M.R. Sadeghi. Evaluation of recursive algorithms for nonstationary disturbance prediction in an adaptive optics experimental setup

Transcription

1 Evaluation of recursive algorithms for nonstationary disturbance prediction in an adaptive optics experimental setup Delft Center for Systems and Control

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3 Evaluation of recursive algorithms for nonstationary disturbance prediction in an adaptive optics experimental setup For the degree of Master of Science in Systems and Control at Delft University of Technology August 15, 2011 Faculty of Mechanical, Maritime and Materials Engineering (3mE) Delft University of Technology

4 The work in this thesis was supported by TNO Science Industry. Their cooperation is hereby gratefully acknowledged. Copyright c Delft Center for Systems and Control (DCSC) All rights reserved.

5 Abstract Adaptive optics (AO) is a technique for correcting the optical effects of atmospheric turbulence in real time, and in this way greatly enhancing the performance of ground-based optical systems beyond the limits imposed by the atmosphere. Optimal control algorithms for AO systems can provide significant additional improvements beyond the standard control approaches presently in use. These optimal control methods depend upon accurate knowledge of atmospheric conditions. The optimal control algorithms proposed in literature, consist of a disturbance predictor, which is designed under the assumption that the influence of the atmospheric turbulence on the wavefront can be modeled by a linear time invariant Auto regressive (AR) or state-space model. However, in practice the turbulence statistics will vary with time, and thus it becomes necessary to constantly update the predictor so that it adjusts to the varying atmospheric statistics. In this thesis time-varying AR model structure and state space model structure based predictors are presented. The parameters of the AR model are identified by the Least Mean squares (LMS), Recursive Least Squares (RLS) and Fast RLS array algorithms at each time instant. For updating the state-space model a recursive subspace identification method, the so called Predictor-Based Subspace IDentification (PBSID), is used. The algorithms are compared in terms of accuracy, rate of convergence, tracking performance and computational complexity on a laboratory setup within a nonstationary turbulence environments. It is shown that the AR model based prediction algorithm with the RLS algorithm posses highest accuracy in comparison to the other algorithms. This algorithm is able to predict the nonstationary turbulence with the accuracy of %99. Moreover it is shown that why, AR model based predictors have higher accuracy compare to that of the state space model based predictors.

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7 Table of Contents Preface ix 1 Introduction 1 2 AO Control and Problem statement AO control loop AO control objective H 2 optimal control for AO system Atmospheric turbulence Problem statement Adaptive algorithms for prediction of nonstationary turbulence Algorithms based on AR model structure Linear estimation problem and steepest decent method LMS algorithm Normalized LMS algorithm RLS algorithm Fast RLS array algorithm Computational cost of stochastic gradient algorithms Algorithms based on state space model structure batch-wise and recursive PBSID algorithm Recursive least square solver for PBSID algorithm Computational cost of PBSID algorithm

8 iv Table of Contents 4 Experimental results Experimental setup Generating nonstationary turbulence Non-stationary turbulence prediction with AR model based algorithms The AR model order selection and prediction accuracy Rate of convergence and tracking performance Nonstationary turbulence prediction by state space model based algorithms Number of states and prediction accuracy Comparing the rate of convergence and tracking performance of the AR and state space model structures Analysis of the results Conclusions and recommendations Conclusions Recommendations A Matlab codes 45 A-1 Maltab codes A-1-1 LMS algorithm A-1-2 Normalized LMS algorithm A-1-3 RLS algorithm A-1-4 Fast RLS array algorithm Bibliography 49

9 List of Figures 2-1 Schematic representation of an AO system, and its main components AO setup at TNO Science and industry Power spectral density of 4 channels (1,11,21,31) in the x and y direction of two data sets for the same temperature (stationary turbulence). Figures in the first row show the power spectral density in the x-direction and figures in the second row show the power spectral density in the y-direction Power spectral density of 4 channels (1,11,21,31) in the x and y direction of two data sets for different temperatures (nonstationary turbulence) Figures in the first row show the power spectral density in the x-direction and figures in the second row show the power spectral density in the y-direction Power spectral density of the first 5000 and the last 5000 samples of slowly varying temperature experiment Power spectral density of the first 5000 and the last 5000 samples of abrupt varying temperature experiment The normalized prediction error with respect to different AR model orders The power spectral density of turbulence (black line) and the prediction error of the second order AR model structure calculated by different algorithms Turbulence of one gradient element of a channel of the WFS Convergence of RLS, NLMS and Fast-RLS algorithms Tracking performance of the RLS, NLMS and Fast-RLS algorithms with respect to the sudden change in temperature Normalized prediction error vs number of states for past window size Normalized prediction error vs number of states for past window size The effect of the size of the past and future windows on the PBSID algorithm accuracy with 20 states The rate of convergence of the AR model structure and the state space model structure by using the RLS algorithm as the least squares problem solver for both model structures Rate of convergence of AR model structure and state space model structure by using RLS algorithm as least squares problem in both model structures

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11 List of Tables 3-1 A comparison of the estimated cost per iteration for several stochastic gradient algorithms for real-valued data in terms of real multiplications, real additions, real divisions, and real square root operation Auto validation and cross validation of state space model for different temperatures Initial value of the Normalized LMS algorithm parameters Initial value of the RLS algorithm parameters Initial value of the fast RLS array algorithm parameters A-1 Algorithm variables in the thesis and their equivalent variables in the matlab files 46

12 viii List of Tables

13 Preface This document is the result of my graduation project at the Delft Center of Systems and Control (DCSC). This project was accompanied with lots of experiences besides its academic goal. First I want to thanks my family, specially my parents, who support me since the first day of my life up to now. Thanks to them for providing me all the supports to study and educate abroad. The only thing I can say is I love them, and do my best to make them happy. Thanks to My girlfriend Nazanin, who support me in difficulties. Thanks to you because of making my life full of energy and happiness. Thanks to my Supervisor Michel Vehrhaegen, who learned me the true way of scientific research. Special thanks to Rufus Fraanje, who supported me as older brother. Thanks for all long and lively discussions, elegant ideas and advices. I am also very grateful to TNO who provide me the lab facilities to enable this research. My special thanks to Niek Doelman, because of the freedom he gave me in my research. Thanks to all colleagues of Precision Motion Systems (PMS) department of TNO Science and Industry. I have spent a great and valuable time in TNO. Thanks for all nice discussions and good ideas. Thanks to Martin the only person who accompanied me late night at TNO. What I have earned during this project is much more than a academical experience. Now I can understand the difference between academic research and industrial projects.

