are also more secure, since LOS technology is fairly difficult to intercept [5]. However, being a point-to-point technology implies there is a need for an unobstructed path, a requirement hard to fulfill in real-life applications. Weather conditions gravely interfere with FSO links, making its availability extremely weather dependent. Fog, rain and snow cause scattering of the laser signal, deflecting part of the light away from the receiver. However, clear weather conditions also present challenges, mainly due to turbulent effects which promote scintillation of the signal at the receiver and result in a phase distorted wavefront [6].

Turbulence in the atmosphere is a non-stationary random process. Hence, aberrations can only be described statistically, where one should consider both statistics of state and evolution of air refractive index fluctuations [7]. There are several studies and models on turbulence. The Kolmogorov-Obukhov law of turbulence is explained in Madèc’s Study of

Control Techniques [7]. Furthermore, Conan et al [8] have studied the wavefront temporal

spectra in regards to the use of adaptive optics systems. An extensive study on atmospheric turbulence was detailed by Wyngaard [9]. The correction of turbulence induced phase distortion of the wavefront would ensure a high bandwidth and consequentially high data throughput, making FSO a very desirable technology for a feeder-link [5]. One way to address the wavefront phase distortion is the use of adaptive optics.

2.2

Adaptive Optics

As described by Tyson, Adaptive Optics (AO) is a scientific and engineering discipline whereby the performance of an optical signal is improved by using information about the environment through which it passes [10].

Figure 2: Graphic representation of correction of phase aberrated wavefront by a deformable mirror [11]

2.2.1 Historical Context

The technique was first proposed by Babcock [12] as a means to compensate distortion and improve astronomical imaging. According to the author, the integrated effect of a number of turbulent elements would be the leading cause for an unsteady enlargement and blurring of the image, and thus with continuous measurement of deviations from all parts of a mirror, amplification and feedback of this information, it would be possible to locally correct the figure of the mirror, as to produce a non distorted beam [12]. This concept is the cornerstone of AO as we know it today. The invention of the laser soon triggered experiments for proof of concept, and defense-oriented research started using mirrors to manipulate laser beams. The launching of satellites brought the need of imaging for surveillance and prompted new attempts to achieve compensation. The first AO system is attributed to Hardy [13], and was able to correct some distortion and sharpen astronomi-cal images. Systems with more than a thousand degrees-of-freedom have now been built [14].

2.2.2 Definition and Principles

AO deals with the control of light in a real-time closed-loop [10]. In the field of physical optics, light is viewed as wave propagation, and is thus best defined by a complex number to describe both its amplitude and phase [14]. This number is called the wave complex amplitude,Ψ, and is defined as:

Ψ =Aeiϕ (2.1)

Where A and ϕ are real numbers representing amplitude and phase of the field fluctuation. A surface on which ϕ takes the same value is named a wavefront surface. An unaberrated wavefront is a flat surface. When the wavefront travels through the atmosphere, the speed of light will vary as the inverse of the refractive index [14]. We consider amplitude to be constant across the field, and thus only take into consideration correction of the wavefront phase. Hence, adaptive optics systems rely on phase conjugation to address distortion [10].

Figure 3: Graphic representation of adaptive optics closed loop system [11].

6 Chapter 2. Background on Adaptive Optics In Figure 3 the basic principle of how an AO system works is summarized. Light from a point source goes through the atmosphere into a telescope which in turn directs it to the deformable mirror and into a beamsplitter, where two optical paths are created: one is the coupling of incoming light into the fibre, and the other is the AO closed-loop. In the AO path, the aberrated wavefront is collected by the wavefront sensor, and the information is fed into a controller, which in turn outputs a control law for the deformable mirror to correct the wavefront phase. The process of sampling, measuring and correcting is done continuously in a closed-loop.

However, this means phase correction takes place based on information from the pre-vious timeframes; without prediction of turbulence, there will be residual temporal error (see section 2.2.6).

Figure 4: Conceptual AO closed-loop operation.

The conceptual AO closed-loop is represented in Figure 4. Light with turbulent phase,

ϕtur, is collected and passed through the system. The residual phase ϕresis sampled by the

wavefront sensor (WFS), which is knowingly a noisy sensor (see section 2.2.3). Thus, the measurement vector ~m includes both the actual measured residual phase ~mres _{and some}

measurement noise, w. These measurements are the input to the controller, which in turn should apply a control law in order to output the deformable mirror (DM) control voltages,

~

d. Hence, the DM is deformed in such a way that the resulting beam is phase corrected, ϕcor. However, sampling the wavefront, processing the measurements, outputting a control

vector and actuating the deformable mirror is bound to cause a delay, which is explicit in the loop by the transport delay e−τ s_{. Therefore, the AO system always produces a beam}

with some residual phase aberrations [7, 15]:

ϕres(x, y, t) =ϕtur(x, y, t)− ϕcor(x, y, t) (2.2)

Consequently, while in closed-loop operation, the DM outputs a minimized residual phase which the WFS captures, meaning that the closed loop operation relies in measuring the residual wavefront phase, ϕres _[7].

