To optimally compensate for time-varying phase aberrations with adaptive optics, a model of the dynamics of the aberrations is required to predict the phase aberration at the next time step. We model the time-varying behavior of a phase aberration, expressed in Zernike modes, by assuming that the temporal dynamics of the Zernike coefficients can be described by a vector-valued autoregressive (VAR) model. We propose an iterative method based on a convex heuristic for a rank-constrained optimization problem, to jointly estimate the parameters of the VAR model and the Zernike coefficients from a time series of measurements of the point-spread function (PSF) of the optical system. By assuming the phase aberration is small, the relation between aberration and PSF measurements can be approximated by a quadratic function. As such, our method is a blind identification method for linear dynamics in a stochastic Wiener system with a quadratic nonlinearity at the output and a phase retrieval method that uses a time-evolution-model constraint and a single image at every time step. (c) 2019 Optical Society of America.

In system identification, input selection is a challenging problem. Since less complex models are desireable, non-relevant inputs should be methodically and correctly discarded before or under the estimation process. In this paper we investigate an input selection extension in least-squares ARX estimation and show that better model estimates are achieved compared to the least-square ssolution, in particular, for short batches of estimation data. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

In this thesis, the use of low-rank approximations in connection with problems in system identification is explored. Firstly, the motivation of using low-rank approximations in system identification is presented and the framework for low-rank optimization is derived. Secondly, three papers are presented where different problems in system identification are considered within the described low-rank framework. In paper A, a novel method involving the nuclear norm forestimating a Wiener model is introduced. As shown in the paper, this method performs better than existing methods in terms of finding an accurate model. In paper B and C, a group lasso framework is used to perform input selection in the model estimation which also is connected to the low rank framework. The model structures where these novel methods of input selection is used on are ARX models and state space models, respectively. As shown in the respective papers, these strategies of performing input selection perform better than existing methods in both terms of estimation and input selection.