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15 The condition of a society wont be changed until they change what is in themselves. Personal perception of the reality

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17 Chapter 1 Introduction Adaptive optics (AO) is a technique for correcting the optical wavefront distortion induced in a light beam as it propagates through a turbulent. An AO imaging system compensates for a wavefront phase error by sensing the perturbation with wavefront sensor (WFS), and adding the estimated conjugated phase by actively adjusting the optical path length differences with the deformable mirror (DM). Several control strategies are proposed for AO system. Series of parallel loops of classical controllers, such as proportional integrator (PI), and proportional integrator and differentiator (PID) were the first class of controllers implemented for AO systems [1, 2]. This class of AO control approach is known as the AO classical control approach. The major problem of the AO classical control approach is that it is based on the implicit assumption that the spatial and temporal dynamics in the turbulence model can be decoupled. Paschall and Anderson were the first to formulate the AO control problem in a Linear Quadratic Gaussian (LQG) optimal control problem [3]. After them, several optimal control strategies were proposed for AO systems [4, 5, 6]. Because of the integration action related to finite exposure time of WFS and time delay caused by data acquisition and processing, the actuator commands must be based on predicted phase distortion. The H 2 optimal control approach proposed by Hinnen [7] is more consistent with reality. This is because of first, it is data driven and second, it does not consider any form of decoupling between the spatial and temporal dynamics in the turbulence model. Moreover it takes the advantage of the spatial temporal correlation of the wavefront distortion. The control input of the H 2 optimal controller consists of a part that is concerned with prediction of the phase distortion. This control strategy is designed based on the stationarity of atmospheric turbulence and it does not cover the nonstationary behavior of the turbulence, however the wind velocity and the strength of atmospheric turbulence can change rapidly, therefore the prediction algorithm should be able to predict the turbulence with variation on the turbulence parameters. By considering the recent researches on the variation of the turbulence parameters [8, 9, 10] and the effect of these parameters on the statistics of the turbulence, it can be concluded that variation in turbulence parameters lead to nonstationary behavior of the atmospheric

18 2 Introduction turbulence. Andreas in 2007 studied the nonstationary behavior of the atmospheric turbulence as result of the variation in temperature and water vapor density. He also pointed out that the nonstationary behavior at night observation is more severe than of day observation [8]. The nonstationary behavior of the turbulence would cause optimal controllers lose their performance. Beside the proposed H 2 optimal control, adaptive control algorithms based on adaptive estimation of optimal reconstructor matrices have also been proposed in recent years[11, 12, 13, 14] to improve the performance of AO systems in applications with strong, time-varying atmospheric turbulence. Gibson has proposed Multi-input a Multi-output controller with lattice structure. The coefficients of the controller are tuned using a recursive least squares algorithm. Doelman et.al in 2009 [15] proposed an adaptive and real-time optimal controller for adaptive optics systems. They modeled the turbulence by state space model structure and exploited the subspace identification algorithm for batch-wise identification of state space matrices. The controller was designed based on the LQG control frame work. The performance of the proposed controller was evaluated with numerically generated wavefront data. Based on this limitation of adaptive control approaches and other control approaches, the goal of these thesis is to improve the existing H 2 optimal control method by adaptive turbulence prediction algorithm in order to obtain better performance to account for nonstationary atmospheric turbulence. To do this the performance of several recursive predictors are evaluated in a laboratory setup. These algorithms are AR model based predictors and state space model based predictors. The remaining chapters are organized as follows: Chapter 2 includes the problem statement and formulation and the main contributories, first the AO control loop will be presented, and then the AO control objective will be formulated. The H 2 optimal control will be reviewed. In the last section of this chapter, the thesis objective is formulated in more detail. Chapter 3 contains the proposed recursive algorithms for prediction of nonstationary turbulence. These algorithms are categorized in two main categories, the prediction methods based on the Auto Regressive model of the turbulence and the prediction methods based on a state space model of the turbulence. In this chapter the proposed algorithms are compared in term of computational cost. Chapter 4 provides the experimental results regarding to the proposed recursive algorithms in Chapter 3. The performance of these algorithms in terms of accuracy, rate of convergence and tracking performance is evaluated in laboratory setup. Chapter 5 contains conclusion and recommendations based on the findings in Chapter 3 and 4.

19 Chapter 2 AO Control and Problem statement In this chapter the goal of the thesis is motivated in more detail. Section 2-1 provides some basics about the AO control loop and it s components. In the Section 2-1 the AO control objectives based on the residual phase error is studied. In the Section 2-3 the H 2 optimal control approach proposed by Hinnen is considered, this is because of the thesis goal is based on the control input of the this optimal control approach. The effect of the change in the turbulence parameters on the statistics of the turbulence is studied in Section 2-4. Finally in the Section 2-5 the goal of the thesis is stated in more detail. 2-1 AO control loop To explain the principle of Adaptive Optics, consider the schematic drawing in the Figure 2-1. When light from a distant star arrives at the outer layer of the atmosphere, it has a perfectly plane wavefront. However, when it enters the atmosphere this flat wavefront will be distorted due to atmospheric turbulence as it introduces time and space varying optical path length differences. This gives rise to a turbulence induced phase profile φ(ρ, t), where ρ R 2 specifies the spatial position in the telescope aperture and t denotes time. The AO system tries to cancel out these wavefront distortions by actively introducing optical path length differences of opposite phase. An AO system is typically composed of the following components which can be seen in Figure 2-1, a wave front sensor (WFS), an active component to influence the optical path length differences or phase and a feedback control which is for example deformable mirror (DM). Light entering the AO system is first directed to the DM. by actively changing the mirror shape; the DM is able to apply a phase correction φ m (ρ, t). The residual phase error is the difference between the turbulence induced wavefront and the applied correction, i.e. ɛ = φ φ m. After applying the wavefront correction, a beam splitter divides the reflected light beam in two parts. The first part of the corrected light beam leaves the AO system and is used by the science camera to form an image of the object of interest. The remainder of the light is directed to the WFS, which provides quantitative information about the residual

20 4 AO Control and Problem statement Figure 2-1: Schematic representation of an AO system, and its main components wavefront. Based on the WFS measurements s(.), the controller has to determine the actuator inputs u(.) to the DM. The controller should adapt the input signal in such a way that the DM cancels out most of the distortions. By counteracting the wavefront distortions, an AO system is able to reduce the devastating effect of atmospheric turbulence on the imaging process, if the AO system is working properly, the light to the science camera should have an almost flat wavefront, as if there were hardly any distortions. In this way, the corrected image can be recorded without being spread out when using long exposure times. By using AO, large ground-based telescopes may reach close to diffraction limited performance in the near infrared [16]. 2-2 AO control objective The quality of a turbulence degraded image is often expressed in terms of the Strehl ratio. The Strehl ratio is defined as the peak intensity of the image of a point source, normalized to diffraction limited peak intensity [17, 18]. This is a useful and sensitive performance measure as any wavefront error is expected to diffract light away from the center of the image, thereby reducing the peak intensity. For an optical system with a (residual) phase distortion ɛ(ρ,.) the Strehl ratio is given by S = 1 A 2 e ikɛ(ρ,.) ddρ 2 (2-1) Where integration extends over the opening aperture and denotes the light collecting area. From Eq. (2-1) it is clear that Strehl ration for an undistorted wavefront, i.e. ɛ(ρ,.) = 0, is

