Optimization and Systems theory
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5B5715 - Linear Stochastic Systems - spring 2007
Information about the course
General information
Welcome to the course. It will be based on some of the material in the book Linear Stochastic Systems: A geometric approach to modeling, estimation and identification , that is beeing written by Anders Lindquist and Giorgio Picci. It is the result of a long going project sponsored by The Swedish Research Council (VR) and the Göran Gustafsson Foundation. This is the first time that this course is given. (Some of the material of the course was part of the course "5B1822 Matematisk systemteori, fortsättningskurs, 4 poäng, F,T" that was once given by the division of optimization and systems theory) Responsible for the course is Per Enqvist.Aim and topics of the course
Anders Lindquist and Giorgio Picci have developed a comprehensive geometric theory for state-space modeling of stochastic processes within the coordinate-free framework of Hilbert space theory. This course will be based on the first part of their forthcoming book and the emphasis will be on systems theoretical concepts such as the state, causality, Markov properties, observability, constructibility, minimality and spectral factorization, which are fundamental for understanding better many methods for prediction, smoothing and system identification. For example, the state of a stochastic process can be described as a subspace making the (generalized) past and future of the process conditionally orthogonal. We will see how this leads to the characterization of the state in terms of splitting subspaces.Deriving the theory we will rely on the spectral representation of stationary processes, innovation representations and spectral factorization and these tools will thus be reviewed. Most of the results holds for both continuous- and discrete-time infinite-dimensional stochastic processes but we will focus on the discrete-time stochastic processes in class.
The second part of the book deals with applications to model reduction, finite interval stochastic realization and subspace identification which will be covered in the next course.