The extremely large telescopes that should see first light in coming years demand so-called adaptive optics systems to overcome the devastating effect of the atmospheric turbulence on the image quality. A sensor measures the incoming distortion of the light and is used for reshaping the latter using a deformable mirror. Processing the large number of sensor channels to operate the actuators at kilohertz frequencies is challenging on the computational point of view. The correction applied by the mirror and based on the sensor measurements should indeed not be already outdated because the turbulence has evolved during the computation time. In order to reduce the memory storage and the computational requirements, prior knowledge on the system is commonly translated into assumptions on the system matrices. When the sensors are regularly spread on a two-dimensional grid as is the case in adaptive optics, and the underlying function that describes the spatial dynamics is separable in its horizontal and vertical coordinates, a particular matrix representation is studied. This parametrization allows to write the matrices with a linear number of parameters (instead of quadratic without) and especially to derive more efficient algorithms for identifying from data the spatio-temporal dynamics of the turbulent atmosphere. This PhD thesis draws pros and cons of such a parametrization of large matrices for Linear Time Invariant systems, especially from an identification perspective. Besides, its close connection with tensors raises new fundamental questions in the analysis of such systems. @en

In this paper, we address the identification of two-dimensional (2-D) spatial-temporal dynamical systems described by the Vector-AutoRegressive (VAR) form. The coefficient-matrices of the VAR model are parametrized as sums of Kronecker products. When the number of terms in the sum is small compared to the size of the matrices, such a Kronecker representation efficiently models large-scale VAR models. Estimating the coefficient matrices in least-squares sense gives rise to a bilinear estimation problem that is tackled using an Alternating Least Squares (ALS) algorithm. Regularization or parameter constraints on the coefficient-matrices allows to induce temporal network properties, such as stability, as well as spatial properties, such as sparsity or Toeplitz structure. Convergence of the regularized ALS is proved using fixed-point theory. A numerical example demonstrates the advantages of the new modeling paradigm. It leads to comparable variance of the prediction error with the unstructured least-squares estimation of VAR models. However, the number of parameters grows only linearly with respect to the number of nodes in the 2-D sensor network instead of quadratically in the case of fully unstructured coefficient-matrices.

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In this paper we consider the identification of matrix state-space models (MSSM) of the following form: <formula><tex>$X(k+1) = A_2 X(k) A_1^T + B_2 U(k) B_1^T Y(k) = C_2 X(k) C_1^T + E(k)$</tex></formula> for all time dependent quantities and matrices of appropriate dimensions. Due to the large size of these matrices, vectorization does not allow the use of standard multivariable subspace methods such as N4SID or MOESP. In this paper the resulting Kronecker structure that appears in the system matrices due to vectorization is exploited for developing a scalable subspace-like identification approach. This approach consists of first estimating the Markov parameters associated to the MSSM via the solution of a regularized bilinear least-squares problem that is solved in a globally convergent manner. Second, a bilinear low-rank minimization problem is tackled which allows to write a three-dimensional low-rank tensor and consequently to estimate the state-sequence and the lower-dimensional matrices <formula><tex>$A_1,A_2,B_1,B_2,C_1,C_2$</tex></formula>. A numerical example on a large-scale adaptive optics system demonstrates the ability of the algorithm to handle the identification of state-space models within the class

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In this paper we propose a data-driven predictive control algorithm for large-scale single conjugate adaptive optics systems. At each time sample, the Shack–Hartmann wavefront sensor signal sampled on a spatial grid of size N × N is reshuffled into a d -dimensional tensor. Its spatial-temporal dynamics are modeled with a d -dimensional autoregressive model of temporal order p, where each tensor storing past data undergoes a multilinear transformation by factor matrices of small sizes. Equivalently, the vector form of this autoregressive model features coefficient matrices parametrized with a sum of Kronecker products between d -factor matrices. We propose an Alternating Least Squares algorithm for identifying the factor matrices from open-loop sensor data. When modeling each coefficient matrix with a sum of r terms, the computational complexity for updating the sensor prediction online reduces from OpN⁴ in the unstructured matrix case to Oprd N^{2d d} 1. Most importantly, this model structure breaks away from assuming any prior spatial-temporal coupling as it is discovered from the data. The algorithm is validated on a laboratory testbed that demonstrates the ability to accurately decompose the coefficient matrices of large-scale autoregressive models with a tensor-based representation, hence achieving high data compression rates and reducing the temporal error especially for a large Greenwood per sample frequency ratio.