2.2.3 Shack-Hartmann Wavefront Sensor

A wavefront sensor (WFS) is an optical instrument designed for measuring aberrations in a wavefront. There are several types of WFS that measure different characteristics of the wavefront. AO systems use mostly the Shack-Hartmann architecture. A Shack-Hartmann WFS is comprised of a microlens array and a detector array. The microlens array is placed in a conjugate pupil plane to section the incoming wavefront into light intensity focus points. A detector array is placed in the focal plane. Each lenslet forms an image of the source at

its focus. The small optical systems formed by each lenslet are called sub-apertures [16]. A Shack-Hartmann WFS measures the position of each focus point in each subaperture.

Figure 5: Graphic representation of a Shack-Hartmann WFS [11].

As is shown in Figure 5, a plane undistorted wavefront generates a focus point at the centroid of each subaperture. If, however, the wavefront is aberrated, the grid will no longer be regular. Hence, the deviation of each spot from its central position can be used to produce a measurement of the gradient of the wavefront over that subaperture [16]. This measurement is however known to include some noise, due to the nature of the Shack-Hartmann wavefront sensor and alignment of the optical path. In order to do so correctly, enough spatial resolution and short exposure time for real-time compensation of atmospheric seeing are required [14]. The WFS spatial resolution is determined by the number of subapertures, while the temporal resolution is given by the sampling frequency [7]. Spatial resolution must match that of the correcting element (i.e. the space between deformable mirror actuators) and must sample faster than the rate of change in atmospheric turbulence, which stands in the millisecond scale, and must be linear over the full range of atmospheric distortions. Further reading on the Hartmann test and working principles of the Shack-Hartmann WFS can be found in Tyson’s Principles of Adaptive Optics [10]. An extensive read on wavefront sensing can be found both in Hardy’s Adaptive Optics for

Astronomical Telescopes [13] and Roddier’s Adaptive Optics in Astronomy [14].

The Shack-Hartmann WFS featured within the THRUST demonstrator is a custom design with 8X8 subapertures arranged on a square grid, giving 52 illuminated subapertures in total. Each of them has 14x14 pixels (196 in total). Thus, it produces 104 measurements at each sampling corresponding to x and y coordinates at each subaperture, with the origin of the axis located at its center. Measurements are in µrad/pixel, to ensure independence from optical path. The detector array is an InGaAs Near Infra-Red Xenics Cheetah [17].

2.2.4 Deformable Mirror

A deformable mirror (DM) is a device that addresses wavefront correction using the same mechanism that originated the distortion, i.e. by physically changing the phase. A DM is comprised of a fine mirror membrane (Figure 6) that is deformed by a grid of actuators, generating a mirrored wavefront in order to cancel phase aberrations in reflected wave,

8 Chapter 2. Background on Adaptive Optics using information from the WFS (Figure 7). The number of actuators defines the degrees of freedom (DoF) that can be corrected in a wavefront.

(a) (b) (c)

Figure 6: Schematic illustrating deformable mirror membrane in three different correction configurations [18].

Albeit a fairly simple concept, a DM is hard to model mathematically - and conse-quentially hard to control - since no actuator movement is effectively independent, as each individual movement will slightly influence nearby deformations, as demonstrated by the interaction matrix in Figure 9. The interaction matrix describes the sensitivity of the WFS to DM actuator’s movement, evidencing dependence of individual deformations on the mirror membrane, as it is a continuous surface moved by a discrete grid of actuators.

Considering this interaction, control of the actuators might be zonal or modal. Zonal control refers to individual actuator control, while modal control refers to controlling the shape of the wavefront as a whole. Modal control has some advantages, such as filtering modes sensitive to measurement noise, separating correction into a DM and a tip/tilt mirror (as the majority of the wavefront phase aberrations are tip and tilt) thus allowing for DM’s with lower strokes, and decoupling the system (see section 3.1.3). Modal control is usually performed via Zernike modes. The Zernike Polynomials are an orthonormal basis set capable of describing any circular image as an infinite sum. They are particularly useful to describe turbulence induced aberrations, as correcting lower order spatial modes accounts for the majority of power in turbulence induced wavefront error [19].

Figure 7: ALPAO Deformable Mirror schematic illustrating voice coil actuators [18].

Figure 8: Rendering of the first 21 Zernike Modes [19].