21 2-3 H 2 optimal control for AO system 5 equal to S = 1. In the presence of any wavefront aberration, the Strehl ratio will be less than 1. The general objective of AO control can hence be formulated as maximizing the Strehl. From a practical point of view, the above expression for the Strehl ratio is not very convenient. Evaluation of Strehl requires accurate knowledge of the wavefront ɛ(ρ,.) over the entire aperture, which is not generally available. Because of the random nature of turbulence, at most the statistical properties of the wavefront are known. However, if the wavefront distortion is not too large, it is possible to relate the Strehl to the variance of the phase error σ 2 ɛ over the aperture. By expanding the exponential in Eq. (2-1) by Taylor series and preserving first two terms, the Strehl can be approximated as [19] S 1 σ 2 ɛ exp( σ 2 ɛ ) (2-2) where σ 2 ɛ = 1 A 2 ɛ 2 (ρ,.)dρ ( ɛ(ρ,.)dρ) 2 (2-3) The above expression provide a reasonable description of S for phase errors with variance up to 0.4rad 2. The latter of the two approximations is sometimes called the extended Marechal approximation. With the Strehl being a strictly decreasing function of σ 2 ɛ, the approximations in Eq. (2-2) suggest that the objective of maximizing the AO imaging quality can be replaced by that of minimizing the residual phase variance. An analysis performed by Herrmann (1992) [20] confirms that even though the above approximations might be rather crude, the minimumvariance wavefront leads indeed to maximum strehl. Since the objective of maximizing the strehl is rather awkward, the AO control objective is reformulated as that of minimizing σ 2 ɛ. 2-3 H 2 optimal control for AO system In this section a summary of the H 2 optimal control will be presented, for more detail the reader can refer to the original publication [7]. In this control approach it is assumed that the atmospheric disturbance is a stationary process and can be modeled as a regular process. This is to say the second order statistics of these signals are modeled as the output of an LTI system with state space model structure: x(k + 1) = A d x(k) + K d e(k) φ(k) = C d x(k) + e(k) (2-4) where e(k) is a zero mean white noise process. Since the turbulence is considered to be stationary, the matrices of the state space model are identified by using subspace identification technique. Also it is assumed that the relation between the actuator inputs u(k) R mu and the DM wavefront correction φ m (k) R my, can be described by a LTI system with state-space realization: H(z) = C m (zi A m ) 1 B m (2-5) There is always one sample delay between measurement and correction, related to finite exposure time of WFS and time delay caused by data acquisition and processing. This delay

22 6 AO Control and Problem statement can be included in DM model. The control objective in this algorithm is to find the optimal controller C(z) that minimize the cost function: J = E{ε T (k)ε(k) + u T (k)qu(k)} (2-6) where Q 0 is a regularization matrix which makes a tradeoff between the expected meansquare residual phase error and expected amount of control effort. By using the the atmospheric turbulence model and the transfer function from control input to the wavefront correction reconstructed from the WFS measurements, H(z), the problem of finding the optimal controller that minimize the cost function of (2-6), can be conveniently expressed in a H 2 optimal control framework. After having formulated the AO control problem in a H 2 optimal control framework, standard H 2 optimal control theory can be used to compute the closed loop optimal controller. This generally involve the numerical solution to two Riccati equations. Due to the special structure of AO control problem it is however possible to simplify the computations. It is also shown that because of minimum phase property of the spectral factor, at least one of the Reccati equations can be avoided. Also the second Riccati equation can be avoided if the model of the DM and WFS dynamics, H(z), is minimum phase or has a known inner-outer factorization. As it is shown that this is the case when the transfer function is of the form H(z) = z 1 (1+z 1 )H. Therefore optimizing controller input after nice analogy is given as: u(k) = (I H Q Hz 1 ) 1 H Q ˆφ(k + 1 k) (2-7) The result of this control algorithm is used to motivate the thesis objective. 2-4 Atmospheric turbulence So far the AO system, AO control objective and H 2 optimal control for AO were introduced. Also it is shown that the H 2 optimal control input consists of a part that is concerned with estimating ˆφ(k + 1 k). ˆφ(k + 1 k) is the prediction of the phase at time k + 1 based on the previous measurement till time k. The question should be answered is the atmospheric turbulence stationary phenomena? To answer this question in this section the statistics of the turbulence based on von karman model, as well as some publication on the atmpspheric turbulence are reviewed. Based on Von-karman model for spatial distribution of the turbulence and its evolution in time by the frozen flow propagation assumption [21], the spatial and temporal correlation phase of the wavefront between two points at distance (δ x, δ y ) can be written R φ (δ t, δ x, δ y ) = E{φ(x, y, t)φ T (t δ t, x δ x, y δ y )} c 2 (L 0 r 0 ) 5/3 ( 2πr L 0 ) 5/6 K 5/6 ( 2πr L 0 ) (2-8) where E{.} is the expected operator and φ(x, y, t) is wavefront phase at time t and position (x, y). δ t, δ x, δ y are shift elements in time, x-direction and y-direction respectively. L 0 is the outer scale of the turbulence, r 0 the Fried parameter.

23 2-5 Problem statement 7 c = 2 5/6 Γ(11/6)π 8/3 (Γ(6/5)24/5) 5/6 where Γ(.) is Gamma function. K 5/6 (.) is modified bessel function of the third type of order 5/6. r = (δ x v x δ t ) 2 + (δ y v y δ t ) 2, where v x is velocity in x direction and v y is wind velocity in y direction. The Eq. (2-8) shows that, temporal and spatial statistics of the atmospheric turbulence depends on the different parameters such as, wind velocity and direction, temperature which influence seeing condition C N and variation in seeing conditions influence the r 0. It can be concluded that variation in each parameter results to the variation in statistics of the turbulence and this resulted to the nonstationary atmospheric turbulence. Beside the these analogies based on the von-karman model, several experiments have been done to investigate the variation of the turbulence parameters. Tubbs (2006) [19] studied the effect of temporal fluctuation in r 0 oh high resolution observation. Also Racin (1996) [20] observed temporal fluctuation in seeing condition over the time scales of minutes. Tokovinin and et. al [10], studied the statistics of turbulence profile. They pointed out the variation in statistics of the turbulence over the scale of minutes. Still several experiments are going to investigate the variation in turbulence profile on smaller time scale. 2-5 Problem statement The preceding sections have given more insight in the scope of the project goals. By considering the recent researche on the variation of the turbulence parameters and the effect of these parameters on the statistics of the turbulence and also Van-Karman model represented in Eq. (2-8), it can be concluded that variation in the turbulence parameters lead to nonstationary behavior of the atmospheric turbulence. The nonstationary behavior of the turbulence cause the optimal controllers lose their perfomance. Therefore, the research objective of this thesis is: Equip the H 2 optimal control approach with an prediction algorithm which is able to account for the nonstationary atmospheric turbulence. Moreover, this prediction algorithm should provide low computational cost, fast convergence rate, and satisfactory tracking performance. Based on the research objective, different prediction algorithms will be introduced in chapter 3. These algorithms will be compared in terms of computational cost. In chapter 4, the performance of these algorithms will be investigated in terms of accuracy, rate of convergence and tracking ability in the laboratory setup.

