|Optimization and Systems theory | KTH ||
SF3881: Lectures on moment problems in signals, systems and control
- A two week intensive mini-course, 3 högskolepoängTeacher: Professor Chris Byrnes
The course will be given during two weeks. People who are only interested in coming to the seminars to listen (and interact) are most welcome to do so. For those who would like to get credits the examination will be in the form of exercises. Note that the credits are equivalent to two weeks full time studies, so it will be expected that the students put all their energy on this course and that they finish it in these two weeks.
Beginning with Chebychev’s use of power moments to prove the Central Limit Theorem in the 19th Century, the moment problem has matured from its various special forms to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. The crossroads of signals, systems and control are no exception, where moment methods have historically been used in circuit theory, model reduction, optimal control, robust control, signal processing, spectral estimation and stochastic realization theory. Indeed, the moment problem as formulated by Krein et al is a beautiful generalization of several important classical moment problems, including the power moment problem, the trigonometric moment problem and the moment problem arising in Nevanlinna-Pick interpolation.
In this course, we first explore an array of examples, starting with the Chebychev’s calcualtions and with the classical use of moments for a form of model reduction. This naturally leads to the interpretation of a broad range of interpolation problems within the context of the generalized moment problem, in the sense of Krein and Nudel'man. We also review the moment problem as formulated by Markov and its application to time optimal control. Each of these formulations involve a natural constraint on the required form of the solution of the corresponding moment problem, but both make essential use of convexity.
Motivated by classical applications and examples, in both finite and infinite dimensional system theory, we recently formulated another version of the monent problem that we call the moment problem for positive rational measures. The formulation reflects the importance of rational functions in engineering appplications. While this version of the problem is decidedly nonlinear, the basic tools still rely on convexity. In particular, we present a solution to this problem in terms of a convex optimization problem that generalizes the maximum entropy approach used in several classical special cases. We conclude with several applications to problems in signals, systems and control.
The schedule for the classes is as follows.
The seminar rooms 3721 and 3733 are located on the 7:th floor at the math department, Lindstedtsv. 25.