The deformable mirror within the THRUST demonstrator is the ALPAO DM97-15, and it has 97 voice-coil actuators, with 11 of them across diameter. It’s dimensions are 52 x 74 x 22 mm3 _{with a main actuator stroke of 60 µm (Peak to Valey) and a pitch (distance}

between actuators) of 1.5 mm. The pupil diameter is 13.5 mm. The settling time is of 0.8 milliseconds for any stroke (+/- 10%), and the first resonance occurs at a frequency of 800Hz. ALPAO DMs have almost no hysteresis (<2%), as well as high linearity (>97%), amounting to a great stability. More technical information can be found in ALPAO’s website [18].

2.2.5 Wavefront Reconstruction

The data collected with the Shack-Hartmann WFS amounts to gradients of the wavefront error in each subaperture, which means the incoming wavefront needs to be reconstructed from that information and converted into DM actuator commands. This relationship between sensing and actuating element is crucial for good performance.

μ

10 Chapter 2. Background on Adaptive Optics The simplest AO reconstruction relies on measuring the system’s Interaction or Cal-ibration Matrix, ¯M (Figure 9). This matrix maps the sensitivity of the WFS to the

DM deformations, and is obtained when calibrating the system, by exciting each DM actuator individually and tracking the WFS response. In order to achieve this, all possible independent patterns the DM is able to create are generated. These independent patterns are known as the DM’s influence functions, and characterize how the mirror membrane deforms with the stroke of one actuator. Each DM has as many influence functions as actuators. Each influence function is measured by the WFS and the response is noted to form an interaction matrix. In Figure 9, it is observable that poking one actuator not only amounts to measurements in several subapertures, but also looks like neighbouring actuators were stroked, thus reflecting the continuous nature of the DM membrane. It should also be noted this matrix can be obtained either by exciting each actuator one at a time, of exciting them in patterns. This method assumes linearity of the the described relationship and can be described as:

~

mi =M ~¯ di (2.3)

where ~mi is the WFS Measurements’ vector (residual error) with size {b}, ¯M is the

calibration matrix of size {b × a}, and ~di is the DM actuator commands vector of size

{a}, where a is the number of actuators in the DM and b the number of subapertures in the WFS. In operation, the DM commands relating to the WFS measurements must be obtained. This is achieved by computing the pseudo-inverse of the interaction matrix, as:

~

di =W ~¯ mi (2.4)

where ¯W is the Reconstruction or Control Matrix. The determination of ¯W assumes

linearity of the servo-system, at least for small values of WFS measurements [7].

2.2.6 Performance Metrics and Error Sources

There are a few important performance metrics to keep in mind regarding optical systems, AO and FSOC systems. In order to better understand those, it is necessary to clarify the Point Spread Function notion.

The Point Spread Function (PSF) refers to the intensity distribution caused by diffrac-tion [10] observed in a detector when imaging a point source, and thus is only dependent on the optical system itself [16]. It is used to characterize the response of an optical system to a point source, and gives information on the optical transfer function. When an optical system is shift invariant, the response to a point object can be estimated through the PSF.

The Strehl Ratio is an adimensional number that evaluates optical quality of a sys-tem, and is the ratio between the intensity of an aberrated beam, Imeas and that of an

unaberrated beam Iperfect. Hence, it assumes a value between 0 and 1, with 1 being a

perfectly unaberrated system [10]. For a low wavefront error, it can be approximated with the incident mean squared phase error, σp, as explicit in equation 2.5 [16]:

S= Imeas

Iperfect

≈ e−σ2p (2.5)

The Fried Parameter (also called coherence length), r0, informs on the quality of optical

transmission through the atmosphere, and is defined as the diameter of a circular area wherein the atmosphere causes a root-mean-squared wavefront aberration of approximately 1 radian [10]. r0 = " 0.42_cosk2 (γ) Z C_N2(h)dh #−3/5 (2.6) where C2

N(h) is the refractive index structure coefficient describing the strength of the

fluctuation, γ is the zenith angle a telescope observes, and κ is given by:

k= 2π

λ (2.7)

where λ is the wavelength. Thus, a telescope with a diameter larger than r0 has no increase

in resolution when compared to a telescope with a diameter of r0 [16].

The Signal-to-Noise Ratio (SNR) is an adimensional measure of quality of transmission (is sometimes expressed in decibel). It is defined as the ratio of the power of a signal Psignal

to the power of the background noise Pnoise [16].

SNR= Psignal

Pnoise (2.8)

The Coherence Time, τ0, otherwise known as the Greenwood time delay is taken as

the time for a one radian wavefront error to appear. It defines the temporal error of the system, which is dominant in FSOC systems, and can be approximated as [16]:

τ0 =0.314

r0

˜ν (2.9)

where ˜ν is a weighted average of the turbulent layer.

The Power Spectral Density describes how the power of a signal is distributed within the frequency spectrum. Since the processes in the atmosphere are time transient, it is useful to assess quality of the coupling and how scintillation is affecting the signal [10].