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25 Chapter 3 Adaptive algorithms for prediction of nonstationary turbulence In this chapter prediction of the nonstationary turbulence is studied. In related publications several models for stationary atmospheric turbulence are introduced, such as Finite Impulse Response (FIR), Auto regressive (AR), Auto regressive and Moving Average model (ARMA) and also state space model. There are limited number of research on modeling the nonstationary process which are mostly related to the modeling of economy. The model exploited in economy is basically parameter varying AR model. In this thesis it is considered that nonstationary atmospheric turbulence can be modeled as time varying AR and state space model. State space model like ARMA model contains both AR and MA part, but the state space model is more optimal in terms of the size of the model. This would be first time that time varying state space model is exploiting to describe a nonstationary turbulence. The models considered for nonstationary turbulence are as follows: 1. Time varying AR model structure (order p) p y(k) = A j (k)y(k j) + e(k) (3-1) j=1 where p indicates an autoregressive model of order p and A 1 (k), A 2 (k),..., A p (k) are the parameters of the model which are time dependent. In this model e(k) is a white noise. If y(k) = s(k) R ns, where s(k) is n s measured slopes at time k, then A j (k) R ns ns and e(k) R ns. 2. Time varying state space model structure The time varying state-space model in the innovation form: x(k + 1) = A(k)x(k) + K(k)e(k) y(k) = C(k)x(k) + e(k) (3-2)

26 10 Adaptive algorithms for prediction of nonstationary turbulence where x(k) R n and y(k) R ns, are the state and output vectors, and e(k) R ns denotes the zero-mean white innovation noise. The state space matrices A(k) R n n, C(k) R ns n and K(k) R n ns are also called system, output and Kalman gain matrices. In this chapter the problem of identification of the model s parameters as well as one stepahead prediction will be studied. This chapter is organized as follows. In the first section the problem of identifying parameters of the AR model structure is considered and it is shown that the problem of finding the AR model parameters can be transformed to a least squares problem. Since these parameters need to be identified at each time instant (online), four recursive stochastic gradient methods, Least Mean squares (LMS), Normalized LMS, Recursive Least Squares (RLS), Fast RLS array algorithms are reviewed. Also these four algorithms are compared in terms of computational cost. In the next chapter Accuracy, rate of convergence and tracking ability of these algorithms will be evaluated in laboratory setup. In the second section, recursive state space identification based on Predictor-Based Subspace Identification (PBSID) method will be discussed. 3-1 Algorithms based on AR model structure In this section it is assumed that a nonstationary atmospheric turbulence can be modeled as a (p-th) order parameters varying AR model. p y(k) = A j (k)y(k j) + e(k) (3-3) j=1 where p indicates an autoregressive model of order p and {A 1 (k), A 2 (k),..., A p (k)} are the parameters of the model which are time dependent. In this model e(t) is a white noise. If y(k) = s(k) R ns, where s(k) is n s measured slops at time k, then A j (k) R ns ns and e(k) R ns. This time dependent parameters result to model which is able to track the changes in turbulence statistics. The problem of finding AR parameters A j (k) can be transformed to the least mean squares problem. The Eq. (3-3) can be written as matrix equation. [ ] y(k) = A 1 (k) A 2 (k) A p (k) y(k 1) y(k 2). y(k p) + e(k) (3-4) y(k) = W (k)y p (k) + e(k) (3-5) [ ] where W (k) = A 1 (k) A 2 (k) A p (k) R ns nsp weighting matrix of past outputs [ ] T and Y p (k) = y T (k 1) y T (k 2) y T (k p) R n sp is a vector of past outputs.

27 3-1 Algorithms based on AR model structure 11 Hence the least mean squares problem for finding the time varying AR s parameters is defined as min E{(y(k) W (k)y p(k))(y(k) W (k)y p (k)) T } (3-6) W (k) Identifying the parameters of the AR model Ŵ (k) at time instant k, the prediction of slops at time k + 1, ŷ(k + 1) is given by: y(k) ŷ(k + 1) = Ŵ (k) y(k 1) (3-7). y(k p + 1) The main part of prediction algorithm is to identify the AR model parameters recursively. Up to now the problem of finding the AR model s parameters was transformed to the least mean square problem. From now the problem of solving the least mean squares problem will be studied in nonstationary environments. In this section stochastic gradient methods will be exploit for solving the introduced least mean squares problem. these methods are obtained from steepest-decent implementations by replacing the required gradient vectors and Hessian matrices by some suitable approximations. Different approximations lead to different algorithms with varied degree of complexity and performance properties (Convergence, tracking performance,...). Stochastic gradient algorithms serve at least two purposes. First, they avoid the need to know the exact signal stochastic (e.g. covariances and cross covariances), which are necessary for successful steepest-decent implementation of but are nevertheless rarely available in practice. Stochastic gradient methods achieve this feature by means of learning mechanism that enables them to estimates the required signal statistics. Second, these methods possess a tracking mechanism that enables them to track variation in the signal statistics. The two combined capabilities of learning and tracking are the main reasons behind the adoption of these methods for solving least mean square problem in nonstationary environment, since in practice there is no priori information available about statistics of the turbulence as well as variation in turbulence statistics. In the body of this section several stochastic-gradient algorithms will be described including: 1. The least-mean-squares (LMS) algorithm. 2. The normalized least-mean-squares (NLMS) algorithm. 3. The recursive least-squares algorithm (RLS). 4. Fast recursive least-squares array algorithm (Fast RLS array algorithm). In this section some of the main results that are useful in motivating the chosen prediction algorithms will be summarized. For more extensive overview, the readers referred to standard publication [22, 23].

