Application of a Global Inverse Function Theorem of Byrnes and Lindquist to a Multivariable Moment Problem with Complexity Constraint

Abstract

A generalized moment problem for multivariable spectra in the spirit of Byrnes, Georgiou and Lindquist is considered. A suitable parametric family of spectra is introduced. The map from the parameter to the moments is studied in the light of a global inverse function theorem of Byrnes and Lindquist. An efficient algorithm is proposed to find the parameter value such that the corresponding spectrum satisfies the moment constraint. * Work partially supported by the MIUR-PRIN Italian grant "New Techniques and Applications of Identification and Adaptive Control".

Multivariable analytic interpolation with complexity constraints: A modified Riccati approach

“…CONCLUDING REMARKS We have extended our previous results [31] for the scalar case to the matrix case. However, multivariable versions of analytic interpolation with rationality constraints have been marred by difficulties to establish existence and, in particular, uniqueness in the various parameterizations [2], [22], [23], [24], [25], [26], [27], [29], [30], and we have encountered similar difficulties here. Our approach attacks these problems from a different angle and might put new light on these challenges.…”

“…On the other hand, in recent years there have been a number of results [25], [26], [27], [29], [30] on the question of existence and uniqueness of the multivariate analytic interpolation problem, mostly for the covariance extension problem (m = 0, n 0 = n + 1), but there are so far only partial results and for special structures of the prior (in our case Σ(z)). Especially the question of uniqueness has proven elusive.…”

“…The multivariable case (ℓ > 1) is much harder, and the nice spectral-zero assignability present in the scalar case appears to be lost or at lease elusive. Restrictive classes of such problems have been considered in large number of papers [2], [22], [23], [24], [25], [26], [28], [27], [29], [30], but the theory remains incomplete, and many problems have been left open.…”

On a Parametric Spectral Estimation Problem

“…We must point out that existence of a bounded and coercive Φ satisfying (3) is not trivial in general. Such feasibility problem was addressed in Georgiou (2002), see also e.g., Ferrante et al (2010Ferrante et al ( , 2012. In this paper, we shall always assume the feasibility in the sense explained next.…”

“…In this work however, we attempt to attack the problem in a direction different from optimization, as a continuation of the work in Ferrante et al (2010), where a parametric family of spectral densities was introduced, and a certain map from the parameter space to the space of generalized moments was studied. The question whether a solution to the parametric spectral estimation problem in fact exists…”