There are two main error sources one should have in mind while dealing with control of a AO system, those being a fitting error and a temporal error. The fitting error is due to the fact that a mechanical DM cannot exactly match the aberration patterns of the eddies of the atmospheric turbulence, nor exactly duplicate Zernike modes, thereby compromising spatial resolution. Therefore, the fitting error shows the wavefront error that resulted after a applying an appropriate fit between the surface and the atmospheric turbulence. For an

12 Chapter 2. Background on Adaptive Optics actuator pitch p, the fitting error for wavefront phase variance, σ2

fit, can be obtained by

[16]: σ2_fit=κ _p r0 5/3 (2.10) where κ depends on the DM’s actuator technology and mirror membrane material, and r0

is the coherence length. Hence, the fitting error is defined by mechanical constraints. Since in AO for FSOC the actuator pitch p is quite small, the temporal error is of higher relevance. The temporal error amounts to limited speed with which is possible to take a WFS exposure, get its read on the detector, compute the DM commands and apply them to the DM, all this regarding how fast the turbulence changes [16]; thus, the temporal error can be seen as a transport delay. The variance of the corrected wavefront due to temporal limits is given by:

σ2_T(τ) =

_τ

τ0

5/3

(2.11) where τ is a time delay, which is physically limited by the minimum time interval between WFS exposure and DM actuator movement. More information and thorough analysis of both the fitting and temporal error can be found in Tyson’s Principles of Adaptive Optics [10, 16].

2.3

Drawing Parallels from Astronomy

AO was motivated by astronomic imaging. While the technology in use is the same intended to use for FSOC, there are some clear differences that should be addressed. In astronomical seeing, systems are usually deployed at night, when atmospheric turbulence is diminished. The object of interest is observed and sampled over a long period of time, but temporal features are usually unimportant and it is assumed that the object is static for the required exposure time. The best sites were chosen, usually places with minimal artificial lightning interference and high up in mountains, where there is also less turbulence, since the air is thinner and thus there is less propagation distance through the atmosphere. As such, viewing conditions occur in optimal circumstances.

Figure 11: Illustration of uplink and downlink in satellite communication [21].

Even though FSOC shares some technological principles of AO with astronomy, the operating conditions are quite different. The system is to be deployed at any given time of the day, and as evidenced in section 2.1, it may require establishing a link with a moving target (LEO and MEO links), with a communication window of about ten to fifteen minutes, while addressing rapidly varying turbulence conditions. As is shown in Figure 11, the downlink to the Optical Ground Station (OGS) requires correction of the wavefront upon arrival, but the uplink requires predistortion of the wavefront as to ensure turbulence effects are dealt with. It also requires propagating the laser in a Point-Ahead Angle (PAA), as the satellite is a fast moving target. It should also be noted that, for a 1 Terabit link, failing to couple the light into the fiber for a second will correspond to a loss of information transmission of about a Terabit.

2.4

Requirements for the Controller

The previous sections explained AO systems and their place in astronomy, further illus-trating how different the working conditions are for FSOC. Hence, requirements for the controller can now be established:

• The AO system is to be deployed at any given time; • Operation needs to withstand fast-changing turbulence; • A steady inbound light into the fibre coupling is required;

• Real-time correction of the wavefront, ideally with wavefront distortion prediction as a means to compensate for the transport delay inherent to the AO servosystem (subsection 2.2.6);

• Controller must adapt the system response to changes in turbulence;

• It must also ensure robustness against sudden changes in working environment and vibrations;

• Scintillation and Fading must be minimized; • A low bit-error rate is required.

The fitting error is unavoidable, as it is a physical constraint. The temporal error might be improved with estimation. It should also be noted that due to the mirror membrane the deformable mirror is extremely hard to model accurately. Hence, a control technique that could both bypass the need to model the system and account for temporal error with estimation would be an appealing solution.

2.5

Fibre Coupling

FSOC links entail lasers traveling through the atmosphere, that need to be coupled into fibres in order to process the information. Coupling the light into the fibre is bound to cause some power loss. The coupling efficiency in single-mode components is affected by the far-field distributions of both the source and receiver [22]. Wavefront aberrations therefore include contributions from fibre misalignments as well as turbulence induced phase shifts.

14 Chapter 2. Background on Adaptive Optics The power loss causes an unstable link and amounts to significantly high loss of infor-mation, due to the high throughput of the data links. The aberrations cause scintillation in the receiver, amounting to peaks and blackouts in the fibre coupling. The main goal of including an AO system for FSOC is optimize the fibre coupling by reducing scintillation, and thereby increasing and stabilizing the power spectrum [22].

2.6

Scintillation and Fading

Scintillation refers to intensity variations, and can be expressed as fluctuations on the log

of the amplitude, otherwise known as log-amplitude fluctuations [10]. Scintillation effects are most noticeable when a point receiving aperture, as the ones in the Shack-Hartmann WFS, is used. In this cases, scintillation is observed as a variation of the irradiance σ2

1,

and can be described as:

σ2₁ =Ahe(4σX2)−1

i

(2.12) where σ2

X is the log amplitude variance, and A is an aperture averaging factor. Since

atmospheric turbulence is dynamic, so is the scintillation it causes, so it should be thought of in terms of temporal power spectra [10]. The major goal of the AO system is to reduce scintillation at the receiver and ensure a stable temporal power spectra.