28 12 Adaptive algorithms for prediction of nonstationary turbulence Linear estimation problem and steepest decent method The parameter (k) for sake of simplicity will be omitted from equations. consider the least mean squares problem, in stationary environment: min W E{(y W Y p)(y W Y p ) T } (3-8) where y is a zero-mean n s 1 random variable with covariance matrix R y = E{yy T }, and let Y p be a zero mean pn s 1 random variable with a positive-definite covariance matrix, R Yp = E{Y p Yp T }. Then n s pn s cross-covariance matrix of y and Y p is denoted by R yyp = E{yYp T }. In stationary environment the weight vector W that solves Eq. (3-8) is given by and that the resulting minimum mean square-error is W o = R yyp R 1 Y p (3-9) m.m.s.e = R y R yyp R 1 Y p R Ypy (3-10) There are several steepest-decent methods that approximate W o iteratively, until eventually converging to it. For example, following recursion W i = W i 1 + µρ (3-11) where we are writing W i 1 to denote a guess for W o at iteration (i 1), and W i, to denote the updated guess at iteration i. The vector ρ is an update direction vector that we should choose adequately, along with the positive scalar µ, in order to guarantee convergence of W i to W o. The scalar ρ is called the step-size parameter since it affects how small or how large the correction term is. introducing the cost function J(W ) = E{(y W Y p )(y W Y p ) T } (3-12) hence the criterion for selecting ρ and µ is to enforce, if possible, the condition J(W i ) < J(W i 1 ). in this way, the value of the cost function at the successive iterations will be monotonically decreasing. Normally the search direction is chosen in such way which is equal to negative conjugate transpose of gradient vector of cost function at W i 1 i.e., R T yy P R Yp W T i 1 = [ W J(W i 1 )] T (3-13) also the search step based on Newton s recursion can be chosen as, where R 1 Y P µ(i) = µr 1 Y p (3-14) resulted from using the inverse of the Hessian matrix of J(W ), namely, R Yp = 2 W J(W i 1 ) = W T [ W J(W i 1 )] (3-15) More generally, when regularization is employed and when the step size is allowed to be iteration dependent, the recursion for Newton s method is replaced by W i = W i 1 + µ(i)(r yyp W i 1 R Yp )(ɛ(i)i + R Yp ) 1 (3-16) Now as it can be seen, the steepest-decent formulations described above, rely explicitly on knowledge of R yyp and R Yp. This fact constitutes a limitation in practice.

29 3-1 Algorithms based on AR model structure Lack of statistical information First the quantities R Yp, R yyp are rarely available in practice. therefore the approximation of these values are used in order to calculate approximation of the gradient vector W J(W i 1 ) and true Hessian matrix 2 W J(W i 1) known as Stochastic-gradient method. the better approximation, the closer it is expected the performance of the resulting method to be to that of the original steepestdecent algorithms. 2. Variation in the statistical information Second, and even more important especially related to the thesis problem statement, the quantities R Yp, R yyp tend to vary with time. In this way, the optimal weight vector W o will also vary with time. This turns out that the stochastic-gradient algorithm provide the mechanism for tracking such variation in signal statistics. In next subsections different stochastic algorithms will be introduced LMS algorithm Assume that we have access to several observations of the random variables {y 0, y 1, y 2...} and {Y p,0, Y p,1, Y p,2,...}. The {Y p,i } refers as regressors. One of the simplest approximations for R yyp, R Yp is to use the instantaneous values then the gradient vector in Eq. (3-13), is approximated by ˆR Yp,i = Y p,i Y T p,i, ˆRyYp,i = y i Y T P,i (3-17) [ W J(W i 1 )] T Y p y T Y p Y T p W T i 1 = Y p (y T Y T p W T i 1) (3-18) and the corresponding steepest-descent recursion Eq. (3-11) becomes W i = W i 1 + µ(y Y p W i 1 )Y T p (3-19) with W 1 =initial guess. The stochastic-gradient approximation introduced above is known as LMS algorithm. LMS algorithm Consider a zero mean random vector y with realization {y 0, y 1, y 2,...}, and zero mean random vector Y p with realization {Y p,0, Y p,1, Y p,2,...}. The optimal matrix W o that solves can be approximated iteratively via the recursion min W E{(y W Y P )(y W Y p ) T } (3-20) W i = W i 1 + µ(y i W i 1 Y p,i )Y T p,i, i 0, W 1 =inial guess where µ is a positive step-size (usually small).

30 14 Adaptive algorithms for prediction of nonstationary turbulence Normalized LMS algorithm The normalized LMS algorithm is originated form the Newton s recursion Eq. (3-16) and assume that regularization sequence ɛ(i) and µ(i) are constants, say, ɛ(i) = ɛ and µ(i) = µ. W i = W i 1 + µ(i)(r yyp W i 1 R Yp )(ɛ(i)i + R Yp ) 1 (3-21) and replacing quantities (ɛi + R Yp ) and (R yyp W i 1 R Yp ) by instantaneous approximation (ɛi + Y P,i YP,i T ) and (y i W i 1 Y p,i )Y p,i, respectively, also Yp,i T (ɛi + Y P,iYP,i T ) 1 = substituting the result in Eq. (3-16), the Normalize LMS algorithm will be W i = W i 1 + Y T p,i ɛ+ Y p,i 2, µ ɛ + Y p,i 2 [y i W i 1 Y p,i ]Y T p,i (3-22) As it can be seen the Normalized LMS employs a time variant step size of the form µ ɛ + Y p,i 2 (3-23) as opposed to the constant step-size µ, which is used in LMS. Now since Normalized LMS was obtained as a stochastic-gradient approximation to NewtonŠs method, and given the superior convergence speed of NewtonŠs recursion Eq. (3-16) as compared to the standard steepest-descent recursion Eq. (3-13)[23], it is expected that Normalized LMS to exhibit a faster convergence behavior than LMS. Normalized LMS algorithm Consider a zero mean random vector y with realization {y 0, y 1, y 2,...}, and zero mean random vector Y p with realization {Y p,0, Y p,1, Y p,2,...}. The optimal matrix W o that solves can be approximated iteratively via the recursion W i = W i 1 + min W E{(y W Y P )(y W Y p ) T } (3-24) µ ɛ+ Y p,i 2 (y i W i 1 Y p,i )Y T p,i, i 0, W 1 =inial guess where µ is a positive step-size and ɛ is small positive variable RLS algorithm Recursive least squares (RLS) employs a more sophisticated approximation for R Yp. Just like Normalized the LMS, RLS is also originated from regularized Newton s recursion Eq. (3-16) by replacing the instantaneous approximation R yyp W i 1 R Yp = (y W i 1 Y p )Y T p, and also replacing R u, by a better estimate for it, which we choose as the exponentially weighted sample average ˆR Yp = 1 i λ i j Y p,j Yp,j T (3-25) i + 1 j=0