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This brief presents an algorithm for the recursive identification of Vector AutoRegressive (VAR) models of large dimensions. We consider a VAR model where the coefficient matrices can be written as a sum of Kronecker products. The algorithm proposed consists of recursively updating the Kronecker factor matrices at each new time step using alternating least squares. When the number of terms in the Kronecker sum is small, a significant reduction in computational complexity is achieved with respect to the recursive least squares algorithm on an unstructured VAR model. Numerical validation of nonstationary atmospheric turbulence data, both synthetic and experimental, is shown for an adaptive optics application. Significant improvements in accuracy over batch identification methods that assume stationarity are observed while both the computational complexity and the required storage are reduced.

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In this paper we address the identification of (2D) spatial-temporal dynamical systems governed by the Vector Auto-Regressive (VAR) form. The coefficient-matrices of the VAR model are parametrized as sums of Kronecker products. When the number of terms in the sum is small compared to the size of the matrix, such a Kronecker representation leads to high data compression. Estimating in least-squares sense the coefficient-matrices gives rise to a bilinear estimation problem, which is tackled using a three-stage algorithm. A numerical example demonstrates the advantages of the new modeling paradigm. It leads to comparable performances with the unstructured least-squares estimation of VAR models. However, the number of parameters in the new modeling paradigm grows linearly w.r.t. the number of nodes in the 2D sensor network instead of quadratically in the full unstructured matrix case.

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In this work we address the identification of (2D) spatial-temporal dynamical systems described by the Vector-AutoRegressive (VAR) form. Modeling large-scale networks has been studied so far assuming different structures to alleviate the computational requirements, for example using sparsity in the interconnection pattern, [1]. A data-sparse structure has been investigated in [2] to model 1D strings of interconnected subsystems. The so-called Sequentially Semi-Separable (SSS) structure represents each of the subsystem in a 1D array with a mixed causal anti-causal linear time varying model which shares unknown interconnections with the closest neighbors. A data-sparse representation may not be achieved as the dimension grow as modeling multi-dimensional systems within the SSS framework requires an exponential number of parameters. To overcome this curse of dimensionality, we present in this work a new class of structured matrices to model 2D spatial-temporal systems. The class of Kronecker networks is first introduced and the identification of Vector Auto-Regressive models is subsequently addressed within this framework. Such a Kronecker structure leads to high data compression and benefits from the efficient linear algebra operations as described in [3]. The challenge lies in deriving algorithms that are, on the one hand, scalable in terms of data storage as well as in terms of computational complexity in identifying and using these models, e.g in subsequent control design, and on the other hand, that still ensures similar prediction performances compared to the unstructured least-squares estimates.@en

This paper will present new developments in the identification of large scale network connected systems in the framework of subspace methods. Special structures based on Kronecker products will be proposed that give rise to bilinear structured low dimensional optimization problem. The problems of interest for these large scale problems are imaging systems where images are blurred by perturbations that are dynamic both in time and space.

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We consider the problem of identifying 1D spatially-varying systems that exhibit no temporal dynamics. The spatial dynamics are modeled via a mixed-causal, anti-causal state space model. The methodology is developed for identifying the input-output map of e.g a 1D flexible beam described by the Euler-Bernoulli beam equation and equipped with a large number of actuators and sensors. It is shown that the static input-output map between the lifted inputs and outputs possess a so-called Sequentially Semi-Separable (SSS) matrix structure. This structure is of key importance to derive algorithms with linear computational complexity for controller synthesis of large-scale systems. A nuclear norm subspace identification method of the N2SID class is developed for estimating these state space models from input-output data. To enable the method to deal with a large number of repeated experiments a dedicated Alternating Direction Method of Multipliers (ADMM) algorithm is derived. It is shown in this paper that a nuclear norm relaxation on the SSS structure can be imposed which improves the estimates of the system matrices.@en