Wavefront propagation is also prone to Fading. Fading refers to variations in at-tenuation of a signal, and is of particular interest for wireless communications, since weather phenomena promotes scattering and fading of signals. The AO system in FSOC is meant to address wavefront correction as to minimize fading and scintillation at the receiver.

Chapter 3

Study of Control Techniques

In this chapter, a summary of studied AO control techniques is presented. First, the current control loop is detailed. Then, for each approach, theoretical concerns are described, an ex-ample of use in AO is detailed (when existing) and adequacy to the system is discussed as a means to compare techniques and make an informed choice. Control techniques that seemed most promising for AO-FSOC were further explored. It is however necessary to keep in mind the results are not comparable, as every paper describes different testing under different conditions with dissimilar systems, in distinct places. In addition, most papers focus AO in astronomy (working conditions described in section 2.3), which makes it hard to extrapolate for FSOC working conditions. The following sections all use notation that refers back to the AO conceptual loop shown in Figure 4 and Section 2.2.2. Currently, the THRUST Demonstrator (Chapter 5) makes use of classical control approach described in section 3.1.3.

3.1

Classical Control

Classical control is normally aimed towards linear time invariant (LTI), input, single-output (SISO) systems. Assuming linear behavior of the AO system, Pièrre Yves Madèc [7] referred to the control as having both a static and a dynamic part, where the static part deals with calculation of the DM actuator commands vector by means of the reconstruction matrix (Chapter 2), while the dynamic part deals with the correction of the phase [7]. Given this concept, Madèc described a solution based on classical control principles, suggesting a PID controller with use of a Smith Predictor.

The controllers hereby referred are thoroughly described in Roddier’s Adaptive Optics

in Astronomy [14], and stand as the basis of AO control when the system is considered

linear.

3.1.1 PID Control

PID control refers to the synthesis of a classic controller that would implement a propor-tional, integral and/or derivative control action to the system, meaning the control action is either proportional to the error, to the integral of the error and/or to the derivative of the error.

Given the MIMO nature of the AO system, and considering PID control is aimed towards SISO systems, applying such a technique requires decoupling. However, the DM

16 Chapter 3. Study of Control Techniques and WFS are spatially coupled, which highly contributes towards the difficult temporal analysis of the system. Thus, transforming a MIMO system into a set of independent SISO systems requires a set of independent inputs. A possible solution is to make use of modal control (see section 2.2.4), allowing the AO system control loop to be seen not as a whole complex MIMO, but as n independent SISO systems working in parallel, that can be analyzed individually [7]. As referred in section 2.2.5, the influence functions of the DM make up such an independent set, that allows for control of each actuator individually. Additionally, input and output space are of different dimensions, which means the controller has to convert sensed data from WFS space and output commands into a DM actuator space. With all these considerations in mind, Madèc [7] explored the temporal behavior of the AO system and its main components (wavefront sensor, mirror drivers and actuators), by modeling within the Laplace frequency domain, with the controller synthesized in discrete-time (Z-transform) to account for the sampling made by the WFS and the zero order hold (ZOH) transfer function for the digital analog converter (DAC), in order to address phase lag, which reduces the frequency domain in which the AO system is efficient [7].

Thus, the overall open loop transfer function G(s) representative of the temporal

behavior of AO system might be described was:

G(s) = ϕ cor ϕres = e−τ s(1 − e−T s)2 T2_s2 C(z=e −τ s_{) (3.1)}

where G(s) characterizes the temporal aspects of each individual components, and as

such doesn’t mention the decoupling, considering it happens within the control computer. Therefore, 1−e−T

T s is the WFS transfer function characterizing both the integrator time of

the detector τ and the inherent time delay; it is also the transfer function for the DAC in the drivers due to its ZOH effect; the drivers’ transfer function was considered to be close to unity; and e−τ s _{is the transfer function for the servo system time delay for read-out}

time between WFS and controller, C(z=e−τ s), and T is the sampling time for incoming

light. Therefore, the decoupling happens only in C(z=e−τ s). Thus, in G(s)the output

ϕcor is the corrected wavefront phase for a measured residual wavefront ϕres. This was

proven as a good approximation for some astronomical AO setups [7]. A possible discrete-time transfer function for a PI controller would be:

CC(z) = Ki

1 − z−1 +Kp, (3.2)

where Ki is the integral gain and Kp the proportional gain. A further analysis on the presented transfer functions can be found in [7].

