Optimization and Systems Theory Seminar
Friday, October 24 2008, 11.00-12.00, Room 3721, Lindstedtsvägen 25

Johan Karlsson, KTH
E-mail: johan.karlsson@math.kth.se

Inverse Problems in Analytic Interpolation for Robust Control and Spectral Estimation

This thesis is divided into two parts. The first part deals with the Nevanlinna-Pick interpolation problem, a problem which occurs naturally in several applications such as robust control, signal processing and circuit theory. We consider the problem of shaping and approximating solutions to the Nevanlinna-Pick problem in a systematic way. In the second part, we study distance measures between power spectra for spectral estimation. We postulate a situation where we want to quantify robustness based on a finite set of covariances, and this leads naturally to considering the weak*-topology. Several weak*-continuous metrics are proposed and studied in this context.
In the first paper we consider the correspondence between weighted entropy functionals and minimizing interpolants in order to find appropriate interpolants for, e.g., control synthesis. There are two basic issues that we address: we first characterize admissible shapes of minimizers by studying the corresponding inverse problem, and then we develop effective ways of shaping minimizers via suitable choices of weights.
These results are used in order to systematize feedback control synthesis to obtain frequency dependent robustness bounds with a constraint on the controller degree.
The second paper studies contractive interpolants obtained as minimizers of a weighted entropy functional and analyzes the role of weights and interpolation conditions as design parameters for shaping the interpolants. We first show that, if, for a sequence of interpolants, the values of the corresponding entropy gains converge to the optimum, then the interpolants converge in H_2, but not necessarily in H-infinity. This result is then used to describe the asymptotic behaviour of the interpolant as an interpolation point approaches the boundary of the domain of analyticity.
A quite comprehensive theory of analytic interpolation with degree constraint, dealing with rational analytic interpolants with an a priori bound, has been developed in recent years.  In the third paper, we consider the limit case when this bound is removed, and only stable interpolants with a prescribed maximum degree are sought. This leads to weighted H_2 minimization, where the interpolants are parameterized by the weights. The inverse problem of determining the weight given a desired interpolant profile is considered, and a rational approximation procedure based on the theory is proposed. This provides a tool for tuning the solution for attaining design specifications.
The purpose of the fourth paper is to study the topology and develop metrics that allow for localization of power spectra, based on second-order statistics. We show that the appropriate topology is the weak*-topology and give several examples on how to construct such metrics. This allows us to quantify uncertainty of spectra in a natural way and to calculate a priori bounds on spectral uncertainty, based on second-order statistics. Finally, we study identification of spectral densities and relate this to the trade-off between resolution and variance of spectral estimates.
In the fifth paper, we present an axiomatic framework for seeking distances between power spectra. The axioms require that the sought metric respects the effects of additive and multiplicative noise in reducing our ability to discriminate spectra.
They also require continuity of statistical quantities with respect to perturbations measured in the metric.  We then present a particular metric which abides by these requirements. The metric is based on the Monge-Kantorovich transportation problem and is contrasted to an earlier Riemannian metric based on the minimum-variance prediction geometry of the underlying time-series. It is also being compared with the more traditional Itakura-Saito distance measure, as well as the aforementioned prediction metric, on two representative examples.

Calendar of seminars Last update: February 4, 2008 by Marie Lundin.