31 3-1 Algorithms based on AR model structure 15 for some scalar 0 λ 1. Choosing a value for λ that is less than one introduces memory into the estimation of R Yp. This is because such a λ would assign larger weights to recent regressors and smaller weights to regressors in the remote past. In this way, the filter will be endowed with a tracking mechanism that enables it to forget data in the remote past and to give more relevance to recent data so that changes in R Yp, can be better tracked by the resulting algorithm. Therefore it is assume that the step size in Eq. (3-16) is chosen as µ(i) = 1/(i + 1) (3-26) whereas the regularization factor is chosen as ɛ(i) = λ i+1 ɛ/(i + 1), i 0 (3-27) Finally the RLS algorithm is explained as follow. RLS algorithm Consider a zero mean random vector y with realization {y 0, y 1, y 2,...}, and zero mean random vector Y p with realization {Y p,0, Y p,1, Y p,2,...}. The optimal matrix W o that solves can be approximated iteratively via the recursion min W E{(y W Y P )(y W Y p ) T } (3-28) P i = λ 1 [P i 1 λ 1 P i 1 Y p,i Y T p,i P i 1 1+λ 1 Y T p,i p i 1Y p,i ] W i = W i 1 + [y i W i 1 Y p,i ]Y T p,i P T i, i 0 with initial condition P 1 = ɛ 1 I and where 0 λ Fast RLS array algorithm RLS algorithm with compare two the LMS and N-LMS has fast rate of convergence, but it needs more computational effort. Therefore in Fast RLS array algorithm, it is tried to decrease the computational effort, by use of array algorithm. The Array algorithms have been widely discussed in the literature. In fast array algorithm by exploiting unitary transformation, RLS parameters are updated. Another advantage is that array methods can be used to exploit any shifting structure in the data. Here just fast RLS array algorithm is represented in order to be short. The reader can refer to the [24] for more details about the algorithm.

32 16 Adaptive algorithms for prediction of nonstationary turbulence Fast RLS array algorithm Consider data {Y P,j, y j } N j=0, where the Y P,j are 1 pn ( ) s where Y p,j = Y P (i) Y p (i 1) Y p (i p + 1) and the y j are 1 n s vectors. Consider also an pn s n s vector W, a scalar 0 λ 1, and pn s pn s positivedefinite matrix Π 1 of the from Π 1 = η.diagonal{λ 2, λ 3,, λ M+1 }, η > 0 (3-29) where M = pn s. when the Y p,j correspond to regressors of a tapped-delay-line implementation. the solution W N of the least-square problem: min W [λn+1 (W W ) Π(W W N ) + λ N j (y j W Y p, j)(y j W Y p, j) T ] (3-30) j=0 can be recursively updated as follows. Start with W 1 = W,γ 1/2 ( 1) = 1,g 1 = ] ] ˆL 1 = ηλ λ M/2 S = [ and repeat for i 0: 1. Find a J-unitary matrix Θ i that annihilates the last two entries in the top row of the post-array below and generates a positive leading entry. Then the entries of the post-array will correspond to [ [ γ 1/2 (i 1) 0 g i 1 γ 1/2 (i 1) ] Y p,i Y pi 1 ˆL i 1 J = ] ˆLi 1 [ 1 S Θ i = 2. Update the weight vector as γ 1/2 (i) [ gi γ 1/2 (i) 0 ] [ ] 0 0 λˆli W i = W i 1 + [g i γ 1/2 (i)][γ 1/2 (i)] 1 [y p,i W i 1 Y p,i ] (3-31) where the quantities g i γ 1/2 (i), γ 1/2 (i) are read from the post-array. Observe that this array algorithm computes the gain vector g i without evaluating the M M matrix P i. Instead, the low-rank factor ˆL i, which is (M + 1) 2, is propagated, resulting in a lower computational complexity.

33 3-2 Algorithms based on state space model structure 17 Algorithm + / LMS 2M + 1 2M Normalized LMS 3M + 1 3M 1 RLS M 2 + 5M + 1 M 2 + 3M 1 Fast RLS array algorithm 6M M Table 3-1: A comparison of the estimated cost per iteration for several stochastic gradient algorithms for real-valued data in terms of real multiplications, real additions, real divisions, and real square root operation Computational cost of stochastic gradient algorithms The computational cost of explained algorithms are summarized in the table below. In this table M = pn s, where p is AR model order and n S is the number of gradient elements. The last four columns in the table count the number of real multiplication, real addition, real division, and the number of square root. LMS algorithm has the lowest computational cost after that Normalized LMS has the lowest computational cost. The RLS algorithm has the highest computational cost. The Fast RLS array algorithm has lower computational cost in comparison to RLS algorithm. 3-2 Algorithms based on state space model structure In this section prediction of the nonstationary turbulence based on state space model structured will be explained. In this section a recursive subspace identification based on the Predictor Based Subspace IDentification method, the so called PBSID will be presented batch-wise and recursive PBSID algorithm The introduced state space model for nonstationary turbulence is as follow: x k+1 = A k x k + K k e k y k = C k x k + e k (3-32) where x k R n, y k R l, are the state and output vector, and e k R l denotes zero mean white innovation process noise. the state space matrices A k R n n, C k R l n and K k R n l are also called the system, output and Kalman gain matrix, respectively. In this algorithm it is assumed that variation in parameters over the particular time window that calculation occurs are negligible, which is equivalent to consider that the process is semi stationary. The size of the window is specified by the past and future window parameters. For sake of simplicity the parameter k is dropped from equations, so one should remember in this section time varying parameters like A k are presented with dropped (k) as A. The Eq. (3-32) can be written in predictor form as: x k+1 = Ãx k + Ky k y k = Cx k + e k (3-33)

34 18 Adaptive algorithms for prediction of nonstationary turbulence with à = A KC. Now the problem is, given the output sequence y k over time k = {0,, N 1}; find all, if they exist, system matrices Ã,C and K up to a global similarity transformation both recursively and batch-wise. by knowing state matrices The one step ahead prediction of the output is given by: ˆx k+1 = ˆÃˆx k + ˆKy k ŷ k+1 = Ĉ ˆx k (3-34) where ˆÃ, ˆK and Ĉ are the identified parameters. The past window is defined by p N +. This window is used to define the following stacked vector: ȳ k = y k y k+1. y k+p 1 In a similar way we can obtain the stacked vectors ȳ k p, ē k and ē k p. The main assumptions are that the system is considered to be observable and that the noise sequence e(k) needs to be white. Before presenting the data equation, first following matrices are introduced. H = CK CÃp 2 K CÃp 3 K 0, Γ = C Cà CÃp 1 [ K = Ãp 1 K ÃK K ] where H is a lower block triangular Toeplitz matrix. Γ is the extended observability matrix, and K is the extended controllability matrix. With these definitions the input-output behavior of the model in Eq. (3-33) is now given by: also state x k is given by: ȳ k = Γx k + Hȳ k + ē k (3-35) x k = Ãp x k p + Kȳ k p (3-36) The other assumption in this algorithm is that we assume that Ãj 0 for all j p. It can be shown that if the system in Eq. (3-33) is uniformly exponential stable the approximation error can be made arbitrarily small by making p large. With this assumption the state x k is approximately given by: x k Kȳ k p (3-37) The output behavior is now approximately given by: y k C Kȳ k p + e k (3-38)