Therefore, the wavefront sensor samples the continuous beam of light into a set number of measurements. The controller receives those measurements and must reconstruct the wavefront into DM space, and control each actuator individually. Hence, while physically the controller is one component, it implements n PID controllers, with n being the number of actuators a= 97 in the case of DM zonal control. Each controller outputs a driver

voltage individually to control the corresponding voice coil, which in turn strokes the mirror to form a corrected wavefront.

Figure 12: AO control loop detailing MIMO decoupling into SISO servo-systems.

The currently implemented control law on the THRUST setup follows the loop consid-ered in Figure 12. It is based on the assumption that the DM actuator commands at the previous timeframe, along with the measurement error they produced, constitute the most accurate estimation of the DM actuator commands at the current timeframe that would amount for a corrected wavefront:

~

di =di−1~ +g ¯W ~mi (3.3)

where ~di is the DM actuator commands vector at timeframe i with dimensions {a} (which

for the DM in the THRUST setup is a=97 given there are 97 actuators; see section 2.2.4),

gis a controller gain scaled between 0 and 1, ¯W is a {a × b} reconstruction matrix (where b=104 given there are 52 subapertures producing two coordinate measurements each; see

section 2.2.3 and 2.2.5),and ~mi is the WFS measurements vector at timeframe i, with size

{b}. For further information on wavefront reconstruction, refer to equations 2.3 and 2.4. The behavior of this control law is akin to that of an integrator described in state-space, since it accounts for rate of change of the error given the previous timeframe outputted command, and its performance stands as a benchmark to improve upon.

As is visible in Section 5.4, Figure 21, data from the measurement campaign has shown that closing the AO loop with this controller increases coupling of light into the fibre; but it also shows some performance disruption. As the data throughput stands in the order of the Terabits per second, a disruption of just a millisecond will amount to a loss of a few Gigabits of data, which is problematic.

While fairly simple, these options are extensive, computationally demanding and don’t account for anything other than phase lag, assuming the system as LTI and therefore ignoring the time varying nature of the disturbances and using a static control law. This configuration overlooks the dynamic nature of vibrations, measurement noise and wind speed variations, making the controlled system neither stable nor robust.

3.1.2 Smith Predictor

The Smith Predictor is a type of predictive controller used for servo systems with a delay in the closed-loop. When a process exhibiting time delay is controlled by an integrator, a ramp control action is generated during deadtime, which leads to overshoot in the actuator

18 Chapter 3. Study of Control Techniques response when it starts moving after deadtime. If overlooked, this behavior can seriously hinder system stability. The Smith Predictor is designed to reduce the effects of time delay by considering it within the controller [7] [23]. It should be noted that this doesn’t remove deadtime from the system response, but only acknowledges it as present within the controller.

The Smith Predictor is defined in the s-domain and aims to simulate the process alongside the control action, thus accounting for deadtime while controlling the system. It requires a priori knowledge of process gain, time constant and delay, as these work as input parameters.

Madèc [7] defined the Z transfer function of an I controller with Smith Predictor as:

CC(z) = Ki (1 − z−1_)(1+K

spz−1) (3.4)

where Kspis the Smith Predictor gain, which cannot be higher than 1, or else the controller

is unstable [7]. Moreover, a more recent study has found the use of a Smith Predictor rarely ever improves PID performance, and in cases it does the improvement comes at the expense of a high sensitivity to time delay errors and stability [24].

3.1.3 Summary

In the previous sections, PID control and a Smith Predictor were considered for control of an AOS. These are discrete-time controllers for modeled LTI systems, meaning the temporal evolution of the wavefront and dynamics of the AO system are not explicitly accounted for. For these classical controllers, the choice of control parameters is a trade-off between disturbance rejection, noise propagation and closed-loop stability. The higher the control bandwidth, the better the disturbance rejection; however this also prompts higher noise propagation and increases risk for instabilities [25].

Moreover, while phase lag and deadtime associated with the system were accounted for, the modeled system is oversimplified which will amount for performance errors. The control laws don’t account for the dynamic nature of external disturbances, and take turbulence as a time invariant process, which is not the case of a real-life application of such a system where vibrations, measurement noise, wind gusts and changes in atmospheric turbulence interfere with the control of an AOS.

3.2

Robust Control

Robust control theory aims to ensure that both performance and stability of a controlled system are robust within certain bounds of an operating point; i.e. that the system is still stable when facing changes in dynamic behavior, and exhibits predetermined response characteristics in the presence of said changes, given that unknown variables are bounded. It requires considerations based on frequency-response analysis and time-domain analysis, making it extremely mathematically complex [26].

3.2.1 H∞ Control

H∞Control is based on Hardy spaces, particularly concerning the H∞ norm, which can be

excitation. If a system transfer function is a member of a Hardy space, it is guaranteed to be stable. Therefore, the method optimizes the stability of the system by forcing a family of models to be stable [26, 27]. It is aimed towards discrete-time linear systems and its theory is highly mathematical [27].

