Structured matrices for predictive control of large and multi-dimensional systems

Abstract

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 The correction applied by the mirror and based on the sensor measurements should indeed not be already outdated because the turbulence has evolved during the computation time.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.