35 3-2 Algorithms based on state space model structure 19 With the approximation given in Eq. (3-38), we can rewrite Eq. (3-35) as: ȳ k ΓKȳ k p + Hȳ k + ē k (3-39) The product ΓK is now given by: CÃp 1 K CÃp 2 K CK. ΓK = CÃp K CÃp 1 K.. C ÃK CÃ2p 1 K CÃp K CÃp 1 K (3-40) With the assumption that Ãj 0 for all j p, this this expression can be approximated by the following upper block diagonal matrix: CÃp 1 K CÃp 2 K CK 0 ΓK CÃp 1 K... C ÃK..... (3-41) CÃp 1 K Due to the introduction of zeros in this matrix, the first block row in Eq. (3-41) can be used to construct the other block rows. Observe now that the product between the state and the observability matrix is approximately given by: under the assumptions on the state it holds that: q k = ΓKȳ k p Γx k, (3-42) lim q k = ΓKȳ k p = Γx k, (3-43) p This implies that we have to find an estimate of C K to construct ΓK. In Eq. (3-38), a linear problem is described in C K and consequently can be used to estimate ΓK batchwise or recursively. To summarize, after the construction of the matrix ΓK we obtain a product between the observability and the state sequence. The approximation of the matrix ΓK described in Eq. (3-41), which can be fully constructed by the product C K and is given by: ΓK the stacked vector Y defined as follow: C K O (l l), C K(:, 1 : l (p 1)). O (l l)(p 1), C K(:, 1 : l) (3-44) [ Y = y p+1,, y N ] in the similar way the stack vector of X can be obtained. Further, the stack matrix Z defined as follow:

36 20 Adaptive algorithms for prediction of nonstationary turbulence [ Z = ȳ 1,, ȳ N p+1 ] If Z has full rank then the C K can be estimated by following linear problem: min Y C KZ 2 C K F. (3-45) The ΓKZ is constructed using relation Eq. (3-41), which equals by definition the extended observability times the state sequence, ΓX. By computing a Singular Value Decomposition (SVD) of this estimate the state sequence and the order of the system is retrieved. Using the following SVD: [ ΓKZ = U U ] [ Σ n 0 0 Σ ] [ V V ] (3-46) where Σ n is the diagonal matrix containing the n largest singular values and V is the corresponding row space. Note that we can find the largest singular values by detecting a gap between the singular values. The state sequence is now estimated by ˆX = Σ 1 2 n V. (3-47) It is well known that when the state and output sequence are known, the system matrices A, C, can be estimated by solving the following linear problem: min θ ( X(:, p + 2 : N) Y ) ( A C ) X(:, p + 1 : N 1) 2 F (3-48) [ ] where θ = A T C T T.In Verhaegen and Verdult (2007)[25], it is also described to find a guaranteed stabilizing estimate of the observer matrix K using the Riccati equation. The recursive solution of described method can be summarized as follow:

37 3-2 Algorithms based on state space model structure 21 Recursive PBSID algorithm Consider the output sequence y k over time k = {0,, N 1}; find all, if they exist, system matrices A,C and K up to a global similarity transformation both recursively. 1. The estimation of C K and construction of q k : The linear problem formulate in Eq. (3-38) derived from the can be written as a recursive least-squares problem, see previous Section for different choice of stochastic gradient algorithms. From the recursive estimate of C K the propagator vector q k can be constructed. 2. The estimation of x k : The state x k can be constructed from the recursive estimate of ΓK. To construct the state at time instance x k in the same state basis as x k 1, the Propagator Method (PM) [25] can be used. Assuming that the system in Eq. (3-32) is observable, then Γ has at least n linearly independent rows. If the order n is known, it is possible to build a permutation matrix S R lp lp such that the extended observability matrix can be decomposed in the following way: [ ] [ ] [ ] qk,1 Γ1 I n = S ΓKȳ q k p = Kȳ k,2 Γ k p = Γ 2 P 1 Kȳk p (3-49) where Γ 1 are the blocks of n independent rows and Γ 2 are the matrices of the l p n others, and P is a unique operator named the propagator. This relation and the approximation in Eq. (3-42) implies that an estimate of the state can be calculated in a particular basis, defined by: ˆx k = q k,1 Γ 1 x k. (3-50) How to find the permutation matrix S, without knowing Γ, such that the first n rows of S Γ are linearly independent is discussed in Mercere et al.(2008) [25]. 3. The recursive estimation of the system matrices: From the estimate of the state update, the system matrices can be updated using recursive versions of the linear least squares problem Eq. (3-48) Recursive least square solver for PBSID algorithm In the previous section four stochastic gradient algorithms were presented. Each algorithm presents different performance besides it s computational cost with respect to the other algorithms. The designer should be consider which property of the algorithm is more desirable. For example RLS algorithm has high rate of convergence but on the other hand has higher computational cost in comparison to the other algorithms Computational cost of PBSID algorithm Computational cost of PBSID algorithm is mostly depends on the least squares solvers. Without considering the specific algorithm as least square solver, in general two least square problems with size of M 1 = p n s and M 2 = (n + n s ) n. where p is the size of past window, n s

38 22 Adaptive algorithms for prediction of nonstationary turbulence is the number of gradient elements and n is the number of states. By refereing to table (3-1) the number of real time operation for the PBSID algorithm can be calculated.

39 Chapter 4 Experimental results In this chapter, the explained adaptive approaches for prediction of nonstationary turbulence are demonstrated in a laboratory setup. In the previous chapter, these algorithms were compared in terms of computational cost. In this chapter, the selected prediction algorithms will be compared in terms of accuracy, rate of convergence and tracking performance. By accuracy, we mean the steady state performance of the algorithm. The rate of convergence is the transient behavior of the algorithm from the time it starts to run until it converges to its steady state performance. The tracking performance is the performance of the algorithm with respect to the change in statistics of the underlying signal. This chapter is organized as follows: In Section 4-1, a brief description of the laboratory setup that has been used to evaluate the performance of the adaptive wavefront prediction algorithms is provided. In Section 4-2, the problem of generating nonstationary turbulence in the experimental setup is considered. In Section 4-3, the performance of three recursive least squares solvers for identification of the coefficient matrices of time varying AR model and prediction of turbulence is investigated. These least square solvers are RLS, Normalized LMS and Fast RLS array algorithm. The performance of the PBSID algorithm for identification of a state space model is investigated in Section 4-4. The prediction of nonstationary turbulence based on these two model structures (AR and state space) is compared. Finally, these results are analyzed in Section Experimental setup This section considers the AO laboratory setup used to test the proposed prediction approaches in an experimental setting. The AO setup in TNO Science and industry, the Netherlands, is depicted in Figure 4-1. In the setup, the laser beam from HeNe laser is focused on pine hole and then collimated by lens L1 to mimic a distant point source. As the laser produces a polarized beam, the intensity of the light source can be adjusted by means of the polarizer. An adjustable aperture A1 is used to increase or decrease the radius of the beam. The atmospheric turbulence is simulated