Minimizing the norm is difficult, since it does not guarantee a unique solution. To tackle this issue, the norm is not brought to an absolute minimum, but rather a value that is slightly greater than the minimum, but ensures a single solution. Hence, this type of controller is often referred to as sub-optimal.

The efficiency of H∞control applied to AO systems has been verified on a test platform

[27], then a reduced order model was developed [28], and a few more variants were explored, all emphasizing the performance of the AO system, but neglecting robust stability. A mixed sensitivity approach to H∞ control showed that this controller is more robust than

an integrator, but doesn’t always present a better performance, and is more adequate for AO systems with a large time-delay uncertainty [29].

3.2.2 Variable Structure Control (Sliding Mode Control)

A widely known solution to the problem of deterministic control of uncertain systems is VSC (variable structure control). VSC is a discrete-time nonlinear method, usually applied to systems with uncertainties or switching dynamic behavior. The characterizing feature of VSC is the sliding motion that takes place whenever the system repeatedly crosses certain subspaces in the state-space. In order to properly accomplish this, the structure of the con-trol law itself varies, and that variation is defined by the position of the state trajectory [30].

The sliding mode control (SMC) is the most common way of synthesizing VSC, and refers to the definition of a sliding surface on which the system would work asymptotically after reaching it. Hence, the system’s behavior is conditioned to converge to a determined hyperplane that is predefined according to desired conditions, which results in a controller that ensures global stability and consistent performance regardless of modeling impre-cision. However, when synthesizing an SMC one should account for chattering effects (finite-amplitude high-frequency oscillations of the controlled structure) [31] and prevent

the existence of limit cycles. SMC is currently in extensive research [30].

Regarding AO control, Senkel et al. [32] have analyzed a combination of an interval sliding mode controller and an interval sliding mode observer for control and state estimate of a system with both bounded and stochastic uncertainty, and concluded one should design both simultaneously by using LMI’s and to perform a joint stability proof with just one common Lyapunov function. It is also suggested to research the choice of sta-bility margin and the maximum switching amplitudes under constraint of a limited step size.

It should be noted there is no data on applicability or performance of an SMC control-ling an AO servosystem. However, this type of control has been effectively simulated for control of a MEMS optical switch, and it proved to be a simple, robust solution against disturbances [33].

20 Chapter 3. Study of Control Techniques

3.2.3 Summary

In the previous sections, two robust control techniques were discussed for control of AO systems. Both solutions account for robust performance, but not optimal response. While robustness is inherent to both controllers, H∞ control doesn’t always perform better than

the integrator.

SMC is still a novel technique, and no research on AO was found. Synthesizing an SMC for a MIMO system may be hard. In addition, time delays cause problems with matching conditions, which are the basis for robustness in this type of controllers. Moreover, chatter-ing effects could be devastatchatter-ing for the deformable mirror.

3.3

Optimal Control

As opposed to classical control in which the aim is to conceive a controller that is stable within a certain bandwidth to satisfy desirable constraints, the system that is the end result of an optimal design is supposed to be the best possible solution to a particular control problem [34].

3.3.1 LMMSE (Linear-Minimum-Mean-Square-Estimator)

Considering the non-stationary nature of the turbulence induced wavefront aberrations, van Kooten et al. [35] modeled the turbulence accounting for wind speed and Fried Parameter variations, and developed a prediction algorithm based on LMMSE. Even though this showed better performance than a simple integrator, results have shown LMMSE is not the optimal solution for varying atmosphere parameters. For further optimization, a focus on wind behavior is required [35].

3.3.2 LQG Control

Linear Quadratic Gaussian (LQG) Control concerns LTI systems described by a linear state-space model with additional white Gaussian noise inputs [36]. The goal of an LQG controller is to compute an output feedback control law to minimize the expected value of a quadratic cost criterion. The Separation Theorem [34] allows the problem to be broken down into estimation and control [37].

For AO systems, the control requires a model of the loop incorporating the deterministic dynamics of its various components, the variance of the measurement noise and the spatial and temporal correlation structure of the turbulent phase [37].

The state vector should be defined with all variables needed at time k to compute the state at time k+1 and fully describe input-output behavior. Therefore, the state vector is not unique and can be further explored for different combination of variables. Furthermore, it should summarize the entire knowledge on the system (including turbulence) while assuring the control law is a function of the state only. Further information on choice for state vector can be found in reference [38]. Moreover, it is required to model turbulent phase dynamics as to completely define a stochastic state-space model [37].

The estimation problem is addressed with a Kalman Optimal Filter by using all mea-surements available until time k to estimate the state vector at time k+1, which would only

require computation of the covariance matrix of the state vector. A state-space stationary model has been defined and tested by [37]. Given the stationary nature of the model, this could be accomplished offline, thus enhancing computational efficiency.

Gaetano Sivo et al. [36] present a steady-state time invariant Kalman filter, with Kalman gain computed offline, and a 2nd order autoregressive stochastic model for the turbulence. Performance surpassed that of the integrators previously analyzed. Moreover, the experiment shows promising on-sky results, where this approach has resulted in a system that is reliable, stable and robust. The LQG controller was able to address vibration mitigation. The on-sky setting showed LQG residual phase variance is 25% lower than the best integrator performance [36].

Whilst an improvement from the previously referred integrator, this solution runs under the assumption that the covariance matrices are stationary. A non-stationary approach would be possible, but demanding, as it would require the Kalman gain to be computed online; optimization techniques to improve computation efficiency would be required. Wind direction and speed were also not accounted for but could eventually be considered. This approach also entirely dismisses previously collected data, that could potentially be useful for a more accurate description of turbulence.

3.3.3 Data-Driven H2 Optimal Control

H2 optimal control aims to minimize the H2 norm of the transfer function, which in turn minimizes the effect of white noise on the power of the output noise, as opposed to LQG control, where the goal is to minimize steady-state covariance of the error. The H2 Optimization problem is similar to LQG Control design, but instead of relying in an au-toregressive model for the turbulence, a more general data-driven atmospheric disturbance model is used. Hence, accurate estimates of physical parameters such as wind speed of frozen layers are not required, and the control problem is conceived as a discrete-time disturbance rejection.

In AO systems, the data-driven approach relies on open-loop WFS data for subspace identification, henceforth producing an atmospheric disturbance model that is then used to compute an optimal controller. Further read on data-driven modeling of wavefront disturbance can be found in reference [39].

Hinnen et al.[25] described the system in an LTI state-space discrete-time approach, under the assumption the uncorrected wavefront can be modeled as a regular process. Therefore, the second-order statistics of these signals are modeled as the output of an LTI system with a zero-mean white noise input. A dedicated subspace identification algorithm that addresses additional computational demands has been developed and tested, and it was able to identify an atmospheric disturbance model for an AO system with up to a few hundred degrees of freedom in a time efficient matter.

The controller was then derived assuming the DM settling time can be neglected with respect to WFS exposure time. This controller acts by decomposing wavefront prediction and statically projecting it on the actuator space. Results have shown reduction of the mean-square residual phase error by more than 70% compared to PI control, assuming ideal DM dynamics. However, when assuming a more realistic DM model, the gain in

22 Chapter 3. Study of Control Techniques performance was reduced to 14% (attributed to large fitting error) [25].

Even though there is a significant increase in efficiency, the presence of unmodeled dynamics, e.i. wind gusts, will affect closed-loop stability and might compromise robust-ness. However, since wavefront distortions act as a disturbance input, inaccuracies in the atmospheric model wont destabilize the loop. Furthermore, continuous operation is only possible if the disturbance model update is performed in closed-loop, which requires constant solving of one or two Riccati equations. For a large-scale AO system, compu-tational efficiency methods should be accounted for as to reduce the closed-loop burden [25].

3.3.4 Summary

In the previous sections, optimal control approaches were discussed for control of AO system. Both LQG and H2 are discrete-time synthesis regarding LTI systems, and both improve upon the use of classic control. However, there are some drawbacks to be considered. While LQG control requires less online processing power, it also doesn’t account for non-stationary covariance of the error, therefore slightly compromising performance. H2 control for AO, however more demanding in terms of processing power, seems to produce a more accurate model for atmospheric turbulence. So, the choice between the two would be a compromise between processing power and residual-phase error. It should also be noted the processing power could induce further delays in the system when implemented in practice.

Improvement and further investigation is still being conducted for both methods. Rémy Juvénal et al. [40] have proposed a modified LQG controller using both wind velocity and direction priors, mixing boiling and frozen flow models, but with no on-sky results yet. The robustness of both control methods appears to be easily affected by external disturbances, that were again unaccounted for, such as wind gusts, changes in wind direction, changes in turbulence strength or loss of lux due to scintillation.

3.4

Adaptive Control

Adaptive control aims to achieve and to maintain acceptable level of performance when parameters from either the plant or disturbance model are unknown or vary. The adaptive controller provides a systematic approach for automatic online tuning of control parameters [41].

The controllers mentioned in the previous sections where LTI controllers, applied to an LTI model of the system. Adaptive controllers use discrete-time LTI model, but augment it with an adaptive filtering loop, making the controlled system non-linear and time-variant [42, 41].

When it comes to AO, adaptive controllers should be able to compensate for loop latency by predicting wavefront error, as well as identify optimal gain in real time. James Gibson et al. [42] developed adaptive algorithms for wavefront prediction based on a multichannel RLS (recursive least-squares) lattice filter, which was chosen for stability and efficiency. The filter accounts for both spatial and temporal wavefront statistics, and the