40 24 Experimental results Figure 4-1: AO setup at TNO Science and industry by the heater H. The temperature of the heater is controlled by the current. A different current results in the change in behavior of the turbulence. This topic will be considered in the Section 4-2. The DM is used to compensate the phase distortion of the beam. The DM consist of 61 low voltage electromagnetic push-pull actuators, which are controlled by Real-time Linux computer. The beam splitter splits the beam, first for the scientific camera (imaging part) and the second Shack-Hartman wavefront sensor (WFS) (sensing part). The imaging part of the beam is focused by lens L4 on the scientific camera. The WFS measures the phase distortion in wavefront. The main component of the Shack-Hartman wavefront sensor is a grid of identical lenses, the so called lens let array that segments telescope aperture into a number of sub-apertures. In the absence of the atmospheric turbulence, a perfect wavefront from a distant star would be flat and could be focused to a point by telescope optics. Turbulence introduces variable distortions to the wavefront, which causes the array of the spots on the WFS detector to be irregularly spaced. By determining the displacement of the spots, the WFS sensor is therefore able to measure the wavefront slopes s(k). The WFS is working with 50 HZ and 36 channels. 4-2 Generating nonstationary turbulence As mentioned before, due to variations in the wind velocity, atmospheric strength and temperature, the atmospheric turbulence has time varying statistics (2-4). Namely, it is a nonstationary process. In order to be able to compare the performance of the proposed predictors in the previous chapter, the nonstationary turbulence should be generated first. The turbulence in the introduced AO setup is generated by a heater. The heater is located in the path of the laser beam. It heats up the air and produces inhomogeneities in the air refractive index. In its path the laser beam is affected by these fluctuations in the refractive index. This procedure leads to random phase abberation in the beam wavefront. The temperature of the heater is controlled by the heater current. The time constant of the response of the temperature to

41 4-2 Generating nonstationary turbulence 25 Figure 4-2: Power spectral density of 4 channels (1,11,21,31) in the x and y direction of two data sets for the same temperature (stationary turbulence). Figures in the first row show the power spectral density in the x-direction and figures in the second row show the power spectral density in the y-direction. the changing current is about 215 seconds, which means that the change of temperature with respect to the current is slow. Therefore, experiments that need variation in the temperature require more time for collecting data. It is expected that change in temperature will lead to change in turbulence statistics. To see the effect of the variation in temperature on the turbulence, the following experiments have been done. In the first experiment, the effect of the temperature on the power spectral density of the turbulence is studied.to do this, two experiments have been performed with two different currents of the heater i = {1A, 2A}. In each experiment, the current of the heater is fixed to the one of the mentioned values. The (open loop) WFS data is recorded by using Shack-Hartmann WFS with 36 lenslets illuminated. This gives rise to a WFS signal s(k) of dimension of n s = samples of WFS are stored. As the temperature is constant during each experiment, it is expected that each experiment is a stationary process. In order to show this, the data set corresponding to the heater current i = 1A is split up in to two data sets of samples. The power spectral density of four selected channels of WFS in the x-y direction of these two sample sets are shown in the Figure 4-2. As it can be seen in Figure 4-2 the power spectral of the turbulence produced by constant temperature of the heater is almost the same for both data sets. In order to show that temperature variation leads to variations in the power spectral density and consequently the auto-correlation coefficients of the turbulence, the power spectral of two sets of measured

42 26 Experimental results Figure 4-3: Power spectral density of 4 channels (1,11,21,31) in the x and y direction of two data sets for different temperatures (nonstationary turbulence) Figures in the first row show the power spectral density in the x-direction and figures in the second row show the power spectral density in the y-direction. turbulence with i = {1A, 2A} are plotted in Figure 4-3. It can be seen that the variation in temperature results in different power spectral densities of the turbulence. From this, it can be concluded that temperature variation results in the variation of the turbulence statistics. This claim also will be supported by comparing the dynamic models identified for the data sets. First, for each data set related to i={1a, 2A}, a state space model based on PI/PO MOESP algorithm [24] is identified regarding to the following model: x(k + 1) = Âx(k) + ˆKe(k) y(k) = Ĉx(k) + e(k) (4-1) Where e(k) is zero mean white noise process and Â, Ĉ and ˆK are the identified state space model matrices. The reader can refer to the book "Filtering and system identification" for more detail about the PI/PO MOESP identification approach. The prediction output is given by: ˆx(k + 1) = (Â ˆKĈ)ˆx(k) + ˆKy(k) ŷ(k) = Ĉ ˆx(k). (4-2) A measure for the model accuracy is given by the Variance Accounted For (VAF) between two measured and predicted output. The VAF is defined as [24] V AF (y j, ŷ j ) = (1 var(y j ŷ j ) ).100% (4-3) var(y j ) Where y j is the measured signal of the channel j and ŷ j is prediction output based on the identified model of the same channel. Now we can validate the identified model using data at

43 4-2 Generating nonstationary turbulence 27 Hearter current i = 1A i = 2A i = 1A 90% 82% i = 2A 83% 90% Table 4-1: Auto validation and cross validation of state space model for different temperatures one current for both the data measured at the same current and also the other current. The resulting VAF values are given in Table 4-1. The Table 4-1 shows that the models of the turbulence at different temperatures are different, since the VAF is dropped by 8% for each cross validation experiment. If the difference between the two temperatures is high, the cross validation will drop by a higher percentage. The power spectral density given by Figure 4-3 and the validation experiments show that variation in temperature leads to a nonstationary turbulence. In order to generate a nonstationary turbulence, the temperature of the heater has been changed during the experiment. Two situations are considered: First, a slow change in the temperature that will be used to evaluate the performance of the predictors regarding to the slow changes in statistics of the turbulence, like accuracy. Second, an abrupt change in temperature which is not the case in reality but it makes it possible to simulate the effect of the sudden changes in turbulence s parameters,(e.g. wind direction and velocity) which lead to fast change in statistics of turbulence. The abrupt change in statistics of the turbulence will be used to investigate the tracking performance of the predictors. For the first case (slowly changing statistics) samples are collected while the temperature is increased gradually from T 1 to T 2 (T 1 and T 2 are the steady state temperatures of the heater currents i1 and i2 respectively). In order to generate an abrupt change in turbulence statistics, samples of the data collected at T 1 is concatenated with that of T 2. The power spectral of the 4 channels of WFS of the first and the last 5000 samples of these turbulences are shown in Figure 4-4 for slow variation in temperature and Figure 4-5 for abrupt change in temperature.

44 28 Experimental results Figure 4-4: Power spectral density of the first 5000 and the last 5000 samples of slowly varying temperature experiment Figure 4-5: Power spectral density of the first 5000 and the last 5000 samples of abrupt varying temperature experiment